DESIGN OPTIMISATION OF SUPERSONIC DIFFUSERS … - B41.pdf · DESIGN OPTIMISATION OF SUPERSONIC...

Proceedings of the Tenth Asian Congress of Fluid Mechanics

May, 17-21, 2004, Peradeniya, Sri Lanka

DESIGN OPTIMISATION OF SUPERSONIC DIFFUSERS USING

ADAPTIVE SIMULATED ANNEALING

S. Ziaei-Rad1, M. D. Emami2, M. Ziaei-Rad3

1,2 Isfahan University of Technology, Isfahan, Iran 3Sharif University of Technology, Tehran, Iran

ABSTRACT: The aim of this paper is to find the optimum shape of a supersonic diffuser subjected to a prescribed pressure, using adaptive simulated annealing. First the behaviour of the compressible flow in a supersonic diffuser was calculated numerically, using a flux splitting method. Then, a fully coupled sequential iterative procedure was used to solve the steady state aeroelastic problem of flexible wall diffuser. Finally, an adaptive simulated annealing algorithm was implemented and used for the shape optimization of the diffuser.

I. INTRODUCTION The design of optimum aircraft and aeroengines or, in general, any optimization problem in

aeronautics are among the most complex ones in engineering. They are multi-objective and often multi-disciplinary, involving competitive disciplines which cannot be handled in isolation. In order to simultaneously face all of the contributing disciplines, the designer should rely on automated optimization methods supported by the necessary analysis tool(s) for each and every discipline. The need for new search algorithms, which are capable of escaping local optima, has led to the development of non-traditional search (and thus optimization) algorithms. Among optimization algorithms, gradient-based methods (GMs) are well-known algorithms, which probe the optimum by calculating local gradient information [1]. Although GMs are generally superior to other optimization algorithms in efficiency, the optimum obtained from these methods may not be a global one, especially in the aerodynamic optimization problems [2-3]. By introducing probabilistic processes in some of the traditional optimisation algorithms or by inventing brand new algorithms, solving complex optimisation problems are now fully operational [4]. A first classification of these methods, all of them being iterative and of stochastic nature, is based on the number of individual solutions processed at the same time. Genetic Algorithm (GA), Simulated annealing (SA) and Tabu Search handle a single current best, which is likely to be improved in the next step. For instance, genetic algorithms (GAs) are known to be robust methods, modelled on the mechanism of natural evolution. In Tabu search methods, previously seen solutions become tabu points in the search space, during the process of selecting the next point to be examined, and the search is usually enhanced with probabilistic elements. Since this paper mainly focused on SA, GA and Tabu Search will not be discussed further and the interested readers are referred to the cited references [5-8]. Shape optimization is, in fact, one of the most frequently faced problems. In fluid mechanics, the search for optimal aerodynamic shapes dates back to Newton. The search for an axisymmetric body with minimum resistance from the surrounding fluid, during its motion with constant speed parallel to the axis

of symmetry, gave rise to the so-called hydrodynamic or aerodynamic shapes. Many researches have been carried out in shape optimisation of aeronautics problems. However, they mostly concentrated on the use of gradient-based [9] or genetic algorithms [10-11]. To the best knowledge of the authors, not many such problems have been solved by the use of simulated annealing or its other variants as an optimisation tool.

This paper discusses the inverse design optimization of a supersonic nozzle with flexible walls using adaptive simulated annealing. In doing so, the behaviour of compressible flow in the supersonic diffuser was modelled numerically using a flux splitting method. To further accomplish our task, a fully coupled sequential iterative procedure was used to solve the steady state aeroelastic problem of flexible wall diffuser. Finally, an adaptive simulated annealing algorithm was implemented and used for the shape optimization of the diffuser. II. DEFINITION OF THE PROBLEM

A schematic of the physical model on coordinate system is shown in Fig. 1. The physical model

consists of an axisymmetric supersonic diffuser with flexible walls. It was assumed that the diffuser has been constrained on both ends (Points A and B in Fig. 1). Steady flow of air stream ( 4.1=γ ) is passing through the diffuser. For simplicity, only the variation of variables in the streamwise direction is considered.

Fig. 1 Structural Model of the Wing III- FLUID-SLOID INTERACTION

For a complete solid-fluid interaction three distinct gradients are necessary. 1- A flow solver, 2- A structural analysis code 3- A coupling interface code. In this paper, the behaviour of compressible flow in a supersonic diffuser was studied numerically in quasi-one dimensional form using a flux splitting method. For structural analysis, a finite element approach was used to compute the nodal displacement of the diffuser wall. Finally, a fully coupled sequential iterative procedure was used to solve the steady state aeroelastic problem of flexible wall diffuser. Each of the above items will be described briefly.

1- The Flow solver

The governing equations in quasi-one dimensional flux vector form may be written as:

( ) 0HxESQ

t=−

∂∂

+∂∂ (1)

Where Q is the vector of conserved variables, E is the invisid flux vector in the streamwise direction, and H denotes the source (load) vector. These may be further expressed by the primary variables as:

axis of symmetryinflow

outflow

Radial displacement

A

B

x

r

[ ]TteuQ ρρρ= (2)

( )[ ]Tt2 upepuuE ++= ρρρ (3)

T

0dxdSp0H

=

(4)

In the above equations, p , ρ and u are the pressure, density and velocity respectively, S is the cross section area and te represents the total energy per unit mass.

To solve the above set of equations, a set of proper boundary conditions is required. At the supersonic inflow boundary, non-dimensionalized variables are specified, namely 5.1M = (Mach number), 1p = and 1=ρ . At the outlet, the flow is subsonic. Therefore, two variables will be determined from the internal computational cells using a second-order implicit extrapolation. The third one, i.e. subsonic pressure, should be specified which is set to 5.2p = for all the cases studied in this paper.

In order to accurately capture the shock wave discontinuity, a finite-difference flux vector splitting method has been used as the flow solver. In particular, we use the flux vector splitting method of Steger and Warming [12]. Steady state solution is obtained in a time-asymptotic sense, using an implicit discretization of governing equations. The computational flow field is divided into N finite-difference meshes, where N indicates the spatial intervals used in the streamwise direction. The implicit discretized form of conservation equation may be expressed as [12]:

( ) ( ) ii1i1ii1i1iiiii1i1i tHEEEExtQA

xtQtBAA

xtSIQA

xt

∆∆∆

∆∆∆

∆∆∆∆

∆∆∆

+−+−−=

+

−−++

− −−

++−

++

−+

−+−

+−

(5)

in which A and B are the derivatives of E and H with respect to Q and t∆ , x∆ are time and spatial steps respectively. I is the unity matrix of order 3. The superscript +/- denotes the diagonal matrices containing the positive/negative eigenvalues of matrices A and E. Moreover, backward finite difference approximation has been used for positive matrices, while forward one is adopted for negative matrices. The system of linear equations (5) was solved for Q∆ and then, the matrix of coefficient is updated using

QQQ n1n ∆+=+ for the next iteration. A steady state solution said to be obtained when the variation of properties (e.g. pressure) is very small between two iterations. 2- The Structural Analysis

Structural analysis has been used to predict the deformation of the field at every global iteration in the coupled-field analysis. The diffuser wall was modelled using 2D beam elements. A computer program was written to calculate the nodal displacement of the beam subjected to the distributed pressure calculated from the flow field solution. In this study, we used the same mesh for both structural and fluid analyses. Because the computational domains share nodes on the common boundary, compatibility conditions on the interface is naturally satisfied. The General equation that has been solved for this case can be written as follow:

FU]K[ = (6) Where [K] is the stiffness matrix computed for the beam element. U is the vector of nodal displacements and F is the external force, i.e. the pressure distribution calculated from the flow solver. 3- The Fluid-Solid Coupling

The solution of equation (6) would require the structural terms on the left hand side to be solved simultaneously with the solution of flow field on the right hand side. However, equation (6) may be solved sequentially, i.e. solve for the flow field and then apply the predicted load (pressure) on the structure and find the new shape of the wall. This procedure will be repeated until the changes in the shape of the walls are smaller than a prescribed tolerance. To summarize, the steps that should be followed to get the final shape of the diffuser are:

1. Start with an initial shape of diffuser wall

2. Solve the flow equations iteratively (inner loop)

3. Solve the structure equations, using the current pressure distribution, and compute the nodal displacements

4. Update the wall profile

5. Repeat steps 2 to 4 until a converged solution is obtained (outer loop) IV. ADAPTIVE SIMULATED ANNEALING

As the name suggests, SA attempts to mimic the physical phenomenon of annealing in which a solid is first melted and then allowed to cool by reducing the temperature. During the cooling process, the particles of the molten solid would form into a structure of minimum energy provided the cooling rate is slow. If the cooling is rapid, formation of a local minimum energy structure may occur, as the particles do not have adequate time to attain thermal equilibrium. Metropolis et al. [13] realized that the thermal equilibrium process could be simulated by introducing the concept of Boltzmann probability distribution. At a fixed temperature (T), the probability that a system would exist at a particular energy state (E) is given by Boltzmann distribution function, P(E) = exp(-E/kT) where k is the Boltzmann constant. At a high temperature, this distribution suggests that the system is equally probable to exist at any energy state; as the temperature decreases, the system shifts towards lower energy states. To determine the equilibrium state of a system, the present configuration with energy Ei is first perturbed to find a new configuration with energy Ei+1. If the difference in the energy of two states, i1i EEE −= +∆ is negative, then the new configuration with lower energy is accepted and replaces the present one. If E∆ is positive (i.e. the new configuration has a higher energy level than the present one), the probability of accepting the new configuration is given by the Boltzmann function, )kT/Eexp()E(P ∆−= . This is known as Metropolis criterion. By randomly generating a sequence of configurations and applying this criterion, the equilibrium state of a system can be eventually obtained. An improved version of SA, referred to as the adaptive SA (ASA) [14-17], is known to provide significant improvement in convergence speed over standard versions of SA. This ASA is also known as the very fast simulated reannealing. In this study, we apply the ASA to inverse supersonic diffuser design. Some examples were solved and used to illustrate the effectiveness of the ASA. Although, no attempt has been made to explicitly compare the convergence speed of the ASA with that of the GA. However, the studies carried out in other research papers [18], indicate that the efficiency of the ASA appears to be on the same order as GA.

Like any other optimization algorithm, we have to define the design variables and the cost function for our inverse design problem. The inverse design of diffuser consists of finding the geometric shape whose pressure distribution along the wall ( )xp , matches a prescribed pressure distribution ( )xp , where x is the Cartesian coordinate measured along the diffuser axis. A discrete objective function may be defined as:

[ ]∑=

−=N

1i

2ii ppI

(7)

In this study, we used the following algebraic formula for describing the shape of the diffuser wall:

( ) ( )dcxtanhbaxs −+= (8) There are four unknown coefficients in this formula, but one of them should be computed in terms of the three others due to constant rate of mass flow during the solution procedure [18]. Therefore, the vector of the design variables can be written as:

d,c,bw =

3i1 UwL iii ≤≤≤≤ (9)

As in a standard SA, the ASA contains two loops. The inner loop ensures that the parameter space is searched sufficiently at a given temperature, which is necessary to guarantee that the algorithm finds a global optimum. The differences with standard SAs are that the ASA uses a much faster annealing schedule and employs a reannealing scheme to adapt itself. The ASA is easy to program, and the user only needs to assign a control parameter c and set two values Naccept and Ngenera. The main steps in the algorithm for our problem are as follow: Step 1: Initialization, an initial w is randomly generated, the initial temperature of the acceptance probability function, Tc(0), is set to the initial value of the cost function J(w), and the initial temperatures of the parameter generating probability functions, Ti(0), 1 ≤ i ≤ 3, are set to 1.0. A user-

defined control parameter c in annealing process is given, and the annealing times, ki for 1 ≤ i ≤ 3 and kc, are all set to 0. Step 2: Point Generation, The algorithm generates a new point in the parameter space with

inewiiiii

oldi

newi UwL) , UL(hww ≤≤−+= (10)

Here Li and Ui are the lower and upper bounds for wi , and hi is calculated as:

−

+−=

−

1)k(T

11)k(T)5.Lsgn(h1v2

iiiiii

i

(11)

Where vi a uniformly distributed random variable in [0, 1]. The value of the cost function I(wnew) is then evaluated and an acceptance probability function is calculated as:

( ) )k(T/)w(I)w(Iexp11P

ccoldnewaccept −+

= (12)

A uniform random variable Punif is generated in [0, 1]. If Punif ≤ Paccept , wnew is accepted; otherwise it is rejected.

Step 3: Reannealing, After every Naccept acceptance points, reannealing takes place by first calculating the sensitivities:

3i1 )w(I)w(I

sbest

ibest

i ≤≤−+

=σ

σδ (13)

where wbest is the best point found so far, σ is a small step size, the dimensional vector iδ has unit ith

element and the rest of elements of iδ are all zeros. Let smax = maxsi , 1 ≤ i ≤ 3. Each Ti is scaled by a factor smax/si and the annealing time ki is reset

3

i

iiiii

i

maxii )0(T

)k(Tlog

c1k), k(T

ss

)k(T

−==

(14)

Similarly, Tc(0) is reset to the value of the last accepted cost function, Tc(kc) is reset to J(wbest), and the annealing time kc is rescaled accordingly:

3

c

ccc )0(T

)k(Tlog

c1k

−= (15)

Step 4: Annealing, After every Ngenera generated points, annealing takes place with

3i1 )ckexp()0(T)k(T

1kk3/1

iiii

ii ≤≤

−=

+=

(16)

and

)ckexp()0(T)k(T

1kk3/1

cccc

cc

−=

+=

(17)

Otherwise, go to step 2. Step 5: Termination, The algorithm is terminated if the parameters have remained unchanged for a few successive reannealings or a preset maximum number of cost function evaluations have been reached; otherwise, go to step 2. V- NUMERICAL IMPLEMENTATION

The problem definition was described in section II of the paper. An algebraic formula in the form of equation (8) was used to generate the surface wall profile of the diffuser. The coefficients b, c and d are set to 0.347, 0.8 and 4 respectively. The value for coefficient ‘a’ has been computed from the fact that mass flow rate during the solution procedure remains constant. If one assumes that the value of inlet area is fixed at 1.0512, then ‘a’ can be calculated from equation (8). The supersonic inlet Mach number for this test case was set to 1.5 and the subsonic exit pressure was set to 2.5 (normalized with respect to the inlet pressure). 1- Diffuser with Rigid Wall

As a preliminary validation case, a supersonic diffuser with rigid wall was considered. The value for calculating the diffuser profile and the flow conditions are as described above. The flow was calculated

and the shock was captured successfully. Figs. 2a and b show the diffuser shape and the pressure distribution along its length. As it is clear from the figure, shock is established at x/L0=0.5.

a- Diffuser shape b-pressure distribution

Fig. 2 The diffuser shape and the pressure distribution along the diffuser length 2- Diffuser with Flexible Wall

In the second case, the same diffuser was considered. However, the walls are not rigid in this case. The stiffness of wall will be controlled through the stiffness of the beam section (i.e. EI). A complete fluid-solid interaction should be solved to find out the final shape of the wall and the ultimate position of the shock inside the diffuser. Fig. 3 depicts the final results for the rigid and flexible walls. The pressure distribution is also plotted on the same figure for better illustration. As the wall deflects towards the inner side (because the ambient pressure is more than the inside pressure), the cross section area of the diffuser reduces and, therefore, the shock moves forward towards the diffuser outlet.

Fig. 3 The profile and pressure distribution for the diffuser with rigid and flexible wall

(Rigid ∞=0)/(EIEI , Flexible 1)/( 0 =EIEI

Fig 4 shows the effect of flexibility on the final shape of the diffuser and the position of shock. It is inferred from the figure that the shock distance from the inlet increases with the increment on the flexibility of the structure.

Fig. 4 Effect of flexibility on the position of shock inside the diffuser

3- Inverse Design Optimization of Diffuser

The idea of inverse design is to calculate the proper shape of a diffuser with flexible walls so that its pressure distribution becomes the same as a diffuser with rigid walls. Generally, the design of diffusers is based on the rigid wall assumptions. However, if the flexibility is high, the modification in the profile should be made in order to produce the same pressure distribution as a rigid wall diffuser. Thus, the idea here is to find the coefficients b, c, and d in a diffuser with flexible wall to have a pressure distribution the same as the one shown in Fig. 2b. In other words, we are searching the variable space to find out the minimum of the quantity I in equation (7), in which ˆp is the pressure distribution along the rigid wall diffuser. The aforementioned case was solved using adaptive simulated annealing and the results for

50 101×== EIEI , is shown in Figs. 5a and b. Figure 5b shows the target and the calculated pressure

distribution. It is clear that the final and the target are in good agreement. The convergence rate for the ASA to reach the optimum of the objective function is depicted in Fig. 6. The value of the objective function becomes flat after 300 iterations. Increasing the number of iterations in the ASA algorithm may further reduce this value. However, this needs more CPU time and computational efforts with little gain in the accuracy of the pressure distribution.

a- Diffuser shape b-Pressure distribution

Fig. 5 Effect of flexibility on the position of shock inside the diffuser ( 1EI/EI 0 = )

Fig. 6 The convergence of the objective function for the inverse design of diffuser

VI. CONCLUDING REMARKS A well known global optimization methods, namely the adaptive simulated annealing (ASA), was introduced and were implemented in a program specifically written for the inverse design of diffusers.

A new simulated annealing method called adaptive simulated annealing which is fast and alleviate the problem of initial tuning was adopted and implemented in the program. It was found that the method could be used in inverse design of the diffusers with flexible walls.

VIII. REFERENCES [1] Gill PE, Murray W, Wright MH. Practical optimization. New York: Academic Press, 1981 [2] Ta'asan S. “Trends in aerodynamics design and optimization: a mathematical viewpoint”, In: AIAA 95-1730, 12th AIAA Computational Fluid Dynamics Conference. 1995. [3] A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic design using the Navier–Stokes equations”, Theoretical Computational Fluid Dynamics, No. 10, Vol. 213, 1998 [4] K.C. Giannakoglou , “Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence”, Progress in Aerospace Sciences, 38, pp. 43–76, 2003 [5] J.H. Holland, “Adaptation in Natural and Artificial Systems”, University of Michigan Press, Ann Arbor, Michigan, 1975. [6] L. Davis (Ed.), Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991 [7] Glover F. and Laguna M. Tabu search, Kluwer Academic Publication, USA, 1997 [8] Siarry, P., & Berthiau, G., “Fitting of tabu search to optimize functions of continuous variables”, International Journal for Numerical Methods in Engineering 40, pp. 2449-/2457, 1997 [9] Jameson, A., “Computational algorithms for aerodynamic analysis and design”, Applied Numerical Mathematics, Volume 13, Issue 5,pp. 383-422, 1993 [10] A. Vicini and D. Quagliarella, “Inverse and Direct Airfoil Design Using a Multiobjective Genetic Algorithm”, AIAA Journal, Vol. 35, pp.1499-1505, 1997 [11] S. Obayashi, Y. Yamaguchi and T. Nakamura, “Multiobjective Genetic Algorithm for Multidisciplinary Design of Transonic Wing Platform”, J. of Aircraft, Vol. 34, pp.690-693, 1997 [12] ] J. L. Steger, R. F. Warming, “Flux Vector Splitting of the Inviscid Gas Dynamic Equations with application to Finite Difference Methods”, NASA TM-78605, July 1979 [13] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, pp1087–1092, 1953 [14] Ingber, L. and Rosen, B., “Genetic algorithms and very fast simulated reannealing: A comparison”, Math. Comput. Mode, Vol. 16, pp87–100, 1992 [15] Ingber, L., “Simulated annealing: Practice versus theory”, Math. Comput. Model, Vol. 18, pp29–57, 1993. [16] Ingber, L., “Adaptive simulated annealing (ASA): Lessons learned”, J. Control Cybernet, Vol. 25, pp33-54, 1996. [17] Chen, S., Luk, B. L., and Liu, Y., “Application of adaptive simulated annealing to blind channel identification with HOC fitting”, Electron. Lett., Vol. 34, pp234–235, 1998 [18] S. Ziaei-Rad and M. Ziaei Rad, “Inverse Design of Supersonic Nozzles with Flexible Walls Using Genetic Algorithm ”, to be published in Journal of Fluid and Structure.

Proceedings of the Tenth Asian Congress of Fluid Mechanics17-21 May 2004, Peradeniya, Sri Lanka

Computation of Supersonic Viscous Flow around a Slender Body at HighAngles-of-Attack

Naresh Kumar, Manoj T. Nair and S. K. SaxenaComputational and Theoretical Fluid Dynamics Division,

National Aerospace Laboratories, Bangalore, India.

ABSTRACT: In the present work, flow past an ogive-cylinder configuration has been computed using a multi-blockRANS solver. There are two objectives of the present study. First is two check the effectiveness of the Spalart-Allmaras model in predicting the vortical flows at high angles-of-attack. The second is to study the effect of variousflow parameters on the flow field. The results are compared with computational and experimental data available inliterature.

1. INTRODUCTION Computational Fluid Dynamics(CFD) is now routinely used in design and analysisof flow past complex aerospace configurations. The ability of the CFD techniques to predict flow pastbodies at low and moderate angles-of-attack are well known. However the design of present generationfighter aircrafts and missiles requires that these techniques should be able to predict the highly vortical flowfields generated by these vehicles at high angles-of-attack during critical maneuvers. At high angles-of-attack the flow field around the body has large separated zones with strong vortices propagating from thenose of the body. At higher angles-of-attack these vortices become asymmetric and can produce large sideforces. With the advancement of the computational techniques it is now possible to predict these flow fieldswith fair amount of confidence.

There has been a number of studies in recent years dealing with vortical flows over ogive-cylinderconfigurations at high angles-of-attack [1–4]. Brich et al [1] have studied laminar and turbulent flow pastthe ogive-cylinder using Parabolized Navier-Stokes equations with Baldwin-Lomax [5] model and Degani-Schiff [3] correction. They studied three different cases varying the Mach number and Reynolds number.Josyula [2] has used the RANS equations. In [2] k−ε turbulence model along with a compressible correctionwas used. Sturek et al [6] have used the OVERFLOW, NPARC and CFL3D codes to study six different casesfor transonic and supersonic velocities at 8 and 14 angles-of-attack.

In the present study, a multi-block Reynolds Averaged Navier-Stokes(RANS) code, MB-EURANIUM [7–9] has been used to compute flow past a 3-caliber ogive with a 10-caliber cylindrical after-body. The aim ofthe present work is to validate the Spalart-Allmaras [10] for high angle-of-attack flows and to study the ef-fect of angle-of-attack and Reynolds number on the flow field. Three different cases have been studied andthe results are compared with experimental and computational studies reported in literature. The physicsof the flowfield around slender bodies at high angles-of-attack is described in Degani & Schiff [3]. Due tolack of space detailed discussion is not provided here.

2. Computational Technique The present computations were done using the multi-block RANS codeMB-EURANIUM developed at National Aerospace Laboratories(NAL), Bangalore, India, to compute com-pressible flow past complex aerospace configurations over a range of Mach numbers. MB-EURANIUMsolves the unsteady conservation law form of the RANS equations. It is a cell-centered finite volume codein generalized body-fitted coordinates. The code can handle multi-block structured grids. It employs theTotal Variation Diminishing (TVD) formulation of Roe’s Riemann [11] solver based on Monotone UpwindScheme for Conservation Laws (MUSCL) and ideal gas assumption to discretize the Euler terms. Theviscous terms are central differenced. The time term is discretized using a multi-stage Runge-Kutta timestepping scheme or a LU-SSOR based implicit scheme. For turbulent simulations two different turbulencemodels namely, Baldwin-Lomax model [5] and one equation Spalart-Allmaras model [10] are available.A number of convergence acceleration techniques are employed to speed up the convergence rate of theexplicit code. The details of the numerical algorithm used in MB-EURANIUM are given in [7, 8].

1

3. RESULTS AND DISCUSSIONS

Figure 1: Sectional view of grid around theogive-cylinder.

The geometry of the ogive-cylinder is shownis Figure 1 along with the 10-block grid used forthe computations. The geometry consists of alength of 13 diameters. The first 3-diameters isthe ogive. The ogive is given by the equationr(x)/d = −0.002615(x/d)3 − 0.03986(x/d)2 +0.3098(x/d), where r(x) is the radius along theaxis(x-direction) of the body. The grid consists of151 points along the axis of the body and 101 eachin the radial and azimuthal directions. The asym-metry in the grid allows for better resolution of vor-tical flow on the leeward side and strong shocks onthe windward side. Flow symmetry is assumed andonly half the grid in the circumferential direction isused. The minimum normal grid spacing is takenas 1 × 10−5 times the diameter.

Since the incoming flow is supersonic, freestreamconditions are imposed at the inflow boundary. Atthe outflow boundary the flow variables are extrap-olated from the interior. On the body no-slip boundary condition is applied. The wall is taken to beadiabatic.

Three different test cases were tested in the present study

1. Case I: M∞ = 2.5, Re = 1.23 ×106, α = 14;

2. Case II: M∞ = 1.8, Re = 0.89 ×106, α = 14; and

3. Case III: M∞ = 2.0, Re = 1.2 ×106, α = 10

For Case I and Case II computations were done on both a fine grid (151×101×101) as well as a coarsegrid (75 × 51 × 51). The coarse grid is obtained by removing alternate points from the finer grid. Thesolution from the coarse grid is used as the initial solution for the fine grid. For Case III the computationswere done only on the coarser grid. Due to limitations of space detailed results are not presented here.

The y+ value at the wall varied from 0.01 to 1.8 for the fine grid and from 0.01 to 2.0 for the coarsegrid. A total of 15 points are in the boundary layer at the end of the ogive on the leeward side and there are25 points in the boundary layer at the end of the body on the windward side.3.1 Case I: M∞ = 2.5, Re = 1.23 ×106, α = 14

The Figure 2 shows the comparison of coefficient of pressure for the present computations with experi-mental and computational results from Birch et al [1] at four different x/d stations. Both fine and coarse gridresults from the present computations are shown. It can be noted that the crossflow separation point is bettercaptured by the present computations compared to the results from Birch et al [1]. The difference betweenthe two grids is small in predicting the separation point, but second peak in the Cp is better captured inthe fine grid. Figure 7(a) shows the density contours at the pitch plane. Figure 3 shows the total pressurecontours at x/d= 3.5, 5.5,8.5 & 11.5. Figure 4 shows the streamlines indicating massive separation. BothFigures 3 and 4 indicates the movement of the primary vortex core away from the body.3.3 Case II: M∞ = 1.8, Re = 0.89 ×106, α = 14

The Figure 5 shows the comparison of coefficientof pressure for the present computations with experimental results from Sturek et al [6] at four differentx/d stations. The experimental results are at a Reynolds number of 0.66 × 105. Both fine and coarse gridresults from the present computations are shown. The results from both the grids are close to each otherexcept at x/d=7.5. The trends in both experiments and computations look similar except at x/d=5.5 wherethe computations fail to capture the second peak. The crossflow separation point is not predicted accuratelytowards the aft of the body. Figure 7(b) shows the density contours at the pitch plane.

2

theta

Cp

0 20 40 60 80 100 120 140 160 180

-0.15

-0.1

-0.05

0

0.05

ExpB.L+D.SB.LFineCoarse

(a) x/d=5.5

Theta

Cp

0 20 40 60 80 100 120 140 160 180-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 ExpB.L+D.SB.LFineCoarse

(b) x/d=7.5

Theta

Cp

0 20 40 60 80 100 120 140 160 180-0.15

-0.1

-0.05

0

0.05

0.1 ExpB.L+D.SB.LFineCoarse

(c) x/d=8.5

theta

Cp

0 20 40 60 80 100 120 140 160 180

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

ExpB.L+D.SB.LFineCoarse

(d) x/d=11.5

Figure 2: Case I: Comparison of present computed Cp with the experimental and computational results of[1]. (BL = Baldwin-Lomax and DS = Degani-Schiff).

Figure 3: Case I: Total pressure contours atx/d=3.5, 5.5, 8.5 & 11.5.

Figure 4: Case I: Streamlines.

3

Theta

Cp

0 50 100 150

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 CoarseFineExp

(a) x/d = 3.5

Theta

Cp

0 50 100 150

-0.2

-0.15

-0.1

-0.05

0

0.05

CoarseFineExp

(b) x/d = 5.5

Theta

Cp

0 50 100 150

-0.15

-0.1

-0.05

0

0.05 CoarseFineExp

(c) x/d = 7.5

Theta

Cp

0 50 100 150

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

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0.06 CoarseFineExp

(d) x/d = 11.5

Figure 5: Case II: Comparison of present computed Cp with the experimental results of [6].

(a) x/d=5.5 (b) x/d=7.5

(c) x/d=8.5 (d) x/d=11.5

Figure 6: Case III: Cp vs. θ.

4

3.3 Case III: M∞ = 2.0, Re = 1.2 ×106, α = 10

(a) Case I

(b) Case II

(c) Case III

Figure 7: Density contours at pitch planefor the three cases.

The Figure 6 shows the coefficient of pressure for thepresent computations at four different x/d stations. The com-putations were done only on the coarse grids. No experimentalresults for Cp are available for the present case. Figure 7(c)shows the density contours at the pitch plane.

Figure 8 shows the comparison of the Cp along the axiallength on the leeward side for the three cases. This figureindicates that the flow on the leeward side in all three casesexpands upto nearly the end of the ogive. At this point there isshock leading to the flow separation. The shock in strongestfor Case II and the weakest for Case III. The vortex formed isthe stronger for the higher angle-of-attack cases (Case I & II).

The density contours for all the three cases in Figures 7(a),7(b) and 7(c) show that the leading edge shock and expansionand the shock before flow separation on the leeward side iscaptured. A comparison of the Cp vs. θ for Case I and Case IIindicates that the crossflow separation starts at a higher θ inCase II, where both the Mach number and Reynolds numbersare lower. The secondary separation is seen as the second peakin Cp which shows that secondary separation is present for alonger length along the body in Case II.

A comparison of Case I and Case III shows that as angle-of-attack is decreased the separated shear layer remains closerto the body (density contours), and the presence of secondaryvortices all along the length of the body. This is due to theprimary vortices remaining closer to the body at lower angleof attack.

A comparison of the density contours at pitch plane forthe three cases indicates presence of a second windward shockdown the cylinder in Case II and Case III. This has also beenindicated by the computations of Prince & Qin [12].

4. CONCLUDING REMARKSThe limited grid independent study done in the present

work shows that the grid size used can capture the flow fieldaccurately. The present computations show that the one-equation Spalart-Allmaras turbulence model [10] is able topredict the overall flow features of the flow field at highangles-of-attack. For Case I, the results of present computa-tions have been compared with computational results obtainedwith Baldwin-Lomax model and its variants. It is observedthat in general the performance of Spalart-Allmaras model isbetter. For lower Reynolds numbers the flow field predictionis not accurate enough. Further investigation is to be donein this aspect. An increase in angle-of-attack is shown to in-crease the strength of the shock leading to separation on theleeward side of the body. The vortex formed due to the roll-up of the shear layer is stronger at higher angles-of-attack. At lower angle-of-attack the vortex is closer to the body inducing a secondary vortex all along the

5

Figure 8: Cp vs. x/d on the leeward side for the threecases.

length. The flow field comparisons show that for Case II and Case III a second windward shock is presentat about x/d=10. A detailed analysis of the flow field has to be carried out to find the cause of this as wellas the influence of Mach number and Reynolds number on this phenomenon.

References

[1] Trevor J. Birch, N. Qin, and X. Jin. Computation of supersonic viscous flows around a slender body at incidence,1994. AIAA 94-1938.

[2] Eswar Josyula. Computational simulation improvements of supersonic high-angle- of-attack missile flows. Jour-nal of Spacecraft and Rockets, 36(1):59–65, 1999.

[3] D. Degani and L.B. Schiff. Computation of turbulent supersonic flows around pointed bodies having crossflowseparation. Journal of Computational Physics, 66:173–196, 1986.

[4] Trevor J. Birch, Ian E. Wrisdale, and Simon A. Prince. CFD predictions of missile flowfields, 2000. AIAA2000-4211.

[5] M. S. Baldwin and H. Lomax. Thin-layer approximation and algebraic model for separated flow, 1978. AIAA78-257.

[6] Walter B. Sturek, Duane Frist, Malcolm Taylor, Hugh Thornburg, and Bharat Soni. Navier-stokes predictions ofmissile body separated flow fields. In M. Hafez and K. Oshima, editors, Computational Fluid Dynamics Review1998, volume 2, pages 734–745. World Scientific, 1998.

[7] S. K. Saxena and K. Ravi. Some aspects of blunt body flow computations with Roe scheme. AIAA Journal,33(6):1025–1031, 1995.

[8] S. K. Saxena and Manoj T. Nair. Implementation and testing of Spalart-Allmaras model in a multi-block code,2002. AIAA 2002-0835.

[9] Manoj T. Nair, Abdul M. Rampurawala, and S. K. Saxena. MB-EURANIUM User’s Manual. Technical ReportPD CF 0110, National Aerospace Laboratories, Bangalore, India, 2001.

[10] P. R. Spalart and S. R. Allmaras. A one equation turbulence model for aerodynamic flows, 1992. AIAA 92-0439.

[11] P. L. Roe. Approximate Riemann solvers, parameter vectors,and difference schemes. Journal of ComputationalPhysics, 43:357–372, 1981.

[12] S.A. Prince and N. Qin. Mechanism of windward vortex shocks about supersonic slender bodies. The Aeronau-tical Journal, 106(1063):507–518, 2002.

6

Proceedings of the Tenth Asian Congress of Fluid Mechanics 17-21 May 2004, Peradeniya, Sri Lanka

THE SUPERSONIC FLOW AND MIXING FIELDS WITH MAINSTREAM ANGLE AND MACH NUMBER OF INJECTOR

Mohammad Ali and S. Ahmed

Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka - 1000, Bangladesh

ABSTRACT: A parametric study of main flow inlet angle and side jet Mach number for Supersonic Combustor has been conducted by solving Two-Dimensional full Navier-Stokes equations. An explicit Harten-Yee Non-MUSCL Modified-flux-type TVD scheme has been used to solve the system of equations, and a zero-equation algebraic turbulence model to calculate the eddy viscosity coefficient. The mainstream is air and gaseous hydrogen is injected from the side wall of the combustor. Air stream angle is varied with a range of 60° ~ 120° with the backward facing step under the inlet port and the Mach number of the injector with a range of 0.7 ~ 1.3. The objective of this study is to find the means of increasing the mixing efficiency in a supersonic flow. On the effect of air stream angle, strong interaction between main and injecting flows can be observed for smaller angle causing sharp increase in mixing efficiency on the top of injector. Also high momentum of air stream towards the side wall causes no recirculation at the upstream of injector and the system becomes unable for flame holding. For the variation of Mach of injector, large and elongated upstream recirculation is found for high Mach, which causes high prenetration and mixing of hydrogen dominated by convection of recirculation. Among the cases, the configuration of moderate air stream angle with high Mach of injector shows higher mixing efficiency and good flame holding capability. 1. INTRODUCTION Good mixing and stable combustion is the crying need to design an efficient combustor for hypersonic vehicles. Particularly, the fuel injection scheme in hypersonic vehicles incorporating Supersonic Combustion Ramjet (Scramjet) engines, requires special attention for efficient mixing and stable combustion. Though a considerable number of researches has been carried out on mixing and combustion of fuel with oxidizer in Scramjet program, still it faces many unresolved problems. The main problems that arise in this regard, concern mixing of reactants, ignition, flame holding, and completion of combustion. More investigations are required to overcome these problems. In fact, in supersonic combustion, high penetration and mixing of injectant with main stream is difficult due to their short residence time in combustor[1]. There exist several methods of fuel injection in the Scramjet propulsion system. Perpendicular injection causes rapid mixing of injectant with main stream and is used to some degree at all flight Mach numbers to promote mixing and reaction, particularly in upstream portion of the combustor. Parallel injection is used when slow mixing process is desired, specially at lower speeds of space vehicles during which perpendicular injection may cause too rapid mixing and combustion, and hence thermally chocking the flow. We used perpendicular injection due to (i) its extensive use in Scramjet program, and (ii) high Mach (M = 5) of main flow that has been selected for the present study. Both experimental and numerical investigations have been performed to analyse the mixing and combustion characteristics, and find out the means of increasing the mixing efficiency. In these investigations the authors showed a number of parameters that can affect on penetration and mixing. In an experiment, Rogers[2] showed the effect of the ratio between jet dynamic pressure and freestream dynamic pressure on the penetration and mixing of a sonic hydrogen jet injected normal to a Mach 4 airstream. In similar flow arrangements, Kraemer et al.[3] found that the relative change in jet momentum was directly proportional to the relative size between the flowfield disturbance and the upstream separation distance. Heister et al.[4] conducted a calculation on the penetration and bow shock shape of a non-reacting liquid jet injected transversely into a supersonic cross flow and obtained a correlation between mass loss, boundary layer thickness, recirculation and related parameters. Ali et al.[5-6] and Ahmed et al.[7] studied the mixing mechanisms and investigated mixing and combustion characteristics for several flow configurations. On the analysis of mixing the author observed that the backward-facing step in finite flow configuration plays an important role to enhance mixing and penetration in both

upstream and downstream of injector. In another study Ali et al.[8] searched the enhancement of mixing by varying the inlet width of air stream and found that the flow inlet configuration of a supersonic combustor can play an important role on mixing in supersonic flow. The present work is a part of investigation conducted by Ahmed[9] and has been searched the effect of air stream angle and Mach Number of the injector on mixing and flame holding capability in supersonic combustor. The geometric configuration of the calculation domain and the inlet conditions of main and injecting flows are shown in Fig.1. For this study, the air stream angle ‘ψ’ is varied by 60° taking as Case 1, 90° (Case 2), 120° (Case 3) when the Mach of injector is 1.0 and injection Mach Number is varied as 0.7 (Case 4), 1.0 (Case 5) and 1.3 (Case 6) when the air stream angle is taken as 90°. The left boundary consists of a backward-facing step of height 5 mm which was found most efficient in mixing by Ali[5]. The inlet conditions of air are used as Weidner et al.[10] except Mach number. We choose Mach 5.0 for the main flow as the test program has been conducted over the flight Mach number range[11] from 3.0 to 7.0. The inlet widths of air and side jet are used as Ali et al.[8] which showed good performance on mixing. 2. MATHEMATICAL MODELING The unsteady, two-dimensional full Navier-Stokes and species continuity equations have been solved to analyse the mixing flow field of hydrogen and air. Body forces are neglected. These equations can be expressed by

yxyxtvv

∂∂+

∂∂=

∂∂+

∂∂+

∂∂ GFGFU

where, U = [ρ,ρu,ρv,E,ρYi]T, F = [ρu,ρu2+p,ρuv,(E+p)u,ρYiu]T, G = [ρv,ρuv,ρv2+p,(E+p)v,ρYiv]T,

Fv = [0,σx,τxy,σxu + τyxv + qx, •m x]T, and Gv = [0,τyx,σy,τxyu + σyv + qy, •m y]T. The details of equations and the calculation of its different parameters are shown by Ali[5]. The fluid dynamics is solved using an explicit Harten-Yee Non-MUSCL Modified-flux-type TVD scheme proposed by Yee[12]. The backward-facing step makes the flowfield turbulent at the present Mach number. Particularly, the recirculations in both upstream and downstream of injector, shocks, and expansion of both main stream and side jet leads us to use a turbulence model. Therefore, to calculate eddy viscosity we selected the zero-equation turbulence model proposed by Baldwin and Lomax [13]. 3. RESULTS AND DISCUSSION 3.1 Effect of airstream angle The present study consists of six cases varying air stream angle and injector Mach number among which case 2 and 5 are common in comparing with the varying parameters. Figure 2 shows the velocity vector in both upstream and downstream of injector for the variation of air stream angle. For Case 1 (ψ=60o), the free air stream strikes the wall near injector with a high momentum. At the region of striking there is a very strong interaction between the main flow and the side jet and the flow is deflected at an angle more than 60o in the upward direction. Due to strike of the main flow at upstream region with high momentum, there is no recirculation in upstream of injector. Case 3 (ψ=120o) shows that the interaction between the main flow and injecting hydrogen is very weak and there is no upstream recirculation for this case as the main flow deflected upward after weak interaction with the side jet. In downstream there is no recirculation in cases 1~3. Another observation is that for Case 2 (ψ=90o) the injecting jet plume expands due to the early separation of boundary layer which inceases the thickness of the boudary layer. Figure 3 shows the penetration and mass concentration of hydrogen in the flow field where different penetration at both upstream and downstream among the configurations can be found. In Case 1 we see that there is no hydrogen in upstream of the injector because a high momentum main flow strikes the wall near the injector resulting in no recirculation at that region. It is then deflected away at an angle more than 60o with the horizontal direction and shows that the mixing of main flow and side jet is maximum at the top of the injector as shown in Fig. 4. We know that recirculation is an important factor for mixing of hydrogen with air in upstream region and as there is no recirculation, no mixing of hydrogen and air in upstream of the injector occurs. Due to primary and secondary upstream recirculations, Fig. 3(b) indicates

that there is good mixing between air and hydrogen in upstream of the injector. Again we know that the flame holding requires longer residence time of flame in the burning range and this residence time strongly depends on the geometric expansion of the recirculation zone[14]. Accordingly Case 2 has a good flame holding capability because it can produce larger and elongated upstream recirculation where most of the region contains good propertion of hydrogen and oxygen. Figure 4 shows that mixing efficiency increases sharply at injector position of respective cases. Generally in upstream region the increasing rate of mixing is moderate and in downstream it is very slow. Individullay Case 1 has the highest increment of mixing efficiency at injector postion due to large gradient of mass concentration and the mixing efficiency up to 0.03 m from left boudnary is zero. Because of high momentum of air stream, there is no hydrogen in upstream of the injector which makes the mixing efficiency zero. Case 3 has not any increment of mixing efficiency in downstream of injector which indicates that the unnecessary increasing of combustor length in far downstream will only increase the material cost for construction. In downstream, the increasing rate of mixing is slower for all cases caused by the supersonic nature of flow. However, among the cases investigated, Case 1 has the maximum increasing rate of mixing in downstream. 3.2 Effect of Injector Mach Number Figure 5 shows the velocity vector in both upstream and downstream of injector for varying Mach number (0.7, 1, 1.3) of the injector. Strong interation is occuring between the main and injecting flow in case 6 (Mach 1.3) shown in Fig. 5 (c). By seeing the slope of vectors just at the top of injetor the strength of interaction can be understood. For all cases (cases 4~6) in upstream there are two recirculations. One is primary caused by the backward facing step and the other is secondary due to primary recirculation and suction of injection. As the upstream recirculation plays an important role in mixing, so better mixing is obtained in upstream for Mach No. 1.3. With the increase in Mach number of the injecting hydrogen the recirculations are increasing in areas and the primary one expands towards left and for case 6 (Mach 1.3) it touches the left boundary. In downsteam two features are noticible: (i) no strong recirculation exists in any case, and (ii) for large Mach (case-6) the injecting jet is bent sharply into upward direction caused by strong interaction between main and injecting flows. Figure 6 shows that there is difference in penetration at both upstream and downstream among the cases. For high Mach (case 6) large and elongated upstream recirculation causes high prenetration dominated by convection of recirculation. At the same time due to strong interaction high gradient of hydrogen mass concentration exists causing high penetration of hydrogen. The mass concentration of hydrogen can be explained separately for upstream and downstream region. Due to primary and secondary recirculations most of the upstream region contains high concentration of hydrogen for all cases (cases 4~6). In downstream due to strong interaction for case 6 the inejcting jet bends more in upward direction resulting in more penetration height of hydrogen (above 0.04m) in downstream indicating more uniform distribution of hydrogen. However, case 6 has good flame holding capability because it can produce larger and elongated upstream recirculation where most of the region contains good proportion of hydrogen and oxygen. In far downstream the hydrogen distribution is seemed to be better (more uniform) in case 6. This uniform distribution of hydrogen is caused by higher expansion of side jet. Figure 7 shows the mixing efficiency along the length of physical model for different cases (case 4~6). Figure 7 shows that mixing efficiency increases very sharply at injector position of respective cases. Generally in upstream region, the increasing rate of mixing is moderate and in downstream it is slow. Individually, case 6 has the highest increment of mixing efficiency at inejctor position due to strong upstream recirculation and high interaction between free stream air and side jet. In downstream, the increasing rate of mixing is slower for all cases caused by the supersonic nature of flow and among the cases investigates, case 4 has the maximum increasing rate of mixing in donwstream. However, the mixing efficiency of case 6 is higher than that of cases 4 and 5 on the top of injetor as well as at the end of the physical domain. 4. CONCLUSIONS The present paper gives some information on mixing of fuel with oxidizer and flame holding capability of the combustion flow field due to variation of air stream angle and Mach of injector. It is

Inlet condition of Air Pressure = 0.101 MPa Temperature = 800.0 K Mach = 5.0

Inlet condition of Hydrogen Pressure = 1.818 MPa Temperature = 1128.0 K Mach (M) = 0.7 ~ 1.3

found that the mainstream with smaller angle shows higher mixing efficiency but has no recirculation in upstream of injector which is much important for flame holding capability. On the other hand, higher stream angle causes higher expansion of the side jet and the upstream of injector is seemed to be stagnant. With the increase of injector Mach Number, mixing efficiency increases continuously without loss of flame holding capability. Therefore, it can be concluded that the parallel (‘ψ’=90º) air stream with high Mach of injector could be a good combustor configuration where efficient mixing and good flame holding are possible. REFERENCES [1] Brown GL and Roshko A: On Density Effects and Large Structure in Turbulent Mixing Layer. J. Fluid

Mechanics, 1974, 64 (4), 775-816. [2] Rogers RC. A Study of the Mixing of Hydrogen Injected Normal to a Supersonic Airstream. NASA TN D-

6114, 1971. [3] Kraemer GO and Tiwari SN: Interaction of Two-Dimensional Transverse Jet with a Supersonic Mainstream.

NASA CR 175446, 1983. [4] Heister SD, Nguyen TT and Karagozian AR: Modeling of Liquid Jets Injected Transversely into a Supersonic

Crossflow. AIAA Journal, 1989, 27 (12), 1727-1734. [5] Ali, M, Fujiwara, T and Leblanc, JE: The Effects of Backward-Facing Step on Mixing and Flame Holding in

Supersonic Combustor, Journal of Energy, Heat and Mass Transfer, 2001, 23, 319-338. [6] Ali, M, Fujiwara, T and Pervez, A: A numerical study on the physics of mixing in two-dimensional

supersonic stream. Indian Journal of Engineering and Materials Sciences, 2002, 9, 115-127. [7] Ahmed, S, Ali, M and Islam, AKMS: The Effect of Injection Angle on Mixing and Flame Holding in

Supersonic Combustor, Int. Journal of Thermal and Fluid Sciences, 2002, 11 (1), 80-91. [8] Ali M and Islam AKMS. Effect of Mainflow Inlet Width on Penetration and Mixing of Hydrogen in Scramjet

Combustor. Proceedings of the Eighth Asian Congress of Fluid Mechanics, December 6-10, Shenzhen, China, 1999, 647–650.

[9] Ahmed, S: A Numerical Study on the Mixing of Hydrogen in Supersonic Air Stream. M.Sc.Engg. Thesis, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, 2001.

[10] Weidner EH and Drummond JP: A Parametric Study of Staged Fuel Injector Configurations for Scramjet Applications. AIAA Paper 81-11468, 1981.

[11] RauschVL, McClinton CR and Hicks JW: Scramjet Breath New Life into Hypersonics. Aerospace America. July, 1997.

[12] Yee HC: A Class of High-Resolution Explicit and Implicit Shock Capturing Methods, NASA TM -101088, 1989.

[13] Baldwin BS and Lomax H: Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows. AIAA Paper 78-257, 1978.

[14] Tabejamaat SJUY and Niioka T: Numerical Simulation of Secondary Combustion of Hydrogen Injected from Preburner into Supersonic Airflow. AIAA Journal, 1997, 35(9).

Fig. 1 Schematic of the physical model; Case 1: ψ=60º, M=1.0; Case 2: ψ=90º, M=1.0; Case 3: ψ=120º, M=1.0; Case 4: ψ=90º, M=0.7; Case 5: ψ=90º, M=1.0; Case 6: ψ=90º, M=1.3.

Case 1 Case 2 Case 3

Fig. 2 Velocity vector around injector; Fig. 3 Mole fraction contour of hydrogen; (a) Case 1, (b) Case 2 and (c) Case 3 φ(0.05,1.0), φ is contour level (a) Case 1, (b) Case 2 and (c) Case 3 Fig. 4 Mixing efficiency along the length of physical model

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AERODYNAMIC COMPUTATIONAL ANALYSIS FOR MACH 6.8 FLOW PAST A BLUNT NOSE WITH FORWARD-FACING SPIKE

S.Zahir 1, M.Arshad2, N.Aizud3, M.A.Khan4 and N.Kamran5

CFD-Chapter, Computational Modelling, Control and Simulation Society of Pakistan, PAKISTAN www.ccsspak.org

ABSTRACT: The current work investigates flow around a standard blunt nose with attached forward facing spike by computation of pressure coefficient, Cp distributions and determination of aerodynamic coefficients using CFD tools for Mach 6.8 flows, focus was maintained to provide Cp distribution and drag comparisons with the published data, while study was extended from supersonic to hypersonic range. Two different spike lengths have been examined to study the forebody flow field and its effects on static aerodynamic coefficients. It has been concluded that spikes reduce the aerodynamic drag due to reduced dynamic pressure on the nose caused by the separated flow on the spikes and that an increase in spike length also causes an increase in normal force coefficient, resulting in an adverse effect on the static stability at the same time subject to an overall predi ction accuracy of the order of ± 10%. NOMENCLATURE M Mach number M ∞ Freestream Mach number L Length of the body D Diameter of the body α Angle of Attack [degree] Re/L Reynold’s number per unit length CN Normal force coefficient Cm Pit ching moment coefficient Xcp/L Centre of pressure per unit length Caf Fore body axial force coefficient Cp Pressure coefficient 1. INTRODUCTION

Recent experimental investigations demonstrated that a nose tipped with spikes has a positive role in reducing aerodynamic drag but also shows an increase in lift for supersonic flows past blunt-nosed bodies [1]. To further investigate these effects computational simulations were made in the range of supersonic to hypersonic velocities [2], presently discussion is focused on features of pressure distribution and estimation of aerodynamic drag for Mach 6.8 flows. In recent past, experimental studies have been conducted to examine the forebody flow field of the spiked blunt body, whereas most of the experimental investigations conducted in the 1950’s concentrated on issues related to high pressures and heating rates around such configurations and possible mechanisms to reduce them. Stadler and Nielsen [3] carried out experiments on a hemisphere-cylinder configuration at freestream Mach numbers of 1.5, 2.67, and 5.0 and a Reynolds number in the range of 0.16 x106 to 0.85 x106 based on the diameter of the cylinder. Their experiments indicated a reduction in surface pressure and consequently, the drag, with the attachment of the forward facing spike. The experiments of Bogdonoff and Vas [4] indicated an initial drop in pressure at the nose of the forebody with increased spike length up to an L/D ratio of 3 (where L is the spike length and D is the cylinder’s diameter). Crawford [5] experimentally found the effect of the spike length on the nature of the flowfield, the surface pressure distribution, and the heat flux variation for a freestream Mach number of 6.8 and Reynolds number 0.12 x106 to 1.5 x106 based on the cylinder’s diameter.

These experimental investigations provide insight into the characteristics of the separated region created by an adverse pressure gradient and shock/boundary-layer interaction over the blunt nose. Unsteadiness of the flow caused by the spike of the blunt body was examined by Maull [6] in 1960. It was found that the shock wave around the body oscillates when the nose has a plain shape. Wood [7] investigated experimentally the flow field over the spiked cone and found that the shape and size of a region of separated flow is controlled primarily by the flow near the reattachment point. Fluctuating pressures in spike-induced flow-separation were observed experimentally by Guenther and Reding [8]. More recently, Shang [9] has provided experimental evidence of drag reduction by plasma injection giving similar treatment as of spikes in hypersonic flows. The features of the supersonic flow field can be delineated through these experimental/numerical investigations, in particular for the present study for Mach number of 6.8, the flow is characterized by a conical shock wave from the tip of the spike, a reattachment shock wave on the blunt body, and a separated flow region ahead of the blunt nose. A schematic of the flowfield over the spiked blunt body is shown in Fig. 1a. This flow field has been numerically investigated and physical aspects of the flowfield have been described using PAK-3D [10] a Computational Fluid Dynamics (CFD) software based on NS solver. However, flow fields around a spiked blunt body associated phenomena and its aerodynamic interpretations that have not been described previously were investigated in earlier studies by Zahir, et al. [11]. In continuation of the previous work, the present study only concerns the aerodynamic coefficient estimation for flow past spiked blunt bodies in supersonic to hypersonic Mach range in general while Cp distribution and drag comparisons at Mach number 6.8 in particular for the two different spike lengths and varying angle of attacks for the purpose of verification of computational results through published numerical/ experimental data comparisons. 2. GEOMETRY The dimensions of the blunt body with different spike lengths considered are shown in Fig. 1b and are the same as investigated by Crawford [5], Mehta [12] and Yamauchi [13]. It is an axisymmetric geometry, with a hemispheric nose cylinder of base diameter D of 7.62x10-2m and cylinder length of 4 D. The spike consists of a conical part and a cylindrical part, with a half-cone angle of 10 degrees and cylinder diameter of 0.1 D. The spike lengths, L, considered are 0.5 D and 1.0 D. 1a . Schematic of flowfield over spiked blunt body 1b. Dimensions of the spiked blunt body

Fig. 1: Geometry and schematic flow field of spiked blunt nose

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0.1 D

M∞

3. GRID GENERATION Grid generation for the current problem was done using the grid generation software PAKGRID [14]. The flow over the configuration is at two degrees of angle of attack and with no side-slip angle, so half the full geometry was modelled, and a symmetry plane was applied through the centre of the geometry. Optimized node distribution was employed and a paraboloid grid was used. The grid was adequately refined near the walls to obtain viscous solutions. For grid independence and to meet satisfactory convergence criterion, three independent grids were generated and finally a grid 132 x 52 x 70 was selected for further studies. For the turbulent viscous computations, a single-block approach was used. The present numerical analyses were performed on a grid containing one block i.e., single block containing cylindrical body and the spike with 0.48 million nodes: in the axial direction, 132 from the tip of the spike to the aft of the cylindrical configuration (52 from the tip of the spike to the nose, 80 from the nose tip to the tail of the cylinder), 52 in the radial and 70 in the circumferential directions. Different algebraic transfinite interpolation (TFI) methods were used on the domains containing the body of the model and on both the symmetric planes, shown in the Fig. 2. 4. BOUNDARY / INITIAL CONDITIONS Practically for all the computational cases, the freestream inflow condition was used on the inflow boundary, a 3-D averaging boundary condition was applied on a plane of symmetry, the extrapolation condition was used on the outflow plane, and no-slip adiabatic wall conditions were applied on the surfaces of the nose-cylinder and the spike. 5. ESTIMATIONS / CFD COMPUTATIONS Estimation of aerodynamic coefficients were first made using preliminary aerodynamic coefficients calculation, engineering analysis based software PAC [15] for supersonic Mach range for a blunt nose without spike to provide a preliminary base-line aerodynamic data only. For both the spikes, aerodynamic coefficient CN, Cm and Xcp/L were estimated and plotted against Mach number for angles of attack of 2 and 6 degrees. Further, CFD computations were conducted at the Mach number of 6.8 with zero angle of attack for both the spike lengths in order to make CD comparison with the published experimental data. To investigate the trends of other aerodynamic coefficients, computations were made at Mach numbers of 5.0, 6.8 and 8.0 with the Reynold’s number per metre of the order of 1 x 106, to study behaviour of CN, Cm and Xcp/L with increasing spike length at different angle of attacks.

Fig. 2: Typical Grid generated for the blunt-spiked nose geometry with spike length of 0.5 D

Axisymmetric compressible 3-D Navier-Stokes equations were solved to get the viscous solutions. Baldwin Lomax turbulence model was used. All computations were carried out using CFD solver PAK-3D

[10]. The post processing was done on the finally converged solutions using post -processing software LOOK [16]. The pressure contours were obtained for different spike lengths and comparisons were made for Mach number 6.8 conditions. Aerodynamic force and moment coefficients were also calculated. a. Aerodynamic Coefficients The static aerodynamic coefficients predicted using PAK-3D are the normal force coefficient, pitching moment coefficient and nondimensional centre of pressure location for Mach 5, 6.8 and 8.0. There is a steady increase of about 30% in normal force coefficient (CN) value with the increase in spike length and angle of attack for M 6.8, but there is also an insignificant decrease with increase in Mach number as shown in Fig.3.

CN VARIATION WITH MACH FOR BLUNT NOSE WITH SPIKES

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3 4 5 6 7 8 9MACH

CN

HEMP-PAC, AOA=2SPIKE-PAK3D, AOA=2, L/D=0.5HEMP-PAC, AOA=6SPIKE-PAK3D, AOA=6, L/D=0.5SPIKE-PAK3D, AOA=2, L/D=1.0SPIKE-PAK3D, AOA=6, L/D=1.0

Fig. 3: Normal Force Coefficient variation with Mach number The pitching moment coefficient (Cm) for both the spike lengths is calculated by using body length as a reference length. The change in pitching moment coefficient remains within around 5% for Mach 6.8 with the increase in spike length and angle of attack, while, a slight decrease is observed with the increase in Mach number, as shown in Fig.4.

Fig. 4: Pitching Moment Coefficient variation with Mach number

Cm VARIATION WITH MACH FOR BLUNT NOSE WITH SPIKES (20% error bar)

-0.25

-0.19

-0.13

-0.07

-0.01

0.05

0.11

0.17

3 4 5 6 7 8 9MACH

Cm


Spike length / Body diameter

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

CrawfordYamauchi et al.

MehtaPAK-3D

The static stability i.e., Xcp/L of the body decreases by about 25% at Mach 6.8 with increase in spike length and angle of attack as shown in Fig. 5. This decrease in static stability is subsequent to an increase in the normal force coefficient.

Fig. 5: Non-dimensional Centre of Pressure variation with Mach number The axial drag coefficient predictions were compared with the experimental data of Crawford [5] and computational data of Mehta [12] and Yamauchi [13] as shown in Fig. 6. The comparison shows that PAK-3D results are within 10% with the experimental data of Crawford [5] and 3 ~ 8 % with the computational results of Mehta [12] and Yamauchi [13].

Fig. 6: Drag comparisons of PAK-3D results with Crawford [5], Mehta [12] and Yamauchi [13] at Mach 6.8

b. Cp Distribution The formation of conical shock at the tip of the spike causes a low pressure region for Mach number 6.8. The length of low pressure region increases with the increase in spike length results in the shifting of centre of pressure location more towards nose of the body. This shifting of centre of pressure location causes the decrease in stability. The strong shock in case of a short spike length of 0.5 D changed into a system of conical waves with the increase in spike length as in 1.0 D case. As a result of this change in flow structure, the total pressure increases with the increase in spike length as shown in Fig. 7.

Xcp/L VARIATION WITH MACH FOR BLUNT NOSE WITH SPIKES (20% error bar)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

3 4 5 6 7 8 9

MACH

Xcp/

L


Spike length / Body diameter

(CD)

spike

/ (C

D) n

on sp

ike

Fig. 7: Cp distribution at Mach 6.8, α = 2.0, spike lengths 0.5 D, 1.0 D

6. RESULTS AND DISCUSSION Pressure contours as shown in Fig. 8 are computed flow fields for the spike lengths of L/D = 0.5 and 1.0 at a Mach number of 6.8 with an angle of attack of 2.0 degrees. Pressure contours represent interaction between the conical oblique shock wave starting from the tip of the spike and the reattachment wave and shear layer from the interaction are shown behind the reattachment shock wave. The pressure contours also represent a separated flow region in front of the blunt body causing low pressures at the blunt nose. Integration of pressure distribution yields normal force coefficient and enables the calculation of pitching moment coefficient for different spike lengths. At Mach 6.8 the normal force coefficient increases by about 30% with the increase in spike length 0.5 D to 1.0 D, refer Fig. 3, whereas the pitching moment coefficient remains within 5%, refer Fig. 4. Centre of pressure location is shifted forward due to this typical flow behaviour and this effect is calculated in terms of Xcp/L, a corresponding decrease in static stability of about 25% is observed with the increase in spike length 0.5 D to 1.0 D, refer Fig. 5.

Fig. 8: Pressure contours plots at freestream Mach Number 6.8 and α = 2.0

Fig. 9: Velocity vectors at freestream Mach number 6.8 and α = 2.0 The velocity vectors were also plotted as shown in Fig. 9 to examine the flow behaviour at the blunt nose due to different spike lengths at Mach number 6.8 and at an angle of attack of 2 degrees. The figure shows that the flow patterns remain similar for different spike lengths. According to these computed flow fields, it was observed that the angle of the conical shock wave is same for both the cases, while the projected angle formed at the hemispheric nose surface with the formation of a recirculation region enhances with the increase in spike length 0.5 D to 1.0 D. 7. CONCLUSIONS The flow fields around forward facing spikes attached to a blunt hemisphere-cylinder nose tip have been calculated using CFD code PAK-3D for a freestream Mach number of 6.8 with different spike lengths. The flow features around the spiked blunt body are characterized by a conical shock wave emanating from the spike tip, a separated region in front of the blunt body and the resulting reattachment shock wave. The aerodynamic coefficients have been calculated in the supersonic to hypersonic range with spike length between 0.5 D and 1.0 D at an angle of attack of 2 degrees, comparisons at Mach 6.8 showed that as the spike length is increased from 0.5 D to 1.0 D, the normal force coefficient increases by 30% while the pitching moment coefficient remains within 5% due to which a corresponding decrease in static stability, Xcp/L is of the order of 25% and axial drag comparisons are within 10% with the experimental data of Crawford [5] and 3 ~ 8 % with the computational results of Mehta [12] and Yamauchi [13]. It is concluded in this study for a hypersonic flow past a blunt nose, there is a reduction in drag with the addition of a forward facing spike, while further decrease in drag was also observed with the increase in spike length. However, at the same instance, significant loss of aerodynamic static stability is observed; practically this loss needs to be considered along with the drag reduction for optimized spike length iterations.

REFERENCES [1] Milicev, S.S., and Pavlovic, M. D., “Influence of Spike Shape at Supersonic Flow Past Blunt-Nosed Bodies: Experimental Study”, AIAA Journal Vol. 40, No. 5, May 2002, pp. 1018-1020. [2] S.Zahir, Nasir, M.Arshad & N.Aizud “Computational Investigation of Aerodynamic Forces Experienced by A Blunt Body with Forward-Facing Short Spikes in Hypersonic Flows”, Second IBCAST CFD-2003, 16-21 June 2003. [3] Stadler, J.R., and Nielson, H. V., “Heat Transfer from a Hemisphere-Cylinder Equipped with Flow-Separation Spikes”, NACA TN 3287, Sept. 1954. [4] Bogdonoff, S.M , and Vas, I. E., “Preliminary Investigations of spiked Bodies at Hypersonic Speed,” Journal of the Aerospace Sciences, Vol. 26, No. 2, 195, pp. 65 -74. [5] Carwford, D. J., “Hypersonic Flow over a spiked-None Hemisphere Cylinder at a Mach Number of 6.8,” NASA TN D-118, Dec. 1959. [6] Maull, D. J., “Hypersonic flow over Axially Symmetric Spiked Bodies,” Journal of Fluid Mechanics, Vol. 8, pt. 4, 1960, pp 584 -592. [7] Wood C. J., “Hypersonic Flow over spiked Cones,” Journal of Fluid Mechanics, Vol. 12, Pt. 4, 1962, pp. 614-627. [8] Reding J.P., Guenther, R. A., and Richter, B. J., “Unsteady Aerodynamic Considerations in the Design of a Drag-Reduction spike,” journal of spacecraft and Rockets, Vol. 14, No.1,1977, pp.54 -60. [9] Shang, J. S. “Plasma Injection for Hypersonic Blunt -Body Drag Reduction”, AIAA Journal, Vol. 40, No. 6, June 2002, pp.1178-1181. [10] User’s Manual PAK-3D, ASAC Pakistan. [11] Nasir, S.Zahir & M.A.Khan, “Computational Investigations of Blunt Body Drag-Reduction Spikes in Hypersonic Flows”, Proceedings of the 11 th Annual Conference of the CFD Society of Canada. Vancouver, Canada, May 28-30, 2003. [12] R.C. Mehta “Numerical heat Transfer Study over Spiked Blunt Bodies at Mach 6.8”, AIAA Paper, May 2000. [13] Yamauc hi, M., Fujii., K. Tamura, Y., and Higashino F., “ Numerical investigation of supersonic Flows Around a

-0887, Jan. 1973. [14] User’s Guide PAKGRID, ASAC Pakistan. [15] User’s Manual PAC, ASAC Pakistan. [16] User’s Manual LOOK, ASAC Pakistan.

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NUMERICAL SIMULATION OF FLOW PAST SINGLE AND TWO SQUARE CYLINDERS IN CROSS FLOW

D. Nachiappan*, T. Sundararajan†and B.H.L. Gowda*

* Department of Applied Mechanics † Department of Mechanical Engineering Indian Institute of Technology Madras, India

ABSTRACT: In the present paper, the vortex shedding behind one and two square cylinders has been simulated numerically. The unsteady, two-dimensional, incompressible Navier-Stokes equations are solved in integral form, on non-orthogonal control volumes. The method employs general bi-directional interpolation schemes for evaluating fluxes through non-orthogonal cell boundaries. A semi-staggered grid is used for domain discretization. The drag and lift coefficients as well as the Strouhal number calculated from our numerical data for Re ≤ 1000 were compared with available experimental and numerical results for flow over an isolated square cylinder and good agreement has been observed. Flow past two square cylinders in staggered arrangement is studied for Re=200. The drag and lift coefficients on both upstream and downstream cylinders together with their vortex shedding frequencies and flow patterns have been predicted for different spacings between the cylinders. 1. INTRODUCTION

The oscillation of chimney stacks and other structures in transverse flows is caused by vortex shedding. Although several studies exist for flows over circular cylinders, published literature with regard to flows across non-circular objects is relatively less; also multi-body configurations where the flow in the wake of one body is influenced by a neighbouring body have not been analyzed in detail. Franke et al. [4] showed that different flow regimes can be distinguished for flow over a square cylinder with increasing Reynolds number. The range of critical Reynolds number values for the onset of a Karman vortex street with periodic vortex shedding from the cylinder was reported by numerous researchers. Davis and Moore [3] studied the flow over a square cylinder with confined walls numerically as well as experimentally, and concluded that shedding frequency strongly depends upon the inlet velocity profile. Breuer et al. [1] used Lattice-Boltzmann and Finite Volume methods for confined flow over a square cylinder at various blockage ratios. A comparison of results from the above studies indicates that a large scatter exists in the data for even integral parameters such as Strouhal number and drag coefficient. Liu and Chen [6] studied the interference effects between two cylinders in tandem arrangement from wind tunnel measurements.

Progress of research on flow interference effects between two square cylinders has been slow due to the fact that obtaining numerical solutions for this problem is very difficult. In the present study, flow past single and two square cylinders has been modeled numerically. The transient Navier-Stokes equations are solved using the Finite Volume Method with a primitive variable formulation in conjunction with non-orthogonal cells. Validation of the numerical prediction for laminar 2-D flow past a square cylinder has been carried out. Global characteristics such as the Strouhal Number of vortex shedding, the drag and lift coefficients were calculated and compared with available experimental and numerical results. The code was later extended to simulate flow around two cylinders in staggered configuration which has not been dealt much in literature. 2. METHOD OF SOLUTION

A Finite Volume based explicit numerical algorithm for solving two-dimensional, unsteady, incompressible Navier-Stokes equations using non-orthogonal cells has been recently developed. As the detailed scheme is reported elsewhere, (Suyambazhahan et al. [10]) only the application of the algorithm for simulating the vortex shedding phenomena behind square cylinders is described here. The method employs a semi-staggered mesh as shown in Fig.1. The nodal pressures are evaluated from the

satisfaction of continuity equation in a quadrilateral cell with velocity nodes as vertices. The mass balance equation is converted into a pressure Poisson equation and solved to update the nodal pressures implicitly. For calculating the cell face fluxes and stress components, a generalized interpolation method is employed as typically done in the Finite Element Method. The flow variables are expanded as

4 4

1,i i i i

i iu N u v N v

= =

= =∑ ∑ and 4

1i i

ip N p

=

= ∑

where Ni are the shape functions used for interpolation. With the help of such generalized interpolation, the gradients of velocity components and pressure are also evaluated wherever required in the quadrilateral cell.

After predicting velocity components and pressure values by applying the momentum and mass balance principles over respective control volumes (Fig .1), post processing has been carried out to calculate the lift and drag forces by integrating the stresses on the cylinder surface. The instantaneous streamlines have also been obtained from the velocity field. Vortex shedding frequency computed from Fast Fourier Transform (FFT) procedure from the time evolution of lift coefficient. 3. NUMERICAL RESULTS FOR A SINGLE SQUARE CYLINDER

The flow solver algorithm was tested by simulating flow past a square cylinder for the Reynolds number (U∞.D/υ, where D is side face of square cylinder) range of 50-1000. The inflow, top and bottom boundaries have been located at ten times the side face of cylinder from the centre of the cylinder. The outlet boundary where wake phenomena are prevalent has been placed at a distance of 23 times the side face of cylinder, for minimizing the effects of outlet boundary conditions on flow field near the cylinder. Uniform velocity profile is used as the inlet boundary condition whereas on top and bottom boundaries, shear free boundary conditions are applied. At the outlet, extrapolation has been used for all the solution variables from the interior flow field. A non-uniform grid has been employed with a fine mesh around the cylinder. The dimensionless distance to the nearest grid point from the cylinder wall was taken as 0.007. A detailed grid independence study was also done with three different grids, to assess the dependence of the predicted solution on the grid employed. Finally a mesh with 328 x 128 grid points was selected for all the simulation.

Figure 2 gives a comparison of different experimental and numerical results reported in literature for the free stream flow over an isolated square cylinder. The simulated results of the present study for the Strouhal number (St) and drag coefficient (CD) compare well with the experimental results of Okajima [8] and Davis and Moore [3]. A gradual increase in Strouhal number has been found with Reynolds number and the maximum value is attained at Re ≈ 500. The occurrence of a peak St value was observed by Davis and Moore [2] at Re ≈ 400. Some of the numerical studies (Suzuki et al. [11], Li et al. [7]) have reported that the peak value occurs at Re ≈ 250. The trend of St variation with Re and also the range of Strouhal number values predicted in the present work agree reasonably with those reported in the literature. The wide scatter in the available data illustrates the level of uncertainty on even gross parameters in unsteady flow. The behaviour of time averaged drag coefficient as a function of Reynolds number is illustrated in Figure 2(b). Initially there is a decrease in the drag coefficient and it reaches a minimum value at a Reynolds number of about 150. The same feature has been reported by Franke et al. [4] also. While

comparing with the experimental results of Davis and Moore (1984), it is observed that the present drag coefficient predictions agree excellently upto the Reynolds number of 500. For Re>500, slight deviation has been found in CD value probably due to the need for finer grids in the boundary layer region. The critical Reynolds number (in which the symmetric pattern of vortices behind the cylinder changing into alternate formation of vortices) for vortex shedding has been found from our calculations as approximately equal to 50. Kelkar and Patankar [5] determined this critical value as Recrit ≈54. Figure 3 shows the time evolution of lift coefficient and its FFT for the Reynolds number of 200. A periodic variation of lift is observed due to the occurrence of alternate vortex shedding from the front corners of the square cylinder. The corresponding FFT (Fig.3. b) indicates a single and sharp frequency peak, implying that vortex shedding is exactly periodic at this Re.

4. NUMERICAL RESULTS FOR TWO SQUARE CYLINDERS IN STAGGERED ARRANGEMENT

For these simulations, the downstream cylinder is located at 45° to the upstream cylinder center and the spacing interval ranges between 1.5 to 4.5 times the side face of the cylinder (Interval Sizes are varied such that L/D=T/D, where D = side face of square cylinder, L = Streamwise spacing between the cylinder centers and T = Transverse spacing between the cylinder centers). The same boundary conditions and computational domain has been employed here as in the single cylinder case. Figures 4 (a,b) show the variations of drag and maximum lift coefficients for both cylinders at a Reynolds number of 200 with longitudinal or transverse spacing (L/D or T/D) between the two cylinders. In both graphs, upstream cylinder values of CD and CL,max are lower than those of the downstream cylinder (CD0 = drag coefficient for isolated cylinder). A strong biased gap flow occurs around the upstream cylinder due to the interference of the downstream cylinder. This biased flow pushes the vortices of the upstream cylinder and a reduced wake occurs (Fig.7.). Even at L/D (T/D) = 4.5, there is a difference between upstream and downstream cylinder drag coefficients. The interference effects appeared to persist almost permanently. The ratio of vortex shedding frequency for the two cylinder case (S) to that for the single cylinder case (S0) at the same Reynolds number has been plotted in Fig.5(a) for different spacings. For dimensionless spacing less than 2.0, the two cylinders function like a single composite body, with vortex shedding frequency less than that for a single cylinder case (S0). For L/D=T/D > 2.0, the shedding frequency for the upstream cylinder tends to increase above S0. This can be attributed to the fact that the space available for vortex formation in the wake region of the first cylinder is less due to the presence of the rear cylinder. Therefore, smaller vortices tend to form at higher shedding frequency (see Fig.7.). For the downstream cylinder, larger vortices tend to form at lower frequency, since there are no space constraints in the wake region. When the dimensionless spacing between the cylinders increases beyond 5, interference effects are significantly reduced and the shedding frequency values for both the cylinders becomes equal to the shedding frequency of isolated cylinder, S0. A plot of CD variation with time shows a beat-type phenomenon (with two distinct frequencies) for the downstream cylinder due to the asymmetry in vortex formation for the upper and lower half regions (Fig .6 & 7.).

5. CONCLUSIONS

In the present paper, numerical calculations of vortex shedding past an isolated square cylinder have been reported in the Reynolds number range of 50-1000 and good agreement is found with experiments for fluid force coefficients and shedding frequencies for all Reynolds numbers. Interference effects between two square cylinders in staggered arrangement are analyzed with the help of fluid force coefficients, shedding frequencies and flow patterns. When dimensionless spacing exceeds 2.0, the shedding frequencies of the two cylinders are different from each other. The shedding frequency of the upstream cylinder is higher than the single cylinder frequency S0, while the shedding frequency of the downstream cylinder is less than S0. The difference between the shedding frequencies of the two cylinders decreases with increase in cylinder spacing and it becomes negligibly small, when dimensionless spacing exceeds a value of about 5.0. Thus, the flow patterns and fluid force coefficients are greatly affected by the spacing between the cylinders.

0 100 200 300 400 500 600 700 800 900 10000.00

0.05

0.10

0.15

0.20

St

Re

Present Study Okajima(1982,exp) Davis&Moore(1982,exp) Davis&Moore(1982,num) Suzuki(1993,num) Li&Humphery(1995,num) Treidler(1991,num) Arnal et al.(1991,num) Franke et al.(1990,num)

Fig. 2. Variation of integral flow parameters with Reynolds number

(a) Strouhal number (b) Drag coefficient

0 100 200 300 400 500 600 700 800 900 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Present Study Franke et al.(1990,num) Davis&M oore(1982,num)Treidler(1991,num) Arnal et al.(1991,num) Li&Humphrey(1995,num) Davis&M oore(1983,Exp)

CD

Re

1.5 2.0 2.5 3.0 3.5 4.0 4.50.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14 Upstream Downstream

CD/C

D 0

L/D & T/D

1.5 2.0 2.5 3.0 3.5 4.0 4.5-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

upstream cylinder downstream cylinder

Max

CL

L/D & T/D

(b) Maximum Lift coefficient

Fig 4. Variation of fluid force coefficients with spacing between cylinders (L/D=Streamwise spacing T/D=Transverse spacing)

(a) Drag coefficient

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00

0.05

0.10

0.15

0.20

0.25

0.30 Re=200

Non

-dim

ensio

nal a

mpl

itude

Non-dimensional frequency

336 338 340 342 344 346 348 350-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4 Re=200

C L

Non-dimensional time

Fig 3. Time evolution of Lift Coefficient and its corresponding FFT at Re=200

L/D=T/D=1.5

L/D=T/D=2.5

L/D=T/D=3.5

t=24.0

t=18.0

t=32.0

t=31.0

t=23.0

t=36.0

Fig. 7. Temporal evolution of streamline patterns for various spacing intervals between cylinders

1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Upstream Downstream

S/S 0

L/D & T/DFig. 5. Variation of shedding frequency with cylinder spacing (Re=200)

220 225 230 235 240 245 2501.6

1.7

1.8

1.9

2.0

Upstream cylinder

Downstream cylinder

C D

Non-dimensional Time

Fig. 6. Time evolution of Drag coefficient for L/D=T/D=2.0

REFERENCES [1] Breuer M Bernsdorf J Zeiser T and Durst F: Accurate computations of the laminar flow past a square

cylinder based on two different methods: Lattice-Boltzmann and Finite Volume. Int J Heat and Fluid Flow. 2000, 21, 186-196.

[2] Davis RW and Moore EF: A numerical study of vortex shedding from rectangles, J Fluid Mech. 1982, 116, 475-506.

[3] Davis RW Moore EF and Purtell LP: A numerical-experimental study of confined flow around rectangular cylinders. Phy Fluids. 1984, 27, 46-59.

[4] Franke R Rodi W and Schonung B: Numerical calculation of laminar vortex-shedding flow past cylinders. J Wind Eng Ind Aero. 1990, 35, 237-257.

[5] Kelkar M and Patankar V: Numerical prediction of vortex shedding behind a square cylinder. Int J Num Meth Fluids. 1992, 14, 327-341.

[6] Liu CH and Chen M: Observations of hysteresis flow around two square cylinders in tandem arrangement. J Wind Eng Ind Aero. 1992, 90, 1019-1050.

[7] Li G and Humphrey JAC: Numerical Modelling of confined flow past a cylinder of square cross section at various orientations. Int J Num Meth Fluids. 1995, 20, 1215-1236.

[8] Okajima A: Strouhal numbers of rectangular cylinders. J Fluid Mech. 1982, 123, 379-398. [9] Okajima A: Numerical simulation of flow around rectangular cylinders. J Wind Eng Ind Aero. 1990, 33,

171-180. [10] Suyambazhahan S Raghavan V Nachiappan D Das SK and Sundararajan T: Development of a novel

algorithm based on non-orthogonal finite volumes for Transient Incompressible Fluid Flow. Int J Num Heat Transfer. Part B, 2002 – Communicated.

[11] Suzuki H Inoue Y Nishimura T Fukutani F and Suzuki K: Unsteady flow in a channel obstructed by a square rod. Int J Heat and Fluid Flow. 1993, 14(1), 2-9.

Proceedings of the Tenth Asian Congress of Fluid Mechanics17-21 May 2004, Peradeniya, Sri Lanka

STUDY OF ENTRAINMENT PROCESSIN A PLANAR JET USING DIFFUSION-VORTEX METHOD

Sreenivas K. R.Engineering Mechanics Unit, Jawaharlal Nehru Center for

Advanced Scientific Research, Bangalore, [email protected]

1. INTRODUCTION:A turbulent jet spreads in a direction normal to its primary flow (jet-axis) by incorporating

irrotational ambient fluid into the turbulent jet-flow; this process is known as entrainment. At sufficientdownstream distance, dynamics of a turbulent jet will be independent of its initial conditions. In this self-similar region, jet-dynamics only depends on the local velocity and length scales (seminal paper ofMorton et. al., 1956; MTT model). According to the MTT model local, inward velocity scale(entrainment velocity, Ue) is proportional to the time-averaged, centerline axial velocity (Vc) at thatlocation. The proportionality constant relating entrainment velocity (Ue) and centerline velocity (Vc) isknown as “entrainment coefficient”, (α); that is, Ue = α VC.

The entrainment relation is universal and valid for many planar and axisymmetric turbulent free-shear flows like wake, plume and jet in the self-similar region. MTT model is highly successful and isapplicable over a wide range of Reynolds numbers from few thousands in the lower limit to 1018 in theupper limit (Turner [1988]). The numerical value of the entrainment coefficient (α) is different fordifferent types of flows. For example, the value of α in an axisymmetric jet is 0.05 and in anaxisymmetric plume it is 0.08 (Turner [1986]). Rate of entrainment and hence the entrainment coefficientalso depends on other factors like off-source buoyancy addition (Bhat and Narasimha [1996],Venkatakrishnan et. al. [1998]), axial acceleration due to applied magnetic field for an electricallyconducting fluid-jet and axial pressure gradient (Choi et. al., [1986]). Ambient viscosity can also affectthe rate of entrainment (Campbell and Turner [1985], Manikandan and Sreenivas [2003]). Even thoughMTT model introduces the concept of entrainment coefficient (α), the model does not suggest the theoryfor variations in the numerical value of α for different types of free shear flows. Experiments are the onlyway to determine the value of α.

There are models proposed by other researchers, for example Priestly and Ball [1955], Lumley[1977] and Hunt [1994], which are used to explain observed higher entrainment in plume compared tothat in jet. However, these models fail to explain observed lower entrainment in off-source buoyancyadded flows or in flows with favourable pressure gradient. Govindarajan [2002] imposed a relativevelocity for a pair of vortices with respect to its ambient and analyzed effect of the relative velocity on theentrainment process. Her simulations indicated that relative velocity could increase or decrease theentrainment based on the distance between two vortex rings. Detailed discussion on various entrainmentmodels can be found in Sreenivas and Prasad [2000], they also proposed that entrainment coefficient canbe modeled based on factors like axial temperature gradient, pressure gradient and acceleration in freeshear flows; however, the proposed model needs to be verified.

In this paper, we demonstrate that even a mild axial acceleration of a planar jet could reduce theentrainment by using transient, two-dimensional, numerical simulations. Numerical method used is

ABSTRACT: We present two-dimensional numerical simulations of a planar jet using diffusionvelocity, discrete vortex method. The relation between large-scale eddy structures and the entrainment isshown. Simulations are also performed for an axially accelerated jet. Entrainment in the case of theaccelerated jet is lower than that for the base-case jet. In this paper we argue how acceleration canstabilize the vortex sheet and reduce the entrainment.

diffusion-velocity, discrete vortex method (Ogami and Akamatsu, [1991]). We also explain themechanism that reduces the entrainment in the axially accelerated jet even though the circulation of thejet increases. In the next section a brief description of the computation method, computational domain andinlet and exit conditions are described. In the third section results and discussion of the simulations arepresented, followed by the conclusions in the forth section.

2. DIFFUSION VELOCITY DISCRETE VORTEX METHOD:Discrete-vortex-method (DVM) offers computational advantage for flows in which vorticity is

confined in certain areas of the flow domain. One can study the dynamics of a fluid-flow with thecomputational economy using DVM; in DVM computation will be restricted to the area having vorticity.DVM is also ideally suited for parallel computation. DVM and its variations have been applied forvarieties of flows including evolution of vortex-sheet, wakes, mixing layers and other shear layers(Ogami and Akamatsu, [1991]).

Vorticity transport equation for a two-dimensional, incompressible flow is given bellow:

ω is the vorticity, t is the time and ν is the kinematic viscosity of the fluid. x and y are the coordinatesnormal to and along the primary flow direction of the jet respectively. U and V are the total velocities in xand y directions. uc and vc are the usual convective velocities in x and y direction due to Biot-Savart law;similarly ud and vd are the diffusion velocities due to non-zero viscosity of the fluid. Equation (2)represents the vorticity transport equation for a two-dimensional incompressible fluid flow. vf is theforcing velocity used to control the axial acceleration of the jet. In the discrete vortex method, thecontinuous vorticity field is discretized into a finite number (N) of vortex blobs having a core radius σ sothat vorticity at any point X

~(x, y) can be given by (for details refer Ogami and Akamatsu, [1991] and

Kempka and Stricland [1993])

convective and diffusion velocities are evaluated as follows

For each vortex particle, new position at the end of a time step is evaluated by computingxj (t+∆t) = xj(t) + ∆t Uj; and yj (t+∆t) = yj(t) + ∆t Vj;

[ ] [ ]

fdcdc vv v uu V U andwhere

)2( 0VU

0c

vc

u

as written re becan (1)Equation

)1(0).V~

(2

++=+

=

=∂∂

+∂∂

+∂

∂

=∂∂

⋅−∂∂

+∂∂

⋅−∂∂

+∂

∂

=∇−∇ +∂

∂

ωωω

ωω

ων

ωω

ωνω

ωνωω

yxt

yyxxt

t

[ ] functionn ditributio core theis and exp1

)()(),( 2

2

21

−=−Γ= ∑

= σπσγγω

SStXXtX j

N

jj

( )( )

( )( )

; v;

~~

exp1~~)x-(x-

2

),( v

; u ;

~~

exp1~~)y-(y

2

),( u

d2

2

2

j

1c

2

2

2

j

1c

y

XX

XXtX

x

XX

XXtX

j

j

N

j

j

dj

j

N

j

j

∂∂

⋅−=

−−−

−∑

Γ=

∂∂

⋅−=

−−−

−∑

Γ=

=

=

ωων

σπ

ωων

σπ

For accelerating the computation, we have adopted fast-multi-pole method of summation (Clark andTutty [1994]).

All the length scales are normalized with the slot width `d` (X=x/d; and Y=y/d) and the velocities byjet exit velocity `vo`. Reynolds number (vod/ν; ν is the kinematic-viscosity of the fluid) for the flow is5000. Computational domain is divided into four regions as shown in Figure 1. First region has the slotthrough which jet enters the ambient medium; its opening width is `d`. Second region is the computationdomain of interest in which jet dynamics is monitored. Even though this region extends from y/d equal to10 to 115, all the statistics are obtained only in the region up to Y=90, beyond which results could beaffected by exit boundary conditions. In the third region, jet structures are convected out. This is a bufferregion that bridges the region of interest and actual end of the computational domain (y/d from 115 to135) up to which vortex particles are tracked. In the fourth region extending from Y=135 to 600, we willassume a distribution of vorticity, from the time averaged, self-similar velocity profile of the jet. Itscontribution to the velocities in other three regions are computed analytically and added.

The multiple region approach we have used in the present simulation is similar to the computation ofplane mixing layer by Basu et. al. [1995]. As the coherent eddy structures get out of the computationaldomain, they tend to sway the jet normal to the direction of primary flow. A uniform axial-velocity of 5 to10% of the jet exit velocity in all three regions will suppresses the swaying instability and the simulationwill corresponds to a co-flowing jet configuration.3. RESULTS AND DISCUSSION:

All the results presented here are obtained after the jet is established in the computationaldomain. Results presented for the statistical quantities are obtained from minimum 350 time-frames after

y/d=0 10 115 135 y/d=>600

I IIIII IV

y/dx/d

Fig. 1: Schematic diagram indicating computational domain. y/d from zero to ten is the nozzle, 10 to115 is the region of interest, 115 to 135 is the region in which flow is convected out and from 135 to600, mean vorticity distribution is imposed and its effect is taken into account in the first 3 regions.

b = 0.13(Y-18)1.05 Vc/vo = 2.9(Y-18)

- 0.5

Fig 2: Axial variation of (a) 1/ewidth of the jet and (b) axialvelocity of the jet. Jet has alinear growth rate in Y and thetime averaged-axial velocitydecreases with a Y-0.5

; both thesebehaviour are evident beyond Yequals to 37.5. Red dots arefrom simulation and blue linesare the best fits.

(a) (b)

the jet is well established. In the Figure 2, we present the axial variations of jet's 1/e –width (Fig. 2a) andaxial velocity (Fig. 2b) for the base-case, planar-jet. Similarly, variations of the normalized axial velocityacross the jet-width, at various axial locations are presented in Figure 3. Results presented in figure 2 and3 clearly indicate that the planar jet in the present simulation has self-similar region extending in thedownstream direction from Y equals to 37.5. Jet-spread rate and the velocity decay presented in Fig. 2,both compare well with the corresponding correlations given in Fischer [1979].

In Figure 4, the panel `a` indicates an instantaneous-fluctuating-vorticity field obtained bysubtracting time averaged vorticity field from the instantaneous vorticity field indicating eddy structures.Arrows indicate velocity at that instant and pink line indicates /dymd & which is the local entrainment (+ve

value) or detrainment (-ve value). Panel `b` represents the net value of fluctuating-vorticity (summed

across the width of the jet) at an Y location and its evolution in time. This clearly shows how eddies (ablue patches; clockwise eddies and red patches; anti-clockwise eddies) move along Y-direction in time.Corresponding entrainment episodes are shown in panel `c`. There is a high correlation betweenentrainment episodes and large eddy structures evolving in time.

Next, we compare the behaviour of the jet subjected to mild-acceleration with that of base case.Decay in VC for the two cases has been shown in Figure 5a. Note that the mild-acceleration imposed only

Fig 4: (a) Instantaneous snap shot of fluctuating-vorticity field (blue negative and red positive)indicating large scale eddy structures, arrows indicate instantaneous velocity field at the edge of thejet and pink line indicates axial variation of /dymd & representing local entrainment and detrainmentin the jet. (b) This panel represents the variation of fluctuating vorticity summed across the width ofthe jet at a given Y location as function of axial distance and time and (c) corresponding variation ofentrainment along the jet-axis and time. Note the high correlation between large-scale structures intop panel and entrainment episodes in the bottom panel.

(a)

(b)

(c)

Y

Y

Y

X

Fig 3: Variation of the axial-velocity at many Y locationsnormalized with local timeaveraged centerline velocity(V/Vc) across the normalizedwidth (x/b) of the jet.

reduces the rate of VC decay. Acceleration is imposed for the region beyond Y greater than 22.Entrainment computed as a function of downstream distance (Y) in the two cases is shown in Figure 5b.Even though acceleration started at Y equals to 22, entrainment shows a notable trend in reduction only

beyond Y equals to 36. Delayed effect of acceleration on the process of entrainment is consistent with theexperimental observations of Bhat and Narasimha [1996]. For the region Y greater than 36, entrainmentin the case of accelerated jet (red line) is consistently lower than the base case (blue line).

In the Figure 5c, organization of positive and negative vortex particles at an instant is shown. In theregion, just after jet comes out of the nozzle vortex sheet is intact. Counter rotating, small eddy structuresform on both edges of the jet. In this laminar region, net-fluctuating vorticity at a given Y location (blackline) is zero. Hence the induction process can not bring much fluid into the jet. As the jet travels furtherdownstream distance, vortex-sheet on both sides of the jet develops instabilities. Vortex-sheet start to roll-up into large convoluted shapes during which positive and negative signed vortex particles startsclumping up into distinct eddies (Y>30). Beyond this, all along the Y, one can see alternate clusters ofpositive and negative vortex particles (Figure 5c). This organization leads to either positive or negativenet-fluctuating vorticity (or circulation) at a given Y location (black line marked +ve/-ve; also see the redand blue regions in Figure 4b). Net circulation of either sign at a given Y location can effectively inducevelocity in the irrotational ambient fluid (“Induction phase”, a kinematic process as proposed by Prof.Narasimha and Prof. Roshko; Dimotakis [1986]).

Acceleration stabilizes the vortex sheet and suppresses the formation of large eddies as described bySreenivas and Prasad [2000]. Even though the accelerated jet will have higher shear and circulationbecause of the absence of large-scale structures the induction process will not be effective. Thusentrainment will be reduced in an accelerated jet. Further, absence of highly convoluted vortex sheet dueto stabilization reduces the area of interaction for the turbulent jet fluid with the non-vortical ambientfluid, this also reduces entrainment by smaller scale eddies. The process is similar to relaminarization ofturbulent flows reported by Narasimha and Sreenivasan [1979]. Parametric study relating the rate of

Fig. 5: Panel a has the comparison of time averaged axial velocity change along Y for base case (blueline) and for the mildly-accelerated jet (red line). Corresponding entrainment coefficient computed forthe two cases are shown in panel b. Panel c indicates the organization of positive vorticity-particles (reddots) and negative vorticity particles (blue dots) at an instant. Black line in the panel c indicatesfluctuating vorticity at a given Y location.

(a) (b) (c) 0.06 0.12 0.19(b) Entrainment Coefficient

acceleration/deceleration of a shear layer and its effect on the entrainment will be highly useful forengineering applications. The limitation of the present study is that the simulations are two-dimensional,however, noting that the entrainment process being largely controlled by the induction phase, which couldbe modeled in two dimensions will justify our simulations.

4. CONCLUSIONS:Two-dimensional numerical simulations based on diffusion-vortex method are used to simulate

the dynamics of planar jets. We have shown, by numerical simulations, how large scale eddy structures ina jet facilitate entrainment process. Axial acceleration will stabilize the vortex sheets and prevent theformation of large-scale structures and thus reduce the entrainment.

5. ACKNOWLEDGEMENTS:Author wishes to thank Prof. R. Narasimha (RN) and Prof. Bhat for introducing this problem and

Prof. Rama Govindarajan and Prof. RN for many useful discussions on the topic, which helped inprogress of this work.

REFERENCE:1. Bhat GS and Narasimha R, Volumetrically heated jet: large eddy structure and entrainment characteristics, J.

Fluid Mech. 329, 303-330, 1996.2. Basu AJ, Narasimha R and Prabhu A, Modelling plane mixing layers using vortex points and sheets, Applied

Mathematical Modelling, 19 (2), 66-75, 1995.3. Campbell IH and Turner JS, Turbulent mixing between fluids with different viscosities, Nature, 313, 39-42,

1985.4. Choi DW, Gessner FB and Oates GC, Measurements of confined, coaxial jet mixing with pressure gradient, J.

Fluid Engg. 108, 39, 1986.5. Clark NR and Tutty OR, Construction and validation of a discrete vortex method for the two-dimensional

incompressible Navier-Stokes Equations, Comput. Fluids, 23, 751-783, 1994.6. Dimotakis, PE, Two-dimensional shear-layer entrainment, AIAA J. 24, 1791-1796, 1986.7. Fischer, HB, List, EJ, Koh, RCY, Imberger, J and Brooks NH, Mixing in inland and coastal waters, (Data for

planar jet; Table 9.2 in the book), Academic, New York, 1979.8. Hunt JCR, Atmospheric jets and plumes, in Recent Research Advances in the Fluid Mechanics of Turbulent

Jets and Plumes, edited by Davies PA and Valente MJ, Kluwer Academic Publishers, Netherlands, pp. 309-334. 1994.

9. Kempka SN and Strickland JH, A Method to Simulate Viscous Diffusion of Vorticity by Convective Transportof Vortices at a Non-Solenoidal Velocity, Sandia Report, SAND93–1763, 1993.

10. Lumley JL, Explanation of thermal plume growth rate, Phys. Fluids 14, 2537, 1971.11. Manikandan MS and Sreenivas KR, Effect of ambient viscosity on the turbulent jet entrainment process

(unpublished), SRFP-2003 report JNCASR, Bangalore, 2003.12. Morton BR, Taylor GI and Turner JS, Turbulent gravitational convection from maintained and instantaneous

sources, Proc. Royal Soc. London A 234, 1 1956.13. Narasimha, R, and Sreenivasan, KR, Relaminarization of fluid flows," Advances in Applied Mechanics 19, 221-

301, 1979.14. Ogami Y and Akamatsu, T. Viscous Flow Simulation using the Discrete Vortex Method – the Diffusion

Velocity Method. Computers & Fluids. 19, 433-441, 1991.15. Priestly CHB and Ball FK, Continuous convection from an isolated source of heat, Q. J. R. Meteorological Soc.

81, 144, 1955.16. Rama Govindarajan, Universal behaviour of entrainment due to coherent structures in turbulent shear flow,

Phys. Rev. Lett., 88, 134503.17. Sreenivas KR and Prasad AK, Vortex Dynamics Model for Entrainment in Jets and Plumes, Phys. Fluids, 12, 8,

2101-2107, 2000.18. Turner JS, Turbulent entrainment: the development of the entrainment assumption and its application to

geophysical flows, J. Fluid Mech. 173, 431-471, 1986.19. Venkatakrishnan L, Bhat GS, Prabhu A, and Narasimha R, Visualization studies of cloud-like flows, Current

Science 74, 597-606, 1988.

1


LES APPLIED TO PREDICT FLOW OVER SMOOTH HILL

S.Vengadesan Department of Applied Mechanics, IIT-Madras, Chennai-600 036, India.

A.Nakayama

Graduate School of Science & Technology, Kobe University, Kobe 657-8501, Japan. ABSTRACT: LES technique is applied to predict flow past an idealized two-dimensional hill with two different slopes to demonstrate its predictive capability. Simulations are performed by conventional Smagorinsky model with non slip and log-law assumption boundary conditions. Results of simulation with log-law assumption applied instantaneously show results closer to the experiment in terms of mean velocity and wake size. 1. INTRODUCTION

The study of wind flow over hill is of great interest in engineering applications like transport and dispersion of pollutants in the atmosphere, agro-meteorological study, construction of wind mills and airport etc.,. Practically, an infinite number of situations are possible due to description of hill geometry, arrangements, and approaching flow conditions and so the flow can experience separation, which is not fixed, small separation bubble or no separation. Various experimental measurements have been reported for different configuration of the hills at different Reynolds numbers [1,2,4,6] and numerical predictions have also been performed [4,6,8,20]. A detailed review is available in [3]. Invariably all the calculations so far are of RANS type and no Large Eddy Simulation (LES) study have been reported yet.

Large-eddy simulation (LES) method has proved very successful in simulating unsteady flows and is considered to be very useful in practical engineering and environmental application. In LES, large-scale motion is resolved by discrete computational grid and directly computed by numerical method and all the small-scales of motions are modeled. This technique has been proved to be successful in simulating simple flows. As fast and large-capacity computers are becoming affordable nowdays, this method has been applied to predict various engineering flows [11,15]. However, its application to flows in natural environment is not quite straightforward, as boundaries are generally undulatory and the geometry is complex.

In the present work, we perform LES of flow past an isolated two-dimensional hill to demonstrate its applicability. The hill is defined by an analytical expression. Experimental measurements are available at moderately high Reynolds number. Standard Smagorinsky model with two wall boundary conditions are considered for simulation. Results are analyzed in terms of mean velocity and turbulence quantities.

2. DESCRIPTION OF THE FLOW

The flow configuration considered is that past an isolated hill (Fig.1), a smooth two-dimensional topography, defined by an analytical expression

( )4/11nHxH

zG

+= , where zG is the elevation of the ground at

horizontal position x, and H is the height of the hill. We consider this test case with two maximum values of the slope angle viz. 15 degree and 25 degree, measured from the horizontal direction, determined by values of n. Table 1 gives details of index n, hill height H and the respective slope angle. This flow has been subjected to a detailed experimental study and experimental results are available in [13]. Mean velocity and turbulent

2

stresses have been measured for the Reynolds number based on the oncoming reference velocity Uref and H of 13000. This is relatively a gentle topography and no flow separation is reported.

3. NUMERICAL METHODS 3.1. Governing equations

The basic equations used in the present LES are three-dimensional, time dependent, Navier-Stokes equations, filtered in order to separate the large scale and the small-scale motions. We consider isothermal incompressible flow and solve the filtered governing equations along with closure subgrid-stress model. Governing equations are not described here and they can be found in [14,15]. LES model chosen is the conventional Smagorinsky model in which the turbulent stress, Rij is modeled as

2 23ij s ij G ijR k Sδ ν= − (1)

where, ks is the subgrid turbulent kinetic energy, δij is the Kronecker delta, νG is the subgrid eddy viscosity and Sij is the strain tensor. The eddy viscosity νG is modeled by

( )1 2

2 ji iG S

j j i

uu uCx x x

ν ∂∂ ∂

= ∆ + ∂ ∂ ∂

(2)

where, ∆ is the grid size defined by the geometric average of the grid spacings in three directions, ( ) 31

321 xxx ∆∆∆ , ui is the spatially filtered velocity component in the streamwise -xi direction, u2 in the spanwise – x2 direction and u3 in the cross-stream –x3 direction, i.e (x1,x2,x3)=(x,y,z); (u1,u2,u3)=(u,v,w). Cs is the model constant for which we use the value of 0.13 in the case of Standard Smagorinsky model. 3.2. Coordinate system, calculation domain and grid

Conventional method to represent geometry like the present one is to use boundary fitted coordinate (BFC) system and applying the boundary conditions at exact locations. However, when the boundary becomes irregular as in real case, it is difficult to generate BFC grid. Alternate way is to use Cartesian grid with various boundary treatments [5,9,17,19]. In the present work, Cartesian coordinate system is used and boundary conditions are applied at the mesh point closest to the real boundary.

The computational region covers the test flow as shown in Figure 1. The computational region for the hill with maximum slope angle of 25 degrees extends from about 8.5H in the upstream and 14H in downstream in the streamwise direction, 7H in the cross stream-wise and 4H in the spanwise direction. The rectangular grid that is used, is uniformly spaced in the spanwise direction. In the streamwise direction, points are closely spaced (90 points) within 4H on either side from the hill summit, stretched with a factor of 1.038. In the cross-streamwise direction, the first point from the ground is placed at 0.03H near the bottom of the wall, stretched with a factor of 1.05 up to 0.5H and then compressed with a factor of 0.95 up to 1.2H and then placed non-uniformly with stretching factor of 1.1 until the end. The computational domain for hill with maximum slope angle of 15 degrees covers the region of 12H and 19H on the upstream and downstream respectively. In the other directions the domain is kept the same as that used for 25 degrees case. The total

-8 -6 -4 -2 0 2 4 6 8

Uref

x/H

Hx

zz

G

Figure.1 Flow configuration

H (cm) n Maximum slope

angle (degrees) 10 23 25 5 20 15

Table 1. Details of Hill Model

3

grid size used is 128x61x21 and 141x61x21 respectively for hill with 25 degrees and 15 degrees. This grid distribution for the test Reynolds number of 13000 gives vertical distance in viscous units, z+=zuτ/ν of the first node about 20 and 15 on the top of hill and at x/H=4 respectively and thus viscous layer are not resolved. So, the mesh is not of high resolution to resolve the laminar sublayer, a key problem encountered in LES of high Reynolds number practical flows when performed with easily accessible computers.

3.3. Numerical Schemes

We solve the governing equations by a finite difference procedure. Non-linear convective terms in the equations are discretised by a third-order upwind differencing, to avoid stability problems [7,14] and viscous terms are discretised by second-order accurate central differencing scheme. Inflow conditions for the streamwise velocities are adopted from experimental data. Radiation outflow condition is applied at the downstream boundary. The periodic boundary conditions are used for the spanwise direction. In the cross flow direction, the nonslip boundary conditions are applied on the ground surface and slip conditions are applied on the top boundary. HSMAC iteration scheme is used for calculating pressure. Time advancing of the momentum equations is done by a second-order accurate explicit, Adams-Bashforth method. Performance of the code had been assessed earlier for flow past a bluff body [14] and for the curved geometry [12]. All the calculations are performed with the non-dimensional time step, dtUref/H of 0.001. All calculations have been allowed to settle down until 40 non-dimensional time units, and then statistical averages over the next 40 non-dimensional time units are obtained that are presented below.

3.4. Wall Boundary Conditions

In wall-bounded flows, the only correct boundary condition at the surface is the no-slip condition, but this requires calculations up to the wall with sufficient grid resolution. However, as the Reynolds number increases, boundary layer thickness decreases, resulting in requirement of large number of grid points. In LES the problem is severe [16] and no definite solution has been proposed yet. We perform calculation for the present test case, with non-slip boundary condition as a baseline solution to compare and this case is referred to as Case A.

As one method of solution, proposed instantaneous two-layer linear-power law velocity distribution was proposed [18]. This two-layer model is modified into three-layer linear-loglaw version in the format given in [21], to specify the boundary conditions for the velocities in the tangential directions, at the first point from the wall. We test this boundary condition and this is referred to as Case B. In this method, an approximation to a wall law given by the following equation is used. u+ = z+, 5 > z+ >0 (3)

u+ = 5 ln z+ - 3.05, 30 > z+ ≥ 5 (4) u+ = 2.5 ln z+ + 5.5, z+ ≥ 30 (5) where, z+=zuτ/ν and u+=u/uτ are the non-dimensionalised vertical distance and velocity respectively. The friction velocity, uτ is calculated from these equations with the velocities at the second point from the wall. 4. RESULTS AND DISCUSSION 4.1. Instantaneous flow field

In order to see the time evolution of the calculation, contours of the instantaneous spanwise vorticity distributions at non-dimensional time interval of 1.0 are plotted in Figure 2 for the hill with maximum slope angle of 25 degrees. For both the computational cases, they are shown at the same time, between tUref/H=41 to tUref/H=45 and to the same scale. In the calculation using the nonslip condition, the unsteadiness is initiated in the thin boundary layer near the top of the hill and grows into large-scale vortex structure

4

downstream. The mean velocity plots (shown in Figure 3) indicate separation around x/H=1 and a very large reverse flow region. The results of calculation with conventional log-law shows more steady behaviour and yet present separation and re-circulation region.

4.2. Mean Velocity

Profiles of time averaged streamwise velocity component,U1 at specified streamwise stations computed using different boundary conditions and experimental results are plotted in Fig.3 (a) for hill with 25 degrees and Fig. 3(b) for hill with 15 degrees. The inflow velocity profile for the calculations at station x/H=-4 and x/H=-6 for 25 degrees and 15 degrees respectively are taken from that of experiment. At x/H=0, experimental results show that the flow accelerates just near the top of the hill. At the same station, calculation with the nonslip boundary condition (Case A) shows the development of the boundary layer and the maximum velocity is drastically under-predicted. Calculations using the log-law boundary condition (Case B) show a thinner boundary layer. As it can be observed, Case A shows separation at x/H=2 and x/H=4 for 25 degrees and 15 degrees hill respectively and predicts a large re-circulation zone at further downstream. Case B predicts mean velocity closer and wake size is reduced. 4.3. Turbulence quantities

Calculated shear stresses are shown in at two streamwise stations – x/H=0 (Fig. 4a) and x/H=6 (Fig. 4b) along with the experimental data for the hill model with 25 degrees. On top of the hill, both cases grossly under-predict the distribution and at farther downstream, there is an improvement in the predictions. At x/H=0, prediction with the non-slip boundary condition case is closer to the experimental values. This can be attributed to the fact that early massive separation caused is responsible for turbulent production. At x/H=6, the results of Case A shows a large negative shear stress which is also due to the large separation. Prediction using the conventional log-law is not satisfactory as very small turbulence is produced.

Contours of constant shear stress are shown in Figure 5 for the hill model with 15 degrees. It can be observed from the experimental data that turbulence stress is significant only in the region downstream of the hill. All the calculations are plotted to the same scale, it is to be mentioned that some of the contours of calculation with, nonslip case could not be shown as they are beyond the common scale that is chosen. Calculation with conventional log-law as boundary condition is also not satisfactory, due to early separation and large re-circulation predicted by it as observed in Figure 3(b).

The above results indicate in order to improve predictions, separation needs to be controlled and a mechanism to generate and sustain turbulence is to be included in the model. One method of solution could be imposing inflow turbulence as against the mean velocity profile at the inlet as adopted here. Further, with present mesh resolution, non-slip boundary condition bridging the first point and wall, energy producing eddies near the wall are probably not captured. Conventional log-law applied though instantaneously, is not appropriate as similarity assumption are made in arriving at them for a mean flow. The results may show improvement if a wall boundary condition is devised which include flow parameters varying spatially and temporally as demonstrated [10] for unsteady calculation by standard k-ε model. 5. CONCLUSIONS

Large Eddy Simulation method has been applied to predict flow past an idealized two-dimensional hill for two different maximum slope angles. Standard Smagorinsky model with two wall boundary condition - non-slip and conventional log-law applied instantaneously, are investigated for their applicability. Results in terms of mean velocity and turbulence quantities are discussed. Their limiting behaviour is elucidated. It is found from the present study that there is an improvement in the mean velocity, but the turbulence predictions are not satisfactory. Further refinements may be possible, if suitable wall boundary conditions which include parameters varying temporally and spatially in addition to the inflow turbulence are incorporated.

5

-8 -6 -4 -2 0 2 4 6 8 10 12x/H

42

43

44

45

TUref/H=41

(a) nonslip

-8 -6 -4 -2 0 2 4 6 8 10 12x/H

TUref/H=41

42

43

44

45

(b) conventional log-lawFigure 2. Time development of lateral vorticity contours for hill model 25 degree

-8 -6 -4 -2 0 2 4 6 8012345

z/H

000 0000 1 2

x/H (b) Hill model with 15 degree

-8 -6 -4 -2 0 2 4 6 80

12

34

5

z/H

00 200000 1U/U

ref

(a) Hill model with 25 degree

Figure 3. Mean Streamvise velocity – filled symbols:experiment; open square: nonslip; open triangle: log-law

0

0.5

1

1.5

2

2.5

3

-5 10-5 0 5 10-5 0.0001 0.00015 0.0002

Expt.non-slipconventional log-law

z/H

-u1u3/Uref2

0

0.5

1

1.5

2

2.5

3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

Expt.non-slipconventional log-law

z/H

-u1u3

2/Uref

(a) at x/H = 0 (b) at x/H = 6

Figure 4. Shear Stress distribution for hill model with 25 degree

6

REFERENCES [1] Almeida GP, Durao DFG, and Heitor MV.: Wake flows behind two-dimensional model hills, Experimental Thermal

and Fluid Science, 1993, 7, 87-101. [2] Baskaran V, Smits AJ, and Joubert PN.: A turbulent flow over a curved hill, Part 1. Growth of an internal boundary

layer, J.Fluid Mech., 1987, 182, 47-83. [3] Belcher SE, and Hunt JCR.: Turbulent flow over hills and waves, Annual Rev. of Fluid Mechanics, 1998, 30, 507-538. [4] Ferreira AD, Lopes AMG, Viegas DX, and Sousa ACM.: Experimental and numerical simulation of flow around two-

dimensional hills, J. Wind Eng and Ind. Aerodyn., 1995, 54/55, 173-181. [5] Forrer H, and Jeltch R.: A higher-order boundary treatment for Cartesian-grid methods, J.Comp. Phys., 1998, 140,

259-277. [6] Kim HG, Lee CM, Lim HC, and Kyong NH.: An experimental and numerical study on the flow over two-dimensional

hills, J.Wind Eng. and Ind. Aerodyn., 1997, 66, 17-33. [7] Kogaki T, Kobayashi T, and Taniguchi N.: Large eddy simulation of flow around a rectangular cylinder, Fluid

Dynamics Research, 1997, 20, 11-24. [8] Kobayashi MH, Pereira JCF, and Siqueira MBB.: Numerical study of the turbulent flow over and in a model forest on

a 2D hill, J.Wind Eng. and Ind. Aerodyn., 1994, 53, 357-374. [9] Majumdar S, Iaccarino G, and Durbin P.: RANS solver with adaptive structured non-boundary conforming grids,

Annual Research Briefs, Centre for Turbulence Research, Stanford University, 2001, 353-366. [10] Mohammadi B, and Pironneau O.: Unsteady separated turbulent flows computation with wall-laws and k-ε model,

Comput. Methods Appl. Mech. Engrg., 1997, 148, 393-395. [11] Moin P.: Advances in large eddy simulation methodology for complex flows, Int. J. Heat and Fluid Flow, 2003, 23,

710-720. [12] Nakayama A, and Noda H.: LES simulation of flow around a bluff body fitted with splitter plate, J. Wind Eng. and

Ind. Aerody., 2000, 85, 85-96. [13] Nakayama A, and Yokota D.: Experimental study of characteristics of rough surface boundary layer past gentle hills,

Annual J. Hydraulics Engg., JSCE, 2001, 45, 43-48. [14] Nakayama A. and Vengadesan SN.: On the influence of numerical schemes and subgrid-stress models on large eddy

simulation of turbulent flow past a square cylinder, Inter. J. for Numer. Meth. Fluids, 2002, 38, 227-253.

-8 -6 -4 -2 0 2 4 6 8

-6 -4 -2 0 2 4 6x/H

-8 -6 -4 -2 0 2 4 6 8

Experiment

conventional log-law

nonslip

x/H Figure 5. Shear Stress distribution for hill model with 15 degree

7

[15] Piomelli U.: Large-eddy simulation: achievements and challenges, Progress in Aero. Sciences, 1999, 35, 335-362. [16] Spalart PR, Jou WH, Strelets M, and Allmaras SR.: Comments on the feasibility of LES for wings, and on a Hybrid

RANS/LES approach, Advances in DNS/LES, Proc. 1st AFOSR Inter. Conf. on DNS and LES, Greyden Press, 1997. [17] Verzicco R, Mohd-Yosof J, Orlandi P, and Haworth D.: Large eddy simulation in complex geometric configurations

using boundary body forces, AIAA, J., 2000, 38, n3, 427-433. [18] Werner, H., and Wengle, H: Large-Eddy simulation of turbulent flow over and around a cube in a plate channel,

Proc. 8th Turbulent Shear Flows, Munich, 19-4-1 – 19-4-6, 1991. [19] Ye T, Mittal R, Udayakumar HS, and Shyy, W.: An accurate Cartesian grid method for viscous incompressible flows

with complex boundaries, J. Comp. Physics, 1999, 156, 209-240. [20] Ying R, and Canuto VM.: Numerical simulation of flow over two-dimensional hills using a second-order turbulence

closure model, Boundary Layer Metrology, 1997, 85, 447-474. [21] Von Karman, T.: The analogy between fluid friction and heat transfer, Trans. ASME, 1939, 61, 705-710.

8

July 7, 2003 Dr.Sajeer Ahmed, Experimental Aerodynamics Division, National Aerospace Laboratories, Bangalore – 560 017. Dear Dr.Sajeer Ahmed, Sub : submission of paper for 10th Asian Congress - reg. Please find herewith two copies of the paper “LES applied to predict flow over smooth hills”, authored by S.Vengadesan and A.Nakayama for consideration to present in 10th Asian Congress on Fluid Mechanics. An electronic version in MS Word titled ‘vnles.doc’ is being sent by e-mail to [email protected]. Kindly acknowledge safe receipt of both and intimate me upon review. With regards, Dr.S.Vengdesan, Assistant Professor, Fluid Mechanics Laboratory, Department of Applied Mechanics, IIT - Madras, Chennai – 36. E-mail: [email protected]

Proceeding of Tenth Asian Congress of Fluid Mechanics

17-21 May 2004, Peradeniya, Sri Lanka

Application of the Multi-mixture fraction Model in Numerical Simulation of Turbulent Reacting Flows

M.D.Emami , S.Ziaei Rad

Mechanical Engineering Department, Isfahan University of Technology

and

H.Afshin

Mechanical Engineering Department, Sharif University of Technology

ABSTRACT: This study is concerned with the numerical simulation of multi-fuel combustion. The Favre-averaged equations of mass, momentum, energy, turbulent kinetic energy and its dissipation, together with the transport equations of the mixture fractions associated with each fuel stream and their variances, are solved numerically. The finite volume method of discretization is used to express the partial differential equations as a discrete system of algebraic equations. The segregated approach and the SIMPLEC algorithm are used. The formulation and the implementation of the multi mixture fraction are validated via a simple numerical experiment. The method is further verified by simulating the combustion of a fuel stream in a furnace fed by air, and flue gases, in which the flue gas is considered as a fictitious fuel stream. The comparison of the current results with those of the measured data and a commercial code results shows good performance of the present formulation. 1. INTRODUCTION

In a majority of turbulent non-premixed flames, the reaction between fuel and oxidizer usually occurs on a length scale that is typically much smaller than the flow characteristics length scales. The effect of turbulence on combustion in such cases is a laminar wrinkled flame, in which the fuel and oxidizer eddies are broken down to the smallest Kolmogorov eddies, where the chemical reaction takes place. Consequently, one might avoid calculating chemical kinetics of the reacting flow in these situations, because the rate-controlling mechanism of combustion is the macroscopic mixing due to turbulence.

One of the most common combustion models suited for the aforementioned mode of combustion, is the conserved-scalar model [3], [19]. The traditional configuration for the formulation of the conserved-scalar model is a two-stream (fuel-oxidizer) mixing and combustion in a homogeneous combustor, first used by Burke and Schuman [6]. The relation between the conserved-scalar and the mass fractions of fuel and oxidizer may be expressed via a suitable thermo-chemical model, such as the flame sheet model [19], and the chemical equilibrium model [12]. The effect of turbulent fluctuations on the mean value of the mass fraction of species is taken into account by the use of a presumed probability density function (pdf) [19]. Such an approach has been used by many researchers, among them [3], [13], [17], to simulate the diffusion flames in various geometries. Extension of the mixture fraction concept to multi-fuel / multi-stream systems has been little studied. This may be attributed to the complexities in the formulation and the code implementation.

Almost all of the researches about the multi-mixture fraction concept and applications have been focused on the two-phase reacting flows [10], [25]. Eddings et al. [10] used different mixture fractions for

devolatilization and char combustion in the study of pollutants resulting from pulverized coal furnaces. In their study, however, the difference between the compositions of volatiles and char-off gases was not taken into account. Brewster et al.[4] presented a generalized theory to describe the devolatilization and combustion of coal particles. A two mixture fraction model was presented by Flores et al.[11] for combustion of pulverized coal. Lockwood and Salooja [20] studied the use of conserved scalar concept for multi-stream mixing of a fuel stream and two oxidant streams. This approach has been followed by others [2], [1] for the simulation of coal-fired furnaces and multi-fuel streams.

The present study is concerned with mixing and combustion of multi-fuel systems. First, the mathematical model of a multi-stream reactive flow is described. Then, the model is verified by a simple numerical experiment. Finally, the predictions of the present model are compared with the published predictions of a commercial CFD code, and the available experimental data [5]. 2. MATHEMATICAL MODELLING

Consider a mixing device, as depicted in Fig.1, where N fuel streams mix and react with an oxidant stream and the combustion products leave the device. A quantity fi (i=1,N) is assigned to each fuel stream, where fi = 1 represents the conditions at the inlet stream i, where other fj (j=1,N ; ji) are zero.

Fig.1: A schematic representation of the mixing/combustion device

As a result of the mixing rule within the device [3], one may write:

1 1

=+=

ox

N

ii ff (1)

In order to consider the quantities fi and fox as mixing parameters, they should satisfy the conditions of the conserved scalar [3] (i.e. the transport equations of these quantities should be source-free). This could be achieved by assuming a one-step reaction for the combustion of each fuel stream.

The transport equation of the ‘mixture fractions’ fi (i=1,N) in the steady state case, may be written as1: [8]

( ) 0~

~

~ =∇Γ−⋅∇ iii ffuρ (2)

1 It is, obviously, implied that the Boussinesq’s hypothesis has been applied to account for the effect of turbulence on the mean quantities.

Fuel stream 1

Fuel stream 2

Fuel stream N

…………….

Oxidant stream

Products

where ρ is the mean mixture density, u~ is the Favre-averaged velocity field, and iΓ is the effective diffusion coefficient of the species i. It should be noted that in the common case of turbulent reacting flows,

iΓ for all the involved streams may be assumed to be equal to the quantity σ

µeff , where effµ is the effective

viscosity and σ is the turbulent Schmidt number. The effective viscosity is the sum of the laminar viscosity and the turbulent viscosity, where the latter is calculated via the standard (k-) model. A value of σ =0.9 is used for all the mixture fraction equations in the present study.

After the calculation of all mixture fractions, the instantaneous mass fraction of a general species j may be calculated from the following formula [3]

( )=

−+=N

koxjkjkoxjunbj YYfYY

1,,,, (3)

where subscript ‘unb’ denotes the unburned state and subscript ‘ox’ refers to the corresponding value in the oxidant stream.

The amount of required oxygen for a fuel stream i is calculated from the following relation:

iiii

reqox SFfY ..., = (4)

in which iF is the mass fraction of the burnable fuel in the fuel stream i, and iS is the amount of oxygen required to burn 1 kg of this part of fuel. The amount of the available oxygen is first calculated via eq. (3), and then is recalculated for each fuel stream after subtracting the required oxygen from eq. (4).

In order to take into account the effect of turbulent field on the mean value of thermo-chemical quantities, the relations presented for the mass fraction of species should be weighted by a pdf, and integrated over the mixture fraction space [8]. It should be noted that, in general, a joint pdf of all fi variables should be used. However, the assumption of statistical independence of mixture fractions, which has been frequently invoked in other studies [7] [18], makes it possible to write the joint pdf in terms of single-variable pdfs. This assumption results in considerable simplification of the solution strategy.

Several presumed pdf have been proposed by researchers, see reference [3], but the incomplete beta-function is adopted for the present study. The incomplete beta-function pdf of mixture fraction fi may be written as:

( ) ( )( )

−−

−−

−

−= 1

0

11

11

1

1

ib

ia

i

bi

ai

idfff

fffβ (5)

in which a and b are parameters that are related to the mean, and variance of the mixture fraction fi via the following relations:

( )i

i

ii ff

ffa

~1~

~1

~

2

−

′′−= (a)

( ) ( )ii

ii ff

ffb

~11~

~1

~

2−

−

′′−= (b)

(6)

The transport equation for the variance 2~ifg ′′= may be derived from the transport equation of the

instantaneous mixture fraction, fi , in steady state form[16]. It may be written as:

( ) kgCxf

gg gm

itiigi

ερσµρ 2 ~

2

, −

∂∂=∇Γ−⋅∇ u (7)

where ig ,Γ is the effective diffusion coefficient of the ith variance, σσσµ ≈=Γ g

g

effig ; ,

, and gC

represents the ratio of time scales for the decay of velocity and scalar fluctuations due to the cascading process. Following Jones and Whitelaw [15], gC =2 is used in this study.

3. THE SOLUTION PROCEDURE

The governing equations are the incompressible, steady state equations of mass, momentum, energy (enthalpy), turbulence kinetic energy and its dissipation, mixture fractions (eq.2) and their variances (eq.7). The following canonical form may express these transport equations:

( ) ϕϕ ϕϕρ S=∇Γ−⋅∇ ~ u (8)

in which ϕ stands for the primary variable in the aforementioned governing equations, ϕΓ is the effective

diffusion coefficient, and ϕS refers to the source term associated to the variable considered. Table 1 summarizes the considered variables and their diffusion coefficients and the associated source terms.

Table 1: The diffusion coefficients and the source terms for the transport equations Equation ϕΓ ϕS

Continuity 1 0 0

Momentum iu effµ ii

jeff

j xp

x

u

x ∂∂−

∂∂

∂∂ µ

Turbulent kinetic energy k k

t

σµµ + ρε−*G

Turbulent dissipation εσ

µµ t+ ( ) kCGC ερεεε 21 −

Mixture fraction fi σµµ t+ 0

Mixture fraction variance gi σµµ t+

kgCxf

gm

i

t

t ερσµ −

∂∂

2

2

Total Enthalpy h σµµ t+

radS **−

* ijijt SSG µ2= where

∂∂

+∂∂=

i

j

j

iij x

u

xu

S21 , 1εC ,

2εC , kσ , εσ , are constants of the model [15]

** radS is the radiation source term, if any.

In addition to these transport equations, the equation of state for a perfect gas, the algebraic relation between the enthalpy and temperature, and the algebraic equations for the mass fraction of species, are used. Appropriate Dirichlet and Neumann boundary conditions are assigned for the variables.

An in-house CFD code, based on the finite volume discretization, is used for the present study. The convective terms are discretized via the NVD method [8], whereas for the diffusion terms the central difference scheme is implemented. The segregated approach basd on the SIMPLEC algorithm [24] is used. For the implementation of the solid wall boundary conditions, the ‘wall function’ method [21] is adopted. The performance of the code for reactive and non-reactive flows has been validated in several works [22][23][9]. 4. CASE STUDIES

As a first step to validate the formulation and implementation of the multi-mixture fraction, a tentative setup is used, in which the combustion of two methane jets in an air stream is numerically simulated. Figure 2 and Table 2 show the geometrical data and the operating conditions of this setup.

Table 2 : Inlet data

Type

Inlet width (mm)

V(m/s)

Fuel

Ch4

5

60

Air

Oxidant

100

20 Fig. 2: Schematic of the geometry

The 2D numerical simulations were performed twice. In the first run, only one mixture fraction was

used (as the two fuel streams are indeed one type of fuel). In the second run, two mixture fractions were adopted for the two fuel inlets. For brevity, only the contour plots of the axial velocity and temperature are presented. Figure 3 shows the contour plots of the axial velocity. The upper part (with respect to the horizontal line y=0) is the simulation results when one mixture fraction is used, and the lower part is the results for two mixture fractions.

Fig. 3: Axial velocity contours Fig. 4: Temperature contours

width

As it is evident from the figure, virtually no difference between the two predictions could be identified.

The symmetry of the temperature contours on Fig 4 is more interesting, as temperature is a secondary variable, which is not derived via a transport equation. It is, therefore, concluded that the present multi-mixture fraction approach is capable of simulating multi-fuel streams in a reacting flow.

The second case study is the combustion of natural gas in a swirl-stabilized burner [5]. The schematic of the furnace and burner details are depicted in Fig. (5-a, b). The dimensions of the burner inlets are normalized with the diameter of the air inlet (see Fig. (5-b) for details). The geometrical and flow characteristics associated to the burner are summarized in Table (3).

Table 3: Scaling burner and furnace

(a) (b)

Fig. 5: (a) Schematic of the furnace, (b) The burner details Bollettini et al.[5] have used the Fluent code to simulate the combustion of natural gas in this furnace. They used two-step reaction for combustion chemistry, k- model of turbulence, and EBU model to account for the turbulence-chemistry interaction. They claim that 2D simulation for this configuration does not yield satisfactory result and, therefore, performed 3D simulations. In the present study, the problem is treated as a 2D axisymmetric case, so half the furnace is considered in the simulation. The present study also considers the flue gas as a fictitious fuel, so two mixture fractions are used in this simulation. The inlet boundary conditions are presented in Table (3). The ‘outflow’ boundary condition is used for the outlet. The ‘no-slip’ boundary condition is considered for the upper wall and the sides’ walls. The symmetric boundary condition is imposed for the lower wall. Due to lack of information about thermal boundary condition on the upper wall, the adiabatic condition is imposed (which, as will be evident later, is apparently the same condition used by [5] ).

Do Dout Df Lf Uo Vg N Inj. D inj-hole NG Velocity mm mm mm m m/s m/s - mm m/s 180 297 1100 3.7 28.533 165 30 3.3 27.3

Figure (6,a-d) shows the radial temperature profiles at different stations downstream of the burner. The experimental data, and the 3D simulation results, taken from [5], are also shown in this figure for comparison. In both of the numerical simulations, the temperature profile near the axis of symmetry is under-predicted compared with the data. This may be attributed to the well-known deficiency of the ε−k model, which cannot predict round jets accurately. Also, far from the symmetry axis, the temperature profiles in both the current study and that of [5] deviate from the experimental data. This is mainly due to the use of adiabatic boundary condition for the enthalpy equation, which is the only choice in the absence of relevant data. However, at other radial locations, the present predictions show as good accuracy, if not better, as the predictions of [5], especially at x/d=4.62 where the present prediction temperature profile is very close to the experimental data.

Fig. 6: Radial temperature profiles at different stations

(a)

(c)

(b)

(d)

5. CONCLUSIONS Based on the results of the this work, and the authors’ experience with the current computer code in

similar cases, the following points may be concluded: • The multi-mixture fraction concept is a very useful tool in modeling different phenomena, such as

isothermal and non-isothermal mixing, and non-premixed flames. Its chief advantage is that the same computer code could be used for all these cases.

• The multi-mixture fraction model can be used for two-phase flow problems, such as evaporation/combustion of fuel sprays and solid-fuel devolatalization or surface combustion. In these cases, usually the gaseous phase is formulated in Eulerian framework, whereas the dispersed phase equations are written in a Lagrangian framework.

• As far as the numerical accuracy and stability are concerned, the mixture fraction method offer better convergence and solution stability ACKNOWLEDGMENT The authors wish to acknowledge the financial support for the current work, provided by Isfahan University of Technology (IUT). REFERENCES

[1] Antifora A., Faravelli T., Kandamby N., Ranzi E., Sala M., and Vigevano L., comparison between two

complementing approaches for predicting Nox Emissions in the Furnaces of utility boilers , 5th int. conf. on Tech.&comb.for a clean Env.12-15 July (1999), Portugal.

[2] Antifora A.,Sala M. ,Perera A. and Vigevano L. , Nox Emissions in combustion systems of coal- Fired Furnaces with a Reducing Environment , 4th int. conf. on Tech.&comb.for a clean Env. (1997), Portugal.

[3] Bilger , R.W., Turbulent diffusion flames, Annual Rev. Fluid Mech , 21:101-135, 1989. [4] Brewster , B.S. , Baxter, L.L., and Smoot L.D. , Treatment of coal Devolatilization in comprehensive Combustion

Modeling, Energy & Fuels 2, 362-370, 1988. [5] Bollettini U., Breussin F,N., and Weber R., A Study On Scaling Of Natural Gas Burners, IFRF Combustion

Journal ,July 2000. [6] Burke, S.P. and Schuman T.E.W. , Diffusion flames, indust. Eng . Chem.20, no. 10-998, 1928. [7] Correa, S.M. and Shyy, W., Computational models and methods for continuous gaseous turbulent combustion,

Prog.Energy Combust .Sci.13, 249-292, 1987. [8] Emami, M.D., Prediction of Finite Rate Chemistry Turbulent Combustion, PhD Thesis, University of London,

1999. [9] Emami, M.D. and Rahmati, A.R., Numerical simulation of Turbulent Premixed Flames based on the Zimont

model, Proceedings of the 4th Aerospace Conference, AKU, pp.407-417, 2003. [10] Eddings, E.G. , Smith, P.J. , Heap, M.P., Pershing, D.W. and Sarofim A.F. , The use of models to predict the

effect of fuel switching on Nox Emissions ,in coal blending and switching of low-sulfer western coals , edited by Bryers and Harding, N.Y, ASME, 169-184 , 1994.

[11] Flores, D.V. and Fletcher, T.H., A two-mixture Fraction Approach for Modeling Turbulent combustion of coal volatiles and char Oxidation Products, paper 95s –120, Spring meeting of central states , April 1995

[12] Gordon, S. and McBride, B.J., Computer program for calculation of complex chemical equilibrium compositions, rocket performance, incident and reflected shocks and Chapman-Jouguet detonations, Tech. Report SP-273, NASA, 1971.

[13] Jones, W.P. and McGurik, J.J., Mathematical modeling of gas-turbine combustion chambers , AGARD Confer.on Comb. Modeling, Proceed. No. 275, 1980

[14] Jones, W.P. and Priddin, C.H., Predictions of the flow field and local gas composition in gas turbine combustors, 17 th Symp.(Int.) on Comb., The Combution Institute, pp.399, 1978.

[15] Jones, W.P., and Whitelaw, J.H., Calculation methods for reacting turbulent flows: A Review , Combustion & Flame, 48, 1-26, 1982.

[16] Jones, W.P., Models for turbulent flows with variable density and combustion, Predication methods for turbulent flows, VKI- Lecture series, 1979-2.

[17] Kent, J.H. and Bilger, R.W., Kolling Report F-41, Department of Mechanical Engineering, The University of Sydney, 1972.

[18] Janicka, J., and Kollmann, W., A prediction model for turbulent diffusion flames including No formation, AGARD Confer. On Comb. Modeling, Proceed. No. 275, 1980.

[19] Libby, P.A. and Williams, F.A., Turbulent flows with non-premixed reactants, Turbulent reacting flows, 1980. [20] Lockwond, F.C. and Salooja, A.P., A Note on the Mixing of Three Stream Diffusion flames, Combustion &

Flame, 1981. [21] Patankar S.V. and Spalding D.B., A calculation procedure for heat, mass, and momentum transfer in three-

dimensional parabolic flows, Int. J. Heat & Mass Transfer 15, 1787-1806, 1972. [22] Sharifi, H., Numerical simulation of non-premixed flame based on the ANN approach, M.Sc. Thesis, Isfahan

University of Technology, 2001. [23] Salafian, R., The influence of Turbulence Modeling on The Numerical Simulation of Non–premixed flame,

Isfahan University of Technology, 2001. [24] Van Doormaal, J.P. and Raithby G.D., Enhancements of the SIMPLE method for predicting incompressible fluid

flows, Numerical Heat Transfer 7, 147-163, 1984. [25] Zhou, L., Theory and Numerical Modeling of Turbulent gas-particle flows and combustion ,CRC press, 1993.

Proceedings of the Tenth Asian Congress of Fluid Mechanics

17-21 May 2004, Peradeniya, Sri Lank

NUMERICAL ANALYSIS OF THE FLOW IN MOCVD REACTION

CHAMBER WITH THREE PLANETARY DISKS

Y. Liu, H. X. Chen, S. Fu Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

ABSTRACT: Computational Fluid Dynamics (CFD) simulation is performed in this study to investigate the

flows in the Metal-Organic Chemical Vapor Deposition (MOCVD) reactor. The numerical code developed in

this paper is based on the SIMPLE algorithm for the velocity-pressure coupling with the non-stagger grids system.

The governing equations are discretized with a finite volume method. The present simulation provided the

detailed flow patterns relating to the orbital and planetary motions in the MOCVD reactor. The effect of

reactor geometry, the reaction flow rate and some working parameters that influence the flow patterns in the

MOCVD reactor have been studied in detail. The CFD analysis demonstrates that it provides a good analysis

tool to the design and optimization of MOCVD reactors.

1. INTRODUCTION

Metal-Organic Chemical Vapor Deposition (MOCVD) is a leading technique for the growth of advanced optoelectronic and high speed compound semiconductor materials. The main challenge of a MOCVD reactor is how to control the produced films with uniform thickness and composition. MOCVD is a very complex process which involves multi-processes of gas-phase surface reactions, mixing of reactant fluids, heat and mass transfer. In a MOCVD reactor with cold walls, the gas mixture temperature increases in the immediate vicinity of a hot susceptor, causing the surface decomposition to occur in the diffusion controlled regime. All the processes of mixing, diffusion, flow, and transfer involve complex problems and mechanisms of fluid dynamics. These factors determine the efficiency of the MOCVD process and the quality of the grown material.

Computational Fluid Dynamics (CFD) analysis is an effective means to guide the MOCVD reactor design and to optimize the MOCVD process. It is able to illustrate the complex flow patterns related to the buoyancy effect, vertical motion, and reactant mixing within the MOCVD reactor. Many CFD simulations have been performed to study the effect of key design parameters, such as inlet velocities, susceptor rotating speed, inlet to susceptor distance, on the film growth rate and uniformity in industrial scale reactors [[1],[2],[3]]. Advanced flow modeling allows the prediction of process conditions and equipment parameters necessary to provide optimum flows. It also identifies regions where instabilities exist, and helps to optimize the thermal-mechanical characteristics of the system.

2. NUMERICAL METHOD 2.1 Basic fluid dynamic characteristics in the MOCVD reactor

According to the fluid dynamics, it is essential that the flow in the MOCVD reactor has very slow inject velocities, corresponding to a low Re number (1-100) to preserve laminar flow in the reactor. Although pressure is often required to be low in the reaction chamber, the reactant gases are still well within the continuum assumption [[4][5], [5],[6]]. The mass fractions of reactant compounds are small compared with those of the carrier gases. Thus, it is assumed that the mass transfer of reaction compounds in the reactor has no effect on the gas dynamics and heat transfer. This enables the simulation to be resolved into two parts: the concentration fields of the MOCVD reactor to be calculated after the velocity and temperature have been determined.

To perform a CFD simulation of the MOCVD reactor flow, there exist some obstacles. First, the buoyancy effect must be accounted for. In order to reduce consumption of reactant and reactor pollution, the surface decomposition should occur on the susceptor instead of the other walls inside the reactor. Thus the other walls should be cold enough to avoid decomposition. This leads to large temperature gradient inside the reactor. The buoyancy effect plays an important role in the flow stability and vortex generation, and it also causes difficulties in the numerical calculation. Secondly, the flow patterns are rather complex in the MOCVD reactor. The newly developed MOCVD reactors often adopt the design of several rotating disks to improve the uniformity of the grown films. The planetary style geometry is often the case that modern MOCVD reactor adopts. The wafer rotates not only itself but also together with susceptor. Such a complex motion calls for advanced numerical algorithm for the CFD simulation, such as grid generation technique, sophiscated coordinate transformation and glide boundaries.

2.2 Governing equations of the CFD simulation

In order to deal with the complex geometry and flow conditions, the CFD simulation is performed under the general coordinate system while the velocity components remain Cartesian. Computation is performed on the non-stagger grid system with the improved pressure-velocity coupling algorithm by Rhie & Chow [[7]]. The curvilinear grid system and unit control volumes are shown in Fig. 1.

The flow inside the MOCVD reactor is regarded as incompressible, laminar and steady flow. The governing equations have the following form:

Continuity equation 0j

j

uxρ∂

=∂

(1)

Momentum equations i j ji

j i j j i

u u uupx x x x x

ρµ ∂ ∂∂∂ ∂

= − + + ∂ ∂ ∂ ∂ ∂ (2)

Species equation CS

s j s

j j j

Y u Yx x x

ρ µ ∂ ∂∂= ∂ ∂ ∂

(3)

Energy equation Pr

p j p

j j j

c Tu c Tx x x

ρ µ ∂ ∂ ∂= ∂ ∂ ∂

(4)

Gas state equation /b

s s

pRT Y M

ρ =∑ (5)

where µ is the molecular viscosity, pb is the atmospheric pressure with pb=1.01325×105 Pa, sY and sW are the mass fraction and molecular weight of species s. R is the universal gas constant. Sc and Pr are Schmidt and Prandtl numbers, which are assumed to be equal to 0.72 in our calculations. 2.3 Boundary conditions

The Dirichlet boundary condition is adopted for the inlet boundary. The von Neumann boundary condition with 0nφ∂ ∂ = is set for the outlet boundary where n is the unit vector normal to the boundary, φ is the unresolved variable. The zero-slip boundary condition is fit for the solid walls.

3. CFD SIMULATION OF A PLANETARY MOCVD REACTOR

The structure of the MOCVD reactor under present investigation is show schematically in Fig. 2. The inlet gas enters the reactor through three coaxial pipes. The relation between the inlet flux and the inlet velocity is shown in table 1. The distance between the end of the pipes and the substrate is denoted by DL. There exist three heated circle wafers on the substrate. The wafers rotate not only themselves but also together with the substrate, forming the planetary style motion.

x1

x2

y1v1

y2v2

P

N

S

E

W

inner middle outer

DL

wafer substrate

Fig. 1 Schematic diagram of the grids Fig. 2 Schematic diagram of the MOCVD reactor

Table 1. Relation between the inlet flux and the inlet velocity Inlet flux (l/min) Inlet velocity (m/s)

inner 1 0.59

1 0.33 middle

3 0.99

1 0.14 outer

3 0.42

3.1 Study of the parameters without wafer heating

In order to study the geometrical parameters influences, the simulation were first conducted considering no wafer rotating and heating. The two key factors studied in this part is the inlet gas flux through the three pipes and the distance DL. Two groups of simulation are performed. The first group is to study the DL influence with simulation parameters as: inner flux=1.0 L/min, middle flux=1.0 L/min, outer flux=1.0 L/min, rotating speed of the substrate=20r/min. The second group is to study inlet flux influence with simulation parameters as: inner flux=1.0 L/min, rotating speed of the substrate=20r/min.

(a) (b)

Fig.3 DL influence: (a). DL=0.02m; (b). DL=0.01m

(a) (b)

(a). middle flux: 3.0 L/min, outer flux: 1 0 L/min; (b).middle flux: 1 0 L/min, outer flux: 3.0 L/min

Fig.4 middle/outer flux influence (inner flux=1.0 L/min)

Fig. 3 shows the simulation results with different DL. It can be seen from Fig. 3 that at the same inlet fluxes, the flow structure in a small DL reactor is obviously better than that in a large DL reactor. In Fig. 3(b), there show vortexes with smaller scales, locating close the center of the circle. The flow streams are smooth near the wafer and few vortexes exist. Such a flow structure condition helps for better deposition qualities. Fig. 3(a) shows the flow patterns in the reactor with large DL, where the flow patterns are more disordered. Large scale vortices exist in most of the flow regions which may have negative effect to the deposition. Fig. 3 shows that decreasing the distance DL may help to improve deposition quality.

Fig. 4 gives the simulation result with different inlet fluxes for the small-distance-reactor case. The simulation result shows that the increase of middle inlet flux make the vortexes apart form the center of the

circle, and the vortex scale increases. However, increase of the outer inlet flux makes the flow structure more complex.

3.2 Study of the parameters under industrial scale

The parameter influences are studied by numerical simulation. Some basic parameters are fixed while in simulation, which are: the inner flux=1.0 l/min; the middle flux=1.0 l/min; the outer flux=1. 0 L/min. Four groups of simulation are performed with different working parameters as follow,

(1) Neither the substrate nor the wafers rotate, wafers are not heated; (2) Only the substrate revolving with a speed of 20 r/min and wafers do not rotate. No heating; (3) The substrate rotate with a speed of 20 r/min, and the wafers rotate with a speed of 120 r/min.

No heating; (4) The substrate rotate with a speed of 20 r/min, and the wafers rotate with a speed of 120 r/min.

Heating temperature is 800K.

0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(a) 0 0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(b) 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(c) 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(d)

Fig.5: 2D streamline contours for different cases: (a). Neither the substrate nor the wafers rotate, wafers not heated; (b).

Substrate revolving of 20r/min, wafers no rotating. No heating; (c). Substrate revolving of 20r/min, wafers rotating of

120r/min. No heating; (d). Substrate revolving of 20r/min, wafers rotating of 120r/min. Heating temperature 800K.

Fig.5 gives the simulation results of the above four groups with 2D streamline contours. For the two cases that the wafer do not rotate, at the end of the middle channels, the streamlines from the outer channels go backward into the middle channels due to the geometrical splay shape. This is quite contrary to original design idea which hopes the flow from the outer channels being a protection layer. Comparing Fig.5b and Fig.5a one can find that the streamlines near the substrate are much smoother in the case with revolution than that in the no-revolution case. With the wafers rotating, the position that the outer channels fluid enters the near substrate region moves outside, but large scale vortex appears at the outboard of the radius. Such vortices may affect the deposition quality. The flow patterns in the near center regions become more complex. A pair of vortices that has opposite directions exists. It may help to increase mixing but its influence on the deposition quality is not clear at current. Further studies are required. Heating the wafers may obviously decrease vortexes as seen from Fig. 5. Reasonable configuration of revolution speed and rotating speed at certain working temperatures may effectively

eliminate vortices and optimize the flow field. CONCLUSIONS

(1) Computational Fluid Dynamics (CFD) simulation has been performed to study the flow patterns in the Metal-Organic Chemical Vapor Deposition (MOCVD) reactor. Simulation results give satisfactory description of the flow patterns in the MOCVD reactor that has complex geometrical configuration and planetary motion style. It is indicated that the CFD simulation is an effective way to direct geometrical design of the equipment parameters that necessary to provide optimum flows. It could also demonstrate regions where instabilities exist, and help to optimize the thermal-mechanical characteristics of the system.

(2) For the MOCVD reactor studied in this paper, the distance DL, the inlet fluxes, the substrate revolving speed and the wafer rotating speed have obvious influence to the flow patterns. A comparably small DL helps to improve flow. Increasing of the rotating speed forces smoother flow stream. Higher heating temperature has the effect of pushing the main large vortex outwards. Reasonable configuration of revolution speed and rotating speed at certain working temperature may effectively eliminate vortices and optimize flow field. The splay shape at the end of the inlet pipes affects flow greatly. Geometrical optimization may help to get better deposition quality.

ACKONWLEDGMENT

This work was supported by The National High Technology Research and Development Program of China (863 Program), No. 2002AA311240.

REFERENCES [1] Knoruzhnikov SE, Robachevsky AM, Segal AS. Simulation of MOCVD-process for Y-Ba-Cu-O film production

in stagnation zone reactor. Materials Science and Engineering, 1994, B22, 317-320.

[2] Tompa GS, Breiland WG, Gurary A, et al. Large area, production MOCVD rotating disk reactor development and

characteristics. Microelectronics Journal, 1994, 25, 757-765.

[3] Won YC, Do HK, Young SC. Modelling of Cu thin film growth by MOCVD process in a vertical reactor. Journal

of Crystal Growth, 1997, 180, 691-697.

[4] Ning Z. CFD simulation of pulsed MOCVD to reduce gas-phase parasitic reaction. SPIE Vol. 3792, 1999, 58-72.

[5] Linda RB. MOCVD of GaAs in a horizontal reactor: modeling and growth. Journal of Crystal Growth, 1991,

109, 241-245.

[6] Nami Z. Computer simulation study of the MOCVD growth of titanium dioxide films. Journal of Crystal Growth,

1997, 171, 154-165.

[7] Rhie CM, Chow WL. Numerical study of the turbulent flow past an airfoil with edge separation. AIAA J., 1983,

21(11), 1525-1532.

1


ANALYSIS OF FLOW PROCESSES IN DI DIESEL ENGINE WITH DIFFERENT PISTON CAVITIES USING CFD

Pramod S Mehta

(National Institute of Technology, Jalandhar, Punjab 144 011) N. Jaipal

(IC Engines Laboratory, Indian Institute of Technology, Madras, Chennai 600 036) ABSTRACT: In-cylinder fluid dynamics in direct injection (DI) diesel engines has significant influence on fuel-air mixing and hence the engine combustion process through its effect on ignition delay, and on premixed and diffusion phases of combustion. The design of combustion chamber is central to achieving proper fuel air mixing. In the present work, a full cycle computational fluid dynamic simulation is attempted by evaluating flat piston, cylindrical bowl and re-entrant bowl geometries in terms of their flow field distribution and turbulence generation. The predicted results of axi-symmetric single valve arrangement show a quantitatively good agreement with the experimental data. It is observed that the re-entrant bowl provides high mean velocities and turbulent kinetic energy near and at compression TDC, where fuel-air combustion occurs. Also, three-dimensional simulation studies concerning the effects of re-entrant combustion chamber shape, engine speed and initial swirl ratio on in-cylinder flow field and turbulence are carried out for the closed part of the engine cycle and discussed here. 1. INTRODUCTION

In DI diesel engine designs, the shaping of piston cavities has attracted considerable attention of developers of modern low emission engines. The shape of the bowl-in-piston is a key parameter controlling the turbulence level and fuel air mixing. One of the earliest attempts to measure the three-dimensional in-cylinder flow field was made by Arcoumanis et al. [1] on a model internal combustion engine motored at 200 rpm with a compression ratio of 6.7 using laser doppler anemometry. They found that the piston-bowl configurations provide a compression-induced squish motion with consequent formation of a toroidal vortex occupying the whole bowl space. McLandress et al. [2] used a modified version of the three-dimensional CFD code KIVA-3 to analyze a heavy-duty, four cycle, dual intake valve, direct injection, diesel engine. Their results show that many complex flow structures develop during intake and some of them survive during compression and contribute to enhanced mixing near TDC. Chen et al. [3] using CFD found that the large-scale flow structure in the engine is dependent on the port geometry, bowl-in-piston geometry and their locations relative to the cylinder and varies linearly with engine speed. Lisbona et al. [4] validated their numerical results with measurements and showed that larger diameter combustion bowls exhibit lower swirl velocities during the expansion stroke. Lin et al. [5] opined that the duration of high turbulence in cylinder can be prolonged and the diffusion combustion during the later period of combustion process can be enhanced by optimizing the combustion chamber shape.

In general, the existing literature deals with detailed experimental and analytical investigation on the effect of combustion chamber geometry on engine combustion and emissions relating to only few specific geometries [4]. There exists a need to make a comparative evaluation of different types of geometries in terms of their turbulence generation and rate of mixing. In this work, a systematic modeling effort using a standard CFD platform STAR-CD for comparative evaluation of combustion chamber shapes on the development of in-cylinder flow field in DI diesel engines has been discussed. 2. COMPUTATIONAL MODEL AND NUMERICAL SCHEME

The computational domain for full cycle three-dimensional simulation includes the axi-symmetric single valve with flat piston and two bowl-in-piston chambers, which represent the

2

cylindrical and re-entrant chambers in DI diesel engine, are shown in Fig. 1 and the specifications of the corresponding engine are given in Table 1. The different re-entrant bowls investigated by closed cycle three-dimensional simulation are shown in Fig. 2. The mesh for bowl-in-piston, inlet port and valves are created using a commercial grid generating package GAMBIT, whereas cylinder cells are generated in PROSTAR in order to control the numbering of cells and vertices. Hexahedral cell layers of 1 mm each are used for meshing after grid independence study. The cells and vertices of different blocks are mapped in order to represent the complete flow domain. Standard k-ε turbulence model is employed to represent the full turbulent flow. The Pressure Implicit with Splitting Operators solution algorithm is used to solve the algebraic finite-volume equations with an upwind differencing scheme as convective flux approximation. At the entrance of the inlet port, constant pressure boundary is specified. This boundary condition is rather simplified assuming that the port is opened to atmosphere unlike in actual engine condition. However, the use of this boundary condition has also been reported earlier in the work of Chen et al. [3]. The vertical edges of the cells above and below the valve and the corresponding cell faces in the cylinder domain were connected together dynamically during the analysis. Attached boundaries were specified on these coincident cell faces where flows are identical. Attachment and detachment of cell boundaries were controlled by the event commands. Cell layers addition and removal are controlled by event and moving grid commands. At one event time step only one cell layer can be removed or restored. All intermediate meshes representing the flow domain at various stages.

Fig. 1: Mesh at BDC for full cycle three-dimensional simulation

Table 1: Engine specifications for full cycle 3-D simulation Bore x Stroke 75.4 × 94.0 mm Compression ratio 6.7:1 Engine speed 200 rpm Valve diameter 34.0 mm Maximum lift 8.0 mm Seat angle 600

Re-entrant factor 1.0 Re-entrant factor 1.155 Re-entrant factor 1.348 Re-entrant factor 1.595

Fig. 2: Various re-entrant geometries investigated

3

3. RESULTS AND DISCUSSION 3.1 Full Cycle 3-D Simulation Initially the results of the simulation are compared with the experimental results of Arcoumanis et al. [1]. The geometries considered are indicated in Fig. 1. The axial mean velocity profiles normalized based on mean piston speed at 36 degrees after intake TDC on a section plane normal to cylinder axis and 26.5 mm from the cylinder head (within the bowl) for re-entrant bowl geometry are compared with the experimental results in Fig. 3. The profiles at 36 degree after intake TDC show that flow field inside the piston bowl follows the piston motion. This is due to the fact that the intake velocity jet has not penetrated into the bowl due to its small entry diameter. Figure 4 shows the normalized axial velocity during late compression stroke at 36 degrees before compression TDC, on a section plane same as that of intake stroke. Towards the end of the compression stroke, the compression-induced squish motion inside the piston bowl results in a flow pattern rotating in the anti-clockwise direction (Fig. 4). Figure 5 represents profiles corresponding to section plane 35 mm from the cylinder head. From profiles between 36 degrees before compression TDC and at compression TDC, it is observed that axial velocity in the bowl increased due to high squish velocity jet penetrating into the bowl through small bowl entry diameter. There is good quantitative agreement between simulation and experiments (within in 10 percent) at the peak velocity location as shown in Fig. 5. 'The discrepancy between experimental and predicted normalized mean axial velocity data near cylinder axis and towards the piston wall in some of the cases can be attributed to the simplified boundary condition chosen in the work and also the assumption of smooth adiabatic wall. Figure 6 shows normalized axial velocity profiles for the cylindrical bowl geometry at compression TDC on a section plane, 35 mm from the cylinder head (within the bowl). Circulating flow patterns as that of the re-entrant bowl have been observed. Figure 7 shows the comparison of axial velocity profiles for different bowls at compression TDC on section plane 25 mm from cylinder head. It is observed that the re-entrant cavity provides strong circulating flow pattern at compression TDC and provides good fuel-air mixing during combustion. The comparison of velocity profiles during intake at 50 degrees after intake TDC, on section plane along cylinder axis for different piston bowl geometries including flat piston are shown in Fig. 8. It shows that an annular jet emanating from valve orifice, flanked by inner and outer toroidal vortices. These jets result in the flow separation on both sides of the orifice and appears as regions of reverse flow near the outer periphery and behind the valve. The magnitude of velocities is same in all three geometries but the flow pattern is varying. In case of re-entrant bowl, the intake jet hits the piston top surface causing a strong inner vortex, whereas in cylindrical bowl due to large cavity diameter, the intake jet is penetrating into the bowl space and hence increasing the velocities inside the bowl. In case of flat piston, the jet is free to penetrate into the cylinder until it hits the cylinder wall and hence a very weak inner vortex is formed compared to piston bowl shapes.

As the piston moves down, the inner vortex grows more than outer vortex and elongates as piston descends (refer Fig. 9). From Fig. 9, it can be observed that the strength of the inner vortex is more in case of re-entrant bowl compared to cylindrical and flat pistons. As the crank angle proceeds, the velocities in the cylinder decay very rapidly from 11.88 m/s at the middle of the intake stroke to 3.77 m/s at the end of intake stroke (refer Fig. 10). At intake BDC, the flow pattern is same in all piston bowl configurations. From Fig. 10, it can be said that flow pattern at the end of intake process is not affected significantly by bowl geometry. This fact has also been observed and reported by Murakami et al. [6].

Figure 11-12 shows the mean velocity distribution and turbulent kinetic energy contours at compression TDC for different bowl shapes. It can be seen that the mean velocity and turbulent kinetic energy is substantially high for bowl-in-piston case compared to flat piston due to high squish velocities. Between the two cavities, re-entrant cavity is providing high velocities and turbulent kinetic energy due to high squish velocity. Hence, it can be concluded that re-entrant cavity will provide better fuel-air mixing and proper combustion. Since the re-entrant cavity shape is a promising

4

geometry in DI diesel engines, it is decided to investigate the effect of various re-entrant geometries on in-cylinder flow field. 3.2 Effect of Re-entrant Geometry

The influence of bowl entry diameter on in-cylinder flow field and turbulence characteristics is studied. Bowl entry diameter is varied for different re-entrant factors (ratio of maximum cross sectional area of the bowl to the minimum cross sectional area) by keeping the bowl volume constant. Figure 13 shows the variation of swirl ratio within the bowl for different re-entrant geometries. It is observed that the initial decrease in swirl ratio during compression is high for high re-entrant factors. This may be due to friction on account of the increased surface area of the bowl. During expansion, the small entry diameter of the re-entrant cavity suppresses the outflow of charge into clearance space from the bowl. The angular momentum of the bowl will increase during expansion stroke, thus the swirl is maintained for a longer time and helps in better mixing during the diffusion combustion process. The re-entrant cavity with small entry diameter provides high turbulent kinetic energy at compression TDC and maintains it for longer time during the expansion stroke and helps in good fuel-air mixing during the diffusion combustion period (refer Fig. 14). 3.3 Effect of Engine Speed and initial Swirl Ratio

Change of speed mainly affects the angular momentum due to wall friction and viscous moments. The angular momentum increases at any crank angle with increase in speed as expected [7]. Due to higher angular momentum, the radial and tangential velocities are higher at higher engine speeds (Figs. 15-16). The effect of initial swirl ratio on peak swirl ratio is shown in Fig. 17. It is observed that at higher initial swirl ratios, peak swirl ratio is higher but the increase in peak swirl ratio is much less compared to increase in initial swirl ratio. It is observed that at higher initial swirl ratio there is an initial decay in swirl ratio.

4. CONCLUSIONS The flow pattern at the end of intake stroke is not significantly influenced by the combustion

chamber geometry shape, which is in conformity with the observations of Murakami et al. [6]. In case of flat piston, there is decay of mean motion and turbulence considerably by the end of compression due to absence of squish. The flow structure at compression TDC depends mainly on the piston bowl geometry rather than intake generated flow field. For re-entrant bowl, the axial flow structure induced by squish seems to be similar to that of cylindrical bowl but of stronger nature. In case of different re-entrant geometries, it is observed that swirl and squish velocities with higher re-entrant factors cause better mixing during diffusion combustion. Both mean velocity and turbulent kinetic energy vary linearly with engine speed as observed in earlier studies. The peak swirl ratio occurring close to compression TDC is found to increase with increase in the initial swirl ratio. However, the increase in peak swirl ratio is much lower compared to the increase in initial swirl ratio.

5. REFERENCES [1] Arcoumanis C, Bicen A F and Whitelaw J H, “Squish and Swirl-Squish Interaction in Motored Model

Engines”, Trans. of ASME, Journal of Fluids Engineering, March 1983, Vol. 105, 105-112. [2] McLandress A, Emerson R, McDowell P, Rutland C, “Intake and In-Cylinder Flow Modeling-

Characterization of Mixing and Comparison with Flow Bench Results”, Trans. SAE, Journal of Engines, 1996, Paper No. 960635.

[3] Chen A, Veshagh A and Wallace S, (1998), “Intake Flow Predictions of a Transparent DI Diesel Engine”, Trans. SAE, SP-1330: “Modeling of SI and Diesel Engines”, 1998, Paper No. 981020.

[4] Lisbona MG, Olmo L and Rindone G. “Analysis of the Effect of Combustion Bowl Geometry of a DI Diesel Engine on Efficiency and Emissions”, Conference on Thermo and Fluid Dynamic Processes in Diesel Engines, Valencia, Spain, Sept. 2000

[5] Lin L, Shulin D, Jin X, Jinxiang W and Xiaohong G, “Effects of Combustion Chamber Geometry on In-Cylinder Air Motion and Performance in DI Diesel Engine”, Trans. SAE, Journal of Engines, 2000, Paper No. 2000-01-0510.

[6] Murakami A, Sakimoto M, Arai M and Hiroyasu H, “Measurement of Turbulent Flow in the Combustion Chamber of D.I. Diesel Engine”, Trans. SAE, Journal of Engines, 1990, Paper No. 900061, 392-401.

[7] Zolver M, Griard C and Henriot S, “3-D Modeling Applied to the Development of a DI Diesel Engine: Effect of Piston Bowl Shape”, Trans. SAE, Journal of Engines 1997, Paper No. 971599, 141-154.

5

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Fig. 8: Velocity vectors (m/s) at 50 degrees after intake TDC for different bowl geometries


Fig. 9: Velocity vectors (m/s) at 90 degrees after intake TDC for different bowl geometries


Fig. 10: Velocity vectors (m/s) at intake BDC for different bowl geometries

6


Fig. 11: Velocity vectors (m/s) at compression TDC for cylindrical bowl

Fig. 12: Turbulent kinetic energy contours (m2/s2) at

compression TDC for different bowls

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MULTIPLE SOLUTIONS IN A DIFFERENTIALLY HEATED CAVITY

M.D.Deshpande and B.G.Srinidhi Computational and Theoretical Fluid Dynamics Division

National Aerospace Laboratories, Bangalore 560 017 India

ABSTRACT: Natural convection in a three-dimensional cavity heated from below has been studied numerically. Results for the case of a long cavity of span-wise aspect ratio three are interpreted in the light of those obtained for a cubical cavity where two distinct solutions have been detected. Recognition of these multiple solutions has been helpful in understanding more complex flows in the long cavity. 1. INTRODUCTION Natural convection in a rectangular parallelepiped is of fundamental interest and is also of practical importance [1]. We consider here mainly the case of a long cavity of span-wise aspect ratio SAR = 3. The bottom wall is heated and the four side walls are insulated leading to buoyancy driven flow. No-slip boundary condition is applied on all the six walls. Below a critical value of Rayleigh number (Ra)crit there is no flow and heat transfer is by pure conduction. When the Rayleigh number is increased above this value convection is set-in and consequently heat transfer is enhanced. Till the Rayleigh number is increased above another critical value the flow remains steady. For the case of a cubical cavity two families of distinct steady solutions have been obtained. Both the families have the same critical Rayleigh number. When the length of the cavity is increased to SAR = 3 we see a more complex flow situation and the individual blocks of flow in the subdomains can be understood better in the light of the results obtained for the cubical cavity. The present procedure has also enabled us to study the classical Rayleigh-Benard convection problem [2] as a nonlinear three-dimensional problem rather than a linear stability problem. In that study the results from the linear stability theory served as a good check for the present computations. 2. MATHEMATICAL DESCRIPTION The geometry of the rectangular parallelepiped and the coordinate axes are shown in figure 1. All the dimensions are normalized with respect to the vertical dimension l′x along the gravity axis. The bottom wall is heated to a non-dimensional temperature θ = 1 and the top wall is kept at θ = 0. The four vertical side walls are insulated and the no-slip boundary condition is applied on all six walls. The cavity is filled with an incompressible, Newtonian fluid of density ρ and viscosity ν. We use l′x and a reference velocity υ′ref = ν / l′x as the length and velocity scales to non-dimensionalize the equations. The Navier-Stokes equations and the energy equation with Boussinesq approximation have been solved to study the natural convection. Rayleigh number Ra = g β ΔT l′x3 / (ν α) and Prandtl number Pr = μ C p / k are the only parameters of interest for a given geometry. We use the well known Marker and Cell method on a staggered grid with a third order upwind scheme to discretize the convective terms. The numerical procedure is time accurate [3]. Extensive grid independence studies and comparison with the results obtained from the stability theory [2] have been done to assure the accuracy of the results. 3. RESULTS AND DISCUSSION Before we give the results for the cavity of SAR = 3 we describe the flow in a cubical cavity where two distinct steady solutions have been obtained.

3.1 Multiple Solutions in a Cubical Cavity In figures 2(a) and 2(b) are shown the flow patterns due to natural convection in a cubical cavity at Ra = 7,000. These two distinctly different steady solutions have been detected for identical boundary conditions and the solution in figure 2(a), named as family A, has a vertical diagonal plane y = z as a separating stream surface dividing the cube into two equal prisms. We can argue that in the same family A, the other diagonal plane y + z = 1 could have been the plane of symmetry instead. Also, the two vortices have to rotate in the same direction but for a given boundary condition both can reverse the sense of rotation. These four possibilities are clubbed together as family A. A distinctly different possibility is seen in figure 2(b) where the mid-plane y = ½ divides the cube into two cuboids with similar flows. It is easy to argue that in this family B also there are four possibilities. Thus one can have any of the eight stable, steady solutions. The pure conduction case without motion is also a solution but is unstable for these supercritical values of Ra. We would like to draw attention specially to two distinct families since others are only trivially different mathematically. But they may turn out to be important and even confusing in a practical problem where measurements are made only at a few locations. In figure 3 is shown the bifurcation diagram where wall temperature at point P(x = y = ½ , z = 0) is plotted as a function of Ra. At (Ra)crit = 3386.75 ± 0.25 the solution bifurcates and any of the eight solutions are possible. In this figure only five solutions are shown since the other three happen to coincide with the ones shown here. 3.2 Long Cavity with SAR = 3 This case with l z = 3 has more complex flow patterns and we can imagine three cubical volumes kept side by side but boundary conditions being applied only at the true boundary. (Ra) crit for this geometry was found to be 2559 ± 2, a substantial decrease from the cubical case due to SAR = 3. In figure 4 are shown the time traces of temperature at x = 0.5, y = 0.41250 & z = 1.5. In frame (a) for Ra = 10 4 we see a nearly monotonic change to steady state. For Ra = 3 x 105 in frame (b), on the other hand, we see an oscillatory behaviour. These oscillations are due to span-wise oscillations of the cells formed that we will see. For Ra = 106 the flow has become turbulent as can be seen from frame (c). Two views of the vortex patterns for Ra = 104 are shown in figure 5. These three pairs of vortices are similar to the ones shown for family B in figure 2(b). Two vortices in each cubical volume in this figure should rotate in the same sense but in the opposite direction as compared to the pair from the neighbouring volume as seen from the bottom frame. For Ra = 105 (see figure 6(a)) flow remains steady but we get in each cubical volume two pairs of vortices instead of one pair shown in figure 5. These are not B-type but have some resemblance to family A. Quite interestingly at Ra = 3 x 10 5 the nature of flow changes completely (see figure 6(b)) and we end up with three pairs of vortices. The vortices we have identified in a cube (figure 2) may act as basic building blocks in a more complex flow as in this long cavity. Hence we can imagine three pairs of family A or nearly family A-type vortices at Ra = 3 x 10 5 in figure 6(b). Thus the pattern with B-type vortices at Ra = 104 has changed to one with A-type vortices at Ra = 3 x 10 5 with a complicated picture at the intermediate Ra of 10 5. As Ra increases we also see a gradual connectivity between the vortices and hence mixing. See that we have two possibilities for the sense of rotation of the vortices. Contours of heat flux q on the bottom wall shown in figure 7 for three values of Ra are interesting. We see from frame (a) for Ra = 104 that the contours roughly divide the rectangle into two parts in the ratio 1:2. At Ra = 105 in frame (b) and at Ra = 3 x 10 5 in frame (c) we see an interchange in the ratio of this subdivision to 2:1. This can be attributed to the reversal of sense of direction of the central vortices from figure 5 to figure 6. From the study made we know that it could have been the other way also. Note that it is not possible to guess the flow field or even the number of vortices from these planar plots. At Ra = 106 flow has become turbulent and we see a more complex heat flux pattern in figure 7(d) as can be

l = 1xl = 1x

lyly

lzlz

zx

g

y

Fig. 1: Geometry of the cavity.

imagined. Average Nusselt number Nu = q l?x /(k ÄT?) for the entire plate has been calculated by integrating these flux values as a function of Ra. 4. CONCLUSIONS Two distinct steady solutions have been obtained for the case of natural convection in a cubical cavity. Further, each family has four possibilities leading to a set of eight stable steady solutions that can occur in a real situation. In the case of a longer cavity with SAR = 3, (Ra)crit decreased as expected and we have three pairs of B-type vortices at Ra = 5,000 to 50,000. At Ra = 105 we have six pairs of vor tices and some pairs resemble family A-type of vortices. Quite interestingly at Ra = 3 x 105 we get only 3 pairs of vortices but this time they are close to A-type. These vortices oscillate in the span-wise direction but finally reach a steady state. At Ra = 106 the flow is turbulent. The patterns of heat flux contours on the hot plate also evolve as Ra is increased but it is not possible to guess the flow patterns inside the cavity from these heat flux contours. REFERENCES [1] Jaluria Y: Fluid flow phenomena in material processing- The 2000 Freeman Scholar Lecture. J. of Fluids Engg.

2001, 123, 173-210. [2] Chandrasekhar S. Hydrodynamic and Hydrom agnetic Stability. Oxford at the Clarendon Press, 1961. [3] Deshpande MD: Natural convection in a cubical cavity: Case of multiple solutions. NAL Project Document PD

CF 0305, 2003.

'x

')

Fig. 2: Two distinct solutions in a cubical cavity. Ra=7,000. (a) Solution from family A. (b) Solution from family B.

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4Fig. 5: Streamline patterns for Ra = 10 resemble family B solution inside a cube.

4(a) Ra = 10 5(b) Ra = 3 x 10

6(c) Ra = 10

Z

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4(a) Ra = 10 5(b) Ra = 10

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ZX

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THE NUMERICAL STUDY OF FLOWFIELD INTERACTION BETWEEN NON-UNIFORM INLET FLOW AND MUTISTAGE COMPRESSORS

W. G. Joo

Associate professor, Dept. of Mechanical Engineering, Yonsei University, Seoul, Korea ABSTRACT: One of the important issues in the assessment of the inlet/engine compatibility in turbofan engines is to predict the coupling effects between the non-uniform inlet and engine. The computational method using an actuator-disk model has been applied to the calculations of the flow through a multistage turbofan engine subjected to non-uniform total pressure inlet flow. The results of a series of calculations are presented and the nature of the interactions between the flow through a fan and non-uniform inlet flow is addressed. 1. INTRODUCTION

Today, the basic trend in the design of a turbofan engine is to increase the engine bypass ratio to improve the propulsion efficiency. However, the increase of nacelle diameter necessary to obtain higher bypass ratios could produce an unacceptable drag penalty because of the additional wetted surface area, and increased weight. A possible solution, maintaining the benefit from an increase in the bypass ratio, is to use shorter and slimmer nacelles with low contraction ratio.

The principal problem associated with low contraction ratio intakes is the high level of flow distortion. Although the engine inlet is designed with the aim of providing satisfactory inlet conditions to an engine, it is difficult to avoid distortion due to internal separation while stationary, in crosswind or high incidence conditions, particularly in low contraction ratio intakes. This distortion can lead to high fan stresses and noise, a reduction in performance and more importantly a decrease in engine surge margin.

The response of a compressor to distorted flow can be divided into two major aspects: the redistribution of inlet flow approaching a blade row and the resulting change in the overall compressor performance. These two subjects are strongly coupled and cannot be considered separately since the change in overall compressor performance will affect the redistribution of inlet flow.

Many theoretical models [1] to predict flow field coupling effects between a compressor and inlet distorted flow have been developed, but most of them are two dimensional methods based on parallel compressor analysis or linearised theory, and the solution of three dimensional linearised equations [2-3] is found for only the case of steady incompressible linearised free vortex flow. These methods predict some important aspects in flow field coupling, but are difficult to use for the practical problems with complicated geometry.

Joo and Hynes [4] developed a three-dimensional computational method using actuator disk models of blade rows for the calculation of non-uniform flow with long length scale, which is capable of predicting non-uniform inlet flow and engine flow field coupling effects in a complicated turbofan geometry.

This paper investigate, with calculation results obtained using their method, the features of the flow field interaction between multistage fans and non-uniform total pressure inlet flow in a military multistage turbofan engine. 2. COMPUTATIONAL METHODS

The calculations of the non-uniform flow through blade row require a computation of the unsteady three-dimensional viscous flow through the whole annulus including all blade passages because the non-uniform flow does not satisfy the periodic boundary condition. This kind of calculations may be performed using a current available supercomputer, but it is not practical for engineering purpose. This leads to a need to model the flow within the blade rows.

Any model for use in this context must comprise two parts: a method for calculating the flow fields outside blade rows and a reliable way of estimating fan performance in non-uniform flow. Flow fields calculation methods are reasonably well developed, but it is the estimation of fan performance that is more pressing issues.

A characteristic feature of the flow field asymmetry associated with intake, pylon, or non-uniform total pressure flow is that the circumferential length scales of interest are long when compared with a blade pitch. Most of the models applicable to low hub-to-tip ratio blade rows exploit this feature and use an actuator disk or a semi-actuator disk blade row model. The essential idea is that the local blade performance at each radius and at each circumferential location can be related to the performance that the blade would exhibit in clean flow when subjected to the same inlet conditions. Simple corrections to account for the effects can also be incorporated.

The present computational method is using an actuator disk blade row model and has been developed by Joo and Hynes [4]. The important issues related to an actuator disk model such as the sensitivity of solution to the disk location are described in details in Joo and Hynes [4] and thus a brief description of the method and the extra details relevant to the current calculation are given here. 2.1 Computational Method for Flow Field Regions

The flow fields outside blade are found from the solutions of three dimensional Reynolds averaged Navier-Stokes equations in conservation form. These equations are written in the absolute frame using a cylindrical coordinate system, because the distorted flow field is assumed to be steady in this frame.

∫ ∫∫ΩΩ

Ω=⋅+ΩA

QddAFUdt∂

∂ (1)

where

+=

+++

=

=

0

000

2

rpV

Q

VHiVVirVVr

iVVV

F

EVVV

U

rr

xx

r

x

θθθθ ρ

ρτρτρ

τρρ

ρρρρρ

with

V V i V i V ix x r r= + +θ θ , Absolute velocity

τ = Stress tensor containing both the static pressure and viscous stresses

2

2VTcE V += , Total internal energy

H = Stagnation enthalpy Ω = Volume of the control volume A = Areas of the control volume surface

These equations are discretised on a set of control volumes, formed by a simple, structured H-grid

construction. Flow variables are stored at cell centers and values on cell faces for flux evaluation are thus found by a simple average of the cell variables on either side of the face with second-order accuracy on smoothly varying grids.

The size of grids near boundaries used for the calculation is usually too large in comparison with a boundary layer thickness to resolve annulus boundary layers. Therefore, only the laminar shear stress, not

turbulence, is considered and the calculation would become effectively inviscid one. The discretised set of equations of motion is solved by time marching. The basic solution algorithm is the same as that developed by Dawes [5] for the calculation of the flow within a single blade row passage. This time marching scheme consists of a two-step explicit and one-step implicit scheme derived as a pre-processed simplification of the Beam-Warming algorithm.

2.2 Actuator Disk Boundary Conditions

The length scales that are likely to characterize the flow interactions between fans and intakes and downstream components are much larger than a blade pitch. In addition, it will emerge that some of the compatibility issues do not involve directly the flow within the blade rows themselves. In these circumstances an actuator disk model for a blade row would seem a useful first approximation.

The flow fields upstream and downstream of the blade row are coupled by boundary conditions imposed across the actuator disk to represent the fan performance. Five matching conditions are required across an actuator disk, corresponding to the five independent flow variables in the equations of motion. The boundary conditions used in the present model, which are applied at each radial location and at each circumferential position, are:

(i) conservation of mass, (ii) conservation of radial momentum, (iii) conservation of rothalpy, (iv) relative exit flow angle specified, (v) entropy rise (or total pressure change) specified.

The last two conditions are associated with blade performance, which are given as input data. The

flow angle at exit to the actuator disk could be taken directly from the values of the measured or calculated flow angle at the blade trailing edge. The flow through the rotor subjected to the non-uniform inlet flow is unsteady to a certain extent, but this unsteady effect is assumed to be neglected.

The conditions described above must be modified when blade sections are choked, since this set of boundary conditions does not contain an allowance for blade blockage. In present calculations, a simple model for choking suggested by Joo and Hynes [4], based on two-dimensional flow into a choked blade section, was incorporated into the actuator disk boundary condition (i) and (v).

The way to integrate these actuator disk boundary conditions into a numerical model depends on the numerical scheme used for calculating flow field regions. The detailed implementation of actuator disk boundary conditions used in the present calculation can be found in Joo and Hynes [4]. 3. CALCULATION RESULTS AND DISCUSSIONS

The computational method applied to the calculation of the flow in a military multistage turbofan engine which is subjected to inlet flow with non-uniform stagnation pressure of square wave type. Fig.1 shows a cross section of calculation domain that is used to model the flow through the multistage turbofan engine and its installation. The engine has an axisymmetric intake and two stage turbofans and a stator.

°180

Two stage blade rows and the stator downstream are contracted to plane actuator disks which are located at mid-chord of each blade row. The flow fields upstream and downstream of the blade row are found using a fully three-dimensional computational method and coupled by boundary conditions imposed across the actuator disk to represent the blade row performance.

The values of relative flow angles and losses at the exit to the fan actuator disk were obtained from a throughflow calculation for multistage fan operating near design in uniform inlet flow. The calculation domain in the core engine section is extended to downstream of pylon where a boundary condition simulating the presence of the core engine is imposed. The calculation domain in the bypass stream also includes the pylon and a throttle-like boundary condition, as in the calculation of the civil turbofan engine, is applied at the outlet.

(a) At rotor 1 inlet

Rotor1 Stator1 Rotor2 Stator2 Stator3 Leading edge of pylon Trailing edge of pylon

Splitter

Rotor1 Stator1 Rotor2

Stator2

Stator3

Pylon

rm θ

x

(a) Cross section (b) Unwrapped annulus at mean radius

Fig.1. Calculation Grids for predictions of the non-uniform flow through multistage turbofan engine

The flow at the inlet boundary to the calculation domain is assumed to be non-swirling, to have non-uniform total pressure of 180 square wave type and constant total temperature and to be roughly aligned with the streamwise surfaces of the calculation grid. The magnitude of total pressure distortion is

7.5% of mean value. The calculations are performed for an engine operating at design fan speed where the total pressure ratio is about 2.52 and the mass flow is 27.9 kg/s and the bypass ratio is 0.84. A

finite volume grid is used.

°

±

37 21150××A percentage variation of static pressure at the first rotor actuator disk inlet is shown in Fig.2.a. The

static pressure at inlet plane to the intake is constant and would remain constant if the fans do not exist in the duct, but the presence of the fans force to redistribute the inlet flow significantly. The calculation result shows 20% variation about the mean value around the annulus at the tip and 13% at the hub and around difference of phase angle in the fan rotation direction with total pressure distortion. This significant variation of static pressure is caused by the interaction between non-uniform inlet flow and the first rotor, and shows that the both flow fields are strongly coupled. The effects of the fan on the upstream flow redistribution appears more strongly in hub region than in the tip region, as in the civil engine, and it would be because of the difference between characteristics of pressure rise of rotor blade at hub and at tip. The strength of this effect decays upstream of the rotor, but it is still felt around 55% of magnitude of rotor inlet distortion at half a mean radius upstream (Fig.2.b), and around 35% at a mean radius upstream (Fig.2.c).

°60

(a) At mean radius upstream of rotor 1 (b) At half mean radius upstream of rotor 1 (c) At inlet of rotor 1

Fig.2. Predicted variations of static pressure

(a) At rotor1 exit (b) At stator1 exit (c) At rotor2 exit

(d) At stator2 exit (e) At stator 3 exit

Fig.8. Predicted variations of total pressure The non-uniform inlet flow produces non-uniform fan performance at each circumferential and

radial position. Figs. 3.a-e show a percentage of stagnation pressure variations at exits to each blade row. Total pressure variation the first rotor exit has a 14% variation about the mean value around the annulus at the tip and 25% at the hub. This result indicates that inlet total pressure distortion is largely amplified in hub region through the first rotor from 15% variation at the first rotor inlet plane by the strong interaction between non-uniform inlet flow and the first rotor. This non-uniform performance can influence the stall margin of the rotor.

The magnitude of total pressure distortion at the first stator exit is similar to one at the first rotor exit, as it could be expected. It can, also, be seen that total pressure distortion is attenuated through the second rotor from 14% to 10% at tip and from 22% to 15% at hub. This magnitude of total pressure variation remains through the second stator and the third stator. Therefore it can be concluded the features of attenuation of total pressure distortion through the blade row is mainly affected by the characteristic of a rotor which produces work.

It should be noted that the calculation results include the effect of pylon downstream, and, therefore, it is needed to investigate quantitatively the each effect of pylon and non-uniform total pressure inlet. 4. CONCLUSIONS

One of the important issues in the assessment of the intake/engine compatibility in civil turbofan engines is to predict the coupling effects between the intake and engine. An actuator disk model has

been applied to the calculation of the flow through a high bypass ratio turbofan geometry including the effects of the presence of the core support pylon, the core engine and an asymmetric inlet flow.

The upstream flow field is significantly redistributed by the interaction between the engine and non-uniform inlet total pressure. The effect of the multistage fan on the upstream flow redistribution takes place to the distance longer than a mean radius. Therefore, the non-uniform total pressure inlet flow and fan flow field coupling seems to be stronger than the non-uniform static pressure inlet flow and fan flow field coupling. Total pressure distortion is amplified or attenuated through the rotor and the variations of total pressure distortion are little through a stator. ACKNOWLEDGMENTS

This research was performed for the Smart UAV Development Program, one of the 21st Century Frontier R&D Programs funded by the Ministry of Science and Technology of Korea.

REFERENCES [1] Mokelke HG: Prediction Method in Distortion Induced Engine Instability. AGARD Lecture Series. 1974 No. 72 [2] Dunham J: Non-axisymmetric Flows in Axial Flow Compressors. Mechanical Engineering Science Monograph .

1965, No. 3, 1-32 [3] Hawthorne WR, Mitchell NA, McCune JE and Tan CS: Non-axisymmetric Flow through Annular Actuator

Discs: Inlet Distortion Problem. ASME Journal of Engineering for Power. 1978, Vol. 100, 604-617 [4] Joo WG and Hynes TP: The Simulation of Turbomachinery Blade Rows in Asymmetric Flow Using Actuator

Disks. ASME Journal of Turbomachinery. 1997, Vol. 119, No. 4, 723-732 [5] Dawes WN: Development of a 3D Navier-Stokes Solver for Application to All Types of Turbomachinery.

ASME Paper 88-GT-70. 1988 [6] Joo WG and Hynes TP: The Applications of Actuator Disks to Calculations of the Flow in Turbofan

Installations. ASME Journal of Turbomachinery. 1997, Vol. 119, No. 4, 733-741

Proceedings of the Tenth Asian Congress of Fluid Mechanics 17-21 May 2004, peradeniva, Sri Lanka

THE APPLICATION OF REDUCED GRID FOR GLOBAL OCEAN MODELING

Mohamad A. Badri

Sub sea R&D center, Isfahan University of Technology, Isfahan, Iran. ABSTRACT A limitation of global climate models with explicit finite-difference procedures, is the time step restriction caused by the decrease in cell size associated with the convergence of meridians near the poles. To keep the longitudinal width of model cells as uniform as possible, a reduced grid is applied to a three-dimensional primitive equation ocean climate model. With this grid, the number of cells in the longitudinal direction is reduced at high latitudes. The grid consists of sub grids with changing in resolution, which interact at interfaces along their northern and southern boundaries. In this paper the finite difference technique to these interfaces has been extended. The reduced grid allows an increased time step while eliminating the need for filtering and reduces execution time per model step about 20%. The reduced grid model has been implemented for parallel computing with two-dimensional domains. Small solution effects and considerable execution time improvements has been shown.

1. INTRODUCTION Global ocean models usually use of finite difference explicit time stepping method in spherical coordinates. At the poles, the coordinate system has a singularity and also the stability conditions of the method, depends on the size of grid cells. So, the convergence of meridians towards the poles in spherical coordinates occurs. In this work, the number of grid cells in the longitudinal direction is decreased near the pole. So, the cells remain uniform in size (Fig. 1). Filtering of high frequency components at higher latitudes allows the use of larger time steps without instability.

Fig 1. Reduced grid

The advantage of using reduced grid, is an increase in the allowable time steps and a decrease in the number of grid cells and grid modifications at small number of latitudes. 2. GOVERNING EQUATIONS Ocean circulation model has been used in this modeling. The total velocities are split into barometric and baroclinic modes. )ˆ,ˆ(),(),( vuvuvu += (1) Barametric velocities are defined as follows, where η is the surface height and H is the depth:

∫∫ −−==

ηη

HHvdzvudz ,

λλρ Fp +)/( 01

φφρ Fp +)/( 01

])([ 11 φλ−− +− vmuma

u (2)

Baroclinic velocities are calculated as : u (3-1) mafvuvnaut −=−−Γ+ −−)(ˆ 1

v (3-2) mufunav ut −=++Γ+ −−)(ˆ 21

(3-3) =wz

whereφ is the latitude,λ the longitude, a the radius of the earth m=secφ , φtan=n 0ρ a reference density and f is the coriolis parameter. (4-1) gpz ρ−=

),,( ZST (4-2) ρρ = T is the temperature and S is the salinity. )(qΓ is the advective operator, and viscous operators are given by:

λF φF

Γ (5-1) zwqvqm )(])( 1 +− φ]2) λmnvu −

uqmaq )[()( 1 += −λ

(5-2) 1[()().( 22λ naAukuAF mzzmhmh −++∇∇= −

(5-3) ]2)1[()().( 22λφ mnuvnaAvkvAF mzzmhmh −−++∇∇= −

mmKA , are the horizontal and vertical viscosity coefficients.

Barotropic velocities are given by: XgHmavf +−= −

λη1ut − (6-1)

YgHaufvt +−=+ −φη

1 (6-2) where X and Y are forcing terms as follows:

∫−

−η

H

dzma 1∫∫ ∫−

− −

−− −+∂∂−∂

∂−=η

λ

η η

φλH

H HdzFuvdzadzumaX 121 )()( (7-1)

∫ ∫− −

−− −+∂∂−∂

∂−=η η

φϕλ H HdzFdzvauvdzma 211 )()( ∫ ∫− −

−η η

H Hdza 1

).() TATk hhhzzh

Y (7-2)

The conservation of the temperature and salinity is given by the tracer transport equations. T ()(Tt =Γ+ (8-1) ∇∇+ ).() SAsk hhhzzh()(sSt =Γ+ (8-2) ∇∇+ where , are the horizontal and vertical diffusion coefficients. hA hk Baroclinic velocities are calculated by integrating of (3-1) to (3-3) then integrating of (6-1) to (6-2) for barotropic equations. 3. REDUCED GRID EXPLANATION The most common arrangements in grid structure in ocean model is for reduction of truncation error and modification of wave speeds accuracy. The ratio of the number of grid cells on adjacent regions has been tested for refinement ratios between two and three. The smaller refinement ratio used in the reduced

grid causes less disparity between the smallest and the largest cells. In this regard, it has been found that the ratio of three is the best to achieve uniformity of the following: - algorithms and cell sizes - solutions between the various regions of the reduced grid In order to simulate the problem to use the reduced grid, four resolution region joined together by three interferences and some advantages have been revealed. The reduced case speedup is closer to the ideal for low processor numbers, but significantly weaker speedup at higher processor numbers. The amount of computation compared to communication is lower for the reduced grid Fig. 2.

Fig 2. Percentage of time used in communication for standard and reduced grid

Fig. 3 shows that at low to moderate numbers of processors, the reduced grid is about more than four times faster and the reduced grid execution speed is three times higher than the standard grid.

Fig 3. Speed of execution for standard and reduced grid 4.CONCLUSIONS In this paper, designation and use of reduced grid in parallel global ocean general circulation model has been investigated. By decreasing the number of grid cells and increasing the length of stability limited time step, computational efficiency is increased.

In this method, a grid with four resolution regions joined together by three interferences and no filtering of variables. The parallel speed up of the model is better than the standard method. ACKNOWLEDGEMENTS This work is part of a numerical study in Sub-sea R&D centre at Isfahan University of Technology. REFERENCES [1] A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion, J. of Computational Physics, pp. 119-143, 1966. [2] G.L. Browning, J.J. Hack, P.N. Swarztrauber, A comparison of three numerical method for solving differential equations on the sphere, Monthly weather review, pp. 1058-1075, 1989. [3] P. B. Duffy, K. Calderia, J. Selvaggi, M. I. Hoffert, Effects of sub grid-scale mixing parametrizations on simulated distributions of natural, temperature and salinity in a three-dimensional ocean general circulation model, J. of physical oceanography, pp. 498-523, 1997. [4] A. A. Mirin, D. E. Shumaker, M.F. Wehner, Efficient filtering techniques for finite-difference atmospheric general circulation models on parallel processors, pp. 729-740, 1998. [5] M. R. Wadley, G.R. Bigg, Implementation of variable time stepping in an ocean general circulation model, pp. 71-80, 1995. [6] P. N. Swarztrauber, D.L. Williamson, J.B. Drake, The Cartesian method for solving partial differential equations in spherical geometry, Dynamics of Atmospheres and Oceans, pp. 679-706, 1997.


Numerical Simulation of Cavitation Inception for Axisymmetric Headforms

M. Zahid Bashir, S. Zahir, S. Bilal, M.A. Khan

CFD-Chapter, Computational Modeling, Control & Simulation Society Of Pakistan, www.ccsspak.org

ABSTRACT: Cavitation inception number for three head forms including hemisphere, cone and flat head is determined numerically and compared with experimental water tunnel results. Multiphase mixture cavitation model is employed. Cavitation inception number is also studied as function of incidence angle. Cavitation inception criteria is established for different types of geometric shapes. CFD results match very well with cavitation tunnel results at zero incidence angle but agreement is not good at intermediate incidence angles. NOMENCLATURE p Free stream pressure [N/m2]

V Free stream velocity [m/s]

Cavitation number Re Reynolds number Density of fluid [kg/m3] Dynamic viscosity of fluid [Pa.s] Angle of attack cp Pressure coefficient

mv Mass averaged velocity [m/s] m Mixture density [kg/m3] k Volume fraction of phase k

m Mass transfer due to cavitation [kg/s]

kdrv , Drift velocity of secondary phase [m/s] R Bubble radius [m] pB Pressure inside bubble [N/m2] pv vapor pressure [N/m2] Volume of individual bubble [m3] Number of bubbles per unit volume of liquid [1/m3] y+ Non dimensional distance of first grid points from body surface 1. INTRODUCTION Cavitation occurs in variety of liquid flows including hydraulic turbo machines, underwater moving vehicles, submarine launched missiles and pipe flow. It has detrimental effects on the performance of hydraulic machinery and objects moving in liquid. For example in hydraulic turbo machines it affects their efficiency and damages the impeller blades severely. In recent years this phenomenon was studied only in laboratories such as in cavitation tunnels. This experimental technique is very expensive and time consuming. Now research is going on to study this phenomenon by means of computational fluid dynamics techniques. A simple technique to study cavitation inception is to determine pressure distribution on the body using single phase flow model. The location where pressure is below saturation vapor pressure is the

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location of cavitation inception and the minimum cavitation number at which pressure drops below saturation vapor pressure is the cavitation inception number [2]. Now a number of cavitation models have been developed by researchers and they are being implemented numerically. In this study a multiphase mixture model with cavitation is implemented. Rayleigh-Plesset equation is used for the modeling of bubble growth and collapse. Cavitation inception for water flowing around three different axisymmetric headforms namely hemispheric nose, a 30 degree cone and a flat nose is studied numerically and compared with experimental results [1]. Cavitation inception is function of cavitation number () defined as:

221

v

ρVPPσ

Cavitation inception also depends on turbulence level so it is also a function of Reynolds number. In this study Reynolds number is kept constant and cavitation number is varied by varying free stream pressure. Effect of turbulence level on cavitation inception is not taken into account. 2. THEORETICAL FORMULATION In this study multiphase mixture model is implemented. It uses single fluid approach. This model allows the phases to be interpenetrating. The continuity equation for the mixture is:

)1(mvρρ mmmt

Where

)2(ρ

vραv

m

n

1k kkkm

and

(3)ραρn

1kkkm

The momentum equation for the mixture is obtained by summing the individual momentum equations for all phases. It is expressed as:

(4)vvραFgρvvµpvvρvρt

n

1kkdr,kdr,kkm

Tmmmmmmmm

Where n is the number of phase

)5(µαµn

1kkkm

)6(vvv mkkdr,

volume fraction equation for secondary phase p is:

(7)vραvραραt pdr,ppmpppp

Following assumptions are made in the cavitation model: The system under investigation involves only two phases. Bubbles are neither created nor destroyed. The population of bubbles per unit volume is known in advance. The volume change of individual bubbles with respect to space and time is denoted by:

)8(πR34 t),rφ( 3

where R is the bubble radius The volume fraction of vapor is defined as

)9(ηφ1

ηφαv

where is the population number of bubbles per unit volume of liquid. The volume fraction of vapor is calculated by:

)10(dt

dρραρ

dtdφ

ηφ1η

ρρvαα

tm

m

v2

m

lmpp

The Rayleigh-Plesset equation relates the pressure and the bubble volume :

)11(dtdR

Rρµ4

Rρ2σ

ρpp

dtdR

23

dtRdR

l

l

ll

B2

2

2

The process of bubble growth and collapse is given by:

(12)

pvp,l3ρ

Bpp2

pvp,l3ρ

pBp2

dtdR

3. GEOMETRY Three axisymmetric head forms are analyzed in this study. Their geometric parameters are given below: Radius of cylindrical portion of the three axisymmetric headforms is same. Radius of cylindrical part = 25 mm Radius of hemisphere = 25 mm Radius of circular arc of flat nose = 12.5 mm Half angle of cone = 30o 4. GRID GENERATION For all three nose shapes two block 2-D grid is generated. The grid is very fine near the wall of the body in order to properly resolve boundary layer and the cavitation inception. The resolution of the grid near solid wall is such that y+<10. Typical number of grid points is 40,000. The grids for three head forms are shown in Figs. 1 – 2. 5. BOUNDARY CONDITIONS At the inlet boundary, velocity components and turbulence parameters are specified. At the outlet, free stream pressure and turbulence parameters are specified. Both at inlet and outlet 0% secondary vapor phase is entering and leaving the domain. At the surface of the body isothermal no slip boundary condition is specified. Since the problem is axisymmetric the symmetry line is taken as axis boundary condition.

6. FLOW CONDITIONS A two phase mixture model with cavitation is implemented. The fluid in the domain of interest is considered to be mixture of liquid water and water vapor. The population density of the bubble nuclei is taken to be 1 x 106 /m3. The size of the bubble nuclei is 1 x 10-5 m. Since Re > 5 x 106, the flow regime is considered turbulent. k- turbulence model with non-equilibrium wall function is used. 7. RESULTS AND DISCUSSION From the point of view of hydrodynamic performance of an underwater moving body, cavitation inception number is an important parameter. In order to make a comparison between different nose shapes from hydrodynamic performance perspective, cavitation inception number is a very important parameter in addition to other parameters such as drag, hydrodynamic stability etc. Due to the importance of cavitation inception number, it is studied numerically for three different head forms. In the case of cavitation tunnel experimental results much experience and judgment are required in order to decide whether cavitation inception has taken place or not. Likewise, CFD also requires an experience and judgment in order to decide the cavitation inception criteria. There are two methods to visualize the inception of cavitation. The first is to plot contours of vapor phase. Presence of vapor phase indicates inception of cavitation. The second method is to plot cp on the surface of the body. If (cp)min <- then cavitation inception has taken place on the surface of the body. Figs 3(a) and 4(a) show the vapor phase contours at two different cavitation numbers for hemispherical head form. Fig 4(a) shows that there is no vapor in the flow field so at this cavitation number cavitation inception has not taken place. Fig 3(a) shows the presence of vapor phase at the junction of nose and cylinder. It means that at this cavitation number cavitation inception has taken place. Figs 3(b) and 4(b) show cp distribution at the corresponding cavitation numbers. In Fig 4(a) (cp)min >- therefore in this case cavitation inception has not taken place. In Fig 3(b) (cp)min <- , therefore in this case cavitation inception has taken place. So in this case both criteria of cavitation inception are compatible. For a conical nose the problem is not so simple. At the junction of cone and cylinder there is an abrupt discontinuity in slopes. At this slope discontinuity CFD is unable to predict the flow behavior accurately on a very small localized region. CFD predicts comparatively low pressure at slope discontinuity and as a result there is a small fraction of vapors generated in this very small region. For these types of geometries a different criterion for cavitation inception is developed on the basis of experience. According to this criterion if the percentage of vapor phase is greater than 10% then cavitation inception has taken place at bodies with abrupt slope discontinuity. Table. 1 Experimental[1] and CFD cavitation inception numbers at various incidence angles. At intermediate incidence angle CFD prediction in not in good agreement with the water tunnel results as shown in Table 1. This is due to the reason that level of complexity of flow increases at incidence angle and multiphase mixture model is unable to handle this situation accurately.

Head Form

α V (m/s)

(σi)exp (σi)CFD

0 6 0.66 0.7 5 6 0.78 0.9

Hemi-spherical

10 6 0.98 1.2 0 6 1.28 1.35 5 6 1.44 1.75

Flat Head

10 6 1.68 2.15 0 6 1.49 1.55 5 6 1.59 1.75

30o cone

10 6 1.66 1.95

8. CONCLUSIONS CFD calculations predict cavitation inception number very accurately for geometries without slope discontinuity and at low incidence angles. At intermediate incidence angles CFD predicted cavitation inception number is not in good agreement with experimental results. For geometries with slope discontinuity the criterion for cavitation inception is that the amount of vapor phase should be greater than 10%. Cavitation inception number is not so sensitive to bubble density and bubble size. Cavitation inception number is independent of advection discretization schemes. 9. REFERENCES [1]. Hua, L., Shiquan, Z. and Yousheng, H. “An experimental study on cavitating axisymmetric headforms.”, Ship building of China, 1995 [2]. Bashir, M. Z., Zahir, S. and Rafi, H., “CFD predictions of axial pressure distribution and flow structure of water and air around hemisphere cylinder configuration at intermediate to high Reynolds numbers and at various incidence angles”, 1st international Bhurban Conference on Applied Sciences and Technology, 2002. Fig 1 Grid of hemispherical head form Fig 2 Grid of flat head form Fig 3 (a) Vapor phase contours for Fig 3 (b) cp distribution on body of hemispherical head form at = 0.70 hemispherical headform at = 0.7

Fig 4 (a) vapor phase contours of Fig. 4(b) Cp distribution on body of hemispherical headform at =0.75 hemispherical headfrom at = 0.75 Fig 5 (a) vapor phase contours at =1.55 Fig 5 (b) cp distribution on the 30o cone head form at = 1.55

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