AI A A-84-0 2 5 3 An Incompressible Navier-Stokes Flow Solver in Three-Dimensional Curvilinear Coordinate System Using Primitive Variables D. Kwak, NASA Ames Research Center Moffett Field, CA; and J.L.C. Chang and S.P. Shanks, Rockwell International, Canoga Park, CA; and S.R. Chakravarthy, Stanford Univ., Stanford, CA

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AlAA 22nd Aerospace Sciences Meeting January 9-12, 1984/Reno, Nevada

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AN INCOMPRESSIBLE NAVIER-STOKES FLOW SOLVER IN THREE-DIMENSIONAL CURVILINEAR COORDINATE SYSTEMS USING PRIMITIVE VARIABLES

Dochan Kwak' NASA Ames Research Center, Moffett Field, California

James L.C. Changt and Samuel P. Shankst Rocketdyne Division, Rockwell International, Canoga Park, California

S u h m a r R. Chakravarthyt Stanford University, Stanford, California

Abstract

An implicit, finite-difference computer code has been developed to solve the incompressible Navier- Stokes equations in a three-dimensional, curvilinear coordinate system. The pressure-field solution is based on the pseudo compressibility approach in which the time derivative pressure term is introduced into the mass conservation equation to form a set of hy- perbolic equations. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds stresses, that are uncoupled from the implicit scheme, are lagged by one time-step to facilitate im- plementing various !evels of the turbulence model. Test problems for external and internal flows are computed, and the results are compared with existing experimen- tal data. The application of this technique for general three-dimensional problems is then demonstrated.

I. Introduction

The development of new solution methodologies is one of the primary pacing items in computa- tional fluid dynamics today.' With the current rate of progress in this discipline, as well as with grid genera- tion techniques, and with the enhancements in com- puter capability, it i s now practical to simulate compli- cated fluid dynamic phenomena associated with realis- tic geometries. To date, large computing times and memory requirements have been major difficulties in producing successful results from full Navier-Stokes codes. Even though the economic aspect should still be a primary concern in developing a fully three- dimensional production code, the time has come to utilize the available algorithms and high-speed com- puters to develop a useful tool for analysts and desig- ners. The present paper presents the development of a

*Research Scientist. tSenior Member of Technic81 StaR. $Senior Research Associate. Member AIAA. Present address, Roekwell Scienw Center, Thousand Oaks,Calilornia.

This paper is declared a work of the US. Government and '4

therefore is in the public domain.

three-dimensional, incompressible, Navier-Stokes sol- ver cast in generalized curvilinear coordinates using primitive variables.

Incompressible flow phenomena are frequently encountered in many engineering applications, espe- cially, in hydrodynamics and in certain classes of aerodynamic problems such as dynamic stall and low- speed wind-tunnel test problems. For most two- dimensional flow simulations, computer time and memory requirements are not major limiting factors, and various numerical techniques have been imple- mented quite successfully. For example, a stream function-vorticity formulation is frequently used for solving lwo-dimensional, viscous, incompressible now problems (for example, see Refs. 2-5). The three-dimensional extension of this method is not straightforward. Various three-dimensional Navier- Stokes codes ha.ve been developed, mainly, for com- pressible flow. A few examples follow: Shang et al! utilbed MacCormack's explicit scheme; Hung and Kordulla7 developed a code based on MacCormack's implicit scheme; Pulliam and Steger8 implemented the Beam-Warming algorithm for a fully implicit code; and Briley and McDonald' independently developed a similar AD1 scheme. Implemeuting these codes for simulating incompressible flows is not efficient and is generally not recommended. Therefore, in the present work, an efflcient three-dimensional Wavier-Stokes sol- ver, using primitive variables, is developed for incom- pressible flow problems.

One of the major problems to be addressed in solving incompressible flows that use primitive vari- ables is making the decision about which pressure solu- tion method should be used to guarantee a divergence- free velocity fleld. The method of solving Poisson's equation for pressure was developed by Harlow and Welch," and has been used frequently for obtaining the pressure field, mostly using explicit methods. The usual computational procedure is to choose the pres- sure fleld such that continuity is satisfied at the next time-level, so that the new flow fleld will be divergence- free. This procedure normally requires a relaxation scheme iterating on pressure until the divergence-free

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condition is reasonably satisfled. This approach can be very time consuming, and thus the computing time re- quired for simulating three-dimensional flows has been prohibitively large. To accelerate the pressure-fleld solution and alleviate the drawback associated with the Poisson's equation approach, Chorin" proposed to use artiflcial compressibility in solving the continuity equation. A similar method was adopted by Steger and KutlerI2 using an implicit approximate factoriza- tion scheme by Beam and Warming.I3 To implemrnt the implicit time-differencing, they fabricated a hyper- bolic time-dependent system of equations by adding a time-derivative of the pressure term to the mass con- servation equation. These hyperbolic equations pos- sess characteristics that are not present in the usual Poisson's equation for pressure (see Ref. 14 for a comprehensive analysis). This approach has been a p plied to simulate laminar, incompressible now within liquid fllled shell^'^. Presently, this procedure has been extended to a three-dimensional flow solber cast in generalized curvilinear coordinates.

In Sec. II of this paper, the governing equa- tions and turbulence model are presented. The finite- difference algorithm is described in Sec. KI, and iu See. IV, results are presented for internal and external flow test problems that verify the accuracy of the code. Additional examples of a more practical nature are in- cluded to show the versatility of the current soher.

lI. Governing Equations and Turbulence Model

Governing Equations

Unsteady, three-dimensional, incompressible flow with constant density is governed by the follow- ing Navier-Stokes equations, written in Cartesian coor- dinates:

To implemeaL an implicit, approximate fac- d torization schemeI3 to the above set of equations, the continuity equaion is modifled according to the procedures of Refs. 11 and 12 as follows:

( 1 4 ap au 8. aw ap* at an - + B(, + - + =

Here, t is time; x, y\ and z are Cartesian coordinates; u,v, and w are corresponding Cartesian velocity com- ponents; p is the pressure; and ~,j is the viscous stress tensor. The parameter 1/,9 is the pseudo compres- sibility, and p' is the value of p at the previous itera- tion. The p and p' terms, and the numerical algorithm are chosen to satisfy the continuity equation (la); they will be discussed in more detail later.

The equations are written in dimensionless form with

. u - v - w u=- , v = - , w=-,

_ Z " # . Z 2'- , &I=-, z = -

t = - ,

UV.1 G.1 U?tJ

Z,.J 27.1 2r.J' - tU,.f - P-P Ie l - (14

Y - B ; = R e - ' = - , B = -

p = -,r;j = 7ii G*I pu:., P U L I '

z 7 c l U r e l

The subscript ref denotes reference quantities and for convenience the tildes ( - ) are dropped from the equa- tions.

i/

The viscous stress tensor can he written in the following form:

Here, R,; is the Reynolds stress, and L/ is the coeflcient of viscosity. Combining Eqs. (Ib) and (Id), the govern- ing equations are written as

where

2

Coord inate Transformation For orthogonal coordinates, Eq. (7) can be further simplified to read

To accommodate fully three-dimensional geometries, the following eneralized independent variables are in- v 2 2 fJD troduced whic 'h transform the physical coordtnates into k. = ;.(& T Cs r C s 2 J I m ~ -r ( & . . . ~ ~ r ~ ~ . s j general curvilinear coordinates:

LJ T = t

(3) F = t(z3 Y, 2, t)

11 = 11(% Y, 2, 0 f = f b > Y, 2, t)

Applying this transformation to Eq (2a), the following form of the governing equations is obtained

aD a(&-&..) a ( P - P d + a(&-&) -+ aT + an (4)

1 a p . = - ( - - ,O,O,O)~

J ST

where J = Jacobian of the transJormation,

D = D I J ,

Here, the contravariant velocities, U, V, and W withdut metric normalization, are deiined as

u = ( I + tdJ + l u v + Czw

y = rlt + n.u+llvv+ 11.w

w = f t + f.U + <uv + c z w

(6)

The viscous terms are

+ (R., ... terms)

where

0 1 0 0 8D I , ==I 1, I,,,-= -,etc (8)

0 0 1 0 BE BC

&bulence Model

Various levels of modeling are available (see Ref. 16 for a review), most of which require considerable experimental inputs. In the present code, however, the turbulence model is conhed to an algebraic model to maximize the cost effectiveness of the flow solver. Even though the accuracy of the result could be limited by the algebraic model, economy is still an overriding factor in many engineering applications.

In the present code, turbulence is simulated by an al ebraic eddy viscosity model, uslng a constitutive e uat ei on involving a "mixing length" that is a measure o?the turbulence length scale. A generalization of this approach is given by the following equation:

1 3 (10) Rij -Rkk6ij - 2utSij

Here, is the turbulent eddy viscosity, and Rkk is the noma1 component of the Reynolds stress. The strain rate tensor is de6ned by

1 sui auj 2 8 X j

sij = 4- + z)

By including the normal stress, R k k , in the pres- sure, u can be replaced by (u + ut). For the turbulent Viscosity, the algebraic model of Baldwin and Lomax17 is implemented in the present code. Following their formulation, the turbulent viscosity for incompressible flow can be written as

(12) y 1 = (V1) .nnrr , Y<Yc

(ut).atrr, Y > Yc where y is the normal distance from the wall, and ye is the smallest value of y a t which values from the inner and outer formula are equal.

In the inner region,

(ut)inncr = PlwI (13)

where (w( is the magnitude of local vorticity. The tur- bulence length scale 1 and the nondimensional distance from wall y+ are deflned as

v L o o o 1 - I

3

In the outer region,

(vt)orier = KGpF,.r.Fdy) (15)

where

Fw.+ = minimum[(ym..Fm~.), ( C , r ~ ~ ~ ~ U ~ d i l / F r n ~ ~ ]

Here, F,,, is the first peak value in a profile given by

+ + F ( d = y lwl [ l - e w - y / A 11 and ymoz is the value of y at that point. The Klcbauofl intermittency factor is written as

F.br.b(y) = (1 + 5 . 5 ( C h r . b y / 3 1 n ~ ~ ) ~ l - ' (16)

and

Udi, = ( J U Z + Y Z + wz)rn.,z - ( J U Z + Y Z + w")m,n (17)

In wakes, the exponential term in F(y) and the second term in Eq. (17) are set equal to zero.

The constants appearing in the model were deter- mined by requiring agreement with the Cebeci formula- tion for constant-pressure boundary layers18. These values are

A = 26, Ccp = 1.6, Cxlpb = 0.3, Cwx = 0.25,

k = 0.4, K = 0.0168

m. Numerical Algorithm

Time Advancing

The numerical algorithm used to advance Eq. (4) in time is an implicit, approximately factored, flnitedifference scheme by Beam and Warming13. By combining trapezoidal-rule time-differencing and the difference form of Eq. (4), the following governing equation in delta-form is obtained:

+ 6,(? - r,)] (D"+' - D") I

where

1La LlW 4 Q = Lo + LIU + LZV f LSW

Lo = ((i)t, L1 = (ti)., L2 = ( t i ) U , La = ( t i ) Z

t i= e,?, ot $ Jot A,B, or C, respecttvely

6< = Jinite diJJerence form of - etc. 8 at '

(lab) The superscript n denotes nth time step, and the vis- cous terms are given from Eq. (9) as

rl - cVc(V(ilmGc,) + (R,, ... terms)

r - uvtp(V(irms,,) + (Ru, ... terms)

rS - -Vc+'(,rm6,,) + (R,, ... terms)

J

(18c) l -7 U J

Approximate Factorization

'The full viscous terms &, kv and 6, in Eq. (7) produce non-tridiagonal elements in the left-hand side of Eq. (18a). Therefore, to implement an approximate factorization scheme, only orthogonal terms are kept on the left-hand side. For steady-state solutions, this em be done since the left-hand side approaches zero as a steady state is approached. For a time-accurate solu- tion, this approximation procedure needs to be further lnvesti ated when a nonorthogonal grid is used. For d the ri %t-hand side, the full viscous terms may be in- cludefj. In the present version, a nearly orthogonal grid is used, and the viscous terms in Eq. (7) are simplified to

E* = -j(VcV<,)imS,D = rJrn6cD

kv = -(Vq.Vq,)Im6,D = 7d,,,S,D

6 - -(V<.V<,)Im6,D = 7slm6,D

After adding smoothing terms to stabilize the com- putation, the approximate-factored form of the govern- ing equation becomes

v

U (19) J

U

" - J

h 2 h

h

[If - J " + ' & ( y - 712-8~) + c.V<A<].

[ I + 2J"+16,(b" - 721-6,) f e,V,,A,].

[r + 2 ~ " + 1 6 , ( Z - 7arms,) + ~ . V , A J ( D " + ~ - D")

E IRHS (1841 - cG[(VcAC)'+ (V,A,)*+ (V7,A,)21Dn (20)

where e , and c e are implicit and explicit smoolhing terms (which are explained in the next section), and

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h = AT = time- step,

V c D = Dj - Dj-1, A q D = DK+I - Dk,

6cD (DI+I - D I - I ) / ( ~ A ~ )

6 , 7 8 4 = I(7*+1 + 7k)(Dk+l- D I )

v - ( 7 k f 'Yk--1)(Dk - DI-I )] / [~(AV) ' I

Analogous terms in the q and { directions are deflned similarly.

Higher Order Smoothing Terms

It has been found that a higher-order smoothing term is required to make the present algorithm stable (see Refs. 12 and 13). In this section, smoothing terms used in the code are described in relation to the usual upwinding scheme as well as their behavior near computational boundaries.

sidered: For simplicity, the following equation is con-

(21) au au -+c-=o at a x

The usual upwind differencing of the convective term results in

C ~ ~ U = c(3uj - 4Uj-1 + uj-z)/(2Az), c > 0 c < 0 ( 2 2 4

~ ( - 3 ~ j f 4 u j + i - ~ j + 2 ) / ( 2 A z ) ,

where 6, is a difference lorm of BulBz. This equation can be writlen, for all c, as

C c8,u = -[(3uj - 4uj-1 + uj-2) + 4Az

W IC1 (-3Uj + 4Uj+l - Uj+z)l + -[(3Uj - 4Uj--1 + U j - 2 )

- (-3Uj f 4Uj+l- U j + z ) ] 4Ax

Replacing the flrst [ ] term of the above by the usual central difference formula results in

C c6,u = -(u 2ax ,+I - U j - 4

IC1 + -(Uj-z - ~ U J - I + 6 ~ 2 - ~ U J + I + U J + Z ) 4Ax

(22b) Therefore, a flnite-difference form of Eq. (21) with a fourth order explicit smoothing term can be written as

,;+I - .; u?+-l - u?+l = 1+1 j-1

At -kc 2Az f*kI @3a)

--(uj-p - 4Uj--1 + 6uj - 4 ~ j + 1 + uJ+z), 4Ax

0 < f,<l

and, in delta form, it can be written as

cht cA t 2Ax

felt At 4Ax

AU + -(Au~+I - Au,--1) = --( 2Ax UJ+I - ui)"

- _L(uj-, - 4uj--1 + 6uj - 4uj+1 + uj+z)"

(23b)

where

A u = u"+' - un

E. = explicit smoothing coeJJicient

In a similar manner, the implicit second-order smoothing term can be obtained by starting from

C I4 C ~ = U -(u,+, 2Ax - u t - , ) - -(u,+, 2A2 - 2u, + u,--1) 124al \- --,

This can be put into a numerical scheme, in delta form, as

cAt 2Ax AIL + -(Auj+i - Auj- I )

( U j - 2 - ~ U J - I + 6uj - 4 U j + i + U J + ~ ) " r.lclAt ~ A X

-- 124b)

where

f , = implicit smoothing coe j j ic ien t

f , = explicit smoothing coe// icient It is interesting to observe how a small perturbation, u', propagates under various smoothing terms. Following the fluid, the second-order smoothing acting on the perturbed quantity is

aut a v f- = 0 _-

at 8%

then u I - - e -coat (ae'*' + be-'"')

For the fourth-order smoothing term,

BU' 3%' - + r - = o at aaX

( 2 5 4

then ul = e-<"4t(aeid" + be-'"")

On the boundary, the flow variables are often ex- trapolated to maintain the same order of differencing. However, a lower order of differencing could be used by backward or forward differencing. And it should be noted that the sign changes when a lower-order smoothing is incorporated. In the present work, the second-order smoothing behaves well on inflow and outflow boundaries, and the fourth-order extrapolated rmoothing term is better on solid boundaries.

Pseudo Compressibility and the Pressure Field Solution

In solving the incompressible Navier-Stokes equations, exact mass conservation is of crucial im- portance in order to obtain a stable solution. In the

5

present study, the continuity equation is modified to a hyperbolic- type [Eq. (Id)], thus, introducing pres- sure waves of flnite speed; the wave speed is inEnity for truly incompressible flow. Wave propagation depends on the magnitude of the compressibility parameter, P, which, if chosen to be too large, would at the Fame time contaminate the accuracy of the approximate fac- torization. Therefore, a correct specification of ,I? is of prime importance to the success of this approach. For inslance, a certain value of lhe pseudo comyressibdily suitable for a particular problem can be totally inade- quate for other geometries and flow speeds, and can even cause instability.

In Ref. 14, wave-propagation characteristics are analyzed using a one-dimensional form of the govern- ing equations. By compwing the velocity cf prpssure- wave propagation and fhr_ rate of vorticity spryading, the following criterion for the artificial compressibility was derived:

where 26 and ZL are the characteristic lengths that the vorticity and the pressure waves have to propagat,e during a given time span. During a duct-flow simula- tion, XL is equal to the total length of the duct, and z6 is half the distance between the two walls of the duct. The computational experiments showed this depen- dency of the pseudo compressibility on the Reyi~olds number, as well as on the characteristic lengths of the geometry.

For the near-field in external flows where pres- sure waves propagate out to inEnity the choice of B is less restrictive. The pressure buildup in the near-field region is only temporary, even if p is not optimum. However, for internal flows, local pressure buildup can be serious when ,!? is not properly chosen (see Rrf. 14 for details).

Boundary Condition

Once the numerical algorithm has been developed the next most important aspect of solving a Ruid dynamics problem is the proper implementation of the boundary conditions. There are several different types of boundaries encountered in numerical simulations: 1) solid surface, 2) far-field, 3) in-fiow and out-flow, and 4) symmetric or reflective boun- daries. All of these are required in the present code.

On a solid surface, the ususal no-slip condit,ion is applied. By taking advantage of the viscous sublayer assumption where the pressure is constant normal to the surface, one obtains

P" = o (27)

The boundary condition can be implemented cither explicitly or implicitly. The latter enhances stability

of the code. For this discussion, a $ = const surface is considered next. On this surface, Eq. (37) can be written as

Expanding this, V f .vp = 0 (283)

+f.(E.PC + 7sPn + I r P c ) = 0 (28L) Since a nearly orthogonal grid is assumed near the surface, this can be further simplified to

This condition is implemented in the c-directional sweep. From Eq. (20), the matrix equation on a solid surface (at L=l ) is written as follows:

where AD = D"+' - D". To implement Eq. (28c) explicitly, one simply sets

then the pressure is updated at the end of each step. The same procedure is used for fixed boundaries such as a free stream boundary. Equation (ZSc) is applied in an implicit manner by setting

j = [ o i , h = l 0 1 0 0 J,q0 0 0 0 0 I, 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0

IV. Computed Results

The flow solver was verified by solving a few simple test problems. To test the code on an ester- nal flow simulation, flow over a circular cylinder was computed. To test this program for internal Bows, a channel-flow problem was computed. Various compli- cated three-dimensional flows were then computed; a couple of examples are described here to demonstrate the capability of the present code.

Flow over a .Circular Cylinder

Flow over a circular cylinder has been a rich source of various fluid dynamic phenomena (see Ref. 19). The Bow over an impulsively started circular cylinder at a Reynolds number of 40 based on the diameter of the cylinder is chosen as a verification case. The three-dimensional coordinate system chosen is shown in Figs. l a and lb. Figs. 2a and 2b show J

the steady-state velocity vectors and stream function

contours. The steady-state pressure eoefflcient on the cylinder surface is then compared in Figs. 3a with that of Mehta (private communication), who used a stream function and vorticity formulation in two dimensions. Various experimental and computztional studies on this bench-mark case have been reported. The quan- tities, that are presently used to verify the physical phenomena are the wake length, L,uvobe; the separation angle, BSep (see Fig. 2b); pressure drag, Cdp, and the pressure coefflcients a t the forward and rear stagnation points, C p ~ and C,,, respectively. The Row field com- puted by the present code INS3D compares quite well with those reported in the literature (see Table 1). For this computation, a nondimensional-time step of 0 1 was used, and the steady state was reached in 300 itera- tions, To simulate the impulsive start, the number of pressure iterations was increased to 6ve at each time- step; the pressure-drag history is compared in Fig. 3b with the time-accurate computation of blehta Even though the present algorithm is geared for a steady- state solution, this example demonstrates the potential for an efflcient time accurate procedure.

v

Channel Flow

Just as the circular cylinder problem is the simplest representation for external flows, channel flow is perhaps the simplest representation for internal flows. However, the chanuel-fiow problem still provides the essential features for testing the present algorithm. Figures l a and 4b show the developing laminar chan- nel 60w. To reduce the channel length for obtaining fully developed flow, a partially developed boundary layer pro6le is used as an inflow condition. 7Jsing a nondimensional time-step of 0.1, fully developed flow is obtained after about 100 steps for a Reynolds num- ber of 1,000, based on channel width. The Baldwin- Lomax turbulence model is tested by incrensing the Reynolds number to 100,000. The velocity vectors for this case are shown in Fig. 5a. The mean velority defect is compared, in Fig. 5b, with Prandtl's universal law, Comte-Bellot's experiment for channel flow,2o and Klebanoffs well-known boundary layer profile. The convergence histories for both laminar and turbulent cases are shown in Fig. 6.

"

Three-Dimensional Flow Examples

One of the prime objectives in developing the present code is t o analyze and verify the Bow field in the Space Shuttle Main Engine (SSME) power head. The examples shown in Figs. 7, 8 and 9 exhibit typical flow characteristics locally encountered in the SSME simulation.

The rectangular duct shown in Fig. 7a is similar t o part of the new 80 x 120 low-speed wind tunnel at Ames Research Center (guide vanes are not shown). The flow, in the form of velocity vectors, is shown at

one station in Fig. 7b. This clean-duct sirnulation shows the nature and magnitude of the secondary Bow, separated region, and substantial variation in mean velocity vectors (Figs. 8a and 8b) from top or bottom wall to the center of the duct.

Figures 9a and 9b show the geometry of an annular duct with a 18Cf' bend. This configuration represents the turn-around duct in the hot-gas manifold of the Shuttle engine. The laminar-flow (Re=1,000) solution in Fig. 9c shows a considerable nonuniformity after the 180' bend. In this case, an adverse pressure gradient is developed after the bend, and subsequently a large area of separation is formed along the inner wall. Further numerical simulation of the SSME will be used to redesign and to optimize the hot-gas manifold. For this computation, a 50 x 21 x 16 mesh was used with a non-dimensional time-step of 0.1. A converged solution was obtained after 300 time-steps, starting from a uniform velocity Geld. The computing time for each time-step was 1.443-04 sec per mesh point on the Cray I S computer a t Ames. Because of the axisymmetric nature of the present configuration, only 16 mesh points were used in the circumferential direction, However, for nonsymmetric flows, more mesh points and proper local elustcring of grid lines will be required. The three-dimensional computation of the hoegas manifold of the SShIE has been performed using the present code. Details of this numerical simulation, as well as comparison with ex- perimental results, will be presented in a future paper.

V. Concluding Remarks

This paper presents the development of an efflcient and, robust computer code for incompressible Navier-Stokes Rows (WS3D). In this work, the basic formulation and the algorithm are described. mS3D has been applied to various geometrically complex flows. Other aspects, such as a multiple-zone computa- tions, implementation of smoothing terms near inter- sections of solid walls, symmetry boundary conditions, and analysis of differencing errors, will be presented in a future paper. The algebraic turbulence model implemented in the present version of the code needs more development for solving the flow with massively separated regions, as in the case of the SSME power head simulation. Higher-level turbulence models are being investigated in an efIort to achieve better es- timates of the turbulence length scales.

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T

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,J

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'*Cebeci, T., " Calculation of Compressible Turbulent Boundary Layers with Heat and Mass Transfer," AIAA Paper 70-741, Los Angel??, Cal i f , 1970

'"Morkovin, M. V., Wow around Circular Cylinder - A Kaleidoscope of Challenging Fluid Phenomena," Symposium on Fully Separated Flows, A. G. Hansm, ed., ASME, New York, 1964, pp. 102-118.

20Comte-Bellot, G., 'Contribution a I'Etude de la Turbulence de Conduite," Doctoral Thesis, University of Greooble, Grenohle, France, 1963.

Flow with Free Suriace," Physics of Fluijs, Vol. 8, No. 12, Dee. 1965, pp. 2182- 2189.

8

v 'dp CPf

0.93-1.05 1.14-1.23 Summary of literature data

Present result 1.03 1.15

'pr gsep Lwake

-0.47-(-0.551 50r53.9' 1.8-2.5

-0.51 52" 1.9

PHYSICAL DOMAIN COMPUTATIONAL DOMAIN

1.75 - - -0.721-0.721 -0.721-

o,&8,-0.481- 0.481 - 0.481 - o.24,---0.241- 0.241-0.241 -

0.967-

1.05 -/

1. J

,I q.K KMAX J=K=L=l

(JMAX.KMAX.IMAX1 = (3.80.41) STRETCHING RATIO= 1.15

a) coordinate system.

.1.25 -.65 -.05 0.55 1.15 1.75 2

b)grid.

Fig. 1 Topology for flow over a circular cylinder. U

I." I

0 .6 1.2 1.8 2.4 3 z

a) velocity vectors,

-1.75 -.75 -.05 .65 1.35 2.05 2.75

* b)stream-function contours.

Fig. 2 Steady state solution for flow over a circular cylinder at Re=40.

0 .2 .4 .6 .a 1 DISTANCE FROM THE LEADING EDGE

a) coordinate system,

- 2D BY MEHTA USING STREAM- FUNCTION VORTICITY Cdp

0 INS3D

8

0 10 20 30 40 TIME, DIAMETERS TRAVELED

b)grid

Fin. 3 Flow over an impulsively started circular

1.2

Y 2h

T - .2

-1.6 I I 0 3 6 9 12 15

x

a) velocity vector.

1.25

1.00 -

c" , ? .75 -

50 -

v//2hI

b) fully developed velocity proflle.

s ~~

cylinder at Re=40. Fig. 4 Developing laminar channel flow at Re = 1000.

(Re based on channel width and average velocity.)

10

-.5 !i - ~ ~ = 1 0 5 (TURBULENTI e Re=103 (LAMINAR1

v

s 4 m L 9 t 4 s ~ :

2 6 -

1 2 -

2h 2 - - - - - - - V

l- - 2 -

-1 6 7 -2 5 0 3 6 9 12 15 0 50 100 150 200 250 300

x NUMBER OF TIME STEPS

a) velocity vectors.

25

20 R PRANOTL'S UNIVERSAL LAW

TURBULENCE MODEL) COMTE-BELLOT (Re=1.14 X 1051, CHANNEL

0 PRESENT RESULT (BALDWIN-LOMAX

x a 0

COMTE-BELLOT ( R ~ 2 . 4 x 1051, CHANNEL KLEBANOFF (Re=1.5 x lo5) , B.L.

v

0 .2 .4 .6 .8 1 v h

b) mean velocity defect comparison

Fig. 5 Developing t.urbulent, channel Bow at, Re = 10'.

Fig. 6 Convergence history comparison for laminar and turbulent channel flow.

FLOW

a) Duct geometry.

x + -.. b) secondary now at z=O plane (view from down

Fig. 7 Flow through a rectangular duct with a 45' bend.

stream).

W

11

__.-

a) Velocity vector at x=0.5 plane.

SEPARATION s ; ~ $ ~ : t t ~ - ~ * ~ - s - - a- -- ............... - - POINT \ ..................

b) velocity vector at x=0.025 plane (top view).

Fig. 8 Flow through a rectangular duct with a 450 bend.

12

2.0

1.3

.6

W V

-0.1

-0.8

-1.5

a) Three-dimensional grid.

4.0 4.7 5.4 6.1 6.8 7.5 X

b)two-dimensional crass section.

3.0 4.2 5.4 6.6 7.8 9.0 x

c) typical flow pattern with separation.

Fig. 9 Flow through a turn-around duct.

13