IJAET International Journal of Application of Engineering and Technology ISSN: 2395-3594

Vol.1 No.1

COMPUTATIONAL ANALYSIS OF ISOLATOR FOR DE-LAVAL NOZZLE FOR SUPERSONIC FLOW

K.M. Pandeya, Aditya. P. Singha Surendra Yadava

aNIT Silchar, (INDIA) Department of Mechanical Engineering

I. INTRODUCTION

Swedish engineer of French descent who, in trying todevelop a more efficient steam engine, designed a turbinethat was turned by jets of steam. The critical component –the one in which heat energy of the hot high-pressuresteam from the boiler was converted into kinetic energy –was the nozzle from which the jet blew onto the wheel.De Laval found that the most efficient conversionoccurred when the nozzle first narrowed, increasing thespeed of the jet to the speed of sound, and then expandedagain. Above the speed of sound (but not below it) thisexpansion caused a further increase in the speed of the jetand led to a very efficient conversion of heat energy tomotion.The theory of air resistance was first proposed by SirIsaac Newton in 1726. According to him, an aerodynamicforce depends on the density and velocity of the fluid, andthe shape and the size of the displacing object. Newton’stheory was soon followed by other theoretical solution offluid motion problems. All these were restricted to flowunder idealized conditions, i.e. air was assumed to possesconstant density and to move in response to pressure andinertia.

Nowadays steam turbines are the preferred power source ofelectric power stations and large ships, although theyusually have a different design-to make best use of thefast steam jet, de Laval’s turbine had to run at animpractically high speed. But for rockets the de Lavalnozzle was just what was needed. But now with theadvancement of computational facility and computationaltools available in present day it becomes easier to simulatethe flow field in a mere personal computer (PC). There arevarious dedicated software now available for flowsimulation such as FLUENT, STAR-CD, LS DYNA,PDETOOL, AFINA, CFX, NUMECA, PHOENICS etc.Out of this FLUENT is the most popular one for itsversatility and user friendly.Since the fluid flow has such an overwhelming impact onindustrial life, we need to be able to estimate it effectively.This ability can result from an understanding of the natureof the processes and from a methodology with which topredict them quantitatively. The prediction of fluid flow ina given situation consists of predicting the values of therelevant variables governing the process of interest, andknowing how these quantities would change in response tochanges in geometry, flow rate, fluid properties, etc.Armed with this expertise, the designer of an engineeringdevice is able to choose the optimum design from amonga number of alternative possibilities and can ensure thedesired performance. Prediction offer economic benefitsand contribute to human well-being.

The investigation of flow processes can be done by twomain methods namely, experimental, theoretical.Experimental investigation offers the most reliable

ABSTRACT

The present study is aimed at investigating the supersonic flow in De-Laval nozzle for Mach 3 with isolator. Andthe result is also validated with mathematical model for de-laval nozzle at mach 3. The flow is simulated usingfluent software. The flow parameters, like pressure ratio, Area of nozzle exit ratio, and the Mach number of theflow at the nozzle exit is defined prior to the simulation. The result shows the variation in the Mach number.,pressure ratio. Detailed flow characteristics like the centerline Mach number distribution and Mach contours ofthe steady flow through the converging – diverging nozzle are obtained to study and assess the suitability of thedesign.

Keywords: - Isolator, supersonic flow, Mach number, Pressure ratio, centerline, Control Volume.

ABSTRACT

This paper presents solutions of supersonic flow fields in two-dimensional De Laval nozzles with a duct. Thephenomenon of sudden expansion of fluid is visualized in most of the day to day activities, which has prompted manyof the researchers to explore in this field. The present study is aimed at investigating the sudden expansion ofsupersonic flow in de Laval nozzle, with 1.74 Mach numbers for various L/D, into a duct. The flow is simulated usinga Fluent. The flow parameters, like pressure ratio, area of duct to area of nozzle exit ratio, and the mach no. of theflow at the nozzle exit is defined prior to the simulation. The flow field of an axi-symmetric flow on sudden expansionis accompanied with flow reversal, flow separation and vortex shedding near the nozzle exit region. The nature offlow development is declared as a smooth flow only when the flow gets attached and the flow is streamlined. Thesuddenly expanded cavity not only causes head losses but also is accompanied by flow oscillations due tophenomenon called vortex shedding.

Keywords: - Supersonic flow, two-dimension, De Laval Nozzle, Mach No., Pressure ratio, flow reversal, stream line, cavity.

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information about a physical process. However, there areserious difficulties of measurements in many situationsand the measuring instruments are not free from errors.Often such measurement itself interferes significantlywith the process being measured, thus making totalexperimental knowledge of the process impossible toobtain. Theoretical investigation works out theconsequence of a mathematical model of the process,which often consists of a set of partial differentialequations for the physical quantities of interest. Theseequations are often of such complexity that if the methodsof classical mathematics were to be used for solving themthere would be a little hope of predicting many cases ofpractical interest.Fortunately, the development of numerical methods andthe availability of large digital computers allowmathematical model to be solved for many practicalproblems. The advantage of theoretical investigation overa corresponding experimental investigation is its low cost,remarkable speed, detailed and complete information ofthe process under different conditions. Even with theremarkable success of numerical solutions, few acceptthem uncritically without some experimental validation.As in the saying by Albert Einstein, “A theory issomething nobody believes except the person proposingthe theory and an experiment is something everybodybelieves except the person doing the experiment”.

II. MATHEMATICAL FORMULATION

Mathematical modeling is usually central to the analysisof engineering systems, which are often very complicated.For a typical fluids system, this complexity arises mainlydue to the time dependent, multidimensional nature of thefluid flow and the complex supplementary conditions thatgovern these systems. In addition, the non-linearity of theflow equations makes the analysis all the morecomplicated. Consequently, a real system is oftensimplified to a computational model resembling theoriginal in shape, geometry and other physicalcharacteristics in the gross features, but not in everydetail. Thus, by the application of fundamental physicallaws, and by incorporating approximations andidealizations, a mathematical model is generally amenableto numerical simulations, which hopefully withoutinvolving exorbitant time and effort in computation givean adequate picture of the physics of the system.

PHYSICAL MODEL

The problem being considered is the supersonic flowthrough de Laval nozzle to numerically simulate the flow[]

Figure 1: Physical model of supersonic flow through De Laval nozzlewith isolator

1. APPROXIMATIONS AND IDEALIZATIONS

The physical model described in the preceding section is asimplified model, with respect to the surface geometry,when compared with typical components actuallyencountered in applications. The further approximationsand idealizations made for the present investigations are asfollows: The fluid is Ideal gas. The flow is Supersonic The flow is assumed to be two-dimensional. K-epsilon model is used for compressible flows. The Nozzle is axi-symmetric.

2. MATHEMATICAL MODEL

A mathematical model comprises equations relating thedependent and the independent variables and the relevantparameters that describe some physical phenomenon.Typically, a mathematical model consists of differentialequations that govern the behavior of the physical system,and the associated boundary conditions.Employing the approximations and the idealizations listedin section 4.2, the physical model described in section 4.1is simulated by an equivalent mathematical modelinvolving the conservation of mass, momentum, withappropriate boundary conditions. The mathematical modelcomprising the partial differential equations, along withtheir boundary conditions is presented in the followingsubsection.

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3. GOVERNING EQUATIONS

The advantage of employing the complete Navier-Stokesequations extends not only o the investigations that can becarried out on a wide range of flight conditions andgeometries, but also in the process the location of shockwave, as well as the physical characteristics of the shocklayer, can be precisely determined. We begin by describingthe three-dimensional forms of the Navier-Stokes equationsbelow. Note that the two-dimensional forms are justsimplification of the governing equations in the threedimensions by the omission of the component variables inone of the co-ordinate directions. Neglecting the presence ofbody forces and volumetric heating, the three-dimensionalNavier-Stokes equations are

Continuity: +( ) + ( ) + ( )

= 0

(1)

x-momentum:( ) + ( ) + ( ) + ( )

= + + (2)

y-momentum:( ) + ( ) + ( ) + ( ) =

+ + (3)

z-momentum:( ) + ( ) + ( ) + ( )

= + + 4)

energy:( ) + ( ) + ( ) + ( ) =( ) + ( ) +( ) + ( )

+( )

+( )

(5)

Assuming a Newtonian fluid, the normal stress σxx, σyy,and σzz can be taken as combination of the pressure p andthe normal viscous stress components τxx, τyy, and τzzwhile the remaining components are the tangential viscousstress components whereby τxy= τyx, τxz= τzx, and τyz= τzy.For the energy conservation for supersonic flows, thespecific energy E is solved instead of the usual thermalenergy H applied in sub-sonic flow problems. In threedimensions, the specific energy E is repeated below forconvenience:

E = e + (u2 + v2 + w2) (6)

It is evident from above that the kinetic energy termcontributes greatly to the conservation of energy becauseof the high velocities that can be attained for flows, whereMa>1. Equations (1)-(6) represent the form of governingequations that are adopted for compressible flows.

The solution to the above governing equations nonethelessrequires additional equations to close the system. First, theequation of state on the assumption of a perfect gas inemployed, that is,

P=ρRT,

where R is the gas constant.

Second, assuming that the air is calorically perfect, thefollowing relation holds for the internal energy:

e= CvT,

where Cv is the specific heat of constant volume. Third, ifthe Prandtl number is assumed constant (approximately0.71 for calorically perfect air), the thermal conductivitycan be evaluated by the following:

k=μ

The Sutherland’s law is typically used to evaluate viscosityµ, which is provided by

µ=µ0

.where µ0 and T0 are reference values at standard sea levelconditions.

Generalized form of Turbulence Equations is as follows :

( ) + ( ) + ( ) + ( ) = +

+ + (Sk=P – D)

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( ) + ( ) + ( ) + ( ) = +

+ + ( = ( −)where = 2 + ++ + ++ + + =

III. GEOMETRY AND GRID ARRANGEMENT

A 2d axi-symmetric computational domain wasconsidered, the initial design parameters for de Lavalnozzle[] and isolator for Mach number 3 . This wasobtained by method of characteristics of nozzle programand trial and error method for isolator. Here the gridarrangement for the both nozzle and isolator at Mach 3 isgiven below. And the length of the isolator from throat is2m.

Figure 2: De-Laval Nozzle for Mach 3

Figure 3: De-Laval Nozzle for With Isolator

IV. RESULT AND DISCUSSION

Mach number 3:-

In this, the nozzle is designed for Mach no. 3. From figure,it is clearly visualized that in the convergent section at inletpoint, Mach number, is in the Sub-sonic region while at thethroat, flow becomes Sonic and at the nozzle exit itbecomes Super-Sonic for which the nozzle is designed.Near the wall, the Mach number is 1.32. This is due to theviscosity and turbulence in the fluid. The flow travels inthe nozzle along the angle of direction. This is the purposeof the nozzle design.

Static pressure:

Static pressure is the pressure that is exerted by a fluid.Specifically, it is the pressure measured when the fluid isstill, or at rest. The above figure reveals the fact that thegas gets over expanded at the nozzle exit. The value ofstatic pressure at nozzle exit is 9.17e+03 Pa. The staticpressure in the nozzle falls. The static pressure in theconvergent section is observed to be 3.22e+05Pa at theinlet which then decreases to 1.66e+05Pa up to the throatand then to 9.17e+03Pa at exit.

Total pressure

From the figure it is observed that, near the throat justbelow the wall, the total pressure increases up to3.68e+05Pa in a patch which is like a disc, while the totalpressure at the nozzle exit at the centre is 3.52e+05Pa and

Figure 4 Mach number

Figure 5 Static Pressure

Figure 6 Total Pressure

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that at the inlet is 3.68e+05Pa. You can easily visualize inthe above figure that, there is decrease in stagnationpressure near the nozzle walls due to viscous effects,whereas the stagnation pressure remains almost constantin the centre.

Velocity Magnitude:-

From the figure, it is clear that the flow is symmetric andflowing across an angle as a characteristic of the De-Lavalnozzle. At the inlet of the nozzle, the velocity is found tobe 2.21e+02 m/s and it rapidly increases up to the throat tonearly 4.85e+02 m/s. At the nozzle exit, the velocity isfound to be around 8.82e+02 m/s. The flow is turbulent, sonear the wall flow separation takes place causing thevelocity to decreases to nearly 8.82e+01m/s while at thecentre the velocity is 7.94e+02 m/s.

Turbulence Intensity:

The above figure reveals the fact that the turbulenceintensity is very low in the convergent section and up tothe throat it is 8.39e+02% while at the centre of the nozzlethe turbulent intensity is maximum up to 5.65e+03%. Theflow is totally symmetric. Due to friction, near the wall,the turbulent intensity is quite higher than that at thenozzle exit. In the divergent section, since there isstabilization of flow, there is a decrease in the turbulentintensity.

Mach number 3 with isolator:-

In this, the Isolator is designed to reduce the Machnumber. From figure, it is clearly visualized that in theconvergent section at inlet point, Mach number, is in theSub-sonic region while at the throat, flow becomes Sonicand at the nozzle exit it becomes Super-Sonic for whichthe nozzle is designed. Near the wall, the Mach number is1.98 because of the viscosity and turbulence in the fluid.The flow travels in the nozzle along the angle ofdirection. When the flow comes in the isolator parts. TheMach number starts decreasing as well as the property offlow has been also changing as can observe from figure.At the exit of the isolator the Mach number is 1.97.

Static pressure:

The above figure reveals the fact that the gas gets overexpanded at the nozzle exit. The value of static pressure atnozzle exit is 4.87e+03 Pa. The static pressure in theconvergent section is observed to be 7.85e+05Pa at theinlet which then decreases to 2.00e+05Pa up to the throatand then to 4.87e+03Pa at exit. From the aboveobservations, it is clear that static pressure reduces in thenozzle as we move from inlet to exit and it starts againincreasing as the flow goes into the isolator andthroughout it is constant.

Figure 7 Velocity Magnitude

Figure 8 Turbulence intensity

Figure 9 Mach Number

Figure 10 Static Pressure

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Total Pressure

The total pressure at the nozzle exit at the centre is2.72e+05Pa and at the inlet is 3.68e+05Pa as given in theboundary condition. The contour of nozzle and nozzle withisolator like disc patch is same in both figure. There is adecrease in stagnation pressure near the nozzle walls due toviscous effects, whereas the stagnation pressure remainsalmost constant in the centre which is the phenomenonobserved in Mach 3 as well. In Mach 3 there was a disclike patch near the wall but here it extends till the exit ofthe nozzle. Overall a major change is not observed in thetotal pressure in the nozzle. While in the isolator there issudden variation in the total pressure and near the wall thestagnation pressure is around 1.29 bar.

Velocity Magnitude:-

As noted from the above figure, at the inlet ofthe nozzle, the velocity is found to be 2.20e+02 m/s and itrapidly increases up to the throat to nearly 5.21e+02 m/s.At the nozzle exit, the velocity is found to be around8.63e+02 m/s, as in case of Mach 3. when the flow comesin isolator the flow characteristics starts changing and thevelocity at the exit is nearly 6.5e+02 m/s. It is clear thatthe flow is symmetric and flowing across an angle as acharacteristic of the De-Laval nozzle.

Turbulence Intensity:

The above figure reveals the fact that the turbulenceintensity is very low in the convergent section and up tothe throat it is 3.51e+02% while at the centre of the nozzlethe turbulent intensity is maximum up to 5.31e+03%. Thisis due to the sudden expansion in area which causes theturbulence just after the throat. Due to friction, near thewall, the turbulent intensity is quite higher than that at thenozzle exit. In the divergent section, as we move towardsthe exit of isolator there is rapidly a change, since there isstabilization of flow, there is a decrease in the turbulentintensity. The flow is totally symmetric.

V. GRID INDEPENDENCE TEST

The grid independence test has been done for Machnumber 3.Mach Number 3 De-Laval Nozzle:-

grid size ( original / adapted / change)cells ( 14400 / 17934 / 3534)faces ( 29100 / 36504 / 7404)nodes ( 14701 / 18571 / 3870)

The grid adaption method is done in respect of Machnumber. The changes in cell, faces and nodes are 3534,7404 and 3870 respectively. As shown in the figure. Thesolution is converging after 919 iteration.

The mass flow rate plot respect to iteration is shown infigure and the mach contour after grid adaption is shownbelow.

Figure 11 Total Pressure

Figure 12 Velocity Magnitude

Figure 13 Turbulence Intensity

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Figure 14

It is observe that there is no change in Mach number. Sothe number of nodes as given in previous was accurate.

RESULT VALIDATION WITH MATHEMATICALMODEL

In this section, the numerical Analysis of nozzle at mach3 has been done on the basis of table 1. This is based ontheoretical formulation.

ACKNOWLEDGEMENT

The authors acknowledge the valuable suggestions fromProf. E. Rathakrishnan, Professor, Department ofaerospace engineering, IIT Kanpur , India. The authorsacknowledge the financial help provided by AICTE fromthe project AICTE: 8023/RID/BOIII/NCP(21) 2007-2008.The Project id at IIT Guwahati is ME/P/USD/4.

VI. CONCLUSION:It is observed that the nozzle which designed for, flow

travel along with the direction and at throat Mach numberis 1 in both nozzles. The result is validating withmathematical procedure of method of characteristics. ForMach 3 the total pressure increases up to 3.68e+05Pa in a

patch which is like a disc, while the total pressure at thenozzle exit at the centre is 3.52e+05Pa and that at the inletis 3.68e+05Pa . turbulence intensity is very low in theconvergent section and up to the throat it is 3.51e+02%while at the centre of the nozzle the turbulent intensity ismaximum up to 5.31e+03%. This is due to the suddenexpansion in area which causes the turbulence just afterthe throat. Due to friction, near the wall, the turbulentintensity is quite higher than that at the nozzle exit. In thedivergent section, as we move towards the exit of isolatorthere is rapidly a change, since there is stabilization offlow, there is a decrease in the turbulent intensity. Theflow is totally symmetric.

VII. REFERENCE

[1] Wick, R.S., (1953), The effect of boundary layer on sonic flow

through an abrupt cross sectional area change, Journal of the Aeronautical

Sciences, Vol. 20, p. 675-682.

[2] Korst, H., (1954), ‘Comments on the effect of boundary layer on the

sonic flow through an abrupt cross sectional area change’, Journal. of

Aeronautical Sciences, Vol. 21, p. 568.

[3] Hall, W.B and Orme, E.M. (1955), ‘Flow of a compressible fluid

through a sudden enlargement in a pipe, Proceedings of Institution of

Mechanical Engineers’, Vol. 169, p. 007-1022.

[4] Benedict.R.P and Steltz, W.G., (1962), ‘ A generalized approach to

one dimensional gas dynamics’, Trans. ASME (power), Vol. 84, p. 44.

[5] Anderson, J. S and Williams, T. T., (1968), ‘Base pressure and noise

produced by the abrupt expansion of air in a cylindrical duct’, Journal of

Mechanical Engineering Science, Vol. 10, No. (3), p. 262-268.

[6] Durst.F., Melling, A and Whitelaw, J.J., (1974), ‘ Low Reynolds

number flow over a plane symmetric sudden expansion’, Journal of Fluid

mechanics, Vol. 12, No.6, p.739.

[7] Cherdon, W., Durst, F. and Whitelaw, J.H., (1978), ‘Asymmetric

flows and instabilities in symmetric ducts with sudden expansions’,

Journal of Fluid Mechanics, Vol. 84, part 1, p.13.

[8] Brady, J.F. and Acrivos, A., (1982), ‘Closed cavity laminar flows at a

moderate Reynolds numbers’, Journal of Fluid Mechanics, Vol. 115, p.

427.

[9] Yang, B.T and Yu, M. H., (1983), The flow field in a suddenly

enlarged combustion chamber, AIAA Journal, Vol. 21, No.1, p. 92-97,

[10] Rathakrishnan, E., Ramanaraju, O. V., and Padmanabhan, K., (1989),

Influence of cavities on suddenly expanded flow field, Mechanics

Research communications, Vol. 16 (3), p. 139-146.

[11] Raghunathan S and Mabey, D. G., (1987), Passive

shockwave/boundary-layer control on a wall mounted model, AIAA

Journal, Vol. 25, No. 2, p. 275-278.

Figure 15 Mach Number

Pandey et al. / International journal of Application of Engineering and TechnologyVol.-1 No.-1

22

[12] Raghunathan, S., (1987), Pressure fluctuation measurements with

passive shock/boundary layer control, AIAA journal, Vol. 25, No. 5, p.

626-628.

[13] Raghunathan, S., (1987), Effect of porosity strength on passive

shockwave/boundary layer control, AIAA journal , Vol. 25, No. 5, p.

757-758.

[14] Wilcox, J.F. Jr., (1988), Passive venting system for modifying cavity

flow fields at supersonic speeds, AIAA Journal, Vol. 26, No. 3, p. 374-

376.

[15] Tanner, M., (1988), Base cavities at angles of incidence, AIAA

Journal, Vol. 26, No. 3, p. 374-377.

[16] Vishwanath P.R. and Patil, S.R., (May-June 1990), ‘Effectiveness

of passive devices for axi symmetric base drag reduction at Mach 2’,

Journal of Spacecraft, p. 234.

[17] Kruiswyk, R.W and Dutton. J. C., (1990), ‘Effect of base cavity on

subsonic near wake flow’, AIAA Journal, Vol. 28, No. 11, p.1885-1895.

[18] Rathakrishnan E., (2001), ‘Effect of ribs on suddenly expanded

flows’, AIAA Journal, Vol. 39, No. 7, p.1402-1404.

[19] Pandey, K.M., and Rathakrishnan E., (2006), ‘Influence of cavities

on flow development in sudden expansion’, International Journal of

Turbo and Jet Engines, Vol. 23, p. 97- 112.

[20] Pandey, K.M. and Rathakrishnan E , (2006), Annular cavities for

base flow control, International journal of Turbo and Jet Engines,

Volume 23, No.2, pp113-127 ISSN- 0334-0082.

[21 Pandey K M , S A Khan and Rathakrishnan E, Passive control of base

flows, Proceedings of the National Seminar on Recent Advances in

Experimental Mechanics, Dept of Aerospace Engg, IIT Kanpur, Mar 15-

16, 2000, Allied Publications Ltd, New Delhi, India 2000 PP-57-71, ISBN

– 81-7764-020-8.

[22] Pandey K M, Base flow in flow though nozzles with sudden

expansion: a study in supersonic regime-2nd National Conference on

Fluid Machinery, Dept of Mechanical Engg, PSG College of Technology,

Coimbatore, India, June 28-29, 1996, pp 256-265, Allied publishers Ltd.

New Delhi, India, 1996, ISBN- 81-7023-543-X.

[23] K.M. Pandey and A.P Singh, “Design and development of De-Laval

Nozzle at Mach 3 and 4 with fluent software”. Accepted and Published in

30 June 2010 IJME.

[24] K.M. Pandey and A.P Singh, “Recent Advances in Experimental and

Numerical Analysis of Combustor Flow Fields in Supersonic Flow

Regime” International Journal of chemical Engineering and Application

,Vol.-1, No. 2, August 2010,ISSN- 2010-0221, pp 132-137.

[25] K.M. Pandey and A.P Singh, “CFD Analysis of Conical Nozzle For

Mach 3 At Degrees Of Angle Of Divergence With Fluent Software”

International Journal of chemical Engineering and Application ,Vol.-1,

No. 2, August 2010,ISSN- 2010-0221, pp 179-185.

[26] Pandey K M, Studies on flow through nozzles in sudden expansion

with passive control, International conference at IISc, Bangalore on recent

advances in Mechanical Engineering, Dec 20-22, 1995, Narosa

publications, New Delhi, India, 1996, Volume 2, pp 1511-1518

[27] Jiyun tu, guan Heng yeoh and chaoqun liu. (2008), “computational

Fluid Dynamics” Elsevier Inc.