Part 13 Compressible Flow - University of California,...

Part 13

Compressible Flow

A Stabilized Finite Element Method forCompressible Turbulent Flows

Christelle Wervaecke, Heloıse Beaugendre and Boniface Nkonga

1 Introduction

On the one hand, all turbulent models include a turbulent viscosity as a parameteror a variable. On the other hand, numerical schemes always induce an artificial dis-sipation. This artificial dissipation is crucial to control, such as to be always lowerto the viscosity obtained by subscale modeling of the turbulence. High order nu-merical approximations provide a framework where the constraint on the numericaldissipation can be achieved. Finite elements are suitable for the design of high orderscheme with compact stencil that is efficient for parallel computing strategies bydomain decomposition and messages passing.

The main weakness of the classical finite element method (Galerkin) is its lackof stability for advection dominated flows. We consider in this work a compressibleNavier-Stokes equations combined with the one equation Spalart-Allmaras turbu-lence model. These equations are solved in a coupling way. The numerical stabilityis achieved thanks to the Streamline Upwind Petrov-Galerkin (SUPG) formulation[2]. Within the framework of SUPG method, artificial viscosity is anisotropic andthe principal component is aligned with streamlines. The aim is to put sufficientviscosity to get rid of instability and unphysical oscillations without damaging theaccuracy of the method. The amount of artificial viscosity is controlled by a stabi-lization tensor τ . Since optimal way to choose τ is still unknown, several ways ofcomputing τ are tested in this paper. Besides SUPG method is also used in combi-nation with a shock-parameter term which supplied additional stability near shockfronts [7].

Christelle WervaeckeINRIA Bordeaux Sud-Ouest, France, e-mail: [email protected]

Heloıse BeaugendreIMB - INRIA MC2 - IPB, France e-mail: [email protected]

Boniface NkongaINRIA PUMAS, Universite de Nice, France e-mail: [email protected]

1

2 Christelle Wervaecke, Heloıse Beaugendre and Boniface Nkonga

2 Spalart-Allmaras turbulence model for compressible flows

Let Ω ⊂ Rd be the spatial domain with boundary Γ . The Navier-Stokes equationsof compressible flows on Ω can be written as

∂U∂ t

+∂Fi

∂xi− ∂Gi

∂xi= 0 (1)

where Fi and Gi are, respectively, the Euler and viscous flux vectors. Appropriatesets of boundary and initial conditions are set for Eq. (1). The turbulent kinematicviscosity νt = µt/ρ is, then, computed using Spalart-Allmaras (S-A) one-equationmodel

∂ρνt

∂ t+ ∇.(ρνtu) = M(νt)νt +P(νt)νt −D(νt)νt (2)

where M(νt)νt represents the diffusion term, P(νt)νt the production source termand D(νt)νt the wall destruction source term. This model is an empirical equationthat models production, transport, diffusion and destruction of the turbulent viscos-ity. See [5], for a complete description of the model. Navier-Stokes equations andSpalart-Allmaras equation are coupled in such a way that, the eddy viscosity is con-sidered as an additional unknown of the system of equations (1).

3 SUPG formulation

The streamline upwind Petrov-Galerkin (SUPG) formulation is one of the most es-tablished stabilized formulations and is widely used in finite element flow computa-tions. SUPG method introduces a certain amount of artificial viscosity in the stream-line direction only. The aim is to prevent numerical instabilities without introducingexcessive numerical dissipation.

3.1 Weak formulation

Consider a discretization of Ω into element subdomains Ωe, e = 1,2, ...,nel , wherenel is the number of elements. Given some suitable finite dimensional trial solutionand test funtion spaces S h and V h, the SUPG formulation can be written as follows: find Uh ∈ S h such that ∀Wh ∈ V h

∫Ω

Wh.

(∂Uh

∂ t+

∂Fi

∂xi− ∂Gi

∂xi− S

)dΩ +

nel

∑e

∫Ωe

τ(

∂Wh

∂xi

). Ai R(Uh) dΩ = 0

(3)where R(Uh) is the residual of the differential equation :

A Stabilized Finite Element Method for Compressible Turbulent Flows 3

R(Uh) =∂Uh

∂ t+

∂Fi

∂xi− ∂Gi

∂xi− S (4)

The variational formulation (3) is built as a combination of the standard Galerkinintegral form and a perturbation-like integral form depending on the residual. τ isthe stabilization matrix, various options to compute this parameter were introducedin the literature. This will be discussed in the next section.

3.2 The stabilization parameter

Stabilization parameter has been a subject of an extensive research over the last threedecades. Nevertheless, the definitions of τ mostly rely on heuristic arguments anda general optimal way to choose τ is still unknown. Therefore, in this paper, threeways of computing τ are tested. By analogy with formula for advection-diffusionproblem, the parameter τ1 is defined by

τ1 =he

||u||+ cId (5)

Where he is a measure of the local length scale, ||u|| is the flow velocity and theacoustic speed is defined as c. The second stabilization parameter proposed reads :

τ2 = |Ωe|( nen

∑k=1

∣∣ A1nix +A2ni

y +A3niz|)−1

(6)

Ai =∂Fi

∂Uin Ωe (7)

Indeed matrices Ai can be considered as a generalization of the scalar advectioncoefficient for the system of equation (1) . Thus, these matrices can be employedin order to define the stabilization parameter. That is done following an expressioncoming from a Residual Distribution schemes, the N-scheme [1]. The third optionhas been proposed by T.E. Tezduyar and M. Senga [7].

τ3 =

( nen

∑k=1

c|j.∇ϕk|+ |u.∇ϕk|)−1

Id (8)

where j is a united vector defined as ∇ρ||∇ρ|| , c is again the acoustic speed and ϕk are

the tests functions.


4 Numerical results

4.1 Flat plate boundary layer

The first problem considered is a compressible high Reynolds number flow over aflat plate. Values of the CTTM are adopted, namely C f = 0.00262 at Reθ = 104 anda Mach number of 0.24. See Spalart and Allmaras1 for more informations on thetest case.

0

10

20

30

40

50

60

0.1 1 10 100 1000 10000 100000

y+ = 0.5y+ = 2y+ = 5

y+ = 10

Fig. 1: Flow over a flat plate. Boundary layer velocity profile

The law of the wall and the log law are plotted with the computed solution.Theoretical and numerical results are in good agreement. The first mesh employedis viewed as fine. The first layer of nodes next to the wall is at y+ = 0.5. To study theeffect of mesh spacing, the closest point to the wall was allowed to vary. Figure (1)illustrates the effect of near wall refinement on the velocity profiles. For the sake ofclarity, curves are shifted to the top on figure (1) while in fact, they overlay. Thereis very little variation in the results. Furthermore, no significant difference in theresults has been observed for the three definitions of the stabilization parameter.

4.2 Backward-facing step

The geometry is a 1.33 m long and 0.10 m tall rectangular inlet duct followed bya 0.0127 m rearward-facing step. The test was performed at a freestream velocityof 44.2 m.s−1 and atmospheric total pressure and temperature. These conditionscorrespond to a freestream Mach number of 0.128, a boundary thickness of 1.5H,and a Reynolds number (based on momentum thikness) of 5000 at a location 4step heights upstream of the step. Figure (2a) gives an overview of the flow fieldobtained, in wich a recirculation is formed downstream of the step.

A Stabilized Finite Element Method for Compressible Turbulent Flows 5

(a) Recirculation zone (b) Velocity profil at x = 2H

Fig. 2: SUPG results for the simulation of the flow behind a backward-facing step

Several computations for this flow have been done. First, the SUPG method isused combined with the stabilization parameters τ1, τ2 and τ3 . Solutions obtainedwith these different parameters are very similar. Then, a finite volume version of thesame code is used to compare the finite volume solution to the SUPG one. Figure2b represents the velocity profile at x = 2H. The fact underlined is that the SUPGmethod brings less numerical diffusion than a finite volume method.

4.3 Transonic flow over a RAE2822 airfoil

The problem considered here is a flow over a RAE2822 airfoil. Fully turbulent flowwas assumed, Reynolds number is 6.5× 106, Mach number is 0.734 and the angleof attack is 2.79 . Here, SUPG formulation is used in combination with a shock-capturing term [7] that provides additional stability near the shock front. Figure (3b)shows result for the coefficient of pressure Cp. The mesh, used for this calculation,includes very stretched elements (aspect ratio up to 2×105). Due to this elongatedcells, we still encounter some numerical instabilities. However Cp obtained by theSUPG calculation is not so far from the experimental one. Some improvement inthe handling of stretched element will certainly lead to better results.

5 Conclusion

A stabilized finite element formulation applied to Navier-stokes equation combinedwith the Spalart-Allmaras model is proposed in this study. Numerical results showthat, not only this method is able to reproduce good turbulent profiles in the case ofa 2D flat plate, but also that it brings less numerical diffusion than a finite volumemethod. Even in the case of a almost incompressible flow, the numerical code is


(a) Pressure isolines (b) Pressure coefficient Cp

Fig. 3: Velocity profiles for finite element and finite volume computations

robust and gives good results. The choice of the stabilization parameter is yet aproblem unsolved, but it seems that the second definition τ2 performed better withthe test cases presented here.

Acknowledgements Authors acknowledge Airbus France for its support through the DESGIVREproject.

References

1. R. Abgrall, Essentially non-oscillatory Residual Distribution schemes for hyperbolic prob-lems, Journal of Computational Physics, Vol. 214, 773-808 (2006)

2. T.J.R. Hughes and T.E. Tezduyar, Finite element methods for first-order hyperbolic systemswith particular emphasis on the compressible Euler equations, Comput. Methods Appl. Mech.Engrg., Vol. 45, 217-284 (1984).

3. T.J.R. Hughes and T.E. Tezduyar , A new finite element formulation for computational fluiddynamics : I. Symmetric forms of the compressible Euler and Navier-Stokes equations andthe second law of thermodynamics, Comput. Methods Appl. Mech. Engrg., Vol. 54, 223-234(1986)

4. D. M. Sommers, Design and experimental results for the S809 airfoil, NREL/SR-440-6918,(1997)

5. P. R. Spalart and S. R. Allmaras, A One-Equation Turbulence Model for Aerodynamic Flows,AIAA-92-0439 (1992).

6. N. Tarchoune, F. Morency and C. Masson, 3D numerical icing simulation on rotating windturbine, CFDSC conference, Kanata, Ontario, May 2009

7. T. E. Tezduyar and M. Senga, Stabilization and shock-capturing parameters in SUPG for-mulation of compressible flows, Comput. Methods Appl. Mech. Engrg., Vol. 195, 1621-1632(2006)

Receptivity of Hypersonic Flow overBlunt-Noses to Freestream Disturbances usingSpectral Methods

Kazem Hejranfar, Mehdi Najafi, Vahid Esfahanian

AbstractThe receptivity of supersonic/hypersonic flows over blunt noses to freestream dis-turbances is performed by means of spectral collocation method. The unsteady flowcomputations are made through solving the full Navier-Stokes equations in 2D. Ashock-fitting technique is used to compute unsteady shock motion and its interactionwith freestream disturbances accurately in the receptivity study. The computationalresults for receptivity of a semi-cylinder at Mach 8 is presented and validated bycomparison with available theoretical and numerical results. The study shows sig-nificant effects of the viscosity on the receptivity process.

1 Introduction

One of the critical points in supersonic/hypersonic design specially for heating anal-ysis of future vehicles is the laminar-turbulent transition prediction. Recently, areview on the effects of laminar-turbulent transitions upon high-speed entry vehi-cles performance revealed that understanding of transition and its origins, is a pre-lude (Lin 2008). The transition process, which is the nonlinear response of laminarboundary layer to freestream disturbances, conceptually has three important stages:receptivity, linear/nonlinear instability and breakdown to turbulence. Receptivity,

Kazem HejranfarAerospace Engineering Department, Sharif University of Technology, Tehran, Irane-mail: [email protected]

Mehdi NajafiAerospace Engineering Department, Sharif University of Technology, Tehran, Irane-mail: [email protected]

Vahid EsfahanianMechanical Engineering Department, Engineering Faculty, University of Tehran, Tehran, Irane-mail: [email protected]

1

2 Kazem Hejranfar, Mehdi Najafi, Vahid Esfahanian

Fig. 1 A schematic of bowshock/freestream disturbancesinteraction in high-speed flowover a blunt body.

which plays a key role in this process, stands for various mechanisms of transfor-mation of external perturbations into internal ones.

The stability and transition of supersonic and hypersonic boundary layers havebeen performed by many researchers. Formation of a shock wave in front of movingobjects at supersonic/hypersonic regimes changes the receptivity of the boundarylayer over them to freestream disturbances (Fig. 1). The bow-shock/disturbance in-teraction problem originates from studies of the receptivity in such flow regimes(Morkovin (1960) and Chiu and Zhong (1996)). While experimental and theoreticalinvestigations are served as reference points, up to now, they are very limited andhardly advisable for the study of different aspects of such problems where numeri-cal tools are appropriate alternatives and are able to surmount. With the increase incomputational power and development of high-order numerical methods, the studyof the fundamental routes of transition to turbulence has become feasible. Such asimulation requires a good resolution of all relevant flow time and length scales,and therefore, highly accurate numerical methods are required.

High-order compact methods have been used for studying the receptivity process(Zhong 1998 & 2003). Although these schemes can provide accurate results forthe receptivity problem, in practice they suffer from some difficulties like boundaryclosure problems. An alternative approach is to use spectral methods that do nothave such deficiencies.

In this work, the receptivity of viscous supersonic/hypersonic flow over bluntnoses to freestream disturbances is studied by means of spectral collocation method.Both steady and unsteady flow computations are made through solving the fullNavier-Stokes equations in 2D. The effects of bow-shock/freestream disturbanceinteractions on the receptivity process are accurately treated by taking the shock asa boundary, governed by the unsteady Rankine-Hugoniot relations. The computa-tional results for receptivity of a semi-cylinder at Mach 8 is presented and validatedby comparison with available numerical and theoretical results.

2 Problem Formulation

The governing equations for this study are the two-dimensional Navier-Stokes equa-tions, which can be written in the following conservation-law form in the Cartesian

Receptivity of Hypersonic Flow over Blunt-Noses to Freestream Disturbances 3

coordinates xi in terms of density, ρ , velocity, ui, pressure, p, temperature, T,

∂∂ t

ρuip

+∂

∂x j

ρ(u j − x j)

ρui(u j − x j)+ pδi j( γ p

γ−1 +ρukuk

2 )(u j − x j)

=

01ρ σxi

γPr ∇ · (k∇T )+(γ −1)Φc

(1)

where

σxi =∂τi j

∂x j, Φc = τi j

∂ui

∂x j, τi j = µ

(∂ui

∂x j+

∂u j

∂xi− 2

3δi j

∂uk

∂xk

). (2)

In this paper, the perfect-gas flow is considered, for which the equation of state isp = ρRT where R is the gas constant. The ratio of specific heats is assumed to beconstant (γ = 1.4). The viscosity coefficient µ is calculated by the Sutherland law.

The extensive application of spectral collocation methods in the last decades forsmooth partial differential equations showed superiority over other traditional meth-ods. Globally accurate solutions and exponential decay of error with the increase incomputational grid points are known to be a specific property of these methods [2].However, the application of global spectral methods is limited to rather simple ge-ometries and obtaining global smooth solutions, as the computational grid pointsare predetermined and the solution should be analytic over the whole computationalregion. Thus, the discontinuous feature of the shock wave in the present study istreated utilizing a shock-fitting technique. Details of the numerical algorithm can befound in [3].

3 Receptivity Simulation

A weak monochromatic planar acoustic wave is introduced in the freestream to ex-cite the instability waves in the shock layer. In this case, the perturbation amplitudesof nondimensional freestream flow variables satisfy the following dispersion rela-tions: ∣∣u′1∞

∣∣= ε,∣∣ρ ′

∞∣∣= εM∞,

∣∣p′∞∣∣= γεM∞,∣∣u′2∞

∣∣= 0,c∞ = (M∞ ±1)√

γ (3)

where k∞ is the wavenumber related to the circular frequency ω by ω = k∞c∞. Also,ε is a small dimensionless number and εM∞ represents the relative amplitude ofthe wave. The plus and minus signs are used for the fast and slow acoustic wavesrespectively. The numerical simulation for the receptivity problem is carried out intwo steps. First, a steady flow-field is computed by advancing the unsteady gov-erning equations, with the time independent flow conditions. Second, a disturbancewave is imposed to the flow conditions by Eq. (3). After a periodic state is achieved,a wave decomposition procedure is performed on perturbed flow variables.


4 Results and Discussion

A cylinder is placed in a hypersonic flow of M∞ = 8.03 with two Reynolds numbersof 2× 103 and 2× 105 based on nose radius. The (35× 25) grid and steady statevelocity vectors are plotted in Fig. 2 for Re∞ = 2×105.

80

70

55

40

3530

25

20

15

Fig. 2 The grid, steady state velocity vectors and pressure contours for Re∞ = 2×105.

After steady state reached, a monochromatic planar fast acoustic wave with ω =101.76 is added to the freestream. The pressure fluctuation contours in the (x− t)

Fig. 3 Instantaneous pressure fluctuation contours along the stagnation line for Re∞ = 2×103 and2×105.

plane along the stagnation line are depicted in Fig. 3. Two neighboring cells intime-direction have different shapes for the higher Reynolds number case and showa nonlinear behaviour, unlike the case for the lower one. This difference is most

Receptivity of Hypersonic Flow over Blunt-Noses to Freestream Disturbances 5

Fig. 4 Pressure variations just behind the bow-shock at the center-line for Re∞ = 2× 103 and2×105.

noticeable near the shock and becomes negligible near the body. The time history ofinstantaneous pressure just behind the bow-shock at the center-line is presented inFig. 4. At the initial moments, there are no reflections from the undisturbed steadyboundary layer and the figure shows a good agreement with the linear predictions ofMorkovin (1960) and the inviscid results of Chiu and Zhong (1996) for the pressureamplitude. After it, there are two distinct regions, for both Reynolds numbers, thatthe pressure amplitude and its time evolution differ considerably from the inviscidone and indicate the damping effect of viscosity on them.

In Fig. 5 the amplitude of the pressure fluctuations along the stagnation line forRe∞ = 2×103 and 2×105 together with the inviscid computation of the developedcode, Chiu and Zhong (1996) and Morkovin (1960) linear theory results are plot-ted. The wavelengths from numerical calculations are slightly larger than those ofthe linear theory. Numerical amplitudes of the pressure waves are also higher thanthose predicted by the linear analysis near the stagnation point. This discrepancy iscaused by the simplified assumption of uniform steady flowfield behind the shockused in the linear analysis, which in reality is not uniform. Comparing the pressurefluctuations amplitude for the viscous cases, the damping effect in the case withlower Reynolds number is obviously clear. As the Reynolds number increases, theviscous and inviscid results in this figure coincide with each other.

5 Conclusions

A highly accurate spectral collocation method is developed for the receptivity studyof viscous supersonic/hypersonic flows over blunt noses. The computational resultsfor receptivity of a semi-cylinder at Mach 8 demonstrate the significant effects of


Fig. 5 Distribution of pressure fluctuation amplitudes along the center-line for Re∞ = 2×103 (left)and 2×105 (right).

viscosity on the receptivity process. The study demonstrates the capability and ef-ficiency of the spectral methods in the receptivity study of supersonic/hypersonicflows over blunt noses.

Acknowledgements The authors would like to thank Sharif University of Technology and Uni-versity of Tehran for financial support of this study.

References

1. Chiu, C., & Zhong, X.: Simulation of transient hypersonic flow using the ENO schemes. AIAAJ. 34:4, 655–661 (1996)

2. Canuto, C., & Hussaini, M. Y., & Quarteroni, A., & Zang, T. A.: Spectral Methods. Funda-mentals in Single Domains. Springer, Berlin (2006)

3. Hejranfar K., Esfahanian V., Najafi M. On the outflow conditions for spectral solution of theviscous blunt-body problem. J. Comp. Phys., 228:11, 3936–3972 (2009)

4. Lin, T. C.: Influence of laminar boundary-layer transition on entry vehicle designs. J. Spac.Rock. 45:2, 165–175 (2008)

5. Morkovin, M. V.: Note on the assessment of flow disturbances at a blunt body traveling atsupersonic speeds owing to flow disturbances in free stream. J. Appl. Mech. 27:6, 223–229(2008)

6. Zhong, X.: High-order finite-difference schemes for numerical simulation of hypersonicboundary-layer transition. J. Comp. Phys. 144:2, 662–709 (1998)

7. Zhong, X., & Tatineni, M.: High-order non-uniform grid schemes for numerical simulationof hypersonic boundary-layer stability and transition. J. Comp. Phys., 190:2, 419–458 (2003)

A Numerical Study of DiffuserPerformance of Gas Dynamic LaserSystem

SD Ravi1, PS Kulkarni2 and NKS Rajan3

Abstract: Based on the earlier CFD analysis of the performance of the gas-dynamically controlled laser cavity it was found that there is a possibilityof optimizing the geometry of the diffuser that can bring about reductionsin both size and cost of the system by examining the critical dimensionalrequirements of the diffuser for two cases i.e. case 1 is 1kg/sec and case 2 is15kg/sec of mass flow rate. Consequently, an extensive CFD analysis has beencarried out for a range of diffuser configurations by simulating the supersonicflow through the arrangement including the laser cavity driven by a bank ofconverging - diverging nozzles and the diffuser. The numerical investigationswith 3D-RANS code are carried out to capture the flow patterns throughdiffusers past the cavity that has multiple supersonic jet interactions withshocks leading to complex flow pattern. Varying length of the diffuser platesis made to be the basic parameter of the study. The analysis reveals thatthe pressure recovery pattern during the flow through the diffuser is criticalfor the performance of the laser device and its dependence on the diffuserlength. If length of the diffuser is weaker beyond a critical lower limit andthe evaluation of this limit would provide a design guideline for a more effi-cient system configuration. The observation based on the parametric studyshows that the pressure recovery transients in the near vicinity of the cavityis not affected for the reduction in the length of the diffuser plates up to its10% of the initial size, indicating the design in the first configuration thatwas tested experimentally has a large factor of margin. The flow stability inthe laser cavity is found to be unaffected since a strong and stable shock islocated at the leading edge of the diffuser plates while the downstream shockand flow patterns are changed, as one would expect. Results of the study forthe different lengths of diffusers in the range of 10% to its full length arepresented, keeping the experimentally tested configuration used in the earlier

(1)Dept. of Aerospace Engg., IISc, Bangalore, India. [email protected] · (2)Dept.of Aerospace Engg., IISc, Bangalore, India. [email protected] · (3)Dept. of

Aerospace Engg., IISc, Bangalore, India. [email protected]

1

2 SD Ravi, PS Kulkarni and NKS Rajan

study as the reference length.

1 Introduction

Computational studies have been carried out for two design configurationsystems involving a parametric study of Gas Dynamics Laser (GDL) using a3D-RANS code [1, 2]. Parametric study is done based on the configurationsselected for the analysis from practical system which involves aerodynam-ically controlled laser cavity which is further used for generation of highpowered laser beams. The parametric study in this arrangement, in orderto produce an active lasing medium, rapid and deep cooling of mixture ofgases from a combustion chamber (CO2-N2-H2O) is achieved by allowingthe gases to expand in a set of convergent - divergent nozzles, allowing thesupersonic jets to drive the cavity used as the zone for the laser activity. Thiscavity would have a distribution of CO2 in non-equilibrium condition withmany quantum levels, providing a favorable condition for the lasing activity.The bank of nozzles followed by the cavity and further by a diffuser sectionfor stable operation, makes the main parts of the aerodynamically controlledcavity which is considered in the present analysis and a 3D view of the con-figuration has been shown in Fig.1.The simulation of flow in a Gas Dynamics Laser system is done using anindustrially standard CFD tool 3D-RANS code. The process of the modelingis taken up in steps of increasing complexity to ensure the numerical stabilityand computational consistency. The different stages have provided clarity ondifferent parameters that include the optimal grid distribution and the opti-mal initial flow profiles to obtain a stable and consistent solution. The detailsof the analysis are explained below.An approach with multiple geometrical blocks that are matching to the geom-etry of the unit considered at different locations, depending on its complexity,is used with their matching surfaces blended so as to generate a good compu-tational mesh. The structured mesh is preferred since the geometry is morecomplex and the flow computations need to capture supersonic, transonicand subsonic zones over the same computational domain. Optimization ofnumber of computational nodes is done by allowing the cluster of grids incritical zones, at places expected to have larger gradients in the flow param-eters and in places where the geometry is more complex. It is observed thatnearly 0.36 to 0.56 millions of computational nodes are required for 10% ofinitial length to full length of the case-1 design configuration and 0.628 to1.146 millions of computational nodes are required for 10% of initial lengthto full length of the case-2 design configuration. Fig. 2 gives view of a sam-ple of the computational mesh generated near cavity at the upstream of the

Performance of Gas Dynamic Laser System 3

diffuser.The computational flow analysis made during the preliminary stage of thestudy includes flow analysis through a single nozzle followed by flow analysisthrough a set of four nozzles and thereafter analysis of flow through a set ofnozzles along with a set of two diffuser vanes. In its final stage the compu-tations were taken up to capture the flow through multiple nozzles and thediffusers. The method of analysis, the results obtained at different stages willbe presented and discussed in the following sections of the paper.

2 The Configuration and the Approach

The basic configuration of the system considered for the analysis, scheme ofcomputation, code validation with experimental data and other basic aspectsof the study have been brought out in detail in an earlier work, and this workis an extension of this reported work. However, for clarity, the arrangementof the system configuration is shown in Fig. 1. In the present analysis, all theother hardware elements except the diffuser plates are kept unchanged, inorder to have a focused parametric study. Further for simplicity of analysis,the downstream element of the diffuser - the acoustic control device is takenoff from computation domain. A 3D-RANS code having upwinding implicitscheme and K−ω approach for turbulence is used for the numerical solution.The RANS Equations are solved for steady, compressible viscous flow. Thegoverning equations used are the conventional standard sets that include:Continuity equation:

∂U j

∂xj= 0 (1)

Momentum equation:

∂

∂t

(ρU i

)+

∂

∂xj

(ρU iU j

)= − p

∂xi− ∂

∂xj

(τij + ρu”

iu”j

)(2)

Energy equation:

∂

∂t

(ρh

)+

∂

∂xj

(ρU jh

)= − p

∂xj

(Qj + ρu”

jh”)

(3)

The boundary conditions and initial conditions used include (a) no slip, im-permeable and adiabatic walls; (b) inlet pressure and temperature of the fluidentering the system to be an assigned boundary value and (c) the outlet fromthe diffuser set to match the ambient conditions. Diminishing residual crite-ria of the variables is used for the convergence with a limit of RMS residualsfalling below 10−4.


3 Results and Discussion

Results from the study given in the Figs. 3(a) and 3(b) belong to a set of acase where the diffuser has the full length, as specified to have used in theexperimental tests built based on empirical designs. The Mach contour plotsshow the distribution of important aerodynamic parameters [2]. It can beobserved that the reflected shocks within the diffuser plates are adequatelystrong enough that the pressure recovery occurs dominantly beyond the trail-ing edge of the diffuser plates, ensuring a ’safe’ operating pressure range inthe aerodynamic lasing cavity. The results shown in Figs. 4(a) and 4(b), Figs.5(a) and 5(b) show the behavior of the aerodynamic characteristics with twoselective sets of diffusers with reduced plate lengths of 60% and 10%. Onecan observe that only the downstream flow structure is affected as comparedto the initial configuration and flow in the laser cavity is not affected. Thisand above data sets are run for the inlet conditions (28 atm.) of the referencedesign set. It could be clearly made out that the length of the diffuser platesis over-designed. To make a sensitivity analysis of the diffuser performancebased on its dependence on the inlet pressure, a set of analysis was madetaking the case of a 10% plate length which is most critical among othercases. A variation in the inlet pressure of about 7.5% was chosen for thisstudy. It was found that the higher side of the pressure did not change theflow pattern categorically, while the lower pressure provided a significantlyunstable flow structure with the solution fairly oscillating. In the Figs. 6(a)and 6(b) that belong to a case with reduced inlet pressure, one can see thatthe flow structure is not fully formed and is unstable. It was found in thiscase that a steady state solution could not be reached numerically. Figs. 7(a)and 7(b) consolidated plot of pressure recovery over the diffuser plates of dif-ferent lengths of Case1 and case 2 study. It was found that the inlet pressureof bank of nozzels play a very importatnt and critical role in the design of dif-fuser length. It was observed that there is a minimum inlet pressure beyondwhich it was not possible to obtain a converged solver for a given length ofthe diffuser. The analysis show that there is a varying range for the deviceassociated with the length of the diffuser and the inlet pressure as shown inverious figures.

4 Conclusions

The CFD Analysis using 3D-RANS code for the complex geometry consid-ered for an aerodynamically driven laser cavity and capturing of the flowdetails have shown a distinct insight of the physical behavior of the config-uration studied. The observations based on the parametric study show thatthe pressure recovery transients in the near vicinity of the cavity, the most

Performance of Gas Dynamic Laser System 5

functionally critical part of the device, are not affected with a reduction inthe length of the diffuser plates up to 10% of the empirical design size. Avalidation of the approach is made by checking the cavity pressure close toexperimental values and other flow parameters agreeing broadly on the theirbehavioral patterns. The analysis makes it clear that the configuration basedon the empirical design has a large factor of margin that can be reducedsignificantly and that there exists a limit in the operating range for the de-vice. The conclusions drawn from the analysis is expected to be significantsince it provides new design considerations, allowing for a hardware optimiza-tion with substantial size reduction in the device with no loss of performance.

References

[1] Complex Flow Analysis through a Multiple Nozzle Driven Laser Cavity.

M.A.Sriram, N. K. S. Rajan and P. S. Kulkarni. Symposium on Applied Aerody-namics and Design of Aerospace Vehicle (SAROD-2007) November 22-23, 2007,

Thiruvananthapuram, India.[2] CFD Analysis of Complex Flow through Multiple Nozzle Driven Aerodynamic

Laser Cavity. S D Ravi, N K S Rajan, P S Kulkarni. National Conference on

Recent Trends in Mechanical Engineering Science RTIMES-2007, Tumkur, Kar-nataka, India.

Fig(1) Fig(2)

Fig. 1 A general view of the initial configuration.

Fig. 2 Computational Mesh Model.

(a) (b)

Fig. 3 Mach Contours for case1 and case2 (full length)


(a) (b)

Fig. 4 Mach Contours for case1 and case2 (60% of the full length)

(a) (b)

Fig. 5 Mach Contours for case1 and case2 (10% of the full length)

(a) (b)

Fig. 6 Mach Contours for case1 and case2 (10% of the full length and 7.5% lower pressure)

Fig(a) Fig(b)

Fig. 7 Consolidated plot of Pressure Recovery over the diffuser plates of different lengths

of Case1 and case2 study

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