[American Institute of Aeronautics and Astronautics 22nd Atmospheric Flight Mechanics Conference - New Orleans,LA,U.S.A. (11 August 1997 - 13 August 1997)] 22nd Atmospheric Flight

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

A97-37258AIAA-97-3495

NUMERICAL SIMULATION OF A SUPERSONIC BASEBLEED PROJECTILE WITH IMPROVED TURBULENCE

MODELLING

Petri Kaurinkoski* and Antti Hellsten*Helsinki University of Technology, Laboratory of Aerodynamics

P.O. Box 4400, FIN-02015 HUT, FINLAND

This paper deals with Navier-Stokes simulations employing Chien's k — e turbulence model for the flow pasta supersonic projectile with base bleed. A new way of avoiding turbulence-related problems near stagnationpoints is suggested and results of simulations with and without the modification are presented. The theoreticalbackground of the modification is briefly described.

NomenclatureCp molar heat capacityD shell diameter; Jacobian matrix of the source

term Q (dQ/dU)E total internal energy per unit volumeF, G, H flux vectors in the x-, y- and ̂ -directionsH total enthalpyM molecular weight in g/molMa Mach numberP production of turbulent kinetic energyPr Prandtl numberK universal gas constant, 8.314 J/mol KR residual; specific gas constant, 1Z/MRe Reynolds number, pooV^D/'p.^S cell face areaT temperature; rotation matrixTu turbulence level, ^/WJWi

U vector of the conservative variablesV cell volume; velocityCf local skin friction coefficient, Tw/(p00V^.1/2)cp, cv specific heat at constant pressure and at con-

stant densitye specific internal energyh specific enthalpyi, j, k unit vectors in Cartesian coordinate systemk thermal conductivity; kinetic energy of turbu-

lence, (u"u" + v"v" + w"w")/2

'Research Scientist, Member AIAA.e-mail [email protected]

t Research Scientiste-mail [email protected]. Copyright © 1997 by P. Kaurinkoski andA. Hellsten. Published by the American Institute of Aeronautics andAstronautics, Inc. with permission.

rh mass flow rate through the inlet holen cell face unit normal vectorp static pressureyn normal distance from the surfacey+ non-dimensional normal distance from the

surface, yn^/pwrw/' p,wa characteristic variable; angle of attack5 boundary layer thickness7 ratio of specific heats cp/cve dissipation of kinetic energy of turbulenceA eigenvaluep densityH dynamic viscosityT normal or shear stress(f> mass fraction of a species; a general scalar

variablea Schmidt number

1 IntroductionNowadays trajectory simulations for artillery projectilescan be carried out quickly on any powerful personal com-puter or workstation. The reliability of these simulationsdepends on the accuracy of the atmospheric model andthe accuracy of the aerodynamic model of the projectile.The trajectory simulations themselves are not too diffi-cult and are performed routinely.

During the past few decades, computational tech-niques for the simulation of different types of jets forcontrolling the flight of projectiles1""3 have been de-veloped. With projectiles, the practical aim is to extendthe firing range, i.e. to reduce the base drag by bleedinggas from the base of the shell. As a part of the design

134American Institute of Aeronautics and Astronautics


process, numerical simulations are widely used for ana-lyzing the different design options. There are, however,many unanswered questions regarding the reliability ofthe numerical simulations: Turbulence modelling alwaysincorporates some uncertainty, and the thermodynamicas well as chemical properties of the bled gas are prob-able sources of errors. In order to become a valuabledesign tool, CFD has to show reliability and robustness.Otherwise, only limited types of cases can be simulatedand the whole potential of CFD is not utilized.

In this paper we present results of Navier-Stokes sim-ulations for a supersonic projectile with base bleed. Thefree-stream Mach number is fixed to 1.2 and the angleof attack is 5°. The same projectile was studied at zeroangle-of-attack by Kaurinkoski4 in varying flow condi-tions. In this work, a slight modification to the widelyused low-Reynolds-number k — e model of Chien5 is pro-posed in order to avoid unphysical anomalies near thestagnation point.

The Navier-Stokes solver used for the simulations isFINFLO. The code has been developed at Helsinki Uni-versity of Technology in the Laboratory of Aerodynam-ics. It is based on the finite volume method and is cap-able of handling 3-D multiblock grids. The solutionmethod is implicit pseudo-time integration with a mul-tigrid acceleration of convergence. In the simulationsRoe's flux-difference splitting is applied for the inviscidfluxes and the thin-layer approximation is used for thefriction terms. The FINFLO code is described in moredetail in Kaurinkoski et al.6 In this work, the effectsof turbulence are taken into account using Chien's low-Reynolds-number k — e model with a slight modification.

The two-equation turbulence models employingBoussinesq's approximation are known to overpredictthe production of turbulent kinetic energy near stagna-tion points. This may lead to dramatically overestimatedturbulence levels near the stagnation point, which in turnmay spoil the whole solution further downstream. In or-der to avoid problems related to this excessive produc-tion, Menter7 suggested an upper limit based on experi-mental results. A modification to that limit is suggestedin this paper. The current modification is an extension tothe upper limit for the production of turbulent kinetic en-ergy. The idea is to utilize a stronger limit outside shearlayers and the original limit inside shear layers. The locallevel of vorticity is employed for determining whetherwe reside in a shear layer or not.

In comparison with results obtained employing the"standard" version of Chien's k — e model, the currentresults are superior in quality near the stagnation region.The base area, however, still remains an area of uncer-tainty. The proposed limit for turbulence production in-creases the robustness of the solver, since developmentof turbulence is slightly damped.

2 Numerical Method

2.1 Governing Equations in DifferentialForm

The Reynolds-averaged Navier-Stokes equations, theequations for the kinetic energy k and dissipation e ofturbulence, and the scalar transport equation can be writ-ten in the following differential form

dU 3(F-FV) 3(G-GV) d(H-Hv)dt+ dx + dy + dz

where U = (p, pu, pv, pw, E, pk, pe,viscid fluxes are

pu

F =

T. The in-

G =

\/

pU2 + p + |/9J

pvupwu

pukpuepu4>pv

puvpV2 + p + |/9/

pwvK — J— Yl 1 — f ) ] c

1 J-J 1 jJ 1^ q fSIV

pvkpve

ife

/

\

c

J

(2)

TT __ pw

\

pwpuwpvw

pwkpwepW(f> J

Here, p is the density, V = ui + vj + wkis the velocity, pis the pressure, and 0 is a scalar variable describing, e.g.,the concentration of a species or the mass fraction of aspecies. More specifically, in this work p$ is the densityof the bled gas.

The total internal energy E is defined as

E = pe + p u2 + v2 + w2

pk (3)

where e is the specific internal energy, and k is the tur-bulent kinetic energy (u"u" + v"v" + w"w")/2. The



viscous fluxes are defined by

Fv =

xy

UTXX + VTxy + WTXZ - qx

fj,c(de/dx)

Gv =

J\

yy

UTxy + VTyy + WTyz ~ I

uk(dk/dy)

\

Hv =VTyz + WTZZ - qz

Hk(dk/dz)

\

where the viscous stress tensor is

\du< dui 1 ,„.. _ mi"-!!11 (^ll P"n "j \~>)j

For the Reynolds stresses we use Boussinesq's approx-imation

dui . dui 1 ,„

where p,T is a turbulent viscosity coefficient. In the mo-mentum and energy equations, the kinetic energy con-tribution has been connected with pressure and appearsin the convective fluxes. The viscous terms contain alaminar and a turbulent part. Similarly, the heat flux iswritten as

(7)

where T is the temperature, and qo is the energy flux dueto diffusion of mass.8 The pressure is calculated from anequation of state p = p ( p , e ) , which, for a caloricallyperfect gas, is written

p=(l-l)(E-p (8)

where 7 is the ratio of specific heats cp/cv. The dif-fusion coefficients of the turbulence quantities and the

scalar quantity are approximated asMT ,. _ .. , Mr .. (9)

where <?k, <rf and a^ are the appropriate Schmidt num-bers and fj,T is the turbulent viscosity of the fluid determ-ined with any turbulence model.

The source term Q has non-zero components for theequations for turbulence and possibly for the scalar equa-tion.

2.2 l\irbulence ModellingIn this study, the solution is extended to the wall insteadof using a wall-function approach. Near the wall, thelow-Reynolds-number model proposed by Chien5 is ad-opted. The source terms for k and e in Chien's model aregiven as

Q= -2yn

where yn is the normal distance from the wall, and y+ isdefined by

V = Vn- = yn- = yn

The production of turbulent kinetic energy is modelledusing Eq. (6) as

(12)

-&In the k — e model the turbulent viscosity is calculatedfrom

Mr = c^ (13)

The equations for k and c contain empirical coefficients.These are given by

ci = 1.44 CTfc = 1.0c2 = 1.92(1-0.CM = 0.09(1-

where the turbulence Reynolds number is defined as

^T = ̂ °5)

Chien proposed slightly different forms for ci and c2.Since the computations performed for the flat-plateboundary layer9 appeared to be insensitive to the modi-fications, the formulas above were based on the mostcommonly used coefficients ci = 1.44 and c2 = 1.92.



2.2.1 Treatment of the Production of Turbulence

A common problem with turbulence models employ-ing Boussinesq's approximation is the inability to accur-ately account for streamline curvature effects. Therefore,production of turbulent kinetic energy is easily over-predicted with Eq.(12) in regions of curved flow.

In order to avoid unphysical growth of the turbulentviscosity HT, e.g., near stagnation points, Menter7 sug-gested a limit for the production of turbulent kinetic en-ergy P as

P = min(P,20/9e) (16)

According to the tests conducted,7 the maximum of theratio P/pe inside shear layers is about two, and thereforethis limit should not affect the well-behaving regions ofthe flow field. Only the problems encountered near thestagnation point should disappear. In addition, this lim-iter has some effect on the solution of shock waves.

We have previously employed Eq.(16) successfully,but in the present simulation this limit was not strongenough. The turbulence production was overpredictedin front of the stagnation point, and as a consequencethe whole solution near the stagnation point was spoiled.Quite evidently, unphysical turbulence production waspredicted also for the inviscid flow field outside theboundary layer on the nose.

In order to differentiate between shear-layer regionsand inviscid regions, we monitor the level of vorticity.A crude estimate for the level of vorticity encounteredinside shear layers is based on laminar boundary-layerflow solution. The boundary-layer thickness is given by

8 = 5.3Lref/-\ (17)and an estimate for the level of vorticity inside the bound-ary layer is

o ~ r> °° — r1 °°v °° ns^5're/ ~ ^n~~£~~ — ̂ iJ r o r——— \i°)0 0.6Lref

where .fteoo = Poo Voo Lref /Voo ar>d C"n is a modelparameter set to 0.03 in the present simulations. Out-side shear layers the limit for production would be 2pcwhereas inside shear layers the original value of 20pe isutilized. For a smooth transition between the two regionsa hyperbolic tangent function is employed as

Pmax = < 2 + 18 tanh|V x V\

n.•ref(19)

Here the exponent nw = 4 is also a selected model para-meter. Later on in this document we will refer to Eq.(19)as the Vorticity-Based Production Limit (VBPL).

The selected model parameters were tested with a ba-sic flat-plate flow, a few 2-D simulations with airfoils and

with the present projectile simulations. The other solu-tions remained unchanged, whereas the projectile sim-ulation was favourably changed. For different types offlows, the model parameters may need some fine-tuning.

This limit may have an effect on the location ofthe transition point, but in the performed 2-D tests, nochange was observed. It is therefore our belief that thefinal steady-state solution in cases where turbulence pro-duction does not cause problems, is not changed, but theiteration time history is altered.

2.3 Finite-Volume Form and Discretizationof the Inviscid Fluxes

In the present solution, a finite-volume technique is ap-plied. The flow equations have an integral form

f (20)-^ / UdV + I F(U}-dS = I QdVV S V

for an arbitrary fixed region V with a boundary S. Per-forming the integrations for a computational cell i yields

(21)dt

faces

where S is the area of the cell face, and the sum is takenover the faces of the computational cell. The flux foreach face is defined by

F = nx(F - ny(G - Gv) + nz(H - Hv) (22)

Here F, FV,G,GV, H and Hv are the fluxes defined byEqs (2) through (4) in the x-, y- and z -directions respect-ively.

The inviscid parts of the fluxes are evaluated withRoe's method.10 The flux is calculated as

F = T~1F(TU) (23)

where T is a rotation matrix which transforms the de-pendent variables to a local coordinate system normal tothe cell surface. In this way, only the Cartesian form Fof the flux is needed. This is calculated from

K

k=l(24)

where Ul and Ur are the solution vectors evaluated onthe left and right sides of the cell surface, r^ the righteigenvector of the Jacobian matrix A = dF/dU —.RAT?"1, the corresponding eigenvalue is \(k\ anda^ is the corresponding characteristic variable obtainedfrom R^AU, where AU = Ur -Ul. A MUSCL-type approach has been adopted for the evaluation ofUl and Ur. In the evaluation of Ul and Ur, primitiveflow variables (p, u, v, w, p), and conservative turbu-lent variables (pk, pe) are utilized.



2.4 Calculation of the Viscous Fluxes andthe Source Term

The viscous fluxes are evaluated using a thin-layer ap-proximation, which is applied in the curvilinear coordin-ate system. In the computer code, the thin-layer modelcan be activated in any coordinate direction. For the de-rivatives in the production term of turbulent kinetic en-ergy (12), however, the thin-layer model is not applied.Instead, the derivatives are calculated exactly.

The possible wall corrections of the turbulent viscos-ity, as well as those of the source terms, are calculatedseparately in the i-, j- and fc-directions. As a result, thesource term may contain several wall correction terms,and the wall-damping of turbulent viscosity is a productof the different wall-damping terms in different coordin-ate directions if several walls are present.

2.5 Boundary ConditionsAt the free-stream boundary, the values of the depend-ent variables are kept as constants. However, in regionswhere the free-stream velocity is directed out from thecomputational domain, the boundary values are extrapol-ated. In the flow field, k and e are limited from below totheir free-stream values. In the calculation of the inviscidfluxes at the solid boundary, the flux-difference splittingis not used. Since the convective speed is equal to zeroon the solid surfaces, the only contribution to the invis-cid surface fluxes arises from the pressure terms in themomentum equations. A second-order extrapolation isapplied for the evaluation of the wall pressure as

3 1Pw = oPl - Til (25)

where the subscript w refers to conditions on the wall,and 1 and 2 refer to the centre of the first and second cellfrom the surface, respectively. A similar formula is usedfor the diffusion coefficients on the wall.

The viscous fluxes on the solid surfaces are obtainedby setting u = v = w = Qon the wall. The centralexpression of the viscous terms is replaced by a second-order one-sided formula.

dyn(26)

where dw is the thickness of the first cell on the surface.The boundary condition for the energy equation can be

determined in two ways: either the wall temperature isset to a predefined temperature, or the wall is assumed tobe adiabatic. In this work, the latter method is employed.The viscous fluxes of k and e, as well as fa, are also setto zero at the wall. In this way there is no need to specifythe surface values of the turbulence quantities.

2.6 Specification of the Inlet BoundaryConditions

The base-bleed boundary is an inlet-type boundary con-dition, since there is flow into the computational domain.However, the boundary condition has to be carefully set,based on the given constraints and the local flow-fieldconditions.

In this study, we specify the mass flow rate m, totalenthalpy H and static pressure of the inlet. In addition,the turbulence level Tu and the turbulent viscosity /J.Tare specified for the k — e model. The turbulent viscosityis needed only for the specification of the dissipation ofturbulent kinetic energy pe at the inlet. In the presentcase, the local velocity is used as the reference velocityfor turbulence level. Naturally, the mass fractions of thespecies are also specified.

For a subsonic inlet, the static pressure p is extrapol-ated from the flow field, whereas in a supersonic inlet, allthe conditions should be set from the inlet side. In prac-tice, the inlet pressure is limited from below with a sonicpressure based on the given m and H. For more details,see Kaurinkoski.11

2.7 Solution AlgorithmThe discretized equations are integrated in pseudo-timeapplying the DDADI factorization.12 This is based onthe approximate factorization and on the splitting of theJacobians of the flux terms. The resulting implicit stageconsists of a backward and forward sweep in every co-ordinate direction. The sweeps are based on a first-orderupwind differencing. In addition, the linearization of thesource term is factored out of the spatial sweeps. Theboundary conditions are treated explicitly, and a spatiallyvarying time step is utilized. The implicit stage can bewritten after factorization as follows

(27)

where / is an identity matrix, e?r. k and d^~- k are first-order spatial difference operators in the i-, j- and k-directions, A, B and C are the corresponding Jacobianmatrices, D = dQ/dU, and Ri is the right-hand side ofEq. (21). The Jacobians are calculated as

A± = * ± kI)R-i (28)



where A1*1 are diagonal matrices containing the positiveand negative eigenvalues, and A; is a factor to ensure thestability of the viscous term. The idea of the diagonallydominant factorization is to put as much weight on thediagonal as possible. In the i-direction, as well as in thej- and fc-directions, the tridiagonal equation set resultingfrom Eq. (27) is replaced by two bidiagonal sweeps anda matrix multiplication.

In order to simplify the linearization of the source termQ, the wall-reflection terms involving wall distances arenot linearized. Also, for stability reasons, only negativesource terms are linearized in the Jacobian matrix D. Inorder to account for the positive source terms (Q+), thefollowing trick is applied

dQ+3U

Q (29)

In this way, the maximum change of U caused by Q islimited to |AC/max|. The actual limit may be evaluatedin many ways. Currently, the contributions of P are in-cluded in a very approximate fashion: The maximumchange of pk is limited with min(|0.1E - pk\, p k ) .For the production term in the e equation, the max-imum change of pe is set to (pe/pk)A(pk)max. A fur-ther simplification is to utilize the trick suggested byVandromme13

-pe = -c. (Pk)2(30)

Another simplification is to ignore the off-diagonalterms in D, and the final form of D becomes

\P\_

du o0

(p*/pk)\P\ (31)

In order to accelerate convergence, a multigrid methodis employed. The multigrid cycling employs a V cycleand is based on the method by Jameson.14 The details ofthe implementation are found in Ref.15

2.8 Modelling of the High-TemperatureEffects and the Mixture Properties

The thermodynamic properties of a mixture of gases canbe determined by analyzing the thermodynamics of thecomponents of the mixture. Each component in turn isa thermally perfect gas and the difficulty of the wholeproblem is divided into smaller ones. For each species i,we can write

Pi = (32)

= cVi dT(33)(34)(35)

where pi is the partial pressure and Ri is the specific gasconstant of species i. For a mixture of n species we have

p =

e =

M =

(36)

(37)

(38)

(39)

where fa = p i / p is the mass fraction of species i. Fora calorically perfect gas, the specific heat capacity cv isconstant. In a thermally perfect gas, however, vibrationis excited and cv is a function of temperature T.

The approach selected in this work is to express cvi :sas algebraically simple expressions and accept the pres-ence of inaccuracies. Knacke16 gives an interpolationequation for Cp = Mcp with coefficients A, B, C and£>as

CP = A + BIO" DIQ-°TZ (40)

where the second and third term account for vibrationalenergy and the fourth term covers the contribution ofelectronic energy. With Eq.(40) the equation of state isclosed and the remaining problem is to solve for T witha given e and p. The details are shown in Ref.4

2.8.1 Transport Properties of a Mixture

The transport properties of a mixture are determined em-ploying Sutherland's formula for viscosity and thermalconductivity for each species, and the mixture propertiesare obtained with Wilke's rule17

_Pmix — (41)

»=i £where

+

lMi ~ N ~ p

(42)

(43)



In Eq. (43) Xt and Ni are the mole fraction and the num-ber of particles respectively of species i in a given sys-tem. An exactly similar formula holds for thermal con-ductivity. Each occurrence of ̂ is simply replaced by

3 Computational Grids

Fig. 1: The grid around the dome-based artillery shell.

All of the presented results are for a 155 mm-diametersupersonic dome-based long-range projectile. A four-block axially symmetric grid with 144 x 64 x 12 cellsin the axial, near-normal and circumferential directions,respectively, in each block is employed for modelling thegeometry of the projectile. The block division was neces-sary only for parallelization purposes.

In total there are 48 cells in the radial direction on thebase surface. The inlet hole was modelled with an 8 x 12patch in each block on the base surface. The y+ valuein the centre of the first cell above the surface was keptbelow 1. The grid around the projectile is seen in Fig. 1.The diameter of the base-bleed hole is 0.0476 m.

4 Computed Test CaseOne flow case is studied: Ma^ = 1.2, Pe^ = 1400 000and a = 5°. The mass flow rate is tin = 0.018kg/s,which is usually expressed with the non-dimensionalquantity / = 0.009 defined by

m(44)

where m is the mass flow rate in kg/s, p^ and Vo-, arethe free-stream density and velocity respectively, and St,is the area of the shell base. The total temperature ofthe bled gas was set to 2 370 K. The specified turbulencelevel Tu and the ratio of turbulent and laminar viscosityA*r/M at the jet inlet were constants, 5% and 50, respect-ively.

The free-stream air is treated as a calorically perfectgas and the composition as well as the model parametersfor the bled gas are the same as those employed in Ref.4

5 Computational ResultsFirst of all, a standard procedure in our simulations is tocalculate an initial guess for the simulations on a coarsergrid employing the multigrid routines. Therefore simu-lations are usually initiated from the third grid level withthe number of cells reduced to 1/64 of the finest grid.Results from level 3 are used as an initial condition forlevel 2, and likewise the solution on level 2 is an ini-tial state for level 1. In this way, the number of itera-tion cycles required on the finest grid level is reduced by20 - 50%, and sometimes a fatal initial transient can beavoided by the coarse-grid initialization. We should alsonote that, for stability reasons and in order to maintaincomparability, the multigrid scheme was not employedto accelerate convergence in any of these computations.

The iteration histories showed some interesting prop-erties. Without VBPL (Vorticity-Based ProductionLimit), the problems at the nose of the body were presenton all grid levels. The convergence properties showed noclear indications of the problems encountered near thestagnation point, but a closer look at the solution revealedsome anomalies from both the turbulence-related quant-ities and the basic flow variables. A very strange sliceof increased density and reduce temperature appeared onthe surface of the nose, and the pk as well as (J,T distri-butions also had strange peaks in the same area.

If the fine-grid simulations were started from an ini-tial guess obtained by employing VBPL, the solution re-mained free of these problems, and a steady solution wasreached in 8 000 cycles. Therefore, it seems the problemis actually only a matter of transient abnormal behaviourof the flow solution. In order to investigate this, compu-tation without VBPL on the second grid level was con-tinued from the essentially converged steady solution.After another 3 000 iteration cycles, the anomalies hadvanished almost completely. With VBPL, however, theproblems never appeared at all.

Another test was to completely omit the coarse-gridsimulations and start the simulations on the finest gridlevel with initially free stream flow. Regardless of thetreatment of turbulence production, however, the itera-tion would fail because of numerical problems behindthe base.

It is not completely certain whether production of tur-bulence is the true origin of these problems, but at leastone solution is to employ Eq.(19), i.e. VBPL. Someproperties of the solution indicate, that the problemswere partially related to the properties of the grid.



Q.CL

300.0-

200.0 -

100.0-

0.0-(

y~-

;'• ''

/

I

I

\\

X'S

\

/=5, K=13pc, No VBPLPE, VBPLP, NoyBPLP7VBPL

^ \'•>. ':

—— i ——— | ————— | ————— | —————3.0000 0.0025 0.0050 0.0075 0.0 00

y[m]

IQ.

0.0100

0.0075

0.0050 -

0.0025 - -

0.00000.0000 0.0025 0.0050

V H

0.0075 0.0100

Fig. 2: Distribution of P and pe (left) and pk (right) in the symmetry plane upstream of the stagnation point after 100 iteration cycles.Here y is distance to the wall.

In order to isolate the effects of VBPL, we computedthe case with and without VBPL starting from the samecoarse-grid initial guess generated without VBPL. Fig. 2shows the distribution of P and pe (left) as a functionof the distance to the wall along a grid-line pointing up-stream in the symmetry plane (Block=4, i = 5, k = 13)after 100 iteration cycles on the finest grid level. Note,that the indices given here refer to grid corner-points andnot cell centre-points. Without VBPL, the solution hasa strong second peak for P at y m 4.1 mm, which isfiltered out with the modified limiter. The pe distributionin the same figure shows slight changes by the VBPL.The main thing here is, that without VBPL there is ahuge imbalance between P and pe, and thus locally muchmore pk is generated than dissipated. For steady state,this would require an equal and opposite imbalance forthe fluxes (convection and diffusion). There is no reasonfor such an imbalance of fluxes on the free-stream sideof the stagnation point.

Fig. 2 shows what has happened to pk (right) after100 iteration cycles. Clearly, the distribution with VBPLis smoother and in that sense seems more reasonable.After all, there should not be too much upstream influ-ence from the stagnation region.

The uncertainty about using VBPL is what happensto the basic flow solution. In order to demonstrate thatthe basic features are not changed at all, Fig.3 shows theCf distribution on the upper surface of the shell in thesymmetry plane after 2 000 iteration cycles. The curvescorresponding to VBPL and no-VBPL are indistinguish-able, which confirms that no dramatic changes in the ba-sic solution have arisen.

Continuing the simulations to 2 000 cycles smoothedout the worst peaks in P and pk, but it is interesting to seein Fig.4 how pe (left) is not affected at all, while the pk

5.00

3.75-

2.50-

1.25-

0.00-

-1.25 -

-2.50 - --0.05 0.10 0.25 0.40 0.55 0.70 0.85

X[m]

Fig. 3: The skin-friction coefficient c/ distributions after 2 000 it-eration cycles.

level (right) without VBPL is approximately 15% higherthan with VBPL. Yet, the P distributions are now quant-itatively similar.

6 ConclusionsWe have solved numerically the supersonic flow past abase-bleed projectile. We have developed a hybrid upperlimit for the production of turbulent kinetic energy in or-der to avoid problems related to stagnation point regions.

The results show that production of turbulent kineticenergy is changed and consequently, the level of pk islower than without the Vorticity-Based Production Limit(VBPL). The final solution is not changed too much, buttransient peaks in pk and P distributions are avoided.

If used with care, the VBPL can solve some of the



wQ.

200.0 -

150.0 -

100.0 -

50.0-

i / \ iM ih Il l l l \ \

P i «^--r---^TY ~*~*^ '̂ \^

1=5, K=13pe, No VBPL

- - - - - JOE, VBPLP, No VBPLP, VBPL

0.0 ———————— : ———————— | ———————— | ———————— |0.0000 0.0025 0.0050 0.0075 0.0100

y[m]

0.0100

0.00000.00 0.04

Fig. 4: Distribution of P and pe (left) and pk (right) in the symmetry plane upstream of the stagnation point after 2 000 iteration cycles.Here y is distance to the wall.

problems in estimating P with the Boussinesq's approx-imations. However, in areas where excessive productionof turbulence is not a problem the solution remains un-changed.

7 AcknowledgementsThis research project has been funded by MATINE, theScientific Commitee of National Defence. Their supportfor this research is gratefully acknowledged.

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3 Sahu, J., Nietubicz, C. J., and Steger, J. L., "Navier-Stokes Computations of Projectile Base Flow withand without Mass Injection," AIAA Journal, Vol. 23,Sep 1985, pp. 1348-1355.

4 Kaurinkoski, P., "Computation of the Flow ofThermally Perfect Gas Past a Supersonic Projectilewith Base Bleed," in 7996 AIAA Atmospheric FlightMechanics Conference, (San Diego, California),pp. 725-734, Jul 1996. AIAA Paper 96-3451-CP.

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Documents

[American Institute of Aeronautics and Astronautics 22nd Atmospheric Flight Mechanics Conference - New Orleans,LA,U.S.A. (11 August 1997 - 13 August 1997)] 22nd Atmospheric Flight