American Institute of Aeronautics and Astronautics1

UNSTRUCTURED MULTIGRID SIMULATIONS OF TURBULENT LAUNCH VEHICLE FLOWSINCLUDING A PROPULSIVE JET

Daniel Strauss*

Instituto Tecnológico deAeronáuticaCTA/ITA/IEAA

12228-900 – São José dos Campos– SP – BRAZILdaniel@tecsis.com.br

João Luiz F. Azevedo†

Instituto deAeronáutica eEspaçoCTA/IAE/ASE-N

12228-904 – São José dos Campos– SP – BRAZILazevedo@iae.cta.br

Abstract

A study of the flowfield around axisymmetriclaunch vehicles in different flight conditions isperformed in this work. Particularly, the VLS secondstage configuration is analyzed considering the casewith and without a propulsive jet in the vehicle base.The study is performed using a cell centered finitevolume formulation on unstructured grids. Differentspatial discretization schemes are compared,including a centered and an upwind scheme. Theupwind scheme is a second-order version of the Liouflux vector splitting scheme. Turbulence effects areaccounted for using two one-equation turbulenceclosure models, namely the Baldwin and Barth andthe Spalart and Allmaras models. An agglomerationmultigrid algorithm is used to accelerate theconverge to steady-state of the numerical solutions.The numerical results obtained are compared withexperimental data as well as with previous structuredgrid simulations.

Introduction

In the context of the development ofcomputational tools for the simulation of launchvehicle flows, previous work[1,2,3,4,5] by the authorshas treated the flowfield around the first Braziliansatellite launcher, the VLS. Viscous turbulent flowsin the forebody region[1,2], as well as in the afterbodyregion[3,4] of the vehicle, were considered using astructured grid approach with centered implicitschemes and an axisymmetric formulation. Moreover,an unstructured grid approach was used in order tosimulate inviscid flows over the VLS forebody in

* Project Engineer at TECSIS Tecnologia e Sistemas Avançados

Ltda. AIAA member†Head, AerodynamicsSubdivision. Senior Member AIAA.

Copyright © 2002 by D. Strauss and J.L.F. Azevedo. Published by

the American Institute of Aeronautics and Astronautics, Inc., with

permission.

Ref. [5]. This unstructured grid capability[5] consistedof an axisymmetric, finite volume, cell centeredsolver, in which a centered and an upwind schemewhere implemented and convergence accelerationwas obtained by an agglomeration multigridprocedure. The present work is, then, an extension ofthe existing capability towards the treatment ofviscous turbulent flows over the VLS, including theforebody region and the afterbody region with andwithout a propulsive jet. Hence, some of the casesstudied in Refs. [3,4,5] are analyzed again with thisnew implementation.

Unstructured grids allow for a much largerflexibility in terms of control of computational meshrefinement than a structured grid approach. They canprovide better resolution of flow features with lowercomputational cost through the use of adaptivemeshes. Moreover, the use of an axisymmetricformulation is indicated due to the axial symmetry ofthe VLS second stage and to the fact that the vehicleis designed to fly at low angles of attack. In addition,the axisymmetric formulation represents a three-dimensional flow but it has a computational costequivalent to a two-dimensional simulation. Sincehigh-speed flows including shock waves are amongthe flows of interest, an upwind spatial discretizationscheme is employed in order to increase to solutionaccuracy. The upwind spatial discretization schemeimplemented is a second-order accurate version ofLiou’s AUSM+ scheme[6,7]. Moreover, the use of one-equation instead of algebraic turbulence closuremodels is necessary for the adequate solution of theflows of interest, as observed in previous work[3,4].The two turbulence closure models used are theBaldwin and Barth[8] and the Spalart and Allmaras[9]

models.The time-stepping method used to advance the

solution of the governing equations in time in thepresent work is a fully explicit, 2nd-order accurate, 5-stage Runge-Kutta scheme[10]. As the problems ofinterest are steady state problems, a variable timestepping option with constant CFL number has been

20th AIAA Applied Aerodynamics Conference24-26 June 2002, St. Louis, Missouri

AIAA 2002-2720

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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implemented in order to accelerate convergence.Experimental data and results from a structured meshsolver[3,4] are used for comparisons with the presentnumerical simulations. Comparisons between theresults obtained with the centered and the upwindspatial discretization schemes and between the resultswith the two turbulence closure models implementedare also presented. Solutions over the forebody andover the afterbody regions were obtained.Furthermore, the afterbody region was treatedconsidering caseswith and without a propulsive jet.

Governing Equations

The governing equations for the aerodynamicflows of interest are the azimuthal-invariant Navier-Stokes equations in cylindrical coordinates, whichcan be written in integral form for the case of no bodyrotation as

( ) 0=+−+∂∂

∫ ∫∫S VV

dVrdzrdrdVt

HFEQ . (1)

In this expression, r is the radial coordinate, z is theaxial coordinate, E = E(i)- E(v) and F = F(i)- F(v).Moreover, Q is the vector of conserved properties,E(i) and F(i) are the inviscid flux vectors, E(v) and F(v)

are the viscous flux vectors and H is theaxisymmetric source term. The definition of theseterms is

( )

( )

( )

( )

,,

,

0

0

0

,

2

2

++

=

+

+=

−=

=

r

r

rz

r

i

z

rz

z

z

i

r

z

upe

pu

uu

u

upe

uu

pu

u

rp

e

u

u

ρρρ

ρρ

ρ

ρρ

ρ

FE

HQ

(2)

( )

( ) .

0

,

0

−+

=

−+

=

∞

∞

rrrrzzr

rr

zrv

zrzrzzz

zr

zzv

quu

Re

M

quu

Re

M

ττττ

ττττ

F

E

In the previous expressions, ρ is the density, ur

and uz are the velocity components, p is the staticpressure, e is the total energy, τij are the componentsof the viscous stress tensor and qi are the heat fluxcomponents. Moreover, a suitable nondimensionali-zation of the governing equations is assumed in order

to write Eq. (2). In the present case, the choice ofreference state proposed in Ref. [11] is adopted. Inthis equation, M∞ is the freestream Mach number andtheReynolds number isdefined in itsusual form as

∞

∞∞=µ

ρ qlRe . (3)

where l is the reference length, ρ∞ is the freestreamdensity, q∞ is the magnitude of the freestreamvelocity vector, and µ∞ is the molecular viscositycoefficient at the freestream temperature.

With the use of a cell centered based finite volumescheme, the discrete conserved variable vector isdefined as an average over the cell of the continuousproperties. Hence, for the i-th volume, the discreteproperty vector is

∫=iVi

i dVV

QQ1

. (4)

The definition of the discrete vector Qi can be used torewrite Eq. (1), resulting in the equation that has to benumerically implemented, that is

( ) ( ) 0=+−+∂∂

∫ iiS

ii VrdzrdrVt

i

HFEQ , (5)

where ( )ii QHH = .

Turbulence Modeling

The correct account for the viscous effects in thepresent case involves the implementation of anappropriate turbulence closure model. The turbulenceclosure models implemented in this work were theBaldwin and Barth[8] and the Spalart and Allmaras[9]

one-equation models. These models attempt to avoidthe need to compute algebraic length scales, withouthaving to resort to more complex two-equation, orκ - ε - type models. The models were implemented inthe present code precisely as described in Baldwinand Barth original work[8] and in Spalart andAllmaras original work[9] for the case with no laminarregions.

The extension of both models for compressibleflows was obtained simply by multiplying thekinematic turbulent viscosity coefficient by the localdensity, as indicated in Refs. [8,9]. Moreover, theturbulence model equation is solved separately fromthe other governing equations in a loosely coupledfashion.

The Baldwin and Barth model partial differentialequation, observing the nondimensionalizationadopted, is

American Institute of Aeronautics and Astronautics3

( ) ( ) +−= PRcfcDt

RDT

T ~~

12 2 ννεε

( ) ( ) ( )

∇⋅∇−∇

++ ∞

TtTt

l RRRe

M ~1~2 ννσ

νσνν

εε. (6)

In the previous equation ( ) ( ) ( )∇⋅+∂∂= VtDt

Dis

the material derivative, which contains the timederivative and convective terms, the first term on theright hand side of the equation is the production termand the terms between the brackets are the diffusion

terms. This equation is solved for the variable ( )TR~ν

and the eddy viscosity is calculated as

( )Tt RDDc~

21 νρµ µ= , (7)

where the damping functions D1 and D2 are designedto allow the model to be used in the near-wall regionand are given by

( )++−−= 11 /exp1 AD η , (8)

( )++−−= 22 /exp1 AD η , (9)

with

νρτ

η d

M

Re

w

w

∞

+ = . (10)

Here, d is the distance to the wall and

2

3

2

∂∂

−

∂

∂+

∂∂

=k

kt

i

j

j

it x

u

x

u

x

uP νν . (11)

In the previous equation, the usual Einstein notationand sum convention are used. The ui and xi symbolsdenote the components of the velocity vector andcoordinate system directions, respectively.Furthermore, ρµν tt = is the equivalent of a

kinematic eddy viscosity.The Spalart and Allmaras model partial

differential equation, observing thenondimensionalization adopted, is

( )( )[ +∇+∇+= ∞ ννν

σνν ~~.

1~~~1 Re

MSc

Dt

Db

( ) ]

−∇+2

12

2

~~

dfcc wwb

νν . (12)

Again, ( )Dt

Dis the material derivative and the first

term on the right hand side of the equation is theproduction term. Moreover, the last term of theequation is the destruction term and the other terms

are the diffusion terms. This equation is solved for thevariable ν~ and the eddy viscosity is calculated as

1~

vt fνρµ = , (13)

where the function fv1 is a damping function used toproperly treat the buffer layer and viscous sublayer.This function is given by

31

3

3

1

v

vc

f+

=χ

χ, (14)

where

ννχ~

≡ . (15)

The production term is based on the vorticity and itcan be written as

222

~~vf

dkRe

MSS

ν∞+≡ , (16)

where

ijijS ΩΩ== ω , (17)

and the rotation tensor Ωij is given by

i

j

j

iij x

u

x

u

∂

∂−

∂∂

=Ω . (18)

Spatial Discretization Schemes

The purpose of the spatial discretization scheme isto numerically evaluate the surface integral in Eq. (5).This approximation of the integral is different for theinviscid and for the viscous flux terms. In this work,while the viscous terms are always treated using acentered scheme, the inviscid terms are treated usinga centered or an upwind scheme. The approximationof the integral of the inviscid flux vectors is calledconvective operator, which is defined as

( ) ( ) ( )( )( ) ( )[ ]∑

∫

=∆−∆=

=−≅

n

kikik

iikikik

iik

S

iii

zrrr

rdzrdri

1FE

FEQC

. (19)

Similarly, the approximation of the integral of theviscous flux vectors is called viscous operator, and isdefined as

( ) ( ) ( )( ) =−≅ ∫iS

vvi rdzrdr FEQV

( ) ( )[ ]∑=

−=n

kikik

vikikik

vik zrrr

1∆∆ FE . (20)

American Institute of Aeronautics and Astronautics4

For the centered scheme, the convective operator canbe written as

( ) ( )( )[∑=

∆=n

kikikik

ii rr

1QEQC

( )( ) ]ikikiki zr ∆− QF , (21)

where ikQ is the arithmetic average of the conserved

properties in the cells which share the ik interface,and n is the number of edges that form the i-th controlvolume. The terms ikr∆ , ikz∆ and ikr are calculated

as

12 nnik rrr −=∆ , 12 nnik zzz −=∆ ,

221 nn

ikrr

r+

= , (22)

where ( )11, nn rz and ( )22 , nn rz are the vertices

which define the interface between cells i and k. Aschematic representation of the nodes and edgenomenclature ispresented in Figure1.

z

r

∆ zik

∆ rik

lik

nik

nz ik

nr ik

n1

n2

Figure1: Nodesand edgenomenclature.

The spatial discretization procedure presented inEq. (21) is equivalent to a central difference scheme.Therefore, artificial dissipation terms must be addedin order to control nonlinear instabilities[12]. In thepresent case, the artificial dissipation operator isformed as a blend of undivided Laplacian andbiharmonic operators[10,13]. Moreover, the scalingterms of the artificial dissipation model wereimplemented following two approaches: Jameson andMavriplis’ work[12] and Mavriplis’ work[13].

The Liou scheme implementation follows thework in Refs. [5,14] for both the first and secondorder version of the scheme. The second orderscheme is obtained by following exactly the sameformulation of the first order version, except that theleft and right states are obtained by a MUSCLextrapolation of primitive variables[15]. In Ref. [14], aone-dimensional stencil normal to the control volumeedge is constructed in order to obtain the extrapolatedinterface properties. In this work, cell averagedproperty gradients are computed and used to calculatethe extrapolated properties[16], which is a differentapproach from that used in Ref. [14]. With the

property extrapolation, the state variables arerepresented as piecewise linear within each cell,instead of piecewise constant. Hence, in order toavoid oscillations in the solution due to the propertyextrapolation, it is necessary to use a limiter. Theminmod limiter was implemented, following thework in Ref. [16]. Moreover, the limiter value is“ frozen” after a certain number of iterations in orderto obtain a better convergence rate, as described inRef. [17].

A centered discretization scheme was also usedfor the viscousoperator, such that it can bewritten as

( ) ( )∑=

∆

∂∂=

n

kikik

ikiik

vi rr

x1,

QQEQV

( )

∆

∂∂− ikik

ikiik

v zrx

QQF , . (23)

In the previous expression, Qik is, again, thearithmetic average of the conserved properties in the

cells which share the ik interface andikix

∂∂Q

is the

arithmetic average of the conserved propertiesderivatives in the cells which share the ik interface.These derivatives are computed in each volumeconsidering that the discrete derivative in a givenvolume is the average on the volume of thederivative, and then using Green’s theorem totransform the computation of the derivative into thecomputation of a line integral.

Multigrid Implementation

The agglomeration multigrid[18] procedure wasselected among the different options of multigridimplementation on unstructured meshes[10,13,18]. Thisapproach does not present the mesh limitations of anested grid approach[10] nor the complexity of thecalculation of the mesh intersections on a non-nestedgrid approach[10,13]. Moreover, it provides a fullyautomatic generation of the coarse meshes in such away that only the fine mesh has to be provided asinput data. Furthermore, the agglomeration multigridapproach maintains the convergence accelerationcharacteristics of theother multigrid procedures.

Since an agglomeration multigrid strategy wasselected, the coarse meshes for the multigridprocedure are generated by agglomerating orgrouping fine mesh volumes to form one coarse meshvolume. A “seed” volume is chosen in the fine meshand, then, all the volumes that have a node or an edgein common with this “seed” volume are grouped andthey form the coarse mesh volume. Another “seed”volume is selected and the agglomeration procedurecontinues grouping all the fine mesh volumes. It

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should be noted that during the agglomerationprocedure only the volumes that have not beenalready agglomerated may be grouped to form acoarse mesh volume. This is a necessary condition inorder to guarantee that there is no volumeoverlapping in thecoarse mesh.

Better coarse mesh quality can be obtained if theselection of the “seed” volumes is not random.Therefore, a list containing all the fine mesh volumesis generated prior to the agglomeration procedure. Inthis work, the list is formed such that the first entriesare the volumes next to a boundary. This approach isvery simple, easy to implement and adds very littlecomputational cost. Although it does not necessarilyprovide the best agglomeration of the interiorvolumes, it results in good quality coarse meshvolumesclose to theboundaries.

As the spatial discretization scheme used in thiswork is linear, a simplification can be made in thecoarse meshes. This simplification consists ineliminating the nodes that belong to only twovolumes. The justification for this procedure comesfrom the fact that the flux passing between the twovolumes is the same whether the boundary separatingthe two volumes is discretized by one or many edges,provided that the discretization scheme is linear.Therefore, a significant amount of storage space canbe saved by performing this mesh simplification.Figure 2 presents an example of such nodeelimination, where the darker lines represent theoriginal boundary separating two coarse volumes andthe dashed line represents the boundary edge after thenodeelimination.

Figure 2: Mesh simplification by the node eliminationprocedure.

Although the mesh simplification previouslydescribed can reduce the total storage space requiredby the code, it brings a complication related to theconnectivity of the nodes in the mesh. As some nodesin the mesh are not used, it is necessary to ensure thatthe remaining nodes are properly counterclockwiseoriented in order to have the normal vectors of eachedge correctly pointing outwards. This isaccomplished using the node orientation in the finemesh volumes to direct the ordering of the nodes inthecoarse mesh volumes.

The agglomeration procedure can be summarized,then, in three steps. The first step consists in definingthe list of volumes of the fine mesh. In the secondstep, the fine mesh volumes are agglomerated to form

the coarse mesh volumes, following the list generatedin the first step. During this step, the meshsimplification described is adopted, and only thenodes that belong to three or more coarse meshvolumes are stored. The third and final step is theverification of the node orientation in each volume ofthe coarse mesh, and the correction of the orientationwhere it is needed. The actual implementation of thisagglomeration procedure was designed to require theminimum amount of storage possible. Therefore, theonly extra information that has to be stored in eachmesh, besides the usual information associated withthe solution procedure, is the number of the coarsemesh volume that contains each of the fine meshvolumes.

In the multigrid procedure, the restriction operatortransfers a variable from a fine mesh to a coarsemesh. The operator used in the present work for theconserved property restriction is the volume weightedaverage. Therefore, the restricted conservedproperties of a coarse mesh volume are equal to thesum of the conserved properties of all the fine meshcells that form this coarse mesh volume, weighted bytheir volumes. On the other hand, the restriction ofthe residuals is accomplished by simple addition ofthe fine mesh residuals. Hence, the residual of acoarse mesh volume is equal to the sum of theresiduals of all the fine mesh volumes that arecontained by this coarse mesh volume. The restrictionoperator for the residuals is different from therestriction operator for the conserved propertiesbecause the residuals can be interpreted as lineintegrals in finite volume schemes. Consequently, asthe residuals of the fine mesh are summed, theinterior edges contribution will cancel each other,leaving only the contribution of the edges that formthecoarse mesh volume.

The prolongation operator, in opposition to therestriction operator, transfers a variable from a coarsemesh to a fine mesh. As usual with typical multigridimplementation, only the conserved propertycorrections have to be prolonged. Hence, only oneprolongation operator has to be defined. In this work,an averaging operator was used to obtain thecorrections in the fine grids. The averaging operatorconsists in, for each edge of the fine mesh,arithmetically averaging the corrections of the coarsemesh volumes corresponding to the two volumes thatcontain the edge. For each volume, then, theseaveraged corrections are summed and the result isdivided by the number of edges of the volume. Thisoperator is very easy to implement, and it has theadvantage of being able to transfer a lineardistribution with less error than a direct injectionprolongation operator.

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Results and Discussion

The computational mesh used in the forebodycases is presented in Figure 3 and the correspondingagglomerated meshes used for the multigridprocedure are also shown in this figure. Although thismesh is composed of quadrilateral cells and could beused in a structured mesh approach, the authorsemphasize that this mesh was treated in a fullyunstructured fashion. Actually, the computationalgrids used in the present work are the same used inRefs. [3,4,5] in order to allow a direct comparisonbetween the present results and the results obtained inthese references.

Figure 3: Computational mesh for the VLS forebodyand three agglomerated meshes obtained from thismesh.

A comparison of pressure coefficient distributionsover the VLS for the cases using the centered schemeand both the Baldwin and Barth[8] and the Spalart andAllmaras[9] turbulence closure models is presented inFigure 4. This case considers M∞ = 0.90 and viscousturbulent flow with Re = 20 × 106. In this figure, theBaldwin and Barth model results are indicated by B-B, while the Spalart and Allmaras model results areindicated by S-A. Moreover, for the simulationsshown in Figure 4, Jameson and Mavriplis’ artificial

dissipation model is used in all cases except for one,in which Mavriplis’ model is used. The pressurecoefficient distribution for the Euler case presented inRef. [5] is also shown in Figure 4. The results forboth the Baldwin and Barth and Spalart and Allmarasturbulence closure models are very similar and theyare also similar to the Euler results. Nevertheless, theviscous turbulent cases do not present theoverexpansion and overcompression shown by theEuler results at the forebody cylinder-boattailintersection region and at the boattail-afterbodycylinder intersection region, respectively. Moreover,the results using Mavriplis’ artificial dissipationmodel instead of Jameson and Mavriplis’ model arein better agreement with the experimental results thantheother resultspresented in Figure4.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3

x/L

Cp

Euler

S-A

B-B

B-B Mavriplis

Experimental

Figure 4: Comparison of pressure coefficientdistributions with the centered scheme(M∞ = 0.90 and Re = 20 × 106). Jameson andMavriplis’ artificial dissipation model is usedwherever nothing is stated.

The results obtained with Liou’s spatialdiscretization scheme and both the Baldwin and Barthand the Spalart and Allmaras models are presented inFigure 5. The corresponding results for the inviscidcase, presented in Ref. [5], as well as the results withthe centered scheme with Mavriplis’ artificialdissipation model, presented in Figure 4, are alsoshown in Figure 5. As observed in the inviscidcases[5], the Liou scheme provides a numericalsolution with better comparison with the experimentaldata than the solution with the centered scheme.However, the solution with the centered scheme andMavriplis’ artificial dissipation model is very close tothe solution with Liou’s scheme. Moreover, similarlyto the observed with the centered scheme, the resultswith the turbulence models do not present theoverexpansion and overcompression observed in theinviscid results.

Another important result obtained for the viscousturbulent case is the turbulent viscosity distributionon the flowfield. Hence, the turbulent viscosity

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contours for the case using Liou’s spatialdiscretization scheme and the Spalart and Allmarasturbulence closure model are presented in Figure 6. Inthis figure, one can observe that meaningful values ofturbulent viscosity are concentrated in the boundary

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3

x/L

Cp

Liou Euler

Liou 2o S-A

Liou 2o B-B

Centered B-B Mavriplis

Experimental

Figure 5: Pressure coefficient distributions withLiou’s second order scheme (M∞ = 0.90 andRe= 20 × 106).

layer close to the body, where the shear stresses arerelevant. Moreover, the boundary layer thickens fromthe forebody to the afterbody. This behavior isexactly the expected one for this case. A result similarto that of Figure 6 was obtained with the Baldwin andBarth turbulence closure model, although someoscillations in the turbulent viscosity were observedat theedgeof theboundary layer.

Figure 6: Turbulent viscosity contours for the casewith Liou’s second order scheme and the Spalart andAllmaras model (M∞ = 0.90 and Re= 20 × 106).

The mesh used to obtain the last results presentedfor the VLS was the mesh presented in Figure 3. Asalready mentioned, this mesh was generated from astructured mesh. This structured mesh was, then, usedto obtain numerical results for the same case with M∞= 0.90 and Re = 20 × 106, but using the structuredgrid code developed in Refs. [3,4,19]. A comparisonbetween the structured and the unstructured results isshown in Figure 7. Both results were obtained usingthe centered spatial discretization scheme and theBaldwin and Barth turbulence closure model. Theresult corresponding to the unstructured mesh shownin Figure 7 is the same result presented in Figure 4

for the case using Mavriplis’ artificial dissipationmodel. From the comparison presented in Figure 7,one can see that the results using both meshapproaches are very similar. This was expected since,besides the mesh topology, a similar formulation wasused in both cases. Moreover, the unstructured meshresults are in a slightly better agreement with theexperimental data.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3

x/L

Cp

Unstructured

Strutuctured

Experimental

Figure 7: Comparison of pressure coefficientdistributions with previous structured results(M∞ = 0.90 and Re= 20 × 106).

The afterbody of the VLS second stage is treatedinitially as a simple base, that is, without a propulsivejet at the vehicle base. This case considers afreestream Mach number of 0.5 and a Reynoldsnumber of 20 million. Moreover, the centered schemewith Mavriplis’ artificial dissipation model is usedand the turbulent effects are accounted for by theBaldwin and Barth turbulence closure model. Themesh used in this case is presented in Figure 8 and itis an extension of the forebody mesh show in Figure3 that also includes the afterbody region. This meshhas a grid point concentration near the base region inorder to capture the boundary layer close to this wall.The pressure contours in the afterbody region for thiscase are presented in Figure 9. These contours arevery similar to the ones obtained in Refs. [3,19] withthe structured grid solver for the same case and usingthesame formulation.

Figure 8: Computational mesh including the VLSafterbody region.

The results shown in Figure 9 reproduce thegeneral trend of the expected behavior of the flow.The flow detaches from the body at the base edge and

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forms a recirculation bubble in the near-wake. Thisseparated region ends in a stagnation pointdownstream of the base. The flowfield described canbe seen in Figure 10, in which a streamline plot of thenear-wake with the separated region is shown. A

Figure 9: Pressure contours in the VLS afterbodyregion without a propulsive jet (M∞ = 0.50 andRe= 20 × 106).

quantitative comparison of the results withexperimental data is presented in Table 1, in terms ofthe axial position of the rear stagnation point. Theposition is measured along the centerline from thevehicle base and is presented nondimensionalized bythe afterbody diameter. The experimental data wasobtained from Ref. [20] for cylindrical afterbodies,and the errors obtained in the numerical results are ofapproximately 13%. These errors are not completelyunexpected as the authors of Ref. [20] indicate thatthe near-wake development is dependent upon theflow conditions approaching the separation point.Therefore, as the forebody in the present work isdifferent from the one used in the experiments of Ref.[20], the boundary layer in the proximity of theseparation point is also different, resulting in thedifferences in the near-wake. Moreover, the resultsobtained with the unstructured calculation werealmost coincident with the ones obtained with thestructured grid approach.

Figure 10: Streamlines in the afterbody region for thecase without a propulsive jet (M∞ = 0.50 andRe= 20 × 106).

The cases including a base with a propulsive jetconsidered in this work are always of underexpandedsonic or supersonic jets, and a sketch of the expectedjet structure is presented in Figure 11. Expansionwaves propagate downstream from the jet exit,resulting in a jet area increase. These expansionwaves reflect at the centerline as compression wavesand cause a jet area reduction further downstream.

The area increase is consistent with the jetacceleration, as the jet is sonic or supersonic.Moreover, the area reduction causes a jetdeceleration, which is achieved by a normal shockwave, also called Mach disk. A system of obliqueshock waves also forms in order to adjust the flowdirection next to the jet boundaries. After the Machdisk, the flow is subsonic and therefore accelerates asa result of the area reduction, until it becomes sonicagain. After that, depending on the jet to freestreampressure ratio, thecomplete jet structure repeats itself.

Table 1: Position of rear stagnation point behind thebody base. Computations consider M∞ = 0.5 and Re =20 × 106 (D is the afterbody diameter and x ismeasured from the body base along the downstreamcenterline).

CaseDescription Position (x/D)

Present unstructured results 1.39

Previous struct. results (Ref. [19]) 1.37

Experimental data (Ref. [20]) 1.21

Figure 11: Sketch of the expected jet structure in theimmediate vicinity of the jet exit.

The parameters indicated in Figure 11 are used tocompare the numerical and the experimental results.These parameters are: diameter of the jet exit (d),initial inclination of the jet boundary (δ), crossflowdiameter of the normal shock wave or Mach disk (S),position of the normal shock wave (l), and length ofthe first periodic structure (w). However, it should benoted that this is the structure of a free jet. Thepresence of a high speed freestream flow can changethis structure as the freestream flow can inhibit the jetexpansion. Since most of the experimental resultsfound on the literature are for jets discharging in stillair, some tests with very low freestream Machnumbers were performed for comparison. However,the real interest of this work is on supersonicfreestream conditions, thus cases with theseconditions will also be shown. The computationalmesh used for the jet cases is presented in Figure 12,which shows only the afterbody region. This mesh issimilar to the one shown in Figure 8, except that ithas a more uniform point distribution, with less

American Institute of Aeronautics and Astronautics9

refinement in thenear-wake in thedirection normal tothebase.

In order to simulate free jet conditions, afreestream Mach number equal to zero should beused. However, the nondimensionalization adopted

Figure 12: Computational mesh in the afterbodyregion for the cases including a propulsive jet on thevehicle base.

yields terms which are divided by the freestreamMach number and, therefore, M∞ cannot be equal tozero to avoid numerical problems. Hence, M∞ = 0.01was chosen for the free jet cases. The Mach numbercontours for a case, with M∞ = 0.01, Mjet = 1.0,pjet = 10 p∞, Tjet = T∞ is shown in Figure 13. In thiscase, as well as in all cases considering a propulsivejet, the Liou spatial discretization scheme was used.Moreover, the Baldwin and Barth turbulence closuremodel was used to properly capture the turbulenceeffects. An excellent qualitative agreement has beenobtained between the numerical solution and theexpected jet structure shown in Figure 11. A verygood quantitative agreement was also obtained, asone can see in Table 2. The experimental resultsshown in this table were obtained from Ref. [21].Discrepancies between the present computationalresults and the experimental data are usually less than13% for all parameters analyzed.

Figure 13: Mach contours in the base region for thefree jet case (M∞ = 0.01, Mjet = 1.0, pjet = 10 p∞,Tjet = T∞ and Re= 20×106).

The freestream conditions considered for the firstcase run including a propulsive jet on the vehicle baseand a non-still freestream were Mach number of 0.5

and Reynolds number of 20 million. The jetconditions adopted in this case where Mach numberof 1.0, static pressure and temperature three and twotimes, respectively, the corresponding freestreamquantities. The Mach number contours in the baseregion for this case are presented in Figure 14. In thisfigure a jet structure very similar to the expected one,presented in Figure 11, can be observed, showing avery good qualitative comparison with experimentalresults. Furthermore, this case was also run in Refs.[4,19] using a structured grid approach and the resultsobtained in the present work were very similar to theones shown in those references.

Table 2: Comparison of the jet parameters withexperimental results for a jet discharging in still air(Mjet = 1.0, pjet = 10 p∞ and Tjet = T∞ ).

CaseDescription w / d l / d S/ d δ (°)

Present unstr. results 3.8 3.1 1.22 35.7

Previous struct. results(Refs. [4,19])

3.7 3.1 1.64 44.6

Exper. data (Ref. [21]) 3.6 2.9 1.40 40.2

Figure 14: Mach number contours in the base regionfor the case including a propulsive jet (M∞ = 0.5,Mjet = 1.0, pjet = 3 p∞, Tjet = 2 T∞ and Re= 20×106).

Flow conditions which are representative of theactual VLS flight regimes were also simulated. In thistest, the corresponding freestream and jet propertiesresult in more complex flowfields and represent amore severe simulation case. The flow propertieswhich are representative for the VLS second stageflight conditions are M∞ = 4.9, Mjet = 4.0, pjet = 112.0p∞, Tjet = 4.6 T∞ and Re = 20 × 106. The Mach numbercontours for this case are shown in Figure 15. Onecan observe that the jet expansion is very large in thiscase, and the plume structure seems to be similar towhat can be observed, for instance, in Figure 14.However the computational domain does not extendfar enough downstream in order to capture the normalshock and all the downstream portions of the plumestructure. Nevertheless, from the point of view of theaccuracy of the solutions here presented, this factshould not cause any problems since the flowfield iscompletely supersonic throughout the entire exitplane. Therefore, whatever flow structures aresupposed to appear downstream of the computational

American Institute of Aeronautics and Astronautics10

exit plane cannot influence the present computationaldomain. Furthermore, as one could expect, the veryhigh jet to freestream static pressure ratio results in astrong expansion downstream of the jet exit, withlocal Mach numbers within the jet core reachingvalues as high as 12. The full range of Mach numberswithin the jet is not represented in the labels of Figure15 in order to allow the reader to see some otherfeatures of the flow. Moreover, the rapid jetexpansion at its exit station causes the formation ofan oblique shock wave in the external flow right atthe afterbody edge. This feature can also be seen inFigure15.

Figure 15: Mach number contours for the VLSsecond stage flight conditions (M∞ = 4.9, Mjet = 4.0,pjet = 112.0 p∞, Tjet = 4.6 T∞ and Re= 20 × 106).

Concluding Remarks

The transonic flowfield around the VLS forebodywas studied using a viscous turbulent formulation.Comparisons between the numerical results obtainedand the experimental data showed a very goodagreement in terms of pressure coefficientdistribution as well as in terms of the flow features.The results with the turbulence models obtained inthe present work do not present the overexpansionand overcompression observed in the inviscid resultsof previous work[5]. Moreover, the solutions obtainedwith the Liou scheme were in better agreement withthe experimental results than the ones obtained withthe centered scheme. Furthermore, the solutions withthe centered scheme and Mavriplis’ artificialdissipation model were closer to the solution with theLiou scheme than the solution with the centeredscheme and Jameson and Mavriplis’ artificialdissipation model.

Afterbody simulations considered cases with andwithout a propulsive base jet. The results without apropulsive jet of the present work were comparedwith previous structured grid results obtained by theauthors and with experimental data for generalcylindrical afterbodies at zero angle of attack. Thepresent unstructured grid results were essentiallyequal to the structured grid results. Moreover, thecomputational results reproduced the experimentaldata in terms of the location of the rear stagnationpoint. The discrepancies between computational and

experimental values for this parameter were at mostof 13 %.

For the cases with a propulsive jet, since themajority of the data in the literature is concerned withfree jets, the simulations initially performedconsidered jets discharging in a still atmosphere. Dueto specific details of code implementation, this wassimulated here considering an external freestreamwith M∞ = 0.01. An extremely good qualitativeagreement was obtained with the availableexperimental results for free jets and the quantitativecomparison of results also indicated good agreementwith the available data. Discrepancies between thepresent computational results and the experimentaldata were less than 13 % for all parameters analyzed.Results with a non-zero freestream Mach number alsoindicated a very good qualitative agreement with thetype of behavior that should be expected in this case.Although there was no experimental data to comparethepresent results, as in the free jet cases, thesolutionstructures were essentially the same ones observed inthe free jet cases. Furthermore, the results obtained,which also included realistic VLS second stage flightconditions, were also compared with the structuredgrid cases of previous work by the authors and, again,very good agreement wasobtained with these results.

Acknowledgments

The authors gratefully acknowledge the partialsupport of Conselho Nacional de DesenvolvimentoCientífico e Tecnológico, CNPq, under the IntegratedProject Research Grant No. 522.413/96-0.

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