AD-A283 688

ARmy REsEARCH LAOpwRTOY •

Numerical Computations ofSupersonic Base Flow With

Special Emphasis onTurbulence Modeling

Jubaraj Sahu

ARL-TR-438 June 1994

LECTE .

DTIC- QUALITY U1SPh•CTYD 5

11ý(94-26903IiHI!ll 94 8 23 1 13

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4. TITLE AND SUBTITLE S. FUNDING NUMBERS

Numerical CO.DpuftumO of Supearsonc Base Flow With Special Emphtasis on PR: 1L161102AH43Turbulance Modeling 61102A-O0-001 M6

6. AUTHOR(S)

Mai S"b

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) B. PERFORMING ORGANIZATIONREPORT NUMBER

U.S. Army Research LaboratoryATTN: A1S~tL-,Wr-PBAberdeen Proving Ground,, MD 21005-506

9. SPONSORING/ MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/IMONITORINGAGENCY REPORT NUMBER

U.S. Army Research LaboratoryATTN. AMSRL-O-AP-L ARL-TR-438Aberdeen Proving Ground, MD 21005-5066

11. SUPPLEMENTARY NOTES

Thi repont supersedes BR.-DMR-971, June 1992

12a. DISTRIBUTION I AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Approved for public release; distribution is unlimited.

13. ABSTRACT (Maximum 200 words)

A zomal, implici, time-marvhig Navier-Stakes computational tecinique hm been used to compute the turbulentsuesoi base flow over cylindrical aftesbodies. A critical element of calculating such flows is the turbulence model.aroseddy viscosity turbulence models have been used In die base region flow computations. These models include two

algebraic turbulence models uad a two-equation k-e model. The k-e equations ar developed In a general coordinatesystem and solved using an implicit, algorithm. C~alculation with the k-c model ame extended up to the walL Flow field

comptatonshave been performedl for a cylindrical aftebody at M - 2.46 and at angle of attack a = 0. Thw results arecopred to the experimental datm for fth sane conditions and the same configuration. Details of the mean flow field as

well Be the turbulence quantities have been presented. In addition. fth computed base pressure distribution bas beencmudwidh the experiment. In general,, the k-e turbulence model performs bette In the near wake than the Algebraic

models and predicts the base pressure much better.

14. SUBJECT TERMS 15. NUMBER OF PAGES34

bas flow, hue pmesse, turbulence models, wake, supersonic flow 16. PRICE CODE

17. SECURITY CLASSIFICATION 115. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE I OF ABSTRACT

Ma~Assml)E I NOaSSmmD UNaLAss1FI uLNSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Psretcbed by ANSI Std Z39.16298-102

INTERN(lOALLY LEff 3LAN..

TABLE OF CONTS

LIST OF FI URES .............................................. v

1. INTRODUClION ............................................... I

2. GOVERNING EQUATIONS AND SOLUtON TECHNIQUE ................ 2

2.1 Gove m l g Equation .3..... ... .. ............... 32.2 Nm erical Te que .......................................... S2.3 Composite oridc me ........................................ 62.4 Twbmience Modelia .......................................... 72.4.1 Badwin-Lam xModel ...................................... 72.4.2 chow Mod .............................................. 82.4.3 Two-Equaton k-e Model ..................................... 9

3. MODELING GEOMETRY AND EXPERIMENT ......................... 12

4. RESULTS ..................................................... 12

S. CONCLUDING REMARKS ........................................ 24

6. REERENCES.................................................. 25

LIST OF SYMBOLS ............................................. 27

DISTRIBUTION LIST ............................................ 29

Sm • • m i | i I

ImsmNnoALLY LEFr BLANK.

iv

LISr OF FRIURES

SPem

1. Schmatc DIApi of Supe ic Bmu FPokw ............................ 2

2. Af.etody Menuem t Locatm .................................... 13

3. Base Region Co p a Grid ..................................... 14

4. Comnputed Wnue Coatows in de Buse Region. M. - 2.46, a = 0,and k-e Model ................................................ 15

5a. Computed Mach Cotmour M. = 2.46. a = 0, and k-e Model ................. 15

5b. Expumak tal S idhlr Photograp .................................... 16

6. Velocity Vectou in the Bus RekM , M- 2.46, a - O, and k-e Model .......... 16

7. SUcamwis Velocity (u) Prolm ML 2.46, a = 0 ........................ 17

S. Nonnal Velocity (w) Pmmefo , M. - 2.46, a = 0 .......................... 1s

9. Tuibulent Kinetic Energy Profiles, M.. = 2.46, a = 0 ....................... 20

10. TudtM Dissipation Rte Profilesm M. - 2.46, a = 0 ...................... 21

11. Tuibulem Shear Sum PmfUles, M1L = 2.46, a = 0 ......................... 22

12. Bae Pasw DistiutitdonM= 2.46, a 0 ............................ 23

Aoae•'- on ?orUTIt]S GR~A&il

DTIC TAB 3Unannounced 0Jigstification

ByD1•-ribution/

Availability Codesjv'3ýil and/or

Dist special

"" • • I II I I I l I I II H

IMIMMONALLY LEFl KALAK

==mo

1. INTRODUCTION

One of the important parameters in shell design Is the tota aerodynamic drag. The toital drag consists

of three components: the pressure drag or the wave drag (excluding the bas). the viscous drag, and the

base drag. The base drag component Is a large part of th total drag and can be as high as 50% ormore

of the total drag. Of these three components of drag, the most difficult one to predict is the base drag

because it depends on the pressure acting on the base. Therefore, it is necessary to predict the base

pressur as accurately as possile

The ability to compute the base region flow field for projectile configurations using Navier-Stokes

computational techniques has been developed over te past few years (Salm. Nietubicz, and Steger 1985;

Sahu 1986, 1987). Recently, improved numerical predictions (Sahu and Steger 1988; Sahu 1990; Sahu

and Nietubicz 1990) have been obtained using the Cray-2 supercomptner ad a more advanced zonal

upwind flux-split algorithm. This zonal scheme preserves the base comer and allows better modeling of

the base region flow. These studies have included base flows for different base geometries. This

capability is very important for determining aerodynamic coefficient data, including the total aerodynamic

drag. As indicated earlier, a number of base flow calculations have been made, and base drag and total

drag have been predicted with reasonable accuracy. However, because available data are lacking, te

predictive capabilities have not been assessed with detailed base pressure distributions, mean flow velocity

components, and turbulence quantities. This is especially true of base flow for axisymmetrical bodies at

transonic and supersonic speeds. Recently, experimental measurements (Herrin and Dutton 1991) have

been made in te base region for supersonic flow over a cylindrical aflerbody. The data include base

pressure distribution (along the base), mean flow, and turbulence quantities.

Figure I is a schematic diagram showing the important features of supersonic base flow. The

approaching supersonic turbulent boundary layer separates at te base comer, and the free shear layer

region is formed in the wake. The flow expands at the base corner and is followed by the recompression

shock downstream from the base that realigns the flow. The flow then redevelops in the trailirg wake.

A low pressure region is formed immediately downstream from the base, which is characterized by a low

speed recirculating flow region. Interaction between this recirctlating region and the inviscid external

flow occurs tdrough the free shear mixing region. This is the region where turbulence plays an important

role.

1

---- Ex ion WSs. v

Fgure I. Shmtc Dia of Suher c Base iow.

The basic configuration used in this study is a cylindrical aflerbody. As mentioned earlier, a simple

composite grid scheme has been used for accurate modeling of the base corner. Numerical flow field

computations have been performed at M,.= 2.46 and at 00 angle of attack. Three ubuilence models (two

algebraic models and a tw-quto model) ar used in the base flow region. All the computaions have

been performed on the Cray-XMP supercomputer. Details of the flow field such as Mach number

contours and base pressure distributions are preseted. Computed base pressure distiutitons are compared

with available experimental data for the same conditions and the same configuration. The algebraic

turbulence models predict a large change in the base pressure distribution over the base. The two-.equatio

k-e model predicts a rather small change in base presure along the base and compares very well with the

experimentally measured base pressure distribution.

2. GOVERNING EQUATIONS AND SOLUTION TEHNQUE

The complete set of time-dpnet Reynold-averaged, thin layer Navier-Stokes equatons is solved

numerically to obtain a solution to this problem. The numerical tecluique used is an implicit, fiute

difference scheme. Although tie-epndn calculations are made, the tranient flow is not of primary

interes at the present time. The steady flow, which is the desired resut, is obtained in a time asymptotic

fasegonn

2

S. . . . .. . . . . . . . .

I

I. Ilbslda h compleft ont of xhree-dimenulcnal (3-D), time-dependent- gauetuizud

gemnetry Raynalds-avraged. thin Mayer. Navier-Stokes equations for general spatial coordiniates i. 1, and

Scan be written as (PWulam and Steger 1982)

+ 8 t + ,.d+ a~? -Re-1D;9(1)

in which

4= t(x. y, z. f) - Imogiudina coordinate; Ti = i1(x~, y, z. t) - circumfemntial coordinate

= ;(xr y, z, t) - nearly norm~4 coordinate; tr - time

and

p~p U

p u ~PuU + p

4-. pv ,-. pvU +47Ppw 7 pwU + t

Le J (e+p)u -4,I

PV PW

pUV + 11J PUW + ýJ

pwV + q,p pWW + ,P

I(e + p) V -1p (e +p)W - ýp (2)

3

and i which

0

2 2ex + *4 .(C;zat + L;Vt + 1W

;2 + .+I~ .)t +; +

C2+ + E(U2jg + V2+ *)

Pr(y- 1)]

L 3 j(3)

In Equation 1, the thin layer approximation is used and the viscous terms involving velocity gradients

in both the longitudinal and cirnumfefential directions are neglected. The viscous trms am retained in

the normal direction, t, and are collected into the vector S. These viscous terms are used everywhere.

However, in the wake or the base region, similar viscous terms are also added in the stream-wise direction.

For this computation, the diffusion coefficients p and K contain molecular ad turbulent pans. The

tubudent contributions are supplied through either algebraic or a two-equation k-e turbulence model.

The velocities in the 1. Y1, and t coordinate directions can be written

V=W + M + VI;+ ,

which represent the contmvaiant velocity components.

4

Tme Caۥt" velocity (xp1mm P ( . w) an reaned is ft depdem vaftabls and areI wirespect to a. (the u stre am speed of ound). The local pessure i dermined

using the relation

p - (Y - l)[e - O.p(U 2 + V2 + W2 )] (4)

in which y is the ratio of specific heats. Density, pis referenced to p. and the total energy, e, to p.a.

The transport coefficiens ar also nondimnuionalized with respect to the corresponding free stream

variables. Thus, the Prandt number that appears in S is defined as Pr = cpg..az,.. In differencing these

equations, it is oftm advantageous to difference about a known base solution denoted by subscript 0 as

8. d- 0)+ 84 (t -to) + 814 (a - do) + 8 A()-.0) - Re 181 (9 - 90)

- - - - ÷ + (5)

in which 8 indicates a general difference operator, and a is the differential operator. If the base state can

be propedy chosen, the differenced quantities can have smaller and smoother variation and therefore less

of a differencing error (Pulliam and Steger 1982).

2.2 Numerical Techniouc. The implicit, approximately factored scheme for the thin layer Navier-

Stokes equations using central differencing in the -q and ý directions and upwinding in • is written in the

following form

[I+ h4 (A +)n + 8 - hRe-1 CJ -1A? J - Dui]

X [I h64'(A-)" + h68ni D - ]A 0

_,&At(8~() b +1: + -(() 11- t. I + y(N-0)

+ 6 R(A" - A.) - Re -3C " -S .)S - D- 0.) (6)

5

in which h = At or (As)2 and the free stream base solution is used. Here, 8 is typically a three point

second order accurate central difference operator, A is a midpoint operator used with the viscous terms,A

and the operators 5•b and 84f are backward and forward three-point difference operators. The flux F has

been cigensplit and the matrices A. B.,an M result from local linearization of the fluxes about the

previous time level. Here J denotes the Jacobian of the coordinate transformation. Dissipation operators

D, and Di are used in the central space differencing directions. The smoothing terms used in the present

study are of the form:

D.I, s(At)J4[&23P(B)P 4 +E41 P( '] i11

Din - (&t)J-[i8p(B)p8 + 2.53 4 8p(B)] I3 j

in which

1(1 + I82)P

and p(B) is the true spectral radius of B. The idea here is that the fourth difference will be tuned near

shocks (e.g., as P gets large, the weight on the fourth difference drops down while the second difference

tines up).

For simplicity, most of the boundary conditions have been inposed explicitly (Sahu 1987). An

adiabatic wall boundary condition is used on the body surface and the no-slip boundary condition is used

at the wall. The pressure at the wall is calculated by solving a combined momentum equation. Free

stream boundary conditions are used at the in-flow boundary as well as at the outer boundary. A

symmetry boundary condition is imposed at the circumferential edges of the grid while a simple

extrapolation is used at the downstream boundary. A combination of symmetry and extrapolation

boundary condition is used at the center line (axis). Since the free stream flow is supersonic, a

nonteflection boundary condition is used at the outer boundary. The flow field is initially set to free

stream conditions everywhere and then advance in time until a steady state solution is obtained.

2.3 Composite Grid Scheme. In the present work, a simple composite grid scheme (Sahu 1990) has

been used in which a large single grid is split into a number of smaller grids so that computations can be

6

perboued oneachgid sepaty. Thmeseidsuetheavab ommM oyoneMid a me. Mie

remaining gidls are stored on am external disk storage device such a the solid state disk device (SSD) of

the Cray X-MP/48 computer. The Cray-2 has a large in-core memory to fit the large single grid.

However, for accurate geometric modeling of complex projectile configuration, which include blunt noses,

sharp comers, and base cavities, it is also desirable to split the large data base into a few smatler zones

on the Cray-2 as well. The use of a composite grid scheme requires special care in storing and fetching

the interface boundary data i.e., the oommuncat among the various zones). In the present scheme,

there is a one to out mapping of the grid points at the interface boundaries. hius, no interpolations ar

required. Details of the data storage, data transfer, and other pertinent information such as metric anddifferencing accuracy at the interfaces are given in the work of Sahu and Steger (1987) and Sahu (1988).

2.4 Tubumlence Modeling. For the base flow calculations, three turbulence models have been used.

Two of these are algebraic eddy viscosity models (Baldwin-Lomax model and Chow model). The third

one is a two-equation k-e turbulence model which is also an eddy viscosity model.

2.4.1 Baldwin-Lomax Model. This model is the one developed by Baldwin and Lomax (1978). It

is a two-layer model in which an eddy viscosity is calculated for a inner and an outer gion, The in

regio follows the Prandtl-Van Driest formulation. In both the inner and outer formulations, the

distibufion of vorticity is used to determine the length scales, thereby avoiding the necessity of finding

the outer edge of the boundary layer. For the inner region,

(p,),. pl2l I (7)

in which

I - icy[ I -exp(-y*1A *)I

Y (p,,Y)/P 1 - .h - Fr, IP,)

and I tol is the absolute magnitude of vorticity. Mwt eddy viscosity for the outer region is given by

(% r - KCC•pFwU&,F k (Y) (8)

7

in which Fw,*g = yx F , or C.* ym u thIFmm, the nmaller of the two values. The quantities y.

and F.. am determined from the funcon F(y) - yl l[I - e.W (-y+ / A+)], in which F,. is the

maximum value of F(y) and y. is the value of y at which it occurs. The function F*U is the

Klebanoff intermittency factor. The quantity udf is the difference between the maximum And minimum

total velocity in the profile and, for boundary layers, the minimum is zero.

The outer formulation can be used in wakes as well as in attached and separated boundary layers. For

free shear layer flow mgions or wakes, the Van Driest damping term [exp(-y+ / A+)] is neglected. Also,

for the base or wake region, the distance y is measured from the center line of symmetry. It is necessary

to specify the following cotants: A+ = 26, Cp = 1.6, CH'* = 0.3. C* = 0.25, i= 0.4. andK= 0.0168.

This type of simple model is generally inadequate for complex flows containing flow separation regions

such as base flow.

2.4.2 Oiow Model. Another algebraic model that has been used in some of our base flow

computations is that of Chow (1985). This model is intended to be used in the base or wake region only.

It is based on the simple exchange-coefficient concept. The turbulent eddy viscosity coefficient is usually

given by

P, = - Xe (9)

in which x is the distance measured from the origin of the mixing region (ie., the base), u, is the velocity

at the edge of the mixing region, and a is the spread rate parameter. It is known that a assumes a value

of 12 for incompressible flow and it increases slightly with Mach number.

a = 12 + 2.76 Me

in which Me is given byI + -7 ]

M.2 2 2

8

I-. A,-

au Pb AP. Is doe avempg baen IPmuu Th 7aipivAlux velocity; a doe edge of te xmixing ugim cm

be as ceutained from

uO• o -1( +-Y,--- / (I + -,-m

As a first app onlmaton, die average value of pt Is asuned to be sune a all poits Afr a constan z

location After reattadhment ttiftlence shoud decay. Since doehumers in tie base flow calculatons

is to obtain die correc base piesomu it is asmamed that die eddy viscosity level at die reattachment stays

the same at other locat downumm. For bae flow withjet, similar algebmaic relatiw - used

for the jet shear layer. This model as well as Baldwin-.omax (1978) model am algebraic nKdels and

depend only on local infonnation. Thl two-equation model contains less empiricism and allows dte flow

history to be taken into accoun.

2.4.3 Two-Eauazlon k-e Model. The two-equation Ubbulence model used her is bChien's (1982) Ic-e

model which is similar to that of Jones and Launder (1972). In this model two vampart equations anr

solved for the two variables, k (tuulent kinetic energy) and e (urbulent dissipation rate).

Dkaa [(± + ' ak 1+ aui(a, a M,

- pe - 2p k (10)Y2

p De + ~~~~~YP a ,I lu u

D 77- 1 ' ) 77J 1) Xj(J- -

" sC2 P -, 2p I e -•(11)

9

Fie. yn Is the distm normal to the surface. The coefficlets In de k and a equatlons me given by

c| I 1.44 C3 - 2 -0 - O k a. 1.0 - a t - 1.3

C2 - 1.92[ 1 -O.3exp(-R; )2 C a 0.09[1 -evp(-O.Oly*)]

in whihR, -k1M~

"7he k-e model employs the eddy viscosity concpt mad relaes th uub*n eddy viscosity to k and a by,

ps = cpp(k 2 /e). (12)

Following the some procedure used for te mean flow equ the bubfltnt e field equations cm be

writm in consevatdon fom and then bmmformed into geralized coodilnses (Ssu and Danb• g 1966).

Te resulting axisymmetric am of nunsformed uubune equatons cmn be written as

a 48+ Us+ ad= I a.4' af'(13)

in which

; [ :] 3"J Pau J L pew

1 (1k 7;C 7;TP

t7 +(,+ e L p

E2 YXC,- a P - 2pe-p*e - 2pkL=p"

2p.

10

and P is ft pod m a bm gvnas

2- 2)t + pjq + VC)(u~ 2 2 +)

2v 2w2 2 ) +p~(CzUt + ty+PI(e + 9 + 4(Ucw+ ) + W%.~~ + + )

Trnis congtitutes a low Reynolds number foanulation of the k-e model Calculaions am extded to the

wall itself, and exac values of the dependent variables at the wall ae used as boumdary conulitons.

Chien's model is beer mahemically behaved near the wall and is, turs, used in this study.

7Ue turbulence field equatiorns solved using the Dean and Wanning (1978), Implicit,

approximately factored, fant dee scheme. A convenient soluion algoddmn for these equatiom has

the following sequem:

[(I - AtD n) + At(kB - 6CC x) I Aq4" - RMnS ( 1) (14a)

[I + At(CA" - NNN)] AqIM -,&#I (14b)

q3A÷1 qjA + Aqx (14c)

in which R•S is ft righ•-band side of Equation 13. A and B are Jacobian matrices resulting from ft

local linearizatio of ft flux Iterms and 6t. 17U source tems are treatd implicitly, Whs results in

the Jacobim matrices C, N, and D, which are included in the and 4 operators as show in quin 14.

The Jacobian matrices are given as

CUB~dP".iw: ]' .. [dP ao]L)

2Cp kP -Re 2pr P~ r 1

CI + C2I2 c2 Re-,p , _v

11

in which

+ P

•re operators &and 6 central difference operator The numerical smoothing is bond on an up-wind scheme, and the details ame given in Sahu (19S4).

3. MODEL GEOMETRY AND RIMENT

The compuaionl accuracy of a numerc scheme can be established through comparison with

available expeimental data. The model used in the experiment and in the computational study is don"

in Figure 2. It is an axisymmetrical cylindrical aftetbody. which has a diameter of 63.5 mm. This figure

also shows the suimons where mean and fluctuating velocity components were measured with a LaserDoppler Velocimeter (LOV) system. The same configuration is used in the numerical simulations for a

direct compuison

Experim tal measrm-ents (Herrin and Dutton 1991) for this model have been made at the University

of Illinois supersonic wind tUrnei The model was tested at 0* ang&e of attack, Mach number of 2.46, and

Reynolds number of 5.21 x Id' per meter. In addition to measuring the velocity components at a few

selected longitudinal positions in the wake or baee reglio the base pressure was measured at 19 positions

along the base. Such detailed base press=ur measuremen-s have not been made in the past and are veryhelpful in the code validation process. The velocity profile is also measured at a Msaton upstrem from

the base, which provides the upstream boundary condition for base region flow field calculations.

4. RESULTS

Numerical computations have been made for the cylindrical aflfody at a Mach number 2.46 and at

0 angle of attack The three-plane vesion of the 3D code was run for the 0 angle of attack case. Two

end planes were used Io specify symmetrical boundary conditions in the circumferential plane.

The solution technique require the discretization of the entire flow region of interes into a suitable

computational grid. The grid outer boundary has been placed I diameter away from the surface of the

aftabody. The downstream boundary was placed 10 diamete away from the base. Since the calculations

12

OD'

40

3D

10

0.0 - -- - - -

-10 1 1-20 0 2D 40 60 OD 100 120 140 160

x (m~m)

Figure 2I Aftbody bkm Luu=i

are in fth supersonic regiume the computatIonal outer boundary was placd close to the body uid a nw

reffection boundary condition used at that boundary. Figure 3 shows an expanded view of ft grid in the

base reglon. 11e surface polzM on ft afkerbody and the bose were obtained first These were then used

as inputs for obtaining the full grid using an algbraic grid generaton pro=a. 1he Wai grid is solit too

two zonesowe upstream, fUND the base, mid the odher one in the bowe region or the wake. Th3en grids

consist of 22 x 60 and 95 x 119 grid points, respectively. Figure 3 show the longitudinal grid clustering

near the bane comaer Grid points am also clustered near the aftebody surface to capture the viscous

effects in ft tiubulait boundary layer. These clustered gri poims ame spad out downstream of the base

in the wake to capture the free shear layer region For the V angle of attack cae conalder4 ed, grid wa

rotated *iraifereui-'ally 50 on eiher side of the mldplane. This provided the three plowe needed In the

code to use carnial finite differences in the crcu'ferim 1 Pal direction. In each case, fte solution was

marched from flee stream condditiom everywhere unil the fina converged solution was obtained. The

results are now presented for both mean and turbulence quantities. Comparison of fth computed results

is made with the available experimental dafta (Hlerrin and Dutton 1991).

13

Figure 3. Base Region Computational Grid.

A few qualitative results are presented next. Figure 4 shows the pressure contwr plot for the base

region. The features to observe are the flow expansion at the base comer followed by the recompression

shock downstream from the base (coalescence of contour lines). Figures 5a and 5b show the comparison

of the computed Mach number contours with experimentally obtained Schliren photograph of the base

region flow field. Both the experiment and the computed results show the flow expansion at the base and

the recompression shock downstream from the base. In addition, Figures 5a and Sb show the free shear

layer in the near wake. Although not indicated in Figure 5a, the flow in the near wake is primarily

subsonic. Figure 6 shows the computed velocity vectors in the base region. The recirculatory flow in thenear wake is clearly evident. Flow reattachment occurs at about three base radii downstream frm the

base. The magnitude of the velocity is shown to be quite small in the immediate vicinity of the base. hie

computed results shown in Figures 4, 5, and 6 were obtained using the two-equation k-e model.

Figures 7 and 8 show the velocity components in the stream-wise and normal directions, respectively.

These velocity profiles are taken at four longitudinal positions in the wake or the base region (X/D = 1.26,

1.42, 1.73, and 1.89). The computed velocity profiles obtained using two algebraic turbulence models and

the two-equation k-e model are compared with the experimental data. Figure 7 shows the comparison of

14

Figure 4. Computed Pressure Contours in the Base Region. M.= 2.46. cx=O. and k-e Model.

Figure 5L. Computed Mach Contours. M,= 2.46. ot =0. adk-c Model.

Figure 5b. Eimenimental Schliren Photomj

Figur 6. Velocity Vectoi in the Base Radion, .N 2.46, az 0. and k-c oe

16

X/D - 1.20 X/D - 1.42

1.4 1.4

1.2 1.2

1.0 1.0

rlR 0.8 rlR 0.8

0.6 -0 0.6

0 __ __L-0. Model0.4 ---.. k9.u.uia. 0.4

0.2 . 0.2

0.0 - 0.0-1.0 o.0 1.0 2.0 3.0 -1. 00 1.0 2.0 3.0

u/u,,0 umu0 .

"X/D - 1.73 X/D - 1.89

1.4 1.4

1.2 - 1.2

1.0 , 1.0

r/R 0.8 r/fR 0.8

0.6 0.6

S-e sado. .... 3- model

0.4 -o .

0.2 ,2

0.0 L 0.0-1.0 0.0 1.0 2.0 3.0 -1.0 o.0D 1.0 2.0 3.0

UlU.= UlU..

Fig= 7. Somm-win Vdocitv (u) PMofiles. M. - 246. a - 0.

17

XID - 1.26 X/D - 1.42

1.4 1.4

1.2 1.2 0.0.8

rlR ~ ~ ~ KjMIL 0.0 o.,.. lA 8 E.,.r/R 08r/R o

0.6 , 00.

0.4 0O0.

0.2 0.2

0.0 0.0-0.50 -0.25 8.00 0.25 0.60 -0.50 -0.25 0.10 0.25 0.50

XID 1.73 XID - 1.89

1.4 111.4 !

1.2 1.2

01.0 1.0

0.8 o.8r/R 0oE,!iLe /

0.6 0.6

0.4 0.4

0.2 0.2 0

0.0 0.0-0.5C -0.25 0.00 0.25 0.50 -0.50 -0.26 0.%0 0.25 0.50

w/uoo w/uoe

FRgwm 8. _Nrmal Velocity (w) Profiles. ?& = 2.46. cz =0O.

18

the u (stream-wise) component of velocity. In general, the profiles obtained with the k-e model agree

much better with th experiment in the shear layer regions for XAD = 1.26 and X/D = 1.42. Tle profiles

are rather poorly predicted by both the algebraic models at these two stations. The reattachment point

estimated from the experimental measuremens is about 1.4 base diameters downstream from the base.

The computed value with the k-c model is 1.5. This small disagreement is also seen in the flow

redevelopment region downstream from the reattachment (X/D = 1.73 and 1.89). The algebraic turbulence

models predict the reattachment point better than that predicted by the k-c model. lie velocity profiles

predicted with these models agree fairly well with the experimentally obtained profiles at these two

stations. Chow model predictions are slightly better than those predicted by the Baldwin-Lomax model

in this flow redevelopment region. Figure 8 shows the comparison of the w (vertical) component of the

velocity. This component of velocity is better predicted by the k-e model than the algebraic models both

in the flow recirculation and redevelopment regions. lie profiles by the algebraic models do not agree

well with the experimental data for radial positions greater than half of the base radius.

Some of t turbulence quantities are presented next Figure 9 shows the tubulent kinetic energy

profiles at the same longitudinal positions in the wake. lhe computed k profiles are obtained using Ue

two-equation k-c tubulence model In the recirculation region (X/D = 1.26) and near the reattachment

(X/D = 1.42), the peak observed experimentally (r/R = 0.5) is poorly predicted by the k-e model

(r/R = 0.4). The location and the magnitude of the peak agree somewhat better at X/D = 1.42 than at

X/D = 1.26. The agreement of the computed profiles with the data is good in the flow redevelopment

region (X/D = 1.73 and 1.89). Figure 10 shows the turbulent dissipation rate (e) profiles at the same

positions in the wake. As seen in this figure, e increases from the center line of symmetry with radius

to about 0.4 of the base height where the peaks occur and then drops quickly to very small values at about

r/R = 0.65. The magnitude and the location of the peaks decrease smoothly with increasing axial distances

downstream from the base.

Figure 11 shows the turbulent shear stress profiles in the wake. The computed values obtained by

both the algebraic models and the k-e model are compared with the experimental data. In general, a small

improvement can be observed in the predicted values with the k-e model over the algebraic models.

Discrepancy exists between the experimentally obtained turbulent shear stess and the predicted shear

stresses with all the turbulence models. This is u'ue especially near the peaks at XID = 1.26 and IA2.

lie magnitude of the peak predicted by the k-e model is about the same as predicted by the Baldwin-

Lomax model at these two positions; however, they both underpredict th experimental peak. The Chow

19

X/D - 1.26 XID -1.42

1.4 1.4

1.2 1.2

1.0 - 1.0 0

rlR O.8 0 r/R 0.80.6 0.6 0o

0 00.4 0.4 0

0.2 0 0.2

8 00.a - 0.0 1°0

-00 .8 0.05 0.10 0.&:/u2 -0.05 0.000 0.06 0.10k/u2

X/D - 1.73 XID " 1.89

1.4 1.4

1.2 1.2

1.0 0 o |sjejjj t 1.0 0 Experiment

0.80.r/R r/R 0.8

0.6 0.60 0

0.4 0 0.4

0.2 0.2

0.0 0.0-0.05 0.0 0.05 0.10 -0.06 O.Ow 0.05 0.10

k/u2 k/u2

Fipure 9. Mubuenet Kineic Enemy Profiles. M = 2.46. = 0.

20

K_ • •:•, •••• •• •• • +7. i. '" - '• •' -, • "••• #' ••. • l . j+

1.4

1.2

1.0X/D = 1.42

0.8 X-D - 1.73

"rI X/D = 1.89

0. 6 k-•--

0.4

0.2

0.00.0 0.2 0.4 0.6

Fgu 10. TIMOeSt Dissntion Rate Profiles. ML = 2.46, a =0.

21

X/D 1.26 XID - 1.42

1.4 1.4

1.2 R 1.20I haffo 0 111;Leceo

1.0 1.0 o

r/R 0.8 r/R 0.6 00

0.6 0.6 00 0

0.4 0.4 0

0.2 0.2

0.0 0.0-0.01 %.00 0.01 0.02 0.03 0.04 -o.o0 0o.00 0.01 0.02 0.03 o.04

-(u'v'l/u2 -(u'v-l/u2

X/D " 1.73 XID 1.89

1.4 1.41.2 zit.; .... iPLa.a

.2 .model~h*L 1 .2 ""I-"imol'

1.0 1.0

rIR O.8 /R 0 .8

0.6 0.6

0.4 - . 0.4

,00.2 - OO

0.

0.0 0 0.0-0.01 0.80 0.01 0.02 0.03 0.04 -0.010&.00 0.01 0.02 0.03 0.04-:(u'v'l/u2 -Fu'--vl/u 2

Figure 11. T'U*lment Shorn Stres Pmfflcs. M 2 = 2 =6. = 0.

22

model muderpredcts ft peak even more. As for th location o the peak, the k-e model does betr than

ft algebraic models. As X/D is Inased f1mom 1.26 to 1.42, die location of the Pea P ldicled by thek-e model moves closer to the cener line similar to tha observed In the experiment. This is not seen in

th prediction by th algebraic models. The k-e model predictions agree bete tn the prdictions bythe algebc models at X/D = 1.73 and 1.89.

Of paricular intenst is the accurate prediction or determination of bas presse and, hence, boe drag.

Figure 12 shows the base pressure disuibution (along the base). The base pressures pedicted by both the

algebraic models and the two-equation k-c turbulenc model are compared with the expermetal data

(Herrin and Dutton 1991). The experimental data are shown in dark dcres, and the computed results are

shown in lines. Here, ZID -= 0.0 corresponds to the center line of symmetry and 71D = 0.5 coresponds

to the base corner. The base pessures predicted by both algebraic turbulence models show a big increase

near the center line of symmetry. The experimental data show almost no change (only 3%) in the base

pressure distribution. The base pressures are very poorly predicted by th algebraic models, not only near

the center line but also near the base comer. A much improved base pressure distribution is predicted by

the k-e model, and its agreement with the measured base pressure is quite good. The k-e prediction shows

a small increase in the base pressure near ft center line, which is not observed in Uh data.

0.5 %

0.4 :

0.3

Z/D"0.2

0.1 Chow model

k-i model00.0 0 Exp ! riment

0.0 0.2 0.4 0.6 0.8 1.0Pb/POO

ligure 12. Base Pressue Distrxion. M = 2.46. a =0.

23

5. CONCLUDING REMARKS

A zonal, implicit, time-marebing Navier-Stokes computational technique has been used to compute

the trbulent supersonic base flow over a cylindrical a1terbody. Flow field computations have been

perfoimed at M,, = 2.46 and at the angle of attack, a = 0.0. Various eddy viscosity turbulence models

(two algebraic and a two-equation k-e) have been used to pwovide the turbulence closure. The k-e

equations were formulated in a generalized coordinate system and were solved using an implicit algorithm.

Numerical results show the details of the flow field such as Mach number cetouns, pressure cotours,

and velocity vector plots. Comparison of both the mean and turbulence quantities has been made with

the available experimental data. Both algebraic atrence models predict the mean velocity components

poorly in the recirzwlatory flow region in the wake. In general, the velocity components predicted by the

two-equation k-e model agree better with the experimental dat than the algebraic models do. Discrepancy

exists between the predicted turbulent shear stress and the experiment for all these turbulecre models.

A small improvement in the predicted location and magnitude of the peak in shear stress exists with the

k-E model. Computed base pressure distributions have been compared with the measured base pressures.

The base pressures predicted by the algebraic models show a much larger variation and do not agree well

with the data, compared to the k-e model. The measured base pressures show a very small change along

the base and are predicted rather well with the k-e uurbulence model.

24

6. REFEENCES

Baldwin, B. S.. and HK Lomax. '"lhin Layer Ap xmaloand Algebraic Model for Separd TumbulenkFlows." AIAA Paper No. 78-257. January 1978.

Beam, R., and R. F. Warming. "An Implicit Factored Scheme for the Compressible Navier-StokesEquations." AIAA Jou , vol 16, pp. 393-402, April 1978.

Chien, K. Y. "Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-NumberTurbulence Model." AIA n vol. 20, pp. 33-38, January 1982.

Chow, W. L "Improvement on Numerical Computation of the Thin-Layer Navier-Stokes EquatoD-WithSpecial Emphasis on the Turbulet Base Presms of a Projectile in Transonic Flight Condition."Contract Report No. DAAG29-81-D-0100, University of Illinois, Urbana-Champalgn, Urbana, IL,November 1985.

Herrin. L L., and L. C. Nuon. "An Exerimental Investigation of the Supersonic Axisymmetric BaseFlow behind a Cylindrical Aferbody." UILU 91-4004, University of Illinois at Urbana-Chpalign,Urban, IL, May 1991.

Jones, W. P., and B. E. Launder. "The Prediction of Laminarization With a Two-Equation Model ofTurbulence." International Journal of Heat and Mass f vol 15, 1972.

Pulliam, T. H., and J. L Steger. "On Implicit Finite-Differen Simulations of Tlhe-Dimensional Flow."AIAA S&mal, vol. 18, no. 2, pp. 159-167, February 1982.

Sahu, J. "Navier-Stokes Comp Study of Axisymmetric Tramonic Turbulent Flows with a Two-Equation Model of Turbulence." Ph.D. Dissertation, University of Delaware, Newark, DE, June 1984.

Sahu, J. "Supersonic Base Flow Over Cylindrical Aflterdies With Base Bleed." AIAA Paper No. 86-0487, Proceedings of the 24th Annual Aerospace Sciences Meeting, Reno, NV, Janua•y 1986.

Sahu, J. "Computatio of Supersonic Flow Over a Missile Aflerbody Containing an Exhaust Jet." AIAAJournal of Spacecraft and Rockets, voL 24, no. 5, pp. 403-410, September-October 1987.

Sahu, J. "Numerical Computaion of Transonic Critical Aerodynamic Behavior." AIAA Jurnal, vol. 28,no. 5, pp. 807-816, May 1990. (also wee BRL-TR-2962, December 1988)

Sahu, J., and J. E. Dmnberg. "Navier-Stokes Computations of Transonic Flows with a Two-EquationTurbulence Model." AIAA •1ml, vol. 24, no. 11, pp. 1744-1751, November 1986.

Sahu, J., and C. J. Nmtubicz. hbree Dimensional Flow Calculation for a Projectile With Standard andDome Bases." BRL-TR-3150, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground,MD, September 1990.

25

Sabu, J., C J. N ulcbicz, and J. L SWser. "Navier-Stokes Computstkot of Projectil Bus Flow Withand Withou Bans Injection." BRL-TR02032, US. Army Balistic Research Labortory. AberleenProving OGund, MD, November 1983. (also see AIAA.Jora vol. 23, no. 9, pp. 1348-1355,September 1985)

Sabu, J., and J. L Sser. "Numerical Simulation of Three-Dimensional Tramonic Flows." AIAA PapsrNo. 87-2293, Atmospheric Flight Mechanics Cmfszence, Montemey, CA, Augut 1987. (also see BRL-TR-2903, March 1988)

26

[Zsr OF SYISOLS

a Wood of soundA+ Van Driest's damping faCtrcjoc2 ,c empizical consat in lth k-e tautmlemi model

CV s~pedifhemuat OAM promeI

Cý ~presaure coefficient 2(P - .* %D body dimeter

e towa mergy per Ufal vehuma,.a.%.E fP. 6 flUx vector of ftransfred lNavleSmn*e equaton

A

H souce term vectoI identity matrixJ Jacobian of lth ifarkmu~matl

k tuibulas kinetic 0Mrgyla,1 m~n lngth

M Mach number

p pruaa sop 2Pr Pradtl numberp..vrAq vector of dependeat varibles In transformed equations

R baseradiusRe Reynolds mnuber, paD/pi..Rt tuftllat Reynolds number, k2/ve

§viscous flux vector or swuce tems in die k-e equations

t physical imeu'vIW cartesian velocity comprw-eiI 1,6U.VW Cxualwar" iai leoclideWS61 .

UT ~ friction velocity, F IP

x9yAz physical Cartsia coordinates

i+law of the Wall coordinate, (pOugy/pw)a angle of attaci

ratio of specific beatscoefficie- of thermal conductivity/IC...

P coffcint of viscosiftyp.tw rmnf ed coordinates in axial, ci rcuferentCiHal and nomldirections

p demlty/p.C tuabulent dissipation MaEd~AA 31D)

I ~ ~ fae timmn- ed w

Itw Shear stres at lime wall

V kinematic viscosity, pip

27

anpblcal coom in the k-e eaadm~

o vordiidy

* ~crdtlca valu

w wall condidomsson smcm candidoes

28

No. of N&of

2 Adidolr I CommandemDefense Teci Info Clob U.S. Amy Missile CommandATIN: DICr-DDA ATM.: AMSMI-RD4-R (DOC)Cam Stadwion Redsloue Arsenal, AL 35B98-5010Alexniia, VA 22304-6145

1 CUmAnderUCommader US. Army Tmk-A-twAordve CommnadU.S. Army Matiel Command ATN: AMSTA-JSK (Armor Eng. Br.)ATINM AMCAM Wvees, h9 483750005001 Eiseahow1i Ave.Alexandria. VA 22333-0001 1 D ior

U.S. Amy TR7WSC Anulyss CommanI Direcor : ATiN: ATRC-WSRU.S. Army Research LSi.orm-- y Whie Sands Missile Ra, NM 002-5502ATMW: AMSRL-OP-Cl-AD,

Tech P•bisidn. U. AIo)1 Commandant2800 Powder Mill Rd. US. Army InH-try SchoolAdepF MD 20783-1145 ATIN: ATSH-CD (Securiy MV- )

Fart Deniiqng GA 31905-5%WU.r• mbcl

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Records Mi xumment ATIN , A ASH-WCB-02800 Powder Mill Rd. Fort Ban-n1g, GA 31905-50Ad i, MD 20M83-1145

1 Wom dI2 C.ommArder Eglin AB, FL 32542-500

U.S. Army ArunetA Reeeaich,DerveNo:menar-A Lnging CenterSbAR-T

ATM. SMeCAR-TDCPicany Arsel Mi 07806-SOOD 2 Dir. USAMSAA

ATI AMXSY-D1 Directr AbMSY-MP, IL CohenBewe Weapons LabMor~ayUS. Army Amnament Reseam, 1 Cdr, USATECOM

DevelopmeMA, and Engineerng Center ATN: AMSITE-TCATIN: SMCAR-CCB-TLWaerviiie NY 12189-4050 1 Dir, USAERDEC

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5 Dir, USARLATIN: AMSRL-OP-AP-L

29

No. of No. of

HQDA (SARD-TRIMs K. Keminos) 6 DirectorWASH DC 20310-0103 National Aeronautics and Space

AdminimsafmHQDA (SARD-TR/Dr. R. COait) Ames Research CenterWASH DC 20310-0103 ATrN: MS-227-8, L. Schiff

MS-258-1, T. Hoist2 USAF Wright Aronautical MS-258-1, D. Chaussee

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Dr. J. Shang MS-258-1, P. BuningMr. N. E. Scaggs Moffen Field, CA 94035

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3 Commander Department of MechanicsNaval Surface Weapmns Center ATFN: LTC Andrew L. DullATTN: Code R44, Dr. F. Priolo West Point, NY 10996

Code R44, Dr. A. WardlawK24, •402-12, Dr. W. Yanta 6 Commander

White Oak Laboratory U.S. Army Armament Research,Silver Spring, MD 20903-5000 Development, and Engineering Center

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National Aernautics and Space R. KlineAdminisration R. Bouicelli

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ATIN: Dr. P. KemmeyDr. James Richardson 3 University of California, Davis

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30

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ATIN: Prof. C. C. I.sGandesville. FL 32611

25 Dir, USARLATTN' AMSRL-WT-P,

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AMSRL-WT, Dr. A. BarowsAMSRL-WT-PD, Dr. B. BumsAMSRL-WT-PA,

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32

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MEMORANDUM FOR SEE DISTRIBUTION

SUBJECT: Correction of ARL-TR-438, Numerical Comoiutations of SuRersonic Base Flow With SnecialEmphasis on Turbulence Modeling, by Jubaraj Sahu, dated June 1994

1. The purpose of this memorandum is to apprise the recipients of subject report of an error in figureplacement (Figures 4, 5a, 5b, and 6), pages 15 and 16. In the subject report the correct figure for Figure4 is shown as Figure 5b; the correct figure for Figure 5b is shown as Figure 4; the correct figure for

SFigure 5a is shown as Figure 6; and the correct figure for Figure 6 is shown as 5a. It is requested thatthe incorrect pages currently in the subject report be removed and replaced with corrected pages 15 and16 attached.

• 2. The U.S. Army Research Laboratory solicits input to improve the quality of reports it publishes.Comments or questions which contribute to that end should be submitted to the Director, U.S. ArmyResearch Laboratory, ATTN: AMSRL-OP-AP-L, Aberdeen Proving Ground, MD 21005-5066.

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End

DISTRIBUTION:

Figure 4. Computed Pressure Contours in the Base Region, M,, 2.46, ot=O. and k-c Model.

Figure 5a. Compvuted Mach Contours. M.= 2.46. a 0. and k-c Model.

15

Figure 5b. Exprenmental Schhiren Photograph.

Figure 6. Velocity Vectors in the Base Region, M,= 2.46. a =0. and k-E Model.

16