AIAA-87-0 1 1 7

CALCULATION OF THREE DIMENSIONAL CAVITY FLOW FIELDS J.J. GORSKI, D.K. OTA, AND S.R. CHAKRAVARTHY ROCKWELL INTERNATIONAL, THOUSAND OAKS, CA

AIM 25th Aerospace Sciences Meeting January 12-15, 19871Ren0, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019

CALCULATION OF T H R E E - D I M E N S I O N A L CAVITY FLOWFIELDS

Joseph J. Gorski', Dale K. Ota", and Sukumar R. Chakravarthy"

Rockwell International Science Center

Abstract

Numerical calculations of laminar and turbulent Row

in a three-dimensional cavity configuration, relative to a weapons bay Rowfield, is presented. Solutions are presented for a simplified version of the F - I l l weapons bay which h z a LID ratio of 6.2. These results were obtained by solving the Navier-Stokes equations with the Baldwin-Lomau t u r - bulence model modified for multiple wall effects. Thc solu- tion procedure makes use of high accuracy T V D schemes, which guarantee oscillation free solutions for the convective terms of the Navier-Stokes equations. The results, pre- sented here, have demonstrated the ability of the rode to predict the complex flowfields associated with these cavity configurations. Further comparison with experimental data is necessary to finalize the validation of this application.

1.0 INTRODUCTION

The flow over cavities has gained increased interest in recent years. One of the reasons for this is that store sep- aration and drag of advanced aircraft is directly linked to how a cavity (bomb bay) affects the flowfield in its vicin- ity. Before the problem of store separation can be studied with all its complexities, an understanding of the more ba- sic cavity flow, which is a complex flow in its own right. needs to be examined. A number of experiments have been done on the cavity Row These experiments have shown that the Row is highly dependent on such parame- ters as; depth, incoming boundary layer thickness, length, Mach number, and Reynolds number. Based on these pa- rameters very different types of flow can occur. Shallow cavities form what is called "closed" cavity flow. In this flow the outer high momentum Ruid enters into the cav- ity and impinges on the cavity floor. This can create two distinct separated regions much like Row over forward and backward facing steps. Because of flow entering such a cav- ity, a store released from it can be drawn back upward into the cGvity along with this incoming h i d . Shallow cavities can also create large amounts of aerodynamic drag on a configuration. For deep, or "open", cavities the outer high speed flow can bridge across the mouth of the cavity cre- ating a large recirculating flow within the cavity. When this happens very little of the high momentum fluid enters the cavity and the outer flow will behave very much as i f

t h r r e were no cavity at all, t h u s allowing a store to scpnratc r l c a n ~ y , This gives a brief discription of the two "eXtTPnleS" of flow. \Vithin these limits there exists a multitude of possitjilities with varying amounts of fluid entering the cavity.

Because the cavity flow field can vary so widely. which in t u r n can drastically affect store separation characteris- tics, it is necessary in high-speed aerodynamics to have a knowledge, apriori, of how a given cavity configuration wil l behave with a given set of flow conditions. For this, a nu- merical scheme capable of accurately calculating the flow is desired. Such a code would give aerodynamicists a quick and efficient means (much more so than by experiment) of

studying a range of cavities with varying flow conditions. It would also be a first step toward creating a code capable of calculating the more complex problem of store separation. Toward this end the present paper will present numerical results for a simplified weapons bay under laminar and tur- bulent conditions. Further work will entail comparisons wi th available experimental data.

As can he seen the numerical scheme used to calculate such flows needs to be highly robust and able to handle a considerable amount of complexity. For this reason high accuracy TVD schemese-O are used to solve the full Navier- Stokes equations for these flows. Another important issue is the turbulence modelling used. The present results were obtained using the Baldwin-Lomax turbulence model with modifications to account for multiple walls.

2.0 N A V I E R - S T O W 0 UATIONS

For such complex flows as these it is necessary to solve the full Navier-Stokes equations in three-dimensions. The Navier-Stokes equations in Cartesian coordinates are given in the following form:

out of this, the subset

Copyrinht @American Ins t i tu te of Aeronautics and .. - - * presently with David Taylor Naval Ship Research and 1 Astronautics, Inc., 1987. All r igh ts reserved Develoment Center. Member AI,AA

" Member Technical Staf f Afcrnhrr \ I 4 4

constitutes the set of Euler equations of inviscid compress-

ible Row. The dependent variable q and the invisrid fluxes

f, , fa, and f3 are given by ‘W‘

In the above, p is pressure, p is density, and the CarteSian

velocity components are u, v, and w ir. the 2, y. and z

directions, respectively. The total energy pe: unit volume,

e, is given by e = p/(7 - 1) +p(ul + v2 + w 2 ) / 2 where 7 is

the ratio of specific heats. The viscous “fiuxes” gl, 92. and

g3 are given by

(2.4)

4

Here, R, is the Reynolds number, P, is the Prandtl number, IC is the thermal conductivity, and T is the sped ic internal energy given by T = p/[(7 - I)p]. The terms rrr, rSip., r,,,

rUz, ruu, rV*, 7*=, 7zu. and r,. are given by

aU 2 r,, = 2p- - - p @ , a2 3

av 2 rvu = 2p- - -pa#

aY 3 aw 2

r,, = 2p- - -pa , az 3

Tzu = r,, = p(- + -1, (2.5) aU av

ay a2 aU aw az at

rr. = 7,. = p( - + -1, aw av

7”. = 7*” = p( - + -) ay az

where aU av aw = ( - + - + -) az ay aZ

and p is the coefficient of viscosity. .Assuming a time invariant grid under the transforma-

tion of coordinates implied hy

r = f , ~ = E ( ~ , Y , z ) , ‘ ~ = t , f z , ~ , z ) , s=<(z ,y9zJ ,

(2.6) eq. 2.1 can be recast in the conservation form given by

where - 9 q = -

J ’

where J is the Jacobian of the transformation given by

3.0 THE TURBULENCE MODELS

An important aspect of computing such Bows is the turbulence model used. The present computation is done with the Baldwin-Lomax turbulence model. f i r t he r study should be done using other turbulence models such as the k-r equations of turbulence. This model is also included in the present code and will be used later.

3 L k E g x a t i o n s of ‘hrbulenca

The k-r model in the code is the standard high Reynolds number form of the equations. Even though the k-r model can take more time to solve than the simpler al- gebraic eddy viscosity models, this is justifiable since the k-r model is generally a9)plicable to a much wider class of Rows. The kinetic energy equation is derived from the Navier-Stokes equations with the main limiting criterion being that it assumes local isotropy. The dissipation equa- tion is not exact but is modelled to represent physical pro. cesses similar to those of the kinetic energy equation. Even

2

with these assumptions the k-r equations have a proven ra-

pability of adequately predicting a large range of c-onrpleu flows, including anisotropic ones.

The k-e equations in Cartesian coordinates can be writ- ten in the following form

1 ' (aM a N ao + - + - + - = - - + - + - + S at az ay at RC az ay aZ

aA a s ac a o

(3.1)

-

Pc = (P + k / 4

where k is kinetic energy, c i s turbulent dissipation, and k

is turbulent eddy viscosity. P represents the production of kinetic energy and the following simplified form of it is used

P = k(u: + u; + u:) (3.2)

The k-e model still employs the eddy viscosity /dif- fusivity concept as it relates eddy viscosity to the kinetic energy and dissipation by

ka k = CWPT (3.3)

This eddy viscosity is then used to creale an effective vis- cosity ( p fk) which replaces p in the Navier-Stokes equa- tions. To solve the above turbulence model the following constants must be specified: uk = 1.0, 0, = 1.3, C1 = 1.44, Ca = 1.92, and C,, = 0.09. In addition eq. (3.1) under- goes a coordinate transformation similar to that shown in eq. (2.8).

The above model constitutes a high Reynolds number form of the k-e equations. This needs to be coupled with a near-wall treatment for calculating the flow in the vicinity of a wall. Most near-wall techniques suffer from the fact that large numbers of grid points are needed for resolv- ing the viscous sublayer, and also, that the equations are very "stiff numerically close to the wall due to the source terms. Conventional law of the wall approaches breakdown

if points are wi th in the viscous sublayer or i f the law of

the wall is not valid which will more than likely be true for flow wi th in the cavity. Because two- and three-dimensional cavities are being computed ZE part of tbis work. thrrr is a wide disparity in the number of grid points bring usrrl re- alitiw to the problem. Consequently, a recently devclrrpd technique wil l be used for computing near-wall turliuknce

levels.

h/

3.2 Sear-WaIl Formitlation

Thr above k-r model is not valid near walls. Therrfore. it becomes necessary to specify values of kinetic energy and dissipation in the fully turbulent region, away from the wall. w h i r h are then used as boundary conditions for rq. (3.1)

To do this it is first assumed that the viscous sublayer is bound by

where the subscripts u and w refer to values at the edge of the viscous sublayer and at the wall respectively and y represents the normal distance to the wall. With this in mind it is now necessary to calculate the friction velocity, u,, at the wall. If the first grid point away from the wall lies within this predetermined region the linear law is used to specify friction velocity

U -

(3.5)

For these situations the friction velocity computed is the same as would be obtained using a first-order difference a t the wall. If the first grid point is outside of this region the following law of the wall formula is used

Here 6 is von Karman's constant (=0.4) and p FY 5.5. .Ye is given by

Uc = ~ a ' ! p / p w ) l / a d D .

which is the van Driest compressibility transformation. It should be mentioned here that the law of the wall equation is never used to set boundary conditions for the Navier- Stokes equations which always use proper no-slip bound- aries with an adiabatic wall condition a t the wall. The law of the wall is only used for specifying kinetic energy and dissipation in the near-wall region. For this reason it is felt that the present approach will outperform conventional law of the wall approaches in complex flow situations where the law of the wall equation is not valid. With the friction ve- locity now known, kinetic energy and dissipatiou can be set

3

at the first grid point outside of the viscous sublayer using the following relations

These values are then used as boundary conditions for eq. (3.1). For points that lie within the viscous sublayer. simple algebraic relations are used to specify kinetic energy and dissipation. These are set by first assuming

k. u ya and c. u Constanf

withir the sublayer. Here the subscript s refers to points insio,. the sublayer. With these assumptions kinetic en- ergy and dissipation are set within the sublayer using the foilowing relations

Y: Pe

!2 P o k. = k.-- (3.5)

This formulation also requires a different equation for eddy viscosity within the sublayer given by

(3.10)

- where C.(y) is a linear interpolation which transitions q. (3.10) smoothly into eq. (3.3) at the sublayer edge. More details of the above k-c and near-wall formulation can be found in Ref. 10 and 11.

In three-dimensions the velocity component used in the law of the wall equations (eqs. (3.6) and (3.7)) is the velocity component tangential to the wall. When a point is in the vicinity of more than one wall a special treatment is needed.

3.2 Corn er Treat-

For the near-wall approach used here, and all other near-wall techniques to the authors knowledge, it is neces- sary to know the normal distance from a point to the wall when a point is in the vicinity of a wall. For many two- and three- dimensional geometries this creates no problems. However, certain points in various flows are influenced by more than one wall, the cavity flow being a case in point. The method used here is to treat each wall separately and then do an inverse average for obtaining the necessary quan- tities as described below.

For the Balwin-Lomax modella, the wall effect is taken into account by using the same inverse averaging method on the computed h. An example of this problem is shown in Fig. 1. Point P is influenced by two different walls. .-

y+y (3.11)

where I* is the desired eddy viscosity. This forniulation w i l l allow the wall with the lower y+ value (usually the closest wall) to haw more importance than the neighboring wall. This same technique is used for more than two walls also. Logic is included to allow for a completely enclosed domain of six walls

4.0 NU%IERICAI, PROCEDURE

For solving the above sets of partial differential equa- tions a finite volume, time-marching, code based on a irn- plicit upwind scheme has been used. The algorithm relies on approximate factorization in each spatial direction for solving the given equations. The convection terms of both the Navier-Stokes and the k-e equations are discretized us. ing a TVD formulation, This implies that the scheme can he up to third-order accurate and guarantees oscillation free solutions. The dissipation terms are treated conventionally by using central difference approximations for all second derivative terms. The Navier-Stokes and k-r equations were solved decoupled from each other with the k-r equations be- ing lagged one step. More information on the above solu- tion procedure can he found in Ref. 9 for the Navier-Stokes equations and in Ref. 10 for the k-e equations.

5.0 COMPUTATION AL GRID

Although the gridding for a problem such as this could he a major area of difficulty, it is not for the present cal- culations. This is because a multi-zone technique is being used. This allows one part of the geometry to he solved more-or-less independently of another wi th the only link being through the boundary conditions. Likewise, the grid can be generated for one section independent of another. This allows simple Cartesian grids to he used with the cav- ity itself being treated as one zone, and the entire outer flow area being treated as another. Representative grid sections for the present calculations are shown in Fig. 2a - 2b.

6.0 RESULTS AND DI SCUSSION

The following three-dimensional results are intended to show that a basic three-dimensional code, capable of cal- culating the flow of interest, is available. Figure 3 shows a schematic of a three-dimensional cavity. As already men- tioned the streamwise flow can drop down into the cavity creating different types of vortex structures depending on various flow quantities and the dimensions of the cavity. In addition flow can enter into the cavity from the side ( Z

direction) rreating a separate vortex strurturr in the crnss- flow plane. 'Therefore. the only way tn arrriratrly rriorlel surh complcx flows may be through a thrrc-~liincn.;ii,n:Il numrrical romputation.

Results are presented here for a siinp!ificil F-l I I weapons bay w i t h LID of about 6.2. Thcsc rt-iults are preliminary ;LS they were done on very coarse grids. Thcsr computations slmw a w r y promising romp,lt;,~lona) caps.

bility, and further validation w i t h e x p ~ r i m ~ ~ t at. trmpted.

G.I Laminar Case

The calculations for this rase were d o n r at a llach number of 2.3G and a Reynolds number of G.6z10fi/m. For

this case the LID ratio is approximately F.2 . F i p r w .~a-.lc!

shows results obtained for this case. Figure .la shows the velocity directions along the centerline. I t ran be s w n that an open cavity flow is obtained with one large vortex taking up most of the cavity and another small countrr-rotating

vortex generated in the lower upstream corner. Fig. 4b shows velocity vectors along the centerline. The flow in

the cavity is of very low momentum Ruid with snme higher momentum fluid impinging on the back wall as shown in Fig. 4b. A point to notice is that very little fluid enters the cavity so there is no expansion into the cavity. Also

from Fig. 4h, it can be seen that the flow impinging on the back wall of the cavity creates a very strong shock. A rep- resentative cross-plane plot of velocity vectors is shown in Fig. 4c. Fig. 4c shows that flow also enters into the cavity from the side making three-dimensional effects important. Fig. 4d shows the surface pressure distribution along the cavity floor. The pressure rise at the back wall reflects the impinging Row at that wall.

6.2 Turbulent Case

The calculations for this case were also done at a Mach number of 2.36 and a Reynolds number of 6.6z1Oe/m. Again the Baldwin-Lomax model has been modified to take in account multiple walls. The turbulent solution is very similar to that of the laminar solution. The main dilierence is that the Row before the cavity is more energetic which keeps the flow from dipping into the cavity region as much as the laminar solution. The main effect of this is to reduce the amount of fluid impinging on the back wall. This has formed an open cavity type of flow like the laminar solution. Fig. 5a - 5d show many of the same features as the laminar solution, with the Row impinging on the hack wall, and flow coming in from t h r sides. One point to notice in Fig. 5b is that the region in the cavity seems to be of lower momentum than t h a t of the laminar

solution. This again could be attributed to Ihe fart that the turbulent flow is hidzing the cavity more cleanly than the rd in three-dimensions. Further study will include refind Snlutions for the three-dimensional turbulent cavities al-

rexly done along with comparisons w i t h esperiniental data.

REFERESCES

t/

111 Charwat, A. F., Rous, J. N . , Dewey, F. C.. and liitz, J. A, , "An Investigation of Srparated Floa.s." Journal OJ .Aerospare Sriences, Vol. 28, Pt. I . June IDGI, pp. 1.57-470, Pt. II , Ju ly 1961, pp. 513-527.

1'2) Ilcihman. T . C. and Sabersky, R. l l . , "1,aminar Flow Over Transverse Rectangular Cavities," lnlernolional Journal UJ N e a l and hlass Transfer, Val. I I , June 1968, pp. 1083-1085.

131 Sarohia, V., "Experimental Investigation d Oscilla tions in Flows Over Shallow Cavities," Al.4.4 Journal, Vol. 15, July 1977, pp. 984-991.

141 Sinha, S . N., Gupta, A. K., and Oberai, M. Sl., "Lami- nar Separating Flow over Backsteps and Cavities, Part I I Cavities," AIAA Journal, Vol. 20, March 1982, pp. 370-375.

151 Stallings, R. L., "Store Separation from Cavities at \J Supersonic Flight Speeds," Journal of SpacccraJt and

Rockets, Val. 20, March-April 1983, pp. 120-132.

[GI Chakravarthy, S. R. and Osher, S . , "High Resolution Applications of the Osher Upwind Scheme for the Eu- ler Equations," AIAA Paper No. 83-1943, Proceed- i n g s of the AIAA Sixth Computational Fluid Dynam- ics Conference, Danvers, h!:ssachusetts, Ju ly 1983, pp. 363.372.

171 Harten, A., "High Resolution Schemes for Hyperbolic Conservation Laws," Journal of Compufational Phy- sics, Vol. 49, 1983, pp. 357-393.

181 Chakravarthy, S. R. and Osher, S . , "A New Class of High Accuracy TVD Schemes for Hyperbolic Conser- vation Laws," AIAA Paper No. 85-0363, 1985.

[SI] Chakravarthy, S. R., Szema, K., Coldberg, U. C., Gorski, J. J., and Osher, S., "Application of a New Class of High Accuracy TVD Schemes to the Navier -Stokes Equations," AIAA Paper No. 85-0165, 1985.

[I01 Gorski, J. J., Chakravarthy, S. R., and Goldberg, U. C., "High Accuracy TVD Schemes for thr-, k.c Equations of Turbulence," A I A A Papcr No. 85- 16G5. 1985.

5

[ I l l Gorski, J. J., ".4 New Near Wa!l Formulation for the k.6 Equations of Turbulence," AIA.4 Paper No. 86- 0556, 1986.

'- (121 Baldwin, 8. S., Lomay H., "Thin Layer Approsirna- tion and A!gehraic \lode1 for Separztcd Turbu- lent Flows," A l A A Paper Yo. 78-25i i , 1978.

SC38824

WALL 2

WALL 1

u

J

Figure 1 Multiple-wall effect

SIDE VIEW IL = 21

IOi 1.8

3 0- 4 3 5 5 0 5 07 19 31

X

Figure 2a F - I l l Weapons Bay grid - Side view

1 8 4

0 6

z

~ 0 5 0 7 19 3.1 4.3 5 5

X

Figure Zh F-l I I \Vcapons Bay grid . Top view

I L I . . l i

SCHEMATIC OF 3 .0 CAVITY

Y Y c, Y."

2.W

Figure 3 Schematic of 3-D Cavity

M, = 2.36 R e = 6.6 x 106/m

0.50

0.24

-0.02

Y

~0 28

~ 0 . 5 4

a80 -0.5 0.7 1.9 3.1 4.3 5 .5

x

Figure $a Velocity Directions (centerline,a=Om)

6

R e 6 6 x 1 0 6 m

, , ,,.",, I I

. I \ / I

, , .I.

050 062 0.74 086 0 9 8 110 z

~ 0 . 1 8

Figure .Lc Velocity Vectors (cross-planr,x=2rn)

R e = 6.6 x 106/m 701 M z z 2 3 6 5 6

4 2

P

2 8

14 4

P

Figure Sb Velocity Vectors (centerline.z=Om)

0 3 , 1 ,ej..O1

0q.I : 1 i . .-

1

m . 1 , , ,

~. 0 10 7

0 5 0 6 2 o n on6 0 9 8 1 1 0 2

Figure 5c Velocity Vectors (cross-plane,u=Zm)

* 5 -

2 0 '

0 5 1

! I

7