A Robust Multi-Block Navier-Stokes Flow Solver for Industrial Applications

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NLR TP 96323

A robust multi-block Navier-Stokes flow solver for industrial applications.

J.C. Kok, J.W. Boerstoel, A. Kassies, S.P. Spekreijse DOCUMENT CONTROL SHEET

ORIGINATOR'S REF. SECURITY CLASS. NLR TP 96323 U Unclassified

ORIGINATOR National Aerospace Laboratory NLR, Amsterdam, The Netherlands

TITLE A robust multi-block Navier-Stokes flow solver for industrial applications.

PRESENTED AT third ECCOMAS Computational Fluid Dynamics Conference, Paris, September 1996.

AUTHORS DATE pp ref J.C. Kok, J.W. Boerstoel, A. Kassies, S.P. 960503 11 29 Spekreijse

DESCRIPTORS Algorithms Navier-Stokes equation Body-wing configurations Pressure distribution Computational fluid dynamics Robustness (mathematics) Convergence Run time (computers) Grid generation (mathematics) Three dimensional flow Jacobi matrix method Turbulence models Multiblock grids

ABSTRACT This paper presents a robust multi-block flow solver for the thin-layer Reynolds-averaged Navier-Stokes equations, that is currently being used for industrial applications. A modification of a matrix dissipation scheme has been developed, that improves the numerical accuracy of Navier-Stokes boundary-layer computations, while maintaining the robustness of the scalar artificial dissipation scheme with regard to shock capturing. An improved method is presented to define the turbulence length scales in the Baldwin-Lomax and Johnson-King turbulence models. The flow solver allows for multi-block grid which are of C -continuous at block interfaces or even only partly continuous, thus simplifying the grid generation task. It is stressed that, for industrial applications, not only a (multi-block) flow solver, but a complete flow-simulation system must be available, including efficient and robust methods for aerodynamic geometry processing, grid generation, and postprocessing. Results are presented which show the applicability of the flow-simulation system for industrial purposes. TP 96323

A Robust Multi-Block Navier-Stokes Flow Solver for Industrial Applications

J.C. Kok ,1 J.W. Boerstoel, A. Kassies, S.P. Spekreijse

Abstract. This paper presents a robust multi-block ¯ow for aerodynamic geometry processing, grid generation, ¯ow solver for the thin-layer Reynolds-averaged Navier-Stokes visualization, and postprocessing. Only then numerically ac- equations, that is currently being used for industrial applica- curate computation results can be ef®ciently produced at ac- tions. A modi®cation of a matrix dissipation scheme has been ceptable investments in man hour and computer costs, and developed, that improves the numerical accuracy of Navier- low turn-around times may be realized. Stokes boundary-layer computations, while maintaining the In order to provide this full CFD functionality, in the robustness of the scalar arti®cial dissipation scheme with re- ENFLOWsystem, the following tools are available (see ®gure gard to shock capturing. An improved method is presented to 1): de®ne the turbulence length scales in the Baldwin-Lomax and

Johnson-King turbulence models. The ¯ow solver allows for the domain modeller ENDOMO, for the subdivision of multi-block grids which are only C0-continuous at block in- ¯ow domains around complex con®gurations into simple terfaces or even only partly continuous, thus simplifying the subdomains called blocks (block decomposition),

grid generation task. It is stressed that, for industrial appli- the grid generator ENGRID, for the generation of grids in cations, not only a (multi-block) ¯ow solver, but a complete edges and faces of blocks, and in the blocks themselves, ¯ow-simulation system must be available, including ef®cient using algebraic and elliptic methods [24, 25],

and robust methods for aerodynamic geometry processing, the ¯ow solver ENSOLV, for the solution of the Euler and grid generation, and postprocessing. Results are presented the thin-layer Navier-Stokes equations in arbitrary multi- which show the applicability of the ¯ow-simulation system block domains,

for industrial purposes. the grid adaptor ENADAP, currently being implemented, for the adaption of grids through grid-point movement, in order to improve the accuracy of computational results 1 INTRODUCTION [8, 9], and

In the past 10 years, progress in computational algorithms, several ¯ow visualization codes. numerical grid generation, and computing power has made it Recently,the ENFLOW system has been used by industry as a possible to solve the Navier-Stokes equations for ¯ows around major ¯ow-analysis tool for CFD analysis of wing-body con- complex con®gurations of engineering interest. In particular, ®gurations, and for the aerodynamic integration of propulsion it has become feasible to compute compressible, turbulent systems with transport-type aircraft. ¯ows around complete aircraft including the aerodynamic ef- The total turn-around time of the ¯ow simulation for a new fects of propulsion systems.In order for industry to apply such complete aircraft con®guration, including the construction of numerical simulations in a routine fashion, speci®cally during a high-quality, Navier-Stokes multi-block grid, is today of the design process of aircraft, it is important that the employed the order of three weeks. However, frequently a total turn- means for ¯ow computations, including grid generation, are around time of one or a few days may be realized, speci®cally robust and ef®cient from industry point of view. when computations have to be performed for a well-known At NLR a Navier-Stokes ¯ow-simulation system, called topology, or after a moderate geometry modi®cation, which ENFLOW, based on multi-block structured grids has been de- often occurs during aerodynamicdesign processes.The actual veloped [5]. This is a collection of CFD (Computational Fluid ¯ow computation needs only a computation time of three to Dynamics) programs for the computation of ¯ows around ®ve hours (for 1 to 2 million grid cells) on the NLR NEC complex aerospacecon®gurationsbasedon the Navier-Stokes SX3 supercomputer. These ®gures concerning turn-around equations. In order for such a system to be useful for industry, times should be contrasted with those of about ®ve to seven it should have full CFD functionality, thus including programs years ago, when multi-block grid generation around complex 1 National Aerospace Laboratory NLR, P.O. Box 90502, 1006 BM con®gurations starting from scratch required three to six man Amsterdam, the Netherlands months [4].

c 1996 ECCOMAS 96. Published in 1996 by John Wiley & Sons, Ltd. TP 96323

aerodynamic and Fourier for the viscous contributions, and the Boussinesq input geometry hypothesis for the Reynolds stresses, while maintaining only

the derivatives in the direction,

EULER / NAVIER-STOKES

FLOW CALCULATION SYSTEM

@~ @~ r

u 1 u

= kr k + r

~ tot (5),

@ kr k geometric modelling block decomposition multi-block @ 3 for CFD work of 3D flow domain grid generation

ICEM-CFD codes ENDOMO code ENGRID code @

kr k

Q = ktot , (6) @

with the temperature T given by the perfect gas law, T = grid flow graphical

adaption computation visualization

( ) p= R . The total viscosity tot, and the total conductivity ktot

ENADAP code ENSOLV code various codes are given by

= +

tot t, (7)

ENFLOW

t + ktot = Cp , (8) results of Pr Prt flow calculation

with the laminar viscosity given by the Sutherland law,

Figure 1. The ENFLOW system and the eddy-viscosity t given by the Baldwin-Lomax (BL) [3], the Cebeci-Smith (CS) [6], or the Johnson-King (JK) The following sections will give a description of the ¯ow turbulence model [16, 13]. solver ENSOLV, and present computational results obtained with the complete ¯ow-simulation system ENFLOW. 3 NUMERICAL MODEL

2 CONTINUOUS MODEL 3.1 Standard scheme

[ 1) Consider as independentvariables the time t 2 0, ,andthe The standard numerical scheme used in ENSOLV consists

T 3

=( ) ~ 2 Cartesian coordinates ~x x,y,z with x D R ,where of the schemes developed by Jameson [12, 11] and Mar- Dis the ¯ow domain. The ¯ow domain is divided in a set of tinelli [18], which may be considered proven technology (e.g. non-overlapping blocks, and each block may be described by references [19, 21, 26]). The continuous equations (1) are

=() 2[ ] a curvilinear coordinate system ,,0,1 .Let discretized in space by a cell-centred ®nite-volume scheme:

the basic dependentvariables be the density , the momentum

dUi,j,k

~ vector u, and the total energy per unit volume E. Assuming R , (9) dt i,j,k

the -direction to be the normal direction in thin boundary c a d

( )= layers or wakes, the thin-layer Reynolds-averaged Navier- Ri,j,k = Di,j,k Di,j,k Di,j,k Vi,j,k, (10)

Stokes equations may be written as ) with Vi,j,k the cell volume, and with the indices (i,j,k indicat-

c c c d ing cell centres, corresponding to the curvilinear coordinates

@ @

F F F F

U c

+ + + =

)

J , (1) ( , , , respectively.The convective ¯ux balanceD , the arti-

@ @ @ @ @ t ®cial dissipative ¯ux balance Da, and the physical dissipative

= ( ~ = )

with J det dx d , and with U the ¯ow-state vector given ¯ux balance D each consist of the summation of the corre- )

by sponding ¯uxes across the six faces of the grid cell (i,j,k .The

convective and physical dissipative ¯uxes at a cell face are

~ U = u . (2) given by equation (3), with A taken equal to the cell-face area

E vector, with ¯ow variables located at the cell face de®ned by c simple averaging of the cell-centre values, and with ®rst-order

The convective ¯ux vector F and the dissipative ¯ux vector d derivatives de®ned by central differencing in computational

F are given by

space.

0 1

! In the standard scheme, the scalar arti®cial dissipation of uA 0

c d

A @ Jameson [12] is used, which is de®ned by a blending of

= = k k

~ +

F uuA p A ,F A , (3)

second-order differences to obtain physically acceptable rep-

~ ~

+ ) ( E p uA u Q

resentations of shock waves and fourth-order differences to ~

~ damp high-frequency modes. The arti®cial dissipative ¯ux at

r = ~

with A = J and uA u A. Assuming a perfect gas, the

) ( + )

pressure p is given by a cell face between cells (i,j,k and i,j 1,k is then given by

) ( )

12 a (2 j 4 j

= ( )

+ = + =

+ =

)( k~k ) =( Fi,j 1 2,k i,j 1 2,k U U i,j 1 2,k (11)

p 1E2u,(4) 3

~ = j + k kj with the ratio of speci®c heats. The shear stress vector with uA c A the spectral radius of the convective and the heat ¯ux Q are expressed using the laws of Newton Jacobian in j direction (multiplied by the cell-face area), and

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1=2 In the standard matrix dissipation model of Swanson and

with c =( pthe speed of sound. The expressions for

( ( )

2) 4 Turkel, the arti®cial dissipative ¯ux is given by

the i and k directions are similar. The factors and are

) ( )

given by a (2 j 4 j

= jAj ( )

+ = + = =

Fi,j+1 2,k i,j 1 2,k U 3U i,j 1 2,k (15)

( ) )

2 1 (2

= f g

min ,k max i,j,k, i,j+1,k , (12) = i,j+1 2,k 2

with A the Jacobian matrix of the convective ¯ux in j direc-

n o

) ) ( (

) ( )

4 (4 1 s 4

k max 0, k , (13) tion, and with similar expressions for the i and k directions.

= + = i,j+1 2,k 64 i,j 1 2,k

Let Λ be the diagonal matrix with the eigenvalues of A along

with the shock sensor, its diagonal,

j + j

pi,j+1,k 2pi,j,k pi,j 1,k Λ ( )

= diag 1, 1, 1, 4, 5 , (16) =

i,j,k . (14)

+ +

pi,j+1,k 2pi,j,k pi,j 1,k = 1 uA, (17)

In the ¯ow solver, the default values for the parameters of the ~

= + k k

4 uA c A , (18)

) ( ) ( )

(2 4 s

= =

arti®cial dissipation are k = 1, k 2,andk 0.5,

= k k u c A , (19) which are used for all computations of section 5. 5 A

The discrete equations (9) are solved by Runge-Kutta time A Let the matrix Q have the eigenvectors of as its column integration, accelerated by local time stepping, implicit resid- vectors. Then, the absolute value of the Jacobian is de®ned ual averaging, and a multi-grid scheme. In order to obtain a by taking the absolute value of its eigenvalues, good damping of high-frequency modes on a grid with cells

= QjΛjQ of high aspect ratio, the high-aspect-ratio scaling of the arti®- jAj , (20)

cial dissipation and the variable-coef®cientresidual averaging Λ j of Martinelli [18] are used. For each stage of the multi-grid where the matrix j is de®ned by taking the absolute value scheme (relaxation, restriction, or prolongation), a complete of the elements of Λ. loop over all the blocks is performed. The relaxation con- In order to avoid that the eigenvalues can become zero sists of performing a complete Runge-Kutta time step for (which is needed in order to satisfy the entropy condition), a

each block, while keeping the ¯ow states in the other blocks lower bound is de®ned using the spectral radius , so that the Λj frozen. diagonal elements of j become

An important aspect of the ¯ow solver is the fact that it al- Ä L

= fj j " g max , , (21) lows for multi-block grids, which are only C0-continuous (and 1 1

1 Ä N

= fj j " g not C -continuous) at block interfaces, or even only partly 4 max 4 , , (22)

Ä N

= fj j " g continuous [17]. The former property greatly simpli®es the 5 max 5 , . (23) multi-block grid generation task, allowing for the generation

L N " of grids in each block independently, once the grids in block Swanson and Turkel [27] at ®rst chose " and to be equal faces have been generated. The latter property makes it pos- to 0.25. This choice, together with a necessary modi®cation sible to locally re®ne the grid in certain blocks by a factor 2 of the shock sensor of equation (14), resulted in an improved or 4 if this is considered necessary for accuracy reasons, with shock resolution. only a moderate increase of computation time. However, to improve the accuracy in boundary layers, L

Swanson and Turkel found it necessary to take " equal to 3.2 Matrix arti®cial dissipation 0.01 for the j direction [28]. In this way, the amount of arti®- cial diffusion in the normal direction is strongly reduced for The standard Jameson-type central differencing scheme with the entropy and shear waves, so that no interference occurs scalar arti®cial dissipation is generally found to provide rather with the physical dissipation. This is speci®cally the case in

poor numerical accuracy in boundary layers, e.g. [2, 20, 28], the lower region of the boundary layer where 1 tends to zero. unless very ®ne grids are used in normal direction, which is Suf®cient damping of the high-frequency modes in normal di- often not feasible for industrial applications. To improve the rection is provided here by the physical dissipative terms. In L accuracy, the matrix dissipation of Swanson-Turkel [27, 28] practice, convergence is not affected with this choice for " , may be used. Since the main interest is to improve the accu- as shown in section 5. racy in boundarylayers, a modi®cation is applied to the matrix We choose to implement a reduced form of the matrix dis- model. This modi®cation essentially consists of applying the sipation model, with as main purpose the reduction of the matrix dissipation only in normal direction, while in tangen- arti®cial dissipation for the normal direction in boundary lay- L tial directions the standard scalar dissipation is used. Thus, ers and wakes. For the i and k directions, the two factors " N the basic Jameson shock-capturing scheme can be retained, and " are set to one, or, similarly, the scalar dissipation model which provides robustness in particular with regard to shock is applied for these directions (which is computationally more capturing, while shock resolution may be improved through ef®cient). For the j direction the two factors are set as grid adaption. Furthermore, this modi®ed matrix dissipation

L = model requires no signi®cant increase in computation time " 0.01, (24)

N = compared to the scalar model. " 1. (25)

3 TP 96323

N Ä Ä 0.025

Taking " equal to one, implies that 4 and 5 are equal actual

to the spectral radius , so that the standard Jameson shock max equation (29)

) ( )

(2 4 capturing scheme can be used. Thus, the factors and 0.02 are de®ned in the same way as in the scalar model (equations (12) and (13)).

N 0.015

= jAj Because of the choice " 1, the ®nal expression for is also simpli®ed with respect to the standard matrix model,

giving a further reduction of the computational effort: 0.01

Ä Ä 1T 1 T

= I +( ) jAj AC BD , (26) 1 12 0.005

~ 2 k ckA

with I the unit 5x5 matrix, and with the column vectors A, B, 0 C,andDgiven by 0 0.2 0.4 0.6 0.8 1

x / c

1 0 ! 1 0

Figure 2. Valu e o f n max for RAE2822 pro®le, case9

A @

~ = A = u ,B A , (27)

H uA that the JK model can give signi®cant improvements (e.g. ref- 1

0 erences [10, 1, 22]). The JK model implemented in ENSOLV !

1 2

~ k

2 ku uA includes the improvements of Johnsonand Coakley [15]. Fur-

@ A

= ~ = C u ,D A , (28) thermore, for the outer viscosity the original Clauser-type for- 1 0 mulation is used (in stead of the BL formulation as done by

Abid et.al. [1]) as suggested by Johnson [14].

+ = in which H = E p is the total enthalpy. For the JK model, the extension to 3D is not trivial. In 2D, the JK model includes an ordinary differential equation R 4 TURBULENCE MODELLING ASPECTS (ODE) for the maximum Reynolds shear stress m .Forthe extension towards 3D, this ODE is often replaced by a 3D A critical aspect of the ¯ow solver with respect to robustness partial differential equation (PDE) [1]. However, the form of was found to be the implementation of turbulence models. A this 3D PDE is not well-de®ned in the literature. We employ straightforward implementation in a 3D ¯ow solver of stan- for 3D ¯ows the following 2D PDE along solid walls:

dard algebraic turbulence models, such as the BL and the JK

@ @

@ g gg

~ r) +(~ r) =

models, usually does not lead to satisfactory results. +(um umRHS, (30)

@ @ @ t

In both turbulence models, the location of maxima of func- p

tions in boundary-layer normal direction must be determined, R

with the variable g = m m, with the subscript m in- while such a location is, in general, not uniquely de®ned. In dicating the location nm, and with and the curvilinear

the BL model [3], the length scale nmax is required, which is coordinates along the solid wall. ! the location of the maximum fmax of the function f = n Dd, This PDE has a non-conservative form and therefore is with n the distance to the solid wall, ! the vorticity magni- discretized by a ®nite differencing scheme in stead of a ®nite tude, and Dd the Van Driest damping term. Straightforward volume scheme. A second-order fully upwind scheme is em- computation of this location often leads to a break down of ployed, to accountfor a proper numerical treatment of regions convergence, in particular when there are two local maxima of in¯uence and dependence near discontinuities in ¯ow so- with values close to each other. For a robust computation of lutions of this PDE. Such discontinuities may be present at

this length scale, the following integral formulation is used: attachment and separation lines. An example is strong sep-

Z aration effects near a wing tip, where vortex sheets and/or

q q

f f vortices leave the surface. = nmax = ndn dn, (29) 0 fmax 0 fmax The discrete equation for the second-order fully upwind scheme is given by

with (a rough estimate of) the boundary-layer thickness. dgs,t JK

From ®gure 2 it may be seen that with a value of q = 8, = Rs,t , (31)

the correct location of the maximum is obtained, when a dt

JK c a c a

+ + + unique maximum exists. The JK model also employs a length Rs,t = Qs,t Qs,t Qs,t Qs,t RHSs,t, (32)

scale nm, which is the location of the maximum Reynolds ) with the indices (s,t indicating cell centres on the solid wall,

shear stress. This length scale may be computed by the same

) and corresponding to the curvilinear coordinates ( , ,re- integral formulation. spectively. The convective and arti®cial dissipative terms are The BL model is known to give unsatisfactory results for given by

¯ows with strong adverse pressure gradients (including shock-

c 1

= ( ( ))

+ inducedseparation). For these type of ¯ows, it has beenshown Qs,t s,t gs+1,t gs 1,t 4 gs 2,t gs 2,t , (33)

TP 96323

d 1 s 0.006

j j Q = g, (34) s,t 4 s,t 4 matrix fine δ* matrix medium

scalar medium

= ~ (r ) with um . 0.005 The discrete equations (31) are solved by a 5-stage Runge-

Kutta scheme, particularly tuned for the fully-upwind scheme 0.004 [29]. For each multi-grid cycle of the main ¯ow equations, a

time step is performed. In practice, this solution method does 0.003 not deteriorate the multi-grid convergence, mainly because the boundary-layer normal direction (with small mesh sizes 0.002 in this direction) is not included in the 2D PDE of equation (30). 0.001 It should be pointed out that the JK model is an intri- cate model (including also four implicit algebraic relations 0 to solve) containing many detailed problems to tackle in or- 0 0.2 0.4 0.6 0.8 1 x / c der to make it applicable in a routine fashion. Only the two main measures we have taken have been described here. Even Figure 4. Displacement thickness for RAE2822, case 9 though these measures have improved the robustness of the implementation of the model in 3D, still the equationsmay not speed,and furthermore with no signi®cantincrease of compu- always converge fully to steady state. In fact, one may state tation time (less than 1%). The main difference between the that algebraic models are usually strongly depended on typi- cal boundary-layer quantities and thus cannot be expected to 1 always work for general 3D con®gurations. Presently, work is scalar model 0.5 matrix model being done on the implementation of more general turbulence model, such as two-equation models. 0 2 || -0.5 / dt ρ 5 RESULTS d -1 log(|| )

The improvement of the boundary-layer resolution using the 10 -1.5 modi®ed matrix dissipation scheme is shown for the RAE2822 -2

= = = 1 airfoil, case 9 (M1 0.73, Re 6.5 10 , 2.8 )[7]. Figures 3 and 4 show the skin-friction coef®cient and the dis- -2.5

placement thickness along the airfoil on a ®ne grid (528 96 -3 grid cells) with the matrix model, and on the corresponding -3.5 medium grid (mesh sizes doubled) with the scalar and the 0 50 100 150 200 250 300 350 400 matrix model. With the matrix model, the solution on the fine-grid relaxations

0.007 Figure 5. Convergence history for RAE2822, case 9 |Cf | matrix fine matrix medium 0.006 scalar medium medium-grid and ®ne-grid solutions with the matrix model is located around the shock. The shock resolution may be 0.005 improved by using the standard matrix model of Swanson- 0.004 Turkel at the cost of a moderate loss of convergence speed and an increase of computation time (in the order of 15%) 0.003 [28]. The shock resolution may also be improved using grid adaptation based on grid-point movement [8, 9]. 0.002 The application of the JK model in 3D is shown for two cases: the ONERA M6 wing and the Aerospatiale AS28G 0.001 wing/body con®guration.For the ONERA M6 wing a 4-block

0 CO-type grid with a total of 0.8 million grid cells has been

0 0.2 0.4 0.6 0.8 1 = x / c used. The test case for the ONERA M6 wing (M1 0.8447,

= =

Re 1 11.78 10 , 5.06 ) has strong shock-induced Figure 3. Skin friction for RAE2822, case 9 separation. Figure 6 shows that the JK model results in a better pressure distribution than the CS model (compared to medium grid is signi®cantly closer to the ®ne-grid solution the experiment [23]) consistent with references [1, 22]. (in particular on the lower side) than with the scalar model. As The ENFLOW system has been used intensively by indus- can be seen from ®gure 5 the improved boundary-layer reso- try for computations around wing/body/pylon/nacelle con®g- lution has been obtained with no degradation of convergence urations, for which the JK model was found to give satis-

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1.5 CS Cp JK experiment 1 -1.4 2 || -1.2 / dt 0.5 ρ

-1 d

-0.8 0 log(|| )

-0.6 10 -0.5 -0.4 η = 0.90

-0.2 η = 0.80 -1 0 JK η = 0.65 -1.5 BL 0.2 η = 0.40 0.4 -2 0 0.2 0.4 x/c 0.6 0.8 1 0 200 400 600 800 1000 1200 1400 fine-grid relaxations Figure 6. Pressure coef®cient for ONERA M6 wing Figure 8. Convergence history for AS28G wing/body

factory results. As an illustration, preliminary results for the =

Aerospatiale AS28G wing/body con®guration (M1 0.80,

= =

Re 1 10.5 10 , 2.20 ) have been included (®gure 7). A 32-blocks grid has been used with a total of 1.2 million grid

BL Cp JK experiment -1.4

-1.2

-1

-0.8

-0.6

-0.4 η = 0.78 -0.2 η = 0.57 0 η = 0.42 0.2 η = 0.23 0.4

0 0.2 0.4 x/c 0.6 0.8 1 Figure 9. Impression of grid around AS28G wing/body/nacelle

Figure 7. Pressure coef®cient for AS28G wing/body one or a few days. Figure 10 compares the computational results for the wing/body and wing/body/nacelle con®gura- cells, and with the no-slip condition only applied on the wing. tions with the experimental results for the wing/body and For this case, besides the shock position, the JK model also the wing/body/pylon/nacelle con®gurations. The computa- has effect on the pressure distribution on the aft part of the tions were performed with the BL model and on medium grid wing. Comparison with the experimental results is not fully levels (0.15 and 0.28 million grid cells). The modi®cation satisfactory, in particular with respect to the shock position. of the pressure distribution on the lower side, as well as the However, uncertainties exist about the geometry de®nition forward shock movement, are consistent between the compu- (smoothed out kink) and wind-tunnel conditions, so that an tational and experimental results, although there is still an in- explanation of the differences cannot (yet) be given. The con- consistency in the actual shock position (as for the wing/body vergence history of ®gure 8 shows comparable convergence case). speeds for the BL and JK turbulence models. The JK model generally requires an increase of the computation time per 6 CONCLUDING REMARKS iteration of the order of 10%, compared to the BL and CS models. A robust multi-block ¯ow solver for the thin-layer Reynolds- The applicability of the complete ENFLOW system to averaged Navier-Stokes equations has been presented. It has complex con®gurations is ®nally shown for an AS28G wing/- been shown that with a modi®ed form of the matrix arti®cial body/nacelle con®guration (without pylon). The complete dissipation model the numerical accuracy in boundary lay- ¯ow simulation, including the generation of a 101-block grid, ers may be improved, while maintaining the standard robust could be performed within three weeks. An impression of shock capturing scheme, and without a signi®cant increase of the grid is given in ®gure 9. Recomputation of the ¯ow computation time. Implementation aspects of algebraic turbu- after repositioning of the nacelle may be performed within lence models have been discussedthat improve the robustness

6 TP 96323

Generation in Computational Fluid Dynamics and Related Fields, 1995. WB (computed) WBN (computed) WB (experiment) [10] T.L. Holst, `Viscous transonic airfoil workshop compendium Cp WBPN (experiment) of results'. AIAA-87-1460, 1987. -1.4 [11] A. Jameson, `Multigrid algorithms for compressible ¯ow cal- -1.2 culations'. Report 1743, Dept. of Mech. and Aerosp. Eng., -1 Princeton University. -0.8 [12] A. Jameson, W. Schmidt, and E. Turkel, `Numerical solution -0.6 of the Euler equations by ®nite volume methods using Runge- -0.4 η = 0.42 Kutta time-stepping schemes'. AIAA-81-1259, 1981. -0.2 η = 0.29 [13] D.A. Johnson, `Predictions of transonic separated ¯ow with an 0 η = 0.23 eddy-viscosity/reynolds-shear-stress closure model'. AIAA- 0.2 η = 0.17 85-1683, 1985. 0.4 [14] D.A. Johnson, `Nonequilibrium algebraic turbulence modeling 0 0.2 0.4 0.6 0.8 1 x/c considerations for transonic airfoils and wings'. AIAA-92- 0026, 1992. Figure 10. Pressure coef®cients for AS28G wing/body and [15] D.A. Johnson and T.J. Coakley, Ìmprovements to a non- wing/body/nacelle equilibrium algebraic turbulence model', AIAA Journal, 28, 2000±2003, (1990). of the solver. Finally, it has been shown that industrial numer- [16] D.A. Johnson and L.S. King, À new turbulence closure model ical ¯ow simulations may be performed ef®ciently with the for boundarylayer ¯ows with strong adverse pressure gradients ¯ow-simulation system ENFLOW. and separation'. AIAA-84-0175, 1984. [17] A. Kassies and R. Tognaccini, `Boundary conditions for Euler equations at internal block faces of multi-block domains using ACKNOWLEDGEMENTS local grid re®nement'. AIAA-90-1590, 1990. This work has been carried out under a contract awarded [18] L. Martinelli, Calculations of Viscous Flows with a Multigrid by the Netherlands Agency for Aerospace Programs, NIVR, Method, PhD dissertation, Princeton University, 1987. [19] L. Martinelli and A. 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