TRANSONIC EULER CALCULATIONS OF A WING-BODY CONFIGURATION USING A HIGH-ACCURACY TVD SCHEME

Abstract

Chung- J in ~oan* Rockwell International

North American Aircraft Operations Los Angeles, California

Sukumar R. Chakravarthy ' Computational Fluid Department

Rockwell International Science Center Thousand Oaks, California

A high-accuracy TVD Euler solver developed by the second author is used to calculate flowfields of an NACA research-type 45-degree swept wing- fuselage configuration at Mach numbers of 0.9 and 1.2 and an angle of attack of 6 degrees. A unique feature in the present calculations is that the flowfield is partitioned into a series of contiguous blocks, each being a nearly rectangular parallelopiped in shape. A grid of H-type is generated for each block independently of others. For efficiency of flow calculation, these blocked grids are combined into a smaller number of solution blocks. The combined grid has large grid- line slope discontinuities within the blocks and at the block interfaces. Calculated results compared with experimental data indicate that the high- accuracy TVD Euler solver can calculate transonic flowfields efficiently and accurately on a such multi-block grid, hence, the flowfield blockings and griddings of realistic aerodynamic configurations can be greatly simplified.

Introduction

Euler equations are the highest level of inviscid flow modeling. A properly designed numerical algorithm, solving the Euler equations, is capable of (1) capturing shocks and calculating the associated entropy and vorticity production, (2) capturing the vortex wakes trailing lifting surfaces, and (3) capturing the interfaces of streams with different total pressure and total temperature, such as the flowfield of a powered nacelle. The first capability is needed for accurate calculation of transonic flows with moderate and strong shocks where rotational effects can not be neglected. Capabilities 2 and 3 can calculate the interference effects among various aerodynamic components without explicit specification of vortex wakes, which must be specifie6 as boundary surfaces in the potential formulations. These capturing capabilities make the Euler equations a very attractive aerodynamic tool for analysis of complex aerodynamic configurations'and complex flows. A unified Euler code can in fact be developed for external, internal, and integrated external/internal flow predictions at all speeds while the programming and code logic can be kept very simple since the shock

"Member of Technical Staff CompuCational Fluid Dynamics Group

'Manag,?r, Computational Fluid Department Member, kIA-4

and vortex wake locations need not be specified.

With the aid of current supercomputers, Euler calculations of realistic aircraft configurations - have become and are becoming practical. However, there is a major obstacle in solving the Euler equations for complex configurations. It is the generation of grids conforming to various body components with appropriate grid cell distributions for flow solvers to calculate the flow variables with sufficient accuracy and convergence rate. Various methods proposed in recent years to overcome this obstacle include (1) multi-block without overlaid grids, (2) multi-block with overlaid grids, (3) adaptive-grid, and (4) random- structure. In the first method, the flowfield of interest is divided into a series of contiguous subdomains (blocks), each being transformed into a rectangular parallelopiped. An individual grid is generated in each block and the global grid is formed by patching together all the individual block grids. The computed grid lines in adjacent blocks may be made to align at the block interface

with complete continuity6 " ' ', or with continuous .

line slope 8'9'10, or a discontinuity in slope1', or perhaps not to align each other at all. Unlike the first method, the second method does not need common boundaries between blocks, but rather, the various block grids are only required to overlay to provide communication among blocks for flow

solvers. The works of Atta and vadvak12 and Benek

et al. 13'14 are typical of this approach. The concept of adaptive-grid approach is to allow the grids to redistribute as the flow solution develops, concentrating in regions of large gradients of flow variables to seek a best use of available grid points. The works of Dwyer et: - A

a1.l5, f2noff16, and Rai and ~nderson" are typical of the adaptive-grid schemes. The random-structure method is based on the standard finite element methods. In Refs. 1 and 2, Jameson et al. described an Euler algorithm based on tedrahedral element and applied to a complete aircraft configuration. Although these methods have been developed to a high degree of sophistication, difficulties in blocking the flow domain and constructing block-face grids still remain'because the geometric complexities of aircraft configurations having components of various topo10,ies. If the numerical convergence of flow solver is sensative to the grid distribution, iterations between flow solver and grid generator may be required to achieve a converged solution. Therefore, the entire flow analysis process can be extremely tedious and time consuming.

In this work, transonic Euler flow calculations on a multi-block grid are presented for an NACA research-type 45-degree swept wing- fuselage configuration at Mach numbers of 0.9 and 1.2 and an angle of attack of 6 degrees. The computational grids are generated by dividing the flowfield into a series of contiguous blocks. Each block is taken to be a nearly rectangular parallelopiped in shape, however a block face may degenerate to a line or a point. A grid of H-type is generated for each block independently of others using che elliptic surface and 3-D grid generators of Refs. 18-19. For efficiency of flow calculation, these individually generated grids are combined into a smaller number of solution blocks. The combined grid has large grid-line slope discontinuities within the blocks and at the block interfaces. This approach to construction of a computational grid greatly simplifies the flowfield blocking and the grid of a nearly rectangular parallelopiped is very easy to generate and control.

The flow solver chosen to perform m1r present calculations is the 3-D upwind-biased high-accuracy Total Variation Diminishing (TVD) Euler solver developed by the second author. In the solver, the highly accurate TVD formulations developed by

Chakravarthy and 0sher20 are embeded in a finite volume framework for general geometries. A steady- state solution is determined as the time asymtotic limit. To advance a time step, the cell-center defined flow variables are preprocessed in terms of characteristic variables to obtain the values of the conservative variables defined just to the right of and just to the left of the cell face using a high-accuracy TVD scheme. Using the left and right states, the Roe's approximate Riemann

solver2' is applied at each cell face to construct the cell-face numerical fluxes. Finally, the discretized Euler equations are solved for the cell-center defined conservative flow variables using an approximate factorization method. Many properties desirable in numerical solution procedure are included. By design, the method avoid numerical oscillations and unphysical expansion shocks and is able to compute discontinuities with ease and high resolution. At a fundamental level, it is based on a Riemann sclver and therefore it closely simulates the signal propagation properties of hyperbolic equations. Dissipation required for numerical stability arises natually in the formulation. Within a five-point bandwith of grid points, the scheme results in third-order accuracy in smooth region for steady-states and one-dimensional uniform grids and it reduces to a first-order accuracy scheme at extrema. The conservative property, the avoidance of numerical oscillation and expansion shock, the adherence to signal propagation principles, the inherent stability, and the achievement of high accuracy have motivated us to calculate flows on multi-block grids with relaxed grid-line slope continuity and led us to believe that the scheme will be able to calculate the transonic flows accurately and efficiently. The present work is mainly intended to establish an effective procedure for accurate and efficient flow analysis of complete aircraft configurations of paramount interest.

In the remaining sections we present coordinate systems, a brief description of the high-accuracy TVD Euler solver, boundary conditions and inter-block communication, numerical results, and conclusions.

Coordinate Svstems

Two Cartesian coordinate systems are used to describe the flow variables and grid geometry. One is a space-fixed (inertial) coordinate system (x,y,z) and time t and another is a grid-fixed (body-fixed) Cartesian coordinate system (X,Y,Z) and time 7 . The computational grid is assumed to

+ be rigid and translate at a constant velocity V

G relative to the space-fixed coordinate system. Then the two coordinate systems are related by the transformation

Here uG, vG, and wG are the x-, y - , and z-

+ components of the grid velocity VG and the two

systems are assumed to coincide with each other initially.

In addition, a grid-fixed curvilinear (computational) coordinate system (E,q,c) and time T is introduced for convenience of the following numerical formulation of the Euler equations. (E,q,c) is related to (X,Y,Z) by the transformation

and related to (x,y,z) by

Applying the chain rule of differentiation, the metric terms are derived as follows:

and

X Y Z + X Y Z S ' I E ' I E S

All the metric terms in Eq.(4) have been expressed in terms of the grid-fixed coordinates and the grid

+ velocity VG since they are known once the

computaional grid is given.

We shall denote the freestream velocity relative to the space-fixed coordinate system by

5 In aerodynamic applications it is often to

-, express Vm in terms of the freestream Mach number

Ma based on the freestream sound speed cm, angle of

attach am, and side slip angle defined such that where

um = cmMWcos /~,COS a,. va = - c m M m sin p,,

where urn, vm, and wm are the x-, y-, and z-

-+ components of Vm, respectively

TVD Semi-Discretization

Similarly, the grid velocity components are expressed as follows:

Let (i,j,k) be the cell-center index system of the block grid considered and i, j, and k are taken in the c - , q-, and 5- directions, respectively. Integrating Eq.(8) over the cell (i,j,k) in the computational space, we have the semi-discrete conservation-law form given by

G, vG = - c M sin j3 uG = c M cos p cos u m G G'

wG = c M cos p.sin a W G G G ( 6 )

where M G, aG, and pG are the grid Mach number based on cm, angle of attack, and side slip angle,

respectively. In our present program, both the freestream velocity and grid velocity relative to the space-fixed coordinate system may be specified to simulate the aircraft flight conditions.

A A

where E, F, and G are numerical or representative fluxes at the cell faces for whi~h discrete

conservation is considered, and Q is the representative conservative quantities (the

Euler Solver

A brief description of the 3-D upwind-biased high-accuracy TVD Euler solver developed by the second author is given in this section to bring out some essences of the scheme on which the solver is based. For details of the scheme, reader is referred to Refs. 20 and 22 and references cited therei.n.

numerical approximation to 0) considered conveniently to be the cell-center value. The half-integer subscripts denote cell faces and the integer subscripts the cell centers.

Let Governine Equations

In the,space-fixed Cartesian coordinate system (x,y,z) and time t, the unsteady 3-D Euler equations for an ideal perfect gas can be written in conservation-law form as

components of the representative surface normal considered and N denotes the set of these normal components. Let the Jacobian matrix of the flux f with respect to the dependent variable Q be denoted by a f / a Q . This Jacobian can also be called the coefficient matrix. Let A be a diagonal matrix

where the dependent variable Q and the fluxes E, F, and G are given by

having the eigenvalues ~'(i-1,2,3,4,5) of the coefficient matrix as its diagonal elements. Let I A l be a diagonal matrix having the absolute

eigenvalues I x ' ~ as its diagonal elements and let

A+= (A + lh/)/2 andh- - (A - lh1)/2 be the positive and negative projections of the eigenvalue matrix A . Finally, let L and R be the left and right eigenvector matrices corresponding to A and chosen such that LR=RL=I, the identity matrix (for explicit expressions of L and R, see Ref.22.) Then

A & A h

the numerical fluxes E,F.G = f are constructed

Q=

where e , P , p, and H denote the total energy per unit volume, density, pressure, and total enthalpy, respectively, and u, v, and w are the x - , y-, and

I

e P pu PV P w

z - omponents of the fluid velocity ?, respectively. The pressure and total enthalpy can be expressed in terms of the dependent variables as follows:

, E=

based on Roe's approximate Riemann Solver" as

I

PHU

PU2 pu + p Pvu PWu

Upon transforming to the computational coordinate system (E,q,C) and time 7 with the aid of the chain rule of differentiation, Eq. (7) becomes

PHW P w puw

P1 PW + P

, G= , F=

PHV PV

;?+ p P W

Here the subscrip m =i,j,k. For convenience of writing, the integer subscripts appearing in Eq.(9)

have been omitted. <+1/2 and &/2 are the

values of the dependent variables defined just to the right of and just to the left of the cell face m+1/2. These values will be defined shortly.

the values of R, L, A+, A - , and 1 ~ 1 evaluated based on the values of the flow variables at the cell

face by the ~oe's~' special averaging scheme using

the left and right states Q- m+1/2

and Q+ m+1/2 ' and

the cell-face normal N m+1/2 '

To define and Q.+~/~, let us define two

characteristic variables a and P, each being a column vector of 5 elements, as follows:

with

Next, we define the slope-limited values given by

where the compression parameter b is related to the accuracy parameter 4 used in Eq.(16) as follows:

The minmod slope-limiter operator is

Then, the left state at the cell face m+1/2 and the right state at the cell face m-1/2 are evaluated as follows :

with

Thz Roe's basic method does not satisfy entropy condition and thus permits stable expansion shocks. The diffusion component of the numerical flux can vanish at sonic rarefactions. In order to help Roe's Riemann Solver avoid expansion shocks,

only at sonic rarefactions (x'(Q m'Nm+1/2) < O < i

A (Qm+l,Nm+l/2)), the corresponding positive and

negative projections are redefined as

Some remarks concerning the numerical dissipation, TVD property, and accuracy of the scheme may be drawn from Eqs.(9)-(16). In Eq.(ll), the last term represents the numerical dissipation: As shown, it arises natually from the upwind differencing. The amount of dissipation is proportional to the absolute eigenvalues. To see the TVD property and accuracy, let us consider the 1-0 linear scalar wave equation

with f = XQ, af/aQ = A = constant. On a uniform grid of spacing Ax, Eq.(9) reduces to

with

and

To evaluate the RHS of Eq.(18), let us express the left and right states of Eq.(16) explicitly in terms of cell-center defined data and classify the data into four regions as follows:

Region 1: < 0 - That is, Qm is an

extremum.

b(am+ll) - That is, a smooth flow region

Region 3: a m+1/2. 'm-1/2 > O and lam+1/21 > b t ~ ~ _ ~ , ~ 1 - That is, a steep gradient-coarse grid

region. Here q4 ( - 1 ~ 4 ~ 1 ) is an accuracy parameter, which will become clear shortly in the following example.

Region 4 : ffm+1/2s > 0 and 18m-1/21 > blum+1,2/ - That is, a steep gradient-coarse grid

region.

am+1/2 = "m+1/2' Pm-1/2 = ba

m+1/2

It is noted that in most cases, regions 3 and 4 can be redefined as region 2 by refining grid distribution such that the neighboring slopes differ no more than a factor of b.

TM Property - Following ~ a d m o r ~ ~ , the time derivative of the total-variation TV[Q(t)] of the data set (4,) based on Eq.(18) is

where s m+1,2 = ~ign(Q,+~ - 4,). Since away from

extrema s m+1/2 = s ~ - ~ , ~ , the only contributions to

the sum of the RHS of Eq.(25) come from extrema

where s m+1/2 6 s ~ - ~ , ~ . At extrema, Qm+1/2 = Qm

given in Eq. (21) for region 1. Q;-~,~ may take the value of Eq.(22), Eq.(23), or Eq.(24). Therefore, for region 2,

for region 3,

and for region 4

It can be shown that the values of the terms enclosfd in the bracketes in Eqs.(26)-(28) are positive in the corresponding regions. Hence, we have

Substituting Eqs.(26)-(28) into Eq.(25) and applying the conditions of Eq.(29), we conclude that the total-variation TV[Q(t)] decreases in time and the scheme is TVD. It is very interesting to note that Eq.(29) implies that the maxima values decrease in time and minima values increase in time.

Accuracy - Consider Eq.(18) in the smooth flow region, i.e., region 2. Substituting Eqs.(22) and (19)-(20) into Eq.(18) and applying the Taylor series expansion about \ , we have the equivalent

differential equation of Eq.(18)

The RHS of Eq.(30) represents the truncation error of the scheme reference to Eq. (17). It is very obvious from Eq.(30) that the choice of $=1/3 results a third-order accuracy TVD scheme. The choice of b=-l results in a second-order accuracy fully.upwind TVD scheme. Fromm's scheme arises when #=O. The scheme reduces locally to a first- order accuracy scheme at extrema as indicated in Eq. (21).

Time Marching

For final update of Eq.(9) in time, several schemes are available in the computer code, including both explicit and implicit methods. For the results presented in this paper, an approximate factorization approach was used. Each dimensional factor was further simplified through diagonalization into a sequence of five scalar tridiagonal inversions rather than a block tridiagonal inversion involving 5x5 block n~atrices. For details of time marching schemes, reader is referred to Ref. 22 and references cited therein.

Boundary Conditions

To accommodate general block-grid geometries and a variety of external, internal, and integrated external/internal aerodynamic flow problems, a set of treatments of block-face boundary conditions has been implemented in the Euler solver described in the previous section in such 2 fashion that specifications of various block-face boundary conditions become a simple matter of choice of a numerical value of an input block-face boundary- condition indicator. This is done by subdividing each block face into a set of sub-regions, each being defined by the interval of IisisI2, JisjSJ2, and KlsksK2, where (i,j,k) are the block grid-point indices with i , j , and k along the f - , 7 - , and - coordinate curves, respectively. Each sub-region is identified with one of the following classifications of block face (BF): 1. degenerate BE, 2. symmetric BF, 3. block-interface BF, 4. periodic BF, 5. farfield supersonic-inflow BE, 6. farfield supersonic-outflow, 7. farfield subsonic- inflow BF, 8. farfield subsonic-outflow BF, 9. general farfield BF, 10. impermeable BF, 11. permeable BE with specified normal velocity, 12. perrneabie BE with specified normal Mach number, 13.

permeable BF with specified pressure, 14. general supersonic-inflow BF, 15. general supersonic- outflow BF, 16. general subsonic-inflow BF, and 17. general subsonic-outflow BE. only the treatments of the block faces used in our present calculations are discussed here. These are BE'S 1-3 and 5-9.

BF 1 - By degenerate BF, it is meant that the block-face area vanishes. In this case, the cell- face areas also vanish and the numerical fluxes of Eq.(ll) are set to zero.

BF 2 - For a symmetric BF, the left and right states of Eq.(16) are calculated by the method of image.

BF 3 - For a block-interface BF, data of the adjacent block are needed to construct the cell- face numerical fluxes as indicated in Eqs.(ll)- (16). To extract information from the adjacent block, we introduce a convenient index transformation matrix between the current and adjacent blocks. Let (i, j , k) be the current block grid-point index system and (I,m,n) the adjacenc block grid-point index system. Let indices of a grid point common to both (i, j ,k)- and (I,m,n)- systems be (io, j o, ko) with respect to the (i, j , k) -

system and (1 ,mo,no) with respect to the (I,m,n)-

system. Finally, let the unit vectors (consider (i,j,k) and (I,m,n) as Cartesian coordinate systems) along the I-, m - , and n-axes be expressed in the (i,j,k)-system as (Ii,I.,l ) , (mi,m. m ) ,

1 k J ' k and (n. ,n. ,nk) , respectively. hen the desired

1 1

transformation matrix takes the following form:

Once the grid-point indices (i,j ,k) and (2,m,n) are known at the cell face of interest, the cell-center defined indices of the adjace.nt block can be easily determined. With the cell-center defined indices of the adjacent block known, the cell-center defined data Q can be extracted for calculations of the left and right states of the cell fhce and the numerical fluxes. In our present program, the transformation matrix T, the common point indices (io, jo,ko) and (Io,mo,no) are

specified by input

BF's 5-9 - For these farfield cell faces, the cell-face state and that just outside of the cell face are calculated by the locally one-dimensional normal Riemann invariants based on the freestream condition and the state at the cell center of the cell considered as described in Ref. 24.

BF 10 - BE 10 is an impermeable surface across which there is no flow. In this case, the locally one-dimension normal Riemann invariants based on the cell-center state and the no flow condition are used to determine the cell-face state and that just outside of the cell considered. The implementation is implicit. That is, the cell-face state is treatei as an implicit function of the cell-center

state, and the function is coupled with Eq.(9) and solved simultaneously in time. This approach has so far provided a very stable numerical solutions.

Numerical Results

To investigate our approach to the Euler flow calculations of general geometries, numerical solutions were obtained for an NACA research-type wing-fuselage configuration at two flight

n conditions: 0.9 Mach and 6- angle of attack and 1.2

0 0

Mach and 6 angle of attack. The wing had 45 sweepback of the 0.25-chord line, an aspect ratio of 4, a taper ratio of 0.6, and NACA 65A006 airfoil sections parallel to the plane of symmetry. The fuselage had circular cross sections and a basic fineness ratio of 12. Very comprehensive pressure measurements are available in Ref. 25 for validating theoretical predictions. In the flow calculations, the grid-fixed (X,Y,Z) coordinate system was chosen with its origin at the nose of the fuselage and with the length of the fuselage stretching in the positive X-direction. The Z-axis was perpendicular to the plane of the wing chord and the Y-axis was chosen so that X+Y+Z completed a right-handed screw.

In grid generation, wing and fuselage surfaces were first represented separately by bi-cubic patch equations. The intersection of wing and fuselage was obtained by simultaneously solving the intersection of the wing and fuselage bi-cubic patch representations. The surface grid of the fuselage about the wing-fuselage intersection and the wing surface grid were generated using the integrated surface patch/surface elliptic grid generator of Ref. 18. Figure 1 shows a 3-D view of

the generated wing and fuselage surface grids. To generate the flowfield grids, the space YrO and ZrO with appropriate farfield distances was divided into 12 blocks, each being a nearly rectangular parallelepiped in shape. The surface grid generators of Refs. 18-19 were used to generate all the block-face grids by specifying the block-face edge grid distributions. Finally, an H-type grid was generated for each block independently of others using the 3-D elliptic grid generator of Ref. 19. To minimize inter-block communication during flow calculations, the 12 individually generated blocked grids were combined into 3 Euler solution blocks. The lower 3 solution blocks in the space ZSO were obtained from the generated upper 3 solution blocks by the method of image since the configuration was symmetric about the Z=O plane. The Y=O plane was treated as a BF 2 block face in the flow calculations since the flowfields we calculated were symmetric about this plane and no grid was needed. The grid sizes of these blocks are: 64x27~11 for blocks 1 and 4, 64x5~11 for blocks 2 and 5, and 74x27~11 for blocks 3 and 6 in the streamwise, spanwise, and vertical directions, respectively. There were 65x17 grid points on the wing surface and 64x30 grid points on the fuselage. The total number of grid points was 89012 including the duplicated grid points at the block interfaces.

Figures 2-6 shows various views of the soultion blocks. Figure 2 shows how the upper part of the flowfield was blocked into 3 solution blocks. As shown in this figure, the upstream block face of block 1 degenerated into a line coinciding with the Y-axis while the upstream block face of block 2 degenerated into a line coinciding with the Z-axis. Block 3 had a horizontal interface with block 1 and a vertical interface with block 2 from the downstream farfield up to the nose of the fuselage.

In front of the nose of the fuselage, block 3 and block 6 (the image of block 3, not shown in this figure) had an interface on the Z=0 plane.

An expanded 3-D view of the fuselage surface grid near the wing-fuselage intersection is given in Fig. 3. Also shown in this figure is a portion of the vertical symmetry-plane grid. Figure 4 is a top view of the grids on the wing-fuselage surface and on the horizontal symmetry-plane. Figure 5 shows a typical spanwise grid surface including the wing and fuselage cross sections. The six flow solution blocks are also identified. An expanded side view of a typical chordwise grid surface near the wing leading edge is shown in Fig. 6. In Fig. 5, the slope discontinuity of the vetical grid lines at the block interfaces can be seen clearly. The spanwise grid lines within the blocks and on the block faces exhibit large slope discontinuity across the vertical plane passing through the wing tip as shown in Figs. 2 and 4.

The present approach to construction of a computational grids greatly simplifies the flowfield blocking and the grid of a nearly rectangular parallelepiped is quite easy to generate and. control. However, how a grid should be constructed will ultimately be decided by the flow solver used in the light of whether flowfield can be calculated accurately and efficiently on the grid constructed. Therefore, in a complete flow analysis, one must simultaneously consfder both the gridding and flow calculation problems. In the following, two solutions calculated on the grid just described are presented.

All the calculations were started impulsively by suddenly setting the computational grid in motion at the flight velocity with the entire flowfield at rest and enforcing the impermeable boundary condition at the wing-fuselage surfaces. Figures 7-8 show the results and compare the prdicted with the experimental data in terms of surface pressure coefficients of the wing and fuselage. For the wing, the chordwise pressure distributions are presented at 20%, 40%, 60%, 80%, and 95% semispan stations. For the fuselage, the longitudinal pressure coefficients are presented at two radial locations as indicated in Figs. 7-(f) and 8-(f). All the symbols in Figs. 7-8 represent the measured data and were read directly from the figures of Ref. 25. The solid lines represent the calculated. As shown in Figs. 7-8, in the regions where the flows are attached and the viscous effects are not important, the agreement with exprement data is very good. Even with the H-type grid used in the present calculations, the peak negative pressure coefficients near the leading edge of the wing upper surface are predicted accurately by the theory. In Fig. 7- (e) , the theory does not agree with the data on the wing upper surface since the flow was separated as indicated by the data. In general, the calculated inviscid shocks are located somewhat behind the measured in the regions where the viscous effects are impcrtant as expected.

Figure 9 shows the solution convergence histories. In both cases CFL = 5 and local variable-time steps were used. In Fig. 9 , the RHS- Ll curves and DQC-L1 curves represent the 1,-norms

of the averaged RHS of Eq.(9) after multiplied by the time step and the corrections of all the five conservative flow variables, respectively. The calculations were performed on the CRAY XMP/14 computer. The computer time per time step is 8.64 seconds.

Conclusions

A multi-block grid of H-type with relaxed grid-line slope continuity within the blocks and at the block interfaces has been constructed for calculations of flowfields of a wing-fuselage configuration at transonic speeds. The correlations of the calculated flowfield solutions with experimental data demonstrate the ability of the high-accuracy TVD Euler solver to calculate the transonic flowfields accurately and efficiently on such a multi-block grid. With this ability of flow solver. the flowfield blockings and griddings of realistic aircraft configurations can be greatly simplified

Acknowledgment

This work was supported by the Rockwell International Independent Research and Development Program.

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21. Roe, P. L.,"Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes," Journal of Computational Physics, Vol. 43, 1981, pp. 357-372.

22. Chakravarthy, S. R. and Ota, D. K., "Numerical Issues in Computing Inviscid Supersonic Flow Over Conical Delta Wings," AIAA Paper NO. 86-0440, January 6-9, 1986.

23. Tadmor, E., "Convenient Total Variation Diminishing Conditions for Nonlinear Difference Schemes," ICASE REport No. 86-74, November, 1986.

24. Jameson, A. and Baker, T. J . , "Solution of the Euler Equations for Complex Configurations," AIAA Paper No. 83-1929, July, 1983.

25. Loviqg, D. L. and Estabrooks, B. B., "Transonic-Wing Investigation in the Langley 8-Foot High-speed Tunnel at High Subsonic Mach Numbers and at A Mach Number of 1.2 - Analysis of Pressure Distribution of.Wing- Fuselage Configuration Having

n a Wing of 45- Sweepback, Aspect Ratio 4, Taper Ratio 0.6, and NACA 65.4006 Airfoil Section," NACA -W L51f07, September 6, 1951.

Fig. 1 View of wing-fuselage configuration showing surface grids.

Fig. 2 3-D view of the blocked grids showing selected grid surfaces.

Fig. 3 Blow-up view of grid on fuselage surface and ver t ica l symmetry plane.

Fig. 4 Top view of grid on wing surface, fuselage surface, and horizontal symmetry plane.

0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X/CHORD (a) 2Y/b = 0 . 2

2YA - 0.4 RRR - 6 DEE. lWX - 0.9

g -0.4

.

0.0 4.0 0.0 12.0 16.0 20.0 Y

:ig. 5 View of a typical spanwise grid surface showing the s i x Euler solution blocks.

21.5 2i.7 2i.9 22.1 22.5 22.5 X

Fig. 6 Side view of a typical chordwise grid surface near the wing leading edge.

F i g . 7 Calculated and measured wing-fuselage surface pressure coeff icients a t Mach 0 . 9 and 6 degrees angle of at tack.

2Y/B - 0.8 RRJR - 6 OEQ. rrYH - 0.9

f I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X K H O R D (d) 2Y/b = 0 .8

0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X/CHORD (e) 2Y/b = 0.95

X/L (f) Fuselage

Fig. 7 (Concluded)

2Y/E - 0.2 RPHR - 5 OEG. M C H - 1.2

0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X K H O R D (a) 2Y/b = 0.2

2Y/B - 0.4 RLPHR - 6 OEG. WCii - 1.2

X/CHORD (b) 2Y/b = 0.4

Fig . 8 Calculated and measured wing-fuselage surface pressure coeff icients a t b!ach 1 . 2 and 6 degrees angle of at tack.

0.8 0.0 0.2 0.4 0.6 0.0 1.0 1.2

X/CHORD (d) 2Y/b = 0.8

2Y/B - 0.95 RLPm - 6 OEG. nmn - 1.2

0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X/CMORD (e) 2U/b = 0.95

0.4 / I I I I I 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2

X/L (f) Fuselage

Fig . 8 (Concluded)

CFL = 5 0

a = 6

- 5 . 0 1 1 0.0 tO.0 80.0 120.0 160.0 200.0

MRRCH ING STEPS

(a) blm = 0 . 9 and a = 6'

0.0 M.0 100.0 150.0 200.0 25'5.0 300.0 350.C MRRCH ING STEPS

(b) Mm = 1 . 2 and a = 6'

F i z . 9 So lu t ion convergence h i s t o r i e s f o r t he so lu t ions presented i n F igs . 7-8.