Pergamon

Gwnpurers & Fluia!s Vol. 24, No. 4, pp. 39HO0, 1995

00457930(94)ooo36-0 Copyright 0 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0045-7930/95 $9.50 + 0.00

AN AUTOMATIC GRID GENERATION PROCEDURE FOR (COMPLEX AIRCRAFT CONFIGURATIONS-f

SUSUMU TAKANASHI’ and MASAMI TAKEMOTO’ ‘Computation,al Aerodynamics Section, National Aerospace Laboratory, 7-44-l Jindaiji-Higashi, Chofu,

Tokyo 182, Japan

*Sanlto Ltd, Harada Bldg. 1105, 2-14-2 Takatanobaba, Shinjuku, Tokyo 169, Japan

(Received I6 March 1994; in revised form 6 October 1994)

Abstract-An automatic grid generation procedure based on a hybrid elliptic-parabolic method for complex aircraft configurations is presented.

The grid system consists of inner and outer grids. The inner grid near the body surface is generated by a parabolic equation with a second-order dissipation term. On the other hand, the outer grid with an arbitrarily specified outer boundary is generated by an elliptic method based on the electro-static theory. This hybrid method is shown to be applicable to any types of surface grids including unstructured (triangular) ones.

The Navier--Stokes simulation for an aircraft configuration was carried out using the block structured grid generated by the present hybrid method.

1. INTRODUCTION

Grid generation te’chniques have made great progress in the last decade [l, 21. Actually, a number of different methods have been developed and utilized for various kinds of numerical flow simulations. However, as far as highly complicated geometries such as a practical aircraft configuration are concerned, it seems to be still difficult to generate structured grids having sufficient smoothness and orthogonality. Thus, a more advanced methodology is desired to be developed for structured grid generation.

In view of this situation, the present authors [3] recently developed a hybrid electric-geometric method of generating block-structured grids for aircraft configurations. The method decomposes the grid into two different grids, namely inner and outer ones. The inner is generated near the body surface by a geometric method with a smoothing function similar to that of Chan and Steger [4], while the outer by the electric method developed by one of the present authors [S]. Although the method is applicable to complex geometries, the smoothing term used there is expressed in a finite difference form, which makes it difficult to automatize the grid generation for complex block-structured surface grids with a number of singular points appearing at the block boundaries.

The objective of this paper is to remove this difficulty from the original hybrid method, and also to simplify the dissipation (smoothing) function used in Ref. [3]. In the new method, the inner grid is generated by a parabolic equation with a second-order dissipation, which is discretized in a non-finite-dzfirence form for full automatic generation. The outer grid, on the other hand, is still generated by the same electro-static method. Since the electro-static field can also be regarded as a solution of an elliptic partial differential equation (Laplace’s equation), we call the present method ‘hybrid elliptic-parabolic method’. Special stress is laid on the point that the method is applicable to any types of su.rface grids, even to unstructured (triangular) ones like that proposed by K. Nakahashi [6].

2. PARABOLIC TYPE OF GRID GENERATION

A new parabolic type of grid generation is proposed here. For simplicity, we first consider the two-dimensional case. The extension to the three-dimensional case can be straightforwardly made as will be described in the latter part of this section.

tThis paper was presented at the 5th International Symposium on Computational Fluid Dynamics, Sendai, Japan, 3 I August- 3 September 1993.

393

394 Susumu Takanashi and Masami Takemoto

The present grid generation scheme is based on the parabolic equation

(1)

where 5 and c are the curvilinear coordinates in the circumferential and normal directions, respectively, r is the position vector of a grid point in space, &-/al is expressed as a specified function, which determines the local grid density in the c-direction as will be shown later, n is the normal unit vector with respect to the grid surface with [ = const., and D is a second-order dissipation for grid smoothing. The exact expression of D is given later. Using the one-side difference formula, we get the discrete form of equation (1):

rik - bk - I rik - rik - I Al = A[ nik - I + Dik (2)

Assuming Ai = 1, equation (2) is rewritten as

with

where

rik = ?ik + D,

i,k = rik _ 1 +Arik-,%-,,

Ar, _ 1 = Yik - Tik _ 1

(3)

If the dissipation term Dik is tentatively omitted from equation (3), this scheme is reduced to the step-by-step geometric grid generation, i.e. the kth grid point rik is recursively obtained from the previous grid point rik _ , starting at the boundary point with k = 1, as long as the local spacing Ar, _ , has been specified. This algorithm is simple, and furthermore it satisfies completely both orthogonality and minimum spacing near the body surface, but a serious difficulty will be encountered in the concave part of the 5 -line, because the grid lines in the C -direction (normal to the body) away from the surface tend to come close to each other and eventually cross over. Once this happens we can not proceed with the generation any more. Hence we need the dissipation term Dik for grid smoothing at each step. Such kind of dissipation has been used in a finite difference form by Chan and Steger [4] and the present authors [3]. However, the finite difference formulation is unsuitable for automatization of grid generation procedure in three-dimensional case as will be discussed later. To overcome this difficulty, first a newly developed dissipation function to meet our demand is presented, though it is still based on the original idea of Chan and Steger, secondly an alternative formulation without use of finite difference equations in three-dimensional case is proposed.

The dissipation D, is defined as

i3*r Dik = gik agZ

( >

ri+ Ik - 2rjk + rj _ ]k

ik = gik At2

(4)

with A4 = 1

where gik is a dissipation coefficient at the (i, k) point. The coefficient g, is defined as

g, =gp.gp.gp.gjp

where

(i) gil) = constant

(5)

nik-l’(rik-rik-I)

(iii) g’~‘=f(l(ri-lk_,-rik_,II+~I’i+(k-l-rik-lII)

for #I 2 0.1

for /3 <O

Automatic grid generation procedure for complex aircraft configurations 395

with

a, =50(&J a,= IO/z, nik-I.(ri-lk-l -rik-1 +Ti+lk-l -rik-1)

P=(IIri-,k-,-rik-,II+ I\ri+Ik-I-rik-III)’

where k,,, is the total number of grid points in the k-direction. The roles of these coefficients are as follows: the first coefficient (i) is a user-specified constant

of O(lO), which controls overall grid smoothness in space. The purpose of (ii) is only to maintain orthogonality as well as minimum spacing near the body surface. The coefficient (iii) is a grid aspect ratio, which serves to prevent the grid spacing in the k-direction from vanishing or becoming negative. The factor /3 in (iv) is the normalized scalar product of the normal vector and the resultant vector of the two adjacent segment vectors along the t-line. The coefficient (iv) plays a primarily important role for grid smoothing, because it has a large value in concave region where fi is positive, while a small value in convex region where /I is negative. It is noted that the value of /I is limited within 0.1 to prevent the dissipation from becoming too strong in an extremely concaved region in the vicinity of the body surface. The values of the constants appearing in the parameters a, and ~1~ have been empirically determined.

The grid spacing Ar, used here is written as

Ar, = Rik eXP(%&+) -eXP(@i&)

eV(ai&+) - exP(@,)

(6)

where Rjk is the shortest distance from the point (i, k) to the outer boundary, and cli is the parameter to control the minimum spacing which plays an important role in the Navier-Stokes simulation.

To efficiently solve equation (3) by iteration, we modify it on the assumption that the coefficient gik including rik is constant at each iteration step, therefore the iteration scheme is written as

where n expresses the iteration step. Let us next consider the three-dimensional case. First we define the body geometry by a

block-structured surface grid as shown in Fig. 1. (N.B. For clear view a coarse grid is given in this figure.) Unlike the two-dimensional case, the grid has several singular points, namely saddle point, accumulated point, and corner point as shown in Fig. 1. In contrast to a regular point, from which four grid lines are emanated, the number of the adjacent grid lines at a singular point are not four except a trivial case, which possibly happens at the accumulated point. A more complicated type of singularity also appears at the juncture of a wing-flap combination as shown in Fig. 2. The triangular gap expressed by dotted lines must be closed with an infinitely thin film to make grid generation easy. As a result, the singular points a and b have six adjacent grid lines, respectively, in this special surface grid topology. Prior to the implementation of grid generation, we need knowledge of these points as well as dummy points if the second-order dissipation term is discretized in a finite difference form in the two different directions at the block boundaries as in Ref. [3]. The work of seeking these points is not easy, but laborious and tedious for given surface grids with many blocks, which are actually needed for complex configurations. This fact prevents grid generation from full automatization, because it is inevitable to specify singular points as well as dummy points by man, not by computer. Furthermore it is difficult to uniquely define two different grid lines for the saddle point with six grid lines, moreover for the wing-flap juncture points which have six Ior more adjacent grid lines. This situation will become much more severe for unstructured surface grids. The primary objective of the present paper is to propose a new parabolic method applicable to a general surface grid topology or structure. (N.B. The electric method has no difficulty with respect to the type of surface grid structure.)

The grid generation scheme in three-dimensional case is written in discretized form as

CAF U,GD r& = f& + Dik (8)

396 Susumu Takanashi and Masami Takemoto

3# regular point

WING :-

\ FLAP Fig. 2. Surface grid topology for a simplified wing-flap

combination.

Fig. 1. Surface grid (coarse grid) of transonic aircraft and its singularities.

where

3, = rik _ 1 + Arik _ 1 nik _ 1 (9)

The dissipation D, consists of the terms concerning all the possible pairs (I$[), r$)) which are made from among the neighbouring points around the point rik. Let us denote the total number of the neighbouring points by I,,,, d$(l=I 2 I

and let us denote the segment vectors of grid lines by 7 ) which are defined as d$ = rjf - r,(Fig. 3) then D, is defined as 3 ’ “ 1 max 2

Dik = & F d!L) 4c:p, gk”“’ (~‘3 9 = 12 23 . * . 9 Lax ) (10)

where (I,,, C,) expresses a binomial coefficient, g$f”’ is the dissipation coefficient with respect to the pair (dj.j’), d$), and it consists of the four factors just as in the two-dimensional case:

Fig. 3. Neighbouring segment vectors around point (i, k). Fig. 4. Typical example of segment vectors and their relative

strengths of dissipation.

Automatic grid generation procedure for complex aircraft configurations 391

where

(i) g{/‘)(l) = constant

bk -- I . (rik - rik - I >

(iii) gtpq”3’ = ;( 11 d;ie)_ , (1 + 11 #_ ,I1 )

f(1 +O.lP for fi 20.1

(iv) gCfq)c4) = I i( > (1 :B)“i for&b <O.l a2

.-

1-B for B <O

with

’ = nik _ 1 ’ (d',f'_ 1 + d$_ 1)

II dk', , I; + II d$i 1 II

The dissipation Dik defined above can be regarded as the weighted average of the contributions of dissipation from the neighbouring points. The value of individual dissipation coefficient g$fq’ depends on the curvature of the kinked line connecting three points, namely rg), rik, I$). For example, if the line is concave with respect to the normal vector nik, g$ ( 34~~) has a large value (see Fig. 4).

We can always evaluate the dissipation coefficient regardless of grid topology or grid structure. Hence the present method is very general compared with the ordinary finite difference formulation for second-order dissipation.

The iterative scheme for solving equation (8) is written as

(12)

The two-dimensional dissipation Dik [equation (4)] is recovered by putting I,,,,, = 2 in equation (10).

3. GRID GENERATION BY HYBRID METHOD

A detailed procedure of grid generation based on the hybrid method for the complete aircraft configuration shown in Fig. 1 is presented here. This configuration is extremely complex, especially near the wing-pylon/pylon-nacelle junctures having severe concave regions. Thus the parabolic method is used for the inner grid generation. Actually, the present parabolic method is more robust than the electric one for strong discontinuous surface slopes in concave regions, since accurate electric charges are difficult to obtain in such regions. However, the application of the parabolic method may be limited to a thin-layer near the body surface, because the slope discontinuity of grid surface tends to be softened rapidly as the grid surface goes away from the body. The resulting inner grid is connected to the outer grid in the next step.

In order to generate the outer grid, we put positive electric charges on the transition points from inner to outer grids, also negative charges on the outer boundary points. The optimum charge distribution can be determined by imposing the condition that the electric force vectors coincide with the vectors which are normal to the transition surface as well as the outer boundary [5]. To preserve sufficient smoothness of grid lines at the transition points and at the few succeeding points in the k-direction, a special measure is taken here, i.e. the normal vector nik _ , in equation (9) is replaced by the weighted vector vi&, defined by

(1 -COo)ni~_,+COfik_,

II(l -w)nik-I +mfik-lII

for k, G k G k,

fik - I for k, -C k (13)

398 Susumu Takanashi and Masumi Takemoto

Fig. 5. Side view of grid.

with k-k,

W=k,

where fik _ , is the unit electric vector, k, denotes the transition surface, and k, denotes a surface, from which the pure elliptic generation starts without the aid of the parabolic equation. The values of k, and k, are determined so as to approximately satisfy the equations

:=0.3 and +=O.5 max max

At the same time, the amount of the dissipation Dik must be rapidly decreased, as the grid surface approaches to the specified outer boundary. More precisely, the dissipation coefficient gjf”’ is multiplied by the factor p defined by

(15)

Next, we show a three-dimensional grid for the transonic aircraft configuration [Fig. 1] generated by the present method. The grid surface is divided into 10 blocks in symmetrical flow case, namely upper and lower fuselages, inboard wing, outboard wing, horizontal-tail, vertical-tail, pylon, nacelle, inlet, and exit of engine. Accordingly, the whole space is divided into the same number of blocks. Figure 5 shows the side view of the space grid at the plane of symmetry. A front view of the grid around the fuselage-wing-pylon-nacelle combination is shown in Fig. 6. Figure 7 shows the surface grid at the outer boundary.

The grid shown above has 0-O topology, but O-C types of grids can be easily generated by imposing the fore-and-aft symmetry condition on both elliptic and parabolic equations. SC grids generated by the present method have been successfully applied to Navier-Stokes simulations for various aircraft configurations. Another important feature of the present hybrid method is that it has the ability to exactly specify the outer boundary configuration. Figure 8 shows a two-dimensional C-type grid with shock-fitted outer boundary. It was generated to obtain highly accurate Navier-Stokes solution for a linear stability analysis of laminar boundary layer about a swept infinite elliptical cylinder at a Mach number of 3.5 [7]. Since orthogonality condition is completely satisfied at the outer boundary, the strong bow shock wave in front of the body can be sharply captured. In this case, the following stretching function to determine the local spacing Ar, was used:

Arik = Rik [

arctan(k - /I) - arctan( - j3) 1 bL~

arctan(k,,. - 1 -/I)-arctan (16) where /I is the additional parameter to control the grid point density near the outer boundary.

Automatic grid generation procedure for complex aircraft configurations 399

Fig. 7. Surface grid at the outer boundary.

Fig. 8. Two-dimensional C grid for supersonic simulation.

In the above procedure of generating the outer grid, only the direction of the electric field vector was used. To improve the method, however, it will be desirable to utilize the magnitude of the electric vector to determine the grid spacing. According to the electro-static potential theory, the grid spacing can be regarded as a difference in potential, which is in inverse proportion to the electric strength. Thus, we propose an advanced technique here. That is, the spacing Ar, in equation (6) or equation (16) is replaced by the alternative spacing Ap, defined by

-

Apik=Arik(l -rr)+oAr,g rk

(17)

with

k-k,

b-h for k,Qk <k,

/z for k,<k<k,,, (‘*)

where Fik is the 1e:ngth of the electric vector at the point (i, k) and Fk is the average length of the vectors at the k-th grid surface. This newly developed technique has been proved to greatly enhance grid quality, especially orthogonality in intermediate regions between the inner and outer boundaries.

4. APPLICATION TO NAVIER-STOKES SIMULATION

A block-structured grid generated by the present method is applied to Navier-Stokes simulation for the transonic aircraft configuration without engine and pylon, for which the wind tunnel test data are available.

The computational algorithm employed to solve the Navier-Stokes equations is the finite volume method, which is based on the upwind TVD scheme with third-order accuracy in space [8]. Solutions are advanced in time implicitly in all the three directions in space inside the domain of each block, while explicitly in both the streamwise and spanwise directions, but still implicitly in the normal direction at the block interface boundaries.

Figure 9 shows .the computed pressure distributions at a Mach number of 0.82, an angle of attack

Susumu Takanashi and Masami Takemoto

CP

I il CP

-0.60’

-0.60’

Computation - Upper

---- Lower Experiment

0 Upper A Lower

Fig. 9. Pressure distributions for transonic aircraft configuration. M, = 0.82, G( = 2”. Re = 1.1 x 106.

of 2”, and a Reynolds number of 1.1 x 106. These pressure distributions are well compared with the experimental data obtained from the NAL 2m x 2m transonic wind tunnel test.

5. CONCLUSIONS

A new hybrid elliptic-parabolic method to automatically generate structured grids for complex aircraft configurations has been developed. The present method has a special feature that it is applicable to any type of surface grids, even to unstructured (triangular) ones.

The application of the structured grid to Navier-Stokes simulation for the transonic aircraft configuration was carried out. The computed pressure distributions are well compared with the wind tunnel test data.

Acknowledgements-The authors would like to thank Mr R. Ito and Mr M. Tachibana for the computational work concerning Navier-Stokes simulation.

The authors would also like to thank MS Y. Hashimoto and MS M. Taira for their help in preparing the manuscript of this paper.

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J. F. Thompson, Z. U. A. Warsi and C. W. Mastin, Numerical Grid Generation. North-Holland (1985). S. Sengupta, J. Hauser, P. R. Eiseman and J. F. Thompson, Numerical Grid Generation in Computational Fluid Mechanics. Pineridge Press (1988). S. Takanashi and M. Takemoto, A method of generating structured-grids for complex geometries and its application to the Navier-Stokes simulation. Comput. Fluid Dynamics J. 2, Japan Society of Computational Fluid Dynamics (1993). W. M. Chan and J. L. Steger, A generalized scheme for three-dimensional hyperbolic grid generation. AIAA Paper 91-1588-CP (1991). S. Takanashi, A simple algorithm for structured-grid generation with application to efficient Navier-Stokes computation. Computers Fluids 19 (1991). K. Nakahashi, Adaptive prismatic grid method for external viscous flow computations. AIAA Paper 93-33 14 (1993). T. Nomura, A linear stability analysis system for compressible boundary layers. Proceedings of the 25th Fluid Dynamics Conference, Japan (1993) in Japanese.

8. M. Tachibana and S. Takanashi, Numerical Simulation of Flow Fields Around an Airplane of Complex Geometry. National Aerospace Laboratory, SP-10 (1989) in Japanese.