An adaptive embedded mesh procedure for leading-edge ... - Deep Blue

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d 370 L'Enfant Promenade, S.W.,

An Adaptive Embedded Mesh Procedure for Leading-Edge Vortex Flows

L/

Kenneth G. Powell' Michael A. Beer+ Glenn W. Law+ Dcpartment of Aerospace Engineering T h e [Jniversity of Michigan

Abstract

resolution is provided where necessary. The redistribution may he done in any of several ways. Methods t,hat. have been used include:

A procedure for solving the conical Euler equations on an adaptively refined mesh is prcsented, along with a a method On first and second Of method for determining which cells to refine. The s o h for example, [l]); physical quantities (see, tion Drocedure is a central-difference cell-vertex scheme. The adaptation procedure is made up of a parameter on a method based on a spring-mass analogy where which the refinement decision is based, and a method the spring constants depend upon flow parameters for choosing a threshold value of the parameter. The re(see, for example [2]); finement parameter is a measure of mesh-converzence. constructed by comparison of locally coarse- and finea method based on variational principles (see, for grid solutions. T h e threshold for the refinement paramexample [3]). eter is based on the curvature of the curve relating the flagged for refinement to the value of These and other adaptive redistribution methods are number of Of Thompson r4i and the refinement threshold. &sults for three test cases Outlined in the review L5]. are presented. The test problem is that of a delta wing In the Other approach, the adaptive refinement at angle of attack in a supersonic free.strem. The re. approach, mesh Points are added by sudividing cells suiting vortices and shocks are captured by that are flagged for division. These methods include: the adaptive code. ones in which entire zones are divided, yielding logically rectangular subgrids (see for example [6,7]);

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d

Introduction

and ones in which single cells may he divided at, random, yielding unstructured meshes (see for example [8,9,101).

One of the primary challenges of computational fluid dynamics is that of disparate length scales. Due to nonlinearities in the equations of motion for a fluid, even the flow about a geometrically simple body can lead to localized high-gradient regions. The Euler equations can give rise to shocks, vortices, slip surfaces and contact discontinuities. More grid points are needed t o resolve these high-gradient regions than would he necessary to resolve low-gradient regions. Since added grid points are expensive, both in t e r m of memory and computation, it is desirable to place the points where they are needed, and, conversely, to avoid placing them where they do not add to the quality of the solution. This is the basis of adaptive-mesh schemes. There are two basic approaches in adaptive-mesh schemes. In one, the number of grid points is held constant, and the grid points are resditributed so that high ~

These methods are outlined in the review article of Berger [6]. The approach taken in this paper is that of adaptive refinement, in which any cell or group of cells may be refined, leading to an unstructured mesh. Since, in general, the location of the high-gradient regions is not known in advance, a method of sensing them must he developed. This gives rise to the need to formulate a refinement parameter. If, for instance, shock waves are to he detected, pressure gradient could be be used as a parameter. That is, the pressure grarlient could be calculated in each cell, and cells with high gradients of pressure would he refined. If, however, slip lines are to be detected, pressure gradient would not be a good parameter, since pressure is constant across a slip-line. Thus the primary goal in designing a refinement parameter is that it should he general enough to detect any region that requires refinement.

'Assistant Professor, A I A A Member tResearch Assistant, AIAA Member

Copyright 0 American Institute of Aeronautics and Astronautics, Inc., 1989. All rights reserved.

1

W'

v

' 7 = -Y Once a refinement parameter has been chosen, this X gives a way of determining to what degree the sohtion would be improved by refining a given cell. In an { = E X adaptive redistribution method, grrd points are redistributed based on this parameter in an analog manner. In an adaptive refinement method, the relation is digi- and assuming conical self-similarity (that the solution tal; a cell is either flagged for refinement or left alone. is independent of r), the Euler equations become For the adaptive refinement scheme, then, a threshold must be set, so that cells with a value of the refinement PUC '1P parameter greater than the threshold value are refined. Determination of this threshold also must be as general as possible. Setting a fixed value of the threshold or PE a fixed percentage of cells t o be refined is not general enough. The only remaining problem is to design a scheme that works on an unstructured mesh that has been adaptively refined. These three components; the SD lution scheme, the refinement criterion and the refinement threshold are described helow for the solution of the conical Euler equations about a delta wing in suwhere personic flow. This problem makes a particularly good test problem for adaptive-refinement schemes, because c = ui'1vicw of the presence of shocks, vortex cores and vortex sheets . in the flows [11,12,13]. v = v-'1u

-

G

Governing Equations

-

= w-{u

and

The governing equations applied here are the conical Euler eauations. They are derived from the threedimensional Euler equations via an assumption of conical self-similarity. The three-dimensional Euler equations may be written in vector form as

K

=J -

These conical Euler equations may be expressed in terms of the state vector U, the Cartesian flux vector F and the conical flux vectors G and H as

The unsteady terms have been included so that an iterative procedure may be employed to reach a conically self-similar steady-state. The equations are solved on the unit sphere by setting r = 1 . They could be solved for any value of r and would give the same solution.

Solution Procedure

where ho is the stagnation enthalpy,

The solution scheme used is a finite-volume, multi-stage scheme in which the state variables are stored at the nodes (Le. a cell-vertex scheme). Similar schemes have been used by other researchers [14,15]. The scheme used here is formulated for an unstructured mesh, and allows for different cells in the mesh to be at different levels of refinement. The grid generation, spatial discretization, added artificial viscosity, temporal discretization and boundary and interface procedures arc described below, along with the data structure t h a t undcrlies the algorithm.

ho = E + PP

The ideal gas law, which may be written as

v

closes the set of equations. Introducing the conical variables

2

Grid Generation

Added Artificial Viscosity

For the cases presented here, the grid generation is carried out by a Joukowsky transformation. The ( q X ) plane is mapped to a complex ,y plane, in which the wing becomes a circle, by the transformation

The added artificial viscosity is a blend of a nonlinear second-difference and a linear fourth-difference. It" is constructed from a weighted Laplacian, Z,and an unweighted Laplacian squared, L', respectively. The weighting function W is a normalized Laplacian of the messure.

where A is the leading-edge sweep of the wing. In the plane, i = constant lines are equiangularly spaced rays emanating from the origin and j = constant lines are concentric rings. Grid points are generated along rays in the x plane by a n exponential stret.ching. This procedure yields near-conformal grids with good resolution near the wing. A grid generated by this procedure forms the base grid, i.e., the grid before adaptive local refinement has taken place. The refinement is handled by sub-dividing a cell into four sub-cells. This is done by adding a node on each face of the cell to be divided, and one in the center of the cell. Bilinear interpolations are used to find the coordinates of tlie new points. A typical grid is shown in Figure 3.

where 11 /Im denotes the L , norm, so that 0 < W < 1. This weighting is chasen for the efficient capturing of shocks and vortices. It causes the second-difference term to be first-order in high-gradient regions and small elsewhere; the fourth-difference term is third-order everywhere. Thus the artificial viscosity is given by

x

D ( 6 ) = r z D g ( 6 ) - ~qDq(fJ)

= €?Z(W,fJ)- €*L2(fJ) where U is a modified state vector, with the energy term p E replaced by pho, so that the discrete equations permit a solution with constant total enthalpy.

Temporal Discretization of Equations

Spatial Discretization of Equations

The spatial d-iaation resulted in a semi-discrete The finite-volume discretization of the partial differen- equation for aU/at, the residual for a cell. The temtial equations is formulated by integrating the conical poral d i s c r e s o n is made up of two process: one t o y . Euler equations over a cell. This gives distribute bU/at to the nodes, and one to integrate the changes at the nodes. For the first process, a simple distribution is used. For the second, a multi-stage scheme is used. The multi-stage scheme is given by, for iteration n, Using Gauss' theorem and tlie mean value theorem, this u(0) = U" may be rewritten as

(i, i, i, a)

where an overbar denotes a cell-average and A is the cell area. The line integral of the fluxes is carried out by a trapezoidal integration about the cell, Le. Unt1

an

[Gdc - H d q ]

=

=

u(k)

where C is the convective operator

and D is the damping operator where the subscripts denote the two nodes that define the face, ordered so that the integral is carried out in a counter-clockwise sense. The cell-average of the source term is calculated by The time-step factor, AtKlrA, is given by averaeine the source term at t,he four nodes defining the cell. This gives the residual at the center of the cell, 8UlOt.

"-

-

-

-

3

L

v

where A is the CFL number and n, is the i t h c o m p e nent of the non-normalized face normal. In the cases presented here, a four-stage scheme was chosen, with coefficients 1 a1 = 4 a3

1

=2

/\

1

a2

=3

a4

=I

Boundary Conditions

Figure 1: Data Structure

There are boundary conditions to he enforced at the physical boundaries; ensuring conservation. The artificial viscosity is constructed in t e r m of face-centered first differences, with 2. Free-sfream conditions upstream of the how-shock, any difference that is added to one node of the face subtracted from the other node. This ensures that the 3. Kutta condition at the leading-edges; artificial viscosity is conservative, even at boundaries and interfaces. and at the numerical boundaries; 1. No flux through the wing,

1. no flow through the symmetry plane for zero-yaw

cases,

Local refinement of cells leads to grids that are inherently unstructured. That is, they cannot be “unwrapped” to give an ( i , j ) grid in computational space. The wing boundary condition is met by retaining only This leads to the necessity of a different data structure the pressure terms in the flux calculation on cell faces than is normally used in CFD codes. The data strucwhich abut the wing. No pressure extrapolation is necture used in this paper is shown in Figure 1. Each cell, essary since the state variables are stored at the nodes. each face and each node are numbered. The following The free-stream boundary condition is implemented arrays describe the connectivity of the grid: by ensuring that the outer boundary of the domain is outside the bow shock and enforcing free-stream condia pointer from each cell t o its nodes; tions there. a pointer from each cell to its faces; The Kutta condition is enforced implicitly by ensuring that the artificial viscosity is present near the a pointer from each cell to its neighbors; leading-edges. This is done by arbitrarily setting the pressure switch W to one at several (four or five) nodes 8 a pointer from each node to its neighbors; in the vicinity of the leading-edges. Numerical expea pointers from each face to its nodes; rience demonstrates that this is adequate to ensure smooth separation [ I l l . a pointer to each interface cell; The symmetry plane is introduced so that the flow a pointer to each boundary node past a wing at zero yaw may be solved on a half-plane. The condition to he enforced at the symmetry line is This choice of data structure is not unique. Many of that the through-flow velocity, zi, is zcro. This is imthe arrays are not “necessary,” but help to speed up plemented by setting 21 to zero initially and zeroing the the computation. There is a trade-off between stordistributed flux for the y-direction momentum equation age and computation in the design of a data structure; at each iteration. the more connectivity arrays available, the quicker the The condition of conservation at the embedding incomputation. terfaces is met by careful treatment of the flux integration and the artificial viscosity calculation there. Proper treatment of the flux integration is ensured by Refinement Parameter Choice treating the midpoint on the face at which a coarse cell abuts a fine cell as a dummy point. That is, the state To develop an adaptive method of deciding which cells vector and flux vector at that point are taken to be the to refine, some parameter must chosen on which t o base average of the two endpoints of the face. In this way, the decision This parameter can be physical or nilthe flux that the coarse cell sees through this face is merical in its hasis. For a physical refinement paramexactly the same as the sum of the two half-faces, thus eter, for instance, cells in which the pressure gradient 2. conservation at the embedding interfaces.

I

i_r

Data Structure

Ma = 1.40 a = 20 Total pressure loss ~~

D =0

A

= 75

128 x 128

1.00

I

I 0.50

tal pressure

M,=1.40

= 20 p = 0 A = 75 128 x 128

~

M,=1.40

12d x 12d

loss

1.00 1 0.0000 1 0.0500 3 0.1000 4 0.1500 5 0.2000 6 0.2500 7 0 3000 8 0.3500 9 0.4000 10 0.4500 11 0.5000 12 0.5500 13 0.6000 14 0.6500 1 5 0 7000 16 0 7500 17 0.8000 18 0.8500 19 0.9000 20 0.9500

0.7!

0.51

(I. 0.25

0.00

, 0.15 , , , , , , , , 0 50 0.75 1.00

,I

tours on Initial Grid

1.01

-0.25~ 0.00

16x16

Figure 30: Case 2 - Cross-Flow Mach Number Con-

Final Grid

= 20 p = 0 A = 75

a = 2 0 p=O A=75

1

A=75

12dx12d

0.0000 0 1000 0.2000 0.3000 0.4000

0.75

0.5000

0.6000 0.7000

0.50

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