A FINITE VOLUME METHOD FOR THE TWO-DIMENSIONAL EULER EQUATIONSWITH SOLUTION ADAPTATION ON UNSTRUCTURED MESHES

Majid Ahmadi Wahid S. Ghaly

Department of Mechanical Engineering, Concordia University,1455 de Maisonneuve W., Montreal, (QC), Canada, H3G 1M8., andCEntre de Recherche en Calcul Applique (CERCA), 5160 Boul. Decarie, Montreal, (QC), Canada, H3X 2H9.

ABSTRACTA cell-vertex finite volume method is used to discretize the Eu-ler equations on unstructured triangular meshes. A five-stageRunge-Kutta pseudo-time integration scheme is used to marchthe solution to steady state. Non-linear artificial viscosity isadded to eliminate pressure-velocity decoupling and to captureshocks. The boundary conditions at inflow and outflow are basedon the method of characteristics. Starting from a Delaunay meshtriangulation, an edge-based mesh adaptation library using meshreorientation and stretching is used to equi-distribute the solu-tion errors as the solution evolves in pseudo-time. To validate themethod, and demonstrate the usefulness of the mesh adaptationlibrary, numerical solutions are presented for some standard testcases.

KEY WORDS: finite volume method, non-linear artificial viscosity,unstructured triangular meshes, edge-based mesh adaptation.

NOMENCLATUREAi maximum eigenvalue of Euler equations

A�B coefficients matricesc speed of soundD discretized AV integralE total internal energy per unit mass

F� G flux vectors in the x and y directionH total enthalpy per unit mass

Ixx� Iyy� Ixy second order momenti� j� k index or unit Cartesian vectors

L pseudo-Laplace operatorM Mach numbern number of edges sharing a node

nxe � nye components of outward normal to an edge ep� stagnation pressurep static pressureQ discretized flux integral

Rx�Ry first order moment

R residual or gas constantSi surface area of the cell i associated with node is entropyT� stagnation temperatureT static temperature or matrix of eigenvectort timeU vector of conservative variables

u� v Cartesian velocity componentsw geometrical weight

x� y Cartesian coordinates� flow angle or coefficients of multistage time step-

ping� ratio of specific heats

�� � difference�T time-step�� boundary of �

�� ��ik � �

� ��ik coefficients for 2nd and 4th order AV terms

�x� �y Cartesian coefficients�� ��� �� �� constants associated with AV terms

density pressure switch

� volume of domain

Subscriptse edge index

i� k nodal indexnew the new conservative valuesold the old time step of conservative values

old � avg the old average valuesp primitive state vector

pred the predicted values

spec the specified values

Superscripts���� ��� the second and fourth order

n time level

1

AcronymsAV Artificial Viscosity

CFD Computational Fluid DynamicsCFL Courant-Fredrick-Lewy numberFVM Finite Volume Method

LIBOM Library of Mesh Optimization

1 INTRODUCTIONSeveral discretization methods have been implemented to simu-late inviscid fluid flow. Among them, the Finite Volume Method(FVM) has proved to be simple yet very efficient in computingsuch flows (e.g. Jameson et al. 1986 and Jameson 1995). Overthe last decade, there has been an increasing trend to use unstruc-tured meshes (as opposed to structured ones) in discretizing thecomputational domain; the main advantages being: the ease ofgenerating an isotropic mesh (using e.g. Delaunay triangulation)for an arbitrarily complex geometry, and the fact that it couplesnaturally and efficiently with solution-adaptive schemes. On theother hand, unstructured meshes inherently require indirect dataaddressing, which makes code vectorization and parallelization avery challenging task.

In this work a time-marching solution method for the two-dimensional Euler equations on purely triangular meshes is pre-sented. The time dependent integral form of the equations arediscretized in space using a cell-vertex FVM and integration inpseudo-time is performed using an explicit five-stage Runge-Kutta time-stepping procedure. Convergence acceleration isachieved by employing local time-stepping (Mavriplis, 1987).The non-linear artificial viscosity formulation advanced by Jame-son et al. (1986) combined with the pseudo-Laplacian discretiza-tion presented by Holmes and Connel (1989) provide a modelwith lower numerical losses and higher accuracy. The inflowand outflow boundary conditions are based on a linearized one-dimensional characteristic method, developed by Giles (1986).

As the flow contains regions of rapid or abrupt variationssuch as shocks whose locations are not known a priori, a meshadaptation library LIBOM with a posteriori local error estima-tor, described by Dompierre et al. (1995), is used to optimize themesh by equi-distributing the error. Hence, the calculations in-volve an optimum mesh (with a minimum number of nodes) andproduce relatively accurate results.

The governing equations, the discretization method and theboundary conditions implementation are first presented, the di-rectional solution-adaptive technique is then described. It isfollowed by the different test cases that validate the numericalmethod and demonstrate the usefulness of the solution-adaptivetechnique.

2 GOVERNING EQUATIONS AND DISCRETIZATIONThe continuity, momentum and energy equations, governing theunsteady two-dimensional flow of an inviscid fluid (loosely re-ferred to as the Euler equations) are written in conservative form

in a Cartesian coordinate system as follows:

�U

�t��F

�x��G

�y� � (1)

where U is a state vector of dependent variables and F andG arethe flux vectors in the x and y directions, and are given by:

U �

����

��u�v�E

���� , F �

����

�u�u� � p�uv�uH

���� , G �

����

�v�uv

�v� � p�vH

���� � (2)

Assuming that the fluid is an ideal gas thermally and calori-cally and, given the definition of total enthalpy H

H � E � p�� (3)

the pressure p can then be written as:

p � �� � �� �

�E � �

�

�u� � v�

(4)

where � is the ratio of specific heats. In what follows, all dis-tances are non-dimensionalized using an appropriate character-istic length and all flow variables are non-dimensionalized usingthe upstream stagnation pressure and temperature as follows:

p �p

p�� T �

T

T�� u,v �

u,vpRT�

� � ��

p� �RT�� (5)

2.1 Spatial DiscretizationA cell-vertex finite volume method is used to discretize the equa-tions of motion, Eqs. (1), on an unstructured triangularmesh. Thecomputational domain is divided into triangles, fixed in time, andthe flow variables are stored at their vertices. For any node, thecontrol volume (surface in 2D) is taken as the union of all trian-gles with a vertex at that node, i.e. the control volumes are over-lapping.

The governing equations, Eqs. (1), are then integrated overeach control volume � (surface in 2D) which is bounded by thesurface �� (curve in 2D), and using Gauss theorem (Green’s The-orem in 2D) one obtains

�

�t

ZZ�

U dx dy �

Z��

�Fdnx � Gdny� � �� (6)

The fluxesF andG along a particular edge of the control volumeare numerically evaluated as the average of the nodal flux valuesat the ends of that edge, which assumes a linear variation alongthe control volume edges and is second order accurate (Lindquist1988).

When the cell-vertex discretization scheme is applied toEq. (6), the following set of coupled ordinary differential equa-tions is obtained for each cell or control volume i surroundingnode i:

�

�t�SiUi� �

nXe��

�Fenxe � Genye� � � (7)

2

where the summation is taken over all edges of cell i, Si is thecell area, Ui is the solution vector, Fe andGe are the componentsof the flux vector on edge e, and nxe , nye are the components ofthe outward normal to edge e. The fluxesFe andGe are functionsof the flow variables at neighboring nodes, thus when Eq. (7) iswritten for all cells i, we get

�

�t�SiUi� � Q �Ui� � � (8)

where the convective operator Q �Ui� represents the discrete ap-proximation to the convective flux integral. It should be men-tioned that an edge-based data structure is used to construct theconnectivityof the unstructuredmesh and the convectivefluxbal-ance is computed in a single loop over the edges using indirectaddressing.

2.2 Artificial ViscosityArtificial viscosity terms are introduced to provide for pressure-velocity coupling, to capture shock waves and to diffuse non-physical numerical oscillations. The discretized Euler equations,Eqs. (8) are augmented by an artificial viscosity term as follows.

�

�t�SiUi� � Q �Ui��D �Ui� � � (9)

where D �Ui� represents the integral of the artificial viscosityterm. D �Ui� is formed of a blend of Laplacian and biharmonicterms (Jameson et al., 1986). The Laplacian is discretized us-ing an undivided pseudo-Laplacian, proposed by Holmes andConnell (1989), where an inexpensive approach using geometricweights is used

L �Ui� �nX

k��

wk�i �Uk � Ui� � (10)

where k represents all neighbors of node i. The weightswk�i arechosen such that the pseudo-Laplacian of a linear functionwill bezero, as would be the case for the true Laplacian. These weightswk�i are defined as

wk�i � � �wk�i � (11)

FollowingHolmes and Connell (1989), thewk�i are of the form

wk�i � �x,i �xk � xi� � �y,i �yk � yi� (12)

where

�x,i ��IxyRy � IyyRx�i�IxxIyy � I �

xy

i

� (13)

�y,i ��IxyRx � IxxRy�i�IxxIyy � I �

xy

i

(14)

and

Rx,i �nX

k��

�xk � xi� � Ry,i �nX

k��

�yk � yi� � (15)

Ixx,i �nX

k��

�xk � xi��� Iyy,i �

nXk��

�yk � yi��� (16)

Ixy,i �nX

k��

�xk � xi� �yk � yi� � (17)

The Laplacian term at each node i is then constructed as:

r�Ui � L �Ui� �nX

k��

wk�i �Uk � Ui� � (18)

The biharmonic AV term is formed by taking the pseudo-Laplacian of r�U , which is done in a second loop over all edges

r�Ui �nX

k��

�r�Uk �r�Ui

� (19)

The final form of the artificial viscosity term at node i is

D �Ui� �nX

k��

�Ak � Ai

�

�h� ��ik �Uk � Ui�

� � ��ik

�r�Uk �r�Ui

i(20)

where Ai is given by (Mavriplis, 1987):

Ai �nX

e��

juenxe � venye j� ce

qn�xe � n�ye � (21)

Physically,Ai represents the integral, over each cell i, of themax-imum eigenvalue of the Euler equations �juj�c� in the directionnormal to each cell edge. By taking the average ofAi andAk, thediffusion term becomes conservative in the U quantities.

To ensure that the AV is significant only in the vicinity ofshocks and oscillations, the coefficients � ��i and � ��i are madeproportional to the second derivative of the pressure and are de-fined as:

� ��ik � � ��max�i� �k� � (22)

� ��ik � max

h�� � �� �

� ��ik

i(23)

where the pressure switch � is given by:

�i �

nX

k��

wk�i �pk � pi�

nX

k��

�pk � pi�

(24)

and � �� and � �� are empirical coefficients taken as ��� and����� (Mavriplis, 1990).

3

2.3 Time IntegrationThe space-discretized Euler equations, given in Eqs. (9), aresolved for the steady state solution by pseudo-time marching.Given some initial guess for the flow field, Eqs. (9) become:

dUi

dt� R �Ui� � �, i � �� �� � � � � (25)

where R �Ui� is the residual

R �Ui� ��

SiQ �Ui��D �Ui�� � (26)

These equations are integrated in time using a fully explicitfive-stage hybrid time-stepping Runge-Kutta scheme, where theconvective flux operator Q �Ui� is evaluated at each stage in thetime step, and the AV term D �Ui� is evaluated only in the firsttwo stages, and is then frozen at that value. The solution is ad-vanced one time step t as follows

U� ��i � U n

i �

U� ��i � U

� ��i � ��

tiSi

hQ�U� ��i

��D

�U� ��i

�i�

U� ��i � U

� ��i � ��

tiSi

hQ�U� ��i

��D

�U� ��i

�i�

U� �i � U

� ��i � �

tiSi

hQ�U� ��i

��D

�U� ��i

�i�

U� ��i � U

� ��i � ��

tiSi

hQ�U� �i

��D

�U� ��i

�i�

U� �i � U

� ��i � �

tiSi

hQ�U� ��i

��D

�U� ��i

�i�

U n��i � U

� �i

(27)

where U ni and U n��

i are the values at the beginning and at theend of the nth time step and the coefficients ��s are:

�� ��

�� �� �

�

�� � �

�� �� �

�

�� � � �� (28)

This scheme represents a particular case of a large class ofhybrid time-stepping schemes, which have been specifically de-signed to produce strong damping characteristics of the high-frequency error modes. To accelerate the convergence to steadystate, the local time step for each cell i is set to the maximum sta-ble value and is given by:

ti � CFLSiAi

(29)

where CFL is Courant-Fredrick-Lewy number andAi is definedby Eq. (21).

2.4 Wall Boundary ConditionsFor an inviscid flow along an impermeable wall, the flow tan-gency condition implies that all fluxes through the wall faces van-ish except for the pressure contribution to the momentum flux.

However, since the flux through any edge on the wall is the aver-age of the two nodal fluxes associated with that edge, it is possiblethat the average flux vanishes while the nodal fluxes do not. Forthis reason, it was found necessary to explicitly impose the tan-gency condition.

2.5 Inflow and Outflow Boundary Conditions

The computation of inflow and outflow boundary conditions isbased upon a linearized one-dimensional characteristic method,described in Giles (1986). At each inflow/outflowboundary thereis a certain number of incoming/outgoing modes. The changesin the outgoing characteristic values are taken from the changespredicted by the flow solver. The changes in the incoming char-acteristics are determined such as to satisfy specified boundaryconditions. The flow angles, stagnation enthalpy and stagnationpressure are specified at inflow, while the static pressure is spec-ified at outflow.

Because the wave propagation normal to the boundary isdominant, variations parallel to the boundary may be neglectedand the linearized one-dimensional Euler equations, in primitivevariables, can be written as:

�Up

�t� A

�Up

�x� � (30)

where Up, the vector of primitive state variables and the coeffi-cient matrix A are

Up �

����

�uvp

���� , A �

�BBB�

u � � �

� u ��

�� � u �� �c� � u

�CCCA

old�avg

(31)

The reference state for evaluating the matrix A will be the stateon the boundary at the old time step. To make the matrix A con-stant, the average value of the state vector on the boundary willbe used to evaluate A. This state will be denoted by the subscript��old�avg.

It is clear that a number of approximation errors are being in-troduced in converting the nonlinear Euler equations into the lin-earized equations. For steady state calculations the error will beproportional to the square of the steady state perturbation at theinflow and outflow. These should be very small and may well beunnoticeable except for the case of an oblique shock at the out-flow.

Subsonic Inflow Boundary. For subsonic inflow, thereare three incoming waves and one outgoing wave. The outgoingcharacteristic wave is obtained from the flow solver whereas thethree incoming waves are specified by prescribing the values oftotal enthalpy H, entropy s, and flow angle �

4

H �

��

� � �

�p

���

�

�u� � v�

(32)

s � ln �p�� � ln ��� (33)

tan� � v�u (34)

The values of H, s and � are first computed by the flowsolver at each time step. Note that specifying the total enthalpyH and entropy s is equivalent to specifying the total temperatureand pressure. The change in H, s and � needed to bring thesecomputed values to the specified ones can be written in terms ofthe changes in the linearized characteristic variables using oneNewton iteration. The linearized characteristic variables are thenrelated to the primitive and/or conservative variables. Hence, tosatisfy the specified inlet boundary conditions, the change in theconservative variables at inlet becomes:

�Unew �

����� ���� ��u�� ��v�� ��E�

���� � M� � M�

uc� u� � v��M (35)

where

M� �

����

� � � �u � � �v � � ��

u� � v��� �u �v ��� � �

����new

�

M��

�����������

�u

cM���

�

���vc

�u�

u� � v�

c�

u�up

�� � �� ���v� u� �u

�

v�vp

�� � �� ��c � u�u�

�v�

�cu�cup�� � ��

���cv� u� u� � v�

�����������old�avg

�

M �

�����

�H�spec � �H�old�s�spec � �s�old

�tan��spec � �tan��old���c�old �upred � �ppred

����� �

M���� �

�pc

��u

� � ��u� � v�

c�

�

where the subscript ��pred refers to predicted values and the sub-script ��spec stands for the value which is specified by the inletflow conditions and the subscripts ��old and ��new refer to the val-ues at the old and new time steps.

Using the Eq. (35), the new value of the state vector U at theinlet nodes becomes

Unew � Uold � �Unew� (36)

Subsonic Outflow Boundary. For subsonic outflow,three outgoing waves are calculated from the numerical solution,while the incoming wave is fixed by specifying the static backpressure, pback. The value of p at each outlet node is calculatedfrom the flow solver and, similar to the subsonic inflow bound-ary, one Newton iteration is used to relate the change in staticpressure to the changes in the linearized characteristic variables.Hence the corresponding change in values of the conservativevariables at the outlet nodes becomes

�Unew �

��������

��c�

� ���c�

� ��

�c

���c

��

�c� �

� � � �

��������old�avg

�

�����

��c�old

��pred � �ppred

��c�old �vpred

��c�old �upred � �ppred

�p�spec � �p�old

����� �

(37)

The update is performed as in Eq. (36).

Supersonic Inflow Boundary. There are four incomingwaves for supersonic inflow which can be specified by inletMachnumber �M �spec, the inflow angle ���spec, and subsequently theflow condition will be determined from these variables usingisentropic relations.

Supersonic Outflow Boundary. There are four outgoingwaves for supersonic outflow which are predicted by the numer-ical solution.

3 MESH ADAPTATIONA mesh adaptation library LIBOM, described in Dompierre etal. (1995), has been used in the present work and is summarizedhere-after.

In any adaptive meshing procedure, there are two basic el-ements: a) an error estimator to quantify the mesh quality andgive directives formesh improvement so as tominimize and equi-distribute the estimated errors, and b) a mesher that would im-plement the directives given by the error estimator and create theadapted mesh.

The ideal error estimator is the discretization error kUexact�Unumericalk, however, as this error is inaccessible, the interpola-tion error kUexact�Uinterpolatedk�which bounds the actual error,will be used as error estimator. When the discretization is basedon a piece-wise constant function, e.g. a cell centered FVM, theinterpolation error is obtained from the solution gradients. How-ever, when piecewise linear functions are used, e.g. cell-vertexFVM, the error based on the gradient becomes less effective, andadaptationmust be based on second derivatives, i.e. themesh will

5

be refined in regionswhere the solution curvature is large. There-fore, the error estimator is based on second derivatives, that is theHessian matrix of the numerical solutionUnumerical. The secondderivatives of a piecewise linear function are then evaluated ina weak sense using integration by parts. The eigenvalues of thisHessian matrix correspond to the error scales and the eigenvec-tors give the principal directions of these errors. Hence, this errorestimator provides directional error estimates, more details can befound in Dompierre et al. (1995).

The next task is to construct a mesh that will equi-distributethe calculated error. To keep the directional character of the er-ror estimator, the error over the edges is evaluated and is used toconstruct an anisotropicmesh with triangles and/or cells stretchedin the direction of maximum error. Starting from the currentmesh, this new mesh is constructed using four simple local op-erations: mesh refinement, mesh coarsening, node relocation andnode reconnection. Each of these operations has the effect ofequi-distributing the error estimate over the edges. Convergenceof this mesh adaptation process is demonstrated in Fortin et al.(1996).

The mesh adaptation library LIBOM is solver-independent.Given a starting mesh, the solver is first called for a certain num-ber of iterations. The solution thus obtained is passed on to themesh adaptor which adapts it as described above, the control isthen passed on to the solver and so on. This process is repeateduntil convergence of both mesher and solver as shown in Valletet al. (1996).

In the present work, the solution was started on a uniformDelaunay mesh and the residual was reduced by two orders ofmagnitude, the solver-mesher loopwas then started and the solverwas allowed to converge by one to two orders of magnitude be-fore remeshing. Typically 5 to 10 mesher-solver iterations wereneeded to obtain a converged solution. The time taken by themesher is negligible compared to that taken by the solver.

4 RESULTSIn this section, the FVM solver described in x 2 is validated andthe mesher described in the previous section are applied to sub-sonic, transonic and supersonic test cases to show the merits ofboth solver and mesher.

4.1 Supersonic compression cornerThe first test case is that of a supersonic compression corner withan inlet Mach number of �� and a wedge angle of ���. Figure 1shows the initialDelaunay mesh composed of 8583 nodes and theiso-Mach lines computed on that mesh, whereas Fig. 2 shows theadapted mesh with 1188 nodes and the corresponding solution.

A close view of the shock region on the initial and adaptedmeshes, given in Fig. 3, shows that the shock is captured on twoto three triangles (i.e. one to two cells). However, on the adaptedmesh, the elements in the shock region have an aspect ratio ofabout ��, i.e. the shock is �� times thinner. The computed shock

angle is ����� compared to the exact value of �����, see e.g. An-derson (1990).

4.2 A ��� wedge in a channel

The supersonic (M � ���) and subsonic (M � ���) flows over a��� wedge in a channel are considered. The results obtained forsupersonic test case using Delaunay triangulation and those ob-tained using directional adaptation are compared in Fig. 4. Thesupersonic flow features, namely the oblique shock at the begin-ning of the ramp, the expansion fan at the end of the ramp andthe shock reflection are well captured. The convergence historyfor the initial mesh is shown in Fig. 5. The convergence rate isunaffected by mesh adaptation.

Figure 6 presents iso-Mach lines for the subsonic case, whereM � ��� at the outflow boundary. The smoothness of the Machisolines reflects the high-accuracy of theAV scheme. As the rampstarts near the inflow section, it was not possible to obtain a con-verged solution when a zero order space extrapolation was usedas boundary condition at inflow, therefore the current boundarycondition implementation has proved to be relatively more ro-bust.

4.3 GAMM-Bump with ���� and ��

The third test case is that of the transonic and supersonic flows ina channel with a ���� and ���� circular arc bumps, respectively.The transonic case is taken from the GAMM-test cases (Rizzi andViviand, 1981). The exit Mach number for the transonic case is���� and inlet Mach number for the supersonic one is ���.

In the case of the transonic bump, the flow accelerates overthe first half bump, becomes supersonic and, as the flow decel-erates over the second half of the bump, a normal shock occursnear the bump trailing edge. The numerical results for Cp alongthe lower wall of the transonic GAMM-test case (����) agreesrather well with the experimental values, as shown in Fig. 7. TheMach contours, presented in Fig. 8, demonstrate the shock cap-turing capability of the FVM solver.

For the supersonic bump, compression shocks form at thebump leading and trailing edges, reflect from the wall and inter-act. Figure 9 shows the shock-shock and shock-wall interactionsfor the original and the adapted meshes. The shocks are capturedon the initial mesh and are much thinner on the adapted mesh.

4.4 Blunt body

The supersonic flow (M � ��) around a circular blunt body isconsidered. The adapted mesh and the Mach contours are shownin Fig. 10. The detached bow shock is accurately captured. Theadapted mesh indicates that the solution second derivative is highonly in the shock region and is more or less uniform downstreamof the shock where theflow decelerates and turns around the bluntbody. It should be mentioned that, for this particular case, thesolution obtained on the initial (unadapted) mesh did capture theshock however, it did not correctly predict the shock strength.

6

5 CONCLUSIONA cell-vertex finite volume method has been devised for solvingthe two-dimensional Euler equations. The steady state solutionis reached by pseudo-time marching the Euler equations usingan explicit five-stage Runge-Kutta scheme. Local time steppingwas used for convergence acceleration. The non-linear blend ofsecond and fourth order AV was found to be successful in cap-turing shocks and eliminating pressure-velocity decoupling withminimal numerical diffusion. The method of characteristics, usedto impose inflow and outflow boundary conditions, allowed forplacing the boundary relatively closer to the body. A new direc-tional solution-adaptive library based on second derivatives hasproved to be very effective in optimizing the solution and themesh. The ability of the FVM solver and the mesh adaptation li-brary to treat subsonic, transonic and supersonic flows has beendemonstrated.

ACKNOWLEDGMENTSThe authors would like to thank Dr. Dompierre who, during hispost-doctoral fellowship at CERCA, provided the authors withthe mesh adaptation library LIBOM and assisted them in us-ing it. This work was supported under the National Scienceand Engineering Research Council of Canada, NSERC Grant#QGP0170337 and the Concordia University Faculty ResearchDevelopment Program.

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Mavriplis D.J., 1990, Accurate Multigrid Solution of the EulerEquations on Unstructured and Adaptive Meshes, AIAAJournal, Vol. 26, no. 2, 1990, pp. 213–221.

Rizzi, A., and Viviand, H., 1981, Numerical Methods forthe Computation of Inviscid Transonic Flows with ShockWaves, Notes on Numerical Fluid Mechanics, Vol. 26.

Vallet, M.-G., Dompierre, J., Bourgault, Y. Fortin, M. andHabashi, W.G., 1996, Coupling Flow Solvers and Gridthrough an Edge-Based Adaptive Grid Method, ASMEFluidsEngineeringConference, July 1996, San Diego, CA.

7

A.

B.

Figure 1. Initial mesh (A) and the corresponding solution (B) for

the supersonic compression corner.

A.

B.

Figure 2. Adapted mesh (A) and the corresponding solution (B)

for the supersonic compression corner.

8

A. B.

C. D.

Figure 3. Close-up view of the shock region on the initial (A) and

adapted (C) meshes with the corresponding solutions (B) and (D)for the supersonic compression corner.

A.

B.

Figure 4. Iso-Mach lines for the original (A) and the adapted mesh(B) for supersonic wedge (M � �).

1e-20

1e-10

1

0 200 400 600 800 1000 1200 1400

L2-N

orm

Iterations

Convergence History

Figure 5. Convergence history for the supersonic wedge with an

inlet Mach number of �.

Figure 6. Iso-Mach lines for subsonic ��� wedge test case.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

-Cp

X/C

Pressure Coefficient on the Lower Wall

Experimental : _______

Numerical : __ __ __

Figure 7. Computed and experimental values of Cp for transonic

GAMM-test case.

9

Figure 8. Mach Isolines for the transonic bump GAMM-test case.

A.

B.

Figure 9. Iso-Mach lines for the original (A) and the adapted mesh

(B) for the supersonic bump (M � ���).

A. B.

Figure 10. The anisotropic adapted mesh (A) and solution (B) for

supersonic blunt body (M � ���).

10