Unstructured High-Order Accurate Finite-Volume ...

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5 - 8 January 2009, Orlando, Florida

AIAA 2009-954

Unstructured High-Order Accurate Finite-Volume Solutions of the Navier-Stokes Equations Chris Michalak

∗

† and Carl Ollivier-Gooch

Advanced Numerical Simulation Laboratory University of British Columbia

A nite-volume scheme utilizing a fourth-order reconstruction for convective and diusive uxes of the Navier-Stokes equations is presented. The reconstruction is obtained eciently by precomputing the pseudo-inverse of the reconstruction matrix. Viscous boundary conditions are handled by introducing additional rows in the reconstruction matrices. Rapid convergence of steady-state problems is made possible by computing the exact ux Jacobian and the use of a Newton-GMRES algorithm. A grid convergence study of cylindrical Couette ow is used to demonstrate the order of accuracy of the scheme. Laminar ow over an airfoil on a sequence of anisotropic grids is also used to qualitatively demonstrate the superior accuracy of the fourth-order scheme. I.

Introduction

Unstructured grid nite-volume methods have seen wide acceptance in recent times due to their ecient handling of complex geometries. Though third-order schemes for the advective uxes have been known1 for almost 20 years, they have seen little use. The number of details which must be properly implemented, such as the correct treatment of curved boundaries, is likely partially to blame. Furthermore, the application of convergence acceleration techniques is also more complicated than for the second-order schemes. Despite this we have recently had success in demonstrating the accuracy of third- and fourth-order schemes for the Euler equations.2 Just as importantly, we have had success in devising convergence acceleration techniques for these schemes.35 The result is that we have successfully demonstrated that third-, and fourth-order schemes can be more computational ecient at solving the Euler equations for a given level of accuracy than second-order schemes. We now aim to extend this work to the Navier-Stokes equations. Though other researchers6, 7 have previously used third-order discretizations for the advective terms, they used traditional second-order discretizations for the viscous terms. The present work is an extension of work previously carried out8 demonstrating the fourth-order accurate solution of the advection-diusion equation. Although the simulation of compressible turbulent ows for aerodynamics applications is our ultimate goal, the present work is restricted to subsonic laminar compressible ows. In Section II we demonstrate how k-exact reconstruction including boundary conditions can be eciently carried out. The evaluation of advective and diusive uxes using this reconstruction is addressed Section III. Finally, in Section IV, we present numerical experiments demonstrating the accuracy of the method and its computational eciency. II.

Solution Reconstruction

The basis of the present work is k-exact least-squares reconstruction9 which has been demonstrated to achieve higher-order accuracy for the solution of the Euler equation.2 This approach has previously been extended to include the diusive uxes of the advection-diusion equation.8 Our current approach is similar to this previous work, but includes some additional eciency considerations, particularly with regards to ∗ PhD student, Student member AIAA. [email protected] † Associate Professor, Member AIAA. [email protected]

1 of 13 American Institute of Aeronautics and Astronautics Copyright © 2009 by Chris Michalak. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

the handling of boundary condition enforcement. Though the ux is evaluated in the conserved variables (ρ, ρu, ρv, E), as is common we reconstruct the primitive variables (ρ, u, v, P ). II.A.

Reconstruction for interior control-volumes

In the nite-volume method, the domain is tessellated into non-overlapping control volumes. Although our approach is equally applicable to vertex- and cell-centered control-volume discretizations, only results using vertex-centered grids will be presented herein. Each control volume Vi has a geometric reference point ~xi . This reference point is taken as the cell centroid for cell-centered control volumes and the vertex for vertex-centered control volumes. For any smooth function u(~x) and its control-volume averaged values ui , the k-exact least-squares reconstruction will use a compact stencil in the neighborhood of control volume i to compute an expansion Ri (~x − ~xi ) that conserves the mean in control volume i and reconstructs exactly polynomials of degree ≤ k (equivalently, Ri (~x − ~xi ) − u(~x) = O ∆xk+1 ) . Conservation of the mean requires that the average of the reconstructed function Ri and the original function u over control volume i be the same: ˆ

1 Vi

Ri (~x − ~xi ) dA = Vi

1 Vi

ˆ

u (~x) dA ≡ ui .

(1)

Vi

The expansion Ri (~x − ~xi ) can be written as: Ri (~x − ~xi )

=

∂u ∂u (x − x ) + (y − yi ) i ∂x ~xi ∂y ~xi ∂ 2 u ∂ 2 u (x − xi )2 + + ((x − xi )(y − yi )) ∂x2 ~xi 2 ∂x∂y ~xi ∂ 2 u (y − yi )2 + ··· + ∂y 2 ~xi 2 u|~xi +

(2)

Taking the control volume average of this expansion over control volume i and equating it to the mean value gives ui

∂u ∂u u|~xi + xi + y ∂x ~xi ∂y ~xi i ∂ 2 u x2 ∂ 2 u ∂ 2 u y 2 + + xy + + ··· ∂x2 ~xi 2 ∂x∂y ~xi ∂y 2 ~xi 2

=

where xn y m

i

1 ≡ Ai

ˆ (x − xi )n (y − yi )m dA.

(3)

(4)

Vi

are control volume moments. k th-order accuracy requires that we compute the k th derivatives by minimizing the error in predicting the mean value of the reconstructed function for control volumes in the stencil of neighboring control-volumes {Vj }i . In other words, we aim to minimizing the dierence between the control volume average uj and the average of the reconstruction Ri over control volume j . The minimum number of control volumes in the reconstruction stencil is equal to the number of derivative terms to be approximated. In practice, to improve conditioning, we increase this minimum to 4, 10, and 16 for second-, third-, and fourth-order schemes respectively. The stencil is constructed by rst considering the rst-neighbors of the control-volume. If additional neighbors are needed they are added in order of topological proximity. All neighbors at given topological distance are added at once. The resulting stencil is therefore topologically centered but may exceed the minimum specied size. The mean value, for a single control volume Vj , of the reconstructed function Ri is 1 Aj

ˆ

Vj

(5)

Ri (~x − ~xi )dA = u|~xi ∂u + ∂x ~xi

1 Aj

ˆ

! (x − xi )dA

Vj

2 of 13 American Institute of Aeronautics and Astronautics

+

+

+

+

! ˆ 1 ∂u (y − yi )dA ∂y ~xi Aj Vj ! ˆ 1 ∂ 2 u 2 (x − xi ) dA ∂x2 ~xi 2Aj Vj ! ˆ 1 ∂ 2 u (x − xi )(y − yi )dA ∂x∂y ~xi Aj Vj ! ˆ ∂ 2 u 1 2 (y − yi ) dA + · · · ∂y 2 ~xi 2Aj Vj

To avoid computing moments of each control volume in {Vj }i about ~xi , replace x − xi and y − yi with (x − xj ) + (xj − xi ) and (y − yj ) + (yj − yi ), respectively. Expanding and integrating, we obtain ˆ 1 ∂u (xj + (xj − xi )) Ri (~x − ~xi ) = u|~xi + Aj V j ∂x ~xi ∂u y j + (yj − yi ) + ∂y ~xi ∂ 2 u x2 j + 2xj (xj − xi ) + (xj − xi )2 + ∂x2 ~xi 2 ∂ 2 u + xy j + xj (yj − yi ) + (xj − xi )y j +(xj − xi )(yj − yi )) ∂x∂y ~xi ∂ 2 u y 2 j + 2y j (yj − yi ) + (yj − yi )2 + + ··· (6) ∂y 2 ~xi 2 This equation is written for every control volume in the stencil. These equations combined with the mean constraint form a constrained least-squares system of the form:           

1 wi1 wi2 wi3

x ¯ wi1 x bi1 wi2 x bi2 wi3 x bi3

y¯ wi1 ybi1 wi2 ybi2 wi3 ybi3

wiN

wiN x biN

wiN ybiN

.. .

.. .

.. .

x2

y2

c2 wi1 x i1 c2 wi2 x i2 c2 w x

xy wi1 x cy i1 wi2 x cy i2 wi3 x cy i3

wi1 yb2 i1 wi2 yb2 i2 w yb2

c2 wiN x iN

wiN x cy iN

wiN yb2 iN

i3

.. .

i3

.. .

i3

.. .

i3

··· ··· ··· ···





         ..   .    ···

u ∂u ∂x ∂u ∂y 1 ∂2u 2 ∂x2 ∂2u ∂x∂y 1 ∂2u 2 ∂y 2

.. .

            



ui u1 u2 u3



          =  ,    .   .   .  uN

(7)

i

where the rst row is the mean constraint, and the geometric terms are n ym xd ij

≡ =

1 Aj

ˆ

((x − xj ) + (xj − xi ))n ((y − yj ) + (yj − yi ))m dA

Vj m X n X l=0 k=0

m! n! (xj − xi )k (yj − yi )l xn−k y m−l j l! (m − l)! k! (n − k)!

and w, the weights, can be chosen to emphasize geometrically nearby data wij =

1 n. |~xj − ~xi |

(8)

where n is typically chose to be 0, 1, or 2. Although the use of n = 0 has been shown to lead to some accuracy issues for highly anisotropic grids,10 in our experience it is the best choice for successful convergence of the scheme and is therefore used in the present work. 3 of 13 American Institute of Aeronautics and Astronautics

Since the matrix in Equation 7 depends purely on geometric terms it is possible to reduce the reconstruction step at each ux evaluation to a matrix-vector multiplication by precomputing a pseudo-inverse during a preprocessing stage. To obtain this form, we rst analytically eliminate the mean constraint from Equation 7. Next, we compute the pseudo-inverse of the resulting unconstrained least-squares problem. For this we use singular-value decomposition (see, for instance,11 for details). The mean-constraint can then be reintroduced to yield a matrix, which we store for each control-volume, which can be used to compute the reconstruction coecients using a single matrix-vector multiplication operation of the form             

u ∂u ∂x ∂u ∂y 1 ∂2u 2 ∂x2 ∂2u ∂x∂y 1 ∂2u 2 ∂y 2

.. .

             

=

ui u1 u2 u3



         . A     ..   . 

(9)

uN i

Since the matrix A is based on purely geometric terms, the same pseudo-inverse matrix can be used to reconstruct all the ow variables in the control-volume which helps minimize memory-use. Furthermore, this reconstruction form helps in constructing the exact ux Jacobian which we have previously shown can be helpful in obtaining rapid convergence of high-order accurate schemes.5 II.B.

Boundary condition enforcement

In general, boundary conditions can be enforced either through special ux functions or by modifying the reconstruction step. In practice, we use special ux functions for boundary conditions of the inviscid ux and for maintaining the adiabatic condition at walls. We modify the reconstruction step to maintain the no-slip condition at the wall by enforcing Dirichlet boundary conditions for the components of velocity u and v . However, our approach to boundary condition enforcement is general enough to be applied to any Dirichlet and Neumann boundary conditions. In previous work with the advection-diusion8 boundary conditions were enforced by introducing additional constraints on the least-squares problem in Equation 7. In the present work we modify this approach by instead augmenting the least-squares problem with additional reconstruction goals. Suppose that the solution must satisfy a Dirichlet boundary condition such that along part of the boundary ∂Ω1 the reconstruction approximately satises R(~x − x~i ) = f1 (~x). Rather than attempting to satisfy this condition everywhere along the boundary, we consider only the solution at each Gauss integration point x~g on the boundary. In terms of the reconstruction polynomial coecients, this reconstruction goal can be expressed a f1 (~xg )

∂u ∂u = u|~xi + (xg − xi ) + (yg − yi ) ∂x ~xi ∂y ~xi ∂ 2 u (xg − xi )2 ∂ 2 u + + (xg − xi )(yg − yi ) ∂x2 ~xi 2 ∂x∂y ~xi ∂ 2 u (yg − yi )2 + + ··· ∂y 2 ~xi 2

The boundary condition can be incorporated into Equation 7 by adding a row to the matrix for each


(10)

boundary Gauss point, this gives               

1 wia wib wi1 wi2 wi3

.. .

wiN

x ¯ y¯ x2 wia (xa − xi ) wia (ya − yi ) wia (xa − xi )2 wib (xa − xi ) wib (ya − yi ) wib (xa − xi )2 c2 wi1 x bi1 wi1 ybi1 wi1 x i1 c2 wi2 x bi2 wi2 ybi2 wi2 x i2 c2 wi3 x bi3 wi3 ybi3 wi3 x i3

.. .

.. .

.. .

wiN x biN

wiN ybiN

c2 wiN x iN 

xy y2 wia (xa − xi )(ya − yi ) wia (ya − yi )2 wib (xa − xi )(ya − yi ) wib (ya − yi )2 wi1 x cy i1 wi1 yb2 i1 wi2 x cy i2 wi2 yb2 i2 wi3 x cy i3 wi3 yb2 i3

ui f1 (x~a ) f1 (x~b ) u1 u2 u3

       =      

.. .

.. .

.. .

wiN x cy iN

wiN yb2 iN

··· ··· ··· ··· ··· ···



              ..  . 

···

u ∂u ∂x ∂u ∂y 1 ∂2u 2 ∂x2 ∂2u ∂x∂y 1 ∂2u 2 ∂y 2

.. .

              

(11)

uN

where the weights for the Gauss point rows can be computed as 1 n. |~xg − ~xi |

wig =

(12)

Neumann boundary conditions can be applied in an analogous way. If the reconstruction polynomial is to x−~ xi ) approximate the boundary condition ∂R(~∂n = f2 (~x) we simply add reconstruction goals at the boundary Gauss points of the form f2 (~xg )

= =

∇R(~xg − ~xi ) · n ˆ ! ∂u ∂ 2 u ∂ 2 u nx ++ (xg − xi ) + (yg − yi ) + · · · + ∂x ∂x2 ~xi ∂x∂y ~xi ! ∂ 2 u ∂u ∂ 2 u ny + (xg − xi ) + (yg − yi ) + · · · ∂y ∂x∂y ~xi ∂y 2 ~xi

(13)

where nˆ is the unit boundary normal. As for the interior case, the pseudo-inverse of the constrained least-squares problem of Equation 11 can be computed to reduce the reconstruction to a matrix-vector product of the form             

u ∂u ∂x ∂u ∂y 1 ∂2u 2 ∂x2 ∂2u ∂x∂y 1 ∂2u 2 ∂y 2

.. .



            

=

h

Ab

     i  Ai       

f1 (x~a ) f1 (~xb ) ui u1 u2 u3

.. .

        ,      

(14)

uN

i

where the component Ab of the pseudo-inverse has as many columns as there are boundary Gauss points. Since the value of f1 (~x) does not change at each iteration, the boundary reconstruction step can be further 5 of 13 American Institute of Aeronautics and Astronautics

             i

optimized by precomputing a vector f1 (x~a ) f1 (~xb )

v b = Ab

! .

The reconstruction can then be found by             

u ∂u ∂x ∂u ∂y 1 ∂2u 2 ∂x2 ∂2u ∂x∂y 1 ∂2u 2 ∂y 2

.. .

             

=

ui u1 u2 u3



          + vb . Ai      ..   . 

(15)

uN i

Therefore the computational cost of reconstruction at the boundaries is almost identical to that of the interior. However since the dierent ow variables do not need to satisfy the same boundary conditions, the pseudo-inverse matrix Ai and vector vb will need to be stored independently for each ow variable. Since only a small fraction of control volumes are adjacent to the boundary, this is not a major issue. Boundary conditions which linearly couple ow variables or their derivatives can also be implemented. This is most easily accomplished by considering the constrained least-squares problem which simultaneously solves the reconstruction polynomial for all ow variables at a control-volume. However this causes the memory and computational cost of boundary control-volume reconstruction to increase quadratically with the number of ow variables. Therefore we elect to only simultaneously solve the reconstruction for the ow variables which are coupled by boundary conditions and independently solve those that are not. III.

Flux Integration and Solution Convergence

Once the reconstruction polynomial coecients are computed as in Section II, the solution and solution gradients can easily be obtained at any Gauss integration point along the control-volume boundary. In the present work, we use a vertex-centered centroidal-dual control-volume which is obtained by connecting mesh face centroids with cell centroids. This results in two control-volume edges per face in the primal mesh. Gauss integration is carried out on these dual edges using one Gauss point per interior edge for the second-order scheme and two Gauss points for the third-, and fourth-order schemes. For the third-, and fourth-order schemes, it is well known that Gauss points must be located on the curved boundary rather than on the straight edge connecting vertices. Recently, it has been shown that for vertex-centered schemes the boundary face centroid used for the construction of the dual-mesh must also be correctly located on the boundary, even for second-order schemes.12 Though using a single Gauss point per dual-edge results in a second-order scheme, our experience is that using two Gauss points on boundary edges results in a signicant reduction of error at little additional computational cost. Therefore we use two Gauss points per edge located exactly on the curved boundary for the all orders of accuracy. The inviscid component of the ux is computed by evaluating the solution at the Gauss point using the reconstruction from both incident control-volumes as input to Roe's approximate Riemann solver.13 The viscous ux requires not only the solution, but also the solution gradients. For this we use the average gradient obtained from the reconstruction of the two incident control-volumes. For a p-order accurate reconstruction scheme, uxes that depend only on the solution will be p-order accurate. Since the solution gradient is only (p − 1)-order accurate, the accuracy of the viscous ux is an order of accuracy lower than that of the inviscid ux. However, previous work8 has analytically demonstrated that the even-order discretization of the Laplace equation on equilateral triangular grids benets from error cancellation leading to an overall scheme that is p-order accurate. Numerical results in the same work conrmed the p-order accuracy of the overall scheme for the advection-diusion equation on an unstructured grid. At the onset of this work it was not known whether this could also be expected of the Navier-Stokes equations on an unstructured grid. The solution is converged to steady state using a Newton-GMRES algorithm.14 As previously reported in our inviscid work, the full-order Jacobian is assembled and used for preconditioning and for explicitly 6 of 13 American Institute of Aeronautics and Astronautics

performing the matrix-vector products needed by the GMRES algorithm.5 The preconditioner used is ILU with one level of ll for the second-order scheme and zero levels of ll for the third- and fourth-order schemes. Globalization is achieved by using a local timestep that is increased as the residual diminishes using the successive evolution-relaxation (SER)15 scheme. IV.

IV.A.

Results

Cylindrical Couette Flow

To numerically evaluate the order of accuracy of the scheme we consider ow around concentric cylinders. The outer cylinder with a radius of 2 is xed and adiabatic while the inner cylinder has a radius of 1 and is rotated clockwise with a tangential Mach number of 0.5. Since isothermal wall conditions for the inner cylinder would be dicult to enforce with the current choice of reconstruction variables, we obtain an equivalent result by augment the xed velocity wall condition with an isobaric condition. To avoid Taylor instabilities a Reynolds number of 10 based on the inner cylinder radius is used. Viscosity is considered constant for this case which allows comparison with the analytic exact solution.16 In particular, in this case the velocity prole reduces to that of the incompressible ow with ro

vθ = Ωi ri rro ri

− −

r ro ri ro

where vθ is the tangential ow velocity, Ωi is the angular velocity of the inner wall, r is the radius, and ri and ro and the radius of the inner and outer cylinders. The numerical solution is carried out on a series of four meshes consisting of 219, 840, 3256, and 12812 vertices. The two coarsest meshes are shown in Figure 1. Since the grids are unstructured and not aligned with the expected ow gradients and since the discretization is carried out in Cartesian coordinates, the error in the numerical solution should not have any unexpected benets from the simplicity of the solution expressed in radial coordinates.

(a) 219 vertex mesh

(b) 840 vertex mesh

Figure 1: Two coarsest grids used in cylindrical Couette ow case The L2 norms of error in the velocity terms for the second-, third-, and fourth-order schemes is shown in Figure 2 and the convergence order of the error norms is shown in Table 1. The results indicate that the eective order of accuracy of the schemes falls between the nominal order of the ux calculation of the solution and gradient terms. IV.B.

Flow Over An Airfoil

Next we consider subsonic laminar compressible ow around an airfoil. Specically, ow around a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5 and Reynolds number of 5000 is simulated on a 7 of 13 American Institute of Aeronautics and Astronautics

0.01

Second-Order Third-Order Fourth-Order

L2 Norm Error

0.001

0.0001

1e-05

1e-06 200

400

800

1600

3200

6400 12800

Num CVs

Figure 2: Convergence of error in velocity for cylindrical Couette ow

Reconstruction Order L1 L2 L∞ 2 1.60 1.59 1.37 3 2.22 2.25 2.38 4 3.54 3.57 3.26 Table 1: Order of error norms in velocity computed using a regression t through data from all grids


series of grids. These grids were generated by adaptive renement using the second-order scheme.17 A total of six grids is considered with the coarsest containing 1345 vertices and the nest containing 37399 vertices. The grids are unstructured, anisotropic, and are not self-similar. The coarsest grid is shown in Figure 3 and a sample ow solution is given in Figure 4. The airfoil surface is considered adiabatic and Sutherland's law is used to compute the viscosity.

Figure 3: Coarsest grid consisting of 1345 vertices

Figure 4: Mach number contours as computed on the 10541 vertex mesh using the fourth-order scheme The convergence process is facilitated by using the converged solution from the previous coarser mesh as the initial solution on the next mesh. The solution is transferred by Gauss integration on the ne grid of the values obtained from the solution reconstruction on the coarse grid. Using this grid-sequencing method the CPU time needed for a solution scales nearly linearly with the number of control volumes. The time required for a solution at each grid level is shown in Figure 5. These results along with the cumulative time required for the grid sequencing are shown in Table 2. In general, the third- and fourth-order schemes are a factor of 2.5 and 2.8, respectively, more costly than the second-order scheme. The majority of this additional cost can be attributed to the doubling in the number of Gauss points required in the higher-order schemes. From the converged ow solution, the pressure and viscous drag coecients are computed and the ow separation point is found. The results are in general agreement with numerical solutions by other re9 of 13 American Institute of Aeronautics and Astronautics

CPU Time (s)

1000


100

10

1 1000

10000

100000

Num CVs

Figure 5: Computational time required for convergence at each grid level

Number of Incremental Time Cumulative Time Control-Volumes 2nd 3rd 4th 2nd 3rd 4th 1345 1.6 4.8 5.3 1.6 4.8 5.3 2681 2.8 8.1 8.8 4.4 12.9 14.1 5363 7.6 17.3 18.8 10.4 25.4 27.6 10541 13.9 31.9 37.1 21.5 49.2 55.9 20016 32.4 83.6 82.3 46.3 115.5 119.4 37399 94.2 190.0 227.4 126.6 273.6 309.7 Table 2: Computational time in seconds required for convergence of airfoil test case using grid-sequencing


searchers.18, 19 We rst consider the convergence of viscous drag with mesh renement in Figure 6 and of pressure drag in Figure 7. We note that the fourth-order scheme exhibits excellent grid-convergence of drag 0.038


Viscous Drag Coefficient

0.037 0.036 0.035 0.034 0.033 0.032 0.031 0

10000

20000 30000 Num CVs Figure 6: Viscous drag coecient grid convergence

40000

coecients. In fact, both drag coecients are within 4 counts of the converged values beginning at the third-coarsest mesh (5363 vertices) and within 2 counts of the converged values beginning at the fourthcoarsest mesh (10541 vertices). The third-order scheme also exhibits good viscous drag convergence but the pressure drag varies by 8 counts between the second-nest and nest grids. Like the third-order scheme, the second-order scheme does not exhibit grid-convergence of drag. Furthermore, the ne grid solution diers signicantly from that of the third- and fourth-order schemes. Next we consider the separation point in Figure 8. The fourth-order scheme exhibits convergence of the separation point to within 1% of cord length beginning at the fourth-coarsest mesh while the third- and second-order schemes continue to see uctuations of up to 5%. Overall, the grid convergence study demonstrates that the fourth-order scheme is remarkably superior to the second- and third-order schemes. The fourth-order solution on the third-coarsest mesh is arguably superior to the second-order solution on the nest mesh and requires only 15 the computational time. V.

Conclusion

The scheme based on a fourth-order reconstruction was shown to obtain an order of accuracy of approximately 3.5 under a mesh renement study of the cylindrical Couette ow. The subsonic laminar airfoil test case demonstrated that this accuracy translates to rapid grid convergence of lift, drag and separation points relative to the second-order scheme. When combined with a matrix-explicit Newton-GMRES convergence acceleration technique, the fourth-order scheme was shown to be only a factor of 2.8 more computationally expensive than a second-order scheme on the same grid. The results indicate that the fourth-order scheme can be less computationally expensive for obtaining a level of accuracy typically desired for aerodynamics simulations. Future work will examine the extension of the present scheme to turbulent ows. The challenges lay in the stability of the high-order discretization of the turbulence model. Furthermore the need for curved boundary treatment for the high-order scheme combined with the highly-anisotropic grids required for ecient discretizations of turbulent ows can lead to malformed control-volumes unless the interior faces are also curved. 11 of 13 American Institute of Aeronautics and Astronautics

0.029


Pressure Drag Coefficient

0.028 0.027 0.026 0.025 0.024 0.023 0.022 0

10000

20000 30000 Num CVs Figure 7: Pressure drag coecient grid convergence

1


0.95 Separation Point (x/c)

40000

0.9 0.85 0.8 0.75 0.7 0

10000

20000 30000 40000 Num CVs Figure 8: Grid convergence of the separation point on the upper airfoil surface


References 1 Barth, T. J. and Frederickson, P. O., Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA paper 90-0013, Jan. 1990.

2 Ollivier-Gooch, C., Nejat, A., and Michalak, C., On Obtaining High-Order Finite-Volume Solutions to the Euler Equations on Unstructured Meshes, Proc. 18th AIAA CFD Conf., 2007, AIAA-2007-4464.

3 Nejat, A. and Ollivier-Gooch, C., Eect of Discretization Order on Preconditioning and Convergence of a High-Order Unstructured Newton-GMRES Solver for the Euler Equations, J. Comp. Phys., Vol. 227, No. 4, 2008, pp. 23662386.

4 Nejat, A. and Ollivier-Gooch, C., A High-Order Accurate Unstructured Finite Volume Newton-Krylov Algorithm for Inviscid Compressible Flows, J. Comp. Phys., Vol. 227, No. 4, 2008, pp. 25922609.

5 Michalak, K. and Ollivier-Gooch, C., Matrix-Explicit GMRES for a Higher-Order Accurate Inviscid Compressible Flow Solver, Proc. 18th AIAA CFD Conf., 2007.

6 Delanaye, M. and Essers, J. A., Quadratic-reconstruction nite volume scheme for compressible ows on unstructured adaptive grids, AIAA J., Vol. 35, No. 4, April 1997, pp. 631639.

7 Geuzaine, P., Delanaye, M., and Essers, J.-A., Computation of High Reynolds Number Flows with an Implicit Quadratic Reconstruction Scheme on Unstructured Grids, Proc. 13th AIAA CFD Conf., Amer. Inst. Aero. Astro., 1997, pp. 610619.

8 Ollivier-Gooch, C. F. and Van Altena, M., A High-order Accurate Unstructured Mesh Finite-Volume Scheme for the Advection-Diusion Equation, J. Comp. Phys., Vol. 181, No. 2, 2002, pp. 729752.

9 Barth, T. J., Recent Developments in High Order K-Exact Reconstruction on Unstructured Meshes, AIAA paper 93-0668, Jan. 1993.

10 Mavriplis, D. J., Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference , 2003.

11 Golub, G. H. and Loan, C. F. V., Matrix Computations , The Johns Hopkins University Press, Baltimore, 1983. 12 Diskin, B. and Thomas, J., Accuracy Analysis for Mixed-Element Finite-Volume Discretization Schemes, Tech. Rep. 2007-08, National Institute of Aerospace, 2007.

13 Roe, P. L., Approximate Riemann Solvers, Parameter Vectors, and Dierence Schemes, J. Comp. Phys., Vol. 43, 1981, pp. 357372.

14 Saad, Y. and Schultz, M. H., GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comp., Vol. 7, No. 3, July 1986, pp. 856869.

15 Mulder, W. and van Leer, B., Experiments with implicit upwind methods for the Euler equations, J. Comp. Phys., Vol. 59, June 1985, pp. 232246.

16 Illingworth, C., Some Solutions of the Equations of Flow of a Viscous Compressible Fluid, Proc. Cambridge Phil. Soc., Vol. 46, 1950, pp. 469478.

17 Pagnutti, D. and Ollivier-Gooch, C., A Generalized Framework for High Order Anisotropic Mesh Adaptation, Comp. Struct., Vol. Submitted, 2008.

18 Mavriplis, D. and Jameson, A., Multigrid Solution of the Two-Dimensional Euler Equations on Unstructured Triangular Meshes, AIAA paper 87-0353, Jan. 1987.

19 Radespiel, R., A Cell-vertex Multigrid Method for the Navier-Stokes Equations, NASA TM-101557, 1989.
