Solutions of High-order Methods for Three-dimensional Compressible Viscous Flowsweb2.utc.edu/~bfc365/welcome_files/LiWang-AIAA-2012-2836... · 2013-08-16 · Solutions of High-order

Solutions of High-order Methods for Three-dimensionalCompressible Viscous Flows

Li Wang ∗ W. Kyle Anderson†

J. Taylor Erwin ‡ and Sagar Kapadia §

SimCenter: National Center for Computational EngineeringUniversity of Tennessee at Chattanooga, Chattanooga, TN

In this paper high-order finite-element discretizations consisting of discontinuous Galerkin (DG) and stream-line/upwind Petrov-Galerkin (SUPG) methods are investigated and developed for solutions of two- and three-dimensional compressible viscous flows. Both approaches treat the discretized system fully implicitly to obtainsteady state solutions or to drive unsteady problems at each time step. The modified Spalart and Allmaras(SA) turbulence model is implemented and is discretized to an order of accuracy consistent with the mainReynolds Averaged Navier-Stokes (RANS) equations using the present high-order finite-element methods. Toaccurately represent the real geometries for viscous flows, high-order curved boundary meshes are generatedvia a CAPRI interface, while the interior meshes are deformed subsequently through a linear elasticity solver.The mesh movement procedure effectively prevents the generation of collapsed elements that can occur dueto the projection of curved physical boundaries and thus allows high-aspect-ratio curved elements in viscousboundary layers. Several numerical examples, including large-eddy simulations of viscous flow over a three-dimensional circular cylinder and turbulent flows over a NACA 4412 airfoil and a high-lift multi-element airfoilat high angles of attack, are considered to show the capability of the present high-order finite-element solversin capturing typical viscous effects such as flow separation and to compare the accuracy between high-orderDG and SUPG discretization methods.

I. Introduction

High-order discretization methods have gained increasing popularity in the past decade for a wide variety of fluiddynamics applications.1–7 More and more physically realistic problems can be tackled through the use of such methodsto obtain high accuracy solution, while reducing the need for extremely high-resolution meshes that are often requiredby lower-order methods. Moreover, a great deal of effort has been devoted to the versatility, robustness and efficiencyof high-order flow solvers, including adaptive mesh refinement techniques,8–12 solution limiting and shock capturingmethods,9,13–16 hybrid methodologies and multigrid solution strategies.2,17–19 To this end, this paper continues on thedevelopment of high-order discretization methods, consisting of discontinuous Galerkin (DG)1–3,6,18,20 and stream-line/upwind Petrov-Galerkin (SUPG)21–24 discretizations, to further expand the capability of high-order schemes insolving a wide range of viscous flow problems for complex geometries and to compare the accuracy of the high-orderDG and SUPG methods. In particular, applications of the present methods for studying the flow around bluff bodiesat a sub-critical Reynolds number and simulations of two-dimensional turbulent flows are considered.

The application of high-order methods to turbulent flows has become an active research topic in the high-ordercomputational fluid dynamics (CFD) community. It is well known that high-order schemes are well-suited for prob-lems with smooth solutions, in which the order of solution accuracy can be generally achieved. However, as thehigh-order methods are applied to turbulent flows using the Spalart-Allmaras (SA) turbulence model,25 robustness be-comes a challenge for high-order methods because the turbulence working variable abruptly decreases at the edge ofturbulent and laminar regions, thus leading to unavoidable oscillations and often negative values, which subsequentlycause the solver to fail. Recent work5,6,26 has investigated a modified SA turbulence model to avoid the stability issuescaused by negative turbulence working variable. In the present work, the modified SA turbulence model is utilized

∗Research Assistant Professor, 701 E. M.L. King Blvd., Chattanooga, TN 37403, AIAA Member, email: [email protected]†Professor, 701 E. M.L. King Blvd., Chattanooga, TN 37403, AIAA Associate Fellow, email: [email protected]‡PhD Candidate, 701 E. M.L. King Blvd., Chattanooga, TN 37403, AIAA Student Member, email: [email protected]§Research Assistant Professor, 701 E. M.L. King Blvd., Chattanooga, TN 37403, email: [email protected]

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American Institute of Aeronautics and Astronautics

42nd AIAA Fluid Dynamics Conference and Exhibit25 - 28 June 2012, New Orleans, Louisiana

AIAA 2012-2836

Copyright © 2012 by Li Wang, W. Kyle Anderson, J. Taylor Erwin and Sagar Kapadia. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

and fully coupled with the other conservation equations. Consistent high-order finite-element discretizations includingboth DG and SUPG methods are applied to the full system of equations.

In addition, it is known that the use of high-order curved boundary elements is essential for high-order schemesto deliver an overall accurate solution.27,28 Poor representation of the actual geometry can result in the productionof a significant amount of artificial entropy along the geometry surface, thus degrading the solution accuracy. Thisis especially the case for high-order methods because relatively coarse meshes are generally utilized where surfacenodes or cells can span a large portion of the geometry surface. On the other hand, more attention is needed whendealing with high Reynolds number viscous flow problems. To allow the use of high-aspect-ratio elements in a thinboundary layer, curvilinear interior elements are required in response to the projection of curved boundary faces andedges into the interior. In this context, the current work makes use of the Computational Analysis PRogrammingInterface (CAPRI)29 mesh parameterization tool30 for evaluating the true positions of additional surface points31 onthree-dimensional configurations, followed by a linear elasticity mesh movement strategy to prevent the generation ofnegative-Jacobian elements.

To numerically solve viscous or turbulent flows, the viscous flux terms must be accurately discretized. In thepresent DG methods, a symmetric interior penalty (SIP) method3,19 is implemented due to the fact that the schemedoes not require introduction of any auxiliary variables, and moreover, it maintains a compact element-based stencilthat simplifies the linearization of the full discretized system for developing implicit methods. In the SUPG methods,on the other hand, the solution is assumed to be continuous across the computational domain and the surface fluxrequired in DG methods vanishes at elemental boundaries. Therefore, stabilization of the scheme is added through astabilization term32 to compensate for lack of dissipation. It should be noted that because of the assumption of solutioncontinunity, the degrees of freedom within a computational mesh decrease greatly in the SUPG methods24 becausethe solution modes or nodes at each edge or face are stored only once. For time-dependent problems, a standardsecond-order backward difference formula (BDF2) is considered. The implicit system is solved using an approximateNewton method, where the linear system is solved via an element Gauss-Seidel scheme or a preconditioned GMRESapproach with ILU(k) preconditioning.33,34 A multigrid approach18,35 is also implemented in the DG flow solver toaccelerate the solution convergence.

The remainder of the paper is structured as follows. In Section II the governing equations are introduced, includingthe modified Spalart and Allmaras turbulence model. Section III describes consistent discretizations of the full systemof equations using discontinuous Galerkin and streamline/upwind Petrov-Galerkin discretizations as well as an implicittime-integration scheme. Section IV briefly reviews the integration of CAPRI with the mesh movement strategy usedin the current work. Several numerical examples are presented in Section V to demonstrate the performance of thecurrent high-order schemes in capturing complex flow structures for three-dimensional viscous flow and turbulentboundary layer separation. Finally, Section VI summarizes the conclusions and discusses the future work.

II. Governing Equations

The compressible Reynolds Averaged Navier-Stokes equations coupled with the modified one-equation Spalart-Allmaras turbulence model5,6,26 can be written in the following conservative form:

¶U(x, t)¶t

+Ñ · (Fe(U)− Fv(U,ÑU)) = S(U,ÑU) in W (1)

where W is a bounded domain. The vector of conservative flow variables U, the inviscid and viscous Cartesian fluxvectors, Fe and Fv, are defined by:

U =

r

rurvrwrErv

, Fx

e =

ruru2 + p

ruvruw

(rE + p)uruv

, Fy

e =

rvruv

rv2 + prvw

(rE + p)vrvv

, Fz

e =

rwruwrvw

rw2 + p(rE + p)w

rwv

(2)

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Fxv =

0txxtxytxz

utxx + vtxy +wtxz +k¶T¶x

1s

µ(1+y) ¶v¶x

, Fy

v =

0txytyytyz

utxy + vtyy +wtyz +k¶T¶y

1s

µ(1+y) ¶v¶y

, Fz

v =

0txztyztzz

utxz + vtyz +wtzz +k¶T¶z

1s

µ(1+y) ¶v¶z

(3)

where the notations r , p, and E denote the fluid density, pressure and specific total energy per unit mass, respectively.u = (u,v,w) represents the Cartesian velocity vector and v represents the turbulence working variable in the modifiedSA model. The pressure p is determined by the equation of state for an ideal gas,

p = (g − 1)(

rE − 12

r(u2 + v2 +w2))

(4)

where g is defined as the ratio of specific heats, which is 1.4 for air. t represents the fluid viscous stress tensor and isdefined, for a Newtonian fluid, as,

ti j = (µ+µT )(

¶ui

¶x j+

¶u j

¶xi− 2

3¶uk

¶xkdi j

)(5)

where di j is the Kronecker delta and subscripts i, j,k refer to the Cartesian coordinate components for x = (x,y,z). µrefers to the fluid dynamic viscosity and is obtained via the Sutherland’s law. µT denotes the turbulence eddy viscosity,which is obtained by:

µT ={

rv fv1 if v ≥ 00 if v < 0 (6)

The source term, S, in Eq. (1) has zero components for the continunity, momentum and energy equations, and takesthe following form for the turbulence model equation:5,6

ST = cb1Sµy − cw1r fw(ny

d)2 +

1s

cb2rÑv · Ñv − 1s

n(1+y)Ñr · Ñv (7)

where n denotes kinematic viscosity that is the ratio of dynamic viscosity to density, µ/r. It should be noted thatthe last term appearing in Eq. (7) results from applying the incompressible form of the SA turbulence model to acompressible form. For problems of low Mach number flow without shocks, the last term will not act as a major termin the turbulence model. The parameters for the production and destruction components of the modified SA turbulencemodel are given as:26

S =

{S + S if S ≥ −cv2S

S + S(c2v2+cv3S)

(cv3−2cv2)S−Sif S < −cv2S

(8)

S =√−→

w ·−→w S =ny

k2T d2

fv2 f v1 =y3

y3 + c3v1

f v2 = 1 − y

1+y fv1(9)

and

r =ny

Sk2T d2

g = r + cw2(r6 − r) fw = g(

1+ c6w3

g6 + c6w3

)1/6

(10)

respectively. −→w denotes the vorticity vector, Ñ × u. The notation d refers to distance to the nearest wall at a specific

location and is computed to account for the curvature of the actual boundaries. The parameter y in the equations isdesigned to remove the effects of negative turbulence working variable on the robustness of the turbulence model as itis discretized by a high-order spatial discretization scheme. This parameter is given by:

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y ={

0.05ln(1+ e20X ) if X ≤ 10X if X > 10

(11)

X =vn

(12)

The parameter y is designed to become zero as the turbulence working variable goes negative, thereby turning off theproduction, destruction, and dissipation terms to prevent instabilities. The constants in the modified SA model to closethe main flow equations are given as: cb1 = 0.1355, s = 2/3, cb2 = 0.622, kT = 0.41, cw1 = cb1/k2

T +(1 + cb2)/s,cw2 = 0.3, cw3 = 2 cv1 = 7.1, cv2 = 0.7 and cv3 = 0.9. k and T denote the thermal conductivity and temperature,respectively, and are related to the total energy and velocity as,

kT = g(µPr

+µT

PrT)(

E − 12(u2 + v2 +w2)

)(13)

where Pr and PrT are the Prandtl and turbulent Prandtl number that are set to be 0.72 and 0.9 respectively. In the caseof laminar flow, the governing equations reduce to the compressible Navier-Stokes equations, where the turbulencemodel equation is deactivated and the turbulence eddy viscosity, µT , in the fluid viscous stress tensor and the thermalconduction term vanishes.

For the purpose of the DG discretization, the Cartesian viscous fluxes are rewritten in the following equivalentform:

Fxv = G1 j

¶U¶x j

, Fyv = G2 j

¶U¶x j

, Fzv = G3 j

¶U¶x j

(14)

where the matrices Gi j(U) are determined by G1 j = ¶Fxv/¶(¶U/¶x j), G2 j = ¶Fy

v/¶(¶U/¶x j) and G3 j = ¶Fzv/¶(¶U/¶x j)

for j = 1,2 and 3 so that they are purely dependent on the conservative flow variables.

III. Discretizations

A. Discontinuous Galerkin discretization

The computational domain W is partitioned into a tessellation of non-overlapping elements (triangular elements for2D and tetrahedral elements for 3D), such that W =

Sk Wk, where Wk refers to the volume of an element k in the

computational mesh. The Galerkin finite-element approximation is expanded as a series of truncated hierarchicalbasis functions,36 {f j, j = 1, · · · ,M}, and solution coefficients as:

Uh =M

åi=1

Uhi fi(x). (15)

The full system of equations, including the main flow (i.e. continuity, momentum and energy) equations and themodified SA turbulence model equation, is discretized using a discontinuous Galerkin method. The DG discretizationis formulated into a weak statement of the governing equations, by multiplying Eq. (1) by a set of test functions, withthe maximum polynomial order of p, and integrating within each element, e.g. k, as:

ZWk

f j

[¶Uh(x, t)

¶t+Ñ · (Fe(Uh)− Fv(Uh,ÑUh))− S(Uh,ÑUh)

]dWk = 0. (16)

Integrating this equation by parts and implementing the symmetric interior penalty method3,19 for the viscous fluxesyields the following weak formulation,

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ZWk

f j¶Uh

¶tdWk −

ZWk

Ñf j · (Fe(Uh)− Fv(Uh,ÑhUh))dWk +Z

¶Wk\¶W

[[f j]]Hc(U+h ,U−

h ,n)dS (17)

−Z

¶Wk\¶W

{Fv(Uh,ÑhUh)} · [[f j]]dS −Z

¶Wk\¶W

{(Gi1¶f j

¶xi,Gi2

¶f j

¶xi,Gi3

¶f j

¶xi)} · [[Uh]]dS +

Z¶Wk\¶W

h{G}[[Uh]] · [[f j]]dS

−Z

¶Wk∩¶W

f+j Fb

v(Ub,ÑhU+h ) · ndS −

Z¶Wk∩¶W

(Gi1(Ub)¶f

+j

¶xi,Gi2(Ub)

¶f+j

¶xi,Gi3(Ub)

¶f+j

¶xi) · (U+

h − Ub)ndS

+Z

¶Wk∩¶W

hG(Ub)(U+h − Ub)n · f

+j ndS +

Z¶Wk∩¶W

f jFe(Ub) · ndS −Z

Wk

f jS(Uh,ÑhUh) = 0

where the unit normal vector n is outward to the boundary. Hc(U+h ,U−

h ,n) represents an approximate Riemann convec-tive flux function, such as Lax-Friedrichs37 or HLLC38 to resolve the solution discontinuities (represented by U+

h andU−

h ) at shared elemental interfaces. The notations {}, [[ ]] and [[ ]] indicate the respective average and jump operators,defined as:

{j} =j+ +j−

2, [[j]] = j

+n+ +j−n−, [[j]] = j

+ − j− (18)

The sixth and the ninth integrals in Eq. (17) are referred to as penalty terms, where the penalty parameter h is explicitlyevaluated by the element geometry and the order of discretization, given by:19,39

h =(p+1)(p+D)

(2D)max(

S+k

V +k

,S−

k

V −k

) (19)

where D represents the space dimensions; Vk and Sk represent the volume and surface of elements k± which sharethe interface. The boundary conditions on ¶W are imposed weakly by constructing a boundary state, denoted byUb. At solid walls, an adiabatic wall with no-slip boundary condition is imposed, which yields ÑT · n = 0 and Ub =(U1,0,0,0,U5,0), and thus the component of the boundary viscous fluxes, Fb

v , associated with the energy equationvanishes. The set of discretized equations is solved in the modal space and the integrals are evaluated using Gaussianquadrature rules,36 which are exact for the polynomial degree 2p in volume integrals and for the polynomial degree2p+1 in surface integrals.17,40

Because the set of basis functions is defined in a master element W spanning between {0 < x,h,z < 1}, a coor-dinate mapping from the reference to a physical element is required for the computation of the first-order derivatives,solution gradients and integrals appearing in Eq. (17). The reference-to-physical transformation and the correspondingJacobian Jk associated with each element k are given by:

xk =M

åi=1

xkifi(x,h,z), Jk =

¶x¶x

¶y¶x

¶z¶x

¶x¶h

¶y¶h

¶z¶h

¶x¶z

¶y¶z

¶z¶z

. (20)

where xk represent the element-wise geometric mapping coefficients. Due to the viscous problems considered in thepresent work that are possibly associated with high Reynolds number flows, high-order curved elements are generallyemployed not only on the physical boundaries41 but also in the interior regions, particularly in the viscous boundarylayer. In such circumstances, the higher-order modes (p > 1) of the geometric mapping coefficients are used todetermine the non-linear mapping and are obtained by the extra information at the quadrature points within eachelement, as:

xk = F−1xpk (21)

where xpk = {xck ,xqk } refers to the coordinates of physical points in the element k, consisting of the element verticesxck as well as additional quadrature points xqk . F denotes the projection mapping matrix which is constituted by thebasis functions evaluated at the aforementioned physical points in the master element (xpk

← xpk ). The additionalsurface points are created using the CAPRI parametric mesh respresention.29,30 To accommodate the properly curvedsurface elements, interior elements in the boundary layer are required to deform to avoid the generation of negativevolumes. Therefore, the current work utilizes a mesh movement strategy based on a linear elasticity theory42 tocompute the deformation of the interior mesh points as well as that of the additional nodes for high-order finite-element meshes.

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B. Streamline/Upwind Petrov-Galerkin discretization

In the streamline/upwind Petrov-Galerkin method, the discretized system of equations is written as the followingweighted residual scheme,

ZW

f

[¶Uh


]dW

+åk

ZWk

[(¶f

¶x[A]+

¶f

¶y[B]+

¶f

¶z[C]

)][t]

[¶Uh


]dWk = 0 (22)

where f is a continuous weighting function defined over the domain. [A], [B] and [C] represent the inviscid fluxJacobians, [¶Fx

e/¶Uh], [¶Fye/¶Uh], and [¶Fz

e/¶Uh], respectively. As seen, the first integral in Eq. (22) corresponds tothe weak statement of a standard Galerkin discretization. Because the solution is assumed to be continuous over thecomputational domain or subdomains24 in the SUPG method, the stabilization is added through the term with thestabilization matrix, [t],32 to compensate for lack of dissipation in the streamwise direction. For inviscid flows, [t] isobtained using the following definitions,23,43

[t]−1 = å j

∣∣∣ ¶f j¶x [A]+ ¶f j

¶y [B]+ ¶f j¶z [C]

∣∣∣ (23)∣∣∣ ¶f j¶x [A]+ ¶f j

¶y [B]+ ¶f j¶z [C]

∣∣∣ = [T][L][T]−1 (24)

where f j denotes the polynomial basis function associated with each node and is the same as the weighting function.[T] and [L] denote the matrix of right eigenvectors and the diagonal matrix of eigenvalues of the left hand side ofEq. (24), respectively. For viscous flows, additional terms are required when the Reynolds number is decreased andviscous terms are dominant.22,44,45 The first volume integral appearing in Eq. (22) is integrated by parts, which yieldsboth volume and surface integrals similar to those of the convection terms in the DG discretization. The system ofdiscretized equations can then be written as:

åk

ZWk

f

[¶Uh

¶t− Ñf · (Fe(Uh)− Fv(Uh,ÑUh))− S(Uh,ÑUh)

]dWk +

Z¶Wk∩¶W

f(Fe(Ub)− Fv(Ub,ÑUh))dS

+åk

ZWk

[(¶f

¶x[A]+

¶f

¶y[B]+

¶f

¶z[C]

)][t]

[¶Uh


]dWk = 0 (25)

It is noted that the surface integrals in the SUPG method are computed only on the domain/subdomain boundariesdue to the continuity of the solution across elemental interfaces. Furthermore, unlike the weak boundary conditions inthe DG discretization, the current SUPG method implements a strong boundary enforcement at solid walls, where thevelocities and the turbulence working variable are enforced to be zero. The solution approximation, Uh, in the SUPGdiscretization is expanded using a similar form as shown in Eq. (15) for the DG discretization. However, due to the useof a series of Lagrange polynomial basis functions, the present SUPG method solves for the solution at a set of nodeswithin each control volume, rather than the modal solution coefficients solved in the DG discretization counterpart.

C. Time-integration scheme

For the high-order DG and SUPG discretization methods discussed previously, the weighted form of the discretizedequations (Eqns. (17) and (25) respectively) can be rewritten into the following ordinary differential equation form as,

MdUh

dt+R(Uh) = 0 (26)

where R represents the discretized spatial residual (including both inviscid and viscous terms) and M denotes themass matrix. A time integration is then performed via an implicit, second-order backward difference formula (BDF2)written as:

Rn+1e (Un+1

h ) =MDt

(32

Un+1h )+R(Un+1

h )− MDt

(2Unh − 1

2Un−1

h ) = 0 (27)

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where Rn+1e represents the unsteady flow residual at time step n + 1. The implicit system is solved using an approx-

imate Newton method,18 where the flow Jacobian matrix is decomposed with element-based (for DG schemes) ornode-based (for SUPG schemes) diagonal and off-diagonal block components. The linearized system is solved viaa preconditioned GMRES approach33,34 with ILU(k) preconditioning. To enhance solution efficiency, the DG solverimplements a p-multigrid solution method,18 driven by a linearized element Gauss-Seidel smoother or a ILU(k) pre-conditioned GMRES algorithm. For steady state problems, the time-dependent term in Eq. (26) vanishes and a localtime-stepping method is incorporated to alleviate the stiffness of the system in the initial stages of the calculation. Theaddition of the local time stepping term has been found to be important to the calculations of turbulent flows and to beessential to the robustness of the flow solvers in the turbulence regimes. In addition, both high-order flow solvers usethe standard MPI message-passing library for inter-processor communication46 and the mesh is partitioned based onthe METIS mesh partitioner.47

IV. CAPRI Mesh Representation and Mesh Movement Strategy

For second-order accurate schemes commonly used to solve the Navier-Stokes equations, the surface of the geom-etry is typically represented by a series of linear elements. To achieve higher-order accuracy, this simple representationmust be replaced with one with increased fidelity to properly account for surface curvature. To this end, an interfacehas been developed for incorporating CAPRI29 into the geometry libraries of the UTC (University of Tennessee atChattanooga) SimCenter to allow communication with CAD software.30 While this interface had been originally de-veloped for purposes of design optimization, it is used in the present context for placing additional nodes or quadraturepoints onto the actual surface geometry as defined by the CAD definition.

An example is demonstrated in Fig. 1 for an analytically defined three-dimensional body of revolution recentlyused in the First international workshop on higher-order CFD methods48 (held together with the Fiftieth AIAAAerospace Sciences Meeting in Nashville, TN in 2012). As seen in Fig. 1 (a), the linear surface representationis clearly inaccurate and using it for higher-order schemes would not allow accurate quadrature on the surface. Incontrast, Fig. 1 (b) shows a surface mesh for the same geometry where quadratic elements are used. After addingadditional surface quadrature points using the CAPRI interface to the elements, the fidelity of the surface is clearlyimproved. Fig. 2 displays the pressure solution for inviscid flow over the 3D analytical body of resolution with free-stream Mach number of 0.5 and 1 degree of angle of attack, obtained by a third-order DG scheme on the mesh withthe quadratic surface representation as depicted in Fig. 2. It is shown that the third-order DG scheme is capable ofachieving a smooth solution on the current mesh with quadratic boundary elements.

As boundary elements are curved to conform to the original geometry configuration, collapsed elements are likelyto be generated, particular when highly stretched elements are applied in the viscous boundary layer. As the Reynoldsnumber of viscous flow increases, the aspect ratio of the elements in the near-wall regions increases. In this context,a robust mesh movement strategy must be employed to accommodate the projection of the surface meshes for two- orthree-dimensional geometries. Here exclusive use is made of a modified linear elasticity theory,42 which assumes thatthe computational mesh obeys the isotropic linear elasticity relations, taken in the following form:

¶

¶x

[d11

¶dx

¶x+d12

¶dy

¶y+d13

¶dz

¶z

]+

¶

¶y

[d44

(¶dx

¶y+

¶dy

¶x

)]+

¶

¶z

[d66

(¶dx

¶z+

¶dz

¶x

)]= 0

¶

¶x

[d44

(¶dx

¶y+

¶dy

¶x

)]+

¶

¶y

[d21

¶dx

¶x+d22

¶dy

¶y+d23

¶dz

¶z

]+

¶

¶z

[d55

(¶dy

¶z+

¶dz

¶x

)]= 0

¶

¶x

[d66

(¶dx

¶z+

¶dz

¶x

)]+

¶

¶y

[d55

(¶dy

¶z+

¶dz

¶y

)]+

¶

¶z

[d31

¶dx

¶x+d32

¶dy

¶y+d33

¶dz

¶z

]= 0 (28)

where d = (dx,dy,dz) denotes the nodal displacement vector in the Cartesian coordinate directions and the coefficients,d, are defined as follows,

d11 = d22 = d33 = E(1−u)(1+u)(1−2u)

d12 = d13 = d21 = d23 = d31 = d32 = Eu

(1+u)(1−2u)

d44 = d55 = d66 = E2(1+u) (29)

where E represents Young’s modulus that is set to be a constant throughout the entire domain. The notation u de-

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(a) Linear surface representation (b) Quadratic surface representation

Figure 1. Surface representations for an analytic 3D body of revolution.

Figure 2. Pressure contours for inviscid flow over an analytic 3D body of revolution using a third-order (P = 2) DG scheme and quadraticsurface representation (the mesh contains a total of 144737 tetrahedrons).

notes Poisson’s ratio, which is set to be 0.25. It is noted that the nodal displacement vector should also include theperturbations at the additional quadrature points for high-order finite-element meshes. Additionally, the perturbationsat the interior mesh points as well as quadrature points are solved with Dirichlet boundary conditions, specified at thesurface nodes and surface quadrature points. The linear elasticity equations are solved by a GMRES algorithm withILU(k) preconditioning.

To illustrate the mesh movement strategy, a NACA 4412 mesh shown in Fig. 3 (a) is used as an example. Fordemonstration purpose, the mesh is generated with a viscous spacing of 1 × 10−3 normal to the wall. Note that thisspacing is not fine enough for the applications presented later and is only used for illustration. Fig. 3 (b) plots contoursof the perturbation magnitude, (d2

x +d2y)

1/2, obtained by the linear elasticity solver. As expected, most perturbations areconcentrated on the airfoil upper surface because of the presence of relatively larger curvatures, and the perturbationmagnitude quickly decays as the distance to the airfoil increases. Figures 3 (c)-(e) depict a close-up view of the viscouslayer near 0.4 chord of length on the upper airfoil surface, with a slightly exaggerated scale in y-axis to view details.As displayed in Fig. 3 (d), collapsed elements are observed when the surface mesh is represented by a high-orderpolynomial interpolation, while holding the interior elements fixed. In contrast, as shown in Fig. 3 (e), the meshmovement strategy effectively produces sufficient deformations for the interior mesh points and quadrature points toprevent negative volumes or Jacobians.

The mesh movement strategy employed in the current work is adequate to obtain valid high-order finite-elementmeshes with high-aspect ratio elements in two dimensions. However, with the current strategy and parameter settings,valid three-dimensional finite-element meshes may be obtained. Extensions of this capability are ongoing.

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(a) Sample mesh for a NACA 4412 airfoil (b) Contours of perturbation magnitude

(c) Close-up view of linear mesh (d) Close-up view of snapped mesh

(e) Close-up view final mesh

Figure 3. Illustration of mesh movement strategy.

V. Numerical Results

In this section, several numerical examples are given to demonstrate the capability of the present high-order finite-element flow solvers and to compare the accuracy between the discontinuous Galerkin and streamline/upwind Petrov-Galerkin methods.

A. Three dimensional viscous flow over a circular cylinder

The first test case aims to examine the capability of the present high-order finite-element methods for capturing three-dimensional unsteady flow structures and viscous boundary effects. This example involves viscous flow over a circularcylinder at a Reynolds number of 2580 based on the diameter of the cylinder.49 The numerical simulation is conductedat a free-stream Mach number of 0.2 using various orders of the DG and SUPG discretizations in conjunction with thesecond-order backward difference time discretization.

The geometry definition and the computational mesh containing 68629 unstructured tetrahedral elements are dis-played in Fig. 4. The x-axis is along the flow streamwise direction and the z-axis is along the spanwise direction. Thedimensions in the streamwise, spanwise and transverse directions correspond to those of Mohammad and Wang.49

Uniform flow is specified at the domain inlet and an outflow boundary condition with fixed pressure is set at the outlet.Periodic boundary conditions are applied on the side walls (i.e. z = 0 and z = p planes) in the spanwise directionwhile symmetry is imposed on the top and bottom walls. The cylinder surface is modeled as a nonslip adiabatic walland the mesh has a viscous spacing of 0.00175 normal to the wall. It is noted that the mesh on the cylinder surface isrepresented by quadratic polynomials and the interior mesh points are moved by the linear elasticity solver discussedin the previous section to allow curved meshes in the viscous boundary layer.

High-order discontinuous Galerkin discretizations consisting of P = 1, P = 2 and P = 3, and streamline/upwind

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(a) Geometry definition (b) Computational mesh

Figure 4. Geometry definition and computational mesh (containing 68629 tetrahedral elements) for viscous flow over a three-dimensionalcircular cylinder.

(a) DG P3 (b) SUPG P2

Figure 5. Isosurfaces of instantaneous streamwise velocity for viscous flow over a three-dimensional circular cylinder at Re = 2580, obtainedby fourth-order DG and third-order SUPG schemes. Isosurface magnitude range from -0.18 to 0.18.

Petrov-Galerkin discretizations consisting of P = 2 are employed for the simulations, along with a fixed time stepsize of 0.02, where the time is nondimensionalized using the free-stream speed of sound. Figures 5 (a) and (b)show snapshots of isosurfaces of the instantaneous streamwise velocity, obtained using a fourth-order DG schemeand a third-order SUPG scheme. Although the contours are not depicted at identical times and are rendered withdifferent visualization software, it is observed that both schemes deliver qualitatively similar solutions. Flow separationoccurring on the upper and lower surfaces of the cylinder as well as interactions of the shedding vortices downstreamof the cylinder with mean flow can be clearly seen. However, more flow features of smaller scales are seen to beresolved in the fourth-order DG solution due to the use of a higher-order spatial scheme.

Fig. 6 plots time histories of lift and drag forces on the cylinder surface using a third-order accurate (P = 2) DGscheme, where the oscillations caused by shedding vortices and flow separation are clearly shown. Fig. 7 displayscontours of the instantaneous streamwise, transverse and spanwise velocities in the wake of the cylinder using thefourth-order DG scheme and third-order SUPG scheme. It shows that the scale of the flow structures grows as thedistance to the cylinder increases, and small-scale structures mostly occur in the wake region near the cylinder. In ad-dition, at the current Reynolds number of 2580, the flow becomes three-dimensional as both small and large structuresoccur in the spanwise direction.49 While the contours are again not at identical times, both the DG and SUPG solu-tions clearly show the unsteady recirculation and the alternating regions of positive and negative transverse velocityresulting from the shedding vortices.

A more quantitative study on capturing the wake statistics is performed next. In particular, the mean velocities andReynolds stresses at various locations in the wake region are compared with the experimental data, provided in thework of Mohammad and Wang.49 Fig. 8 (a) displays the mean streamwise velocity profiles normalized by free-stream

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Figure 6. Lift and drag time histories for viscous flow over a three-dimensional circular cylinder using a third-order DG discretization andthe BDF2 scheme with a time step size of 0.02 (bottom curve: lift, upper curve: drag).

(a) Instantaneous streamwise velocity (DG P3) (b) Instantaneous streamwise velocity (SUPG P2)

(c) Instantaneous transverse velocity (DG P3) (d) Instantaneous transverse velocity (SUPG P2)

(e) Instantaneous spanwise velocity (DG P3) (f) Instantaneous spanwise velocity (SUPG P2)

Figure 7. Fourth-order discontinuous Galerkin solution and third-order streamline/upwind Petrov-Galerkin solution for instantaneousstreamwise, transverse and spanwise velocities in x-z plane (y = 0) in the wake of the cylinder (Re = 2580). Velocity magnitude is normalizedby free-stream velocity and scales range from -1 to 1.

velocity, using various orders of DG schemes. As seen, the second-order scheme cannot capture the mean streamwisevelocity accurately on the current mesh, which is evident of the deviation of the profile shapes and the dissipationfurther downstream. In contrast, the third-order and fourth-order DG schemes provide much improved agreement withthe experimental data, where the magnitude of the mean u-velocity and the shape of the profiles both agree well. Fig. 8(b) shows the mean transverse velocity at various stations in the wake of the circular cylinder, obtained by the third- andfourth-order DG schemes. The profile shapes are captured well as a whole, with slight under-prediction at the locationof x/c = 2 in the cylinder wake. Improved results are observed as the discretization order increases from third order tofourth order on the same mesh. For example, at the location of x/c = 2.5, the fourth-order DG solution shows betteragreement with the experiment than the third-order DG solution. Figures 8 (c)-(e) plot the normalized streamwiseReynolds stress, transverse Reynolds stress, and shear stress, respectively. Both the third-order and fourth-order DGschemes show fairly good agreement with the experiments, considering the fact that the current computational meshis relatively coarse. Slight over-predictions of the cross-stream Reynolds stress are observed at locations of x/c = 2and x/c = 2.5 in the cylinder wake.

Fig. 9 shows the mean flow quantities obtained using a third-order streamline/upwind Petrov-Galerkin scheme.The third-order SUPG solution is in good agreement with the experimental data, and furthermore, it is observed thatthe third-order SUPG scheme delivers qualitatively similar profiles as those obtained by the third-order DG scheme.

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(a) Normalized u (b) Normalized v

(c) Normalized u′u′ (d) Normalized v′v′ (e) Normalized u′v′

Figure 8. Discontinuous Galerkin solutions for normalized mean flow quantities in three-dimensional viscous flow over a circular cylinder(Re = 2580). Circle symbols represent experimental data.

However, the profiles of the cross-stream Reynolds stress at x/c = 2.5 and x/c = 3, as displayed in Fig. 9 (d), forthe third-order SUPG scheme are seen to be captured slightly more accurately than those in the DG schemes. Theagreement of the mean transverse velocity obtained by the third-order DG scheme is slightly better than that of thethird-order SUPG scheme, in locations near the circular cylinder. The solution difference, however, is generally smallbetween these two schemes.

B. Turbulent flow over a NACA 4412 airfoil

The second example describes an application of the modified SA turbulence model for flow around a NACA 4412airfoil using the present high-order finite-element methods. It is designed to investigate the accuracy of differentspatial discretization schemes in capturing the structure of the turbulent flow and boundary-layer separation. For thiscase, the free-stream Mach number is 0.2, the Reynolds number is 1.52×106, and the angle of attack is 13.87 degrees.

The effects of grid density and discretization order are assessed for this test case and the computational resultsare compared with experimental data as well as the CFL3D results.50 Here, the CFL3D results have been obtainedusing very fine meshes and serve as benchmark solutions in the studies below. A set of three computational meshes,consisting of 5338, 11133 and 27537 triangular elements, is utilized to evaluate the boundary layer profiles of thepresent DG and SUPG schemes of discretization orders ranging from P = 1 to P = 3. The viscous spacing normal to thewall associated with the respective coarse, medium and fine meshes is 4 × 10−5, 2 × 10−5 and 1 × 10−5, respectively.Each computation starts with uniform flow and continues until a steady-state solution is reached, where the residualof the turbulence model equation is required to drop at least 8 orders of magnitude. The airfoil surface is modeled asa non-slip adiabatic wall in the DG schemes while constant wall temperature is used in the SUPG schemes. However,it is noted that the different wall boundary conditions should not have much influence on the solution due to the lowMach number flow.

Figures 10 (a)-(b) show the respective medium computational mesh and Mach number contours around the airfoil,

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-5

-4

-3

-2

-1

0

1

2

-3 -2 -1 0 1 2 3

<u>

/U

y/D

x/D = 0.7

x/D = 1

x/D = 1.5

x/D = 2

x/D = 2.5

x/D = 3

(a) Normalized u

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-3 -2 -1 0 1 2 3

<v>

/U

y/D

x/D = 0.7

x/D = 1

x/D = 1.5

x/D = 2

x/D = 2.5

x/D = 3

(b) Normalized v

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-3 -2 -1 0 1 2 3

<u

′ u′ >

/U

2

y/D

x/D = 0.7

x/D = 1

x/D = 1.5

x/D = 2

x/D = 2.5

x/D = 3

(c) Normalized u′u′

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-3 -2 -1 0 1 2 3

<v

′ v′ >

/U

2

y/D

x/D = 0.7

x/D = 1

x/D = 1.5

x/D = 2

x/D = 2.5

x/D = 3

(d) Normalized v′v′

-2.4

-2

-1.6

-1.2

-0.8

-0.4

0

0.4

-3 -2 -1 0 1 2 3

<u

′ v′ >

/U

2

y/D

x/D = 0.7

x/D = 1

x/D = 1.5

x/D = 2

x/D = 2.5

x/D = 3

(e) Normalized u′v′

Figure 9. Third-order Petrov-Galerkin solution for normalized mean flow quantities in three-dimensional viscous flow over a circularcylinder (Re = 2580). Circle symbols represent experimental data.

obtained by a fourth-order DG scheme. As seen, a very smooth solution is obtained on the current mesh. In addition,the turbulent boundary layer is very thin in the front portion of the airfoil and, as shown in Fig. 10 (c), the flow separatesin the region close to the trailing edge, at about the 0.84 chord length location. Although not shown, contours obtainedusing the SUPG scheme are virtually identical. It should also be noted that the NACA 4412 airfoil presented in thistest case has a small blunt trailing edge at x/c = 1 with a thickness of 0.0026c. This is in contrast to the CFL3D results,which were obtained on a very slightly different geometry, modified to close the trailing edge.

Figures 11-13 compare the computed velocity profiles via various orders of discontinuous Galerkin solution againstthe experimental data and CFL3D results50 for several locations along the NACA 4412 airfoil upper surface depictedin Fig. 10 (d). On the coarse mesh, the second-order DG scheme cannot accurately capture the velocity profiles, inwhich the turbulence boundary layer is seen to be much thicker and the flow separates at an earlier station. Both thethird-order and fourth-order DG schemes provide better agreement with the CFL3D results on the coarse mesh. Onthe medium and fine meshes, there is excellent agreement between the third- and fourth-order DG solution and theCFL3D results. In addition, as the computational mesh becomes finer, the second-order DG scheme shows improvedagreement with CFL3D and on the fine mesh the velocity profiles obtained by the second-order DG scheme agree wellwith CFL3D.

A similar study is performed next for various orders of the streamline/upwind Petrov-Galerkin methods, as dis-played in Figures 14-16. Similarly, the SUPG schemes provide excellent agreement with CFL3D as the computationalmesh is refined and the solution profiles between the third-order (or fourth-order) SUPG scheme and third-order (orfourth-order) DG scheme have no significant differences. The agreement of the second-order SUPG solution profileswith CFL3D on the coarse mesh is better than that of the second-order DG counterparts. However, as the computa-tional mesh is further refined, the solution difference between these two second-order schemes becomes very small.The discrepancy between the DG and SUPG solutions is likely due to the Lax-Friedrichs flux in the DG schemes,

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X

Y

0 0.5 1

-0.5

0

0.5

(a) Medium mesh (b) Mach number solution

(c) Streamlines near the trailing edge (d) Illustration of profile locations

Figure 10. (a) Medium computational mesh (containing 11133 triangular elements). (b) Mach number solution using a fourth-order DGscheme for turbulent flow over a NACA 4412 airfoil at Re = 1.52×106, M¥ = 0.2, and a = 13.87

◦ . (c) A close-up view in the region near theblunt trailing edge and streamlines. (d) Illustration of the profile locations.

which is known to be overly dissipative for boundary layers.Fig. 17 depicts contours of the turbulence working variable computed using fourth-order DG and SUPG schemes

on the fine mesh. The turbulence working variable ranges from approximately -500 to 1440 and the regions with highvalues are mainly concentrated on the wake of the airfoil. As seen, the DG solution is qualitatively very similar tothe SUPG solution. Fig. 18 shows convergence of the computed lift and drag forces as a function of the numberof elements as the order and mesh increases. The computed lift on the fine mesh is resolved with a 0.3% relativedifference between the fourth-order DG and SUPG schemes, while a 0.5% relative difference (within 1.8 drag counts)for the computed drag is obtained. Note also that the variation of the lift and drag coefficients is much less for thehigher-order schemes as the mesh is refined.

Fig. 17, discussed previously, clearly demonstrates that the turbulence working variable is negative at several lo-cations in the field. This occurs in the regions at the edge of the boundary layer where the working variable decreasesfrom potentially very large values back to freestream levels over a short distance corresponding to only a few meshcells. This rapid variation can cause oscillations similar to those often observed at shock waves for other flow vari-ables. As in the case of shock waves, these oscillations can be eliminated by discretizing the convective terms for theturbulence model using a first-order methodology. To investigate the differences between a consistent discretizationand a first-order discretization for the convection term, velocity profiles discussed previously are compared using thesetwo approaches. Unlike the consistent discretization approach, in which the turbulence working variable has negativevalues, the turbulence working variable in the first-order convection approach maintains positive values throughoutthe entire domain. While the remainder of the terms in the model are discretized to a higher order, the larger dis-

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(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 11. Velocity profiles of discontinuous Galerkin solutions at various stations of NACA 4412 airfoil upper surface on coarse mesh(Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 12. Velocity profiles of discontinuous Galerkin solutions at various stations of NACA 4412 airfoil upper surface on medium mesh(Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 13. Velocity profiles of discontinuous Galerkin solutions at various stations of NACA 4412 airfoil upper surface on fine mesh(Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

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(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 14. Velocity profiles of streamline/upwind Petrov-Galerkin solutions at various stations of NACA 4412 airfoil upper surface oncoarse mesh (Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 15. Velocity profiles of streamline/upwind Petrov-Galerkin solutions at various stations of NACA 4412 airfoil upper surface onmedium mesh (Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 16. Velocity profiles of streamline/upwind Petrov-Galerkin solutions at various stations of NACA 4412 airfoil upper surface on finemesh (Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ ).

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(a) DG (b) SUPG

Figure 17. Contours of turbulence working variable, v, on the fine mesh (containing 27537 triangular elements) using fourth-order DG andSUPG schemes for turbulent flow over a NACA 4412 airfoil at Re = 1.52 × 106, M¥ = 0.2, and a = 13.87

◦ .

Figure 18. Lift and drag convergence for turbulent flow over a NACA 4412 airfoil at Re = 1.52 × 106, M¥ = 0.2, and a = 13.87◦ .

(a) x/c = 0.675 (b) x/c = 0.842 (c) x/c = 0.953

Figure 19. Comparison of velocity profiles on the medium mesh obtained using a third-order SUPG discretization and a first-order convec-tion discretization to the turbulence model equation, in the case of turbulent flow over a NACA 4412 airfoil at Re = 1.52 × 106, M¥ = 0.2,and a = 13.87

◦ .

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cretization error resulting from the first-order convection scheme negatively effect the accuracy of the overall scheme.As evidenced in Fig. 19, the low-order discretization of the convective terms in the turbulence model leads to pooreragreement with the CFL3D solution, demonstrating that the consistent discretization model provides better agreementand should be utilized.

(a) Computational mesh (b) Contours and isolines of Mach number (DGP1)

(c) Contours and isolines of Mach number (DGP3) (d) Streamlines around 30P30N

(e) Streamlines around slat (f) Streamlines around flap-cove and flap

Figure 20. Computational mesh, Mach number contours and steamlines for turbulent flow over a 30P30N multi-element airfoil at M¥ = 0.2,a = 16.2

◦ and Re = 9,000,000.

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C. Turbulent flow over a high-lift multi-element airfoil

The final numerical example consists of turbulent flow over a multi-element airfoil configuration at a Reynolds numberof 9,000,000, Mach number of 0.2, and 16.2 degrees of angle of attack. This configuration has been widely utilizedand had originally been used for a code-validaition workshop.51 The capability of the present high-order finite-elementmethods is examined for capturing various flow structures for this complex geometry.

The geometry of the multi-element airfoil consists of a leading edge slat, a main center element and a trailing edgeflap. The computational mesh employed in this test case is shown in Fig. 20 (a), which contains 31434 triangularelements. Figures 20 (b)-(c) depict the computed Mach number solution using the respective second and fourth-orderdiscontinuous Galerkin schemes; contours obtained using the SUPG method are not shown but are very similar. It isnoted that the slat wake persists all the way over the flap, and in addition, a much smoother solution is observed asthe higher-order scheme is employed. This implies that increasing discretization order results in improved solutionaccuracy. Streamlines around the multi-element airfoil configuration are displayed in Fig. 20 (d), with the close-upviews on the slat and the flap as well as the flap cove on the main element, as depicted in Figures 20 (e)-(f). As seenin the figures, the streamlines show high curvatures near the gap between the slat and the main element. Recirculationzones near the slat lower surface and the flap cove are clearly shown.

Figure 21. Computed turbulence working variable using a fourth-order DG scheme for turbulent flow over a 30P30N multi-element airfoilat M¥ = 0.2, a = 16.2

◦ and Re = 9,000,000.

(a) DG (b) SUPG

Figure 22. Computed surface pressure using various orders of DG and SUPG schemes for turbulent flow over a 30P30N multi-elementairfoil at M¥ = 0.2, a = 16.2

◦ and Re = 9,000,000.

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(a) Illustration of profile stations

(b) P1 at x/c = 0.45 (main) (c) P1 at x/c = 0.8982 (Flap) (d) P1 at x/c = 1.1125 (Flap)

(e) P2 and P3 at x/c = 0.45 (Main) (f) P2 and P3 at x/c = 0.8982 (Flap) (g) P2 and P3 at x/c = 1.1125 (Flap)

Figure 23. Comparison of computed velocity profiles for various orders of DG and SUPG schemes with experimental data for turbulentflow over a 30P30N multi-element airfoil at M¥ = 0.2, a = 16.2

◦ and Re = 9,000,000.

Contours of the computed turbulence working variable for the fourth-order DG scheme are depicted in Fig. 21,where the maximum turbulence working variable is distributed in the regions of the flap wake. The relatively highvalues in the slat wake and slat recirculation region as well as the flap-cove can also be observed. It should be notedthat as the triangular mesh elements downstream of the multi-element airfoil increase in size, the turbulent working

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variable exhibits large oscillations, which leads to negative values. In such cases, the production, destruction, anddissipative terms of the modified SA model become inactive to maintain the stability of the scheme.

The computed surface pressures for this test case are displayed in Fig. 22 for various orders of the DG and SUPGschemes. There are some over-predictions on the main element and the flap using the second-order DG scheme,however they are not present in the solution of the second-order SUPG scheme. Both the third-order (or fourth-order)DG and SUPG schemes provide very good agreement on the distributed surface pressure compared to the experimentalresults.

Computed velocity profiles at several locations along the main element and the flap are shown in Figure 23 and arecompared with experimental data. Also shown is an illustration (Fig. 23 (a)) that indicates the locations where the dataare obtained. Figures 23 (b)-(d) first compare the second-order DG and SUPG schemes at various stations along themain element and flap. At the locations of x/c = 0.45 and x/c = 0.8982, the velocity profiles near the wall are capturedfairly well, however, some over-predictions are shown in the slat wake near the upper surface of the main element aswell as the flap immediately downstream of the main element (the behavior is more evident in the second-order DGsolution). This is because more dissipation is produced in the slat wake as it is resolved by the lower-order schemes.At the location near the flap trailing edge (x/c = 1.1125), the second-order schemes cannot resolve an accurate profile,as more deviations from the experimental results are shown. The computed velocity profiles using the third and fourth-order DG and SUPG schemes are plotted in Figures 23 (e)-(g). The third-order DG and SUPG solutions agree verywell with each other as well as with the the corresponding fourth-order solutions. In addition, enhanced agreement isseen between the present third or fourth-order solutions with the experimental results.

VI. Conclusions

This paper presents high-order finite-element discretization methods, consisting of discontinuous Galerkin andstreamline/upwind Petrov-Galerkin methods, for solutions of three-dimensional viscous flow and two-dimensionalturbulent flows. High-order curved meshes at physical boundaries are employed to accurately represent the actual twoand three-dimensional geometries. To prevent the generation of collapsed elements, the interior elements must be de-formed accordingly. This is particularly important for high Reynolds number flows, where high aspect-ratio elementsare required in viscous boundary layers to accurately resolve the flow features. The present work utilizes a modifiedlinear elasticity theory for this mesh movement procedure to obtain the perturbations required at the interior meshpoints as well as the additional quadrature points needed for a high-order finite-element mesh. Consistent discretiza-tion for the modified turbulence model equation is carried out for both the DG and SUPG methods investigated in thiswork. Solutions between the DG schemes and SUPG schemes agree very well for the present three-dimensional vis-cous flow and two-dimensional turbulent flow cases. However, slightly better solution using the second-order SUPGscheme than the second-order DG counterpart is observed in the two-dimensional turbulent flow cases, and this islikely related to the fact that a Lax-Friedrichs flux function is currently applied in the discretization of the convectiveflux terms, which could lead to more dissipation when the computational mesh is coarse. Future work will concentrateon the efficiency and robustness of the high-order solvers and extending the three-dimensional codes for turbulentflows.

VII. Acknowledgments

The work was supported by the Tennessee Higher Education Commission (THEC) Center of Excellence in AppliedComputational Science and Engineering (CEACSE). The support is greatly appreciated.

References

1F. Bassi, S. Rebay, Numerical evaluation of the two discontinuous Galerkin methods for the compressible Navier-Stokes equations, Int. J.Numer. Meth. Fluids. 40 (2002) 197–207.

2K. J. Fidkowski, T. A. Oliver, J. Lu, D. Darmofal, p-multigrid solution of high-order discontinuous Galerkin discretizations of the compress-ible Navier-Stokes equations, J. Comput. Phys. 207 (2005) 92–113.

3R. Hartmann, P. Houston, Symmetric interior penalty DG methods for the compressible navier-stokes equations II: Goal-oriented a posteriorierror estimation, International Journal of Numerical Analysis and Modeling 3 (1) (2005) 141–162.

4Z. Wang, Adaptive High-Order Methods in Computational Fluid Dynamics, World Scientific Publishing Co. Pte. Ltd., 5 Toh Tuck Link,Singapore 596224, 2011.

5D. Moro, N. C. Nguyen, J. Peraire, Navier-stokes solution using hybridizable discontinuous galerkin methods, AIAA Paper 2011-3407(2011).

21 of 23


6N. K. Burgess, D. J. Mavriplis, Robust computation of turbulent flows using a discontinuous Galerkin method, AIAA Paper 2012-0457 (Jan2012).

7H. Luo, H. Segawa, M. R. Visbal, An implicit discontinuous Galerkin method for the unsteady compressible NavierStokes equations,Computers and Fluids 53 (2012) 133–144.

8K. J. Fidkowski, D. Darmofal, A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equa-tions, J. Comput. Phys. 225 (2) (2007) 1653–1672.

9L. Wang, D. J. Mavriplis, Adjoint-based h-p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations, J. Comput.Phys. 228 (20) (2009) 7643–7661.

10T. Leicht, R. Hartmann, Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations, J. Comput. Phys.229 (2010) 7344–7360.

11K. J. Fidkowski, D. Darmofal, Review of output-based error estimation and mesh adaptation in computational fluid dynamics, AIAA J.49 (4) (2011) 673–694.

12D. L. D. M. Yano, An optimization framework for anisotropic simplex mesh adaptation: Application to aerodynamic flows, AIAA Paper2012-79 (Jan 2012).

13L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J. Flaherty, Shock detection and limiting with discontinuous Galerkin methods forhyperbolic conservation laws, Appl. Numer. Math. 48 (3) (2004) 323–338.

14P.-O. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, AIAA Paper 2006-112 (Jan 2006).15G. E. Barter, D. L. Darmofal, Shock capturing with higher-order, PDE-based artificial viscosity, AIAA Paper 2007-3823 (Jun 2007).16N. K. Burgess, D. J. Mavriplis, An hp-adaptive discontinuous Galerkin solver for aerodynamic flows on mixed-element meshes, AIAA Paper

2011-490 (Jan 2011).17C. R. Nastase, D. J. Mavriplis, High-order discontinuous Galerkin methods using an hp-multigrid approach, J. Comput. Phys. 213 (1) (2006)

330–357.18L. Wang, D. J. Mavriplis, Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations, J.

Comput. Phys. 225 (2) (2007) 1994–2015.19N. K. Burgess, C. R. Nastase, D. J. Mavriplis, Efficient solution techniques for discontinuous Galerkin discretizations of the Navier-Stokes

equations on hybrid anisotropic meshes, AIAA Paper 2010-1448 (Jan 2010).20L. Wang, D. J. Mavriplis, Adjoint-based h-p adaptive discontinuous Galerkin methods for the compressible Euler equations, AIAA Paper

2009-0952 (Jan 2009).21D. L. Bonhaus, A higher order accurate finite element method for viscous compressible flows, phD Dissertation,Virginia Polytechnic Institute

and State University (1998).22Venkatakrishnan, Higher order schemes for the compressible Navier-Stokes equations, 16th AIAA Computational Fluid Dynamics Confer-

ence, Orlando, FL, AIAA Paper 2003-3987 (June 2003).23J. T. Erwin, W. K. Anderson, S. Kapadia, L. Wang, Three dimensional stabilized finite elements for compressible navier-stokes, AIAA Paper

2011-3411 (June 2011).24W. K. Anderson, L. Wang, S. Kapadia, C. Tannis, B. Hilbert, Petrov-Galerkin and discontinuous Galerkin methods for time-domain and

frequency-domain electromagnetic simulations, Journal of Computational Physics 230 (23) (2011) 8360–8385.25P. Spalart, S. Allmaras, A one-equation turbulence model for aerodynamic flows, Le Recherche Aerospatiale 1 (1994) 5–21.26T. A. Oliver, A high-order, adaptive, discontinuous Galerkin finite element method for the Reynolds averaged Navier-Stokes equations, phD

Dissertation, Massachusetts Institute of Technology, Boston (2008).27F. Bassi, S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys. 138 (1997)

251–285.28J. ven der Vegt, H. ven der Ven, Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas

dynamics, in the Fifth World Congress on Computational Mechanics, Vienna, Austria (2002).29R. Haimes, G. Follen, Computational Analysis PRogramming Interface, proceedings of the 6th international conference on numerical grid

generation in computaitonal field simulation (1998).30W. E. Brock, C. Burdyshaw, S. L. Karman, V. C. Betro, B. Hilbert, W. K. Anderson, R. Haimes, Adjoint-based design optimization using

CAD parameterization through CAPRI, to appear in the proceedings of the 50th AIAA Aerospace Sciences Meeting (Jan 2012).31L. Wang, D. J. Mavriplis, W. K. Anderson, Adjoint sensitivity formulation for discontinuous Galerkin discretizations in unsteady inviscid

flow problems, AIAA Journal 48 (12) (2010) –.32A. N. Brooks, T. J. R. Hughes, Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on

the incompressible navier-stokes equations, Computer Methods in Applied Mechanics and Engineering 32 (1-3) (1982) 199–259.33Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput.

7 (3) (1996) 856–869.34Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, 2003.35C. R. Nastase, D. J. Mavriplis, A parallel hp-multigrid solver for three-dimensional discontinuous Galerkin discretizations of the Euler

equations, AIAA Paper 2007-0512 (Jan 2007).36P. Solin, K. Segeth, I. Dolezel, High-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman and Hall, 2003.37C.-W. Shu, Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws, ICASE Re-

port No. 97-65, NASA/CR-97-206253, 1997.38P. Batten, N. Clarke, C. Lambert, D. M. Causon, On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput. 18 (2)

(1997) 1553–1570.39K. Shahbazi, D. J. Mavriplis, N. K. Burgess, Multigrid alorithms for high-order discontinuous galerkin dicretizations of the compressible

navier-stokes equations, J. Comput. Phys. 228 (21) (2009) 7917–7940.40B. Cockburn, S. Hou, C.-W. Shu, the Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV:

The multidimensional case, Math. Comput. 54 (545).

22 of 23


41L. Wang, W. K. Anderson, Sensitivity analysis for the compressible Navier-Stokes equations using a discontinuous Galerkin method, AIAAPaper 2011-3408 (Jun 2011).

42O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, 30 CorporateDrive, Suite 400, Burlington, MA, USA, 2005.

43T. J. Barth, Numerical Methods for Gasdynamic Systems on Unstructured Meshes, Vol. 5, edited by D. Kroner, M. Ohlberger and C. Rhode,1998.

44L. P. Franca, S. L. Frey, T. J. R. Hughes, Stabilized fnite element methods: I. application to the advective-diffusive model, Computer Methodsin Applied Mechanics and Engineering 95 (2) (1992) 253–276.

45F. Shakib, T. Hughes, Z. Johan, A new finite element formulation for computational fluid dynamics: X. the compressible Euler and Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 89 (1-3) (1991) 141–219.

46W. Gropp, E. Lusk, A. Skjellum, Using MPI: Portable Parallel Programming with the Message Passing Interface, MIT Press, Cambridge,MA, 1994.

47G. Karypis, Metis, university of minnesota, department of computer science, http://www-users.cs.umn.edu/ karypis/metis (2003).48Http://zjw.public.iastate.edu/hiocfd.html.49A. H. Mohammad, Z. Wang, Large eddy simulation of flow over a cylinder using high-order spectral difference method, Advances in Applied

Mathematics and Mechanics 2 (4) (2011) 451–466.50Http://turbmodels.larc.nasa.gov/.51S. M. Klausmeyer, J. C. Lin, Comparative results from a CFD challenge over a 2D three-element high-lift airfoil, NASA Technical Memo-

randum 112858 (1997).

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Solutions of High-order Methods for Three-dimensional Compressible Viscous Flowsweb2.utc.edu/~bfc365/welcome_files/LiWang-AIAA-2012-2836... · 2013-08-16 · Solutions of High-order