high-order methods for solutions of three-dimensional turbulent...

High-order Methods for Solutions of Three-dimensionalTurbulent Flows

Li Wang∗, W. Kyle Anderson†, J. Taylor Erwin‡ and Sagar Kapadia§

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

This paper presents a high-order discontinuous Galerkin (DG) method for three-dimensional turbulentflows. As an extension of our previous work, the paper further investigates the incorporation of a modifiedSpalart-and-Allmaras (SA) turbulence model with the Reynolds Averaged Navier-Stokes (RANS) equationsthat are both discretized using a modal discontinuous Galerkin approach. The resulting system of equations,describing the conservative flow fields as well as the turbulence variable, is solved implicitly by an approximateNewton approach with a local time-stepping method to alleviate the initial transient effects. In the context ofhigh-order methods, curved surface mesh is generated through the use of a CAPRI mesh parameterizationtool, followed by a linear elasticity solver to determine the interior mesh deformations. The requirementsfor the wall coordinate and viscous stretching factor used for viscous mesh generation are studied on a two-dimensional turbulent flow case. It has been concluded that, for attached turbulent flows, the conventionalparameters often used in low-order methods can be somewhat less stringent when a higher-order method isconsidered. Several other numerical examples including a direct numerical simulation of the Taylor-Greenvortex and turbulent flow over an ONERA M6 wing are considered to assess the solution accuracy and to showthe performance of high-order DG methods in capturing transitional and turbulent flow phenomena.

I. Introduction

Despite the current dominance of second-order accurate methods in computational fluid dynamics (CFD), chal-lenges persist in the accurate numerical simulation for problems with a broad spectrum of dynamic scales, such as areasof the large-eddy and direct numerical simulation of turbulence, fluid and structure interactions, and aeroacoustics. Incomputational aeroacoustics as an example, the scales of unsteady fluctuations vary over many orders of magnitude,making it impossible to accurately resolve the smallest scales on a reasonable computer. The prevailing approach forobtaining engineering solutions is to model the small-scale fluctuations using a wall model while attempting to cap-ture the larger scales directly using locally very fine mesh spacing. Unfortunately, sufficient resolution of these flowfeatures using a second-order accurate method requires meshes that often exceed 100 million mesh points for evensimple configurations. To offset the severe computational limitations, many high-order (greater than second-order)formulations1–9 have been developed and investigated. The high-order spatial discretizations resolve enhanced flowfeatures within each element using high-order polynomial basis functions, and hence, potentially significant advancesand dramatic reduction on mesh sizes can be achieved. Because the stencils remain compact, these approaches becomeespecially beneficial in parallel computing environments. In our previous work, we have focused on the developmentof the high-order methods, particularly discontinuous Galerkin (DG)4,8 and stabilized upwind Petrov-Galerkin5,10

discretizations, for flow problems in inviscid, laminar and turbulent regimes. The present research efforts continue inthe development of the DG methods, while the emphasis is placed on the application of three-dimensional turbulentflows with curved surface mesh representations.

To numerically solve turbulent flow problems, the Reynolds Averaged Navier-Stokes (RANS) equations coupledwith the one-equation of a modified Spalart and Allmaras (SA) turbulence model7,11,12 are considered. The DGdiscretizations of the convective and viscous fluxes are carried out through the implementation of a Riemann fluxfunction and a symmetric interior penalty (SIP) method8,13,14 respectively. The SIP method does not require anauxiliary variable in the discretization of the viscous flux terms (i.e. second derivatives) and is capable of maintaining

∗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

51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition07 - 10 January 2013, Grapevine (Dallas/Ft. Worth Region), Texas

AIAA 2013-0856

Copyright © 2013 by Li Wang, W. Kyle Anderson, Taylor Erwin and Sagar Kapadia. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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the optimal error convergence rate for high-order methods. In addition, the modified SA model is particularly designedto make the original SA model15 insensitive to negative values of the turbulence working variable. Because theturbulence working variable often drops several orders of magnitudes at the edge of the turbulent boundary layer, thisabrupt change can lead to large solution oscillations or negative values when the mesh is not sufficiently fine. Toprevent the stability issue, an auxiliary parameter is defined in the modified SA model to inactivate the production,destruction and dissipative terms of the model in the “troubled”elements, while keeping only the advection terms. Thisprocedure has a very effective improvement on the robustness of the model as it is discretized by a high-order method.Moreover, a tightly coupled system is implemented in this work for the discretized RANS equations and the turbulencemodel equation. The solution is solved implicitly based on an approximate Newton method, where the flow Jacobianmatrix is constructed in an element-wise manner and is stored using block compressed row storage. To make theproposed algorithm efficient and competitive, we consider a multigrid approach,2,4,16 driven by a linearized elementGauss-Seidel smoother or a Generalized Minimal Residual (GMRES)17 algorithm with ILU(k) preconditioning.

Furthermore, for second-order accurate CFD schemes, the geometry surface is typically represented by a seriesof linear elements. In the context of high-order methods, for which the associated computational mesh is generallycoarse, this simple representation must be replaced with one that has increased fidelity to properly account for surfacecurvature.8,18,19 In application of the inviscid flows, it is common to use high-order surface representation only forelements coinciding with the surface boundary, while keeping the interior mesh linear. This process, however, be-comes nontrivial when dealing with turbulent flow problems of high Reynolds numbers. Specifically, highly distortedelements are required in a thin turbulent boundary layer, and as a result, the projection of the curved surface meshcan lead to a substantial number of collapsed or negative Jacobian elements. In the present work, we adopt a linearelasticity theory20 to determine the necessary perturbations for the interior mesh points and the additional quadraturepoints to avoid the generation of collapsed elements. Additionally, the CAPRI mesh parameterization tool describedin References21,22 is utilized in the mesh generation procedure for obtaining the true positions of surface quadraturepoints23 on arbitrary three-dimensional configurations.

The remainder of the paper is structured as follows. In Section II the governing equations are introduced. SectionIII describes the spatial discontinuous Galerkin discretization method as well as the formulation for an implicit time-integration scheme. Section IV briefly reviews the use of the CAPRI mesh parameterization interface and the meshmovement strategy employed in the present work. Several numerical examples are presented in Section V to assess thesolution accuracy and to show the performance of the high-order DG methods in capturing transitional and turbulentflow phenomena. Finally, Section VI summarizes the conclusions and discusses the future work.

II. Governing Equations

The compressible Reynolds Averaged Navier-Stokes equations coupled with the one equation of the modifiedSpalart and Allamars turbulence model7,11 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 =

rrurvrwrErṽ

, Fxe =

ruru2 + p

ruvruw

(rE + p)uruṽ

, Fye =

rvruv

rv2 + prvw

(rE + p)vrvṽ

, Fze =

rwruwrvw

rw2 + p(rE + p)w

rwṽ

(2)

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Fxv =

0txxtxytxz

utxx + vtxy +wtxz +k ¶T¶x1s µ(1+y)

¶ṽ¶x

, Fyv =

0txytyytyz

utxy + vtyy +wtyz +k ¶T¶y1s µ(1+y)

¶ṽ¶y

, Fzv =

0txztyztzz

utxz + vtyz +wtzz +k ¶T¶z1s µ(1+y)

¶ṽ¶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 ṽ 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

− 23

¶uk¶xk

di 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 ={

rṽ fv1 if ṽ ≥ 00 if ṽ < 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:7,11

ST = cb1S̃µy − cw1r fw(nyd

)2 +1s

cb2rÑṽ · Ñṽ −1s

n(1+y)Ñr · Ñṽ (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

S̃ =

{S + Ŝ if Ŝ ≥ −cv2SS + S(c

2v2+cv3Ŝ)

(cv3−2cv2)S−Ŝif Ŝ < −cv2S

(8)

S =√−→w ·−→w Ŝ = ny

k2T d2fv2 fv1 =

y3

y3 + c3v1fv2 = 1 −

y1+y fv1

(9)

and

r =ny

S̃k2T d2g = r + cw2(r6 − r) fw = g

(1+ c6w3g6 + c6w3

)1/6(10)

respectively. −→w denotes the vorticity vector, Ñ × u. The notation d refers to distance to the nearest wall at a specificlocation and is computed to account for the curvature of the actual boundaries. The parameter y in the equations isdesigned to remove the effects of a negative turbulence working variable on the robustness of the turbulence model asit is 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 =rṽµ

(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/k2T +(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(µ

Pr+

µTPr T

)(

E − 12(u2 + v2 +w2)

)(13)

where Pr and PrT are the Prandtl and turbulent Prandtl numbers that are set to be 0.72 and 0.9 respectively. In the caseof a direct numerical simulation, the governing equations reduce to the compressible Navier-Stokes equations, wherethe turbulence model equation is deactivated and the turbulence eddy viscosity, µT , in the fluid viscous stress tensorand the thermal conduction term vanishes.

For the purpose of the DG discretization, we rewrite the Cartesian viscous fluxes in the following equivalent form:

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 = ¶Fyv/¶(¶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. Spatial discretization

The computational domain W is partitioned into a tessellation of non-overlapping tetrahedral elements such thatW =

Sk Wk, where Wk refers to the volume of an element k in the computational mesh. The discontinuous Galerkin

finite-element approximation is expanded as a truncated series of basis functions,24 {f j, j = 1, · · · ,M}, and solutioncoefficients as,

Uh =M

åi=1

Ũhi 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:Z

Wkf 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 method3,14 for the viscous fluxesyields the following weak formulation,

ZWk

f j¶Uh¶t

dWk −Z

WkÑf 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\¶WJ{G}[[Uh]] · [[f j]]dS

−Z

¶Wk∩¶Wf+j F

bv(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∩¶WJG(Ub)(U+h − Ub)n · f

+j ndS +

Z¶Wk∩¶W

f jFe(Ub) · ndS −Z

Wkf jS(Uh,ÑhUh) = 0

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where the unit normal vector n is outward to the boundary. Hc(U+h ,U−h ,n) represents an approximate Riemann con-

vective flux function, particularly the HLLC25 function, 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 J is explicitlyevaluated by the element geometry and the order of discretization,14,26 given by:

J =(p+1)(p+D)

(2D)max(

S+kV +k

,S−kV −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, Fbv , associated with the energy equationvanishes. The set of the discretized equations is solved in modal space and the integrals are evaluated using Gaussianquadrature rules,24 which are exact for polynomial degree 2p in volume integrals and for polynomial degree 2p+1 insurface integrals.27,28

Because the set of basis functions is defined in a master element W̄ 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

x̃kifi(x,h,z), Jk =

¶x¶x¶y¶x¶z¶x

¶x¶h¶y¶h¶z¶h

¶x¶z¶y¶z¶z¶z

. (20)where x̃k represents 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 boundaries29 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 quadrature points within each element,as:

x̃k = F−1x̂pk (21)

where x̂pk = {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 (x̂pk ← x̂pk ). The additionalsurface points are created using the CAPRI parametric mesh respresention.21,22 To accommodate the properly curvedsurface elements, interior elements in the boundary layer must be deformed to avoid the generation of negative vol-umes. Therefore, the current work utilizes a mesh movement strategy based on a linear elasticity theory20 to computethe deformation of the interior mesh points as well as that of the additional nodes for high-order finite-element meshes.

B. Time-integration scheme

For time-dependent flows, the temporal scales are resolved using a second-order backward difference formula (BDF2).4

We rewrite the weighted form of the discretized equations in the following ordinary differential equation form,

MdŨhdt

+R(Ũh) = 0 (22)

where R represents the discretized spatial residual (including both inviscid and viscous terms) and M denotes the massmatrix. Then, the time advancement is performed using the BDF2 temporal scheme, written as:

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Rn+1e (Ũn+1h ) =

MDt

(32

Ũn+1h )+R(Ũn+1h )−

MDt

(2Ũnh −12

Ũn−1h ) = 0 (23)

where Rn+1e represents the unsteady flow residual at time step n+1. The implicit system is solved using an approximateNewton method,4 where the flow Jacobian matrix is decomposed with element-based diagonal and off-diagonal blockcomponents. The linearized system is solved via a multigrid approach,2,4,16 driven by a linearized element Gauss-Seidel smoother or an ILU(k) preconditioned GMRES approach.17,30 For steady state problems, the time-dependentterm in Eq. (22) vanishes and a local time-stepping method is incorporated to alleviate the stiffness of the system in theinitial stages of the calculation. The size of the local time step is determined by a gradually increasing CFL number,typically ranging from 1 to 200. The addition of the local time stepping term has been found to be exclusively importantto the calculations of turbulent flows and to be essential to the robustness of the flow solver in turbulence regimes. Inaddition, the high-order flow solver operates on parallel computers using the standard MPI message-passing library31

and the mesh is partitioned based on the METIS mesh partitioner.32

IV. 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 that has increased fidelity to properly account for surface curvature. To this end, an inter-face has been developed for incorporating CAPRI21 into the geometry libraries of the UTC (University of Tennessee atChattanooga) SimCenter to allow communication with CAD software.22 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.

As boundary elements are curved to conform to the original geometry configuration, collapsed elements arelikely to be generated, particularly when highly stretched elements are applied in the viscous boundary layer. Asthe Reynolds number increases, the aspect ratio of the elements in the near-wall regions increases. In this context, arobust 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,20,33,34 which assumesthat the 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 (24)

where d = (dx,dy,dz) denotes the nodal displacement vector in the Cartesian coordinate directions and the coefficients,di j, 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) (25)

where E represents Young’s modulus that is taken as inversely proportional to the distance from the closest wall.The notation u denotes Poisson’s ratio, which is set to be 0.3. It is noted that the nodal displacement vector shouldalso include the perturbations at the additional quadrature points for high-order finite-element meshes. Moreover, theperturbations at the interior mesh points as well as quadrature points are solved with Dirichlet boundary conditions,specified at the surface nodes and surface quadrature points. The linear elasticity equations are solved by a GMRESalgorithm with ILU(k) preconditioning.

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. For a computational mesh in three dimensions, the spec-ified perturbations at the deformed boundaries are realized with a sequence of small steps such that the strains or

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(a) Computational mesh (y+ = 1) (b) Mach number contours

Figure 1. Computational mesh (containing 9671 triangular elements) with y+ of 1 and Mach number contours for turbulent flow over aNACA0012 airfoil at M¥ = 0.15, a = 0◦ and Re = 6,000,000 using a fifth-order DG discretization.

deformations in each linear elasticity problem become small. In addition, our previous research5 shows that the useof a quadratic (p = 2) finite-element mesh guarantees that the optimal error convergence rate for high-order schemescan be achieved without influence from utilizing a curved interior mesh. Therefore, in this paper, we first use thelinear elasticity solver to determine the interior perturbations as a quadratic mesh. Then, the quadratic mesh wouldbe used for a high-order DG scheme (p ≥ 2) for solution approximations. Also, it should be noted that in the meshmovement process for three-dimensional highly stretched elements, difficulties may be encountered at critical surfaceareas, such as places having sharp and large curvatures. In the current results that will be discussed in Section V.C, afew elements near the trailing edge on the tip of the ONERA M6 wing have negative Jacobians and the surface in thisarea is therefore held fixed to that corresponding to linear elements. Further research has been undergoing to make thelinear elasticity solver perform more robustly.

V. Numerical Results

In this section, we present several steady and unsteady turbulent flow cases to examine the requirements for keymesh parameters, such as the wall coordinate and viscous stretching factor for high-order accurate methods, and todemonstrate the performance of the present DG schemes for capturing transitional and turbulent flow features.

A. Turbulent flow over a NACA0012 airfoil

The first example considers turbulent flow over a NACA0012 airfoil at a free-stream Mach number of 0.15, 0 degreeangle of attack and a Reynolds number of 6,000,000. This test case aims to provide some guidelines for higher-ordermethods (p > 1) in selecting key parameters used in the viscous mesh generation. Particularly, the wall coordinate(y+) and the stretching factor (b) in the viscous boundary layer are investigated for various orders of the DG schemesranging from third to fifth-order of accuracy. The conventional setting for these parameters in lower-order methodssuggests a wall coordinate of 1 for placing the closest point to the wall and a stretching factor of 1.15 for the viscousmesh near boundary to be stretched. However, due to the fact that higher-order methods provide additional resolutionwithin each element by increasing the orders of polynomial basis functions, this setting may lead to excessive meshpoints in the viscous boundary layer. The numerical simulation is conducted in the following using various ordersof the DG discretizations ranging from p = 2 to p = 4. The two-dimensional meshes employed in this example aregenerated based on an advancing front algorithm35 and the airfoil surface is represented by high-order polynomialfunctions with non-slip and adiabatic boundary conditions. The convective flux is resolved using the HLLC fluxfunction.25

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1. Effect of wall coordinate

The effect of the wall coordinate is first studied using four unstructured triangular meshes with y+ of 1, 5, 10 and15 respectively, while keeping the stretching factor fixed to be 1.15. As the wall coordinate increases the mesh sizedecreases, and for this case, the four meshes contain 9671, 8573, 7775 and 7269 triangular elements respectively. Fig.1 displays the computational mesh corresponding to y+ of 1 and Mach number contours near the airfoil using thefifth-order accurate DG scheme. It is seen that a smooth solution is obtained around the airfoil and the present DGscheme resolves a very thin turbulent boundary layer. Fig. 2 depicts profiles of the horizontal velocity in the turbulentboundary layer obtained at various stations of the airfoil upper surface (h = 0.55, 0.8, 0.9 and 0.97) for the meshes ofy+=1, 5, 10 and 15. It is clearly shown that the present variation in the wall coordinate has no visible influence on thesolution profiles for all the DG schemes: the solution profiles from larger wall coordinates show good agreement withthe one from y+ = 1 at all stations.

To better quantify the influence of wall spacing, Fig. 3 displays profiles of the dimensionless velocity as a functionof the wall coordinate obtained at 0.4 chord-length location on the upper airfoil surface. It is observed that the profileson the mesh of y+ = 1 agree well with the exact solution for all different orders of the DG schemes, and this is also thecase for y+ = 5. Deviations can be seen for the third-order (p = 2) DG scheme as the wall spacing increases to 10 and15. However, for the fourth and fifth-order accurate DG schemes, the law-of-the-wall profiles for y+ of 10 still agreewell with those obtained from the mesh of y+ = 1. As the wall coordinate increases to 15, some deviations appear inthe profile of the fourth-order DG scheme. However, the profile for the fifth-order DG scheme on the mesh of y+ = 15exhibits improved agreement compared to the DG p = 3 case. The results in Fig. 3 imply that, for the present ordersof DG schemes, it is necessary to have at least one or two mesh points distributed in the viscous sublayer (y+ < 10) tocapture an accurate profile of the law of the wall.

Finally, distribution of the skin friction coefficients on the airfoil upper surface is plotted in Fig. 4 and is comparedto the CFL3D solution,36 which is obtained on a substantially bigger mesh and is used to serve as the referencesolution. In general, the profiles of the skin friction on the meshes of y+ = 1 and y+ = 5 agree reasonably well withthe CFL3D solution for all the DG schemes. On the mesh of y+ = 10 or y+ = 15, the third-order DG scheme showsover-predicted skin friction on the airfoil upper surface. This explains that the law-of-the-wall profiles of y+ = 10 andy+ = 15 shown in Fig. 3(a) are below the curve of y+ = 1. In the case of the fourth and fifth-order DG schemes, theagreement has greatly improved for the profiles of y+ = 10, while to some extent the fifth-order DG scheme providesa more accurate skin friction solution for y+ of 15 as compared to the fourth-order DG counterpart. Based on theseries of comparisons discussed above, it is concluded that for a higher-order scheme (p > 1) on attached flow, thecomputational mesh can be generated with a wall distance of y+ = 5. For a spatial scheme of an order that is higherthan third-order, this requirement of the wall spacing can increase to 10 with acceptable solution accuracy.

2. Effect of viscous stretching factor

The stretching factor is another parameter that determines the density of a mesh in the viscous layer and further thesize of the overall computational mesh. Therefore, in this section, the effect on the variation of the viscous stretchingfactor is studied for the DG discretizations ranging from p = 2 to p = 4. Here considered are a sequence of fourunstructured triangular meshes with the stretching factor of 1.15, 1.2, 1.3 and 1.4 (and a fixed wall spacing of y+ = 5),which results in meshes containing 8573, 6965, 5409 and 4581 triangular elements respectively. It is noted that a 22%increase in the stretching factor (from 1.15 to 1.4) leads to a mesh that contains nearly half the number of elements asthe original mesh.

Fig. 5 displays profiles of the horizontal velocity at several stations of the NACA 0012 airfoil upper surface usingthe meshes with various stretching factors. It is shown that the variation in the stretching factor has no visible influenceon the solution profiles for all the DG schemes. Next, in Fig. 6, the dimensionless velocity in the turbulence boundarylayer is plotted as a function of the wall coordinate and is compared to the exact solution. Similarly, the profile obtainedon a mesh with a bigger stretching factor agrees very well with the one corresponding to the conventional setting of1.15. Finally, distribution of the skin friction coefficients on the airfoil upper surface is depicted in Fig. 7 for all thepresent DG discretizations. It can be observed that no deviations are made in the solution of the skin friction as thestretching factor increases from 1.15 to 1.4, especially for the DG p = 3 and p = 4 schemes. To this end, we concludethat, for a DG scheme of an order higher than second-order, the stretching factor of 1.4 can be selected for stretchinga viscous boundary mesh with acceptable solution accuracy. Note that this result is obtained for a fully attached flowand may not be definitive for separated flows.

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u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97

y+=1y+=5y+=10y+=15

(a) p = 2

u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97

y+=1y+=5y+=10y+=15

(b) p = 3

u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97

y+=1y+=5y+=10y+=15

(c) p = 4

Figure 2. Solution profiles obtained on various meshes with y+ of 1, 5, 10 and 15, using different DG schemes from p = 2 to p = 4 for theturbulent NACA0012 airfoil case.

y+

u+

100 101 102 1030

5

10

15

20

25

Law of the wally+ = 1y+ = 5y+ = 10y+ = 15

(a) p = 2

y+

u+

100 101 102 1030

5

10

15

20

25


(b) p = 3

y+u

+

100 101 102 1030

5

10

15

20

25


(c) p = 4

Figure 3. Profiles of the dimensionless velocity as a function of wall coordinate obtained on various meshes with y+ of 1, 5, 10 and 15, usingdifferent DG schemes from p = 2 to p = 4 for the turbulent NACA0012 airfoil case.

X

Cf

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

y+ = 1y+ = 5y+ = 10y+ = 15CFL3D

(a) p = 2

X

Cf

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

y+ = 1y+ = 5y+ = 10y+ = 15CFL3D

(b) p = 3

X

Cf

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

y+ = 1y+ = 5y+ = 10y+ = 15CFL3D

(c) p = 4

Figure 4. Comparison of skin friction coefficients on the airfoil upper surface using various meshes with y+ of 1, 5, 10 and 15 and differentDG schemes from p = 2 to p = 4 for the turbulent NACA0012 airfoil case. Circle symbols represent the CFL3D solution as the benchmarksolution.

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u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97

Factor=1.15Factor=1.2Factor=1.3Factor=1.4

(a) p = 2

u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97


(b) p = 3

u

y

0 0.2 0.4 0.6 0.8 1 1.20

0.005

0.01

0.015

0.02

0.025

=0.55

=0.8

=0.9

=0.97


(c) p = 4

Figure 5. Solution profiles obtained on various meshes with the viscous stretching factor of 1.15, 1.2, 1.3 and 1.4, using different DG schemesfrom p = 2 to p = 4 for the turbulent NACA0012 airfoil case.

y+

u+

100 101 102 1030

5

10

15

20

25

Law of the wallFactor = 1.15Factor = 1.2Factor = 1.3Factor = 1.4

(a) p = 2

y+

u+

100 101 102 1030

5

10

15

20

25


(b) p = 3

y+

u+

100 101 102 1030

5

10

15

20

25


(c) p = 4

Figure 6. Profiles of the dimensionless velocity as a function of wall coordinate obtained on various meshes with the viscous stretchingfactor of 1.15, 1.2, 1.3 and 1.4, using different DG schemes from p = 2 to p = 4 for the turbulent NACA0012 airfoil case.

X

Cf

0 0.2 0.4 0.6 0.8 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Factor = 1.15Factor = 1.2Factor = 1.3Factor = 1.4

(a) p = 2

X

Cf

0 0.2 0.4 0.6 0.8 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


(b) p = 3

X

Cf

0 0.2 0.4 0.6 0.8 10

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


(c) p = 4

Figure 7. Comparison of skin friction coefficients on airfoil upper surface using various meshes with the viscous stretching factor of 1.15,1.2, 1.3 and 1.4 and different DG schemes from p = 2 to p = 4 for the turbulent NACA0012 airfoil case.

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XY

Z

(a) Computational mesh (b) t∗ = 0 (c) t∗ = 3

(d) t∗ = 6 (e) t∗ = 9 (f) t∗ = 15

Figure 8. Computational mesh (containing 48,000 tetrahedral elements) and some isosurfaces of the vorticity magnitude for the Taylor-Green vortex at Re = 280 at different times using the fifth-order DG discretization and the BDF2 temporal scheme.

B. Direct numerical simulation of Taylor-Green vortex at Re = 280

This test example considers a direct numerical simulation of the Taylor-Green vortex at Re = 28037 with a simpleinitial condition in the field as follows,

u = U0 sin( x

L

)cos

( yL

)cos

( zL

)v = −U0 cos

( xL

)sin

( yL

)cos

( zL

)w = 0

p = p0 +r0U20

16

(cos

(2xL

)+ cos

(2yL

))(cos

(2zL

)+2

)r =

pr0p0

(26)

where the notations of L and r0 denote the respective length and density for nondimensionalization. The velocity fieldin the code is non-dimensionalized by the speed of sound, c0, and therefore, the reference velocity U0 is determinedby U0 = M and the reference pressure p0 is determined by p0 = 1/g. The Mach number, M, for the simulation is set tobe 0.1 and the initial conservative variables such as momentum and total energy can be obtained using the expressionsgiven in Eq. (26).

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t*

Kin

etic

En

erg

y

0 3 6 9 12 150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

DG p2DG p3DG p4Reference

(a) Kinetic energy

t*

Kin

etic

En

erg

y

4 6 8 10 12

0.04

0.06

0.08

0.1


(b) Close-up view of kinetic energy

t*

Dis

sip

atio

n R

ate

0 3 6 9 12 150.002

0.004

0.006

0.008

0.01

0.012

0.014


(c) Dissipation rate

t*

En

stro

ph

y

0 3 6 9 12 150.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


(d) Enstrophy

Figure 9. Evolution of kinetic energy, dissipation rate and enstrophy for simulations of the Taylor-Green vortex at Re = 280 using the BDF2temporal scheme and various orders of DG discretizations (p = 2,3 and 4).

This test case aims to assess the accuracy and the performance of the present DG methods for three-dimensionalperiodic and transitional flow. The flow is confined in a cube box defined as −pL ≤ x,y,z ≤ pL with periodic boundaryconditions in all directions. The length scale L in the description is chosen to be 1 and the computational mesh, asdisplayed in Fig. 8(a), contains 21 points in each direction and a total of 48,000 tetrahedral elements. An order ofconvergence study is carried out using the DG discretizations of third, fourth and fifth-order precision, which resultsin 480,000, 960,000 and 1,680,000 degrees of freedom, respectively. The time evolution is resolved using the second-order backward difference scheme (BDF2) with a fixed time-step size of 0.02. It should be noted that the time (t) in thepresent code is a non-dimensional quantity based on the length and the speed of sound. Therefore, the employed time-step size is in fact equivalent to 0.002 for the case where the nondimensional time (t∗) is obtained using the referencevelocity. It is also noted that halving the time-step size does not produce improvements on the solution accuracy. Theconvective flux is solved by the HLLC Riemann flux function, and moreover, the implicit problem at each time step issolved by a p-multigrid approach4 driven by a linearized element Gauss-Seidel smoother. This approach sufficientlyconverges the L2 density residual to 10−14 within 2 p-multigrid iterations.

Figures 8(b)-(f) illustrate isosurfaces of the vorticity magnitude at different times using the fifth-order DG scheme.As seen, the vortex structure is relatively large at the initial time and becomes quite small at the later times as the dis-sipation rate increases. At t∗ ≈ 6, the flow appears to reach the maximum of the dissipation rate and thus the strongestdynamic motions of the flow. In addition, the fifth-order DG scheme delivers smooth vortex representations withoutvisible discontinuities at elemental interfaces. To further assess the accuracy of the high-order DG discretizations,

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the temporal evolution of the averaged turbulence kinetic energy, Ek, the kinetic energy dissipation rate, e, and theaveraged enstrophy, j, is studied. These quantities are computed at each time step involving volume integration in theentire field, given as:

Ek =1

r0W

ZW

ru · u

2dW

e = −dEkdt

j =1

r0W

ZW

r−→w ·−→w

2dW (27)

where u and −→w denote the respective velocity and vorticity vector. The volume integrations given in Eq. (27) areevaluated using Gaussian quadrature rules that are exact for the polynomial degree of 2p. In addition, the turbulencekinetic energy dissipation rate is computed with a first-order finite difference scheme, as en = (En+1k − Enk )/Dt∗.

Fig. 9 depicts evolutions of the turbulence kinetic energy, the dissipation rate and the enstrophy, obtained usingthe DG p = 2, p = 3 and p = 4 schemes. The Fourier pesudo-spectral solution digitized from Reference37 is used inthe comparisons to serve as the reference solution. For the evolution of the turbulence kinetic energy displayed in Fig.9(a), one can observe that the high-order DG schemes are capable of capturing the basic dynamics of turbulent flowsand the difference between all the DG solutions and the reference solution is very small. Fig. 9(b) further provides aclose-up view of the turbulent kinetic energy in the time period of 4 < t∗ < 12, where it shows that the third-order DGscheme contains less energy as time evolves, while both the fourth and fifth-order DG solutions are in good agreementwith the reference solution. Fig. 9(c) depicts the evolution of the energy dissipation rate, e, obtained from all the DGdiscretizations. It is clearly shown that the third-order DG solution predicts higher dissipation rates when the smallscales are of strong dominance (t∗ ≈ 6) in the field, while abrupt decay appears after the turbulent flow passes themaximum rate of dissipation. In contrast, significant improvements are observed in the solutions of the fourth andfifth-order DG counterparts resulting from the extra resolutions offered by the higher-order schemes. Finally, the timehistory of the enstrophy is provided in Fig. 9(d). For incompressible flow, the behavior of the enstrophy evolutionshould be similar to that of the dissipation rate based on the relation, j = r02µ e. However, the numerical simulationfrom the third-order DG scheme shows substantial under-prediction near the enstrophy peak on the current mesh.The prediction improves greatly with the use of the higher-order DG schemes as the shape of the enstrophy curve iscaptured more consistently and accurately compared to the reference solution.

C. Turbulent flow over an ONERA M6 wing

The final numerical example consists of turbulent flow over an ONERA M6 wing configuration at a Reynolds numberof 11,720,000 per mean aerodynamic chord (MAC), Mach number of 0.3 and 3.08 degrees of angle of attack. Thisconfiguration has been widely utilized in experimental and numerical studies for transonic flows, however, the currentinvestigation focuses on subsonic and turbulent flow.

The geometry definition and the computational mesh containing 255,654 tetrahedral elements are displayed in Fig.10(a). The wing surface is modeled as a non-slip adiabatic wall. Based on the previous two-dimensional turbulentinvestigation, the mesh comprises a viscous spacing approximately as y+ of 10, along with a stretching factor of 1.4 forstretching the viscous mesh normal to the wall. The mesh surface is represented by a series of quadratic polynomialsand the interior mesh is deformed using the linear elasticity method. Although most of the elements are moved andcurved in a proper manner to adjust the projection of the curved surface boundary, the surface near the wing tip isheld fixed to avoid forming a few isolated elements with negative Jacobians. Further research has been undergoing toexamine the issue with the mesh movement.

A close-up view of the leading edge area is illustrated in Fig. 10(b), where the curvatures of the wing surface canbe clearly seen. Fig. 10(c) plots contours of the perturbation magnitude, D = (d2x +d2y +d2z )1/2, obtained by the linearelasticity solver at the spanwise location of h = 0.5. Because of the presence of relatively large curvatures, the frontportion of the wing and the leading edge regions exhibit the largest perturbations. The perturbations in the trailing edgeareas are significantly smaller due to the sharp trailing edge configuration. Moreover, it is seen that the perturbationmagnitude decays quickly as the distance to the wing increases.

The third and fourth-order DG discretizations are employed to capture the three-dimensional turbulent flow fea-tures, while the modified SA turbulence model discussed in Section II is applied with a discretization order that isconsistent with the main flow equations. The convective flux is solved by the HLLC Riemann flux function and the

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(a) Computational mesh (b) Close-up view around the leading edge

(c) Magnitude of mesh perturbations at h = 0.5 (d) Contours of Mach number near the wing

Figure 10. Solution of a turbulent ONERA M6 wing at M¥ = 0.3, a = 3.08◦ and Re = 11,720,000 per MAC. (a) Computational meshcontaining 255654 unstructured tetrahedral elements. (b) Close-up view of the mesh around the leading edge. (c) Contours of the meshperturbation magnitude at half the spanwise extent. (d) Contours of Mach number solution around the ONERA M6 wing using a fourth-order DG discretization.

Figure 11. Contours of pressure solution on the wing surface and turbulence working variable at certain spanwise locations using a fourth-order DG discretization for the turbulent ONERA M6 wing test case.

implicit problem is solved using the ILU(0) preconditioned GMRES algorithm associated with a local time-steppingmethod. Steady state solution is reached with the L2 norm of the density residual dropped to 10−14 and that of theturbulent working variable residual dropped to 10−11. Fig. 10(d) illustrates contours of the Mach number solution nearthe ONERA M6 wing surface, obtained using the fourth-order DG discretization. As seen, a very smooth solution isobtained on the current mesh and the turbulent boundary layer is quite thin especially in the front portion of the wingdue to the high Reynolds number. In addition, the trailing edge wake appears to dissipate as the element size increasesquickly in the downstream.

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X

Cp

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-1.5

-1

-0.5

0

0.5

1

CFL3DDG p2

=0.2 =0.44 =0.65 =0.8 =0.9 =0.96

(a) p = 2

X

Cp

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-1.5

-1

-0.5

0

0.5

1

CFL3DDG p3

=0.2 =0.44 =0.65 =0.8 =0.9 =0.96

(b) p = 3

Figure 12. Distribution of pressure coefficients on the turbulent ONERA M6 wing at certain spanwise locations using third-order andfourth-order accurate DG discretizations. Lines represent the CFL3D solution and symbols represent the DG solutions. Different colorsindicate different locations.

X

Cp

0.1 0.2 0.3 0.4 0.5 0.6 0.7

-2

-1.5

-1

-0.5

0

0.5

1

DG p3 (Q1 mesh)DG p3 (Q2 mesh)

Figure 13. Comparison of pressure distribution at h = 0.2 using linear and quadratic meshes and the fourth-order DG scheme for theONERA M6 wing test case.

Contours of the pressure solution on the wing surface as well as the computed turbulence working variable atcertain spanwise locations (i.e. constant z-planes) are depicted in Fig. 11 for the case of the fourth-order accurateDG scheme. It is observed that at each constant z-plane the solution contours for the turbulence working variableare similar to a two-dimensional case.10 The maximum turbulence working variable is distributed in the region of thetrailing edge wake. It should be noted that as the tetrahedral elements downstream of the ONERA M6 wing increasein size, the turbulent working variable exhibits large oscillations, which leads to negative values. In such cases, theproduction, destruction and dissipative terms of the modified SA model become inactive to maintain the stability ofthe scheme.

The computed surface pressure coefficients obtained using the third and fourth-order DG discretizations are dis-played in Fig. 12 for various locations (h = 0.2, 0.44, 0.65, 0.8, 0.9 and 0.96) along the spanwise direction. TheCFL3D solution obtained from a substantially fine grid is shown to serve as the benchmark solution in the compar-isons. It is observed that for both third and fourth-order DG schemes very good agreement is achieved as comparedto the CFL3D benchmark solution. Slight over-predictions appear near the leading edge on the upper wing surface,but they are very minor. In addition, the pressures on the lower side of the wing as well as those at the trailing edgeare well predicted and the overall profiles are captured very accurately in both schemes. Figure 13 further provides a

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comparison of the pressure distribution at h = 0.2 using the fourth-order DG scheme computed on a linear (Q1) meshand the quadratic (Q2) mesh from the previous study. Note that the linear mesh corresponds to the one before the meshmovement process, and therefore, it contains the same number of elements as the quadratic mesh but having the origi-nal piecewise linear representations for the wing surface as well as linear interior elements. Although in both cases theDG schemes have converged to the same level, large differences in the surface pressures can be observed between thesolutions obtained on the linear and quadratic meshes. Because of the non-smooth surface geometry, the DG solutionon the linear mesh shows significant over-predictions near the leading edge region on both the upper and lower wingsurfaces, and furthermore, discontinuities are apparent at the elemental interfaces all along in the direction of airflow.On the contrary, the distribution of the pressures is very smooth in the case of the quadratic surface representations.

VI. Conclusions

This paper presents a high-order discontinuous Galerkin discretization method for solutions of three-dimensionalturbulent flows. The modified Spalart and Allmaras turbulence model is exclusively considered for this application,in which a consistent high-order DG discretization to the turbulence model equation performs very well regardingaccuracy and robustness. In the context of high-order methods, for which the surface geometry often comprisessignificantly fewer elements than the case of low-order methods, the need of a high-order (greater than linear) surfacerepresentation becomes apparent for ensuring the overall solution accuracy. However, the projection of the surfacemesh can lead to issues of collapsed interior elements, particularly as highly distorted elements are required in theboundary layer for flows under high Reynolds number conditions. To overcome this problem, a linear elasticitymethod is utilized to determine the necessary perturbations at the interior nodes and additional quadrature points, anda successful procedure results in a high-order curved finite element mesh.

Studies have been conducted for selecting certain key parameters in the mesh generations for different orders ofthe DG methods. It has been shown that, for attached turbulent flow, the conventional settings of y+ of unit 1 and thestretching factor of 1.15 can result in excessive elements in the viscous boundary layer, and furthermore, this situationmay aggravate the difficulties in the mesh movement process. To this end, a viscous mesh with y+ of 5 or 10 anda stretching factor of 1.4 is seen to be capable of delivering accurate solutions for a DG discretization that is higherthan second-order accuracy. Several other three-dimensional numerical examples such as the Taylor-Green vortex andthe turbulent ONERA M6 wing are presented to demonstrate the capabilities and accuracy of higher-order methodsin three-dimensional turbulent flows. Future work will concentrate on the robustness of the mesh movement strategyand the development of high-order DG methods on hybrid meshes for turbulent flow in the transonic and supersonicregimes.

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. The authors would also liketo thank Dr. Christopher Rumsey for providing with the CFL3D solutions for the ONERA M6 wing test case.

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.

4L. 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.

5J. T. Erwin, W. K. Anderson, S. Kapadia, L. Wang, Three dimensional stabilized finite elements for compressible Navier-Stokes, AIAAPaper 2011-3411 (June 2011).

6Z. Wang, Adaptive High-Order Methods in Computational Fluid Dynamics, World Scientific Publishing Co. Pte. Ltd., 5 Toh Tuck Link,Singapore 596224, 2011.

7D. Moro, N. C. Nguyen, J. Peraire, Navier-Stokes solution using hybridizable discontinuous Galerkin methods, AIAA Paper 2011-3407(2011).

8L. Wang, W. K. Anderson, Shape sensitivity analysis for the compressible Navier-Stokes equations via discontinuous Galerkin methods,Computers & Fluids 69 (2012) 93–107.

16 of 17


Dow

nloa

ded

by L

i Wan

g on

Jan

uary

10,

201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

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856

9H. Luo, H. Segawa, M. R. Visbal, An implicit discontinuous Galerkin method for the unsteady compressible Navier-Stokes equations,Computers and Fluids 53 (2012) 133–144.

10L. Wang, W. K. Anderson, T. Erwin, S. Kapadia, Solutions of high-order methods for three-dimensional compressible viscous flows, AIAAPaper 2012-2836 (Jun 2012).

11N. K. Burgess, D. J. Mavriplis, Robust computation of turbulent flows using a discontinuous Galerkin method, AIAA Paper 2012-0457 (Jan2012).

12T. A. Oliver, A high-order, adaptive, discontinuous Galerkin finite element method for the Reynolds Averaged Navier-Stokes equations, PhDDissertation, Massachusetts Institute of Technology, Boston (2008).

13R. Hartmann, P. Houston, An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equa-tions, J. Comput. Phys. 227 (22) (2008) 9670–9685.

14N. K. Burgess, C. R. Nastase, D. J. Mavriplis, Efficient solution techniques for discontinuous Galerkin discretizations of the Navier-Stokesequations on hybrid anisotropic meshes, AIAA Paper 2010-1448 (Jan 2010).

15P. Spalart, S. Allmaras, A one-equation turbulence model for aerodynamic flows, Le Recherche Aerospatiale 1 (1994) 5–21.16C. 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).17Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput.

7 (3) (1996) 856–869.18F. Bassi, S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys. 138 (1997)

251–285.19J. 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).20O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, 30 Corporate

Drive, Suite 400, Burlington, MA, USA, 2005.21R. Haimes, G. Follen, Computational Analysis PRogramming Interface, proceedings of the 6th international conference on numerical grid

generation in computaitonal field simulation (1998).22W. 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, aIAA-2012-968 (Jan 2012).23L. Wang, D. J. Mavriplis, W. K. Anderson, Adjoint sensitivity formulation for discontinuous Galerkin discretizations in unsteady inviscid

flow problems, AIAA Journal 48 (12) (2010) 2867–2883.24P. Solin, K. Segeth, I. Dolezel, High-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman and Hall, 2003.25P. 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) 1553–1570.26K. 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) 7917–7940.27B. 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).28C. R. Nastase, D. J. Mavriplis, High-order discontinuous Galerkin methods using an hp-multigrid approach, J. Comput. Phys. 213 (1) (2006)

330–357.29L. Wang, W. K. Anderson, Sensitivity analysis for the compressible Navier-Stokes equations using a discontinuous Galerkin method, AIAA

Paper 2011-3408 (Jun 2011).30Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, 2003.31W. Gropp, E. Lusk, A. Skjellum, Using MPI: Portable Parallel Programming with the Message Passing Interface, MIT Press, Cambridge,

MA, 1994.32G. Karypis, Metis, University of Minnesota, Department of Computer Science, http://www-users.cs.umn.edu/ karypis/metis (2003).33E. J. Nielsen, W. K. Anderson, Recent improvements in aerodynamic design optimization on unstructured meshes, AIAA J. 40 (6) (2002)

1155–1163.34Z. Yang, D. J. Mavriplis, A mesh deformation strategy optimized by the adjoint method on unstructured meshes, AIAA Paper 2007-557 (Jan

2007).35D. J. Mavriplis, An advancing front delaunay triangulation algorithm designed for robustness, J. Comput. Phys. 117 (1) (1995) 90–101.36Langley research center, turbulence modeling resource, http://turbmodels.larc.nasa.gov/naca0012 val.html (2011).37J. Chapelier, M. D. L. L. Plata, F. Renac, E. Martin, Final abstract for ONERA Taylor-Green DG participation, abstract at 1st International

Workshop on High-Order CFD Methods, Nashville, TN (Jan 2012).

17 of 17


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