Simulation of the Compressible Taylor Green Vortex using ...aero-comlab.stanford.edu/Papers/AIAA-Paper_2014-3210.pdfthe turbulent spectrum. One of the optimized FR schemes (the OFR

Simulation of the Compressible Taylor Green Vortex

using High-Order Flux Reconstruction Schemes

J. R. Bull ∗ and A. Jameson

Stanford University, Stanford, CA 94305, USA

In this paper, we investigate the ability of high-order Flux Reconstruction (FR) nu-merical schemes to perform accurate and stable computations of compressible turbulentflows on coarse meshes. Two new FR schemes, which are optimized for wave dissipationand dispersion properties, are compared to the nodal Discontinuous Galerkin and Spec-tral Difference methods recovered via the Energy-Stable FR method. The compressibleTaylor-Green vortex benchmark problem at Re = 1600 is used as a simple a priori test of thenumerics. Dissipation rates computed from kinetic energy, vorticity and pressure dilata-tion are plotted against reference solutions. Results show that at low mesh resolution theFR schemes are highly accurate and stable across a range of orders of accuracy, althoughoscillations can appear in the solution at orders of six and above. While the FR methodhas a built-in stabilization mechanism, an additional means of damping these instabilitiesis required. The schemes vary in the amount of numerical dissipation and resolution ofthe turbulent spectrum. One of the optimized FR schemes (the OFR scheme) is shown tohave greater spectral accuracy than any of the others tested, motivating its future usagefor high-order, high-fidelity CFD.

Nomenclature

Ω domainu solution vectorf flux vectoru, v, w velocity componentsp pressureξ local coordinatep polynomial orderN number of solution points per direction per elementgL, gR left and right correction functionsli ith Lagrange polynomialLp degree p Legendre polynomialc free parameter in Flux Reconstruction methodJ Jacobiank wavenumberEk kinetic energyζ enstrophyω vorticityS rate of strain tensorε1 directly computed dissipation rateε2 vorticity-based dissipation rateε3 strain-based dissipation rateε4 bulk viscosity-based dissipation rateε5 dilatation-based dissipation rate

∗Postdoctoral Scholar, Dept. of Aeronautics and Astronautics, Stanford University, AIAA Student Member.

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American Institute of Aeronautics and Astronautics

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7th AIAA Theoretical Fluid Mechanics Conference

16-20 June 2014, Atlanta, GA

AIAA 2014-3210

Copyright © 2014 by Jonathan Bull. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA Aviation

I. Introduction

The well-established CFD techniques of second-order numerical methods and Reynolds-Averaged NavierStokes (RANS) turbulence models are capable of predicting steady attached flows at cruise conditions, butthey are incapable of predicting conditions at the fringes of the flight envelope which are often characterizedby turbulent separated flows. Many other aerodynamic problems of central importance also feature complexturbulent flows, including combustion, acoustic noise prediction and the design of hypersonic vehicles. Inorder to improve the performance, efficiency and safety of future generations of aircraft, CFD must movebeyond the current plateau of second-order RANS methods and establish a new norm of high-order accurate,high-fidelity simulation.

High-order accurate (p > 2) numerical methods offer significantly better wave and vortex propagationproperties than second-order accurate (p = 2) schemes, thanks in large part to their lower numerical dissi-pation. The development of high-order accurate finite difference (FD) schemes brought about new levels ofaccuracy in aeroacoustic problems,1 however the extension to unstructured meshes remains a major road-block to their use for flows over and through complex geometry. Over the past two decades, DiscontinuousGalerkin (DG) methods have proved to be highly successful for high-order accurate simulations in complexgeometry owing to their formulation on simplex elements. Recently the DG methodology, which was firstconceived as a finite-element style integral formulation, has been recast in differential form as the SpectralDifference (SD)2,3 and the Flux Reconstruction (FR) methods4 using high-order polynomial basis functionsto discretize the solution within an element. Flux Reconstruction is a general high-order framework which,in the case of linear fluxes, recovers the collocation based nodal DG method and the SD method as well asallowing for the definition of new schemes.4 The FR framework is conceptually simple, easy to implementon unstructured meshes and cheap to compute owing to the lack of integration and mass matrix inversionprocedures.

A family of Energy-Stable Flux Reconstruction (ESFR) schemes have been developed by the AerospaceComputing Lab (ACL) at Stanford University.5 The ESFR schemes were proven to be stable in an energynorm of Sobolev type for the 1D linear advection equation for all orders of accuracy on an arbitrary mesh.5

Hence the ESFR schemes can be formulated on quadrilateral and hexahedral elements by taking tensorproducts of 1D operators. Subsequently the stability proof was extended to the linear advection-diffusionequation in 1D,6 on triangles7 and on tetrahedra.8 The energy norm contains a non-negative coefficient cwhich allows for the recovery of the nodal DG (with c = 0) and SD schemes, as well as the g2 scheme ofHuynh and an infinite variety of new stable schemes.5

For nonlinear fluxes, high-order methods are well known to be susceptible to aliasing instabilities causedby inexact representation of the true flux in a finite-dimensional polynomial subspace.9 The aliasing errorassociated with the ESFR schemes arises from the use a collocation projection of the flux at the solutionpoints.10 It was shown for the ESFR schemes in 1D that the error is minimized by choosing the solutionpoints to be the Gaussian quadrature points.10 Recently, enhanced nonlinear stability has also been achievedin simplex elements by devising new quadrature schemes.11 However, the error does not disappear unlessthe flow is well-resolved, i.e. all the energetic modes of the flux are exactly represented in the polynomialapproximation. Considering the simulation of high Reynolds number turbulent flows, it is impractical toresolve all the scales of motion and therefore some additional control over aliasing errors is sought.

Many techniques have been proposed for controlling nonlinear instabilities in high order numerical meth-ods. Perhaps the simplest method, commonly utilized with second-order schemes, is upwinding of theinterface fluxes to add numerical dissipation, for example using Roe’s method.12 Upwinding was shown toimprove the stability of the ESFR schemes (compared to central flux) in 1D by Vincent, Castonguay andJameson.5 It was shown for the ESFR schemes by Jameson and Lodato13 that the amount of dissipationadded by the interface flux is proportional to a high-order derivative of the solution, thus providing anefficient damping of the energy in the high-order modes. A stabilization technique developed specifically forhigh-order methods is to apply a low-pass filter to the polynomial basis in order to reduce or remove destabi-lizing high wavenumber components.14–16 An equivalent approach is to include a high-order derivative term,for example the spectral vanishing viscosity (SVV) method.17,18 Other techniques include over-integration,also known as polynomial de-aliasing, but this significantly increases computational cost.16,19 The coeffi-cient c in the ESFR schemes offers another means of control over the numerical properties. Setting c > 0was shown to have a stabilizing effect compared to the baseline case of c = 0 by Vincent, Castonguay andJameson.5 However, accuracy was reduced at the same time, implying that the choice of c is a compromisebetween stability and accuracy. It was shown by Allaneau and Jameson20 that a nonzero value of c is iden-

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tical to applying a low-pass filter to the residual in the case of a linear flux. With all of these techniques, itis vital to maintain the advantage of using high order methods over second order methods by only addingenough dissipation to stabilize the scheme – not so much that the propagation of turbulence and other wavephenomena is adversely affected.

Investigation of the spectral properties of FR schemes provides an insight into their ability to resolvemultiscale phenomena such as turbulence. Vincent, Castonguay and Jameson21 carried out a von Neumannanalysis of the ESFR schemes, finding that the SD method recovered via ESFR had the lowest dispersionerror (they also identified a value of c which maximized the allowable CFL condition). Recently, Asthanaand Jameson22 conducted a full modal analysis of the FR method in 1D to obtain dissipation and dispersionrelations for each mode. They solved an optimization problem to identify a value of c in the ESFR schemesthat minimized errors associated with wave dissipation and dispersion; the optimized scheme is henceforthreferred to as the OESFR scheme. They went further to carry out a multidimensional constrained opti-mization of the general FR method, identifying a scheme outside the ESFR family that is optimal in termsof wave propagation, henceforth referred to as the Optimized FR (OFR) scheme. We hypothesize that, byvirtue of their superior resolution of the energy-containing modes, these optimized schemes will be moreaccurate that other FR schemes. It is expected that the benefits of the optimized schemes will be carried tohigher dimensions by taking tensor products.

In this paper, we analyze the behavior of the Flux Reconstruction method in under-resolved compressibleturbulent flow. The compressible Taylor-Green Vortex (TGV) benchmark problem at Re = 1600 is an idealtest case due to the deterministic nature of the flow, yet it contains many of the features of real turbulent flowsincluding vortex stretching and interaction as well as dilatation (compressibility) effects. It has been usedby many authors for high-order method validation including Beck and Gassner,16 Diosady and Murman,19

Chapelier et al.,23 Don, Gottlieb and Shu,24 Johnsen, Varadan and van Leer25 and Carton de Wiart et al.26

The TGV was identified as a challenging problem for high-order methods in the 1st International Workshopon High-Order CFD Methods.27 Bull and Jameson28 simulated the TGV problem with the SD schemerecovered via ESFR, matching high-resolution reference data on relatively coarse hexahedral and tetrahedralgrids. Results using more schemes and polynomial orders are presented in this paper and new details of theability of the FR schemes to represent compressible turbulent flows emerge. The simulations are carried outusing HiFiLES, the ACL’s unstructured GPU-accelerated Flux Reconstruction solver. Details of the codeand its verification and validation can be found in Lopez et al.29

It is shown that the FR method accurately predicts the mean turbulent energy cascade and the importantflow structures on relatively coarse grids, thanks to the high order basis functions and to low dissipative anddispersive errors. The stabilization provided by the FR method sufficiently damps aliasing instabilitiesat polynomial orders of five or less, with the amount of damping depending on the particular scheme.At higher than fifth order, all schemes display instabilities at low mesh resolution (sometimes leading toresidual divergence) which may require further stabilization, for example in the form of a filter. The OESFRscheme developed by Asthana and Jameson22 displays nearly identical behavior to the DG scheme recoveredby ESFR. The OFR scheme22 is as stable as, but more accurate than, ESFR recovering the SD and DGschemes. Energy spectra show that the OFR scheme provides superior resolution of the energy in the higherwavenumbers, confirming that the analysis of Asthana and Jameson22 is applicable to the Navier-Stokesequations in three dimensions.

These results lend support to the further use of high-order FR schemes – and in particular the newlydeveloped OFR scheme – for large eddy simulation (LES) of high Reynolds number turbulent flows. Theirturbulence-resolving abilities and low numerical dissipation make them ideal for applications involving far-field propagation of vortices and waves, including aircraft noise prediction and boundary layer ingestion.Future work will include the development of dealiasing filters to improve stability at higher orders andinvestigation of the suitability of the OFR scheme for more complex high Reynolds number turbulent flows.

II. High-Order Flux Reconstruction

A. General Formulation

The compressible Navier-Stokes equations are discretized using the high-order Flux Reconstruction scheme.We write the equations in conservative form in a 3D domain Ω with spatial coordinates x = x1, x2, x3 and

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time t:∂u∂t

+∂f∂x

= 0 , (1)

where u = (ρ ρu ρv ρw ρe)T are the conservative variables and f is the flux. The domain is split intonon-overlapping elements Ωj . For simplicity, we consider the one-dimensional case. Inside the element adegree p polynomial, defined on a set of N = p+1 solution points, is used to represent the solution, resultingin an Nth-order accurate scheme. The Gauss-Legendre quadrature points (shown in blue in Figure 1) arechosen as the solution points as they were found to minimize aliasing errors for nonlinear problems.10 Ap+ 1th-order polynomial is used to represent the flux. The Gauss-Lobatto points are used as the flux points(shown in red in Figure 1).

1 2 3 4 5

1 2 3 4

Solution Points Flux Points

Figure 1. Location of solution and flux points in 1D for p=3.

The piecewise-continuous pth-order solution polynomial u is defined as

u(x) =N∑i=1

uili(x) , (2)

where ui(x) are the nodal solution values and li(x) is a set of basis functions, in this case the Lagrange polyno-mials. A similar expression is used to obtain the (p+1)th-order flux polynomial f . The flux is discontinuousacross element boundaries and the common interface fluxes are found using a two-point upwind-biased fluxformula such as Roe’s method.12 The next step is to construct a globally continuous flux polynomial. Inthe FR method this is achieved by adding a flux correction polynomial ∆f to the discontinuous flux f . Thecorrection satisfies: (a) f + ∆f equals the common interface fluxes, and (b) the corrected flux optimallyrepresents the discontinuous flux in the element interior. ∆f is given by

∆f(x) = [f∗L − f(−1)]gL(x) + [f∗R − f(1)]gR(x) , (3)

where f∗L, f∗R are the common interface fluxes at left and right interfaces and gL(x), gR(x) are order p + 2

polynomial correction functions satisfying gL(−1) = gR(1) = 1, gL(1) = gR(−1) = 0, gL(x) = gR(−x). Thecorrected, globally C0-continuous flux fC is given by fC = f + ∆f . An isoparametric mapping from thephysical domain x ∈ Ωj to the reference domain ξ ∈ [−1, 1] is introduced:

ξ|Ωj (x) = 2x− xj

xj+1 − xj− 1. (4)

Now, denoting uδj as the discrete solution in element Ωj and fδj as the discrete flux, the update step is writtenin semi-discrete form as

∂uδj∂t

= −J−1j

[Djfδj + (f∗(xj)− fδj (xj))gL,ξ + (f∗(xj + 1)− fδj (xj + 1))gR,ξ

], (5)

where Jj is the Jacobian in element Ωj , gL,ξ and gR,ξ are the derivatives of the correction functions at thesolution points and Dj = ∂lj

∂ξ is the discrete derivative operator. The time derivative is discretized by anexplicit fourth order Runge-Kutta scheme, thus avoiding the need to construct and invert large matrices.

B. Energy-Stable Flux Reconstruction Schemes

We consider the 1D conservation equation:

∂u

∂t+

∂

∂x

(f

(u,∂u

∂x

))= 0 , (6)

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where in general f is a nonlinear function. The second-order PDE is written as a system of first-order PDEsby introducing an auxiliary variable q:

∂u

∂t+

∂

∂x(f(u, q)) = 0 , (7a)

q − ∂u

∂x= 0. (7b)

Now, the linear advection-diffusion equation is given by Eq. (7) with

f = au− b∂u∂x

, (8)

where a and b are constant scalars. It was proven by Vincent, Castonguay and Jameson30 that the FRschemes are energy-stable for the linear advection-diffusion equation Eq. (8) using an ‘energy method’, aswas used to prove stability of the linear advection problem.5,31 The schemes are energy-stable if the followinginequality is satisfied:

12d

dt‖UD‖2p,c + b‖QD‖2p,κ ≤ 0, (9)

where ‖UD‖p,c and ‖QD‖p,κ are broken Sobolev norms of the solution u and the auxiliary variable q whichare defined as follows:

‖UD‖p,c =

N∑n=1

∫ xn+1

xn

[(uDn)2

+c

2(Jn)2p

(duDndx

)2]dx

1/2

, (10a)

‖QD‖p,κ =

N∑n=1

∫ xn+1

xn

[(qDn)2

+κ

2(Jn)2p

(dqDndx

)2]dx

1/2

. (10b)

Here, the constants c and κ parameterize the schemes. For c ≥ 0 and κ ≥ 0, ‖UD‖p,c and ‖QD‖p,κ are norms,and the schemes are guaranteed to be stable in accordance with equation (9). The proof of stability is quitegeneral, as it ensures boundedness of the solution for all orders of accuracy, independent of the locationof solution points within the 1D element. It can then be shown that to satisfy the stability condition, thecorrection functions gL and gR are given by

gL =(−1)p

2

[Lp −

(ηp(c)Lp−1 + Lp+1

1 + ηp(c)

)], (11a)

gR =12

[Lp −

(ηp(c)Lp−1 + Lp+1

1 + ηp(c)

)], (11b)

where Lp is the degree p Legendre polynomial, ηp(c) = (c(2p+ 1)(app!))2)/2 and 0 ≤ c ≤ ∞ is the stabilityparameter in equation (10). If c = cDG = 0, then ηp = 0, implying

gL =(−1)p

2(Lp − Lp+1) , (12a)

gR =12

(Lp + Lp+1) , (12b)

which are the left and right Radau polynomials respectively, hence c = 0 recovers a particular nodal DGscheme as shown by Huynh.4 The recovered scheme uses a collocation projection of the flux onto a polynomialspace of degree k, which has significant implications for the nonlinear stability. The spectral difference (SD)scheme can be recovered (for a linear flux) if the flux correction ∆f is zero at a set of p points in the interiorof the standard element.5 The only way to satisfy this requirement is if c = cSD where cSD is given by

cSD =2p

(2p+ 1)(p+ 1)(app!)2. (13)

A third scheme, identified by Huynh as being particularly stable, is referred to as the g2 scheme4 and isrecovered by choosing c = cg2 where cg2 is given by

cg2 =2(p+ 1)

p(2p+ 1)(app!)2. (14)

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In the general case of a nonlinear flux, it is well documented that high-order schemes suffer from aliasing-driven instabilities resulting from the projection of the flux onto a polynomial space of finite (p) dimensions.9

It was shown by Allaneau and Jameson20 that setting c > 0 in the ESFR schemes corresponds to dampingof the highest-order polynomial mode by the application of a filter to the residual. In fact, filtering is acommonly used stabilization technique with DG schemes.9,15,32 Therefore, the FR formulation implicitlyincludes a stabilization mechanism. Furthermore, Jameson, Vincent and Castonguay10 showed that thealiasing error associated with the ESFR schemes could be minimized in 1D by choosing the solution pointsto be the Gaussian quadrature points. Williams, Castonguay, Vincent and Jameson11 devised new quadratureschemes in order to enhance nonlinear stability in simplex elements (triangles and tetrahedra).

C. Spectral Properties

Vincent, Castonguay and Jameson21 performed a von Neumann analysis of the ESFR formulation, identifyingdissipation and dispersion relations and calculating the order of accuracy as a function of c. Accuracy inthis sense is correlated with the fraction of resolvable wavenumbers for which waves are propagated withnegligible dissipation and dispersion. Their analysis is summarized here. Consider the 1D linear advectionequation, Eq. (6) with f = au, where a is the advection speed, in coordinates x′, t′. Let the grid be ofuniform resolution h and non-dimensionalize the equation with x = x′/h and t = t′a/h, so that we have

∂u

∂t+∂u

∂x= 0. (15)

Since we know the direction of information propagation we can fully upwind the flux, i.e. f∗(xj) = uδ(xj−1).Then the update step Eq. (5) is rewritten as

∂uδj∂t

= −J−1j

[C0uδj + C−1uδj−1

], (16a)

C0 = D− gL,ξlTL, (16b)C−1 = gL,ξlTR, (16c)

where gL,ξ is the gradient of the left correction function and lL and lR are vectors containing the values ofthe Lagrange polynomials on the left and right interfaces. An initial condition u(x, 0) = eikx is specifiedwhich admits a solution of the form u(x, t) = eik(x−t), where k is the wavenumber. The solution can beexpressed in the parent domain using the mapping in Eq. (4):

u(x ∈ Ωj , t) = eik(j−t)eik(ξ+1)

2 , (17)

This infinite-dimensional exact solution must be projected to the finite-dimensional polynomial space toobtain the numerical solution:

uδj(t) = eik(j−aδ(k)t)v, (18)

where aδ(k) is the numerical wavespeed as a function of wavenumber and v is the projection vector. Byintroducing this numerical solution into the update step Eq. (5) we arrive at the semi-discrete dispersionrelation

Mv = aδv, (19a)

M =−2ik

(C0 + e−ikC−1), (19b)

which is a p+ 1-dimensional eigenvalue problem for each wavenumber k. The solution of Eq. (19) providesp+ 1 numerical modes for each k with the complex eigenvalues

aδp(k) = aδpr (k) + iaδpi(k), p = 1, 2, . . . , p+ 1, (20)

where aδpr and aδpi are the real and imaginary numerical wavespeeds respectively. The analytical solution hasthe exact dispersion relation ar = 1, ai = 0, therefore the errors associated with numerical dispersion anddissipation are eik(1−aδpr t) and eik(1−aδpi t) respectively. The usual interpretation of the existence of multiplenumerical modes for each wavenumber is that one mode is ‘physical’ in the sense that it most closely follows

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the analytical mode, while the remaining p of the p + 1 admissible modes are ‘spurious’. These spuriousmodes are often neglected by assuming that they contain only a small fraction of the energy and are fairlydissipative.21 The reader is referred to Asthana and Jameson22 for a comprehensive discussion of the spuriousmodes.

Figure 2 plots the real and imaginary components of the physical mode against the normalized wavenum-ber k/(p+ 1) for the DG scheme recovered via ESFR for polynomial orders from one to five (p1 to p5). Thereal part is plotted as the effective wavenumber keff = aδrk/(p + 1) and the imaginary part aδpi is plotteddirectly. The components of exact wavespeed, ar and ai, are plotted for reference. As p is increased, theexact solution is approximately followed over a larger proportion of the range of resolvable wavenumbers.However, the overshoot becomes more pronounced, implying a lower CFL limit. Dissipation is reduced athigher p, which translates as better resolving efficiency. Figure 3 plots the real and imaginary components

22 K. Asthana, A. Jameson

k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3 P = 1

P = 2

P = 3

P = 4

P = 5

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-2

-1.5

-1

-0.5

0

P = 1

P = 2

P = 3

P = 4

P = 5

Exact

(b)

Fig. 3 Effect of polynomial order: (a) Effective wavenumber ke f f = k adr /(P+1) and (b) Imaginary part ad

iof the numerical wavespeed for the physical mode for DG scheme via FR for P = 1 to 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)

22 K. Asthana, A. Jameson

k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3 P = 1

P = 2

P = 3

P = 4

P = 5

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-2

-1.5

-1

-0.5

0

P = 1

P = 2

P = 3

P = 4

P = 5

Exact

(b)

Fig. 3 Effect of polynomial order: (a) Effective wavenumber ke f f = k adr /(P+1) and (b) Imaginary part ad

iof the numerical wavespeed for the physical mode for DG scheme via FR for P = 1 to 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(b)

Figure 2. Effect of polynomial order on (a) effective wavenumber keff and (b) imaginary part aδi for thephysical mode of the DG scheme via ESFR for p1 to p5 (from Asthana and Jameson22)

of the physical mode for DG, SD and Huynh’s g2 scheme4 (all via ESFR) at order p3. The effect of changingc from c = cDG = 0 to c = cSD is that the numerical wavespeed remains closer to the exact wavespeed forlonger, but the dissipation starts increasing (i.e. aδpi < 0) at a lower normalized wavenumber, suggestingthat an optimal scheme might exist in between DG and SD. The g2 scheme is inferior to SD in terms of botherrors.

FR schemes with minimal dispersion and dissipation 23

k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3

DG via FRSD via ESFRHuynh’s g

2

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-1

-0.5

0


2

Exact

(b)

Fig. 4 Effect of correction function: (a) Effective wavenumber ke f f = k adr /(P + 1) and (b) Imaginary part

adi of the numerical wavespeed for the physical mode for DG, SD and Huynh’s g2 scheme (via FR) for P = 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)


k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3


2

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-1

-0.5

0


2

Exact

(b)

Fig. 4 Effect of correction function: (a) Effective wavenumber ke f f = k adr /(P + 1) and (b) Imaginary part

adi of the numerical wavespeed for the physical mode for DG, SD and Huynh’s g2 scheme (via FR) for P = 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(b)

Figure 3. (a) effective wavenumber keff and (b) imaginary part aδi for the physical mode of the DG, SD andg2 scheme (all via ESFR) at order p3 (from Asthana and Jameson22)

D. Optimized FR Schemes

Recently, Asthana and Jameson22 carried out an optimization of the ESFR schemes in spectral space usingc as the free parameter. They identified an optimal value of c at each order p for which the dissipationand dispersion errors were minimized over the range of resolvable wavenumbers, denoting the scheme asthe OESFR (Optimal ESFR) scheme. Optimizing with respect to both errors balanced the competingeffects described above, finding a minimum close to c = cDG = 0. They then tackled the more complex

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multidimensional optmization problem of finding the zeros of the correction functions which minimized thedissipation and dispersion errors. A general form of the left correction function was considered:

gL(ξ) = Πpn=1

ξ − ξn1 + ξn

ξ − 12

, (21)

which ensured a unity value on the left interface; the right correction function gR is simply a mirror of gL.The p-dimensional solution space of free zeros ξ1, ξ2, . . . , ξp+1 contains the family of ESFR schemes as asubspace. The so-called Optimal FR (OFR) schemes could then be identified subject to the constraint thatthey are linearly stable. For p=1 the OESFR scheme was recovered owing to the single degree of freedom, butfor p > 1 the schemes were outside the ESFR family. Figure 4 plots the dispersion and dissipation relationsfor DG via ESFR, OESFR and OFR for p4. The OESFR scheme has a slightly lower dispersion error thanDG and an almost identical dissipation error, while for the OFR scheme both errors are significantly lowerthan DG.


k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3

DG via FR

OESFR

OFR

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-1

-0.5

0

DG via FR

OESFR

OFR

Exact

(b)

Fig. 9 Optimal schemes: (a) Effective wavenumber ke f f = k adr /(P + 1) and (b) Imaginary part ad

i of thenumerical wavespeed for the physical mode for DG, OESFR and OFR schemes for P = 4. Note that all threeschemes are of the same formal order.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a)


k / (P+1)

(k/(

P+

1))

ar

0 0.5 1 1.5 2 2.5 3

0

1

2

3

DG via FR

OESFR

OFR

Exact

(a)

k / (P+1)

ai

0 0.5 1 1.5 2 2.5 3

-1

-0.5

0

DG via FR

OESFR

OFR

Exact

(b)

Fig. 9 Optimal schemes: (a) Effective wavenumber ke f f = k adr /(P + 1) and (b) Imaginary part ad

i of thenumerical wavespeed for the physical mode for DG, OESFR and OFR schemes for P = 4. Note that all threeschemes are of the same formal order.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(b)

Figure 4. (a) effective wavenumber keff and (b) dissipation error ai for the physical mode of the DG schemevia ESFR, OESFR and OFR for p4. From Asthana and Jameson22

E. FR Schemes for Turbulent Flow Simulations

The ESFR schemes have been used successfully in a number of challenging turbulent flows, including transi-tional flow over an SD7003 airfoil at ReC = 60, 000,33 transitional flow over a pitching and plunging NACA0012 wing section34 and unsteady flow over a flapping wing-fuselage configuration.35 Equipped with anLES model, they have also been used to accurately simulate the turbulent flow over a square cylinder atRe = 21, 400 on relatively coarse hexahedral36 and tetrahedral28 meshes. Nevertheless, there remain openquestions about the schemes’ behavior in under-resolved turbulent flows. It is not known which are the mostaccurate schemes in terms of faithfully capturing as much of the turbulent spectrum as possible when thegrid resolution is larger than the Kolmogorov lengthscale. Unfortunately, it is likely that the most accuratescheme will not be the most stable, and so the question needs to be rephrased as ‘which scheme has the bestbalance of accuracy and stability?’ The results presented in the previous section encourage the use of thenewly developed OFR scheme for turbulent flow simulations, where the improvements in spectral accuracyover DG and SD should be visible, particularly on coarse meshes where the full spectrum can not be fullyresolved. It is also important to examine the role of polynomial order in simulations of turbulent flows. Inorder to take full advantage of high-order schemes over second order schemes including computational effi-ciency, we wish to use a high polynomial order. The above spectral results imply that higher orders providebetter spectral resolution and lower numerical dissipation. Yet as the order is increased, it was found thataliasing instabilities grew large in simulations of the Taylor Green Vortex.28 Furthermore, the reduced CFLlimits at higher orders might reduce the computational efficiency by enforcing longer simulation times.

III. The Compressible Taylor-Green Vortex

The Taylor-Green Vortex (TGV) problem is a canonical flow which provides a convenient stepping stonetowards simulating real flows, requiring the solution of the Navier Stokes equations in 3D at moderate

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Reynolds numbers. From a simple initial datum, vortex stretching mechanisms cause the flow to decay alonga well-defined trajectory, generating a detailed turbulent spectrum over a period of 20 seconds, as shown inFigure 12. The TGV was one of the problems in the 1st and 2nd International Workshops on High-OrderCFD Methods27) and is considered to be a challenging test for high-order methods. We employ it to testthe ability of the various FR schemes described above to accurately represent the turbulent spectrum.

A. Problem Setup

The geometry is a triply-periodic box of dimension 0 ≤ (x, y, z) ≤ 2π and the initial condition is given bythe following:

u(t0) = u0 sin(x/L) cos(y/L) cos(z/L), (22)v(t0) = −u0 cos(x/L) sin(y/L) cos(z/L), (23)w(t0) = 0, (24)

p(t0) = p0 +ρ0

16

[cos(

2xL

)+ cos

(2yL

)][cos(

2zL

)+ 2], (25)

where L = 1, u0 = 1, ρ0 = 1 and p0 = 100. The Mach number is set to 0.08 (consistent with the initialpressure p0) and the initial temperature is 300K. A Reynolds number of 1600 is prescribed by adjustingthe viscosity. Three meshes are used: a coarse mesh of 163 hexahedral elements, a medium mesh (323

elements) and a fine mesh (643 elements). We compare our results to several others submitted to thehigh-order workshop. A high resolution reference solution was computed by Debonis37 using Bogey andBailley’s 13-point dispersion-relation-preserving (DRP) scheme38 on a mesh of 5123 elements. Debonis usedthe same scheme on meshes of 643, 1283 and 2563 elements and we also compare our results to these. Theclosest published results in terms of using a similar method are those of Beck and Gassner, who used afiltered fourth-order accurate DG method on a 643 element mesh.16 These are also plotted in some figuresfor comparison. Energy spectra are compared to a pseudo-spectral computation on a 5123 element meshcomputed by Carton de Wiart et al.26

B. Diagnostics

Several diagnostic quantities can be computed from the flow as it evolves in time, allowing the characteristicsof the numerical scheme to be observed. Firstly, the volume-averaged kinetic energy is given by

Ek =1ρ0V

∫V

12ρu.udV, (26)

where V is the volume. We can directly compute the rate of dissipation from the kinetic energy:

ε1 = ε(Ek) = −dEkdt

. (27)

In incompressible flow, it can be shown that the dissipation rate is related to the integrated enstrophy ζ bya constant:24

ε2 = ε(ζ) =2µρ0ζ, (28)

ζ =1ρ0V

∫V

12ρω · ωdV, (29)

where ω is the vorticity and µ is the dynamic viscosity. We refer to ε2 as the vorticity-based dissipation. Incompressible flow, the dissipation rate is given by the sum of three components, ε3, ε4 and ε5, given by

ε3 = ε(S) = 2µ

ρ0V

∫V

S : SdV, (30)

ε4 = ε(µv) =µvρ0V

∫V

(∇.u)2dV, (31)

ε5 = ε(p) = − 1ρ0V

∫V

p∇.udV, (32)

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where µv = 0 is the bulk viscosity (thus ε4 is neglected) and S is the rate-of-strain tensor.24 The pressuredilatation-based dissipation rate, ε5, drops out in the incompressible limit and is expected to be small atMach 0.08. Therefore, the strain-based dissipation rate, ε3 is expected to be almost identical to ε2.

C. Results

1. Effect of Polynomial Order

Figure 5 shows the volume-averaged kinetic energy Ek for the SD scheme recovered via ESFR on the coarseand medium meshes at orders p1−p4, equivalent to second to fifth order accuracy. Orders p1−p4 use N = 2−5solution points or degrees of freedom (DoF) per direction per element respectively. The reference solution,computed by Debonis using the DRP scheme on a 5123 element mesh,37 is plotted as open circles. On bothmeshes, Ek computed by SD at p1 and p2 decays too fast, indicating an excess of numerical dissipation. Asthe order increases, Ek gets closer to the reference solution, matching it exactly on the medium mesh at p5.

0 5 10 15 20Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

Norm

aliz

ed tu

rbul

ent k

inet

ic e

nerg

y

SD-16x2SD-16x3SD-16x4SD-16x5DRP-512

(a) coarse mesh

0 5 10 15 20Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

Norm

aliz

ed tu

rbul

ent k

inet

ic e

nerg

y


(b) medium mesh

Figure 5. Volume-averaged kinetic energy Ek on (a) coarse mesh and (b) medium mesh using SD at p1 to p4(=2–5 DoF per direction per element). ‘NxM ’ refers to N3 mesh, p = M − 1 order scheme; ‘DRP-512’ = DRPscheme on 5123 mesh.37

Figure 6 shows the directly computed dissipation rate ε1 for the SD scheme at orders p3 to p6 (a) onthe coarse mesh and (b) on the medium mesh. At p3 and p4 on the coarse mesh, ε1 grows too fast. Atp3 on the medium mesh the growth is approximately correct but the peak rate at around t = 9 is under-predicted, while at p4 the peak is quite well predicted. This would suggest that the important flow dynamicsare being resolved correctly and that they evolve and decay at the correct rate. At p5 on both meshes,oscillations appear near the peak, which are thought to originate from large variations present in the highorder polynomials within each element. On the coarse mesh these lead to divergence of the solution at t ≈ 9,but interestingly, at p6 the solution is more stable.

Figure 7(a) shows ε1 using SD at p3 on coarse, medium and fine meshes compared against the referenceDRP solution37 and against the fourth order accurate filtered DG on a 643 element mesh by Beck andGassner.16 The SD scheme on the fine mesh almost exactly matches both the DG and DRP solutions.Figure 7(b) shows ε1 using SD with a constant 1283 degrees of freedom on different meshes: p7 on the coarsemesh, p3 on the medium mesh and p1 on the fine mesh. Here we can observe the competing effects of stabilityand accuracy: the low order solution on the fine mesh is stable but inaccurate due to excessive dissipation,while the high order solution on the coarse mesh is unstable, diverging at t ≈ 9. From Figures 6(a) and7(b) it is notable that p5 and p7 are unstable and p6 is more stable, suggesting that the behavior of oddand even orders may differ. More research is needed to ascertain the reasons for the oscillations and theirpossible dependence on order. The order p3 solution on the medium mesh has a good balance of accuracyand stability. This result suggests that very high orders are not a good choice for under-resolved turbulentflow simulations unless the instabilities can be controlled. It also highlights the advantage of using high orderversus second order accurate methods, namely, greater accuracy thanks to lower numerical dissipation.

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0 5 10 15 20Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Dis

sipa

tion

Rate


(a) coarse mesh

0 5 10 15 20Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Dis

sipa

tion

Rate


(b) medium mesh

Figure 6. Directly computed dissipation rate ε1 using SD on (a) coarse mesh at p1 − p4, and (b) medium meshat p1, p3, p4 and p5. ‘DRP-512’ = DRP scheme on 5123 mesh37

0 5 10 15 20Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Dis

sipa

tion

Rate

SD-16x4SD-32x4SD-64x4Beck DG-64x4DRP-512

(a) p3

0 5 10 15 20Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Dis

sipa

tion

Rate

SD-16x8SD-32x4SD-64x2DRP-512

(b) constant DoF

Figure 7. ε1 using SD (a) at p3 on coarse, medium and fine meshes, and (b) with constant 1283 degrees offreedom (p7 on the coarse mesh, p3 on the medium mesh and p1 on the fine mesh). ‘DRP-512’ = DRP schemeon 5123 mesh37

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2. Effect of FR Scheme

Figure 8 shows a close-up of the peak of ε1 computed using the DG, SD and OFR schemes at orders p3 andp5 on the medium mesh. The OESFR scheme obtained identical results to DG and is therefore not shown.At p3 all the schemes behave in a similar manner, under-predicting the peak. At p5 all the schemes capturethe peak very well but with some small oscillations. As discussed above, these are thought to be caused byrepresenting the flow structures with high order polynomials, which naturally contain large variations. Ina real turbulent flow, one expects these variations to disappear when considering average quantities over asufficiently large volume. However, the TGV flow has several symmetries (see Figure 12) so that variationsoccurring simultaneously in different parts of the flow may add up.

Figure 9 shows the vorticity-based dissipation rate ε2 computed using the DG, SD and OFR schemes atorders p3 to p6. This quantity is a measure of how well the vorticity-carrying small scales (i.e. the inertialrange of turbulence in a real flow) are resolved. Numerical dissipation reduces the sharpness with whichvelocity gradients are approximated;24 its effect is visible as a reduction in peak dissipation rate comparedto the reference solution, as was observed by Debonis37 and Beck and Gassner.16 The reference DRPsolution on a 5123 element mesh and a coarser DRP solution on a 1283 mesh from Debonis37 are plottedfor comparison. Ar p3 all the schemes miss the peak; at higher orders they all capture the peak dissipationrate quite accurately due to reduced numerical dissipation. All the FR schemes do better than the DRPscheme on the 1283 mesh despite being computed on a coarser mesh. OFR tends to slightly over-predict ε2at p4 and p5 while the other schemes, particularly DG, tend to under-predict it. At p6 OFR lies exactly onthe reference DRP solution and DG and SD just below it. The slight over-prediction of ε2 (i.e. vorticitystrength) by the OFR scheme may indicate that energy contained in the small resolved scales, which containmost of the vorticity, is over-predicted. Additionally, in the case of a nonlinear flux it is known that thealiasing error causes energy to pile up in the scales near the grid scale due to inadequate draining by thenumerical scheme or by molecular viscosity when the grid is coarse.39 In Section C the OFR scheme wasshown to have reduced numerical dissipation at high wavenumbers than DG or SD, implying a smaller drainof the energy. The divergent case in Figure 7(b) (SD at p7 on the coarse mesh) is possibly an example ofthe excess energy destabilizing the simulation. Additional stabilization of the FR schemes at low resolutionand high order may be necessary, for example by filtering.14

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130

Dis

sipa

tion

Rate

DG-32x4SD-32x4OFR-32x4DRP-512

(a) ε1, p3

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130

Dis

sipa

tion

Rate

DG-32x6SD-32x6OFR-32x6DRP-512

(b) ε1, p5

Figure 8. ε1 using DG, SD and OFR at (a) p3 and (b) p5 on medium mesh.

3. Compressibility Effects

The dissipation rate due to pressure dilatation, ε5, measures the effect of compressibility on the dissipationof turbulent energy. Figure 3 plots ε5 for the SD scheme at p3 on the coarse, medium and fine meshesversus the DRP scheme on a 643 mesh. As the Mach number is so low, the effects of compressibility shouldnot be very strong and ε5 is expected to be close to zero (as is the case with the DRP scheme). The SDscheme greatly over-predicts ε5, although it clearly converges to the correct solution as the mesh is refined,as observed by Chapelier et al.23 with a DG method.

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7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130D

issi

patio

n Ra

teDG-32x4SD-32x4OFR-32x4DRP-128DRP-512

(a) p3

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130

Dis

sipa

tion

Rate

DG-32x5SD-32x5OFR-32x5DRP-128DRP-512

(b) p4

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130

Dis

sipa

tion

Rate


(c) p5

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

0.0125

0.0130

Dis

sipa

tion

Rate


(d) p6

Figure 9. ε2 using DG, SD and OFR at (a) p3, (b) p4, (c) p5 and (d) p6 on medium mesh. ‘DRP-128’ and‘DRP-512’ = DRP scheme on 1283 and 5123 element meshes37

4 6 8 10 12 14 16Time (s)

0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Dis

sipa

tion

Rate

SD-16x4SD-32x4SD-64x4DRP-64

Figure 10. Pressure dilatation-based dissipation rate ε5 using SD at p3 on coarse, medium and fine meshes.‘DRP-64’ = DRP scheme on a 643 element mesh37

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4. Energy Spectra

Figure 11 displays energy spectra computed from the DG, SD and OFR results at p5 on the medium meshcompared to the reference solution computed using a pseudo-spectral method on a 5123 mesh by Carton deWiart et al.26 It is clear that the OFR scheme achieves much better spectral resolution at higher wavenumbersthan either DG or SD. All three schemes – and in particular the OFR scheme – have a pronounced bump atthe highest resolved wavenumbers, indicating the pile-up of energy at the grid scale. Also visible in 11(a) isa slight under-prediction of energy by all three schemes in the lowest wavenumbers.

100 101 102 10310−10

10−8

10−6

10−4

10−2

100

k

E(k)

DG−32x6SD−32x6OFR−32x6spectral

(a) DG/SD/OFR, medium mesh

102

10−8

10−6

10−4

k

E(k)

DG−32x6SD−32x6OFR−32x6spectral

(b) Subsection of (a)

Figure 11. Energy spectra of the DG (red), SD (blue) and OFR (green) schemes at t = 9 seconds at p5 on themedium mesh compared to the reference spectral solution26 (black)

5. Flow Visualization

Figure 12 shows the Q criterion colored by velocity magnitude at four times, computed from the DG solutionat p3 on the fine mesh. The evolution flow structure is evident, starting from large vortices and decayinginto finer and more complex structures as the peak dissipation rate at t = 9 is passed. Figure 13 shows aportion of Figure 12(c) compared to results from Beck and Gassner40 using a collocation-based nodal DGmethod at p3 on a 643 element mesh. The structures are almost identical.

IV. Conclusion

In this paper the performance of the Energy-Stable Flux Reconstruction (ESFR) schemes and the op-timized OESFR/OFR schemes in simulations of under-resolved turbulent flow was investigated. The com-pressible Taylor-Green Vortex problem at Re = 1600 was simulated on a range of coarse meshes and at arange of polynomial orders using DG and SD recovered via ESFR as well as Optimized ESFR (OESFR) andOptimized FR (OFR). At low polynomial orders (p1, p2), numerical dissipation impacted on the accuracyof predictions of the turbulent kinetic energy and directly computed dissipation rate. At moderate orders(p3, p4), there was a good balance between accuracy and stability. The third order (p3) SD scheme on thefinest mesh matched the reference solution. At orders p5 and above, instabilities began to appear in thesolution which are thought to originate from variations in the high order polynomials within each elementcoupled with accumulation of energy in the small scales. Further work is needed to find ways (e.g. filtering)of controlling the instabilities on coarse meshes.

Differences between the schemes showed up in their predictions of the vorticity-based dissipation rate ε2,which measures how accurately the small scales are resolved. All the FR schemes tested were able to predictthis quantity more accurately than Bogey and Bailley’s dispersion relation-preserving scheme on a much finermesh.37 The fact that ε2 is so well predicted by the FR schemes at low resolution is strongly encouraging

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(a) t = 2.5, Q = 0.5 (b) t = 5, Q = 1.5

(c) t = 8, Q = 1.5 (d) t = 10.75, Q = 1.5

Figure 12. TGV solution on the fine mesh using 3rd order DG method, showing isosurfaces of Q criterioncolored by velocity magnitude at time t = 2.5 to 10.75 seconds.

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(a) HiFiLES

Time

Dis

sip

ati

on

Ra

te -

dk

/dt

0 2 4 6 8 10 12 14 16 18 200

0.002

0.004

0.006

0.008

0.01

0.012

0.014

DNS128x264x432x816x16

Time

Dis

sip

ati

on

Ra

te -

dk

/dt

6 8 10 12

0.01

0.011

0.012

0.013 DNS128x264x432x816x16

2 = 1.5 N = 1 N = 3 N = 152563

2563

t = 8s N = 1

N = 3

N = 15

643

h

Re = 1600

2

(b) Beck and Gassner

Figure 13. Upper left quadrant of Figure 12(c) compared to collocation-based nodal DG at p3 on a 643 meshby Beck and Gassner.40

for their use in simulations of high Reynolds number turbulent flows. The DG and OESFR schemes, whichbehaved an an identical manner in all tests, were the least accurate. The SD scheme was slightly moreaccurate and the OFR scheme was the most accurate, achieving good agreement with the reference solutionon the medium mesh at p6. The dissipation due compressibility effects, ε5, was over-predicted by all theschemes at low resolution and/or low polynomial order, but the prediction was significantly improved onthe finest mesh. Energy spectra (plotted in Section 4) showed that the OFR scheme is in closer agreementwith the reference spectral solution of Carton de Wiart et al.26 at high wavenumbers than DG or SD,demonstrating that the results of Asthana and Jameson22 are applicable to the Navier-Stokes equations inthree dimensions.

These results show that the FR method is an ideal basis for simulating turbulent flows on coarse meshes,thanks to a good balance of stability and accuracy and the ability to tune the spectral characteristics via theparameter c or by optimizing the correction functions. Furthermore, the ESFR schemes can be solved onunstructured tetrahedral meshes, enabling high-order LES of flows over complex geometry. More researchis needed to increase stability at high orders by draining energy from the highest resolved wavenumbers.Future work will employ the FR schemes for LES of complex high Reynolds number flows.

Acknowledgments

This research was made possible by the support of the NSF under grant number 1114816, monitored byDr Leland Jameson, and the Air Force Office of Scientific Research under grant number FA9550-10-1-0418,monitored by Dr Fariba Fahroo.

References

1Lele, S., “Compact Finite Difference Schemes with Spectral-Like Resolution,” Journal of Computational Physics, Vol. 103,1992, pp. 16–42.

2Kopriva, D. A. and Kolias, J. H., “A Conservative Staggered-Grid Chebyshev Multidomain Method for CompressibleFlows,” Journal of Computational Physics, Vol. 125, 1996, pp. 244–261.

3Liu, Y., Vinokur, M., and Wang, Z. J., “Spectral difference method for unstructured grids I: basic formulation,” Journalof Computational Physics, Vol. 216, 2006, pp. 780–801.

4Huynh, H., “A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods,” AIAAP., 2007, 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, Jun 25–28.

5Vincent, P. E., Castonguay, P., and Jameson, A., “A new class of high-order energy stable flux reconstruction schemes,”

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Journal of Scientific Computing, Vol. 47, No. 1, 2011, pp. 50–72.6Castonguay, P., Williams, D. M., Vincent, P. E., , and Jameson, A., “Energy Stable Flux Reconstruction for Advection-

Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, 2012, Submitted.7Williams, D. M., Castonguay, P., Vincent, P. E., and Jameson, A., “Energy Stable Flux Reconstruction for Advection-

Diffusion Problems on Triangles,” Journal of Computational Physics, 2012, Submitted.8Williams, D. M. and Jameson, A., “Energy Stable Flux Reconstruction for Advection-Diffusion Problems on Tetrahedral

Elements,” Journal of Scientific Computing, 2013, Accepted.9Hesthaven, J. and Warburton, T., Nodal Discontinuous Galerkin methods: Algorithms, Analysis, and Applications,

Springer Verlag, 2007.10Jameson, A., Vincent, P. E., and Castonguay, P., “On the Non-Linear Stability of Flux Reconstruction Schemes,” Journal

of Scientific Computing, 2011.11Williams, D. and Jameson, A., “Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow

Problems on Tetrahedral Meshes,” AIAA P., 2013, 43rd AIAA Fluid Dynamics Conference.12Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors and Difference Schemes,” Journal of Computational

Physics, Vol. 43, 1981, pp. 357–372.13Jameson, A. and Lodato, G., “A note on the numerical dissipation from high-order discontinuous finite element schemes,”

Preprint submitted to Computers and Fluids.14Bogey, C. and Bailly, C., “Large eddy simulations of round free jets using explicit filtering with/without dynamic

Smagorinsky model,” International Journal of Heat and Fluid Flow , Vol. 27, 2006, pp. 603–610.15Blackburn, H. and Schmidt, S., “Spectral element filtering techniques for large eddy simulation with dynamic estimation,”

Journal of Computational Physics, Vol. 186, 2003, pp. 610–629.16Beck, A. and Gassner, G., “On the accuracy of high-order discretizations for underresolved turbulence simulations,”

Theoretical and Computational Fluid Dynamics, Vol. 27, No. 3-4, 2012, pp. 221–237.17Karamanos, G. and Karniadakis, G., “A spectral vanishing viscosity method for large-eddy simulations,” J. Com-

put. Phys., Vol. 163, No. 1, 2000, pp. 22–50.18Pasquetti, R., “Spectral vanishing viscosity method for LES: sensitivity to the SVV control parameters,” J. Turbul.,

Vol. 6, No. 12, 2005, pp. 1–14.19Diosady, L. and Murman, S., “Design of a Variational Multiscale Method for Turbulent Compressible Flows,” AIAA

paper 2013-2870 , 2013.20Allaneau, Y. and Jameson, A., “Connections between the Filtered Discontinous Galerkin Method and the Flux Recon-

struction approach to High Order Discretizations,” Computer Methods in Applied Mechanics and Engineering, 2011, pp. 3628–3636.

21Vincent, P. E., Castonguay, P., and Jameson, A., “Insights from von Neumann analysis of high-order flux reconstructionschemes,” Journal of Computational Physics, Vol. 230, No. 22, 2011, pp. 8134–8154.

22Asthana, K. and Jameson, A., “High-order Flux Reconstruction schemes with Minimal Dispersion and Dissipation,”Accepted.

23Chapelier, J.-B., Plata, M., and Renac, F., “Inviscid and viscous simulations of the Taylor-Green vortex flow using amodal Discontinuous Galerkin approach,” AIAA paper 2012-3073 , 2012.

24Don, W., Gottlieb, D., and Shu, C., “Numerical Convergence Study of Nearly- Incompressible, Inviscid Taylor-GreenVortex Flow,” J. Sci. Comput., Vol. 24, No. 1, 2005, pp. 1–27.

25Johnsen, E., Varadan, S., and van Leer, B., “A Three-Dimensional Recovery-Based Discontinuous Galerkin Method forTurbulence Simulations,” AIAA Paper 2013-0515 , 2013, 51st AIAA Aerospace Sciences Meeting, Grapevine, Texas.

26Carton de Wiart, C., Hillewaert, K., Duponcheel, M., and Winckelmans, G. S., “Assessment of a discontinuous Galerkinmethod for the simulation of vortical flows at high Reynolds number,” Int. J. Numer. Meth. Fl., Vol. 74, 2014, pp. 469–493.

27Wang, Z., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K.,Huynh, H., Kroll, N., May, G., Persson, P.-O., van Leer, B., and Visbal, M., “High-Order CFD Methods: Current Statusand Perspective High-Order CFD Methods: Current Status and Perspective,” International Journal for Numerical Methods inFluids, Vol. 72, No. 8, 2013, pp. 811–845.

28Bull, J. and Jameson, A., “High-Order Flux Reconstruction Schemes for LES on Tetrahedral Meshes,” Notes on NumericalFluid Mechanics and Multidisciplinary Design, edited by W. Haasse, Springer, May 2014, final manuscript submitted.

29Lopez, M., Sheshadri, A., Bull, J., Economon, T., Romero, J., Watkins, J., Williams, D., Palacios, F., Jameson, A.,and Manosalvas, D., “Verification and Validation of HiFiLES: a High-Order Navier-Stokes unstructured solver on multi-GPUplatforms,” AIAA P., 2014, Accepted to 44th AIAA Fluid Dynamics Conference, Atlanta, Georgia, June 16-20, 2014.

30Castonguay, P., Williams, D., Vincent, P. E., and Jameson, A., “Energy stable flux reconstruction schemes for advection–diffusion problems,” Comput. Methods Appl. Mech. Engrg., Vol. 267, 2013, pp. 400–417.

31Jameson, A., “A proof of the stability of the spectral difference method for all orders of accuracy,” Journal of ScientificComputing, Vol. 45, No. 1, 2010, pp. 348–358.

32Fischer, P. and Mullen, J., “Filter-based Stabilization of Spectral Element Methods,” Comptes Rendus de l’Academiedes Sciences-Series I-Mathematics, Vol. 332, No. 3, 2001, pp. 265–270.

33Castonguay, P., Liang, C., and Jameson, A., “Simulation of Transitional Flow over Airfoils using the Spectral DifferenceMethod,” AIAA P., Vol. 2010-4626, 2010.

34Ou, K. and Jameson, A., “Towards Computational Flapping Wing Aerodynamics of Realistic Configurations usingSpectral Difference Method,” AIAA Paper 2011-3068 , 2011, 20th AIAA Computational Fluid Dynamics Conference, Honolulu,HI, June 27-30, 2011.

35Ou, K., Castonguay, P., and Jameson, A., “3D Flapping Wing Simulation with High Order Spectral Difference Methodon Deformable Mesh,” AIAA P., 2011, 49th AIAA Aerospace Sciences Meeting, Orlando, FL, Jan 4-7.

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36Lodato, G., Castonguay, P., and Jameson, A., “Discrete filter operators for large-eddy simulation using high-order spectraldifference methods,” Int. J. Numer. Meth. Fl., Vol. 72, No. 2, 2013, pp. 231–258.

37Debonis, J., “Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods,”AIAA Paper 2013-0382 , 2013.

38Bogey, C. and Bailly, C., “A family of low dispersive and low dissipative explicit schemes for flow and noise computations,”J Comp Phys, Vol. 194, 2004, pp. 194–214.

39Colonius, T. and Lele, S. K., “Computational aeroacoustics: progress on nonlinear problems of sound generation,”Progress in Aerospace Sciences, Vol. 40, No. 6, 2004, pp. 345–416.

40Beck, A. and Gassner, G., “Numerical Simulation of the Taylor-Green Vortex at Re=1600 with the DiscontinuousGalerkin Spectral Element Method for well-resolved and underresolved scenarios,” AIAA P., 2012, 50th AIAA AerospaceSciences Meeting, Nashville, TN, 2012.

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Documents

Simulation of the Compressible Taylor Green Vortex using ...aero-comlab.stanford.edu/Papers/AIAA-Paper_2014-3210.pdfthe turbulent spectrum. One of the optimized FR schemes (the OFR