adaptive high-order methods in computational fluid dynamics

Implicit time integration is applied to the fully coupled RANS and k-ω equations, both for steady and unsteady computations.. In this context, the purpose of this chapter is to describe

Vol 2 Adaptive High-Order Methods in Computational Fluid Dynamics

edited by Z J Wang (Iowa State University, USA)

Forthcoming

Vol 1 Computational Methods for Two-Phase Flows

by Peter D M Spelt (Imperial College London, UK), Stephen J Shaw (X'ian Jiaotong – University of Liverpool, Suzhou, China) & Hang Ding (University of California, Santa Barbara, USA)

N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

Computational Fluid Dynamics

Editor

Z J Wang

Iowa State University, USA

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4313-18-6

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All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ADAPTIVE HIGH-ORDER METHODS IN COMPUTATIONAL FLUID DYNAMICS Advances in Computational Fluid Dynamics — Vol 2

To My Family

vii

experts on adaptive high-order methods in computational fluid dynamics (CFD) It covers several widely used, and still intensively researched methods, including the discontinuous Galerkin (DG), residual distribution, differential quadrature, k-exact finite volume, spectral volume/spectral difference, PNPM, and correction procedure via reconstruction methods The reasons for including such a wide coverage

of methods are to: (1) provide a single source of reference, (2) present a snapshot of the state-of-the-art, and (3) facilitate the observation of similarities and differences as well as pros and cons of these methods

In the present context, adaptive high-order methods refer to numerical methods that are capable of handling unstructured adaptive meshes with accuracy higher than second-order These methods are compact, scalable, capable of handling both complex physics and geometry, and suitable for modern parallel supercomputers and graphics processing units (GPUs) They are widely considered the next major breakthrough in CFD, and have already found applications in computational aeroacoustics, computational electromagnetics, vortex dominated flows, and large eddy simulation and direct numerical simulation of turbulent flows

A concerted effort was made to minimize overlaps among the chapters For example, the first 7 chapters describe different aspects of the DG methods, while the last 8 chapters are devoted to other high-order methods Main topics covered include innovative formulations, analyses, efficient solution and time marching algorithms, parallel implementation, turbulence modeling, discontinuity-capturing techniques, error estimates, hp-adaptations, and dynamic mesh techniques, etc

The book requires a graduate student level of understanding It should serve as an excellent source of information for CFD developers, educators, researchers, users, and students who are interested in the state- of-the-art and the remaining challenges in adaptive high-order methods

without their hard work Finally, I’d like to thank Ying Zhou for producing the color cover graphic, and Varun Vikas for help with Latex

Z.J Wang Ames, Iowa June 30, 2011

ix

Francesco Bassi, Lorenzo Botti, Alessandro Colombo, Antonio Ghidoni And Stefano Rebay

Higher-Order Finite-Element Discretizations in CFD

Laslo T Diosady and David L Darmofal

for Discontinuous Galerkin Methods

Tobias Leicht and Ralf Hartmann

with Time Accurate Local Time Stepping

Gregor J Gassner, Florian Hindenlang and Claus-Dieter Munz

for CFD

Jaime Peraire and Per-Olof Persson

Discontinuous Galerkin Methods

Jianxian Qiu

Schemes for Diffusion Revisited

Bram van Leer, Marcus Lo, Rita Gitik and Shohei Nomura

Chapter 9: High-Order Finite-Volume Discretization of the 235

Euler Equations on Unstructured Meshes

Carl Ollivier-Gooch and Chris Michalak

Schemes for Hyperbolic Problems

Rémi Abgrall

(RBF-DQ) Method and Its Applications

Chang Shu

Discretizations

Chris Lacor and Kris Van den Abeele

Discretization of Steady Problems

Georg May and Antony Jameson

CHAPTER 1 DISCONTINUOUS GALERKIN FOR TURBULENT

FLOWS

Dipartimento di Ingegneria Industriale,Universit`a degli studi di Bergamo,Viale Marconi 5, 24044 Dalmine (BG), Italy

∗francesco.bassi@unibg.it

†lorenzo.botti@unibg.it

Dipartimento di Ingegneria Meccanica e Industriale,

Universit`a degli Studi di Brescia,Via Branze 38, 25123 Brescia, Italy

¶stefano.rebay@ing.unibs.it

The purpose of this chapter is to present all the relevant features of a

high-order DG method developed over the years for the numerical

solu-tion of the RANS and k-ω equasolu-tions The method has been implemented

using orthogonal and hierarchical modal shape functions defined in the

real space The code can handle hybrid grids consisting of tetrahedra,

prisms, pyramids and hexahedra Implicit time integration is applied to

the fully coupled RANS and k-ω equations, both for steady and unsteady

computations A directional shock-capturing term, proportional to the

inviscid residual, is employed to control oscillations around shocks Most

of the numerical results presented in this chapter have been computed

within the EU-funded ADIGMA project to investigate the capability of

the method for aeronautical applications

1 Introduction

In recent years several high-order methods have been emerging as

practi-cal tools to go beyond the second-order accuracy of standard finite volume

1

discretizations of PDEs on general unstructured grids For aerospace

ap-plications this is of particular importance to further increase the impact of

Computational Fluid Dynamics (CFD) on the aerodynamic design of new

generation aircraft The Discontinuous Galerkin (DG) method, in

particu-lar, has been gaining popularity as one of the most promising approaches

to the accurate and robust numerical solution of ever more complex

physi-cal models and has attracted great efforts of many research groups into its

development

In this context, the purpose of this chapter is to describe several

devel-opments of the DG method implemented over the years in a fully parallel

DG code, named MIGALE, that we have used for the numerical solution

of the Euler, Navier-Stokes and the coupled RANS and k-ω turbulence

model equations These developments include: i) a proposal for adapting

the smooth-wall treatment of the variable ω to the degree of the polynomial

approximation, ii) the adoption of orthonormal and hierarchical modal

ba-sis functions defined in the real space for arbitrary shape elements, iii) a

shock-capturing technique based on the inviscid residual and applied in the

direction of the pressure gradient, iv) an implicit time integration technique

suited both for steady and unsteady problems

The capabilities of the present version of the code will be demonstrated

by computing several fairly complex problems taken from the suites of test

cases proposed within the EU-funded project ADIGMA In the conclusions

we will give a brief account of other recent implementations which are

already quite mature and will outline future directions of development

This section describes relevant implementation aspects of the DG

discretiza-tion applied to the coupled set of RANS and k-ω equadiscretiza-tions, including

approach and implicit time integration

∂t(ρe0) +

∂

∂xj(ρujh0) = ∂

∂xj[uiτbij− qj] − τij

∂ui

∂xj+ β∗ρkee ω r, (3)

∂xj

+ τij

where the pressure, the turbulent and total stress tensors, the heat flux

vector and the eddy viscosity are given by

τij= 2µt



Sij−13

and turbulent Prandtl numbers and

Sij =12

is the mean strain-rate tensor The closure parameters α, α∗, β, β∗, σ, σ∗

Notice that the RANS and k-ω equations here employed are not in

in Eqs (3), (4), (5) Motivations for this choice have been discussed in

fulfills suitably defined “realizability” conditions, which set a lower bound

on eω in such equations This limitation substantially improves the stability

and robustness of turbulent flow computations because there is numerical

evidence that too small, though positive, values of ω = eω e can lead to

sudden breakdown of computations

Realizability conditions, which guarantee that the turbulence model

pre-dicts positive normal turbulent stresses and satisfies the Schwarz inequality

for shear turbulent stresses, lead to the following inequalities

eω e

α∗ − 3



Sii−13

realizable turbulent stresses is given by

eω e r0

The solution of Eq (14) is trivial for the high-Reynolds number k-ω

2.1.1 Surface boundary condition for eω

2

where y is the local coordinate normal to the wall Eq (18) admits the

following near wall solutions

e

ω = log

6νβ

βeω e w



where eωwis the value of eω at the wall (y = 0) Of course, these solutions are

nothing but the logarithm of the viscous sublayer solutions for ω reported

appropriate solution for smooth walls, whereas non-singular solutions are

effects of surface roughness through surface boundary conditions

It has been shown in Ref 1 that singular and non-singular solutions for

smooth wall solution is recovered when the surface roughness tends to zero

In the rough-wall method surface values of ω can be simply set by means

of the correlation

ωw= Sr

u2 τ

the grid density or the degree of polynomial approximation should be high

enough to provide accurate solutions

On the other hand, the implementation of the smooth wall boundary

condition for ω requires special care in the numerical treatment of the

Relying on the rough-wall method, the approach recommended by

Wilcox is simply to skip the issue of the numerical treatment of the

sin-gularity by replacing the perfectly smooth surface with an hydraulically

smooth surface In this so-called “slightly-rough-wall” boundary condition,

r < 5, i.e., it should ensure that the surface is hydraulically smooth with

roughness peaks lying within the viscous sublayer

The approach proposed by Menter consists in setting at the wall a finite

ωw= 106νw

by the factor 10, or, put another way, the analytical solution computed at

10

to find the equivalent slightly-rough-wall roughness implied by Menter’s

opti-mize the factor 10 of Menter’s formula by means of an accurate near-wall

numerical study of the ω solution and by comparing skin friction

distribu-tions of flat plate flows computed on differently refined grids The value of

the factor proposed by Hellsten is 1.25 instead of 10

In the framework of the DG method, an approach like that of Hellsten

was presented in Ref 2 where it was found that a good agreement between

experimental and numerical skin friction distributions of flat plate flows

6/β/50 As the solutions

keeping α not dependent on the polynomial degree of the solution did work

accurately However, as higher degree polynomials can follow closer and

closer the exact near wall distribution of ω, it seems reasonable to make α

dependent on the degree k of the polynomial approximation

A possible approach in this direction has been outlined in Ref 5, where

with the value at the wall of the L2projection of the singular solution onto

the basis of the polynomial approximation

Here we propose an alternative way which consists of setting at the wall

the value eωk

around y = h, truncated to k terms, i.e.,

e

ωk

w= eωh− ∂ e∂yω

h

h1!+

∂2ωe

∂y2

h

h22! − · · ·

= eωh+

kXn=1

1

The finite values eωk

wat the wall can again be related to the exact solution

To actually apply Eq (25), h needs to be specified In the flat plate

computations presented below h has been set equal to the distance from

the wall of the centroid of the elements next to the wall As Eq (25) holds

for hydraulically smooth surfaces, we remark that the slightly-rough-wall

r < 5 If

polynomial solutions computed on relatively coarse grids targeted at

high-degree polynomial approximations

The boundary condition for ω has been tested on the flat plate flow

wall The Figure 1 displays the skin-friction distribution along the plate and

the profiles of u velocity component and of turbulence quantities at x/L =

between DG results and “average” experimental data is an effect produced

by the high-Reynolds number k-ω model here employed, that disappears

Wieghardt law of the wall

300

law of the wall P 3

law of the wall P 4

law of the wall P 5

law of the wall P 6

DG - P 3

DG - P 4

DG - P 5

DG - P 6

using the modified coefficients of the low-Reynolds number version of the

model

The governing equations can be written in compact form as

∂u

source terms, Fc, Fv∈ RM⊗ RN denote the inviscid and viscous flux

func-tions, respectively, and N is the space dimension

The weak form of Eq (27) reads

Z

+ZΩφs(u, ∇u) dx = 0, (28)where φ denotes any arbitrary, sufficiently smooth, test function and

Ω, consisting of a set of non-overlapping hybrid-type elements The

follow-ing space settfollow-ing of discontinuous piecewise polynomial functions for each

component uh i = uh 1, , uh M of the numerical solution uh is assumed:

of global degree at most k on the element K

The discontinuous approximation of the numerical solution requires

in-troducing a special treatment of the inviscid interface flux and of the viscous

flux For the former it is common practice to use suitably defined numerical

flux functions which ensure conservation and account for wave propagation

For the latter we employ the BR2 scheme, presented in Refs 8, 9 and

the-oretically analyzed in Refs 10, 11 (where it is referred to as BRMPS), to

obtain a consistent, stable and accurate discretization of the viscous flux

Accounting for these aspects, the DG formulation of problem (28) then

requires to find uh 1, , uh M ∈ Φh such that

Γ h

[[φh]] · bfu±h, (∇huh+ ηere([[uh]]))±

dσ+

Z

Ω h

φhs(uh, ∇huh+ r([[uh]])) dx = 0, (30)

average trace operators

[[q]]def= q+n++ q−n−, {·}def= (·)++ (·)−

where q denotes a generic scalar quantity and the average operator applies

to scalars and vector quantities By definition, [[q]] is a vector quantity

These definitions can be suitably extended to faces intersecting ∂Ω

ac-counting for the weak imposition of boundary conditions The local lifting

operator re, which is assumed to act on the jumps of uhcomponentwise, is

defined as the solution of the following problem

Z

Ω h

φh· re(v) dx = −

Ze{φh} · v dσ, ∀φh∈ [Φh]N, v ∈L1(e)N

, (32)and the global lifting operator r is related to reby the equation

e∈E h

operators on the two sides of any edge e have support on the two elements

sharing the edge e Hence, the global lifting operator for any element

inde-pendently For the former we usually employ the Godunov flux or,

than the number of faces of the elements The BR2 viscous flux

discretiza-tion is as compact as possible because, for each element K, it only couples

the nearest neighbor elements This feature is obviously very attractive for

the implicit implementation of the method

2.2.1 Orthonormal and hierarchical basis functions

The actual implementation of Eq (30) requires specifying the test and trial

several aspects of the DG discretization, namely, i) numerical efficiency,

ii) conditioning of the DG discrete operators, iii) capability of easily

han-dling complex-shape elements

Modal expansion bases defined in the physical space can be used for

irregular and polyhedral elements in a very straightforward manner

Fur-thermore, it is quite easy to construct hierarchical and orthonormal sets

of shape functions that overcome the ill-conditioning of element mass

ma-trices that becomes evident for high-degree polynomial approximations on

highly stretched and curved elements Complex applications presented in

the following are in fact based on this type of approximation The main

drawback of such modal polynomial approximations is the cost of numerical

integration for elements with non-constant Jacobian mapping

h Defining on K a set {ϕi}, i = 1, , NDOF, of

is the number of degrees of freedom of complete polynomials of degree k

Simple choices for {ϕi}, such as the set of monomials {xlymzn: l+m+n ≤

k}, are not advisable in general and for the sake of improving stability and

efficiency a set of orthogonal polynomial basis functions is highly preferable

The procedure to produce a set of orthonormal basis functions on a generic

element K relies on the modified Gram-Schmidt (MGS) orthogonalization

algorithm The sole requirement of this procedure is the capability to

com-pute the integral of polynomial functions on the desired element shapes

Let us denote with {ϕi} and {bi}, i = 1, , NDOF, the set of orthonormal

basis functions we wish to construct and a starting set of linearly

inde-pendent basis functions defined on K, respectively The MGS procedure

with re-orthogonalization can be simply setup as shown in the following

Line 2 indicates that orthogonalization is applied twice As reported in

Ref 13, this is enough to get a set of basis functions which are orthonormal

up to machine precision It can be shown that the above MGS algorithm

following system of equations

ϕi=

i−1Xj=1

orthogonal to the i − 1 already orthonormalized basis functions, whereas

created ϕi For i = 1, , NDOF these coefficients are given by

aij

aii = − (bi, ϕj)K, j = 1, , i − 1,1

aii

=

sZK



bi−Xi−1j=1(bi, ϕj)K

2dx,

rij = −aij

aii,

rii= 1

aii

hierarchical In fact, increasing the degree of polynomial approximation

changing the already existing orthonormal basis functions

As regards the starting set of basis functions {bi} for a generic element

K, we have found that a simple and effective choice is the set of monomials,

up to the prescribed degree, expressed in a local frame of reference having

its origin in the centroid of K and the coordinate axes coincident with the

principal axes of the element

Finally we remark that the MGS algorithm outlined above is also used

to compute the values of basis functions (and of their spatial derivatives, if

necessary) at any location other than those needed to compute the integrals

of lines 4 and 6 In such cases the symbols b and ϕ at lines 5, 7 and 8

denote values of starting and orthonormalized basis functions (or of their

derivatives) at the desired location

2.3 Time integration

The DG space discretization of Eq (30) results in the following system of

(nonlinear) ODEs in time

where U is the global vector of unknown degrees of freedom, M is a global

block diagonal matrix and R (U) is the vector of “residuals”, i.e., the vector

of nonlinear functions of U resulting from the integrals of the DG discretized

space differential operators in Eq (30) We remark that using the DOFs of

approxima-tion, then the matrix M represents the global block diagonal mass matrix,

which, using orthonormal basis functions, reduces to the identity matrix

as unknowns of the polynomial expansion of the solution, then the block

ZK

2.3.1 Linearly implicit Runge-Kutta schemes

Implicit time integration of Eq (36) can be efficiently performed by means

of linearly implicit Rosenbrock-type Runge-Kutta schemes The class of

methods here considered can be compactly written as

sXj=1

αijKj



 − J

i−1Xj=1

γijKj, i = 1, , s,

(37)

where s is the number of stages, bi, αij, γij are real coefficients and J =

the Euler scheme and for the schemes proposed in Refs 14 and 15 are

summarized in Table 1

Table 1 Coefficients of some linearly implicit Runge-Kutta schemes.

An implementation of Eq (37) that saves at each stage the cost of the

j=1γijKj can be written as follows

sXj=1

aijWj



i−1Xj=1

cijWj.The coefficients of the transformed scheme are given by

(m1, , ms) = (b1, , bs) Γ−1, (aij) = (αij) Γ−1,

C = diag γ−1, , γ−1

− Γ−1,where Γ−1 def= (γij)−1 denotes the inverse of the coefficient matrices of

The matrix-explicit or the matrix-free GMRES algorithm can be used

to actually solve Eq (38) at each time step In both cases system

precondi-tioning is required to make the convergence of the GMRES solver acceptable

in problems of practical interest The Jacobian matrix implemented in our

code has been derived analytically and takes full account of the dependence

of the residual on the unknown vector and on its derivatives, including the

Table 2 Inverse of the (γ ij ) matrices of Table 1.

implicit treatment of the lifting operators and of the boundary conditions

Using a suitably accurate time integration scheme, this allows to employ

the implicit solver also for accurate unsteady computations

The choice of the time step can significantly affect both the efficiency

and the robustness of the method For steady computations we have

im-plemented the pseudo-transient continuation strategy with the local time

step given by

c + d,where

hK

SK,define convective and diffusive velocities and the reference dimension of

robust strategy to increase the CFL number as the residual decreases is not

an easy task, especially for turbulent computations The rule here proposed

is essentially the result of intensive numerical experimentation and aims at

Fig 2 Streamlined body: Mach number contours of P 4

solution and residuals

(

x = min (xL 2, 1) if xL ∞ ≤ 1

number, the maximum CFL number of explicit schemes and the exponent

(usually ≤ 1) governing the growth rate of the CFL number, respectively

been found useful to prevent sudden breakdown of computations once the

CFL number has already reached quite high values For relatively simple

steady test cases, such as the flow around a streamlined body (Figure 2,

232969 DOFs), the implicit time integration combined with the above CFL

number evolution rule provides quadratic Newton convergence to machine

accuracy

The shock-capturing approach consists of adding to the DG discretized

equations an artificial viscosity term that aims at controlling the high-order

modes of the numerical solution within elements while preserving as much as

possible the spatial resolution of discontinuities The shock-capturing term

is local and active in every element, but the amount of artificial viscosity is

proportional to the (inviscid) residual of the DG space discretization and

thus it is almost negligible except than at locations of flow discontinuities

The shock-capturing term added to Eq (30) reads

XK

ZK

p(u±h, uh) (∇hφh· b) (∇huh· b) dx, (40)with the shock sensor and the pressure gradient unit vector defined by

∂p(uh)

∂uh i

si(u±h), dp(uh) =

MXi=1

are actually the lifting of the interface jump in normal direction between

the numerical and internal inviscid flux components The further factor

fp(uh) in Eq (41) is a pressure sensor defined by

fp(uh) =|∇hp(uh)|

p(uh)



hKk



which improves the accuracy of solutions in regions with high but

other-wise smooth gradients and allows using the same value of the user-defined

parameter C (typically C = 0.2) for different degrees of polynomial

1 (∆x) 2+(∆y)1 2 +(∆z)1 2

where ∆x, ∆y and ∆z are the dimensions of the hexahedral enclosing K,

scaled in such a way that their product matches the volume of K The

shock-capturing technique outlined above is highly non-linear and

residu-als convergence of steady state solutions can be quite difficult, even

imple-menting a fully (linearized) implicit discretization of the shock-capturing

term (40) This is in fact the case for the solution of the transonic flow

p T u k ln( ω ) CFL

solution and residuals convergence

80860 DOFs), shown in Figure 3, that requires quite a large number of

Newton iterations for convergence

3 Numerical Results

In this section we present the results of high-order DG solutions of several

complex turbulent flows of aeronautical interest All the computations

freestream conditions and the higher-order solutions from the lower-order

ones Solutions have been advanced in time by means of the linearly implicit

backward Euler method and the linear system (38) has been solved using

the default iterative solver available in PETSc, i.e., the restarted GMRES

algorithm preconditioned with the block Jacobi method with one block per

process, each of which is solved with ILU(0)

The flow around the three elements airfoil has been computed with a farfield

elements with curved, four-node edges, see Figure 4 The main difficulties

of this test are due to highly distorted elements shapes and to the flow

complexity of strongly interacting wakes, see Figure 5 Figure 6 displays

Fig 4 L1T2: pressure and Mach contours of P 6

solution.

iterations and performance index units, which is a relative measure of CPU

for external flows that combines a simple geometry with complexities of

transonic flow, i.e., local supersonic flow, shocks, and turbulent

bound-ary layers separation The flow conditions are those of Test 2308, i.e.,

Performance Index Unit

solution.

namic chord The grid consists of 215632 hexahedral elements with curved,

eight-node faces, shown in Figure 7 superimposed to the pressure contours

with 60 Krylov subspace vectors and 120 maximum iterations These

approximations All the computations have been run in parallel using 512

cores of the CINECA BCX/5120 cluster

In Figure 8 the pressure coefficient distributions of P2solution are

com-pared with the experimental data at seven sections along the span of the

wing The shock-capturing technique proves capable of providing accurate

Fig 8 ONERA M6: pressure coefficient of P 2

solution (◦ 2156320 DOFs) compared with the experimental data (4).

resolution of the lambda shock structure all along the suction surface of the

wing and, unlike many results presented in the literature, the shocks can

still be clearly distinguished at section y/b = 0.8 Despite the quite coarse

sepa-ration near the wing tip, as shown in Figure 9 Table 3 reports the force

coefficients of P0→2solutions, showing that at least a one-degree higher P3

solution would be useful to assess the convergence of force coefficients

Fig 9 ONERA M6: flow separation near the wing tip of P 2

solution.

approx-imation on a grid of 188928 hexahedral elements with curved, eight-node,

faces shown in Figure 11(a) The same Figure 11 displays the pressure

convergence history in terms of Newton iterations and performance index

algorithm with 60 Krylov subspace vectors and 120 maximum iterations

All the computations have been run in parallel using 512 cores of the DLR

p u w k ln( ω ) CFL

Performance Index Units

10

p u w k ln( ω ) CFL

those computed by the TAU and FUN3D codes The FUN3D and TAU

solutions have been computed on a grid with 11459041 nodes and on an

employs 1889280 DOFs

The DLR-F6 wing-body transport configuration has been the object of

several wind-tunnel tests and computational studies, see Ref 19, and also

solution (◦ 1889280 DOFs) compared with TAU (– – – 17053510 DOFs) and FUN3D (– · – 11459041 DOFs).

se-ries with the aim of assessing the state of the art of computational methods

as practical aerodynamic tools for aircraft force and moment prediction In

grids with 50618 and 404944 hexahedral elements with curved, eight-node

The parameters of the restarted GMRES solver have been set to 60 Krylov

subspace vectors and 120 maximum iterations for the coarse grid solutions,

and to 120 vectors and 480 iterations for the fine grid solutions Figure 16

shows the residuals convergence history of the coarse-grid solutions in terms

of Newton iterations and performance index units The coarse and fine grid

computations have been run in parallel using respectively 128 and 512 cores

Table 4 DLR F6: force and pitching moment coefficients of DG solutions.

(a) coarse grid

of the DLR CASE cluster facility Figure 15 highlights the capability of the

P3solution to capture the detail of flow separation at the wing-root junction

coefficients computed on the coarse and fine grids There is a discrepancy

between the more accurate results on the two grids that needs to be further

investigated and no clear conclusion can be drawn about the asymptotic

values of the aerodynamic coefficients One issue could be the poor

geomet-rical approximation of surfaces when using only quadratic mappings for the

faces of very coarse meshes Finally, Figures 17, 18 and 19 give an overview

of the pressure coefficient and skin friction distributions of DG solutions

compared with reference results of the TAU and CFL3D codes taken from

Ref 18

solutions.

Fig 15 DLR F6: wing-root juncture flow separation and turbulence intensity contours

10

p T u w k ln( ω ) CFL

Performance Index Unit

8

p T u w k ln( ω ) CFL

exper-imentally within the second international Vortex Flow Experiment

(VFE-2) The geometry here considered is the delta wing with large-radius

lead-ing edge, for which experimental pressure data are available in Ref 20

614770 tetrahedral and prismatic elements The prisms fill a few layers

of elements within boundary layers on the delta wing and on the sting

The available grid points allowed to define only a linear mapping of

el-ement faces and this resulted in inaccurate pressure distributions on the

wing and sting surfaces All the computations have been run in parallel

using 512 cores of the CINECA HPC cluster facility The convergence of

residuals for this test case was quite difficult and, more importantly,

observed vortex breakdown on the wing suction surface This issue could

be related to the poor representation of the wing and sting surfaces and

needs to be further investigated Figure 20 shows the pressure coefficient

Figure 20 clearly highlight the better resolution of vortices provided by the

solution (◦ 1012360 DOFs) compared with TAU (—— 5102446 DOFs) and CFL3D (– – – 2256896 DOFs, – · – 7689088 DOFs, – ··

– 26224640 DOFs).

solution (◦ 1012360 DOFs) compared with TAU (—— 5102446 DOFs) and CFL3D (– – – 2256896 DOFs, – · – 7689088 DOFs,

– ·· – 26224640 DOFs).

4 Final Remarks

In this chapter we have presented and demonstrated several well-tried

fea-tures of the DG code MIGALE, that has been developed over the years

for the numerical solution of the coupled RANS and k-ω turbulence model

equations

Open issues of the proposed DG method are mainly related to its

com-putational cost and this has motivated our most recent research efforts in

two directions

On the one hand, we have developed a spectral DG method, with a

cou-ple of choices for the sets of collocation and integration points, to improve

the computational efficiency and a p-multigrid strategy to reduce the RAM

required by the fully coupled implicit solver The p-multigrid algorithm has

been analyzed in Ref 21 and applied to the solution of the compressible

solution (◦ 4049440 DOFs) compared with TAU (—— 5102446 DOFs) and CFL3D (– – – 2256896 DOFs, – · – 7689088 DOFs, – ··

– 26224640 DOFs).

Euler and Navier-Stokes equations in Refs 22 and 23 First applications

of the p-multigrid strategy to shockless turbulent flows around complex 3D

geometries have already provided encouraging results

On the other hand, we are working on exploiting the flexibility of the

modal DG discretization, with shape functions defined in the real space, to

improve the computational efficiency by means of agglomeration strategies

The agglomeration technique provides also the natural setting for the

devel-opment of h-multigrid solution strategies for high-order DG discretizations

First results of this research activity have already been reported in Ref 24

Finally, even if the shock-capturing approach turned out to be robust

and accurate, further research is needed to make its formulation fully

consis-tent with a residual-based artificial viscosity Moreover, the adverse impact

on the regularity of convergence of residuals needs to be further

investi-gated

its origin in the centroid of K and the coordinate axes coincident with the

principal axes of the element

Finally we remark... to define only a linear mapping of

el-ement faces and this resulted in inaccurate pressure distributions on the

wing and sting surfaces All the computations have been run in parallel... a starting set of linearly

inde-pendent basis functions defined on K, respectively The MGS procedure

with re-orthogonalization can be simply setup as shown in the following