High order simulation of transitional flow over the SD7003airfoil with the Discontinuous Galerkin Spectral Element

Method

Hannes Frank, Andrea Beck, Thomas Bolemann, Florian Hindenlang,Claus-Dieter Munz

Institute of Aerodynamics and Gasdynamics, Universitat Stuttgart, Germany

Koln, May 28, 20132nd High Order Workshop, 2013

Motivation

I Research interest: Applicability of high-order Discontinous Galerkinmethods for LES and DNS

I Qualifying features of DG methodsI High quality of numerical approximation (low dissipation/dispersion)I Complex geometries feasible through unstructured meshesI Efficient massively parallel computations possible

DGSEM I: Discontinouous Galerkin Spectral Element Method

I Code: FLEXI

I Compressible Navier-Stokes equations

I Hexahedral elements

I Nodal Galerkin Approach: polynomial basis and test functions

I Inter-element coupling via FV flux functions f ∗ (for this case: RoeRiemann solver)

〈J Ut , ψ〉E +(

f ∗ nξ, ψ)∂E−⟨F,∇ξψ

⟩E

= 0 (1)

I We use Gauss Integration to evaluate the inner products:

I Inexact integration of the nonlinear terms can cause instability (aliasing)

I Occurs especially for underresolved problems (→LES)

I We use higher integration accuracy to ensure stability (polynomialde-aliasing, Kirby and Karniadakis)

Q

Q

Q1

2

i

UQ1

UQi

N = 7

y

x

U

Case C3.3 SetupI SD7003 airfoil, Re = 60 000, Ma = 0.1, α = 8 degI Calculation without LES modelI Accuracy: O(4) in space and time (explicit RK4)I Time step ≈ 10−5 convection times (C/U∞)I Unstructured mesh, curved at airfoil surfaceI Farfield boundary: 100 chords radiusI spanwise extent: ∆z = 0.2, 12 cellsI sharp cut-off trailing edge

Mesh DOF·10−6 no. cells u.surface ∆(s/c)max ∆(s/c)LE ∆(n/c) n+x=0.8

M1 3.25 82 0.018 0.0036 0.0025 6.8M2 4.26 104 0.0054 0.0036 0.0021 5.4

Results: Mesh M1 I

Time evolution of CL

Time

CL

0 5 10 15 20 25 30

0.6

0.8

1

1.2averaging period

CL

(1) [15,25]

(2) [20,30]

Chord-wise velocity (t ∈ [20, 30])

CoordinateX

­cp

0 0.25 0.5 0.75 1­1

­0.5

0

0.5

1

1.5

2

2.5

3

t=15­25

t=20­30

CoordinateX

cf

0 0.25 0.5 0.75 1­0.02

­0.01

0

0.01

0.02

t=15..25

t=20­30

Results: Mesh M1 II

CoordinateX

­cp

0 0.25 0.5 0.75 1­1

­0.5

0

0.5

1

1.5

2

2.5

3

DGSEM, N=3

Garmann & Visbal

Galbraith & Visbal

Catalano & Tognaccini

Boom & Zingg

CoordinateX

cf

0 0.25 0.5 0.75 1­0.02

­0.01

0

0.01

0.02

DGSEM, N=3

Garmann & Visbal

Galbraith & Visbal

Catalano & Tognaccini

Boom & Zingg

CL CD xs xrDGSEM, Mesh M1 0.923 0.045 0.027 0.310Garmann & Visbal [1] 0.970 0.039 0.023 0.259Galbraith & Visbal [2] ≈ 0.91 ≈ 0.043 0.040 0.280Catalano & Tognaccini [3] ≈ 0.94 ≈ 0.044 0.030 0.290Boom & Zingg [4] 0.968 0.034 0.037 0.200

Table: Mean aerodynamic loads and seperation (xs) and reattachment (xr ) locations.

Effects of Refinement I

I Mesh M2: better resolution of the seperating shear layerI Increased resolution in streamwise and wall-normal direction

Mesh DOF·10−6 no. cells u.surface ∆(s/c)max ∆(s/c)LE ∆(n/c) n+x=0.8

M1 3.25 82 0.018 0.0036 0.0025 6.8M2 4.26 104 0.0054 0.0036 0.0021 5.4

M1

M2

Effects of Refinement II

I Velocity Magnitude Contours (t = 25)

M1 M2

CoordinateX

­cp

0 0.2 0.4 0.6 0.8­1

­0.5

0

0.5

1

1.5

2

2.5

3

M1

M2

CoordinateX

cf

0 0.2 0.4 0.6 0.8­0.02

­0.015

­0.01

­0.005

0

0.005

0.01

0.015

0.02

M1

M2

I No major differences between M1 and M2

Effects of Refinement IIII Aerodynamic loads

CL CD xs xrMesh M1 0.923 0.045 0.0274 0.310Mesh M2 0.922 0.046 0.0276 0.315

I Profiles of x velocity and mean squared fluctuations u′u′ (suction side)

U

Su

rfa

ce

No

rm

al D

ista

nc

e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.01

0.02

0.03

0.04

0.05

M1

M2

Garmann & Visbal

x=

1

.025

<u’u’>

Su

rfa

ce

No

rm

al D

ista

nc

e

0

0.01

0.02

0.03

0.04

0.05 0.1

Computational Cost

I Calculations were performed on the Cray XE6 cluster at HLRS, Stuttgart(Taubench 9.1s)

Mesh M1 M2# DOF 3.25 · 106 4.26 · 106

# time steps / conv. time 1.03 · 105 0.91 · 105

# Procs 2822 5548Wall time [h] 0.86 0.57CPU time [WU] 0.96 · 106 1.25 · 106

CPU time/ DOF [WU] 0.3 0.29CPU time/(DOF timestep) [WU] 2.88 · 10−6 3.19 · 10−6

Table: Computational cost for one convection time (C/U∞)

I Total wall time for mesh M2 (30 convection times): 17 hours

ConclusionI First results look promisingI DGSEM code is capable of highly parallel computationsI Outlook:

I Refinement in the nose region (seperation point!)I Investigations on higher order approximationsI Effects of SGS models

Thank you for your attention!

Numerics Research Group at the Institute of Aerodynamics andGasdynamics

→ www.iag.uni-stuttgart.de/nrg ←

Literature

D. J. Garmann and M. R. VisbalHigh-order solutions of transitional flow over the SD7003 airfoil usingcompact finite-differencing and filtering.High Order Workshop, 2012

M.C. Galbraith and M. R. VisbalImplicit Large-Eddy Simulation of Low Reynolds Number Flow past theSD 7003 airfoil.46th AIAA Aerospace Sciences Meeting and Exhibit, 2008

P. Catalano and R. TognacciniLarge Eddy Simulations of the Flow around the SD7003 airfoil.AIMETA Conference, 2011

P. Boom and D. ZinggTime-Accurate Flow Simulations Using an Efficient Newton-Krylov-SchurApproach with High-Order Temporal and Spatial Discretization.51st AIAA Aerospace Sciences Meeting including the New Horizons Forumand Aerospace Exposition, 2013