testcase c 3.1 mda 30p-30n · slide 3 > 2d high lift case> tobias leicht & marcel...
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www.DLR.de • Slide 1 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
Testcase C 3.1MDA 30P-30NTobias Leicht & Marcel WallraffDLR Braunschweig (AS - C2A2S2E)
www.DLR.de • Slide 2 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
DG discretization
Basis functions
non-parametric ortho-normal basis functions
directly formulated in physical space
also referred to as Taylor-DG
need to be evaluated for each mesh element
RANS equations
SA turbulence model (negative SA)
second scheme of Bassi and Rebay (BR2) for viscous terms
Roe flux as a convective flux, based on an
eigen-decomposition of the full jacobian
www.DLR.de • Slide 3 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
2D high lift airfoil MDA 30P 30N
Mach number M = 0.2,
Reynolds number Re = 9 · 106,
angle of attack α = 16◦.
Figure : Pressure for a p = 2 solution on 33 728 elements mesh.
www.DLR.de • Slide 4 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
Testcase 3.1Mesh hierarchy with own meshes (DLR):
(structured) quadrilateral meshes with piecewise quarticboundaries
farfield distance approx. 50 chord lengths
2 108, 8 432, 33 728 and 134 912 elements
x
y
20 10 0 10 20 30
20
10
0
10
20
x
y
0 0.2 0.4 0.6
0.4
0.2
0
0.2
x
y
0.4 0.45 0.5 0.55
0.05
0
0.05
Figure : Coarsest mesh with 2 108 elements.
www.DLR.de • Slide 5 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
Numerical algorithms: Multigrid
p-MG
h-MG based onunstructured agglomeration
www.DLR.de • Slide 6 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
Numerical algorithms
possible solver choices
single grid Backward-Euler
start up strategy in mesh or order sequencingfor improved initial conditions
linear MG as preconditioner
non-linear MG (FAS) to accelerate process in pseudo-time
non-linear MG with linear MG on each level
www.DLR.de • Slide 7 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
0 100 200 300 400
10−9
10−7
10−5
10−3
10−1
101
nonlinear BWE iterations
resi
dual
com
pone
nts
ρ (density)ρv1ρv2
ρEρν̃
Figure : Convergence of all residual components for an MDA 30P-30NSA-computation with p = 2 on the 134 912 element mesh.
www.DLR.de • Slide 8 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
0 100 200 300 40010−12
10−9
10−6
10−3
100
nonlinear BWE iterations
dens
ityre
sidu
al
single-gridh-FAS+LMG
Figure : Convergence of the density component for an MDA 30P-30NSA-computation with p = 2 on the 134 912 element mesh.
www.DLR.de • Slide 9 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
·104
10−12
10−9
10−6
10−3
100
work units
dens
ityre
sidu
al
single-gridh-FAS+LMG
Figure : Convergence of the density component for an MDA 30P-30NSA-computation with p = 2 on the 134 912 element mesh.
www.DLR.de • Slide 10 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
10−3 10−23.2
3.4
3.6
3.8
4
4.2
DoF−12
lift
10−3 10−2
4 · 10−2
6 · 10−2
8 · 10−2
0.1
0.12
0.14
DoF−12
drag
p=1p=2p=3
102 103 104 1053.2
3.4
3.6
3.8
4
4.2
work units
lift
102 103 104 105
4 · 10−2
6 · 10−2
8 · 10−2
0.1
0.12
0.14
work units
drag
p=1p=2p=3
www.DLR.de • Slide 11 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
Reference values
Assuming idealized error behavior
CL = CrefL + ε · N−α
d
the p=2 results on the finest three meshes have been exploited toobtain
CrefL = 4.1719
CrefD = 0.04665
observed order α = 2.5
www.DLR.de • Slide 12 > 2D high lift case > Tobias Leicht & Marcel Wallraff • January 5th , 2014
10−3 10−2
10−2
10−1
100
DoF−12
lift
erro
r
10−3 10−2
10−3
10−2
10−1
DoF−12
drag
erro
r
p=1p=2p=3
102 103 104 105
10−2
10−1
100
work units
lift
erro
r
102 103 104 105
10−3
10−2
10−1
work units
drag
erro
r
p=1p=2p=3