Skew-symmetric matrices and accurate simulations of compressible turbulent flow
description
Transcript of Skew-symmetric matrices and accurate simulations of compressible turbulent flow
Skew-symmetric matrices and accurate simulations of compressible turbulent
flow
Wybe RozemaJohan Kok
Roel VerstappenArthur Veldman
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A simple discretization
( ๐ ๐๐ ๐ฅ )๐=๐ ๐+1โ ๐ ๐โ 12h
+๐(h2)
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The derivative is equal to the slope of the line
๐ ๐โ 1
๐
๐ ๐+1
h
๐+1๐โ1
The problem of accuracy
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How to prevent small errors from summing to complete nonsense?
๐ ๐+1๐โ1
exact
2 nd order
Compressible flow
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Completely different things happen in air
shock wave
acoustics
turbulence
Itโs about discrete conservation
Skew-symmetric matrices
Simulations ofturbulent flow
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ยฟ๐ถ๐=โ๐ถ &
Governing equations
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๐๐ก ๐๐+๐ป โ (๐๐โ๐)+๐ป๐=๐ป โ๐๐๐ก ๐ ๐ธ+๐ป โ (๐๐๐ธ )+๐ป โ (๐๐)=๐ป โ (๐ โ๐ )โ๐ป โ๐
๐๐ก ๐+๐ป โ (๐๐ )=0
๐
๐ญ ๐convective transport
pressure forces
viscous friction
๐ ๐ฆ๐ฅ๐
heat diffusion
Convective transport conserves a lot, but this does not end up in standard finite-volume method
๐ ๐ธ= 12 ๐๐ โ๐+๐๐
Conservation and inner products
Inner product
Physical quantities
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Square root variables
Why does convective transport conserve so many inner products?
โ๐ โ๐๐โ2 โ๐๐ โจ โ๐ ,โ๐ โฉ
โจโ๐ , โ๐๐ขโ2 โฉ
โจ โ๐๐ ,โ๐๐ โฉ
โจ โ๐๐ขโ2
, โ๐๐ขโ2 โฉ
kinetic energy
density internal energy
mass internal energy
momentum kinetic energy
Convective skew-symmetry
Skew-symmetry
Inner product evolution
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Convective terms
Convective transport conserves many physical quantities because is skew-symmetric
โจ๐ (๐ )๐ ,๐ โฉ=โ โจ๐ ,๐ (๐ )๐ โฉ
๐๐ก๐+๐ (๐ )๐=โฆ๐ (๐ )๐=
12 ๐ป โ (๐๐ )+ 12๐ โ๐ป๐
+... =
0 +...
โ๐โ๐๐โ2
โ๐๐
Conservative discretizationDiscrete skew-symmetry
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Computational grid
The discrete convective transport should correspond to a skew-symmetric operator
โจ๐ ,๐ โฉ=โ๐ฮฉ๐๐๐๐๐
(๐ (๐)๐ )๐=1ฮฉ๐
โ๐๐จ๐ โ๐ ๐
๐๐๐(๐ )
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Discrete inner product
ฮฉ๐๐จ ๐๐
โ๐โ๐๐โ2
โ๐๐
๐ถ=12 ฮฉ
โ1 ยฟ
Matrix notationDiscrete conservation
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Discrete inner product
The matrix should be skew-symmetric
โ๐โ๐๐โ2
โ๐๐Matrix equation
Is it more than explanation?
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โ๐โ๐๐โ2
โ๐๐
A conservative discretization can be rewritten to finite-volume form
Energy-conserving time integration requires square-
root variables
Square-root variables live in L2
Application in practice
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NLR ensolv multi-block structured
curvilinear grid collocated 4th-order
skew-symmetric spatial discretization
explicit 4-stage RK time stepping
Skew-symmetry gives control of numerical dissipation
๐๐
๐ (๐)
โ ฮพ
Delta wing simulations
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Preliminary simulations of the flow over a simplified triangular wing
test section
coarse grid and artificial dissipation outside test section
ฮฑ = 25ยฐM = 0.3 = 75ยฐ
Re = 5ยท104
27M cells ฮฑ
transition
Itโs all about the grid
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Making a grid is going from continuous to discrete
๐๐
๐ (๐)
conical block structure
fine grid near delta
wing
The aerodynamics
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ฮฑ
๐๐ฅ
๐
The flow above the wing rolls up into a vortex core
bl sucked into the vortex core
suction peak in vortex core
Flexibility on coarser grids
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Artificial or model dissipation is not necessary for stability
skew-symmetricno artificial dissipation
sixth-order artificial dissipation
LES model dissipation (Vreman, 2004)
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preliminary finalM 0.3 0.3 75ยฐ 85ยฐฮฑ 25ยฐ 12.5ยฐRec 5 x 104 1.5 x 105
# cells 2.7 x 107 1.4 x 108
CHs 5 x 105 3.7 x 106
23 weeks on 128 cores
preliminary
final (isotropic)
ฮx = const.ฮy = k x
ฮx = ฮy
x
y
ฮxฮy
The final simulations
The glass ceiling
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what to store? post-processing
Take-home messages The conservation
properties of convective transport can be related to a skew-symmetry
We are pushing the envelope with accurate delta wing simulations
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โ๐โ๐๐โ2
โ๐๐
[email protected]@rug.nl
๐ถ๐=โ๐ถ