A simple discretization
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Transcript A simple discretization
Skew-symmetric matrices and
accurate simulations of
compressible turbulent flow
Wybe Rozema
Johan Kok
Roel Verstappen
Arthur Veldman
1
A simple discretization
ππ+1
ππβ1
ππ
ππ₯
β
πβ1
π
π
ππ+1 β ππβ1
=
+ π(β2 )
2β
π+1
The derivative is equal to the slope of the line
2
The problem of accuracy
πβ1
π
π+1
How to prevent small errors from summing to
complete nonsense?
3
Compressible flow
shock wave
turbulence
acoustics
Completely different things happen in air
4
Itβs about discrete conservation
Skew-symmetric
matrices
Simulations of
turbulent flow
5
Governing equations
ππ‘ π + π» β ππ = 0
ππ‘ ππ + π» β ππ β π + π»π = π» β π
ππ‘ ππΈ + π» β πππΈ + π» β ππ = π» β π β π β π» β π
ππ¦π₯
π
convective transport
π
ππ
pressure forces
viscous friction
heat diffusion
Convective transport conserves a lot, but this does not
end up in standard finite-volume method
6
Conservation and inner products
Square root variables
ππ
π
density
ππ
2
kinetic
energy
internal
energy
Inner product
π, π =
ππ β
π₯
Physical quantities
π, π
mass
π,
ππ, ππ
internal energy
ππ’
ππ’
2
2
momentum
,
ππ’
2
kinetic energy
Why does convective transport conserve so many inner
products?
7
Convective skew-symmetry
Convective terms
Inner product evolution
ππ‘ π + π π π = β¦
ππ‘ π, π
=
ππ‘ π, π + π, ππ‘ π
=
β π π π, π β π, π(π)π +...
=
0 +...
1
1
π π π = π» β ππ + π β π»π
2
2
Skew-symmetry
π π π, π = β π, π π π
Convective transport conserves many physical
quantities because π(π) is skew-symmetric
8
Conservative discretization
Computational grid
π
Ξ©π
π¨π
Discrete skew-symmetry
π(π)π
π
1
=
Ξ©π
0
Discrete inner product
π, π =
π
Ξ©π ππ ππ
1 β1 βπΉ 1
πΆ= Ξ©
πβ
2
2
πππ(π)
π¨π β ππ
2
π
πΉ
1
πβ2
0
βπΉ
1
π+2
πΉ
π+
0
1
2
The discrete convective transport π(π) should
correspond to a skew-symmetric operator
9
Matrix notation
Matrix equation
Discrete conservation
ππ‘ π + πΆπ = β―
πΆπ, π + π, πΆπ
=
ππ πΆ π Ξ©π + ππ Ξ©πΆπ
=
ππ (Ξ©πΆ)π +Ξ©πΆ π
=
0
ππ‘ π + πΆπ = β―
Discrete inner product
π, π = ππ Ξ©π
The matrix Ξ©πΆ should be skew-symmetric
(Ξ©πΆ)π = βΞ©πΆ
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Is it more than explanation?
ο½
A conservative discretization can be rewritten to
finite-volume form
οΎ
Energy-conserving time integration requires squareroot variables
οΎ
Square-root variables live in L2
11
Application in practice
NLR ensolv
π
π
βΞΎ
ο§ multi-block structured
curvilinear grid
ο§ collocated 4th-order
skew-symmetric
spatial discretization
ο§ explicit 4-stage RK
time stepping
Skew-symmetry gives control of numerical dissipation
12
Delta wing simulations
transition
coarse grid and
artificial dissipation
outside test section
test section
Preliminary simulations of the flow over a simplified
triangular wing
13
Itβs all about the grid
conical block
structure
fine grid
near delta
wing
π
π
Making a grid is going from continuous to discrete
14
The aerodynamics
ππ₯
π
bl sucked into the
vortex core
Ξ±
suction peak in
vortex core
The flow above the wing rolls up into a vortex core
15
Flexibility on coarser grids
skew-symmetric
no artificial dissipation
sixth-order artificial
dissipation
LES model dissipation
(Vreman, 2004)
Artificial or model dissipation is not necessary for
stability
16
The final simulations
preliminary
Ξx = const.
Ξy = k x
Ξy
preliminary
final
M
0.3
0.3
ο
75°
85°
Ξ±
25°
12.5°
5 x 104
1.5 x 105
2.7 x 107
1.4 x 108
5 x 105
3.7 x 106
Ξx
Rec
# cells
final (isotropic)
CHs
Ξx = Ξy
y
x
23 weeks on
128 cores
17
The glass ceiling
what to store?
post-processing
18
Take-home messages
ο§ The conservation
properties of convective
transport can be related
to a skew-symmetry
ο§ We are pushing the
envelope with accurate
delta wing simulations
[email protected]
[email protected]
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