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
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A simple discretization
𝑓𝑖+1
π‘“π‘–βˆ’1
πœ•π‘“
πœ•π‘₯
β„Ž
π‘–βˆ’1
𝑖
𝑖
𝑓𝑖+1 βˆ’ π‘“π‘–βˆ’1
=
+ 𝑂(β„Ž2 )
2β„Ž
𝑖+1
The derivative is equal to the slope of the line
2
The problem of accuracy
π‘–βˆ’1
𝑖
𝑖+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
It’s about discrete conservation
Skew-symmetric
matrices
Simulations of
turbulent flow
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Governing equations
πœ•π‘‘ 𝜌 + 𝛻 βˆ™ πœŒπ’– = 0
πœ•π‘‘ πœŒπ’– + 𝛻 βˆ™ πœŒπ’– βŠ— 𝒖 + 𝛻𝑝 = 𝛻 βˆ™ 𝝈
πœ•π‘‘ 𝜌𝐸 + 𝛻 βˆ™ πœŒπ’–πΈ + 𝛻 βˆ™ 𝑝𝒖 = 𝛻 βˆ™ 𝜎 βˆ™ 𝒖 βˆ’ 𝛻 βˆ™ 𝒒
πœŽπ‘¦π‘₯
𝒖
convective transport
𝒒
𝑭𝑝
pressure forces
viscous friction
heat diffusion
Convective transport conserves a lot, but this does not
end up in standard finite-volume method
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Conservation and inner products
Square root variables
πœŒπ’–
𝜌
density
πœŒπ‘’
2
kinetic
energy
internal
energy
Inner product
πœ‘, πœ— =
πœ‘πœ— β…†π‘₯
Physical quantities
𝜌, 𝜌
mass
𝜌,
πœŒπ‘’, πœŒπ‘’
internal energy
πœŒπ‘’
πœŒπ‘’
2
2
momentum
,
πœŒπ‘’
2
kinetic energy
Why does convective transport conserve so many inner
products?
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Convective skew-symmetry
Convective terms
Inner product evolution
πœ•π‘‘ πœ‘ + 𝑐 𝒖 πœ‘ = …
πœ•π‘‘ πœ‘, πœ—
=
πœ•π‘‘ πœ‘, πœ— + πœ‘, πœ•π‘‘ πœ—
=
βˆ’ 𝑐 𝒖 πœ‘, πœ— βˆ’ πœ‘, 𝑐(𝒖)πœ— +...
=
0 +...
1
1
𝑐 𝒖 πœ‘ = 𝛻 βˆ™ π’–πœ‘ + 𝒖 βˆ™ π›»πœ‘
2
2
Skew-symmetry
𝑐 𝒖 πœ‘, πœ— = βˆ’ πœ‘, 𝑐 𝒖 πœ—
Convective transport conserves many physical
quantities because 𝑐(𝒖) is skew-symmetric
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Conservative discretization
Computational grid
𝑓
Ξ©π‘˜
𝑨𝑓
Discrete skew-symmetry
𝑐(𝒖)πœ‘
π‘˜
1
=
Ξ©π‘˜
0
Discrete inner product
πœ‘, πœ— =
π‘˜
Ξ©π‘˜ πœ‘π‘˜ πœ—π‘˜
1 βˆ’1 βˆ’πΉ 1
𝐢= Ω
π‘˜βˆ’
2
2
πœ‘π‘›π‘(𝑓)
𝑨𝑓 βˆ™ 𝒖𝑓
2
𝑓
𝐹
1
π‘˜βˆ’2
0
βˆ’πΉ
1
π‘˜+2
𝐹
π‘˜+
0
1
2
The discrete convective transport 𝑐(𝒖) should
correspond to a skew-symmetric operator
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Matrix notation
Matrix equation
Discrete conservation
πœ•π‘‘ 𝝋 + 𝐢𝝋 = β‹―
𝐢𝝋, 𝝑 + 𝝋, 𝐢𝝑
=
𝝋𝑇 𝐢 𝑇 Ω𝝑 + 𝝋𝑇 Ω𝐢𝝑
=
𝝋𝑇 (Ω𝐢)𝑇 +Ω𝐢 𝝑
=
0
πœ•π‘‘ 𝝑 + 𝐢𝝑 = β‹―
Discrete inner product
𝝋, 𝝑 = 𝝋𝑇 Ω𝝑
The matrix Ω𝐢 should be skew-symmetric
(Ω𝐢)𝑇 = βˆ’Ξ©πΆ
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Is it more than explanation?

A conservative discretization can be rewritten to
finite-volume form
οƒΎ
Energy-conserving time integration requires squareroot variables
οƒΎ
Square-root variables live in L2
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Application in practice
NLR ensolv
𝝃
𝒙
βˆ†ΞΎ
 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
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
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It’s all about the grid
conical block
structure
fine grid
near delta
wing
𝝃
𝒙
Making a grid is going from continuous to discrete
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The aerodynamics
πœ”π‘₯
𝑝
bl sucked into the
vortex core
Ξ±
suction peak in
vortex core
The flow above the wing rolls up into a vortex core
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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
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The final simulations
preliminary
Ξ”x = const.
Ξ”y = k x
Ξ”y
preliminary
final
M
0.3
0.3

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
Ξ”x
Rec
# cells
final (isotropic)
CHs
Ξ”x = Ξ”y
y
x
23 weeks on
128 cores
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The glass ceiling
what to store?
post-processing
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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
[email protected]
[email protected]
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