skew-symmetric matrices and accurate simulations of compressible turbulent flow wybe rozema johan...
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Skew-symmetric matrices and accurate simulations of compressible turbulent
flow
Wybe RozemaJohan Kok
Roel VerstappenArthur Veldman
1
A simple discretization
(𝜕 𝑓𝜕 𝑥 )𝑖
=𝑓 𝑖+1− 𝑓 𝑖− 12h
+𝑂(h2)
2
The derivative is equal to the slope of the line
𝑓 𝑖− 1
𝑖
𝑓 𝑖+1
h
𝑖+1𝑖−1
The problem of accuracy
3
How to prevent small errors from summing to complete nonsense?
𝑖 𝑖+1𝑖−1
exact
2 nd order
Compressible flow
4
Completely different things happen in air
shock wave
acoustics
turbulence
It’s about discrete conservation
Skew-symmetric matrices
Simulations ofturbulent flow
5
¿𝐶
𝑇=−𝐶&
Governing equations
6
𝜕𝑡 𝜌𝒖+𝛻 ∙ (𝜌𝒖⊗𝒖)+𝛻𝑝=𝛻 ∙𝝈𝜕𝑡 𝜌 𝐸+𝛻 ∙ (𝜌𝒖𝐸 )+𝛻 ∙ (𝑝𝒖)=𝛻 ∙ (𝜎 ∙𝒖 )−𝛻 ∙𝒒
𝜕𝑡 𝜌+𝛻 ∙ (𝜌𝒖 )=0
𝒖
𝑭 𝑝convective transport
pressure forces
viscous friction
𝜎 𝑦𝑥
𝒒
heat diffusion
Convective transport conserves a lot, but this does not end up in standard finite-volume method
𝜌 𝐸= 12 𝜌𝒖 ∙𝒖+𝜌𝑒
Conservation and inner products
Inner product
Physical quantities
7
Square root variables
Why does convective transport conserve so many inner products?
√𝜌 √𝜌𝒖√2 √𝜌𝑒 ⟨ √𝜌 ,√𝜌 ⟩
⟨√𝜌 , √𝜌𝑢√2 ⟩
⟨ √𝜌𝑒 ,√𝜌𝑒 ⟩
⟨ √𝜌𝑢√2
, √𝜌𝑢√2 ⟩
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
⟨𝑐 (𝒖 )𝜑 ,𝜗 ⟩=− ⟨𝜑 ,𝑐 (𝒖 )𝜗 ⟩
𝜕𝑡𝜑+𝑐 (𝒖 )𝜑=…𝑐 (𝒖 )𝜑=
12𝛻 ∙ (𝒖𝜑 )+ 1
2𝒖 ∙𝛻𝜑
+... =
0 +...
√𝜌√𝜌𝒖√2
√𝜌𝑒
Conservative discretizationDiscrete skew-symmetry
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Computational grid
The discrete convective transport should correspond to a skew-symmetric operator
⟨𝜑 ,𝜗 ⟩=∑𝑘
Ω𝑘𝜑𝑘𝜗𝑘
(𝑐 (𝒖)𝜑 )𝑘=1Ω𝑘
∑𝑓
𝑨𝑓 ∙𝒖 𝑓
𝜑𝑛𝑏(𝑓 )
2
Discrete inner product
Ω𝑘𝑨 𝑓
𝑓
√𝜌√𝜌𝒖√2
√𝜌𝑒
𝐶=12Ω−1 ¿
Matrix notationDiscrete conservation
10
Discrete inner product
The matrix should be skew-symmetric
√𝜌√𝜌𝒖√2
√𝜌𝑒Matrix equation
Is it more than explanation?
11
√𝜌√𝜌𝒖√2
√𝜌𝑒
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
12
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
𝝃
𝒙
𝒙 (𝝃)
∆ ξ
Delta wing simulations
13
Preliminary simulations of the flow over a simplified triangular wing
test section
coarse grid and artificial dissipation outside test section
α = 25°M = 0.3 = 75°
Re = 5·104
27M cells α
transition
It’s all about the grid
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Making a grid is going from continuous to discrete
𝝃𝒙
𝒙 (𝝃)
conical block structure
fine grid near delta
wing
The aerodynamics
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α
𝜔𝑥
𝑝
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 75° 85°α 25° 12.5°Rec 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)
Δx = const.Δy = k x
Δx = Δy
x
y
ΔxΔy
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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√𝜌√𝜌𝒖√2
√𝜌𝑒
[email protected]@rug.nl
𝐶𝑇 =−𝐶