skew-symmetric matrices and accurate simulations of compressible turbulent flow
DESCRIPTION
Skew-symmetric matrices and accurate simulations of compressible turbulent flow. Wybe Rozema Johan Kok Roel Verstappen Arthur Veldman. A simple discretization. The d erivative is equal to the slope of the line. The problem of accuracy. exact. 2 nd order. - PowerPoint PPT PresentationTRANSCRIPT
Skew-symmetric matrices and accurate simulations of compressible turbulent
flow
Wybe RozemaJohan Kok
Roel VerstappenArthur Veldman
1
A simple discretization
( π ππ π₯ )π=π π+1β π πβ 12h
+π(h2)
2
The derivative is equal to the slope of the line
π πβ 1
π
π π+1
h
π+1πβ1
The problem of accuracy
3
How to prevent small errors from summing to complete nonsense?
π π+1πβ1
exact
2 nd order
Compressible flow
4
Completely different things happen in air
shock wave
acoustics
turbulence
Itβs about discrete conservation
Skew-symmetric matrices
Simulations ofturbulent flow
5
ΒΏπΆπ=βπΆ &
Governing equations
6
ππ‘ ππ+π» β (ππβπ)+π»π=π» βπππ‘ π πΈ+π» β (πππΈ )+π» β (ππ)=π» β (π βπ )βπ» βπ
ππ‘ π+π» β (ππ )=0
π
π πconvective transport
pressure forces
viscous friction
π π¦π₯π
heat diffusion
Convective transport conserves a lot, but this does not end up in standard finite-volume method
π πΈ= 12 ππ βπ+ππ
Conservation and inner products
Inner product
Physical quantities
7
Square root variables
Why does convective transport conserve so many inner products?
βπ βππβ2 βππ β¨ βπ ,βπ β©
β¨βπ , βππ’β2 β©
β¨ βππ ,βππ β©
β¨ βππ’β2
, βππ’β2 β©
kinetic energy
density internal energy
mass internal energy
momentum kinetic energy
Convective skew-symmetry
Skew-symmetry
Inner product evolution
8
Convective terms
Convective transport conserves many physical quantities because is skew-symmetric
β¨π (π )π ,π β©=β β¨π ,π (π )π β©
ππ‘π+π (π )π=β¦π (π )π=
12 π» β (ππ )+ 12π βπ»π
+... =
0 +...
βπβππβ2
βππ
Conservative discretizationDiscrete skew-symmetry
9
Computational grid
The discrete convective transport should correspond to a skew-symmetric operator
β¨π ,π β©=βπΞ©πππππ
(π (π)π )π=1Ξ©π
βππ¨π βπ π
πππ(π )
2
Discrete inner product
Ξ©ππ¨ ππ
βπβππβ2
βππ
πΆ=12 Ξ©
β1 ΒΏ
Matrix notationDiscrete conservation
10
Discrete inner product
The matrix should be skew-symmetric
βπβππβ2
βππMatrix equation
Is it more than explanation?
11
βπβππβ2
βππ
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
ππ
π (π)
β ΞΎ
Delta wing simulations
13
Preliminary simulations of the flow over a simplified triangular wing
test section
coarse grid and artificial dissipation outside test section
Ξ± = 25Β°M = 0.3 = 75Β°
Re = 5Β·104
27M cells Ξ±
transition
Itβs all about the grid
14
Making a grid is going from continuous to discrete
ππ
π (π)
conical block structure
fine grid near delta
wing
The aerodynamics
15
Ξ±
ππ₯
π
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
16
Artificial or model dissipation is not necessary for stability
skew-symmetricno artificial dissipation
sixth-order artificial dissipation
LES model dissipation (Vreman, 2004)
17
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)
Ξx = const.Ξy = k x
Ξx = Ξy
x
y
ΞxΞy
The final simulations
The glass ceiling
18
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
19
βπβππβ2
βππ
[email protected]@rug.nl
πΆπ=βπΆ