Skew-symmetric matrices and
accurate simulations of
compressible turbulent flow
Wybe Rozema
Johan Kok
Roel Verstappen
Arthur Veldman
1
A simple discretization
𝑓𝑖+1
𝑓𝑖−1
𝜕𝑓
𝜕𝑥
ℎ
𝑖−1
𝑖
𝑖
𝑓𝑖+1 − 𝑓𝑖−1
=
+ 𝑂(ℎ2 )
2ℎ
𝑖+1
The derivative is equal to the slope of the line
2
The problem of accuracy
𝑖−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
5
Governing equations
𝜕𝑡 𝜌 + 𝛻 ∙ 𝜌𝒖 = 0
𝜕𝑡 𝜌𝒖 + 𝛻 ∙ 𝜌𝒖 ⊗ 𝒖 + 𝛻𝑝 = 𝛻 ∙ 𝝈
𝜕𝑡 𝜌𝐸 + 𝛻 ∙ 𝜌𝒖𝐸 + 𝛻 ∙ 𝑝𝒖 = 𝛻 ∙ 𝜎 ∙ 𝒖 − 𝛻 ∙ 𝒒
𝜎𝑦𝑥
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
6
Conservation and inner products
Square root variables
𝜌𝒖
𝜌
density
𝜌𝑒
2
kinetic
energy
internal
energy
Inner product
𝜑, 𝜗 =
𝜑𝜗 ⅆ𝑥
Physical quantities
𝜌, 𝜌
mass
𝜌,
𝜌𝑒, 𝜌𝑒
internal energy
𝜌𝑢
𝜌𝑢
2
2
momentum
,
𝜌𝑢
2
kinetic energy
Why does convective transport conserve so many inner
products?
7
Convective skew-symmetry
Convective terms
Inner product evolution
𝜕𝑡 𝜑 + 𝑐 𝒖 𝜑 = …
𝜕𝑡 𝜑, 𝜗
=
𝜕𝑡 𝜑, 𝜗 + 𝜑, 𝜕𝑡 𝜗
=
− 𝑐 𝒖 𝜑, 𝜗 − 𝜑, 𝑐(𝒖)𝜗 +...
=
0 +...
1
1
𝑐 𝒖 𝜑 = 𝛻 ∙ 𝒖𝜑 + 𝒖 ∙ 𝛻𝜑
2
2
Skew-symmetry
𝑐 𝒖 𝜑, 𝜗 = − 𝜑, 𝑐 𝒖 𝜗
Convective transport conserves many physical
quantities because 𝑐(𝒖) is skew-symmetric
8
Conservative discretization
Computational grid
𝑓
Ω𝑘
𝑨𝑓
Discrete skew-symmetry
𝑐(𝒖)𝜑
𝑘
1
=
Ω𝑘
0
Discrete inner product
𝜑, 𝜗 =
𝑘
Ω𝑘 𝜑𝑘 𝜗𝑘
1 −1 −𝐹 1
𝐶= Ω
𝑘−
2
2
𝜑𝑛𝑏(𝑓)
𝑨𝑓 ∙ 𝒖𝑓
2
𝑓
𝐹
1
𝑘−2
0
−𝐹
1
𝑘+2
𝐹
𝑘+
0
1
2
The discrete convective transport 𝑐(𝒖) should
correspond to a skew-symmetric operator
9
Matrix notation
Matrix equation
Discrete conservation
𝜕𝑡 𝝋 + 𝐶𝝋 = ⋯
𝐶𝝋, 𝝑 + 𝝋, 𝐶𝝑
=
𝝋𝑇 𝐶 𝑇 Ω𝝑 + 𝝋𝑇 Ω𝐶𝝑
=
𝝋𝑇 (Ω𝐶)𝑇 +Ω𝐶 𝝑
=
0
𝜕𝑡 𝝑 + 𝐶𝝑 = ⋯
Discrete inner product
𝝋, 𝝑 = 𝝋𝑇 Ω𝝑
The matrix Ω𝐶 should be skew-symmetric
(Ω𝐶)𝑇 = −Ω𝐶
10
Is it more than explanation?

A conservative discretization can be rewritten to
finite-volume form

Energy-conserving time integration requires squareroot variables

Square-root variables live in L2
11
Application in practice
NLR ensolv
𝝃
𝒙
∆ξ
 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
13
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
14
The aerodynamics
𝜔𝑥
𝑝
bl sucked into the
vortex core
α
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
Δx = const.
Δy = k x
Δy
preliminary
final
M
0.3
0.3

75°
85°
α
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
Δx
Rec
# cells
final (isotropic)
CHs
Δx = Δy
y
x
23 weeks on
128 cores
17
The glass ceiling
what to store?
post-processing
18
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
wyberozema@gmail.com
w.rozema@rug.nl
19