### 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
+1
−1

ℎ
−1

+1 − −1
=
+ (ℎ2 )
2ℎ
+1
The derivative is equal to the slope of the line
2
The problem of accuracy
−1

+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
Skew-symmetric
matrices
Simulations of
turbulent flow
5
Governing equations
+  ∙  = 0
+  ∙  ⊗  +  =  ∙
+  ∙  +  ∙  =  ∙  ∙  −  ∙

convective transport

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

density

2
kinetic
energy
internal
energy
Inner product
,  =
ⅆ
Physical quantities
,
mass
,
,
internal energy

2
2
momentum
,

2
kinetic energy
Why does convective transport conserve so many inner
products?
7
Convective skew-symmetry
Convective terms
Inner product evolution
+    = …
,
=
,  + ,
=
−   ,  − , () +...
=
0 +...
1
1
=  ∙  +  ∙
2
2
Skew-symmetry
,  = − ,
Convective transport conserves many physical
quantities because () is skew-symmetric
8
Conservative discretization
Computational grid

Ω

Discrete skew-symmetry
()

1
=
Ω
0
Discrete inner product
,  =

Ω
1 −1 − 1
= Ω
−
2
2
()
∙
2

1
−2
0
−
1
+2

+
0
1
2
The discrete convective transport () should
correspond to a skew-symmetric operator
9
Matrix notation
Matrix equation
Discrete conservation
+  = ⋯
,  + ,
=
Ω +  Ω
=
(Ω) +Ω
=
0
+  = ⋯
Discrete inner product
,  =  Ω
The matrix Ω should be skew-symmetric
(Ω) = −Ω
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
 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
conical block
structure
fine grid
near delta
wing

Making a grid is going from continuous to discrete
14
The aerodynamics

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
[email protected]
[email protected]
19
```