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

8

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

9

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

14

Making a grid is going from continuous to discrete

𝝃𝒙

𝒙 (𝝃)

conical block structure

fine grid near delta

wing

The aerodynamics

15

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

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)

Δx = const.Δy = k x

Δx = Δy

x

y

ΔxΔy

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

√𝜌√𝜌𝒖√2

√𝜌𝑒

wyberozema@gmail.comw.rozema@rug.nl

𝐶𝑇 =−𝐶