Sub-grid scale model with structure effects incorporated

through the helicityNobumitsu YOKOI

Institute of Industrial Science, University of Tokyo

In collaboration with Akira YOSHIZAWA

turbulent electromotive force

Reynolds (+ turb. Maxwell) stress

: mean velocity strain : mean magnetic strain

FlowLarge-scale structure

Turbulence

Transportcoefficients

Productionrates

Intensity of turbulence:energy

Structure of turbulence:helicities

Transport suppression

Eddyviscosity

Velocityshear

Magneticshear

Turbulent magneticdiffusivity

Electriccurrent

Vorticity

1

Mean velocity

: Time scale of turbulence

laminar turbulent turbulent mean

Swirling flow and turbulence (Yokoi & Yoshizawa, PoF, 1993)

2

Mean velocity

: Time scale of turbulence

laminar turbulent turbulent mean

Uz

Uz

Uθ

Uθ

r

r

z

θ

θ

turbulent swirling

Swirling flow and turbulence (Yokoi & Yoshizawa, PoF, 1993)

2

Mean velocity

: Time scale of turbulence

laminar turbulent turbulent mean

Uz

Uz

Uθ

Uθ

r

r

z

θ

θ

turbulent swirling

Swirling flow and turbulence (Yokoi & Yoshizawa, PoF, 1993)

2

Mean velocity

: Time scale of turbulence

laminar turbulent turbulent mean

Uz

Uz

Uθ

Uθ

r

r

z

θ

θ

turbulent swirling

Swirling flow and turbulence (Yokoi & Yoshizawa, PoF, 1993)

2

Mean velocity

: Time scale of turbulence

laminar turbulent turbulent mean

Uz

Uz

Uθ

Uθ

r

r

z

θ

θ

turbulent swirling

Swirling flow and turbulence (Yokoi & Yoshizawa, PoF, 1993)

2

Smagorinskyconstant CS

Isotropic flow

Mixing-layer flow

Channel flow

Classical Smagorinsky model

Problem of constant adjustment

SGS viscosity

: Smagorinsky constant

Structure effect

3

Smagorinskyconstant CS

Isotropic flow

Mixing-layer flow

Channel flow

Dissipativenature

more

less

Classical Smagorinsky model

Problem of constant adjustment

SGS viscosity

: Smagorinsky constant

Structure effect

3

Smagorinskyconstant CS

Isotropic flow

Mixing-layer flow

Channel flow

Fluctuating vorticity(Robinson, Kline & Spalart 1988)

Vorticalstructures

weak

intermediate

strong

Dissipativenature

more

less

Classical Smagorinsky model

Problem of constant adjustment

SGS viscosity

: Smagorinsky constant

Structure effect

3

Smagorinskyconstant CS

Isotropic flow

Mixing-layer flow

Channel flow

Fluctuating vorticity(Robinson, Kline & Spalart 1988)

Vorticalstructures

weak

intermediate

strong

Coherent vortical structures may be related to the less dissipative nature

Dissipativenature

more

less

Classical Smagorinsky model

Problem of constant adjustment

SGS viscosity

: Smagorinsky constant

Structure effect

3

Grid-Scale (GS) velocity equation

SGS stress

SGS viscosity Helicity-related coefficient

Two-equation model

SGS K equation

SGS HS equation

Helicity SGS model

4

One-equation model

SGS viscosity

Helicity-related coefficient

SGS turbulent energy KS

SGS turbulent helicity (HS) equation

SGS Reynolds stress

Local equilibrium of the SGS energy

Production ~ Dissipation

Production ~ Dissipation

5

Zero-equation model (Same level as Smagorinsky model)

Local equilibrium of the SGS helicity Production ~ Dissipation

with

requires testusing DNS

6

Grid-Scale (GS) velocity equation

SGS stress

GS strain rate

Magnetohydrodynamic Case

Grid-Scale (GS) magnetic-field equation

SGS electromotive force

7

MHD Counterpart of the Smagorinsky Model

Local equilibrium of the SGS energy

Production ~ Dissipation

Production

Dissipation

Turbulent magneticPrandtl number

8

Smagorinsky model for MHD

Local equilibrium

9

Local equilibrium of the SGS cross helicity

Production

Dissipation

Production ~ Dissipation

10

Smagorinsky model for MHD

Turbulent magneticPrandtl number

Local equilibrium

11

Helicity-related Termsin MHD Case

Turbulent MHD residual helicity

HR equation

12

Local equilibrium

Equipartition

13

14