Direct and large-eddy simulation of rotating turbulence

rotating turbulence

Bernard J. Geurts, Darryl Holm, Arek Kuczaj

Multiscale Modeling and Simulation (Twente)

Anisotropic Turbulence (Eindhoven)

Mathematics Department, Imperial College (London)

IMS Turbulence Workshop

London, March 26, 2007

connect DNS to experiment - rotating decaying turbulence

show accuracy of regularization modeling - LES

apply regularization modeling to high Re flow

quantify cross-over from 3D to 2D dynamics as Ro

↓

connect DNS to experiment - rotating decaying turbulence

show accuracy of regularization modeling - LES

apply regularization modeling to high Re flow

quantify cross-over from 3D to 2D dynamics as Ro

↓

connect DNS to experiment - rotating decaying turbulence

show accuracy of regularization modeling - LES

apply regularization modeling to high Re flow

quantify cross-over from 3D to 2D dynamics as Ro

↓

connect DNS to experiment - rotating decaying turbulence

show accuracy of regularization modeling - LES

apply regularization modeling to high Re flow

quantify cross-over from 3D to 2D dynamics as Ro

↓

illustrate low Ro compressibility effects

connect DNS to experiment - rotating decaying turbulence

show accuracy of regularization modeling - LES

apply regularization modeling to high Re flow

quantify cross-over from 3D to 2D dynamics as Ro

↓

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

_{Rotation induced compressibility effects}

Outline

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

Rotation induced compressibility effects

Taylor-Proudman: columnar structuring

Strong rotation/small friction:

“Motion of a homogeneous fluid will be the same in all

planes perpendicular to the axis of rotation”

Taylor’s experiment:

When an object moves in a rotating flow, it drags along

with it a column of fluid parallel to the rotation axis

Taylor-Proudman: columnar structuring

Strong rotation/small friction:

“Motion of a homogeneous fluid will be the same in all

planes perpendicular to the axis of rotation”

Taylor’s experiment:

When an object moves in a rotating flow, it drags along

with it a column of fluid parallel to the rotation axis

Taylor-Proudman: columnar structuring

Strong rotation/small friction:

“Motion of a homogeneous fluid will be the same in all

planes perpendicular to the axis of rotation”

Taylor’s experiment:

When an object moves in a rotating flow, it drags along

with it a column of fluid parallel to the rotation axis

Guadalupe Islands

Rotating turbulence in a container

Experiment:

Morize, Moisy, Rabaud, PoF, 2005

Ro

=

6.4

Ro

=

0.6

Ro

=

0.3

Vertical structuring

of flow as Ro

=

u

r

/

(

2

Ω

r

ℓ

r

)

decreases

Rotating turbulence in a container

Experiment:

Morize, Moisy, Rabaud, PoF, 2005

Ro

=

6.4

Ro

=

0.6

Ro

=

0.3

Vertical structuring

of flow as Ro

=

u

r

/

(

2

Ω

r

ℓ

r

)

decreases

Modulation:

Competition 3D turbulence – 2D structuring

Rotating turbulence in a container

×

Ro

=

5.5

+

Ro

=

1.2

*

Ro

=

0.35

Rotating Rayleigh-B ´enard problem

Rotating Rayleigh-B ´enard problem

Temperature:

buoyancy dominated (Ta

=

0 - left) and

rotation dominated (Ta

>

Ra)

Rotating Rayleigh-B ´enard problem

Temperature:

buoyancy dominated (Ta

=

0 - left) and

rotation dominated (Ta

>

Ra)

Outline

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

Rotation induced compressibility effects

Governing equations

Incompressible formulation:

rotating frame of reference

∇ ·

u

=

0

∂

t

u

+

∇ ·

(

u u

) +

∇

p

−

1

Re

∇

2

_u

₌

u

×

Ω

Ro

Characteristic numbers:

Re

=

u

r

ℓ

r

ν

r

;

Ro

=

u

r

2

Ω

r

ℓ

r

x z y

Numerical method:

Pseudo-spectral discretization:

full de-aliasing, parallelized

Helical-wave decomposition:

Coriolis diagonalization

Governing equations

Incompressible formulation:

rotating frame of reference

∇ ·

u

=

0

∂

t

u

+

∇ ·

(

u u

) +

∇

p

−

1

Re

∇

2

_u

₌

u

×

Ω

Ro

Characteristic numbers:

Re

=

u

r

ℓ

r

ν

r

;

Ro

=

u

r

2

Ω

r

ℓ

r

x z y

Numerical method:

Pseudo-spectral discretization:

full de-aliasing, parallelized

Helical-wave decomposition:

Coriolis diagonalization

Benefit of helical wave decomposition (HWD)

Decay of kinetic energy:

0

0.05

0.1

0.15

0.2

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

t

E

Rotation: Ro

=

10

−2

RK4 with HWD

(solid)

at

∆

t

=

3.10

−3

RK4 without HWD at

∆

t

=

7.10

−3

₍₊₎

∆

t

=

5.10

−3

₍

_⊲

₎

∆

t

=

5.10

−

4

(

◦

)

Vertical structuring - vorticity

(a)

(b)

(c)

(d)

Decaying turbulence at R

λ

=

92.

Snapshot t

=

2.5: Ro

=

∞

Geometrical statistics - alignment

Eigenvalues/vectors S

ij

:

(

λ

i

,

z

i

)

(

positive

)

λ

1

> λ

2

> λ

3

(

negative

)

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.8 2 2.2 2.4 2.6 2.8 3 3.2x 105 p df cos(ω, z1)

(1)

−1−0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14x 105 p df cos(ω, z2)

(2)

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7x 10 5 p df cos(ω, z3)

(3)

cos

(

ω

,

z

_i

)

i

=

1

Ro

→ ∞

– aligned or perpendicular, rotation reduces

parallel alignment

i

=

2

alignment

ω

and z

₂

– reduced by rotation

i

=

3

Ro

→ ∞

– mainly perpendicular, reduced preference with

rotation

Geometrical statistics - vortex stretching

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12x 10 5

p

df

cos(

ω, W

)

cos

(

ω

,

W

)

Ro

→ ∞

vortex stretching dominant over vortex compression

Depleted skewness at strong rotation

Experimental observation - reduced skewness at high 1

/

Ro:

Cambon et al.

S

=

A

(

1

+

2Ro

−2

₎

1

/

2

;

S

∼

Ro

∼

Ω

Depleted skewness at strong rotation

Simulated skewness:

DNS at 512

3

0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t S

(a)

10 −1 100 101 10−1 100 Roω S

(b)

Velocity skewness versus t (a) and versus Ro (b)

Vorticity skewness

(a)

10 0 102 10−2 10−1 100 (Ωt)0.6 Ωt/(2π) Sω

(b)

Experimental (a) and simulated (b) vorticity skewness at

various rotation rates

Ω

Algebraic decay of kinetic energy

DNS at 256

3

:

0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t E

(a)

10 −1 100 101 10−2 10−1 t E

(b)

Algebraic decay exponent

0 20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1/Ro n

(a)

10 0 101 102 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1/Ro n

(b)

Outline

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

Rotation induced compressibility effects

DNS and LES in a picture

Classical problem: wide dynamic range

capture both large and small scales: resolution problem

N

∼

Re

9

/

4

;

W

∼

Re

3

If Re

→

10

×

Re then N

→

175N and W

→

1000W

Coarsening/mathematical modeling instead: LES

Filtering Navier-Stokes equations

Convolution-Filtering:

filter-kernel G

u

i

=

L

(

u

i

) =

Z

G

(

x

−

ξ

)

u

(

ξ

)

d

ξ

Application of filter:

∂

t

u

i

+

∂

j

(

u

i

u

j

) +

∂

i

p

−

1

Re

∂

jj

u

i

−

1

Ro

(

u

×

Ω

)

i

=

−

∂

j

τ

ij

Turbulent stress tensor:

τ

ij

=

u

i

u

j

−

u

i

u

j

Shorthand notation:

Closure problem

NS

(

u

) =

0

⇒

NS

(

u

) =

−∇ ·

τ

(

u,

u

)

⇐ −∇ ·

M

(

u

)

Closed LES formulation

Find v

:

NS

(

v

) =

−∇ ·

M

(

v

)

Filtering Navier-Stokes equations

Convolution-Filtering:

filter-kernel G

u

i

=

L

(

u

i

) =

Z

G

(

x

−

ξ

)

u

(

ξ

)

d

ξ

Application of filter:

∂

t

u

i

+

∂

j

(

u

i

u

j

) +

∂

i

p

−

1

Re

∂

jj

u

i

−

1

Ro

(

u

×

Ω

)

i

=

−

∂

j

τ

ij

Turbulent stress tensor:

τ

ij

=

u

i

u

j

−

u

i

u

j

Shorthand notation:

Closure problem

NS

(

u

) =

0

⇒

NS

(

u

) =

−∇ ·

τ

(

u,

u

)

⇐ −∇ ·

M

(u)

Closed LES formulation

Find v

:

NS

(

v

) =

−∇ ·

M

(

v

)

Filtering Navier-Stokes equations

Convolution-Filtering:

filter-kernel G

u

i

=

L

(

u

i

) =

Z

G

(

x

−

ξ

)

u

(

ξ

)

d

ξ

Application of filter:

∂

t

u

i

+

∂

j

(

u

i

u

j

) +

∂

i

p

−

1

Re

∂

jj

u

i

−

1

Ro

(

u

×

Ω

)

i

=

−

∂

j

τ

ij

Turbulent stress tensor:

τ

ij

=

u

i

u

j

−

u

i

u

j

Shorthand notation:

Closure problem

NS

(

u

) =

0

⇒

NS

(

u

) =

−∇ ·

τ

(

u,

u

)

⇐ −∇ ·

M

(u)

Closed LES formulation

Find v

:

NS

(

v

) =

−∇ ·

M

(

v

)

Regularization models

Alternative modeling approach

‘first principles’ – derive implied subgrid model

achieve unique coupling to filter, e.g., Leray, LANS-α

Example:

Leray regularization - alteration convective fluxes

∂

t

u

i

+

u

j

∂

j

u

i

+

∂

i

p

−

1

Re

∆

u

i

=

0

LES template:

∂

t

u

i

+

∂

j

(

u

j

u

i

) +

∂

i

p

−

1

Re

∆

u

i

=

−∂

j

m

L

_ij

Implied Leray model

m

L

_ij

=

L

u

_j

L

−1

(

u

_i

)

−

u

_j

u

_i

=

u

_j

u

_i

−

u

_j

u

_i

Rigorous analysis available (

∼

1930s)

Provides accurate LES model (

∼

03)

Regularization models

Alternative modeling approach

‘first principles’ – derive implied subgrid model

achieve unique coupling to filter, e.g., Leray, LANS-α

Example:

Leray regularization - alteration convective fluxes

∂

t

u

i

+

u

j

∂

j

u

i

+

∂

i

p

−

1

Re

∆

u

i

=

0

LES template:

∂

t

u

i

+

∂

j

(

u

j

u

i

) +

∂

i

p

−

1

Re

∆

u

i

=

−∂

j

m

L

_ij

Implied Leray model

m

L

_ij

=

L

u

_j

L

−1

(

u

_i

)

−

u

_j

u

_i

=

u

_j

u

_i

−

u

_j

u

_i

Rigorous analysis available (

∼

1930s)

Provides accurate LES model (

∼

03)

Regularization models

Alternative modeling approach

‘first principles’ – derive implied subgrid model

achieve unique coupling to filter, e.g., Leray, LANS-α

Example:

Leray regularization - alteration convective fluxes

∂

t

u

i

+

u

j

∂

j

u

i

+

∂

i

p

−

1

Re

∆

u

i

=

0

LES template:

∂

t

u

i

+

∂

j

(

u

j

u

i

) +

∂

i

p

−

1

Re

∆

u

i

=

−∂

j

m

L

_ij

Implied Leray model

m

L

_ij

=

L

u

_j

L

−1

(

u

_i

)

−

u

_j

u

_i

=

u

_j

u

_i

−

u

_j

u

_i

Rigorous analysis available (

∼

1930s)

Provides accurate LES model (

∼

03)

Regularization models

Alternative modeling approach

‘first principles’ – derive implied subgrid model

achieve unique coupling to filter, e.g., Leray, LANS-α

Example:

Leray regularization - alteration convective fluxes

∂

t

u

i

+

u

j

∂

j

u

i

+

∂

i

p

−

1

Re

∆

u

i

=

0

LES template:

∂

t

u

i

+

∂

j

(

u

j

u

i

) +

∂

i

p

−

1

Re

∆

u

i

=

−∂

j

m

L

_ij

Implied Leray model

m

L

_ij

=

L

u

_j

L

−1

(

u

_i

)

−

u

_j

u

_i

=

u

_j

u

_i

−

u

_j

u

_i

Rigorous analysis available (∼

1930s)

Provides accurate LES model (∼

03)

NS-

α

_{regularization}

Kelvin’s circulation theorem

d

dt

I

Γ(

u

)

u

j

dx

j

−

1

Re

I

Γ(

u

)

∂

kk

u

j

dx

j

=

0

⇒

NS

−

eqs

t

₁

t

₂

u

u

NS-

α

_{regularization}

α

-principle: Fluid loop:

Γ(

u

)

→

Γ(

u

)

⇒

Euler-Poincar ´e

∂

t

u

j

+

u

k

∂

k

u

j

+

u

k

∂

j

u

k

+

∂

j

p

−

∂

j

(

1

2

u

k

u

k

)

−

1

Re

∂

kk

u

j

=

0

Rewrite into LES template: Implied subgrid model

∂

t

u

i

+

∂

j

(

u

j

u

i

) +

∂

i

p

−

1

Re

∂

jj

u

i

=

−∂

j

u

j

u

i

−

u

j

u

i

−

1

2

u

j

∂

i

u

j

−

u

j

∂

i

u

j

=

−∂

j

m

ij

α

Cascade-dynamics – computability

k

k

∼ 1/∆

−13/3

E

k

−3

M

k

k

k

k

L

DNS

−5/3

NSa

NS-α,

Leray

are dispersive

Regularization alters spectrum –

controllable cross-over as

Confined and nonconfined decay

Decay exponent n and spectral exponent p

Nonconfined

Confined

Nonrotating

p

=

5/3

n

=

6/5

n

=

2

Rotating(I)

p

=

2

n

=

3/5

n

=

1

Rotating(II)

p

=

3

n

=

0

n

=

0

Rotating I – energy transfers time scale

Ω

−

1

LES: Algebraic decay at R

_λ

₌

92

DNS at 256

3

– LES at 64

3

:

0 20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1/Ro n

(a)

10 0 101 102 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1/Ro n

(b)

Decay exponent well captured:

DNS,

Leray

,

LANS-

α

LES: Algebraic decay at R

_λ

₌

200

LES at 128

3

:

0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 1/Ro n

(a)

10 0 101 102 0 0.5 1 1.5 2 1/Ro n

(b)

Leray - dashed

,

LANS-

α

– solid

Leray closer to ‘nonconfined’, LANS-α

closer to ‘confined’

Evidence Rotating II - Kraichnan

LES: Leray spectra at R

_λ

₌

200

LES at 128

3

:

totally inhibited energy transfer regime

100 101 102 −1 0 1 2 3 4 5x 10 −3 k/(2π) Π

(a)

10 0 101 102 1 1.5 2 2.5 3 1/Ro p

(b)

(a):

Average transport-power spectra

(b):

Energy-spectrum exponent showing transition

-5/3 (Ro

→ ∞) to -3 (Ro

→

0) – Kraichnan

Outline

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

_{Rotation induced compressibility effects}

Compressible rotating mixing layer

z

y

Ω

x

Consider range of Ro

Vorticity along axis of rotation:

ω

₂

₌

∂

₃

_u

₁

₋

∂

₁

_u

₃

Ro

=

∞

Ro

=

10

Ro

=

2

Kinetic energy decay

0 10 20 30 40 50 60 70 80 90 100 7.5 8 8.5 9 9.5 10x 10 4

(a)

00 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 8 9 10x 10 4

(b)

E

E

t

t

(a):

Ro

=

∞

(solid),

Ro

=

10 (dashed),

Ro

=

5 (dash-dotted)

(b):

Ro

=

1 (solid),

Ro

=

0.5 (dashed)

and

Ro

=

0

.

2

(dash-dotted)

Contributions to decay rate

Total decay:

dE

dt

=

D

−

W

;

D

=

h

p

∇ ·

ui

;

W

=

1

2Re

hS

:

Si

D: pressure/velocity divergence, W dissipation

incompressible:

D

=

0, W

≥

0

→

monotonous decay of E

compressible:

W

≥

0, D

6

=

0

→

non-monotonous decay

Effect of rotation:

no direct influence

Contributions to decay rate

Total decay:

dE

dt

=

D

−

W

;

D

=

h

p

∇ ·

ui

;

W

=

1

2Re

hS

:

Si

D: pressure/velocity divergence, W dissipation

incompressible:

D

=

0, W

≥

0

→

monotonous decay of E

compressible:

W

≥

0, D

6

=

0

→

non-monotonous decay

Effect of rotation:

no direct influence

Contributions to decay rate

Total decay:

dE

dt

=

D

−

W

;

D

=

h

p

∇ ·

ui

;

W

=

1

2Re

hS

:

Si

D: pressure/velocity divergence, W dissipation

incompressible:

D

=

0, W

≥

0

→

monotonous decay of E

compressible:

W

≥

0, D

6

=

0

→

non-monotonous decay

Effect of rotation:

no direct influence

Slow rotation – incompressible limit

0 10 20 30 40 50 60 70 80 90 100 −300 −250 −200 −150 −100 −50 0 50

D

W

t

Dissipation and pressure/velocity divergence:

Ro

=

∞,

Ro

=

10,

Ro

=

5

,

Ro

=

2

Rapid rotation – compressibility

0 10 20 30 40 50 60 70 80 90 100 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5x 10 4

(a)

−25000 10 20 30 40 50 60 70 80 90 100 −2000 −1500 −1000 −500 0

(b)

D

W

W

t

t

Dissipation W and pressure/velocity divergence D:

Ro

=

1,

Ro

=

0.5,

Ro

=

0

.

2

,

Ro

=

0

.

1

dominance D over W as Ro

↓

non-monotonous decay

Outline

1

Rotation modulates turbulence

2

_{Rotation of incompressible fluid}

3

_{Rotation at high Reynolds numbers}

4

Rotation induced compressibility effects

Concluding remarks

Established general correspondence DNS and

experiments MMR

Illustrated relevance helical wave decomposition

Accurate predictions – Leray and LANS-α

at R

λ

=

92

Leray at R

λ

=

200 – exponent spectrum from -5/3 to -3 as

Ro

↓: 3D - 2D

Compressibility effects induced by Coriolis force:

non-uniform decay, indirect rotation effect