Pierre Ricco1, Claudio Ottonelli, Yosuke Hasegawa ... · NEAR-WALL ENSTROPHY GENERATION IN A...

NEAR-WALL ENSTROPHY GENERATION IN A DRAG-REDUCED

TURBULENT CHANNEL FLOW WITH SPANWISE WALL

OSCILLATIONS

Pierre Ricco 1, Claudio Ottonelli,Yosuke Hasegawa, Maurizio Quadrio

1 Sheffield, 2 Onera Paris,

3 Tokyo, 4 Politecnico Milano

ERCOFTAC/PLASMAERO Workshop, Toulouse, 10 December 2012

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 1-1

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE

Energy input into system

Pre-determined forcing

NUMERICAL APPROACH

Direct numerical simulations of wall turbulence

Fully-developed turbulent channel flow (Reτ = uτh/ν = 200)

Compact finite-difference scheme along wall-normal direction

Spectral discretization along streamwise and spanwise directions

SPANWISE WALL OSCILLATIONS

New approach: Turbulent enstrophy

Transient evolution

CONSTANT DP/DX

τw is fixed in fully-developed conditions

GAIN: Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 2-1

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE

Energy input into system

Pre-determined forcing

NUMERICAL APPROACH

Direct numerical simulations of wall turbulence

Fully-developed turbulent channel flow (Reτ = uτh/ν = 200)

Compact finite-difference scheme along wall-normal direction

Spectral discretization along streamwise and spanwise directions

SPANWISE WALL OSCILLATIONS

New approach: Turbulent enstrophy

Transient evolution

CONSTANT DP/DX

τw is fixed in fully-developed conditions

GAIN: Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 2-1

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE

Energy input into system

Pre-determined forcing

NUMERICAL APPROACH

Direct numerical simulations of wall turbulence

Fully-developed turbulent channel flow (Reτ = uτh/ν = 200)

Compact finite-difference scheme along wall-normal direction

Spectral discretization along streamwise and spanwise directions

SPANWISE WALL OSCILLATIONS

New approach: Turbulent enstrophy

Transient evolution

CONSTANT DP/DX

τw is fixed in fully-developed conditions

GAIN: Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 2-1

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE

Energy input into system

Pre-determined forcing

NUMERICAL APPROACH

Direct numerical simulations of wall turbulence

Fully-developed turbulent channel flow (Reτ = uτh/ν = 200)

Compact finite-difference scheme along wall-normal direction

Spectral discretization along streamwise and spanwise directions

SPANWISE WALL OSCILLATIONS

New approach: Turbulent enstrophy

Transient evolution

CONSTANT DP/DX

τw is fixed in fully-developed conditions

GAIN: Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 2-1

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE

Energy input into system

Pre-determined forcing

NUMERICAL APPROACH

Direct numerical simulations of wall turbulence

Fully-developed turbulent channel flow (Reτ = uτh/ν = 200)

Compact finite-difference scheme along wall-normal direction

Spectral discretization along streamwise and spanwise directions

SPANWISE WALL OSCILLATIONS

New approach: Turbulent enstrophy

Transient evolution

CONSTANT DP/DX

τw is fixed in fully-developed conditions

GAIN: Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 2-1

SPANWISE WALL OSCILLATIONSGEOMETRY

Mean flow

x

y

zLx

Ly

LzWw = A sin(

2πT

t)

R =

Cf ,r−Cf ,oCf ,r

=

U2b,o−U2

b,r

U2b,o

Why does the skin-friction coefficent decrease?

Cf = 2τw/(ρU2b ) decreases→ study why Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 3-1

SPANWISE WALL OSCILLATIONSGEOMETRY

Mean flow

x

y

zLx

Ly

LzWw = A sin(

2πT

t)

R =

Cf ,r−Cf ,oCf ,r

=

U2b,o−U2

b,r

U2b,o

Why does the skin-friction coefficent decrease?

Cf = 2τw/(ρU2b ) decreases→ study why Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 3-1

SPANWISE WALL OSCILLATIONSGEOMETRY

Mean flow

x

y

zLx

Ly

LzWw = A sin(

2πT

t)

R =

Cf ,r−Cf ,oCf ,r

=

U2b,o−U2

b,r

U2b,o

Why does the skin-friction coefficent decrease?

Cf = 2τw/(ρU2b ) decreases→ study why Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 3-1

SPANWISE WALL OSCILLATIONSGEOMETRY

Mean flow

x

y

zLx

Ly

LzWw = A sin(

2πT

t)

R =

Cf ,r−Cf ,oCf ,r

=

U2b,o−U2

b,r

U2b,o

Why does the skin-friction coefficent decrease?

Cf = 2τw/(ρU2b ) decreases→ study why Ub increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 3-1

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

Lx Lz

∫ Lx

0

∫ Lz

0

f (x , y , z, t)dzdx

PHASE

f (y , τ) =1

N

N−1∑

n=0

f (y , nT + τ)

TIME

〈f 〉 (y) =1

T

∫ T

0

f (y , τ)dτ

GLOBAL

[f ]g =

∫ h

0

〈f 〉 (y)dy

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 4-1

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

Lx Lz

∫ Lx

0

∫ Lz

0

f (x , y , z, t)dzdx

PHASE

f (y , τ) =1

N

N−1∑

n=0

f (y , nT + τ)

TIME

〈f 〉 (y) =1

T

∫ T

0

f (y , τ)dτ

GLOBAL

[f ]g =

∫ h

0

〈f 〉 (y)dy

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 4-1

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

Lx Lz

∫ Lx

0

∫ Lz

0

f (x , y , z, t)dzdx

PHASE

f (y , τ) =1

N

N−1∑

n=0

f (y , nT + τ)

TIME

〈f 〉 (y) =1

T

∫ T

0

f (y , τ)dτ

GLOBAL

[f ]g =

∫ h

0

〈f 〉 (y)dy

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 4-1

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

Lx Lz

∫ Lx

0

∫ Lz

0

f (x , y , z, t)dzdx

PHASE

f (y , τ) =1

N

N−1∑

n=0

f (y , nT + τ)

TIME

〈f 〉 (y) =1

T

∫ T

0

f (y , τ)dτ

GLOBAL

[f ]g =

∫ h

0

〈f 〉 (y)dy

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 4-1

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

Lx Lz

∫ Lx

0

∫ Lz

0

f (x , y , z, t)dzdx

PHASE

f (y , τ) =1

N

N−1∑

n=0

f (y , nT + τ)

TIME

〈f 〉 (y) =1

T

∫ T

0

f (y , τ)dτ

GLOBAL

[f ]g =

∫ h

0

〈f 〉 (y)dy

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 4-1

MEAN FLOW

0 25 50 75 100 125 150 1750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 101 1020

5

10

15

20

25

y

⟨ U⟩

R

T

Scaling by viscous units

Mean velocity increases in the bulk of the channel

Mean wall-shear stress is unchanged

Optimum period of oscillation T≈75

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 5-1

MEAN FLOW

0 25 50 75 100 125 150 1750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 101 1020

5

10

15

20

25

y

⟨ U⟩

R

T

Scaling by viscous units

Mean velocity increases in the bulk of the channel

Mean wall-shear stress is unchanged

Optimum period of oscillation T≈75

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 5-1

MEAN FLOW

0 25 50 75 100 125 150 1750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 101 1020

5

10

15

20

25

y

⟨ U⟩

R

T

Scaling by viscous units

Mean velocity increases in the bulk of the channel

Mean wall-shear stress is unchanged

Optimum period of oscillation T≈75

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 5-1

MEAN FLOW

0 25 50 75 100 125 150 1750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 101 1020

5

10

15

20

25

y

⟨ U⟩

R

T

Scaling by viscous units

Mean velocity increases in the bulk of the channel

Mean wall-shear stress is unchanged

Optimum period of oscillation T≈75

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 5-1

MEAN FLOW

0 25 50 75 100 125 150 1750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 101 1020

5

10

15

20

25

y

⟨ U⟩

R

T

Scaling by viscous units

Mean velocity increases in the bulk of the channel

Mean wall-shear stress is unchanged

Optimum period of oscillation T≈75

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 5-1

TURBULENCE STATISTICS

100 101 102-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

100 101 102-1

0

1

2

3

4

5

6

7

y

⟨ u2

⟩ ,⟨v

2

⟩ ,⟨w

2

⟩ ,⟨ uv⟩

y

vw

φ=0

φ=π4

φ=π2

φ= 3π4

Turbulence kinetic energy decreases

Streamwise velocity fluctuations are attenuated the most

New oscillatory Reynolds stress term vw is created,⟨vw⟩

= 0

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 6-1

TURBULENCE STATISTICS

100 101 102-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

100 101 102-1

0

1

2

3

4

5

6

7

y

⟨ u2

⟩ ,⟨v

2

⟩ ,⟨w

2

⟩ ,⟨ uv⟩

y

vw

φ=0

φ=π4

φ=π2

φ= 3π4

Turbulence kinetic energy decreases

Streamwise velocity fluctuations are attenuated the most

New oscillatory Reynolds stress term vw is created,⟨vw⟩

= 0

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 6-1

TURBULENCE STATISTICS

100 101 102-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

100 101 102-1

0

1

2

3

4

5

6

7

y

⟨ u2

⟩ ,⟨v

2

⟩ ,⟨w

2

⟩ ,⟨ uv⟩

y

vw

φ=0

φ=π4

φ=π2

φ= 3π4

Turbulence kinetic energy decreases

Streamwise velocity fluctuations are attenuated the most

New oscillatory Reynolds stress term vw is created,⟨vw⟩

= 0

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 6-1

TURBULENCE STATISTICS

100 101 102-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

100 101 102-1

0

1

2

3

4

5

6

7

y

⟨ u2

⟩ ,⟨v

2

⟩ ,⟨w

2

⟩ ,⟨ uv⟩

y

vw

φ=0

φ=π4

φ=π2

φ= 3π4

Turbulence kinetic energy decreases

Streamwise velocity fluctuations are attenuated the most

New oscillatory Reynolds stress term vw is created,⟨vw⟩

= 0

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 6-1

ENERGY BALANCE: A SCHEMATIC

DU

DW

DT

Puv

Pvw

Ubτw

Ew

MKE-x

MKE-z

TKE

+3.5 15.9

+13.2

+12.9

+2.79.4

+0.86.5

+0.3

+1.1 6.5

Energy is fed through Px (→ Ubτw ) and wall motion (→ Ew )Energy is dissipated through:

Mean-flow viscous effects (streamwise→ DU , spanwise→ DW )Turbulent viscous effects (→ DT )

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 7-1

ENERGY BALANCE: A SCHEMATIC

DU

DW

DT

Puv

Pvw

Ubτw

Ew

MKE-x

MKE-z

TKE

+3.5 15.9

+13.2

+12.9

+2.79.4

+0.86.5

+0.3

+1.1 6.5

Energy is fed through Px (→ Ubτw ) and wall motion (→ Ew )Energy is dissipated through:

Mean-flow viscous effects (streamwise→ DU , spanwise→ DW )Turbulent viscous effects (→ DT )

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 7-1

ENERGY BALANCE: A SCHEMATIC

DU

DW

DT

Puv

Pvw

Ubτw

Ew

MKE-x

MKE-z

TKE

+3.5 15.9

+13.2

+12.9

+2.79.4

+0.86.5

+0.3

+1.1 6.5

Energy is fed through Px (→ Ubτw ) and wall motion (→ Ew )Energy is dissipated through:

Mean-flow viscous effects (streamwise→ DU , spanwise→ DW )Turbulent viscous effects (→ DT )

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 7-1

ENERGY BALANCE: A SCHEMATIC

DU

DW

DT

Puv

Pvw

Ubτw

Ew

MKE-x

MKE-z

TKE

+3.5 15.9

+13.2

+12.9

+2.79.4

+0.86.5

+0.3

+1.1 6.5

Energy is fed through Px (→ Ubτw ) and wall motion (→ Ew )Energy is dissipated through:

Mean-flow viscous effects (streamwise→ DU , spanwise→ DW )Turbulent viscous effects (→ DT )

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 7-1

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

⟨A∂W

∂y

∣∣∣∣y=0

⟩

︸ ︷︷ ︸Ew

= −

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

−

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[(∂U

∂y

)2]

g︸ ︷︷ ︸DU

+

[(∂W

∂y

)2]

g︸ ︷︷ ︸DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

+

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[∂ui

∂xj

∂ui

∂xj

]

g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU +DW +DT

TURBULENT DISSIPATION

DT =[ωiωi

]g

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 8-1

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

⟨A∂W

∂y

∣∣∣∣y=0

⟩

︸ ︷︷ ︸Ew

= −

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

−

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[(∂U

∂y

)2]

g︸ ︷︷ ︸DU

+

[(∂W

∂y

)2]

g︸ ︷︷ ︸DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

+

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[∂ui

∂xj

∂ui

∂xj

]

g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU +DW +DT

TURBULENT DISSIPATION

DT =[ωiωi

]g

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 8-1

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

⟨A∂W

∂y

∣∣∣∣y=0

⟩

︸ ︷︷ ︸Ew

= −

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

−

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[(∂U

∂y

)2]

g︸ ︷︷ ︸DU

+

[(∂W

∂y

)2]

g︸ ︷︷ ︸DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

+

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[∂ui

∂xj

∂ui

∂xj

]

g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU +DW +DT

TURBULENT DISSIPATION

DT =[ωiωi

]g

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 8-1

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

⟨A∂W

∂y

∣∣∣∣y=0

⟩

︸ ︷︷ ︸Ew

= −

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

−

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[(∂U

∂y

)2]

g︸ ︷︷ ︸DU

+

[(∂W

∂y

)2]

g︸ ︷︷ ︸DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

[uv∂U

∂y

]

g︸ ︷︷ ︸Puv

+

[vw∂W

∂y

]

g︸ ︷︷ ︸Pvw

+

[∂ui

∂xj

∂ui

∂xj

]

g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU +DW +DT

TURBULENT DISSIPATION

DT =[ωiωi

]g

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 8-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease?

Why does Ub increase?

THREE POSSIBILITIES

1 Stokes layer acts on DU directly

→ excluded because W does not work directly on(∂U/∂y

)2

2 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

]g

directly

→W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi

The term enstrophy was coined by G. Nickel and is from Greek στρoφη, which means turn

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 9-1

TURBULENT ENSTROPHY EQUATION

1

2

∂ωiωi

∂τ︸ ︷︷ ︸1

= ωxωy∂U

∂y︸ ︷︷ ︸2

+ ωzωy∂W

∂y︸ ︷︷ ︸

3

+ωj∂u

∂xj

∂W

∂y︸ ︷︷ ︸

4

−ωj∂w

∂xj

∂U

∂y︸ ︷︷ ︸5

− vωx∂2W

∂y2

︸ ︷︷ ︸6

+ vωz∂2U

∂y2︸ ︷︷ ︸7

+ωiωj∂ui

∂xj︸ ︷︷ ︸8

−1

2

∂

∂y

(vωiωi

)︸ ︷︷ ︸

9

+1

2

∂2ωiωi

∂y2︸ ︷︷ ︸10

−∂ωi

∂xj

∂ωi

∂xj︸ ︷︷ ︸11

.

Stokes layer influences dynamics of turbulent enstrophyThree terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 10-1

TURBULENT ENSTROPHY EQUATION

1

2

∂ωiωi

∂τ︸ ︷︷ ︸1

= ωxωy∂U

∂y︸ ︷︷ ︸2

+ ωzωy∂W

∂y︸ ︷︷ ︸

3

+ωj∂u

∂xj

∂W

∂y︸ ︷︷ ︸

4

−ωj∂w

∂xj

∂U

∂y︸ ︷︷ ︸5

− vωx∂2W

∂y2

︸ ︷︷ ︸6

+ vωz∂2U

∂y2︸ ︷︷ ︸7

+ωiωj∂ui

∂xj︸ ︷︷ ︸8

−1

2

∂

∂y

(vωiωi

)︸ ︷︷ ︸

9

+1

2

∂2ωiωi

∂y2︸ ︷︷ ︸10

−∂ωi

∂xj

∂ωi

∂xj︸ ︷︷ ︸11

.

Stokes layer influences dynamics of turbulent enstrophyThree terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 10-1

TURBULENT ENSTROPHY EQUATION

1

2

∂ωiωi

∂τ︸ ︷︷ ︸1

= ωxωy∂U

∂y︸ ︷︷ ︸2

+ ωzωy∂W

∂y︸ ︷︷ ︸

3

+ωj∂u

∂xj

∂W

∂y︸ ︷︷ ︸

4

−ωj∂w

∂xj

∂U

∂y︸ ︷︷ ︸5

− vωx∂2W

∂y2

︸ ︷︷ ︸6

+ vωz∂2U

∂y2︸ ︷︷ ︸7

+ωiωj∂ui

∂xj︸ ︷︷ ︸8

−1

2

∂

∂y

(vωiωi

)︸ ︷︷ ︸

9

+1

2

∂2ωiωi

∂y2︸ ︷︷ ︸10

−∂ωi

∂xj

∂ωi

∂xj︸ ︷︷ ︸11

.

Stokes layer influences dynamics of turbulent enstrophyThree terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 10-1

TURBULENT ENSTROPHY EQUATION

1

2

∂ωiωi

∂τ︸ ︷︷ ︸1

= ωxωy∂U

∂y︸ ︷︷ ︸2

+ ωzωy∂W

∂y︸ ︷︷ ︸

3

+ωj∂u

∂xj

∂W

∂y︸ ︷︷ ︸

4

−ωj∂w

∂xj

∂U

∂y︸ ︷︷ ︸5

− vωx∂2W

∂y2

︸ ︷︷ ︸6

+ vωz∂2U

∂y2︸ ︷︷ ︸7

+ωiωj∂ui

∂xj︸ ︷︷ ︸8

−1

2

∂

∂y

(vωiωi

)︸ ︷︷ ︸

9

+1

2

∂2ωiωi

∂y2︸ ︷︷ ︸10

−∂ωi

∂xj

∂ωi

∂xj︸ ︷︷ ︸11

.

Stokes layer influences dynamics of turbulent enstrophyThree terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 10-1

TURBULENT ENSTROPHY EQUATION

1

2

∂ωiωi

∂τ︸ ︷︷ ︸1

= ωxωy∂U

∂y︸ ︷︷ ︸2

+ ωzωy∂W

∂y︸ ︷︷ ︸

3

+ωj∂u

∂xj

∂W

∂y︸ ︷︷ ︸

4

−ωj∂w

∂xj

∂U

∂y︸ ︷︷ ︸5

− vωx∂2W

∂y2

︸ ︷︷ ︸6

+ vωz∂2U

∂y2︸ ︷︷ ︸7

+ωiωj∂ui

∂xj︸ ︷︷ ︸8

−1

2

∂

∂y

(vωiωi

)︸ ︷︷ ︸

9

+1

2

∂2ωiωi

∂y2︸ ︷︷ ︸10

−∂ωi

∂xj

∂ωi

∂xj︸ ︷︷ ︸11

.

Stokes layer influences dynamics of turbulent enstrophyThree terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 10-1

TURBULENT ENSTROPHY PROFILESFIXED WALL

100 101 102-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

y

25

7

8

9

10

11

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 11-1

TURBULENT ENSTROPHY PROFILESOSCILLATING-WALL PROFILES

100 101 102-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

y

23

4

5

6

7

8

9

10

11

Term 3, ωzωy∂W/∂y → turbulent enstrophy production is dominant

Other oscillating-wall terms are smaller

Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 12-1

TURBULENT ENSTROPHY PROFILESOSCILLATING-WALL PROFILES

100 101 102-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

y

23

4

5

6

7

8

9

10

11

Term 3, ωzωy∂W/∂y → turbulent enstrophy production is dominant

Other oscillating-wall terms are smaller

Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 12-1

TURBULENT ENSTROPHY PROFILESOSCILLATING-WALL PROFILES

100 101 102-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

y

23

4

5

6

7

8

9

10

11

Term 3, ωzωy∂W/∂y → turbulent enstrophy production is dominant

Other oscillating-wall terms are smaller

Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 12-1

TURBULENT ENSTROPHY PROFILESOSCILLATING-WALL PROFILES

100 101 102-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

y

23

4

5

6

7

8

9

10

11

Term 3, ωzωy∂W/∂y → turbulent enstrophy production is dominant

Other oscillating-wall terms are smaller

Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 12-1

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet

Key: transient from start-up of wall motion

0 100 200 300 400 5000

2.5

5

7.5

10

12.5

15

t

USEFUL INFORMATION

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 13-1

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet

Key: transient from start-up of wall motion

0 100 200 300 400 5000

2.5

5

7.5

10

12.5

15

t

Term 3

USEFUL INFORMATION

RED: term 3 increases abruptly , then decreases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 14-1

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet

Key: transient from start-up of wall motion

0 100 200 300 400 5000

2.5

5

7.5

10

12.5

15

t

Term 3

Enstrophy

USEFUL INFORMATION

RED: term 3 increases abruptly , then decreases

BLACK: turbulent enstrophy increases , then decreases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 15-1

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet

Key: transient from start-up of wall motion

0 100 200 300 400 5000

2.5

5

7.5

10

12.5

15

t

Term 3

Enstrophy

USEFUL INFORMATION

RED: term 3 increases abruptly , then decreases

BLACK: turbulent enstrophy increases , then decreases

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 16-1

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet

Key: transient from start-up of wall motion

0 100 200 300 400 5000

2.5

5

7.5

10

12.5

15

t

Term 3

Enstrophy

TKE

USEFUL INFORMATION

RED: term 3 increases abruptly , then decreases

BLACK: turbulent enstrophy increases , then decreases

BLUE: TKE decreases monotonically

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 17-1

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction

→ drag reduction

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 18-1

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction

→ drag reduction

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 18-1

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction

→ drag reduction

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 18-1

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction

→ drag reduction

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 18-1

DRAG REDUCTION MECHANISM

Initial state

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 19-1

DRAG REDUCTION MECHANISM

Shortt < 50

Initial state

ωzωy∂W∂y ↑ ωiωi ↑

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 20-1

DRAG REDUCTION MECHANISM

Shortt < 50

Initial state

ωzωy∂W∂y ↑ ωiωi ↑

DT ↑

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 21-1

DRAG REDUCTION MECHANISM

Shortt < 50

Intermediate50 < t < 400

Initial state

∂uv∂y ↓ωzωy

∂W∂y ↑ ωiωi ↑

DT ↑

TKE ↓

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 22-1

DRAG REDUCTION MECHANISM

Shortt < 50

Intermediate50 < t < 400

Initial state

∂U

∂t> 0

∂uv∂y ↓ωzωy

∂W∂y ↑ ωiωi ↑

DT ↑

TKE ↓

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 23-1

DRAG REDUCTION MECHANISM

Shortt < 50

Intermediate50 < t < 400

Longt > 400

Initial state ‘Drag reduction’

∫ h

0Udy ↑

∂U

∂t> 0

∂uv∂y ↓ωzωy

∂W∂y ↑ ωiωi ↑

DT ↑

TKE ↓

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 24-1

PHYSICAL INTERPRETATION OF ωzωy∂W/∂y

ωzωy∂W/∂y is key term leading to drag reduction

ωzωy∂W/∂y → ∂W/∂y acts on ωzωy

ωzωy ≈∂u∂y∂u∂z

∂u∂y → upward eruption of near-wall low-speed fluid∂u∂z → lateral flanks of the low-speed streaks

0 200 400 600 8000

200

400

x

z

∂u∂y∂u∂z located at the sides of high-speed streaks

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 25-1

MODELLING TURBULENT ENSTROPHY PRODUCTION

y

z

xn

xs

α

ωyz

SIMPLIFIED TURBULENT ENSTROPHY EQUATION

1

2

∂

∂t

(ω

2y + ω

2z

)= ωzωy G −

(∂ωy

∂y

)2

−

(∂ωz

∂y

)2

Rotation of axis

1

2

∂ω2n

∂t= Snnω

2n −

(∂ωn

∂y

)2

Integration by Charpit’s method

ωn = ωn,0 esinα cosαGt︸ ︷︷ ︸stretching

e−β2 t e−βy︸ ︷︷ ︸

dissipation

, β =∂ωn/∂t

∂ωn/∂y∼λy

λt

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 26-1

OSCILLATION PERIOD VS. TERM 3

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

[ωzωy∂W/∂y

]g

R

T=8

T=21

T=42 T=100

Drag reduction grows monotonically with global production term

This happens up to optimum period

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 27-1

THANK YOU!

REFERENCE

Ricco, P. Ottonelli, C. Hasegawa, Y. Quadrio, M.Changes in turbulent dissipation in a channel flow with oscillating walls

J. Fluid Mech., 700, 77-104, 2012.

5 DECEMBER 2012 WALL-OSCILLATION DRAG-REDUCTION PROBLEM 28-1