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

NUMERICAL APPROACH

Transient evolution

CONSTANT DP/DX

GAIN: Ub increases

NUMERICAL APPROACH

Transient evolution

CONSTANT DP/DX

GAIN: Ub increases

NUMERICAL APPROACH

Transient evolution

CONSTANT DP/DX

GAIN: Ub increases

NUMERICAL APPROACH

Transient evolution

CONSTANT DP/DX

GAIN: Ub increases

SPANWISE WALL OSCILLATIONSGEOMETRY

Mean flow

LzWw = A sin(

Cf ,r−Cf ,oCf ,r

U2b,o−U2

Why does the skin-friction coefficent decrease?

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

Mean flow

LzWw = A sin(

Cf ,r−Cf ,oCf ,r

U2b,o−U2

Mean flow

LzWw = A sin(

Cf ,r−Cf ,oCf ,r

U2b,o−U2

Mean flow

LzWw = A sin(

Cf ,r−Cf ,oCf ,r

U2b,o−U2

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f (y , t) =1

∫ Lx

∫ Lz

f (x , y , z, t)dzdx

f (y , τ) =1

N−1∑

f (y , nT + τ)

〈f 〉 (y) =1

f (y , τ)dτ

GLOBAL

[f ]g =

〈f 〉 (y)dy

AVERAGING OPERATORS

f (y , t) =1

∫ Lx

∫ Lz

f (y , τ) =1

N−1∑

f (y , nT + τ)

〈f 〉 (y) =1

f (y , τ)dτ

GLOBAL

[f ]g =

〈f 〉 (y)dy

AVERAGING OPERATORS

f (y , t) =1

∫ Lx

∫ Lz

f (y , τ) =1

N−1∑

f (y , nT + τ)

〈f 〉 (y) =1

f (y , τ)dτ

GLOBAL

[f ]g =

〈f 〉 (y)dy

AVERAGING OPERATORS

f (y , t) =1

∫ Lx

∫ Lz

f (y , τ) =1

N−1∑

f (y , nT + τ)

〈f 〉 (y) =1

f (y , τ)dτ

GLOBAL

[f ]g =

〈f 〉 (y)dy

AVERAGING OPERATORS

f (y , t) =1

∫ Lx

∫ Lz

f (y , τ) =1

N−1∑

f (y , nT + τ)

〈f 〉 (y) =1

f (y , τ)dτ

GLOBAL

[f ]g =

〈f 〉 (y)dy

MEAN FLOW

0 25 50 75 100 125 150 1750

100 101 1020

⟨ U⟩

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

MEAN FLOW

0 25 50 75 100 125 150 1750

100 101 1020

⟨ U⟩

MEAN FLOW

0 25 50 75 100 125 150 1750

100 101 1020

⟨ U⟩

MEAN FLOW

0 25 50 75 100 125 150 1750

100 101 1020

⟨ U⟩

MEAN FLOW

0 25 50 75 100 125 150 1750

100 101 1020

⟨ U⟩

TURBULENCE STATISTICS

100 101 102-0.2

100 101 102-1

⟨ u2

⟩ ,⟨v

⟩ ,⟨w

⟩ ,⟨ uv⟩

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⟩

100 101 102-0.2

100 101 102-1

⟨ u2

⟩ ,⟨v

⟩ ,⟨w

⟩ ,⟨ uv⟩

φ=π4

φ=π2

φ= 3π4

100 101 102-0.2

100 101 102-1

⟨ u2

⟩ ,⟨v

⟩ ,⟨w

⟩ ,⟨ uv⟩

φ=π4

φ=π2

φ= 3π4

100 101 102-0.2

100 101 102-1

⟨ u2

⟩ ,⟨v

⟩ ,⟨w

⟩ ,⟨ uv⟩

φ=π4

φ=π2

φ= 3π4

ENERGY BALANCE: A SCHEMATIC

+3.5 15.9

+2.79.4

+0.86.5

+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 )

+3.5 15.9

+2.79.4

+0.86.5

+1.1 6.5

+3.5 15.9

+2.79.4

+0.86.5

+1.1 6.5

+3.5 15.9

+2.79.4

+0.86.5

+1.1 6.5

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

⟨A∂W

∣∣∣∣y=0

︸︷︷︸Ew

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[(∂U

g︸︷︷︸DU

[(∂W

g︸︷︷︸DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[∂ui

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU +DW +DT

TURBULENT DISSIPATION

DT =[ωiωi

Ubτw +

⟨A∂W

∣∣∣∣y=0

︸︷︷︸Ew

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[(∂U

g︸︷︷︸DU

[(∂W

g︸︷︷︸DW

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[∂ui

DT =[ωiωi

Ubτw +

⟨A∂W

∣∣∣∣y=0

︸︷︷︸Ew

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[(∂U

g︸︷︷︸DU

[(∂W

g︸︷︷︸DW

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[∂ui

DT =[ωiωi

Ubτw +

⟨A∂W

∣∣∣∣y=0

︸︷︷︸Ew

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[(∂U

g︸︷︷︸DU

[(∂W

g︸︷︷︸DW

[uv∂U

g︸︷︷︸Puv

[vw∂W

g︸︷︷︸Pvw

[∂ui

DT =[ωiωi

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 Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3 Stokes layer acts on DT =[ωiωi

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

KEY QUESTIONS

THREE POSSIBILITIES

directly

KEY QUESTIONS

THREE POSSIBILITIES

directly

KEY QUESTIONS

THREE POSSIBILITIES

directly

KEY QUESTIONS

THREE POSSIBILITIES

directly

KEY QUESTIONS

THREE POSSIBILITIES

directly

KEY QUESTIONS

THREE POSSIBILITIES

directly

TURBULENT ENSTROPHY EQUATION

∂ωiωi

∂τ︸︷︷︸1

= ωxωy∂U

∂y︸︷︷︸2

+ ωzωy∂W

∂y︸︷︷︸

+ωj∂u

∂y︸︷︷︸

−ωj∂w

∂y︸︷︷︸5

− vωx∂2W

︸︷︷︸6

+ vωz∂2U

∂y2︸︷︷︸7

+ωiωj∂ui

∂xj︸︷︷︸8

(vωiωi

)︸︷︷︸

∂2ωiωi

∂y2︸︷︷︸10

−∂ωi

∂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

∂ωiωi

∂τ︸︷︷︸1

= ωxωy∂U

∂y︸︷︷︸2

+ ωzωy∂W

∂y︸︷︷︸

+ωj∂u

∂y︸︷︷︸

−ωj∂w

∂y︸︷︷︸5

− vωx∂2W

︸︷︷︸6

+ vωz∂2U

∂y2︸︷︷︸7

+ωiωj∂ui

∂xj︸︷︷︸8

(vωiωi

)︸︷︷︸

∂2ωiωi

∂y2︸︷︷︸10

−∂ωi

∂ωi

∂xj︸︷︷︸11

∂ωiωi

∂τ︸︷︷︸1

= ωxωy∂U

∂y︸︷︷︸2

+ ωzωy∂W

∂y︸︷︷︸

+ωj∂u

∂y︸︷︷︸

−ωj∂w

∂y︸︷︷︸5

− vωx∂2W

︸︷︷︸6

+ vωz∂2U

∂y2︸︷︷︸7

+ωiωj∂ui

∂xj︸︷︷︸8

(vωiωi

)︸︷︷︸

∂2ωiωi

∂y2︸︷︷︸10

−∂ωi

∂ωi

∂xj︸︷︷︸11

∂ωiωi

∂τ︸︷︷︸1

= ωxωy∂U

∂y︸︷︷︸2

+ ωzωy∂W

∂y︸︷︷︸

+ωj∂u

∂y︸︷︷︸

−ωj∂w

∂y︸︷︷︸5

− vωx∂2W

︸︷︷︸6

+ vωz∂2U

∂y2︸︷︷︸7

+ωiωj∂ui

∂xj︸︷︷︸8

(vωiωi

)︸︷︷︸

∂2ωiωi

∂y2︸︷︷︸10

−∂ωi

∂ωi

∂xj︸︷︷︸11

∂ωiωi

∂τ︸︷︷︸1

= ωxωy∂U

∂y︸︷︷︸2

+ ωzωy∂W

∂y︸︷︷︸

+ωj∂u

∂y︸︷︷︸

−ωj∂w

∂y︸︷︷︸5

− vωx∂2W

︸︷︷︸6

+ vωz∂2U

∂y2︸︷︷︸7

+ωiωj∂ui

∂xj︸︷︷︸8

(vωiωi

)︸︷︷︸

∂2ωiωi

∂y2︸︷︷︸10

−∂ωi

∂ωi

∂xj︸︷︷︸11

TURBULENT ENSTROPHY PROFILESFIXED WALL

100 101 102-0.025

-0.015

-0.005

TURBULENT ENSTROPHY PROFILESOSCILLATING-WALL PROFILES

100 101 102-0.025

-0.015

-0.005

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

Other oscillating-wall terms are smaller

Turbulent dissipation of turbulent enstrophy increases

100 101 102-0.025

-0.015

-0.005

100 101 102-0.025

-0.015

-0.005

100 101 102-0.025

-0.015

-0.005

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

USEFUL INFORMATION

INTERESTING, BUT...

0 100 200 300 400 5000

Term 3

USEFUL INFORMATION

RED: term 3 increases abruptly , then decreases

INTERESTING, BUT...

0 100 200 300 400 5000

Term 3

Enstrophy

USEFUL INFORMATION

BLACK: turbulent enstrophy increases , then decreases

INTERESTING, BUT...

0 100 200 300 400 5000

Term 3

Enstrophy

USEFUL INFORMATION

INTERESTING, BUT...

0 100 200 300 400 5000

Term 3

Enstrophy

USEFUL INFORMATION

BLUE: TKE decreases monotonically

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

SHORT STAGE

INTERMEDIATE STAGE

LONG STAGE

→ drag reduction

SHORT STAGE

INTERMEDIATE STAGE

LONG STAGE

→ drag reduction

SHORT STAGE

INTERMEDIATE STAGE

LONG STAGE

→ drag reduction

DRAG REDUCTION MECHANISM

Initial state

Shortt < 50

Initial state

ωzωy∂W∂y ↑ ωiωi ↑

Shortt < 50

Initial state

ωzωy∂W∂y ↑ ωiωi ↑

DT ↑

Shortt < 50

Intermediate50 < t < 400

Initial state

∂uv∂y ↓ωzωy

∂W∂y ↑ ωiωi ↑

DT ↑

TKE ↓

Shortt < 50

Initial state

∂t> 0

∂uv∂y ↓ωzωy

DT ↑

TKE ↓

Shortt < 50

Longt > 400

Initial state ‘Drag reduction’

0Udy ↑

∂t> 0

∂uv∂y ↓ωzωy

DT ↑

TKE ↓

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

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

MODELLING TURBULENT ENSTROPHY PRODUCTION

SIMPLIFIED TURBULENT ENSTROPHY EQUATION

2y + ω

)= ωzωy G −

(∂ωy

(∂ωz

Rotation of axis

∂ω2n

∂t= Snnω

2n −

(∂ωn

Integration by Charpit’s method

ωn = ωn,0 esinα cosαGt︸︷︷︸stretching

e−β2 t e−βy︸︷︷︸

dissipation

, β =∂ωn/∂t

∂ωn/∂y∼λy

OSCILLATION PERIOD VS. TERM 3

0 0.02 0.04 0.06 0.08 0.10

[ωzωy∂W/∂y

T=42 T=100

Drag reduction grows monotonically with global production term

This happens up to optimum period

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.

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

Documents

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