Skin-friction drag reduction via steady streamwise ... · Skin-friction drag reduction via steady...

Skin-friction drag reductionvia steady streamwise oscillations

of spanwise velocity

Claudio Viotti1, Maurizio Quadrio1 & Paolo Luchini2

1Dip. di Ingegneria Aerospaziale, Politecnico di Milano (I)2Dip. di Ingegneria Meccanica, Universita di Salerno (I)

11th European Turbulence Conference 2007

25-28 June 2007

Porto, Portugal

Viotti-Quadrio-Luchini () Longitudinal standing waves ETC11 1 / 22

Skin friction drag-reductionSpanwise-based methods

Obivious requirements

Simple ⇒ feasible

Energy efficient ⇒ net saving

Additional requirement

Steady ⇒ convertible into passive device ??


Temporal-oscillating wallJung, Mangiavacchi & Akhavan (PoF, 1992)

Ww = A sin

(

2π

Tt

)

zy

x

Flow

Stokes layer interacts with wallstructures

DR up to 40%

optimal period T+opt ≈ 100− 125

net energy saving up to 7%

unsteady!!


Spanwise forcingFurther studies and variants

Ww Temporal-oscillating wall

A sin( 2πT

t) Jung et al. (1992), Laadhari et al. (1994),J. I. Choi et al. (2002), Quadrio & Ricco (2003, 2004)

A sin( 2πλ z − 2π

Tt) Zhao et al. (2004)

Fz Spanwise body force

Ae−y/∆ sin( 2πT

t) Du et al. (2002), Berger et al. (2000),Breuer et al. (2004)

Ae−y/∆ sin( 2πλ z) Du et al. (2002)

Ae−y/∆ sin( 2πλ z − 2π

Tt) Du et al. (2002), Zhao et al. (2004),

Pang & K.S. Choi (2004)

Ae−y/∆ sin( 2πλ x) Berger et al. (2000)


Time vs. space

Question

Can a temporal oscillation be converted into a spatial oscillation?

Possible answer

Yes, by exploiting the convective nature of the flow


Convection velocity in turbulent wall flowsQuadrio & Luchini, Pof 2003

Channel flow,DNS at Reτ = 180

convection velocity ofturbulent fluctuations

convection is not zero atthe wall!

near wall: U+c ≈ 10

0

5

10

15

20

1 10 100

mean velocity

convection velocity

U+ c,〈

U+〉

y+


Our control law: streamwise steady waves

Ww = A sin

(

2π

λx

x

)

zy

x

Flow

TWO PARAMETERS

Amplitude A

Wavelength λx


Numerical study with DNS

Turbulent channel flow at Reτ = 200

Domain size: Lx = 6πh, Ly = 2h, Lz = 3πh

Spatial resolution: Nx × Ny × Nz = 320 × 160 × 320

Averaging time: 104 viscous time units

Number of simulations: 35


Computing tool

DNS pseudo-spectral code

Parallel computing oncommodity hardware (Luchini &Quadrio JCP-2006)

The “Personal Supercomputer”:Powerful dedicated system with128 Opteron CPUs, 100GBRAM, 4TB disk


Spanwise flow (1)The laminar case

Contour of spanwise velocity W

Temporal oscillation

0 0.2 0.4 0.6 0.8 1

0

0.25

y/h

t/T0.40.30.20.1 0.1

-0.1-0.1 -0.2-0.3-0.4

Spatial oscillation

0 0.2 0.4 0.6 0.8 1

0

0.25

y/h

x/λ0.40.30.20.1

-0.1 -0.2-0.3-0.4


Spanwise flow (2)Laminar solution vs. turbulent DNS data

δT = ν1/2T 1/2 δX = ν1/3λ1/3x

dUdy

−1/3details

0

15

30

-1 0 1 0

2.5

5

-1 0 1

Laminar solution (analytical)

0

2.5

5

-1 0 1 0

2.5

5

-1 0 1

DNS W /AW /AW /AW /A

y/δ T

y/δ X

y/δ X

y/δ X

Spanwise-averaged turbulent flow cohincides with analytical solution inboth cases


A natural question

TEMPORALOSCILLATION

T+opt ≈ 100 − 125

U+c = 10

=⇒

SPATIALOSCILLATION

λ+opt ≈ 1000 − 1250 ??


A natural question

TEMPORALOSCILLATION

T+opt ≈ 100 − 125

U+c = 10

=⇒

SPATIALOSCILLATION

λ+opt ≈ 1000 − 1250 ??


Drag reduction (1)

0

10

20

30

40

50

0 1000 2000 3000 4000 5000

spatialA+=12

A+=6A+=2

0

10

20

30

40

50

0 1000 2000 3000 4000 5000

temporalA+=12

0

10

20

30

40

50

0 1000 2000 3000 4000 5000

temporalA+=12

0

10

20

30

40

50

0 1000 2000 3000 4000 5000

λ+,U+c T+λ+,U+c T+λ+,U+c T+λ+,U+c T+

%DR%DR%DR%DR


Drag reduction (2)

0

10

20

30

40

50

60

0 5 10 15 20 25

λ+=300λ+=1250

0

10

20

30

40

50

60

0 5 10 15 20 25

T+=30T+=125

0

10

20

30

40

50

60

0 5 10 15 20 25

T+=30T+=125

0

10

20

30

40

50

60

0 5 10 15 20 25

A+A+A+A+

%DR%DR%DR%DR


Power budget: definitions

Power required to drive the flow:

Pdrive = µ〈∂U

∂y

Qx

h〉

Power required to control the flow:

Pcontr = µ〈∂W

∂yW 〉

Net power saved after DR:

Pnet = ∆Pdrive + Pcontr


Power budget (1)

-100

-80

-60

-40

-20

0

0 1000 2000 3000

A+=12

A+=6

A+=12-100

-80

-60

-40

-20

0

0 1000 2000 3000-100

-80

-60

-40

-20

0

0 1000 2000 3000

λ+,U+c T+λ+,U+c T+λ+,U+c T+

%Pcontr%Pcontr%Pcontr

αλ+x

−1/3

βT+−1/2


Power budget (2)

-60

-30

0

30

0 1000 2000 3000

A+=12

A+=6

A+=12-60

-30

0

30

0 1000 2000 3000

-60

-30

0

30

0 1000 2000 3000

λ+,U+c T+λ+,U+c T+λ+,U+c T+

%Pnet%Pnet%Pnet


Isosurfaces of streamwise velocity: u+ = −4

Uncontrolled

0

1

2

3

0 50 100 150 200

y+

u+ rm

s


Isosurfaces of streamwise velocity: u+ = −4

Controlled, DR 45%, actual wall units

0

1

2

3

0 50 100 150 200

y+

u+ rm

s


Mean velocity profiles

0

5

10

15

20

25

1 10 100

uncontrolledtemporal

spatial

y+

〈U+〉


Conclusions

1 Uc translates temporal into spatial forcing2 Spatial forcing is more efficient

Higher DR (up to 52%)Higher net saving (up to 23%)

3 Turbulent spanwise mean flow is based on laminar dynamics

Passive device?


Spanwise flow (3)Spatial oscillation - analytical results

Governing equation for z-component of velocity

U(y)∂W

∂x= ν

(

∂2W

∂x2+

∂2W

∂y2

)

Hypothesis (δX thickness of the “spatial Stokes layer”)

δX ≪ h, λ = O[h] ⇒∂2W

∂x2≪

∂2W

∂y2, U(y) ∼

dU

dy

∣

∣

∣

∣

w

y

Simplified governing equation:

dU

dy

∣

∣

∣

∣

w

y∂W

∂x= ν

∂2W

∂y2


Spanwise flow (4)Spatial oscillation - analytical results

It is possible to look for a solution in the form

W (x , y) = ℜ[

e i2π/λxF (y)]

obtaining a complex Airy equation for F (y)

idU

dy

∣

∣

∣

∣

w

yF (y) = νλd2F (y)

dy2

solution=⇒ F (y) = CA

e−i2π/3iy

ν1/3λ1/3 dUdy

∣

∣

−1/3

w

δX = ν1/3λ1/3x

dUdy

∣

∣

−1/3

w


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