Turbulence Characteristics Near A Porous Surface - KIT fileTurbulence Characteristics Near A Porous...

Turbulence Characteristics Near

A Porous Surface

M. Uhlmann†

&J. Jimenez, G. Kawahara‡, A. Pinelli

U. Politecnica Madrid, Spain‡ Ehime University, Japan

† Now at:Potsdam Institute for Climate Impact Research

February, 2000

Near-Wall Turbulence: Regeneration & Control

Outline

1 The turbulence regeneration mechanism:

streak/vortex cycle

2 A passive control experiment:

flow over a porous wall

3 Simple active control:

forced wall-transpiration

Uhlmann, Jimenez, Kawahara & Pinelli 1


Objectives

Fundamental: understanding near-walldynamics

−→ regeneration of turbulence

Passive control: porous wall

−→ simple strategy for drag increase

−→ guideline for active control

−→ didactic purposes

Active control: forced transpiration

−→ MEMS-type strategy

−→ similar effect as porous wall

−→ simplifies statistical analysis

(phase relations)



Configuration / definitions

• focus on incompressible, plane channel

flow

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τw....................

Flow

y

xz

2h

• wall units uτ =

√

τwρ

Reτ =huτ

ν

u+ = u/uτ ω+ = ων

u2τ

x+ =xuτ

ν

• concentrate on near-wall zone: y+ ≤ 100

−→ majority of production

(autonomous!)


Near-Wall Turbulence: (1) Regeneration

Buffer-layer structures

definition: localized flow patterns which

have a high coherence over a significant

time

• high/low-speed streaks (u′)

length λ+x ≈ 1000

lateral spacing λ+z ≈ 100

• quasi-streamwise vortices (ω′x)

length λ+x ≈ 400



Mechanisms of turbulence regeneration

◦ interaction with outer flow structures

◦ wall cycle (’mirror’ vorticity)

• streak/vortex-cycle

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of mean shear

instability mechanism

deformation



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ωx....................................................................................................................

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.........................................................................................................................................................................................................................................................................

y

u′



Linear stability analysis

• base flow: U(y, z) = U(y) + Us(y) cos(γz)

mean profile

& streaks+U0

∆Us ys

• perturbations (normal modes):

sinusoidal

(spanwise bending).........................................................................................................................................................

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z

• eigensystem analysis of linearized

Navier-Stokes



Eigenfunction: streamwise vorticity

’typical’ parameters: λ+z = 100, ∆U+

s = 4

most unstable sinusoidal perturbation:

0

0.5

1

1.5

2

2.5

3

x/h

0 0.25 0.5z/h

0

0.5

1

1.5

2

y/h

Reτ = 180, λ+x = 540


Near-Wall Turbulence: (2) Porous Wall

Channel flow over a porous wall

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pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppp ppppppppppppppppppppppppppppppppppppppppp

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ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppp ppppppppppppppppppppppppppppppppppppppppp

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ppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp



pppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppp

ppppppppppppppppppppppppppp

......................

ppppppppppppppppppppppppppp

p + p′

ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp

ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppp

2h

Lx Lz

U

p(x)

u = v = w = 0

u = w = 0

Darcyv′

w = −β p′w

• standard method (Kim et al. 1987)

Fourier in (x, z), but B-splines O(h6) in y

• box size/spatial resolution

L+x ≈ 1700, Nx = 256 → ∆x+ ≈ 9.5

L+z ≈ 700, Nz = 192 → ∆z+ ≈ 5.

Ny = 192 → ∆y+ = 0.2 . . . 2.2 (tanh)

• Reynolds Re = 3250 (nominal−→ Reτ ≈ 180)



Permeable wall boundary

~v = −β∇pporous medium

(Darcy)

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• allow for transverse motion only

• pressure drop: fluctuations in channel

v(y=0) = −β p′(y=0)

⇒ no net mass flux

• possible realization:

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

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ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppp

..........................

p(x) + p′

..........................

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp





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ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppp..........................

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...........................

............................................................................................................................................................................................................................................................................................................................................................................

y

x

p(x)

perforated plate, plenum chamber, spanwise compartments



Effect on skin friction

fully developed

cf

Re

2

9

10

11

12

13

14

15

16

17

0 20 40 60 80 100 120 140 160

40%

porous wall

t U0/h

∆cf [%]

0

20

40

60

0.06 0.1 0.14

(Wagner & Friedrich 98)

rms wall-transpiration/uτ



Mean velocity

U

Umax

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

porous wall

y/h

U+

0

5

10

15

20

25

1 10 100 1000

porous

2.5 log(y+) + 5.9

y+



Vorticity fluctuation intensity

ω′+x

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0 5 10 15 20 25 30 35 40 45 50

porous

y+

ω′+y

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40 45 50

porous

y+



Point vortex model for streak formation

potential

flow

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• • • • •

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp

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ppppppppppppppppppppp xppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp

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..................

.................. ..................

H

λ

Γ

z

• linearize for small porosity β � 1

v′ =√

A(λ) · (y/H)2 +B(λ) · β2 y/H � 1

(— - - model; • ◦ DNS)

v′

ωxcyc

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 = H

porousimpermeable

y+



Two-point auto-correlations: spanwise

Ruu

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350

porous wall

impermeable wall

-

FLOW

-

-

b

bz′ u

u′

∆z+

−→ spanwise organization of the flow

• origin

• intensity

• consequences



Linear stability analysis: inviscid

piecewise-linear mean profile

constant vorticity

layer

(impermeable wall) ................

................

................

................

................

................

..........................................................................................................pppppppppppppppppppppppppp

U

pppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppppppppppppppppp

neutrally

stable

constant vorticity

layer

(porous wall)................

................

................

..........................................................................................................

................

U

................

................

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppp

unstable

wake................

................

................

................

................

..........................................................................................................

..........................................................................................................

U

................

................

................

................

................

unstable



Two-point auto-correlations:

streamwise

R2Dvv

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

porous wall

impermeable case

-

FLOW

66

b bx′

v v′

∆x+

−→ consistent with “rollers”



Energy of two-dimensional modes

q2D

q

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2

porous

y/h

−→ large part of energy in 2d motion

• “rollers” account directly for most of the

increase in turbulence energy near the wall



“Rollers”, transpiration & friction

(snapshot)

y

h

(2D) 13 Apr 1999 2-d plot

0 2 4 6 80

1

2

3

4

5

6

7

(2D) 13 Apr 1999 2-d plot

��-

�� (ψ)

v+w

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8x/h

c f·Re

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8



Correlation: friction – transpiration

joint probability density histogram

cfcf 0

0

0.5

1

1.5

2

2.5

-3 -2 -1 0 1 2 3 4vw

symbols are from uniform injection/suction cases:

•,4 Andersen et al. 1975, Mariani 1993 (exp.)

◦ Sumitani & Kasagi 1995 (DNS)



Effects of porosity – summary

Turbulence regeneration cycle

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppp

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pppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

streaks

advection

streamwisevortices

sinusoidal

instability

“rollers”higher

induced v′

“rollers”

Spanwise rollers

• cyclic up-lift and down-wash of structures

• high local cf variations → larger mean

• propagation speed ≈ 12uτ

Conflict in realization

size of rollers ↔ length of compartments



Exp. measurement: boundary layer

Stanislas et al., Ecole Centrale de Lille

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.......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .....

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.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

D

x

y

LzLx

plenum chamberporous wall element


Near-Wall Turbulence: (3) active control

Large scale wall-actuation experiment.....................

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x

vw

vm

vm

λ

Uc6

?

� -

-

6-

• wavelength λ+ ≈ 500

• intensity vm = 0.14uτ

• phase velocity Uc = 10uτ

⇒ drag increase

porosity •

actuation ◦ ∆c f

(%)

0

10

20

30

40

0.1 0.2

v′+w



Actuation results: phase-locked avg.

streamfunction of spanwise constant modes

y

h

0

0.2

0.4

0.6

0.8

1

1.2

0 90 180 270 360

φ

c f·Re/

2

0

5

10

15

20

25

30

0 90 180 270 360

φ



Linear 2D analysis

streamfunction

y

h

0

0.2

0.4

0.6

0.8

1

1.2

0 90 180 270 360φ


Near-Wall Turbulence:

Conclusions

∗ streak/vortex cycle

• porous wall: suitable for drag increase

• spanwise-oriented “rollers” responsible for

the effect

• realization problematic in non-zero

pressure gradient flows

◦ large-scale actuation: similar effect as

porosity

◦ selection of phase velocity is crucial

◦ realization using MEMS devices


Turbulence Characteristics Near A Porous Surface - KIT fileTurbulence Characteristics Near A Porous...

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