P OISEUILLE + GSL Q UADRIO , P HIL .T RANS .R.S OC .A 2011 Stability - - PowerPoint PPT Presentation

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P OISEUILLE + GSL Q UADRIO , P HIL .T RANS .R.S OC .A 2011 Stability - - PowerPoint PPT Presentation

Sep 08, 2023 383 likes •599 views

Stability of Poiseuille flow + GSL M.Quadrio L INEAR STABILITY Introduction OF PLANE P OISEUILLE FLOW Formulation OVER A G ENERALIZED S TOKES LAYER Results .Martinelli 2 & P M.Quadrio 1 , 2 , F .Schmid 2 1 Dip. Ing. Aerospaziale,

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SLIDE 1

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

LINEAR STABILITY

OF PLANE POISEUILLE FLOW OVER A GENERALIZED STOKES LAYER

M.Quadrio1,2, F .Martinelli2 & P .Schmid2

  • 1Dip. Ing. Aerospaziale, Politecnico di Milano (I)

2LadHyx, École Polytechnique (F)

ETC 13, Warsaw, September 2011

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SLIDE 2

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OUTLINE

1

INTRODUCTION

2

FORMULATION

3

RESULTS

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SLIDE 3

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OUTLINE

1

INTRODUCTION

2

FORMULATION

3

RESULTS

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SLIDE 4

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

POISEUILLE + GSL

QUADRIO, PHIL.TRANS.R.SOC.A 2011

w = Acos(κx−ωt)

y x z Flow δ 2h λ c

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SLIDE 5

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

THE SPANWISE OSCILLATING BOUNDARY LAYER

QUADRIO & RICCO, JFM 2011

w(y,t) TSL w(y,x) SSL w(y,x−ct) GSL

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Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

TURBULENT DRAG REDUCTION

QUADRIO ET AL., JFM 2009

  • 20
  • 2
  • 10
  • 1
  • 10

10 10 10 10 20 2 20 2 20 30 30 40 4 4 40

ω kx

  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5

33 45 24 33 42 29 38 13 47 3 32 31

  • 3
  • 9

41 37 34 19 6

  • 18

7

  • 9

10 47 8 35 24 1 1

  • 8
  • 10
  • 7

2 24 16 38

  • 7
  • 18
  • 15

46 47 45 8 16 40 33 30 31 29 24 20 13 23 16 21 44 43 5

  • 17

21

  • 14

48

  • 1

41 45 38 26

  • 16
  • 17

36 18 15 15 31 34 33 19 4

  • 2

45 16

  • 16

46 44

  • 20
  • 23
  • 22
  • 10
  • 2
  • 23
  • 20
  • 14

45 39 18 3

  • 6
  • 1

14 26 36 14 1

  • 21

31 34 27 18

  • 3 5

21 32 36 37 36 1 24 48 44 32 34 29

  • 8

28 20 36 40 42 17 42 45 47 15 37 46 40 46 45 46 45 47 46 41 45 46 46 21 40 42 45 43 36

  • 15

41

  • 8

8 36 33 22 5

  • 9

4 35 34 27 32

  • 6
  • 7

3

  • 9
  • 7

33 16 31 34 27 18

  • 3

5 21 32 34 0 -6

  • 7

3

  • 9
  • 7

22 32 33 33 27 5 22 32 33 33 27 5 0

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SLIDE 7

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

QUESTION: DO WAVES AFFECT TRANSITION?

A PRELIMINARY SURVEY BY DNS

Temporal problem (plane channel flow) by DNS, Re = 2000 Transition scenario: oblique waves (Reddy et al., JFM 1998) Optimal i.c. for α = 1,β = ±1, 1% random noise Initial energy = 2 × transition threshold No generality

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SLIDE 8

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OBLIQUE WAVES

AFTER-TRANSIENT DRAG REDUCTION, A = 0.25

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SLIDE 9

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OBLIQUE WAVES

Gmax/Gmax,re f , A = 0.25

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SLIDE 10

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OUTLINE

1

INTRODUCTION

2

FORMULATION

3

RESULTS

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SLIDE 11

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

LINEAR STABILITY: FORMULATION

MODAL AND NON-MODAL CHARACTERISTICS

Similar to Orr-Sommerfeld-Squire problem, but... Additional transversal stationary base flow W(x,y) =

1

Ai(0)ℜ

  • ejκxAi
  • − jy

δx e−j4/3π

with δx = (ν/κuy,0)1/3 Streamwise-varying coefficients!

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SLIDE 12

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

FOURIER TRANSFORM IN x

W(x) is sinusoidal:

  • ejp κ

n xejκxe−jαxdx = 0

for pκ n +κ −α = 0 q(x,y,t) =

+M

∑

i=−M

ˆ qi(y,t)e j(i+m)κx The problem becomes global in x Block-tridiagonal matrix Each block is like a standard Orr-Sommerfeld-Squire problem Size of full problem is (2M +1)2× a single OSSq

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Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

A HUGE PARAMETRIC STUDY

Large number of parameters: Spanwise wavenumber β of the perturbation Base flow wave number κ Base flow amplitude A Reynolds number Re Wall-normal resolution N Modal truncation (streamwise resolution) M

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SLIDE 14

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OUTLINE

1

INTRODUCTION

2

FORMULATION

3

RESULTS

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SLIDE 15

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

CHANGES IN LONG-TERM STABILITY

κ = 1, β = 1.5

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SLIDE 16

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

CHANGES IN MAXIMUM TRANSIENT GROWTH

κ = 1, β = 1.5

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SLIDE 17

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

CONCLUSIONS, PERSPECTIVES

Large effects on at least one (temporal) transition scenario Formulation of the linear stability problem Least-stable eigenvalue reduced Transient growth weakened Energy transfer among wavenumbers?

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SLIDE 18

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OPTIMAL INPUT

κ = 1, β = 1.5, A = 0 (TOP) VS A = 1 (BOTTOM)

u v w 0.11 - 0.74 2.68 - 2.63 3.28 - 3.79

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SLIDE 19

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

OPTIMAL OUTPUT

κ = 1, β = 1.5, A = 0 (TOP) VS A = 1 (BOTTOM)

u v w 481x - 44X 0.47x - 0.51X 0.40x - 0.46X

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SLIDE 20

Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

MODAL TRUNCATION (ADAPTIVE RESOLUTION)