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

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, Politecnico di Milano (I) 2 LadHyx, École Polytechnique (F) ETC 13, Warsaw, September 2011

O UTLINE Stability of Poiseuille flow + GSL M.Quadrio Introduction I NTRODUCTION 1 Formulation Results F ORMULATION 2 R ESULTS 3

O UTLINE Stability of Poiseuille flow + GSL M.Quadrio Introduction I NTRODUCTION 1 Formulation Results F ORMULATION 2 R ESULTS 3

P OISEUILLE + GSL Q UADRIO , P HIL .T RANS .R.S OC .A 2011 Stability of Poiseuille flow + GSL M.Quadrio w = A cos ( κ x − ω t ) Introduction Formulation Results c 2h λ y z x Flow δ

T HE SPANWISE OSCILLATING BOUNDARY LAYER Q UADRIO & R ICCO , JFM 2011 Stability of Poiseuille flow + GSL w ( y , t ) M.Quadrio TSL Introduction Formulation Results w ( y , x ) SSL w ( y , x − ct ) GSL

T URBULENT DRAG REDUCTION Q UADRIO ET AL ., JFM 2009 Stability of Poiseuille flow + GSL 5 20 40 -10 M.Quadrio 36 41 43 45 45 46 44 20 5 -20 -20 -23 -23 -22 -17 -10 -2 40 0 10 0 20 23 8 0 Introduction 4 15 38 41 44 46 45 36 6 -15 -18 0 0 Formulation 2 38 46 -16 -21 4 - -10 Results 31 42 45 47 -20 24 45 13 10 3 40 46 -15 -18 2 0 15 41 -8 -17 8 15 30 k x 47 30 45 47 33 -16 -2 17 0 0 0 1 0 4 - 2 2 18 21 29 35 43 45 46 46 32 -7 -14 3 0 16 0 0 2 20 44 46 48 48 34 10 -14 21 4 30 33 40 0 45 46 47 40 8 1 -8 -10 13 24 0 31 1 21 34 37 41 45 45 47 39 31 18 10 3 -3 -6 -9 -9 -1 7 14 19 26 24 16 33 36 40 42 42 42 36 14 1 -7 1 24 28 20 0 10 32 36 37 38 37 36 26 1 -8 -1 19 29 29 24 16 34 36 35 33 22 5 -9 4 27 32 0 16 18 22 27 32 34 33 34 33 33 33 32 31 27 21 5 0 3 5 0 -6 -3 -7 -9 -7 -7 -9 -7 -6 -3 5 0 0 5 3 21 27 31 32 33 34 33 34 32 27 22 18 16 -3 -2 -1 0 1 2 3 ω

Q UESTION : D O WAVES AFFECT TRANSITION ? A PRELIMINARY SURVEY BY DNS Stability of Poiseuille flow + GSL M.Quadrio Temporal problem (plane channel flow) by DNS, Introduction Formulation Re = 2000 Results 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

O BLIQUE WAVES A FTER - TRANSIENT DRAG REDUCTION , A = 0 . 25 Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

O BLIQUE WAVES G max / G max , re f , A = 0 . 25 Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

O UTLINE Stability of Poiseuille flow + GSL M.Quadrio Introduction I NTRODUCTION 1 Formulation Results F ORMULATION 2 R ESULTS 3

L INEAR STABILITY : FORMULATION M ODAL AND NON - MODAL CHARACTERISTICS Stability of Poiseuille flow + GSL M.Quadrio Introduction Similar to Orr-Sommerfeld-Squire problem, but... Formulation Additional transversal stationary base flow Results � � δ x e − j 4 / 3 π �� e j κ x Ai − jy with 1 W ( x , y ) = Ai ( 0 ) ℜ δ x = ( ν / κ u y , 0 ) 1 / 3 Streamwise-varying coefficients!

F OURIER TRANSFORM IN x Stability of W ( x ) is sinusoidal: Poiseuille flow + GSL M.Quadrio p κ � e jp κ n x e j κ x e − j α x dx � = 0 for n + κ − α = 0 Introduction Formulation Results + M q i ( y , t ) e j ( i + m ) κ x ∑ q ( x , y , t ) = ˆ i = − M The problem becomes global in x Block-tridiagonal matrix Each block is like a standard Orr-Sommerfeld-Squire problem Size of full problem is ( 2 M + 1 ) 2 × a single OSSq

A HUGE PARAMETRIC STUDY Stability of Poiseuille flow + GSL M.Quadrio Large number of parameters: Introduction Spanwise wavenumber β of the perturbation Formulation Base flow wave number κ Results Base flow amplitude A Reynolds number Re Wall-normal resolution N Modal truncation (streamwise resolution) M

O UTLINE Stability of Poiseuille flow + GSL M.Quadrio Introduction I NTRODUCTION 1 Formulation Results F ORMULATION 2 R ESULTS 3

C HANGES IN LONG - TERM STABILITY κ = 1, β = 1 . 5 Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

C HANGES IN MAXIMUM TRANSIENT GROWTH κ = 1, β = 1 . 5 Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results

C ONCLUSIONS , PERSPECTIVES Stability of Poiseuille flow + GSL M.Quadrio Introduction Large effects on at least one (temporal) transition Formulation scenario Results Formulation of the linear stability problem Least-stable eigenvalue reduced Transient growth weakened Energy transfer among wavenumbers?

O PTIMAL INPUT κ = 1, β = 1 . 5, A = 0 ( TOP ) VS A = 1 ( BOTTOM ) Stability of u v w Poiseuille 0.11 - 0.74 2.68 - 2.63 3.28 - 3.79 flow + GSL M.Quadrio Introduction Formulation Results

O PTIMAL OUTPUT κ = 1, β = 1 . 5, A = 0 ( TOP ) VS A = 1 ( BOTTOM ) Stability of u v w Poiseuille 481x - 44 X 0.47x - 0.51 X 0.40x - 0.46 X flow + GSL M.Quadrio Introduction Formulation Results

M ODAL TRUNCATION ( ADAPTIVE RESOLUTION ) Stability of Poiseuille flow + GSL M.Quadrio Introduction Formulation Results