Stability and Transition in Shear Flows Bangalore 2010g

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Reynolds pipe flow experimentReynolds pipe flow experiment

• Original 1883 apparatus

• Dye into center of pipeDye into center of pipe

• Critical Re=13.000

• Lower today due to traffic

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History of shear flow stability and transitionHistory of shear flow stability and transition

• Reynolds pipe flow experiment (1883)

• Rayleigh’s inflection point criterion (1887)

• Orr (1907) Sommerfeld (1908) viscous eq.

• Heisenberg (1924) viscous channel solution

• Tollmien (1931) Schlichting (1933) viscous ( ) g ( )BL solution

• Schubauer & Skramstad (1947) experimental TS-wave verificationp

• Klebanoff, Tidstrom & Sargent (1962) 3D breakdown

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Bypass transitionBypass transition

• High and low speed streaks in the streamwise direction

• Transition due to free-stream turbulence

Kl b ff (1977) d • Klebanoff (1977) modes, Tu > 0.5% in BL

• Subcritical transition in (Matsubara & Alfredsson 2000)Poiseuille and Couette flows

( )

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Linear temporal stability theory

Summary of chapter 2 3 and 4Summary of chapter 2, 3 and 4

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Disturbance equationsDisturbance equations

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Parallel shear flows:Parallel shear flows:

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Parallel shear flows, contParallel shear flows, cont

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Orr-Sommerfeld and Squire equationsOrr Sommerfeld and Squire equations

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Interpretation of modal resultsInterpretation of modal results

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Interpretation of modal results contInterpretation of modal results, cont.

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Inviscid disturbances Inviscid disturbances

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Rayleigh’s inflection point criterionRayleigh s inflection point criterion

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Inviscid algebraic instabilityInviscid algebraic instability

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Lift-up effectLift up effect

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Plane Poiseuille flowPlane Poiseuille flow

Neutral curve and spectrum

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A P S- Eigenfunctions for PPFA, P, S Eigenfunctions for PPF

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Blasius boundary layerBlasius boundary layer

• Re = 500

• 0.2

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• TS-mode

Continuous spectrum: OS in free-streamContinuous spectrum: OS in free stream

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Critical Reynolds numbersCritical Reynolds numbers

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Initial value problem:model with non-orthogonal eigenfunctionsmodel with non orthogonal eigenfunctions

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The intial value problem (IVP) for OS-SQThe intial value problem (IVP) for OS SQ

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General formulation of viscous IVPGeneral formulation of viscous IVP

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Disturbance measureDisturbance measure

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Discrete formulationDiscrete formulation

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Discrete formulation, cont.Discrete formulation, cont.

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Maximum amplificationMaximum amplification

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2D PPF: envelope and selected IC2D PPF: envelope and selected IC

Re=1000 Re=5000, 8000

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2D PPF: dependence on N2D PPF: dependence on N

Eigenvalues Re=3000 G(t)

coefficients Gmax(n)

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3D PPF and Blasius flow Re=10003D PPF and Blasius flow, Re 1000

12060

180

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Optimal disturbances PPF Re=1000Optimal disturbances PPF, Re 1000

2D disturbance 3D disturbance2D disturbance 3D disturbance

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Re-dependence of growth and responseRe dependence of growth and response

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Re-dependence of Gmax for PPFRe dependence of Gmax for PPF

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Re-dependence of Rmax for PCFRe dependence of Rmax for PCF

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Re-dependence of Gmax and RmaxRe dependence of Gmax and Rmax

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The forced problem and the resolventThe forced problem and the resolvent

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Discrete formulationDiscrete formulation

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Maximum response to forcingMaximum response to forcing

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Pseudospectra resolvents and sensitivityPseudospectra, resolvents and sensitivity

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Re-dependence of Gmax and RmaxRe dependence of Gmax and Rmax

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Spatial linear stability theory

Summary of chapter 7Summary of chapter 7

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Model problemModel problem

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Burger’s eq contBurger s eq., cont.

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Spatial OS-SQ systemSpatial OS SQ system

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Spatial OS-SQ system, cont.Spatial OS SQ system, cont.

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Boundary layer flowBoundary layer flow

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Optimal disturbances in spatial BLOptimal disturbances in spatial BL

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Spatial optimals, cont.Spatial optimals, cont.

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The optimization problemThe optimization problem

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Derivation of the action of the adjointDerivation of the action of the adjoint

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Optimal growth and disturbanceOptimal growth and disturbance

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Non-linear interactions

Summary of chapter 5 and 8Summary of chapter 5 and 8

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Quadratic non-linear interactionsQuadratic non linear interactions

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Non-linear v-eta formulationNon linear v eta formulation

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Fourier-transformed equationsFourier transformed equations

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Convolution sums and triad interactionsConvolution sums and triad interactions

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ExampleExample

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Non-linear equilibrium statesNon linear equilibrium states

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2D finite amplitude states in PPF2D finite amplitude states in PPF

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Secondary instability of 2D wavesSecondary instability of 2D waves

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Form of the solutionForm of the solution

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Classification of modes

Classification of modes

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Secondary instability equationsSecondary instability equations

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Secondary instability of 2D TS wavesSecondary instability of 2D TS waves

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Transition to turbulence

Summary of chapter 9Summary of chapter 9

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Transition to turbulence: 3 scenariosTransition to turbulence: 3 scenarios

Streak breakdown2nd instability of TS-waves Oblique transition

y q

TS-wave oblique mode streak

subharmonic mode

fundamental mode

induced streak

fundamental mode

fundamental mode

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Secondary instability of TS-wavesSecondary instability of TS waves

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Streak breakdownStreak breakdown

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Oblique transitionOblique transition

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Transition times vs disturbance energyTransition times vs disturbance energy

Blasius boundary layer Pl P i ill flBlasius boundary layer Plane Poiseuille flow

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Threshold for transition in PPFThreshold for transition in PPF

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DNS of becondary breakdown of TS-wavesDNS of becondary breakdown of TS waves

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(Rist & Fasel 1995)

Free-stream turbulence and streak breakdownFree stream turbulence and streak breakdown

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Klebanoff mode and optimal growthKlebanoff mode and optimal growth

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DNS of oblique transitionDNS of oblique transition

(B li t l 1994)(Berlin et al. 1994)

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DNS of Wiegel experimentDNS of Wiegel experiment

(Berlin 1998)

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Transition types in Wiegel experimentTransition types in Wiegel experiment

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Thank you!

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