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Stability and Transition in Shear Flows Bangalore 2010g
KTH Mechanics
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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