Course on Stability and Transition

Reynolds pipe flow experiment

History 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 BL solution
	Schubauer & Skramstad (1947) experimental TS-wave verification
	Klebanoff, Tidstrom & Sargent (1962) 3D breakdown

Bypass transition

	High and low speed streaks in the streamwise direction
	Transition due to free-stream turbulence
	Klebanoff (1977) modes,    Tu > 0.5% in BL
	Subcritical transition in Poiseuille and Couette flows

Disturbance equations

Stability definitions

Stability definitions

Critical Reynolds numbers

Critical Reynolds numbers

Reynolds-Orr equation

Linear growth mechanisms

Evolution of disturbances in shear flows

Parallel shear flows:

Parallel shear flows, cont

Orr-Sommerfeld and Squire equations

Interpretation of modal results

Interpretation of modal results, cont.

Squire’s transformation

Squire’s theorem

Inviscid disturbances

Rayleigh’s inflection point criterion

Inviscid algebraic instability

Lift-up effect

Critical Layer

Thickness of critical layer

Plane Poiseuille flow

A, P, S- Eigenfunctions for PPF

Blasius boundary layer

Continuous spectrum

Critical Reynolds numbers

The intial value problem (IVP)

A simple eigenfunction expansion

Transient growth

Model with non-orthogonal eigenfunctions

General formulation of viscous IVP

Disturbance measure

Adjoint OS-SQ system

Biorthogonality

Component form of adjoint

Solution of IVP using eigenfunction expansions

Discrete formulation

Discrete formulation, cont.

Maximum amplification

2D PPF: envelope and selected IC

2D PPF: dependence on N

3D PPF and Blasius flow, Re=1000

3D PPF: G(t), Re=1000

Optimal disturbances PPF, Re=1000

The forced problem and the resolvent

Discrete formulation

Maximum response to forcing

Pseudospectra, resolvents and sensitivity

PPF: resolvents, growth and forcing

PPF: resolvents, growth and forcing

Re-dependence of growth and response

Re-dependence, cont.

Re-dependence of Gmax for PPF

Re-dependence of Rmax for PCF

Re-dependence of Gmax and Rmax