Multimedia files -3/13 Instability of plane parallel flows Contents: 1.Canonical basic velocity profiles 2.Critical Reynolds numbers for the canonical flows 3.Neutral stability surface for the plane Poiseuille flow 4.Stability results for the plane Poiseuille flow 5.Further reading Canonical (classical) basic velocity profiles introduction Orr-Sorrmerfeld equation Squire equation with boundary conditions at solid walls and in free stream, Main characteristic of the canonical profiles is their dependence only on one coordinate. U,U',U '' play role of parameters in OS and Squire equations Significance of the second mean velocity derivative Canonical (classical) mean velocity profiles, cont. Plane Couette flow (moving walls, Orr 1907) Plane channel flow (plane Poiseuille flow, Heisenberg 1924) Exact solutions of the stationary two-dimensional Navier-Stokes equations: typically closed flows A subset of the profiles: Pipe flow (Hagen- Poiseuille flow) Critical Reynolds numbers for the canonical flows FlowRe E Re G Re T Re L Pipe (Hagen-Poiseuille flow) Channel (Poiseuille flow) Moving walls (Plane Couette flow) Neutral stability surface for the plane Poiseuille flow Re=hU/ Neutral stability curve for the plane Poiseuille flow Spectrum of eigenmodes for the plane Poiseuille flow (temporal formulation) Schensted Pekeris even odd Symmetric profile: U=U 0 (1-y 2 /h 2 ) Discrete spacing between the eigenvalues (the spectrum is discrete) Airy Discrete spacing a lot of negative values Spectrum of eigenmodes for the plane Poiseuille flow (spatial formulation) A, P, S eigenfunctions for PPF symmetric antisymmetric real and imaginary parts absolute value Experimental difficulties channel flow typical wind tunnel test section range Mean velocity deviations Conditional stability Transient effects Visualization of laminar-turbulent transition triggered by a TS-wave in the plane Poiseulle flow Linear region Nonlinear distortions of the wavefront Plane view Comparison of experimental and theoretical results Basic velocity profile Streamwise disturbance velocity Amplification rates for Re=4000 Neutral stability curve Further reading Betchov R. and Criminale W. O. (1967) Stability of parallel flows, NY: Academic. Drazin P. G. and Reid W. H. (1981) Hydrodynamic Stability, Cambridge University Press Schmid P.J., Henningson D.S. (2000) Stability and transition in shear flows, Springer, p