(1) Transition from one to another laminar flow. (a) Thermal instability: Bernard Problem

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1 Professor Fred Stern Fall Chapter 6: Viscous Flow in Ducts 6.2 Stability and Transition Stability: can a physical state withstand a disturbance and still return to its original state. In fluid mechanics, there are two problems of particular interest: change in flow conditions resulting in (1) transition from one to another laminar flow; and (2) transition from laminar to turbulent flow. (1) Transition from one to another laminar flow (a) Thermal instability: Bernard Problem Raleigh #: A layer of fluid heated from below is top heavy, but only undergoes convective cellular motion for 4 gαγd gαγd Ra = = > Ra 2 υw/ d kν α = coefficient of thermal expansion = Γ= = ρ = ρ ( 1 α T ) T / d dt dz d = depth of layer k, ν =thermal, viscous diffusivities 0 cr bouyancy force viscous force 1 ρ T ρ P

2 Professor Fred Stern Fall w=velocity scale: convection (wγ) = diffusion (kγ/d) from energy equation, i.e., w=k/d Solution for two rigid plates: Ra cr = 1708 for progressive wave disturbance α cr /d = 3.12 ^ i ( x ct) ct w we = = e i [cos( x ct) + sin( x ct)] λ cr = 2π/α = 2d ^ i ( x ct) T= Te α α = α r c=c r + ic i α r = 2π/λ=wavenumber c r = wave speed For temporal stability c i : > 0 unstable = 0 neutral < 0 stable Ra > 5 x 10 4 transition to turbulent flow Thumb curve: stable for low Ra < 1708 and very long or short λ.

3 Professor Fred Stern Fall (b) finger/oscillatory instability: hot/salty over cold/fresh water and vise versa. (Rs Ra) cr = 657 Rs = gβd ( ds dz) ρ = ρ (1 α T 0 4 / νk S + β S)

Professor Fred Stern Fall 2014 4 (c) Centrifugal instability: Taylor Problem Bernard Instability: buoyant force

4 Professor Fred Stern Fall (c) Centrifugal instability: Taylor Problem Bernard Instability: buoyant force > viscous force Taylor Instability: Couette flow between two rotating cylinders where centrifugal force (outward

5 Professor Fred Stern Fall from center opposed to centripetal force) > viscous force. Ta = = 2 2 ( Ω Ω ) rc i i o 2 ν centrifugal c = r r force/ viscous force 0 i << r i Ta cr = 1708 α cr c = 3.12 λ cr = 2c Ta trans = 160,000 Square counter rotating vortex pairs with helix streamlines

6 Professor Fred Stern Fall

7 Professor Fred Stern Fall (d) Gortler Vortices Longitudinal vortices in concave curved wall boundary layer induced by centrifugal force and related to swirling flow in curved pipe or channel induced by radial pressure gradient and discussed later with regard to minor losses. For δ/r >.02~.1 and Re δ = Uδ/υ > 5

8 Professor Fred Stern Fall (e) Kelvin-Helmholtz instability Instability at interface between two horizontal parallel streams of different density and velocity with heavier fluid on bottom, or more generally ρ=constant and U = continuous (i.e. shear layer instability e.g. as per flow separation). Former case, viscous force overcomes stabilizing density stratification. 2 2 ( ρ ρ ) αρ ρ ( ) 2 g U U c i < > (unstable) U U 1 2 large α i.e. short λ always unstable Vortex Sheet 1 ρ1 = ρ2 ci = ( U1+ U2 ) > 0 2 Therefore always unstable

9 Professor Fred Stern Fall

10 Professor Fred Stern Fall (2) Transition from laminar to turbulent flow Not all laminar flows have different equilibrium states, but all laminar flows for sufficiently large Re become unstable and undergo transition to turbulence. Transition: change over space and time and Re range of laminar flow into a turbulent flow. Uδ Re = ~ 1000 δ = transverse viscous thickness cr υ Re trans > Re cr with x trans ~ x cr Small-disturbance (linear) stability theory can predict Re cr with some success for parallel viscous flow such as plane Couette flow, plane or pipe Poiseuille flow, boundary layers without or with pressure gradient, and free shear flows (jets, wakes, and mixing layers). Note: No theory for transition, but recent DNS helpful.

11 Professor Fred Stern Fall Outline linearized stability theory for parallel viscous flows: select basic solution of interest; add disturbance; derive disturbance equation; linearize and simplify; solve for eigenvalues; interpret stability conditions and draw thumb curves. ^ u = u+ u ^ v= v+ v ^ p= p+ p u, v = mean flow, which is solution steady NS ^ ^,v u = small 2D oscillating in time disturbance is solution unsteady NS ^ ^ ^ ^ ^ 1 ^ ^ 2 ut+ uux+ uux + vuy+ vuy = px + υ u ρ ^ ^ ^ ^ ^ 1 ^ ^ 2 vt+ uvx+ uvx + vvy+ vvy = px + υ u ρ ^ ^ u + vx = 0 Linear PDE for x ^ u, ^ v, ^ p for (u,v, p ) known. Assume disturbance is sinusoidal waves propagating in x direction at speed c: Tollmien-Schlicting waves. ^ iα ( x ct) Ψ (, xyt,) = φ( ye ) ^ ^ ψ iα ( x ct u = = φ e ) y ^ ^ Ψ iα ( x ct v = = iαφe ) x Stream function y =distance across shear layer

12 Professor Fred Stern Fall ^ ^ u v = 0 Identically! x + y α = α r + iα i = wave number = 2 π λ c = c r + ic i = wave speed = ω α Where λ = wave length and ω =wave frequency Temporal stability: Disturbance (α = α r only and c r real) c i > 0 unstable = 0 neutral < 0 stable Spatial stability: Disturbance (cα = real only) α i < 0 unstable = 0 neutral > 0 stable

13 Professor Fred Stern Fall Inserting ^ u, ^ v into small disturbance equations and eliminating ^ p results in Orr-Sommerfeld equation: inviscid Raleigh equation i u c φ αφ u φ = φ αφ+ αφ α Re 2 IV 2 4 ( )( '' ) '' ( 2 '' ) u u / U = Re=UL/υ y=y/l 4 th order linear homogeneous equation with homogenous boundary conditions (not discussed here) i.e. eigen-value problem, which can be solved albeit not easily for specified geometry and (u,v, p) solution to steady NS.

14 Professor Fred Stern Fall Although difficult, methods are now available for the solution of the O-S equation. Typical results as follows (1) Flat Plate BL: + Uδ = crit υ Re = 520 (2) αδ * = 0.35 λ min = 18 δ * = 6 δ (smallest unstable λ) unstable T-S waves are quite large (3) c i = constant represent constant rates of damping (c i < 0) or amplification (c i > 0). c i max =.0196 is small compared with inviscid rates indicating a gradual evolution of transition.

15 Professor Fred Stern Fall (4) (c r /U 0 ) max = 0.4 unstable wave travel at average velocity. (5) Re δ*crit = 520 Re x crit ~ 91,000 Exp: Re x crit ~ 2.8 x 10 6 (Re δ*crit = 2,400) if care taken, i.e., low free stream turbulence

16 Professor Fred Stern Fall Falkner-Skan Profiles: (1) strong influence of β Re crit β > 0 fpg Re δ*crit : 67 sep bl 520 fp bl 12,490 stag point bl Re crit β < 0 apg

17 Professor Fred Stern Fall

18 Professor Fred Stern Fall

19 Professor Fred Stern Fall Extent and details of these processes depends on Re and many other factors (geometry, pg, free-stream, turbulence, roughness, etc).

20 Professor Fred Stern Fall Rapid development of spanwise flow, and initiation of nonlinear processes - stretched vortices disintigrate - cascading breakdown into families of smaller and smaller vortices - onset of turbulence Note: apg may undergo much more abrupt transition. However, in general, pg effects less on transition than on stability

21 Professor Fred Stern Fall

22 Professor Fred Stern Fall Some recent work concerns recovery distance:

1. Introduction, tensors, kinematics

1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and