058:0160 Professor Fred Stern Chapter 6- part2 1 Fall 2009 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 A layer of fluid heated from below is top heavy, but only undergoes convective “cellular” motion for bouyancy force viscous force 1 α = coefficient of thermal expansion = T T / d dT 1 T dz Raleigh #: Ra gd gd 4 Racr w / d 2 k 0 d = depth of layer k, ν =thermal, viscous diffusivities P 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 2 w=velocity scale: convection (w) = diffusion (k/d) from energy equation, i.e., w=k/d Solution for two rigid plates: Racr = 1708 for progressive wave disturbance ^ i ( x ct ) ct αcr/d = 3.12 w we e i [cos( x ct ) sin( x ct )] ^ i ( x ct ) T T e λcr = 2π/α = 2d α = αr c=cr + ici For temporal stability αr = 2π/λ=wavenumber cr = wave speed ci : > 0 =0 <0 unstable neutral stable Ra > 5 x 104 transition to turbulent flow Thumb curve: stable for low Ra < 1708 and very long or short λ. 058:0160 Professor Fred Stern Chapter 6- part2 3 Fall 2009 (b) finger/oscillatory instability: hot/salty over cold/fresh water and vise versa. Rs gd ds dz /k 4 (Rs – Ra)cr = 657 S (1 T S ) 0 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 4 (c) Centrifugal instability: Taylor Problem Bernard Instability: buoyant force > viscous force Taylor Instability: Couette flow between two rotating cylinders where centrifugal force (outward 058:0160 Professor Fred Stern Chapter 6- part2 5 Fall 2009 from center opposed to centripetal force) > viscous force. Ta r c i 2 2 i o 2 c r r r 0 i i centrifugal force/ viscous force Tacr Tatrans = 1708 = 160,000 αcrc = 3.12 λcr = 2c Square counter rotating vortex pairs with helix streamlines 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 6 058:0160 Professor Fred Stern Chapter 6- part2 7 Fall 2009 (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 Reδ = Uδ/υ > 5 058:0160 Professor Fred Stern Chapter 6- part2 8 Fall 2009 (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. g 22 12 12 U1 U 2 U U 1 2 2 ci 0 (unstable) large α i.e. short λ always unstable Vortex Sheet 1 2 1 2 ci (U1 U 2 ) >0 Therefore always unstable 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 9 058:0160 Professor Fred Stern Chapter 6- part2 10 Fall 2009 (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. Re cr U δ = transverse viscous thickness ~ 1000 Retrans > Recr with xtrans ~ 10-20 xcr Small-disturbance (linear) stability theory can predict Recr 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. 058:0160 Professor Fred Stern Chapter 6- part2 11 Fall 2009 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 vv ^ p p p u, v mean flow, which is solution steady NS ^ ^ u,v small 2D oscillating in time disturbance is solution unsteady NS ^ ^ ^ ^ ^ ^ 1^ u t u u x u u x v u y v u y p x 2 u ^ ^ ^ ^ ^ ^ 1^ v t u v x u v x v v y v v y p x 2 u ^ ^ ux v x 0 ^ Linear PDE for ^ ^ u, v, p for ( u , v , p ) known. Assume disturbance is sinusoidal waves propagating in x direction at speed c: Tollmien-Schlicting waves. ^ i ( xct ) ( x, y , t ) ( y ) e ^ ^ i ( xct ) u e y ^ ^ i ( xct ) v i e x Stream function y =distance across shear layer 058:0160 Professor Fred Stern ^ Fall 2009 ^ u v 0 x Chapter 6- part2 12 Identically! y i = wave number = 2 c c ic = wave speed = r r i i Where λ = wave length and ω =wave frequency Temporal stability: Disturbance (α = αr only and cr real) ci >0 =0 <0 unstable neutral stable Spatial stability: Disturbance (cα = real only) αi <0 =0 >0 unstable neutral stable 058:0160 Professor Fred Stern ^ Chapter 6- part2 13 Fall 2009 ^ Inserting u , v into small disturbance equations and ^ eliminating p results in Orr-Sommerfeld equation: inviscid Raleigh equation (u c)( '' 2 ) u '' u u /U i ( IV 2 2 '' 4 ) Re Re=UL/υ y=y/L 4th 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. 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 14 Although difficult, methods are now available for the solution of the O-S equation. Typical results as follows (1) Flat Plate BL: U Re 520 crit (2) αδ* = 0.35 λmin = 18 δ* = 6 δ (smallest unstable λ) unstable T-S waves are quite large (3) ci = constant represent constant rates of damping (ci < 0) or amplification (ci > 0). ci max = .0196 is small compared with inviscid rates indicating a gradual evolution of transition. 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 15 (4) (cr/U0)max = 0.4 unstable wave travel at average velocity. (5) Reδ*crit = 520 Rex crit ~ 91,000 Exp: Rex crit ~ 2.8 x 106 (Reδ*crit = 2,400) if care taken, i.e., low free stream turbulence 058:0160 Professor Fred Stern Chapter 6- part2 16 Fall 2009 Falkner-Skan Profiles: (1) strong influence of Recrit > 0 fpg Recrit < 0 apg Reδ*crit : 67 sep bl 520 fp bl 12,490 stag point bl 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 17 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 18 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 19 Extent and details of these processes depends on Re and many other factors (geometry, pg, free-stream, turbulence, roughness, etc). 058:0160 Professor Fred Stern Fall 2009 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 Chapter 6- part2 20 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 21 058:0160 Professor Fred Stern Fall 2009 Chapter 6- part2 22 Some recent work concerns recovery distance: