u 0t cos AU Thus vel distribution where u yt co k 2 s ky amplitude phase i t u

U 0t cos au thus vel distribution where u yt co k 2 s

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u 0,t cos A=U Thus, vel. distribution where u y,t co k 2 s = ky amplitude phase i t u y t e U t Ue t ky ω = ω ω ν = ω − ±²²³²²´
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( ) velocity of fluid decreases exponentially as the distance from the plate y increases. rate of decrease k= , will be faster for higher frequency and smaller viscosity. 2 fluid oscillates in time ω ν ( ) ( ) -ky with the same frequency as the input freq. in boundary. amplitude of oscillation Ue max. amp. at y=0 U phase shift cos t-ky g y = ω specific time instantaneous time U y
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Can define thickness of oscillating shear layer, δ as again where velocity amplitude drops to 1% of U. u/U=0.01. 4.6 4.6 0.01 = k 6.5 2 ky k y u e e e U k δ δ δ δ δ = = = = = ω ν = ν ν ω remember characteristic of laminar flows ( ) w 0 0 : For air at 20 , with a plate frequency of 1 Hz =2 rd/sec 10 The wall shear stress at the oscillating plate sin 4 lags the max. vel. y y Ex C mm u U t y shear stress π δ π τ τ µ µ = = ω = = = ρω ω − D by 135 2 4 π π + D Since governing eq. is linear, the method of superposition is applicable. Hence, superposing several oscill. of diff. freq. & ampl. the sol. for arbitrary periodic motion of plate can be obtained.
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UNIFORM FLOW OVER A POROUS WALL ( ) non-linear inertia terms are not zero V. 0 but linearized to permit a closed form solution V JG JG Example: steady, fully-dev. flow over a plate with suction y fully dev. u=u(y) p=const. δ • suction on surface is sometimes used to prevent boundary layers from separating.
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I. solution with primitive variables, u, v, p u x 2 2 0 v=const. : u v y u u u x v x y x + = + = ν N ( ) ( ) 2 2 0 2 2 2 2 2 1 v= V uniform : 0 0 0 =0 & V y u p y x p y y du d u V dy dy d u V du V V dy d A y u y Be α α α α ν + ρ ∂ = = ν + = + = = − ν + ν ν = ( ) ( ) . . u y=0 0 u y B C s U = → ∞ =
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( ) V note if blowing instead of suction v=V u y y u y A Be ν = + → ∞ → → ∞ not physically possible u would be unbounded at large y ( ) ( ) ( ) 0 0 A+B=0 u y B.L. thicknes 1 s V y u U A V u y U e δ ν = = ν → ∞ = = II. solution using Vorticity transport eq . Vorticity Transport in 2-D ( ) 2 2 2 2 2 . z z z z z V t u v t x y x y ω ω υ ω ω ω ω ω ω υ + = + + = + JG JG JG JG
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( ) ( ) 2 z 2 2 2 of diffusion vorticity toward away fr plate 0 & 0 fully-dev. flow & v= V=const. from continuity z z z z z z convection viscous steady t x v y y y d y dy d d V dy dy ω ω υ ω ω υ ∂ω = = ω = ω = ±²³²´ om plate 2 2 1 z 1 Integrate once, 0 at y . . 0 0 at y z z z z z d d V dy dy d V c dy B C c d dy ω ω ω ω ω = ν = + ν ω = → ∞ = = → ∞ ±³´
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z 2 2 z z 2 yields, 0 at y , 0=c c ??
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