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Unformatted text preview: Tony Burden’s Lecture Notes on Turbulence, Spring 2006 Wallbounded shear flows version 1: channel flow Fully developed channel flow See the example in the section of the lecture notes which present Reynolds equation and the Reynolds stress. For fully developed, statistically twodimensional channel flow Reynolds equation reduces to the thinshearlayer equation which in turn reduces to, 0 = d P w d x + d d y ρ h u v i + μ ∂U ∂y . In the righthand side we can identify the total mean shear stress, τ ( y ) = ρ h u v i + μ ∂U ∂y , which consists of two contributions; the Reynolds shear stress and the mean viscous shear stress. Integrating Reynolds equation across the channel ( h = 2 δ ), the balance of forces for a section of the channel is found to be, d P w d x h = 2 τ w , where, τ w = τ (0) = μ ∂U ∂y y =0 is the (total) mean shear stress at the wall. Reynolds equation can now be rewritten as, d τ d y = d d y ρ h u v i + μ ∂U ∂y = d P w d x = 2 τ w h , and integrated to yield the linear relationship, τ ( y ) = ρ h u v i + μ ∂U ∂y = τ w 1 2 y h = τ w 1 y δ , where δ = 1 2 h is the halfwidth of the channel. This equation can be divided by the mass density, ρ , to yield,h u v i + ν ∂U ∂y = u 2 τ 1 y δ , 1 where u τ is the friction velocity defined by, τ w = ρu 2 τ . This velocity scale and the simple linear expression for τ ( y ) are central to the under standing of channel flow and boundary layers. For practical purposes the Reynolds number of channel flow is defined to be, Re = ρ ¯ Uh μ = ¯ Uh ν , where, ¯ U = 1 h Z h U ( y ) d y, is the bulk velocity which characterizes the mass flow through the channel. The intro duction of the friction velocity, u τ , above allows us to define another Reynolds number, Re τ = u τ δ ν , which in some ways is closer to the basic fluid mechanics of the flow. In particular we note that we can rewrite this Reynolds number as a ratio of length scales, Re τ = u τ δ ν = δ ν/u τ = δ l * , where, l * = ν u τ . In the analysis below we will see that l * is a small length scale which characterizes the influence of the viscosity on the flow very close to the wall. High Reynolds number is now equivalent to large scale separation; Re τ 1 → δ l * . This is essential for the analysis presented below. Empirically the friction factor, f = τ w / 1 2 ρ ¯ U 2 , in channel flow is found to vary only very slowly with the Reynolds number when Re 10 3 . This implies that the velocity ratio u τ / ¯ U = p f/ 2 also only varies very slowly with Re and in turn that the two Reynolds numbers, Re and Re τ , are equivalent from the point of view of the ‘highReynoldsnumber’ property of the flow....
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
 Spring '11
 Staff
 Shear, Stress

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