bdrylayers - Tony Burden’s Lecture Notes on Turbulence...

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Unformatted text preview: Tony Burden’s Lecture Notes on Turbulence, Spring 2006 Wall-bounded 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 two-dimensional channel flow Reynolds equation reduces to the thin-shear-layer 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 right-hand 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 half-width 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 ‘high-Reynolds-number’ 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.

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bdrylayers - Tony Burden’s Lecture Notes on Turbulence...

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