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Unformatted text preview: Tony Burdens 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- h u v i + U y . In the right-hand side we can identify the total mean shear stress, ( y ) =- h u v i + 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 ), the balance of forces for a section of the channel is found to be,- d P w d x h = 2 w , where, w = (0) = U y y =0 is the (total) mean shear stress at the wall. Reynolds equation can now be rewritten as, d d y = d d y- h u v i + U y = d P w d x =- 2 w h , and integrated to yield the linear relationship, ( y ) =- h u v i + U y = w 1- 2 y h = w 1- y , where = 1 2 h is the half-width of the channel. This equation can be divided by the mass density, , to yield,-h u v i + U y = u 2 1- y , 1 where u is the friction velocity defined by, w = u 2 . This velocity scale and the simple linear expression for ( 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 = Uh = Uh , 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 , above allows us to define another Reynolds number, Re = u , 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 = u = /u = l * , where, l * = u . 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 1 l * . This is essential for the analysis presented below. Empirically the friction factor, f = w / 1 2 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 / U = p f/ 2 also only varies very slowly with Re and in turn that the two Reynolds numbers, Re and Re , are equivalent from the point of view of the high-Reynolds-number property of the flow....
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