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Unformatted text preview: Tony Burdens Lecture Notes on Turbulence, Spring 2007 3c Plane jet Compared to the general analysis of a thin shear layer, the freestream velocity U = 0, and the freestream pressure is constant, so, U U and d P d x = 0 . Constant momentum flux Within the freeshearlayer approximation, i.e. ten to twenty nozzle diamters down stream, a controlvolume analysis or the thinshearlayer equation itself shows that the total momentum flux, Z  U 2 d y = constant . The mathematical problem The thinshearlayer equation, U U x + V U y = y h u v i , together with the continuity condition, U x + V y = 0 , under the freestream condition, U ( x, y ) when y , and assuming symmetry, U y ( x, 0) = 0 and V ( x, 0) = 0 , on the centreline. 30 Selfsimilar profiles U ( x, y ) = U c ( x ) F ( ) , and, h u v i = U 2 c g ( ) , where, U c ( x ) = U ( x, 0) , is the centreline velocity and, = y ` ( x ) . The choice of F rather than f in the profile for the streamwise velocity is related to the streamfunction formalism and the continuity condition. The lengthscale can be chosen to be the halfwidth, ` = ` 1 / 2 ( x ) defined by, U ( x, ` 1 / 2 ) = 1 2 U c ( x ) = 1 2 U ( x, 0) . Preservation of selfsimilarity requires u T U c . In fact, measurements show that, [ u ( x, 0)] 2 1 / 2 U c = 0 . 24 . 01 for x 40 D. Constant flux of momentum implies that, U 2 c ` = constant , i.e. independent of x . Following the details in the appendix, the thinshearlayer equation becomes, U c U c F 2 ( U c ` ) U c ` F F 00 = U 2 c ` g , which can be rewritten in the form, U c ` U c F 2 ( U c ` ) U c F F 00 + g = 0 . Separation of variables requires now, U c ` U c = constant and ( U c ` ) U c = constant , so that, ` = constant , and, ` x while U c ` 1 / 2 x 1 / 2 . These results agree well with measurements for x 20 D . 31 The selfsimilar profile of U/U c as a function of is now determined by, 1 2 ` n F 2 + F F 00 o = g , once g ( ) has been modelled as a function of F ....
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 Spring '11
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