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Unformatted text preview: Lecture 18- Marine Hydrodynamics Lecture 18 4.9 Turbulent Flow Reynolds Stress Assume a ow v with a time scale T . Let denote a time scale << T . We can then write for each component of the velocity u i = u i + u i (1) where by definition 1 u i = u i dt It immediately follows that u i u = u i u i = u i u i = 0 , also u i etc. = i x x Substitute Eq. (1) into continuity and average over , i.e., take ( ) + u i = 0 , = u i u i u i = 0 x i = x i x i x i u i u i u i , = u i = 0 x i but = 0 = + x i x i x i , just shown 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20 ( ) Substitute Eq. (1) into the momentum equations and take ( ) u i u i 1 ij 1 p 2 + u j = = + u i t x j x j x i 2 = 2 u i u i u i u i u i ; similarly + = t t t p = ( p + p ) = p x i x i etc....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.
- Spring '05