Vorticity_production_at_wall - Vorticity Production at a...

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Vorticity Production at a Wall – How Does Vorticity get Produced? Author: John M. Cimbala, Penn State University Latest revision: 11 October 2007 Note : Consider steady, incompressible, Newtonian flow in which gravity is neglected and the wall is stationary. Equations of motion at the wall Start with the Navier-Stokes equation (differential momentum equation), with gravity ignored, 2 ii j i j ij uu p u tx x x ρ ⎛⎞ ∂∂ += + ⎜⎟ ⎝⎠ j u x μ . ( 1 ) Invoke the two vector identities discussed in class when the Bernoulli equation was derived. The result is () 1 2 1 j i k jj ji j k k i j k i j u u p u x x x x j ω εω ε ρρ +− = + . ( 2 ) Simplify Eq. (2) by using incompressible continuity, steady flow, and the fact that at the wall, u i = 0 by the no slip condition. Right at the wall, then, all terms with u i go to zero; rearranging, 1 2 1 k ijk j j i p xx x μμ =− at the wall . ( 3 )
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