1-7vorti - 1 Notes on 1.63 Advanced Environmental Fluid...

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1 Notes on 1.63 Advanced Environmental Fluid Mechanics Instructor: C. C. Mei, 2001 [email protected], 1 617 253 2994 September 19, 2002 1-7vorti.tex 1.7 Vorticity Theorem for a viscous fl uid Consider an incompressible fl uid. · ~ q = 0 (1.7.1) ~ q t + ~ q · ~ q = ~ f p ρ + ν 2 ~ q (1.7.2) where μ = ρν , ν = kinematic viscosity . (1.7.3) Note the following identity in index form: q j q i x j = q j à q i x j q j x i ! + q j q j x i = 2 q j ij + x i μ 1 2 q j q j = ³ ~ ζ × ~ q ´ i + à ~ q 2 2 ! i . where ~ ζ = × ~ q . Hence, in vector form, the convective inertia term can be written as ~ q · ~ q = ~ q 2 2 + ~ ζ × ~ q (1.7.4) We can now rewrite the momentum equation (1.7.2) by using (1.7.4) ~ q t + ~ q 2 2 + ~ ζ × ~ q = ~ f p ρ + ν 2 ~ q. (1.7.5) The equation governing the transport of vorticity follows by taking the curl of (1.7.5). For a conservative body force such as gravity, × ~ f 0 (1.7.6) Recall the vector identities: × S = 0 (1.7.7)
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2 for any scalar S , · × ~ T = 0 (1.7.8) for any vector ~ T , and
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