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Navier-Stokes Equation &acirc;€“ dimensional form25

# Navier-Stokes Equation &acirc;€“ dimensional form25...

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12.005 Lecture Notes 25 Navier-Stokes Equation – dimensional form 2 ii i ij j vD v p f x xx D t ηρ −+ + = ∂∂ Assume: Characteristic velocity 0 v Characteristic length L Characteristic stress 0 / vL η Characteristic time 0 / Choose non-dimensional variables 0 0 0 '/ ' ' v L vv vxx L p p t t === = or 0 0 0 '' ' ' ' v L v xx Lp pt t Lv = 2 0 0 1 ' v L ff x Lx v t Lt Substitute into Navier-Stokes equation 2 00 0 22 1' 1 1 ' ' ' i j v D v p f v LL x L x x L LD t ηη ρ + = or 2 0 ' ' ' ' i j L D v p f x D t + = where 0 = Re

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Re gives importance of inertial terms relative to viscous terms. Re ± 1 viscous forces balance acceleration negligible Re ² 1 inertia dominates Note: in dimensionless form, Re is the only parameter in the Navier-Stokes equation. for given geometry (boundary conditions) ALL equivalent (non-dimensional) problems at same Re give same result! Examples: 1. Low Reynolds number flow past a cylinder. Re ± 1 Symmetry, like in the sphere problem. Figure 25.1 Figure by MIT OCW. v i 2. Re = 10 v t i = 0 (steady) v j x j 0

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Navier-Stokes Equation &acirc;€“ dimensional form25...

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