Day19-slides - Atms 502 Numerical Fluid Dynamics Thu., Oct....

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1 Atms 502 Numerical Fluid Dynamics Thu., Oct. 26, 2006 Nondimensional numbers and scaling 10/31/06 Atms 502 - Fall 2006 - Jewett 3 Nondimensional numbers Richardson number Ratio of thermal/mechanical KE Ratio of static stability to shear 2 High: stable, low-shear flow Low: unstable / highly sheared Ri = g " #" # z $ % & ( ) u z $ % & ( ) 2 Pure Convective Instability Ri = - Movies from: fluid.stanford.edu/~fringer/movies/main.html Shear 0 < Ri < 1/4 Dominated Ri = -0.04 Buoyancy Ri = -1.34 10/31/06 Atms 502 - Fall 2006 - Jewett 4 Nondimensional numbers Reynolds number Ratio of nonlinear terms to friction PBL: U~10 m/s, L~1 km, ν ~10 -5 m 2 /s Re ~ 10 9 Large Re: turbulent flow ν = kinematic viscosity, = μ / ρ Rossby number Ratio of nonlinear terms to Coriolis: Midlatitudes: ~0.1 <<1 in ocean (geostrophic flow) Re = u " # u $ # 2 u % UL Ro = u "# u fU $ U fL
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2 10/31/06 Atms 502 - Fall 2006 - Jewett 5 Nondimensional numbers Prandtl number Ratio of kinematic viscosity to thermal diffusivity approx. 0.7 in air 0.33 - 1.0 in cloud models Low: rolls; high: turbulent convection, no cells Froude number Ratio of inertial to gravity terms Rotational Froude number Densimetric Froude #, k Used to characterize outflows; k~.75-1.5 Pr = " k Fr = u 0 2 gL or = u " g h Fr = L 2 L R 2 , L R = " f 10/31/06 Atms 502 - Fall 2006 - Jewett 6 Nondimensional numbers Example: scaling These nondimensional numbers appear in, e.g., scale analysis of the equations.
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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Spring '09 term at University of Illinois at Urbana–Champaign.

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Day19-slides - Atms 502 Numerical Fluid Dynamics Thu., Oct....

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