A layer of water deep has a top-to-bottom temperature difference of

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Convection, rotation and geophysical flows

Fluid Dynamics, Ch Thermal convection and the Rayleigh-Benard problem

Problem

A layer of water 2.0 cm2.0\,\mathrm{cm} deep has a top-to-bottom temperature difference of 5.0 K5.0\,\mathrm{K}. Use αT=2.07×10−4 K−1\alpha_T = 2.07\times10^{-4}\,\mathrm{K^{-1}}, ν=1.00×10−6 m2 s−1\nu = 1.00\times10^{-6}\,\mathrm{m^2\,s^{-1}}, κ=1.43×10−7 m2 s−1\kappa = 1.43\times10^{-7}\,\mathrm{m^2\,s^{-1}} and g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. The viscous diffusion time is τν=d2/ν\tau_\nu = d^2/\nu, the thermal diffusion time is τκ=d2/κ\tau_\kappa = d^2/\kappa, and the free-fall time is τff=d/(gαTΔT)\tau_{\rm ff} = \sqrt{d/(g\alpha_T\Delta T)}. ((a)) Find τν\tau_\nu, τκ\tau_\kappa and τff\tau_{\rm ff} for this layer. ((b)) Form τντκ/τff2\tau_\nu\tau_\kappa/\tau_{\rm ff}^2 and compare it with the Rayleigh number computed directly from Ra=gαTΔTd3/(νκ)Ra = g\alpha_T\Delta T d^3/(\nu\kappa).

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