Scaling of heat flux and energy spectrum for very large Prandtl number convection

Abstract

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ rms . We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra γ with γ ≈ ( 0.30 – 0.32 ) , Pe ∼ Ra η with η ≈ ( 0.57 – 0.61 ) , and θ rms ∼ const . The Nusselt number is observed to be an intricate function of Pe , θ rms , and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E u ( k ) ∼ k − 13 / 3 , which is in a very good agreement with our numerical results. The entropy spectrum E θ ( k ) , however, exhibits dual branches consisting of k − 2 and k 0 spectra; the k − 2 branch corresponds to the Fourier modes ˆ θ ( 0 , 0 , 2 n ) , which are approximately − 1 / ( 2 n π ) . The scaling relations for Prandtl number beyond 10 2 match with those for infinite Prandtl number.