title: 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 θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.
authors.affiliation: Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
authors.forename: Ambrish
authors.initials: A
authors.lastname: Pandey
authors.affiliation: Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
authors.forename: Mahendra K
authors.initials: MK
authors.lastname: Verma
authors.affiliation: Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
authors.forename: Pankaj K
authors.initials: PK
authors.lastname: Mishra