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 ${\theta }_{\mathrm{rms}}$. We complement the analytical work with direct numerical simulations, and show that $\mathrm{Nu}\sim {\mathrm{Ra}}^{\gamma }$ with $\gamma \approx (0.30-0.32)$, $\mathrm{Pe}\sim {\mathrm{Ra}}^{\eta }$ with $\eta \approx (0.57-0.61)$, and ${\theta }_{\mathrm{rms}}\sim \mathrm{const}$. The Nusselt number is observed to be an intricate function of $\mathrm{Pe}$, ${\theta }_{\mathrm{rms}}$, and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum $E_u(k)\sim k^{-13/3}$, which is in a very good agreement with our numerical results. The entropy spectrum $E_{\theta }(k)$ however exhibits dual branches consisting of $k^{-2}$ and $k^0$ spectra; the $k^{-2}$ branch corresponds to the Fourier modes $\hat{\theta }(0,0,2n)$, which are approximately $-1/(2n\pi )$. The scaling relations for Prandtl number beyond $10^2$ match with those for infinite Prandtl number.

14 pages, 14 figures

References in corpus (5)

• The unifying theory of scaling in thermal convection: The updated prefactors
• A Comparison of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux
• Benchmarking and scaling studies of a pseudospectral code Tarang for turbulence simulations
• Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View
• Role of Bulk flow in Turbulent Convection

Cited by in corpus (21)

• Phenomenology of buoyancy-driven turbulence: recent results
• Energy Spectrum of Buoyancy-Driven Turbulence
• Scaling of a Fast Fourier Transform and a Pseudo-spectral Fluid Solver up to 196608 cores
• Scaling of large-scale quantities in Rayleigh-Bénard convection
• Near isotropic behaviour of turbulent thermal convection
• Dynamos at extreme magnetic Prandtl numbers: Insights from shell models
• Large eddy simulations of turbulent thermal convection using renormalized viscosity and thermal diffusivity
• Steady Rayleigh--Bénard convection between stress-free boundaries
• Thermal boundary layer structure in low-Prandtl-number turbulent convection
• Flow reversals in turbulent convection with free-slip walls
• Variable energy flux in turbulence
• Combined particle image velocimetry and thermometry of turbulent superstructures in thermal convection
• Dynamics of large-scale quantities in Rayleigh-Bénard convection
• Revisiting Reynolds and Nusselt numbers in turbulent thermal convection
• Similarities between 2D and 3D convection for large Prandtl number
• Prandtl number dependence of the small-scale properties in turbulent Rayleigh-Bénard convection
• Scaling and spatial intermittency of thermal dissipation in turbulent convection
• Non-Boussinesq convection at low Prandtl numbers relevant to the Sun
• On the Applicability of Low-Dimensional Models for Convective Flow Reversals at Extreme Prandtl Numbers
• Similarities between characteristics of convective turbulence in confined and extended domains
• Reversals in infinite-Prandtl-number Rayleigh-Bénard convection