arXiv subjects

Philip Winchester

Publications and source records attributed to Philip Winchester.

3 recordsLinked to original sources

arXiv2023

Two-dimensional Rayleigh-Bénard convection without boundaries

We study the effects of Prandtl number $Pr$ and Rayleigh number $Ra$ in two-dimensional Rayleigh-Bénard convection without boundaries, i.e. with periodic boundary conditions. In the limits of $Pr \to 0$ and $\infty$, we find that the dynamics are dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large scale dissipation. For finite Prandtl number in the range $10^{-3} \leq Pr \leq 10^2$, the Nusselt number tends to follow the `ultimate' scaling $Nu \propto Pr^{1/2} Ra^{1/2}$, and the viscous dissipation scales as $ε_ν\propto Pr^{1/2} Ra^{-1/4}$. The latter scaling is based on the observation that enstrophy $\langle ω^2 \rangle \propto Pr^0 Ra^{1/4}$. The inverse cascade of kinetic energy forms the power-law spectrum $\hat E_u(k) \propto k^{-2.3}$, while the direct cascade of potential energy forms the power-law spectrum $\hat E_θ(k) \propto k^{-1.2}$, with the exponents and the turbulent convective dynamics in the inertial range found to be independent of Prandtl number. Finally, the kinetic and potential energy fluxes are not constant in the inertial range, invalidating one of the assumptions underlying Bolgiano-Obukhov phenomenology.

physics.flu-dyn↗

arXiv2021

The onset of zonal modes in two-dimensional Rayleigh-Bénard convection

We study the stability of steady convection rolls in 2D Rayleigh--Bénard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and five orders of magnitude in the Rayleigh number $(8π^4 < Ra \leq 3 \times 10^7)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $Γ= 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $\Pr \to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-$\Pr$ limit, we observe that the stability boundary reaches an asymptotic value $Ra = 2.54 \times 10^7$ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.

physics.flu-dyn↗

arXiv2020

Zonal flow reversals in two-dimensional Rayleigh-Bénard convection

We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Bénard convection in a two-dimensional, rectangular geometry of aspect ratio $Γ$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $Γ= 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $Γ= 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.

physics.flu-dyn↗