Numerical calculations of two-dimensional large Prandtl number convection in a box

Abstract

Convection from an isolated heat source in a chamber has been previously studied numerically, experimentally and analytically. These have not covered long time spans for wide ranges of Rayleigh number Ra and Prandtl number Pr. Numerical calculations of constant viscosity convection partially fill the gap in the ranges $\mathit{Ra}= 1{0}^{3} {{\unicode{x2013}}}1{0}^{6} $ and $\mathit{Pr}= 1, 10, 100, 1000$ and $\infty $. Calculations begin with cold fluid everywhere and localized hot temperature at the centre of the bottom of a square two-dimensional chamber. For $\mathit{Ra}\gt 20\hspace{0.167em} 000$, temperature increases above the hot bottom and forms a rising plume head. The head has small internal recirculation and minor outward conduction of heat during ascent. The head approaches the top, flattens, splits and the two remnants are swept to the sidewalls and diffused away. The maximum velocity and the top centre heat flux climb to maxima during head ascent and then adjust toward constant values. Two steady cells are separated by a vertical thermal conduit. This sequence is followed for every value of $Pr$ number, although lower Pr convection lags in time. For $\mathit{Ra}\lt 20\hspace{0.167em} 000$ there is no plume head, and no streamfunction and heat flux maxima with time. For sufficiently large Ra and all values of Pr, an oscillation develops at roughly $t= 0. 2$, with the two cells alternately strengthening and weakening. This changes to a steady flow with two unequal cells that at roughly $t= 0. 5$ develops a second oscillation.

Constraining the source of mantle plumes

“…The numerical simulations of Whitehead et al (2013) show that when the Prandtl number is greater than ⇠ 10 3 , the dynamics of thermal plumes are essentially identical to those calculated at an infinite Prandtl number, confirming that our experiments are relevant to the mantle.…”

Convection driven by temperature and composition flux with the same diffusivity

“…Whether the perturbation grows or decays is easily viewed by subtracting out the original linear distributions of T and C. In all cases, flows have the same qualitative structure irrespective of grid size and only the quantitative values change a little. This result is also found for plume studies (Whitehead et al 2013). Benchmark calculations with different grid sizes (in parenthesis) and a time step of 10 −6 with Raf = 0 give the four critical values Rac = 805 (32), 791 (64), 782 (128), and 780 (256).…”

“…The rectangular grid is two-dimensional with 128 points on a side giving an accuracy of 1 part in 10 6 . Resolution was tested as in an earlier study (Whitehead et al 2013) and all extrema of the fields are accurate to better than 1%. Since C has only flux boundary conditions, there is a slow random drift in the spatially averaged value of concentration C within the chamber over long times.…”

The continental drift convection cell

“…For most runs the continent starts on the far right, but trial runs with other initial locations produce no change in properties reported here. The procedure has equation advanced numerically in time [ Whitehead et al ., ] using a leapfrog‐trapezoidal scheme. Then equation is solved by inverting the Poisson equation, and equation is solved the same way.…”