Onset of thermal convection in a rectangular parallelepiped cavity of small aspect ratios
Abstract
Onset of thermal convection of a fluid in a rectangular parallelepiped cavity of small aspect ratios is examined both numerically and analytically under the assumption that all walls are rigid and of perfect thermal conductance exposed to a vertically linear temperature field. Critical Rayleigh number R _{ c } and the steady velocity and temperature fields of most unstable modes are computed by a Galerkin spectral method of high accuracy for aspect ratios A _{ x } and A _{ y } either or both of which are small. We find that if A _{ x } is decreased to 0 with A _{ y } being kept constant, R _{ c } increases proportionally to {A}_{x}^{4}, the convection rolls of most unstable mode whose axes are parallel to the shorter side walls become narrower, and their number increases proportionally to {A}_{x}^{\tfrac{1}{2}}. Moreover, as A _{ x } is decreased, we observe the changes of the symmetry of most unstable mode that occur more frequently for smaller A _{ x }. However, if {A}_{x}={A}_{y}=A is decreased to 0, although we again observe the increase in R _{ c } proportional to {A}^{4}, we obtain only one narrow convection roll as the velocity field of most unstable mode for all A. The expressions of R _{ c } and velocity fields in the limit of {A}_{x}\to 0 or A\to 0 are obtained by an asymptotic analysis in which the dependences of R _{ c } and the magnitude and length scale of velocity fields of most unstable modes on A _{ x } and A _{ y } in the numerical computations are used. For example, R _{ c } is approximated by {π }^{4}{A}_{x}^{4} and 25{π }^{4}{A}^{4} in the limits of {A}_{x}\to 0 and A\to 0, respectively. Moreover, analytical expressions of some components of velocity fields in these limits are derived. Finally, we find that for small A _{ x } or A the agreement between the numerical and analytical results on R _{ c } and velocity field is quite good except for the velocity field in thin wall layers near the top and bottom walls.
 Publication:

Fluid Dynamics Research
 Pub Date:
 April 2018
 DOI:
 10.1088/18737005/aaa194
 Bibcode:
 2018FlDyR..50b1402F