Boundary layer instability of the natural convection flow on a uniformly heated vertical plate

Direct numerical simulation is employed to investigate the two-dimensional boundary layer instability of a natural convection flow on a uniformly heated vertical plate submerged in a homogeneous quiescent environment. A Boussinesq fluid with Prandtl numbers of Pr = 0.733 (air) and 6.7 (water), in the local Rayleigh number range 0 ≤ Ra _x ≤ 2.4 × 10 ¹⁰, is studied. Controlled low amplitude numerical disturbances introduced into the base flow excite unstable travelling waves, with the resulting waves tracked and analyzed as they travel up the boundary layer. The numerical simulation readily reproduced what is predicted by the parallel linear stability theory for the two dimensional mode relatively short wave spectrum, but not for some parts of the long wave spectrum. Critical Rayleigh numbers have been obtained separately for both the temperature and velocity signals using the numerical results, and shown to be in good agreement with each other provided the data is renormalized using the boundary layer scalings of Sparrow and Greg [1]. It has been shown that the disturbance behavior depends on the Prandtl and Rayleigh numbers, the excitation frequency and to a lesser extent the prescribed thermal coupling at the plate.