Three-dimensional convective cooling in a vertical channel with flush-mounted heat sources

Three-dimensional free convection in a vertical channel with spatially periodic, flush-mounted heat sources is investigated by a spectral element method. All numerical solutions are obtained using a time-accurate finite-difference integration scheme capable of capturing temporal instabilities that spontaneously appear at large values of Grashof number, Gr. In addition, the leading order approximation of the 3-D solution for small Gr is derived and compared with the numerical solutions. The agreement is excellent for sufficiently small Gr. Computations are carried out for a Boussinesq fluid, Prandtl number, Pr = 0.71, non-dimensional reference temperature, Θ_b* = 0.12 and values of Grashof number in the range 0.1 ≤ Gr≤ 5 x 10⁴. For given aspect ratios, and for sufficiently small values of Grashof number, the solution evolves to a unique, time-independent state that exhibits the maximum symmetry consistent with the boundary conditions. At Gr* ≃ 28,000, self-sustained oscillations appear spontaneously in the flow and thermal fields. For time-dependent solutions (Gr ≥ Gr*) the symmetry of the flow and temperature fields breaks down. Temperature and velocity distributions as well as maximum temperature, maximum velocity and local Nusselt number distributions are presented for the values of Grashof number studied. For time-dependent flows, instantaneous as well as averaged-in-time solutions are discussed.