Linear stability analysis of horizontal convection under a Gay-Lussac type approximation

A linear stability analysis is conducted for horizontal natural convection under a Gay-Lussac (GL) type approximation in a relatively shallow enclosure cavity. The GL type approximation is developed based on extending density variations to the advection term as well as gravity term through the momentum equation. Such a treatment invokes the GL parameter (Ga=βΔθ) as the non-Boussinesq parameter with a physical value ranging 0≤Ga≤2, characterising deviation from the classic Boussinesq approximation. Results are compared against the Boussinesq approximation in terms of the Nusselt number and skin friction. Extreme values of Ga are found to produce a counter-rotating convection cell at the hot end of the enclosure at higher Rayleigh numbers - a feature absent from Boussinesq natural convection modeling. For stability analysis purposes, linearized perturbation equations under the GL type approximation are derived and solved to characterise the breakdown of the steady two-dimensional solution to infinitesimal three-dimensional disturbances. Stability results predict that the flow remains stable up to Ra_cr1=6.46×10⁸ for the Boussinesq case (Ga=0), and then with increasing Ga the flow briefly stabilises to Ga≅0.2, then becomes progressively more unstable with futher increases in Ga, yielding a critical Rayleigh number Ra_cr2=4.23×10⁸ at Ga_max=2. Three-dimensional transition is predicted to be via an oscillatory instability mode of the steady base flow having a spanwise wavelength that increases as Rayleigh number increases. 3D-DNS simulations verify the linear stability analysis predictions in terms of growth rate, and elucidate the mode shapes achieved at saturation.