Linear stability analysis of horizontal convection under a Gay-Lussac type approximation. (January 2022) - Mayeli, Peyman; Tsai, Tzekih; Sheard, Gregory J.
Highlight: Horizontal convection is studied under a Gay-Lussac type approximation. Linear stability analysis is conducted to predict stability threshold of the pertinent parameters. Two critical Rayleigh numbers corresponding to the minimum and maximum Gay-Lussac parameter are introduced. At any Rayleigh number between two critical Rayleigh numbers, there is a critical Gay-Lussac parameter that beyond which the horizontal convection becomes unstable. Linear stability analysis results are verified against the 3D-DNS simulations. Abstract: 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 = β Δ θ ) 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 R a cr 1 = 6.46 × 10 8 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 R a cr 2 = 4.23 × 10 8 at G a 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.