Stability of a temporally evolving natural convection boundary layer on an isothermal wall

Abstract

The stability properties of a natural convection boundary layer (NCBL) adjacent to an isothermally heated vertical wall with Prandtl number 0.71 are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first invesitgated by solving the eigenvalue problem with a ‘frozen’ base flow. The critical point is found to be Grδ = 454.2 with the most unstable wavenumber of k = 0.0544, where Grδ is the Grashof number based on the velocity integral boundary layer thickness δ. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearised disturbance equations using a ‘frozen’ base flow as an initial-value problem at various Grδ. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three dimensional full direct numerical simulations of the temporally evolving flow. The selective amplification that is commonly found in the spatially developing NCBL is also observed in the temporally evolving case. The amplification rate predicted by the linear theory compares well with the direct stability analysis from Grδ ∼ 8500 to a transition point of Grδ ∼ 1.3×104, confirming the temporally evolving NCBL shares very similar instantaneous stability properties to the ‘frozen’ steady base flow in this range. The transition Grashof number also coincides with the sudden change in the base flow and the mean flow statistics. The direct simulations show the value of the transition Grashof number depends on the initial perturbation amplitude. After the transition point, the direct stability results diverge from the linear stability predictions as the non-linear mechanisms become important.