A slot with applied temperature stratification is considered when mean gravity is directed along its length and weak quasistatic jitter is applied in the spanwise direction, but when there is no component of gravity in the vertical. The behavior of the slot is governed by a number of factors: The sense of the mean gravity with respect to the applied stratification, the spanwise and lengthwise Rayleigh numbers, the Prandtl and Biot numbers, and the spanwise–lengthwise aspect ratio of the slot. A perturbation expansion of the governing equations is performed for weak spanwise jitter. At the first order of perturbation there is a circulation around the slot, producing an advected temperature field with spanwise gradients. At second order there are inflows or outflows in both the spanwise and lengthwise directions, along with a vertical redistribution of fluid. There is also a temperature field with lengthwise gradients, which typically competes with the applied temperature gradient. Equations are derived governing the vertical structure of all these fields and are solved in terms of a set of special basis functions. A parametric study is performed for the solutions. When lengthwise buoyancy forces are absent (the lengthwise Rayleigh number is zero), it is comparatively easy to deduce the required fields. However, finite lengthwise Rayleigh numbers couple the momentum and thermal equations thereby affecting the structure of the fields. Interesting behavior is predicted for small Biot numbers, when convected heat is effectively trapped in the slot: Infinitessimal flows can produce finite advected temperatures. The limits of small Biot number and small lengthwise Rayleigh number are found to be noninterchangeable. At large lengthwise Rayleigh number, boundary layers occur for stable applied stratification and layered cellular structures occur for unstable stratification. For the stable case at moderately small Biot number, the temperature jump across the boundary layer is small compared with the depth independent temperature in the bulk. Then by exploiting the boundary layer nature of the solutions, it becomes simple to predict the bulk fluid temperatures, interfacial heat fluxes and the circulations associated with the buoyant flows. Turning to the unstably stratified case, it is demonstrated that runaways can occur at first order in the spanwise jitter, and these correspond to resonant excitation of three-dimensional, stationary, long wave Rayleigh–Bénard modes. It is demonstrated how the Biot number and the spanwise–lengthwise aspect ratio of the slot influence the lengthwise Rayleigh number at which these resonances occur. There is in addition a set of two-dimensional Rayleigh–Bénard modes, which can potentially become excited at second order. When the Biot number and the spanwise–lengthwise aspect ratio are not too large, the Rayleigh numbers corresponding to the two sets of modes are nearly coincident. The second-order system will then be strongly forced near resonance, causing it to have a disproportionately large response.
- fluid equations
- stratified flows