The evolution of general three-dimensional disturbances in a thermally stratified couette flow

The evolution of general three-dimensional disturbances in a thermally stratified Couette flow is investigated with the help of initial-value problem approach. Using Boussinesq approximation, the disturbances satisfy a system of linear first-order partial differential equation which is solved subject to appropriate initial and boundary conditions using two-dimensional Fourier transformation and squire transformation. The governing stability equation is solved using both perturbation method and Fourier method. In the Fourier approach, the solution is obtained in terms of Bessel function. Perturbation solution is obtained for small values of Brunt Väisälä frequency. The terms in the perturbation series are obtained in terms of two representations of Green's function, one in the form of sine hyperbolic functions and the other in the form of Fourier sine series which represent the total energy and the energy of a single component respectively. In order to quantify disturbances amplitudes, a single measure which is basically the sum of the square of the L₂–norm of the Fourier transforms of the disturbances is used to analyze the temporal evolution of disturbances in the case of square wave pulse for velocity and temperature. Graphically it is found that the total energy and the sum of first five components of energy are almost same.