Unsteady inhomogeneous convective Couette flow

Abstract

The paper is devoted to constructing an exact solution of the unsteady Oberbeck–Boussinesq system that describes the development of a spatially inhomogeneous convective flow of a viscous incompressible fluid in a plane horizontal layer. The solution extends the well-known stationary class of Lin–Sidorov–Aristov solutions to the case of time-dependent velocity and temperature fields. The class under study corresponds to layered flows characteristic of large-scale geophysical processes, where the vertical velocity is negligible and the horizontal components depend linearly on the coordinates. For this class, a closed system of evolution equations is derived that takes into account the nonlinear interaction of the vorticity component with the transverse velocity. Applying the Duhamel integral reduces the problem to integral equations and yields explicit expressions for the unsteady expansion coefficients in Fourier series, ensuring high accuracy of computations. With specific boundary conditions (constant velocity and temperature at the upper boundary, zero initial conditions), an exact solution in the form of infinite series is constructed. Detailed numerical analysis reveals a non-monotonic change in the counterflow region: at the initial stage, the counterflow exists due to inertial effects, then vanishes during viscous relaxation, and reappears under the action of the nonlinear term. The relaxation times of the velocity and temperature fields, the dynamics of shear stresses, and the displacement of the extremum of the longitudinal velocity are examined. It is shown that in the unsteady regime, dynamic unloading – the shear stress vanishing at certain instants – is possible. The regularities observed are important for understanding the dynamics of vortex structures in the ocean and atmosphere. The results obtained can serve as a benchmark for verifying numerical methods for simulating convection and turbulence, as well as for interpreting large-scale geophysical flow.