Stationary Regimes and Parametrization of Ekman Friction in the Karman Model of Flow Induced by External Vortical Body Force

Abstract

Using numerical simulation of the Karman model of viscous fluid flow under the action of an external vortex body force, two different stationary modes were extracted and studied in detail: with small (Batchelor regime) and with substantial (Stewartson regime) secondary circulation. The diagram of stationary regimes is plotted in the space of flow parameters—Rossby and small Ekman numbers. For the flow decaying to the stationary one in the Batchelor regime, a theoretical model is proposed based on which a stationary solution to the problem can be obtained, as well as a parameterization of the Ekman friction coefficient, the Ekman pumping velocity, and the stationary pressure in terms of average flow characteristics (vorticity and divergence). For the Stewartson regime, a parameterization of the stationary flow is proposed and a numerical investigation of the decay rate is conducted. The results of the theoretical analysis have been compared with numerical calculations and found to be in good agreement.