Kinetic energy conservation in 2D vortical flow

Abstract

In this study, distribution of vorticity in 2D vortical flow is evaluated by using energy conservation law. A variational formula named “SPKE functional” (singular part of kinetic energy functional) is introduced, which explain the relation between the rate of kinetic energy transferred to the fluid by a moving body and the production and distribution of vorticity in the fluid. As an example, high Reynolds number flow past a circular cylinder is considered and numerical calculation for velocity, pressure field, drag force and rate of kinetic energy is provided. Using these results, two important consequences are deduced: first, drag force obtained from solving the momentum equation by using conventional vortex method, is compared with the equivalent drag force caused by the energy transferred to the fluid. This is done by using SPKE functional, and validity of vortex method to satisfy the conservation of energy is verified and approved. Second, a functional for 2D vortical flow is derived which has a physical meaning of equivalent energy transferred to the fluid due to the generation of vorticity. This is a function of kinematical parameters of fluid flow such as stream function and vorticity field. Two aspects of this formulation contain great importance: (i) derivation of a functional for 2D vortical flow and (ii) drag force being calculated directly from the vorticity field. The method presented here is capable of obtaining total drag force directly from the discrete vorticity field, as well as calculating either the pressure field. This is where first derivatives of vorticity (or even second derivatives of velocity) is not required. SPKE functional can be used in variational approach for studying 2D vortical flow field numerically.