Reynolds stress and the energy balance of a localized two-dimensional vortex in a uniform shear flow

摘要

Consideration is given to the kinetic energy balance of a localized two-dimensional vortex in unbounded space, subject to a uniform background shear flow. For this problem, a quadratic invariant based on the total flow can be constructed that consists of the sum of the vortex self-energy and the energy of interaction with the background flow. It is shown that an energy equation also may be written for the rate of change of vortex self-energy, relating this to the rate of working by the Reynolds stress. The stress integral is demonstrated to converge for a localized vortex of finite circulation, in contrast to the total kinetic energy. The two approaches to the energy balance are shown to be complementary, and the relation between the Reynolds stress and interaction energy is discussed. As an example, the integrated Reynolds stress is evaluated for a uniformly sheared elliptical (Kirchhoff) vortex. The stress integral includes far field contributions, indicating that appreciable exchange of energy with the external flow occurs well beyond the boundary of the vortex.