The Horizontal Far Wake Behind a Heated or Cooled Body

Abstract

Buoyancy affects the horizontal wake far downstream of a heated or cooled body, especially a horizontal plate, in an indirect manner via the hydrostatic pressure perturbation. Plane (2D) flow at very large Reynolds and Péclet numbers is considered in the present paper. Both laminar and turbulent flows are investigated, with the aim of providing asymptotic solutions that are suitable as outflow boundary conditions for numerical solutions of the equations of continuity, momentum, thermal energy and, in case of turbulent flow, the balance of turbulent kinetic energy. Dimensionless variables are introduced, using the total heat flow from, or towards, the plate as a parameter. It turns out that the buoyancy effects in the momentum equations and in the turbulent kinetic energy balance, respectively, are of the same order of magnitude and can be characterized by a Richardson number. The asymptotic expansions for large distances from the plate lead to a set of ordinary differential equations for a self-similar flow field. The interaction between the wake and the potential flow is taken into account by applying Bernoulli’s equation as a boundary condition to the momentum equation of the wake. As the thermal energy equation as well as the boundary conditions for the temperature perturbation are homogeneous, the solution of the temperature field contains a free coefficient, which is determined from the over-all thermal energy balance. The results of the analysis are in remarkable contrast to the classical solutions for the wake flow without buoyancy. In particular, driven by the hydrostatic pressure disturbance, the flow does not decay with increasing distance from the plate. Furthermore, the flow is governed by the total heat flow at the plate, whereas the effect of the drag force acting on the plate is negligible. The set of ordinary differential equations is solved numerically. For laminar flow, two kinds of solutions are found. One of them describes a flow field containing a region of reversed flow. In case of turbulent flow a turbulence model based on the turbulent kinetic energy balance is applied. In addition the limit of very weak buoyancy effects is considered, leading to power laws in terms of the Richardson number.