Mixed convection flow over a horizontal plate and the horizontal wake far downstream

Abstract

The laminar mixed convection flow over a heated or cooled horizontal plate of finite length is a surprisingly intriguing problem. The hydrostatic pressure jump at the trailing edge and across the wake is to be compensated by induced circulation in the outer potential flow, similar to a plate at a non-zero angle of attack. The mixed convection flows over semi-infinite and finite horizontal plates, respectively, are thus substantially different. As the hydrostatic pressure jump remains finite along the infinitely extended wake, the induced outer potential flow around the plate would grow beyond bounds in a domain extending to infinity. Thus, boundary conditions must be imposed at the boundaries of a finite domain. However, the numerical solution of the Navier-Stokes equations remains a challenging problem due to the lack of appropriate outflow conditions. The traditional outflow condition cannot comply with the finite hydrostatic pressure jump across the wake. Thus, an asymptotic expansion for the wake far downstream from the plate is performed. Remarkably, the perturbation of the flow does not decay with increasing distance from the plate as in a classical wake. The asymptotic solution is implemented as an outflow boundary condition for the numerical solution of the Navier-Stokes equations. The results are compared with a boundary-layer solution obtained in previous work for the limiting case of vanishing Prandtl number.