Stratified non-diffusive flow over a horizontal flat plate

Abstract

The effects of inertia on the upstream-growing boundary layer over a finite horizontal flat plate of length b moving uniformly with speed U0 in a linearly stratified \[ [(dp/dy)_{-\infty}- = - \rho_0\beta)], \] viscous, non-diffusive fluid under the Boussinesq approximation are studied. The nonlinear inertia terms are linearized by the Oseen approximation, but no boundary-layer approximation is required. The flow is governed by two parameters, namely the internal Froude number Fr[= U0/(βgb2)½] and a parameter L3[= βgb3/U0ν], where is proportional to the ratio of boundary-layer thickness to plate length for the case Fr = 0. Large values of L3 and Fr2 = 0 correspond to the case of an upstream boundary layer. By increasing the Froude number gradually, a transition occurs from an upstream boundary layer accompanied by an upstream wake to a downstream boundary layer with a downstream wake. The upstream boundary layer and wake are characterized by a balance of viscous and buoyancy forces, whereas the downstream boundary layer and wake are characterized by a balance of inertia and viscous forces. In the so-called critical-boundary-layer case, Fr4L3 = O(1), inertia, viscous and buoyancy forces are all important and this boundary layer is accompanied by both upstream and downstream wakes. Complete transition occurs when Fr4L3 increases from 10·0 to 1000·0. The drag on the plate is also calculated.