THE UNSTEADY VISCOUS FLOW OVER A GROOVED WALL: A COMPARISON OF TWO NUMERICAL METHODS (BIOT-SAVART, NAVIER-STOKES).

Abstract

Unsteady two-dimensional laminar flow of an incompressible viscous fluid over a periodically grooved wall is investigated by numerical simulation using two independent finite-difference methods. One is the vorticity-stream function method, and the other involves the vorticity-velocity induction law formulation. The fluid motion is initiated impulsively from rest and is assumed to be spatially periodic in the streamwise direction. The flow field, which includes the time development of the shear layer and the recirculating flow in the zone of separation, is examined in detail during the transient phase to the steady-state condition. The analytical and numerical formulations, which include the implementation of the boundary conditions, are derived in detail. The generation of vorticity at the solid surfaces is modelled differently in the two approaches. This vorticity production plays an important role in determining the surface-pressure distribution and the drag coefficient. Characteristics of the transient solution for a moderate Reynolds number in the laminar range are presented. Included with the graphical results are the temporal development of the constant stream function contours, including the dividing contour between the zone of separation and the main flow, and the constant vorticity contours. These latter contours show the interactions of separated vortices. The flow is found to approach a steady-state condition comprising an undisturbed uniform flow, a nonuniform irrotational flow, a shear layer adjacent to the grooved wall, and a recirculating vortex flow in the groove. Results also include the time development of the surface shear stress, surface pressure, drag coefficient and several typical velocity profiles, which characterize the flow in the recirculating region. Comparisons of the results obtained by the two numerical methods are made during the major development of the flow. The results showing the general features of the flow development including the time development of the shear layer, free shear layer and recirculating vortex flow are in good agreement. However, a significant deviation does exist at early times for the distribution of surface pressure, which accordingly has noticeable effect on the drag coefficient. Nevertheless, the gap between the distributions of surface pressure and drag coefficients dies out gradually as time progresses. The form of the stream function and vorticity contours at the steady state agrees well with those obtained from a recent numerical investigation of the steady flow in grooved channels.