A spectral/finite‐difference approach for narrow‐channel flow with inertia

Abstract

SUMMARYA hybrid spectral=ÿnite-di erence scheme is proposed to determine the inertial ow inside narrow channels. The ow ÿeld is represented spectrally in the depthwise direction, which together with the Galerkin projection lead to a system of equations that are solved using a variable step ÿnite di erence discretization. The method is particularly e ective for non-linear ow, and its validity is here demonstrated for a ow with inertia. The problem is closely related to high-speed lubrication ow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing with the fully two-dimensional ÿnite-volume solution (FLUENT), and the widely used depth-averaging method from shallow-water theory. It is found that a low number of modes are usually su cient to secure convergence and accuracy. Good agreement is obtained between the low-order description and the ÿnite-volume solution at low to moderate modiÿed Reynolds number. The depth-averaging solution is unable to predict accurately (qualitatively and quantitatively) the high-inertia ow. The in uence of inertia is examined on the ow.