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

A hybrid spectral/finite-difference scheme is proposed to determine the inertial flow inside narrow channels. The flow field 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 finite difference discretization. The method is particularly effective for non-linear flow, and its validity is here demonstrated for a flow with inertia. The problem is closely related to high-speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing with the fully twodimensional finite-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 sufficient to secure convergence and accuracy. Good agreement is obtained between the low-order description and the finite-volume solution at low to moderate modified Reynolds number. The depthaveraging solution is unable to predict accurately (qualitatively and quantitatively) the high-inertia flow. The influence of inertia is examined on the flow.