Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

The solution of fluid flow problems exhibits a singular behaviour when the conditions imposed on the boundary display some discontinuities or change in type. A treatment of these singularities has to be considered in order to preserve the accuracy of high‐order methods, such as spectral methods. The present work concerns the computation of a singular solution of the Navier–Stokes equations using the Chebyshev‐collocation method. A singularity subtraction technique is employed, which amounts to computing a smooth solution thanks to the subtraction of the leading part of the singular solution. The latter is determined from an asymptotic expansion of the solution near the singular points. In the case of non‐homogeneous boundary conditions, where the leading terms of the expansion are completely determined by the local analysis, the high accuracy of the method is assessed on both steady and unsteady lid‐driven cavity flows. An extension of this technique suitable for homogenous boundary conditions is developed for the injection of fluid into a channel. The ability of the method to compute high‐Reynolds number flows is demonstrated on a piston‐driven two‐dimensional flow.