Exponential boundary-layer approximation space for solving the compressible laminar Navier-Stokes equations

Abstract

In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional steady compressible Navier-Stokes equations under some restrictions, we show that the flow variables actually vary in the form of an exponential function instead of a polynomial one inside the boundary layer. Then, we propose an exponential space for approximating the solutions inside the boundary layer and numerically implementing it in the frame of direct discontinuous Galerkin (DDG) method. We show that the DDG methods based on the exponential boundary-layer space give much better numerical results for both conservative variables and wall-normal gradients than those with the standard polynomial space. Generally only 1–2 grid points inside the boundary layer are demanded to resolve the wall boundary layer to obtain satisfactory wall-normal gradients under the exponential space. Preliminary extension to two-dimensional laminar boundary-layer flow shows a similar performance of the proposed exponential boundary-layer space, exhibiting its potential applications in high dimensions.