Handling solid–fluid interfaces for viscous flows: Explicit jump approximation vs. ghost cell approaches

The ghost cell approaches (GCA) for handling stationary solid boundaries, regular or irregular, are first investigated theoretically and numerically for the diffusion equation with Dirichlet boundary conditions. The main conclusion of this part of investigation is that the approximation for the diffusion term has to be second-order accurate everywhere in order for the numerical solution to be rigorously second-order accurate. Violating this principle, the linear and quadratic GCAs have the following shortcomings: (1) restrictive constraints on grid size when the viscosity is small; (2) susceptibleness to instability of a time-explicit formulation for strongly transient flows; (3) convergence deterioration to zeroth- or first-order for solutions with high-frequency modes. Therefore, the widely-used linear extrapolation for enforcing no-slip boundary conditions should be avoided, even for regular solid boundaries. As a remedy, a simple method based on explicit jump approximation (EJA) is proposed. EJA hinges on the idea that a velocity-derivative jump at the boundary reduces to the value of the velocity-derivative at the fluid side because the velocity of the stationary boundary is zero. Although the time-marching linear system of EJA is not symmetric, it is strictly diagonal dominant with positive diagonal entries. Numerical results show that, over a large range of viscosity and grid sizes, EJA performs much better than GCAs in terms of stability and accuracy. Furthermore, the second-order convergence of EJA does not depend on viscosity and the spectrum of the solution, as those of GCAs do. This paper is written with enough details so that one can reproduce the numerical results.