Implementation of slip boundary conditions in the finite volume method: new techniques

Abstract

SUMMARY Two different techniques for the implementation of the linear and nonlinear slip boundary conditions into a finite volume method based numerical code are presented. For the linear Navier slip boundary condition, an implicit implementation in the system of equations is carried out for which there is no need for any relaxation, especially when handling high slip coefficients. For three different nonlinear slip boundary conditions, two different methods are devised, one based on solving a transcendental equation for the boundary and the other on the linearization of the slip law. For assessment purposes, comparison is made between these new methods and the usual iterative process. With these new methods, the convergence difficulties, typical of the iterative procedure, are eliminated, and for some of the test cases, the convergence rate even increased with the slip velocity. The details of these implementations are given first for a simple geometry using orthogonal meshes and Cartesian coordinates followed by their generalization to non‐Cartesian coordinates and nonorthogonal meshes. The developed code was tested in the benchmark slip‐stick and 4:1 contraction flows, evidencing the robustness of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.

Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction

“…A detailed study of the flow characteristics, focusing on the influence of wall slip velocity coefficient on the vortex growth, intensity and sizes is provided. For such purpose, we use an efficient procedure that calculates the slip velocity along the iterations of the numerical procedure, by incrementally increasing the slip velocity, so that a smaller slip velocity than the velocity at the center of the nearest computational cell, is obtained (a necessary condition to avoid numerical divergence [31]). This introduction is followed by Section 2 where the governing equations together with the wall slip boundary condition employed are presented.…”

Numerical Analysis of Rarefied Gas Flows Using the Academic CFD Code Galatea

“…The incorporated second-order accurate model didnt produce improved results; this lack of difference is attributed to the relatively fine grid utilized. In addition, the incorporated normalization scheme [18] was revealed a considerable enhancement, as the Dirichlet-type of the slip/jump boundary conditions was found to cause excessive oscillations without it, especially during the initial steps of the iterative solution procedure. The proposed solver was validated against a benchmark test case concerning rarefied laminar flow over a wing with a NACA0012 airfoil in different angles of attack [13,19,29].…”

“…It actually represents the vorticity flux into the surface divided by the vorticity of the flow field on the surface, obtained by the no-slip approximation; in this study it was set equal to -1.0 [14]. The velocity slip and temperature jump boundary conditions are implemented in a Dirichlet way, hence they are susceptible to produce oscillations during the iterative procedure (especially during the initial steps of an explicit iterative scheme), or even lead simulation to fail [18]. Therefore, a normalization scheme was incorporated in Galatea solver, allowing for the gradual increase of the slip velocity and temperature values; mathematically, it is described as follows [18] U i s = αU i−1 s…”

“…Among them the model of Beskok and Karniadakis appears to be especially attractive, as it avoids the numerical difficulties, entailed by the evaluation of the second derivative of slip velocity when complex geometries along with unstructured hybrid grids are encountered [14,15]. As the aforementioned boundary conditions are imposed in a wall-function mode, they are susceptible to produce residual oscillations, especially during the initial steps of an explicit iterative solution; a remedy to this drawback was proposed by Ferras et al [18], where a normalization scheme is considered, allowing for the gradual increase of the slip/jump values. Finally, few quite many studies include comparisons of the DSMC and the modified by the slip/jump boundary conditions Navier-Stokes PDEs approaches (as in this work), demonstrating the potential of such methodologies to effectively simulate rarefied gas flows at slip flow regime [1,14,19].…”

“…Particularly, the slip model of Beskok and Karniadakis [14,15] was adopted due to its relatively easy implementation on unstructured, tetrahedral or hybrid grids; it can overcome the numerical difficulties, entailed by the evaluation of the second derivative of slip velocity [14]. Furthermore, the normalization scheme, reported in [18], was included to mitigate the excessive oscillations, caused by the Dirichlet-type of the slip/jump boundary conditions, especially during the initial steps of the iterative solution procedure. The proposed solver was validated against a benchmark test case concerning rarefied laminar flow over a wing with a NACA0012 airfoil in different angles of attack [13,19,29].…”