Algebraic-volume meshfree method for application in finite volume solver

Abstract

Nowadays the finite volume (FV) method remains the most popular choice for Computational Fluid Dynamics (CFD). But few meshfree methods have been developed based on the framework of existing FV codes which can boost the development of the meshfree method. An algebraic volume (AV) concept for the meshfree method is proposed to mimic the geometric control volume (CV) of the FV method via a meshfree point cloud. Similar to the CV, each AV is closed by algebraic faces which are constructed via the weighted least-squares method. Only the bias fluxes are solved at the midpoint of the current point and its neighbor. The other flow variables of Navier–Stokes (NS) equations are solved in the same way as the traditional FV method. So it automatically solves the compressible flow with shock wave without additional treatment. The AV meshfree method only requires to modify the calculation method of volume and face area vector from a geometric way to an algebraic solution in pre-processing, therefore it is straightforward to implement in a traditional FV solver. The numerical methods of upwind schemes for convective flux, corrected central difference for viscous flux and implicit temporal discretization are identical for both the mesh-based FV method and AV meshfree method. The AV meshfree method has been integrated into an FV code and applied to high Reynolds number, transonic, viscous flow simulations. The accuracy and convergence of both the FV method and AV meshfree method are verified via 2D low-speed flow and transonic 3D flow. The results of AV meshfree method agree quite well with those of the FV method and experimental data. And the comparison shows good point size convergence. Furthermore, a demo of hybrid FV method and AV meshfree method is presented to investigate the potential of this new meshfree method. The trajectory results obtained by CFD and experimental data are in good agreement.