A novel Hermite Taylor least square based meshfree framework with adaptive upwind scheme for two dimensional incompressible flows.

*(English)*Zbl 1390.65131Summary: A new meshfree framework based on Hermite Taylor least square finite difference method is proposed. The conventionally used least square finite difference (LSFD) scheme with ghost point method for Neumann boundary conditions is known to have shortcomings especially for irregular nodal distributions. In this work, the performance of the LSFD scheme is augmented by incorporating a novel Hermite Taylor least square (HTLS) method for easy and efficient implementation of the Neumann boundary conditions. The method is initially validated by solving a Poisson equation with both Dirichlet and Neumann boundary conditions. With its promising numerical performance, the method is extended to the full Navier-Stokes equations in two dimensions. An innovative adaptive upwind scheme is adopted to handle the convective terms in the momentum equations by modifying the support domain in the upstream direction. By using a modified Euclidean distance function according to the local flow direction and the value of parameter that controls the convection effect (mesh Peclet number), the local support domain can be shifted towards the upstream direction thereby naturally incorporating the upwind effect while computing the coefficients for the LSFD method. The Navier-Stokes equations are solved in a primitive variables (velocity and pressure) approach by using a first order semi-implicit projection method. In order to validate the developed framework, three flow problems (lid driven cavity, channel flow and flow over a circular cylinder) are considered. All of these problems are well documented because of their benchmarking relevance. It is observed that the new framework produces results which match qualitatively as well as quantitatively with earlier established theory and observations and hence demonstrate its ability to successfully simulate flows of practical interest in an entirely meshfree approach.

##### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

##### Software:

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\textit{M. P. Borthakur} and \textit{A. Biswas}, Comput. Fluids 130, 37--48 (2016; Zbl 1390.65131)

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