Numerical solution of non-steady flows, around surfaces in spatially and temporally arbitrary motions, by using the MLPG method

The Meshless Local Petrov Galerkin (MLPG) method is used to solve the non-steady two dimensional Navier-Stokes equations. Transient laminar flow field calculations have been carried out in domains wherein certain surfaces have: (i) a sliding motion, (ii) a harmonic motion, (iii) an undulatory movement, and (iv) a contraction-expansion movement. The weak form of the governing equations has been formulated in a Cartesian coordinate system and taking into account the primitive variables of the flow field. A fully implicit pressure correction approach, which requires at each time step an iterative process to solve in a sequential manner the equations which govern the flow field, and the equations that model the corrections of pressure and velocities, has been used. The temporal discretization of the governing equations is carried out by using the Crank-Nicolson scheme. The moving Least Squares (MLS) scheme is used to generate, in a local standard domain, the shape functions of the dependent variables. The integration of the entire set of flow equations, including those equations of an elliptic elastostatic model which is used to update the position of the MLPG nodes in domains with moving surfaces, is carried out in the local standard domain by using the Gauss-Lobatto-Legendre quadrature rule. The weight function used in the MLS scheme, and in the weighted residual MLPG process, is a compactly supported fourth order spline. We conclude that the MLPG method coupled with a fully implicit pressure-correction algorithm, is a viable alternative for the solution of fluid flow problems in science and engineering, particularly those problems characterized by non-steady fluid motion around flexible bodies with undulatory or contraction-expansion movements.