Shaaban Abdallah
University of Cincinnati, USA
Title: Review of the state of the art and new mathematical formulations for solutions of the incompressible Navier-Stokes equations in velocity and pressure derivatives
Biography
Biography: Shaaban Abdallah
Abstract
Numerical simulations of the incompressible Navier-Stokes equations in primitive variables employ the velocity and pressure as dependent variables. The lack of an equation for the pressure causes numerical difficulties in calculating the pressure. Two formulations have been developed for solving the pressure problem. The Artificial Compressibility method modifies the continuity equation by adding a time dependent pressure term, and the pressure Poisson method enforces the continuity equation through the divergence of the momentum equation. The pressure equations are solved iteratively and upon convergence of the numerical solution, the continuity equation is satisfied. Mostly these techniques are developed and tested using explicit schemes. An implicit approach for solving the incompressible Navier Stokes equations known as the Fractional Step method derives an approximate pressure Poisson equation. It is important to state here that the velocity field of incompressible flows does not depend on the pressure but it depends on the pressure derivatives. Recently, we developed two methods that employ the velocity and pressure derivatives as dependent variables. The pressure derivatives increase the dependent variables to four in two-dimensions and six in three dimensions. Additional governing equations are obtained from the identity that the Curl Gradient of the pressure is zero. In the second method, the continuity equation is replaced by its spatial derivatives and enforces the continuity equation at the boundary. These techniques are developed and tested using explicit time-marching schemes. They do not require boundary conditions for the pressure and they satisfy the continuity equation to machine zero. In this study, we re-derived the above mentioned methods using lower and upper (LU) factorization of the discretization matrix of the governing equations. In addition, we developed the implicit technique of the velocity and Pressure derivatives formulation. Exact LU factorization of the matrix representing the discrete Navier-Stokes equations is obtained. The main advantage of this approach, in addition to the above mentioned advantages of the velocity and pressure derivatives explicit methods, is that no matrix inversion of the implicit operator is required.