Implicit Solutions of the Incompressible Navier-Stokes Equations Using the Pressure Gradient Method

A modified pressure gradient method is developed for solving the incompressible twoand three-dimensional Navier-Stokes equations in primitive variables. Velocity and pressure gradient vectors, rather than the velocity and pressure, are considered as dependent variables. This choice of the dependent variables is consistent with the flow physics as the velocity field of incompressible flows is not dependent on the pressure, but rather the pressure gradient. This introduces one and two additional dependent variables for twoand three-dimensions respectively. To close the system of equations, additional equations are obtained from the identity that the curl of the pressure gradient is zero. This choice of dependent variables also leads to Dirichlet boundary conditions for the pressure gradient, which improves numerical stability and convergence rates. The modified pressure gradient method includes a pressure gradient averaging term in the momentum equations to stabilize the solution. This term is extended here to general curvilinear coordinates. The system of equations is discritized using a conservative finite volume formulation. Time integration is achieved with a fully implicit scheme, including implicit boundary conditions. The implicit scheme guarantees that continuity is satisfied to machine zero. Numerical results compare well with solutions available in the literature for the cavity flow problem and flow about a circular cylinder at Reynolds number 40.