CFD Online Discussion Forums

(1). I am not going to talk about this 'not so simple SIMPLE METHOD'. (2). As you know that the Navier-Stokes equations are non-linear. To make the thing more complicated, let's take the steady-state equation for incompressible flow for example. You have a first-order continuity equation, momentum equations and energy equation of second-order. (3). Since we don't know where and how to obtain the solution for this set of steady-state equations, the iterative method is normally used to obtain the solution from the discretized equations. (4). In the incompressible flow, the pressure field is a function of the velocity field. In other word, you don't have to solve the velocity field and the pressure field at the same time. But, in the primitive variable formulation, velocity and pressure are used to be the primitive variable at the same time. (5). This is the case with the SIMPLE METHOD, which deal with the velocity and the pressure directly at the same time. Therefore, in order to solve the momentum equations with two unknown primitive variables ( velocity vector and the pressure), one has to first guess at the value of the pressure field ( guessing is basically iterative) and solve the momentum equations for the velocity field ( How?) (6). The steady-state momentum equation with known pressure term is still non-linear. So this part has to be solved iteratively in order to obtain the converged final velocity field consistent with the boundary conditions of the particular problem you are trying to solve. (7). But since the pressure field is the identical twin of the velocity field, the guessed initial pressure values are not the right answer. So, you have to update the pressure field through a separate route. For this reason, a separate pressure route, or pressure equation is derived in such a way that an updated pressure can be computed. This is saying that the pressure field ( the identical twin of the velocity field) also has to be iteratively updated. (8). If you are lucky, if your initial guess is good (I don't know how to define it yet), and if the iterative method parameters are properly selected ( I don't know how to set those either), the posibility of obtaining the answer is relatively high. (9). Since the pressure can be decoupled from the original momentum equations easily, the separate route is to use the "vorticity-velocity" formulation, or in 2-D case the "vorticity-stream function" formulation. These methods can also be used for variable density, turbulent flows. In this case, the pressure field is computed after the converged velocity field is obtained. I have been using the vorticity related approach for turbulent, variable density flow with low Reynolds number two-equation models for over 25 years. It is relatively easy to use with this approach. Unfortunately, you are not going to find my published papers in this area. (10). Since the pressure-based methods also are widely used in commercial codes, it might be a good idea to learn how these pressure-related methods actually work. good luck.