The Projection Method for the Navier-Stokes Equations
A well-known approach to solve the Navier-Stokes equations is the pressure-correction method. This type of method is a so-called segregated method, and it generally requires far less memory than the COMSOL Multiphysics default formulation. Several versions of the original method have been developed (see Ref. 12, for example). COMSOL uses incremental pressure-correction schemes.
This formulation is only available for time-dependent problems and requires the time discrete solver. It is available for the Laminar Flow and Turbulent Flow, k-ε interfaces.
This method reformulates the Navier-Stokes equations so that it is possible to solve for one variable at a time in sequence. Let u and p be the velocity and pressure variables and uc and pc the corrected velocity and pressure variables, respectively. The pressure-correction algorithm solves the Navier-Stokes equations using the following steps:
1
Solve in sequence for all u components following equation:
where the superscript denotes the time-step index, and is discretized using a BDF method up to second order where the u values from previous time steps are replaced by uc values. To first order it is discretized as:
2
Solve Poisson’s equation to adjust the pressure:
(3-59)
3
Update the corrected velocity:
For incompressible flows, the
term in Equation 3-58 and the term in Equation 3-59 are excluded.
Due to the specific time discretization scheme, this algorithm is only available with the time discrete solver.
Because the velocity components and the pressure are solved in a segregated way, some boundary conditions have a different implementation or might not be available with the projection method. In such cases, this is mentioned in the documentation for each boundary condition.
When the projection method is used for turbulent flows or with multiphysics couplings, the same algorithm is used for the velocity and pressure variables. Extra steps are needed to solve the other variables. By default the equation form used for these variables is the time-dependent form, and the time derivative is automatically discretized using a second-order BDF method.