1. Themomentum equations(11)couplethe velocityvectortothe pressurefield. Thevelocity-pressurecoupling becomes a key issue when reaching the incompressible limit. A decoupling of both fields can be achieved by applying a decomposition to the velocity field into an interim and a final solution leading to a fractional step method. This is the basic idea of projection-based schemes. The projection method was originally introduced by Chorin4,5and Temam54in the late 1960's for constant density flows. They perform a Helmholtz decomposition by which an arbitrary vector field Ψ is split up into a solenoidal and an irrotational part. In a more general approach, an arbitrary vector field Ψ can be decomposed into a vector field with a so-called divergence-constraint Ψdcand an irrotational vector field Ψir. Without loss of generality, the irrotational part can further on be defined as the gradient of an arbitrary scalar field Φ. Taking the divergence of Ψ leads to:
2. The interim velocity vector is obtained through the so-called predictor step by solving the momentum equations and first ignoring the pressure gradient term. This formulation of the predictor step leads to the so-called non-incremental projection scheme. Incremental projection schemes add the pressure gradient term obtained from the last time step as an explicitly treated term to the left hand side of the momentum equations. In this case, Φ is set to the difference between the pressure from the new and old time steps, multiplied with a constant coefficient. The incremental projection method for constant density flows has been introduced by Goda19in 1979. Both approaches realize a decoupling of the velocity and pressure variables as part of the predictor step.
3. The first step can be seen as the predictor step of the CPM algorithm. The steps 2-7 represent the corrector step of the CPM scheme. The steps 8 and 9 form the thermodynamic closure of the CPM method.