Projection method (fluid dynamics)

The projection method

The projection method is an effective means for computing accurate solutions to time-dependent incompressible fluid-flow problems. Typically the method requires that a set of, in most cases, fluid dynamics equations are solved using standard numerical means for compressible flow, and that the resulting velocity field is 'projected' onto a divergence-free vector field. This allows the divergence restraint imposed on incompressible and psuedo-incompressible flow to be maintained, whilst at the same time allowing for the use of efficient, explicit numerical techniques.

Equation set

The variable density, inviscid, incompressible equation set can be written in the following form: $\rho _{t}+\left({\vec {U}}\cdot \nabla \right)\rho =0$

$\nabla \cdot {\vec {U}}=0$

${\vec {U}}_{t}+\left({\vec {U}}\cdot \nabla \right){\vec {U}}=-{\frac {1}{\rho }}\nabla p+S_{U}$

where ${\vec {U}}$, $\rho$ and $p$ represent the velocity field, the density and the pressure respectively, and $S_{U}$ represents any external momentum source (e.g. gravitational acceleration). Additional equations, e.g. for temperature, internal energy, enthalpy, multi-phase concentration quantities etc. may be added as passive scalar equations under this construction. (NOTE: It is important to maintain equation of state coupling between thermodynamic variables and so addiational transport maybe redundant in the numerical sense.)

This set of transport equations must be solved in time whilst retaining the divergence restraint on the velocity field.

Numerical construction

Typically the projection method operates as a two-stage fractional step scheme, a method which uses multiple calculation steps for each numerical time-step. In many projection algorithms, the steps are split as follows:

1. First the system is progressed in time to a mid-time-step position, solving the above transport equations for mass and momentum using a suitable advection method. This is denoted the predictor step.
2. At this point an initial projection maybe implemented such that the mid-time-step velocity field is enforced as divergence free.
3. The corrector part of the algorithm is them progressed. These use the time-centred estimates of the velocity, density, etc. to form final time-step state.
4. A final projection is then applied to enforce the divergence restraint on the velocity field. The system has now been fully updated to the new time.