Are there any numerical advantages of the vorticity equation over the original momentum equation in the study of 2D incompressible flows?
The originar N-S equation have the pressure term in gradient form. Hence pressure poison equation is also needs to be solved for correcting pressure while solving original N-S equation in incompressible flow for imposing mass conservation. Vorticity formulation avoids that Pressure correction process, which is a critical step in solving in incompressible flows. But providing boundary condition for Vorticity formulation is a tricky one since it is a transformed equations, that is why Vorticity formulations are used for simple problems like Lid-Driven cavity, Backward facing steps etc. But for a complex problem and flow over complex geometry it is better to use original N-S equations.
The basic idea is that pressure term is eliminated and stream function and vorticity can be computed easily from the transformed equations.
The originar N-S equation have the pressure term in gradient form. Hence pressure poison equation is also needs to be solved for correcting pressure while solving original N-S equation in incompressible flow for imposing mass conservation. Vorticity formulation avoids that Pressure correction process, which is a critical step in solving in incompressible flows. But providing boundary condition for Vorticity formulation is a tricky one since it is a transformed equations, that is why Vorticity formulations are used for simple problems like Lid-Driven cavity, Backward facing steps etc. But for a complex problem and flow over complex geometry it is better to use original N-S equations.
Vorticity-based algorithms are computationally powerful because they lend themselves to grid-free calculations. They are very useful if you can assume that the vorticity is concentrated in space. However, as suggested above, boundaries present a particular challenge to this type of approach.
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