sandbox/hasansh/README

README

Constant density method of [Dodd and Ferrante, 2014]

Projection method is the most common method for solving incompressible navier stokes equations. However, solving a poisson equation for pressure significantly reduces the speed of calcualtions. Dodd and Ferrante, 2014 proposed a constand density method for increasing the effciency of solving poisson equation.

In this method instead of solving a variable coefficient version of poisson equation:

\displaystyle \nabla \cdot \left(\frac{1}{\rho^{n+1}}\nabla p^{n+1}\right)= \frac{1}{\Delta t} \nabla \cdot u^{*}

we solve a constant coefficient method as following:

\displaystyle \nabla ^2 p^{n+1}= \nabla \cdot \left[\left(1-\frac{rho_0}{\rho^{n+1}}\right)\nabla \hat{p}\right]+\frac{\rho_0}{\Delta t} \nabla \cdot u^{*}

You can see details of implementation of this method in fast-poisson.h

There are two version of constant density method which are called FASTP* and FASTPn. The difference between these methods is calcualtion of \hat{p}. For two-phase problems with low density ratios these methods yield accurate results as can be seen in fast-capwave-density.c.

By increasing the density ratio (air-water) we can see that these methods have stability problems and we need to reduce CFL number and TOLERANCE to get acceptable results as it is shown in fast-capwave-air-water.c.

The stability problem of this method becomes apparent for oscillating droplet test case where we used density ration of 1/10 and decreased TOLERANCE to improve the restuls, as can be seen in fast-oscillation.c. This method however have serious problems for density ration of 1/1000 and results are unstable and even by reducing CFL and TOLERANCE we can’t obtain accurate results.