Working Pressure in Momentum Equation

Momentum equation in (Navier-Stokes equations) can be written as

Pressure in the above equation is the thermodynamic/mechanical (assuming Stokes hypothesis). But CFD codes in general don't use thermodynamic pressure in momentum equation discretization. They instead define a working pressure and use that. Here are the steps that take us to the final equation that is used for discretization.

Step 1

Introduce a constant reference density.

Step2

Introduce a constant reference pressure such that

Then

and

or

Notes

• Step 1 helps in decreasing the magnitude of gravity source term that can otherwise have some destabilizing effects. Step 2 is done to avoid precision issues. In general, the variation in pressure is orders of magnitude smaller than the mean value of pressure. If we work with absolute magnitude of pressure in discretization, we will waste most of the significant digits and may not capture variation in pressure to desired accuracy.
• In a stagnant (constant density) liquid column, if we chose reference density to be liquid's density then p_therm will increase linearly as we go deeper but p_working will be constant.