The Pressure Equation of Incompressible Flow

The standard mathematical model for incompressible flow of unit density with vanishing viscosity takes the form of the incompressible Euler equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u + \nabla p = 0$,
• $\nabla\cdot u=0$,

where $u(x,t)$ is flow velocity, $p(x,t)$ is pressure and $(x,t)$ are space-time coordinates. 

The first equation expresses Newton's 2nd Law (conservation of momentum), while the second equation is not a physical law but a side condition as a stipulation that the velocity is to be divergence free. 

When solving the incompressible Euler equations computationally the second equations is typically replaced by an elliptic equation for the pressure of the form 

• $-\delta*\Delta p = - \nabla\cdot u$,      (1)
• where $\delta \approx h$ with $h$ mesh size.

In a variational setting this equation results from regularisation of the pressure as Lagrange multipler for the side condition $\nabla\cdot u=0$, which can be seen as a form of relaxation. The choice of $\delta =h$ brings $\delta\Delta$ in parity with $\nabla\cdot$. The equation (1) can be seen as a physical law connecting pressure to the divergence of velocity. 

Let us now give more perspective on the elliptic pressure equation by extending to compressible ideal flow where the equation for the internal energy $e$

• $\frac{\partial e}{\partial t}+\nabla\cdot (eu)+p\nabla\cdot u = 0$,

can be seen as an equation (expressing a law of physics) for the pressure $p=\gamma e$ with $\gamma >0$ a gas constant:

• $\frac{Dp}{Dt}=-\gamma p\nabla\cdot u$    (2)
• with $\frac{Dp}{Dt}=\frac{\partial p}{\partial t}+\nabla\cdot (pu)$,

With suitable normalisation the Mach number is $\sqrt{\gamma}$ with a switch from supersonic over transonic to subsonic flow as $\gamma$ increases beyond 1. 

This equation suggests a parabolic variant of the incompressible pressure equation of the form

• $\frac{\partial p}{\partial t}-\delta*\Delta p = - \nabla\cdot u$  (3)

where the right hand side of the compressible pressure equation $-\gamma p\nabla\cdot u$ is replaced by $-\nabla\cdot u$ for the incompressible case (and (2) is correspondingly regularised). Again, (3) can be seen as a law of physics connecting pressure to the divergence of velocity. It allows direct time stepping of the flow equations as an expression of physicality with finite speed of propagation/sound , compared with the elliptic equation (1) with infinite speed of pressure variation/sound.  

Choosing $\gamma$ small gives compressible flow with large variation of $\nabla\cdot u$ corresponding to moderate variation of density, while choosing $\gamma$ large will keep the variation in density small. 

We can thus view the compressible solver as a unified flow solver for both compressible and incompressible flow depending on the choice of the single parameter $\gamma$, with a natural connection to standard incompressible solvers. 

The unified flow solver can be explored as item 98 in the Model Workshop of DigiMat BodySoul including comparison a with standard incompressible solver.

Remark. Integrating (2) along streamlines gives

•  $p \sim \exp(-\gamma\int \nabla\cdot u\,dt)$ 

which expresses an exponential connection between the pressure $p$ and $-\gamma\nabla\cdot u$ compatible with $\nabla\cdot u$ being small for $\gamma >>1$.