CFD Online Discussion Forums - Physical meaning of pressure in pressure eqn

There is a great deal of myth and misunderstanding regarding the incompressible pressure. A little research into published literature will support the view I will describe.

We start with the compressible Navier-Stokes equation (NSE). The incompressible limit is singular, but we can develop the limit as an asymptotic expansion. To do so, we define a reference Mach number M, which is the ratio of a reference velocity to the speed of sound in the compressible medium. The definition is not exact, but is used for scaling like the reference length or reference velocity or temperature or etc. used to develop dimensionless forms of the NS equations. We then write the dependent variables such as velocity, pressure, density, etc. as asymptotic series in powers of M. In particular we write P=P0+P1*M+P2*m^2+... . Note that the incompressible limit corresponds to the limit M -> 0.

The two lowest order terms in M, P0 and P1, appear as grad P0=0 and grad P1=0, from which we conclude that P0 is constant, and is regarded as the background pressure. P0 is taken to be the thermodynamic pressure, a function of the volume, temperature, density, and material of the fluid. P1 is referred to as the accoustic pressure, and is involved with the propagation of sound waves. Proper investigation of this term involves a more complicated dual-scale asymptotic expansion, so we take P1=0 (it must be different from P0) in this simpler one-scale expansion.

The coefficient of next power of M yields the incompressible Navier-Stokes equation. It involves the zero-order velocity, density, etc., but the second order pressure P2. From the point of view of the incompressible asymptotic limit, the incompressible pressure is the second order term in the asymptotic expansion.

Now the incompressible NSE can be decomposed into solenoidal and irrotational parts, referred to as the Leray decomposition. The solenoidal part describes a pressureless governing equation for the divergence-free velocity, and the irrotational part gives the gradient of P2 as a function of the divergence-free velocity. For computational purposes, the pressure equation is just a form of the pressure-Poisson equation.

Now if one expands the velocity in terms of solenoidal/divergence-free basis functions, the velocity will necessarily be divergence-free (computation involves a projection operator in the differential form or solenoidal weight functions in the weak form, but the latter comes naturally in the Galerkin finite element method). If one does not use a solenoidal basis, the velocity cannot be divergence-free without some further measures. A general velocity can be separated into the sum of a divergence-free part and the gradient of a potential (the Helmholtz decomposition). We can find a solenoidal velocity by calculating this potential gradient and subtracting it out. When this potential gradient is introduced, the governing equation looks like we have reintroduced the pressure. But this is not the pressure, and it does not need pressure-like boundary conditions. NOT EVERYTHING CALLED P IS A PRESSURE.

The projection potential is tied very closely with the representation of the velocity. The two must satisfy a condition called the LBB condition, and this usually means that acceptible projection potentials must be of lower order than the velocity. The computed projection potentials often show checkerboarding, which does not contaminate the velocity, but is not the behavior one would expect from a pressure.

If one first computes the divergence-free velocity (I prefer to use divergence-free bases for this), the pressure can be computed as a function of the velocity from the pressure Poisson equation using whatever basis you like, without regard to any LBB condition.

The conceptual problem you have with the meaning of things like trial pressure and the like will disappear (I hope) if you consider that it is really not a pressure at all, but rather a projection potential.