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 dontknow November 18, 2005 03:11

Physical meaning of pressure in pressure eqn

Pls pardon me if this is a very simple question....

In the fractional step scheme, we need to solve the pressure poisson eqn to make the velocity divergence free. However, what is the physical meaning of this pressure? say in a simulation of an airfoil in an external flow with initial p=0 (gauge pressure 0, absolute pressure 10100pa)

I am using the non-dimensionalized NS eqns, where the pressure is non-dimensionalized by rho*(vel_inf)^2. How can we relate this pressure (and velocity) to get pressure, lift and drag coefficient?

Is there any good references which discuss this? Most papers seem to stop at getting the pressure.

Thank you very much.

 Jonas Holdeman November 18, 2005 14:09

Re: 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.

 Mehr August 30, 2010 10:12

Relation to Fractional step method for solving incompressible Navier-Stokes

Hello Jonas,
I found the explanation for pressure really interesting. Still it is not clear for me if the pressure which you call projection potential is the same as P2 in the expansion?

For making a general velocity field divergence-free, you have written about calculating the potential gradient and subtracting it. Do you mean the same procedure as in Fractional step method that we first find an intermediate velocity field and then we solve Poisson equation and update the velocity field. So what you mean by that it doesn't need pressure-like boundary conditions!?

If I have understood correctly then in fractional step method we don't need LBB condition.