Hi all,

I was wondering if someone could explain the difference between a compressible flow and an incompressible flow given that a variable density flow can be considered incompressible. Also, if someone could explain the related numerical techniques for each flow. i.e. Pressure-based, density based etc.

Thank you all.

Incompressible flow is defined as flow where the density is constant. It is generally accepted that flow at speeds below .3 times the Mach number can be treated (on average) as incompressible. But even at higher Mach number, the flow away from shocks is constant density.

Different fluids have different equations of state. The equation of state of a fluid involves the density. When the density is constant, all fluids (except some wierd ones) at a given Reynolds number behave in the same way, a sort of idealized behavior. One can imagine an idealized fluid with constant density, an incompressible fluid, which is used as a model for incompressible flow.

In principle, an incompressible flow has fewer variables to solve for and should be easier to compute. It is simple if you use divergence-free basis functions, because then you have one less (incompressibility) equation to satisfy. Unfortunately, most people don't use divergence-free basis functions and things get rather messy.

Incompressibility implies constant density of the fluid parcel when followed in lagrangian coordinate (D(rho)/Dt = 0....it's the material derivative...that's zero)....people often make the mistake of setting rho as constant for incompressible fluid. You can always have immiscible mixture of several incompressible fluids...that would make it a variable density problem (in eulerian coordinates)...but note if solved in lagrangian coordinates, each fluid parcel would retain it's original density.

one more point: fewer variables don't make incompressible flow easier to solve....I must agree that use of divergence-free basis function will make it easier, but that comes at a cost too... most incompressible solver need to solve a pressure-poisson equation that is computationally very expensive....other techniques like artificial compressibility have problems of their own too.... In compressible flows, pressure can be solved from equation of state (simple algebraic expression !!)... for incompressible flow, the pressure loses it's thermodynamic role...but is merely a lagrange multiplier in momentum equation that enforces the contuity constraint.

Can You further explain by what you mean by " a Lagrangian multiplier in momentum equation that enforces the continuity constrant". Thank yuo.

Think of momentum equation as a cost function subjected to the continuity constraint....and the balance is achieved through the "pressure" in momentum equations via minimization. actually, it has been shown by several researchers already...not at all a new interpretation or result. see, papers by Gresho and Sani to elaborate on the role of "pressure" in incompressible flows...also, Holdeman (yes...he responded to your query first!!) has few downloadable preprints on divergence free basis functions for incompressible flows....some sections in these papers discuss the role played by "pressure"

Thank you. Where are these located?

The papers are located at http://j.t.holdeman.att.net/research.htm

I disagree with MT's statement that there is an additional cost in using divergence-free functions, at least in the computation of the FLOW. There is no need for a Poisson solver in computing the FLOW. If one is interested in the pressure, then that is an additional cost incurred when it is needed, but not in any nonlinear iterations or in intermediate time steps.

If one formulates the equation for the flow as a variational problem in a space of solenoidal and non-solenoidal functions, then there is a Lagrange multiplier associated with the incompressibility constraint. This is associated with projecting out the non-solenoidal component of the trial solution. The equation for the Lagrange multiplier looks like an equation for the pressure because the non-solenoidal part can be written as the gradient of the Lagrange multiplier. BUT, it is not a pressure and needn't behave like a pressure. In particular it need not be smooth. This is why projection-type computations can give good velocities even when this function shows checkerboard-type behavior. Unfortunately, as MT's remarks demonstrate, the Lagrange multiplier is too often incorrectly associated with the necessarily smooth pressure.

Regardless of which side you may take in a debate over the role of the Lagrange multiplier, the issue is irrelevant to, and may be avoided by, the use of the divergence-free basis functions/finite elements.

The URL I typed is incorrect. The correct URL is: http://j.t.holdeman.home.att.net/research.htm I left out the ".home". Sorry.