Solving the World’s Toughest CFD Problems

# The Incompressibility Assumption

All materials, whether gas, liquid or solid exhibit some change in volume when subjected to a compressive stress. The degree of compressibility is measured by a bulk modulus of elasticity, E, defined as either E=δp/ (δρ/ρ ), or E=δp/(-δV/V), where δp is a change in pressure and δρ or δV is the corresponding change in density or specific volume. Since δp/δρ =c2, where c is the adiabatic speed of sound, another expression for E is E =ρc2. In liquids and solids E is typically a large number so that density and volume changes are generally very small unless exceptionally large pressures are applied.

If an incompressible assumption is made in which densities are assumed to remain constant, it is important to know under what conditions that assumption is likely to be valid. There are, in fact, two conditions that must be satisfied before compressibility effects can be ignored. Let us define “incompressibility” as a good approximation when the ratio δ ρ/ρ is much smaller than unity. To determine the conditions for this approximation we must estimate the magnitude of changes in density.

In steady flow, the maximum change in pressure can be estimated from Bernoulli’s relation to be δp=ρu2. Combining this with the above relations for the bulk modulus, we see that the corresponding change in density is δρ/ρ = u2/c2.

Thus, the assumption of incompressibility requires that fluid speed be small compared to the speed of sound,

(1)     $latex \displaystyle u\ll c.$

(2)  $latex 1\ll ct$