Compressible Flow

By definition, compressibility of a fluid is

τ = (1/v)(dv/dp)

 

Where:

τ  = compressibility

v = specific volume (volume per unit mass)

p = pressure

The role of the compressibility in determining the properties of a fluid in motion is seen as follows. Define v as the specific volume (volume per unit mass). Hence, v = 1/ρ. Substituting this definition to the earlier equation, we obtain

τ = (1/ρ)(dρ/dp)

Thus, whenever the fluid experiences a change in pressure dp, the corresponding change in density from thi equation is

dρ = ρ x τ x dp.

Consider a fluid flow, if the fluid is a liquid where the compressibility is very small, the for a given pressure change dp from one point to another in the flow, this equation states that dρ will be negligibly small. In turn, we can reasonably assume that ρ is constant and that the flow of a liquid is incompressible. On the other hand, if the fluid is a gas, where the compressibility is large, then for a given pressure change dp from one point to another in flow, this equation states that dρ can be large. Thus, ρ is not constant, and in general, the flow of a gas is a compressible flow.

The exception to this is the low-speed flow of a gas; in such flows, the actual magnitude of the pressure changes (dp) throughout the flow field is small compared with the pressure itself. Thus, for a low-speed flow, dp in the equation above is small, and even though τ is large, the value of dρ can be dominated by the small dp. In such cases, ρ can be assumed to be constant, hence allowing us to analyze low-speed gas flows as incompressible flows.

Mach number is defined as:

M = V/a

Where:

M = Mach number

V = local flow velocity

a = local speed of a sound

It has been shown that when M larger than 0,3, the flow should be considered as compressible.

Ref.: 

 J. D. Anderson Jr., Fundamentals of  Aerodynamics, Third Edition, New York: The Mc-Grawhill Companies Inc., 2001.