Distinction between an Incompressible and a Compressible Flow

• In order to know, if it is necessary to take into account the compressibility of gases in fluid flow problems, we need to consider whether the change in pressure brought about by the fluid motion causes large change in volume or density.
  
  Using Bernoulli's equation
  
  p + (1/2)ρV²= constant (V being the velocity of flow), change in pressure, Δp, in a flow field, is of the order of (1/2)ρV² (dynamic head).
  Invoking this relationship into
  
  

   we get ,  
  

  (2.12)

          
  So if Δρ/ρ is very small, the flow of gases can be treated as incompressible with a good degree of approximation.
  
  

• According to Laplace's equation, the velocity of sound is given by
  
  

  Hence               
  
  

        
  where, Ma is the ratio of the velocity of flow to the acoustic velocity in the flowing medium at the condition and is known as Mach number. So we can conclude that the compressibility of gas in a flow can be neglected if Δρ/ρ is considerably smaller than unity, i.e. (1/2)Ma²<<1.

• In other words, if the flow velocity is small as compared to the local acoustic velocity, compressibility of gases can be neglected. Considering a maximum relative change in density of 5 per cent as the criterion of an incompressible flow, the upper limit of Mach number becomes approximately 0.33. In the case of air at standard pressure and temperature, the acoustic velocity is about 335.28 m/s. Hence a Mach number of 0.33 corresponds to a velocity of about 110 m/s. Therefore flow of air up to a velocity of 110 m/s under standard condition can be considered as incompressible flow.