## Wednesday, October 16, 2019

### Incompressible Flow

Incompressible Flow Definition

Continuity equation can be written as

If the material derivative of density is 0 (or negligible) i.e. if the density of a material particle does not change (or does not change significantly) as it moves through the flow domain, then the divergence of velocity vector is equal to 0 (or close to 0). Divergence free velocity is generally taken as the definition of incompressible "flow".

Note that constant density fluid implies divergence free velocity but the reverse is not true. Flows involving constant density fluid are always incompressible. Flows involving non-constant density fluid can also be assumed to be incompressible or divergence free under appropriate operating conditions. There is a difference between incompressible "fluid" and incompressible "flow" and that must always be kept in mind. Incompressible "fluid" implies incompressible "flow" but incompressible "flow" does not always imply incompressible "fluid".

Compressible / Incompressible "Fluid" and Incompressible "Flow"

For any fluid, equation of state (eos) can be prescribed to express density in terms of other thermodynamic variables - pressure and temperature. Fluids with dependence of density on pressure are called compressible "fluids" and those with no dependence of density on pressure are called incompressible "fluids". Strictly speaking, no fluid is incompressible. There is always some amount of compressibility, however small that may be. But for many fluids, it is fair to neglect compressibility for a wide class of problems. For example, water's density does not change appreciably when subjected to a wide range of pressures. For these fluids, we can simply neglect the dependence of density on pressure. And in many cases, we can go a step further and assume density to be constant.
• Constant Density "Fluid" As mentioned above, any flow involving constant density incompressible "fluid" can be termed incompressible "flow" (i.e. divergence free velocity).
• Ideal Gas "Fluid" At low Mach numbers, density variation in flows involving ideal gas can be very small. So the assumption of incompressible "flow" (i.e. divergence free velocity) is fairly accurate for low Mach number flows involving ideal gas.
• Other Compressible "Fluids" For any general eos, we have to look for flow regime where the density variation over the solution domain is negligible. Over that flow regime, the assumption of incompressible "flow" would hold in general.

OK! Now we can identify incompressible "flow" based on eos and operating condition but how does it help?
• Simplified Theoretical Analysis With the assumption of incompressible flow, simple formula can be derived that are easy and simple to use for theoretical analysis e.g. Bernoulli's equation and simple definitions of total/stagnation temperature & pressure.
• Faster/Robust Solution Approach For a compressible "fluid", all the flow + energy variables (rho, p, V, T) are tightly coupled via the set of Euler (or Navier-Stokes) equations and equation of state. In general, this set of tightly coupled equations are solved numerically using what is generally termed as density-based approach. The approach has inherent coupling of both convective and acoustics velocity scales. At moderate and high Mach numbers, the two velocity scales are comparable but low Mach numbers the convective velocity scale is much smaller than acoustic velocity scale. This wide separate between convective and acoustic scales presents several problems with density-based approach. So people started looking for alternatives and pressure-based approach was derived with the assumption of incompressible flow. Pressure-based approach decouples the solution of momentum equation (based on convective scales) from that of pressure (based on acoustic scales) and provides a much better (faster & robust) alternative to density-based approach for incompressible flows. So if we have identified that the problem under consideration involves incompressible "flow", the choice of which numerical approach to use is simple.