# Equations of Compressible and Incompressible Flow in Fluid Dynamics

### Key Takeaways

• Fluid dynamics are often differentiated into compressible and incompressible flows, each of which may be viscous or inviscid.

• Incompressible flow reduces the continuity equation for conservation of mass to a divergenceless equation, and this greatly simplifies the Navier-Stokes equations.

• Compressible flow is more complex, and a pair of equations must be solved to determine the flow velocity field as well as the density in space and time. The equations shown in the above image come in many different forms, and it can be difficult to keep them all straight. These equations can describe any number of flow situations, such as viscous, inviscid, compressible vs. incompressible, solenoidal, turbulent, and many other flows. In this article, we will focus specifically on compressible and incompressible flow in fluid dynamics as well as the basic equations that describe these types of flow. As we’ll see, these sets of equations generally require a CFD simulation package to determine a flow solution.

## Compressible and Incompressible Flow in Fluid Dynamics

The difference between compressible and incompressible flow in fluid dynamics is conceptually simple: a compressible fluid can experience a density change during flow while an incompressible fluid does not experience such a change. In terms of the material mechanics for an incompressible fluid, shear forces on the fluid due to viscosity or external body forces do not cause changes in density during flow. In other words, any derivatives of the density terms in the Navier-Stokes equations can be ignored.

This important distinction is what simplifies the incompressible fluid dynamics equations compared to the full Navier-Stokes equations. In either case, the goal in CFD problems is to calculate the flow velocity field as well as the density in compressible flow problems. The important equation relating density and flow velocity is the continuity equation: Fluid continuity equation

Here, u is the fluid flow vector and ⍴ is the fluid density. The flow rate is generally expressed in Cartesian coordinates, although many systems can be simplified by transforming the Navier-Stokes equations into an alternative coordinate system (cylindrical, linearly scaled, etc.). This equation effectively expresses the conservation of mass flow rate flowing into and out of any region of the system.

The above equation will be an important tool as we continue to analyze compressible and incompressible flow in fluid dynamics problems. First, we’ll look at compressible flow (which may be viscous) and we’ll show the reduction to incompressible flow.

### Compressible Flow Equations

The main equations for compressible flow include the above continuity equation and the momentum equation from the Navier-Stokes equation. The main equation of motion is: Navier-Stokes momentum equation for compressible flows

In this equation, μ and λ are proportionality constants that define the viscosity and the fluid’s stress-strain relationship. The value of λ is generally a function of viscosity. In textbooks or other fluid mechanics guides, these values are related linearly to account for cases where fluids only undergo elastic deformation, which is typically the case for compressible fluids experiencing low body forces. This may not be the case for low-pressure gases, which could be easily compressed during flow.

From here, we can immediately account for viscous or inviscid flows. An inviscid flow is defined by setting μ = 0 in the above momentum equation, which eliminates the viscous terms in the above equation and simplifies the resulting system. Typically, fluids with sufficiently low viscosity, which then exhibit flow with very high Reynolds numbers, can be approximated using the inviscid form of the above momentum equation.

### Incompressible Flow

Incompressible flow relies on two approximations to the continuity and momentum portions of the Navier-Stokes equations. Incompressible fluids have constant density in space and time, so all derivatives of the density are zero. The resulting continuity equation reduces to: Continuity equation for incompressible flows

This then simplifies the momentum equation by removing any divergence terms, specifically the term proportional to λ. We would then arrive at the following form of the momentum equation for incompressible viscous flows: Momentum equation for incompressible viscous flows

An inviscid approximation with μ = 0 can now be applied to the above equation as usual, giving the textbook form of the Euler equation of motion for incompressible inviscid flows.

There is one additional relation that relates streamline velocity, pressure, and density in a flow, known as Bernoulli’s equation. The following equation is valid for streamlines in incompressible flow: Bernoulli’s equation for incompressible flows

This equation is effectively a statement of conservation of energy in incompressible flows. This equation is also system-specific, meaning different systems will have different values for the constant on the right-hand side.