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 NavierStokes 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 NavierStokes equations can be ignored.
This important distinction is what simplifies the incompressible fluid dynamics equations compared to the full NavierStokes 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 NavierStokes 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 NavierStokes equation. The main equation of motion is:
NavierStokes momentum equation for compressible flows
In this equation, μ and λ are proportionality constants that define the viscosity and the fluid’s stressstrain 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 lowpressure 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 NavierStokes 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 systemspecific, meaning different systems will have different values for the constant on the righthand side.
Additional Fluid Dynamics Equations
The above equations don’t tell the whole story; there is also an enthalpy equation from thermodynamics that includes a dissipation term and accounts for temperature changes during flow. Thermal considerations are the basis for fluid behavior such as convective heat transfer, where a fluid is to facilitate heat transport. Stressstrain relations are also important for describing compressible fluids and would need to be incorporated into the above equations, further complicating descriptions of compressible flow.
Systems designers can solve any of the above systems with a CFD simulation package. The complete set of fluid dynamics analysis and simulation tools in Omnis from Cadence are ideal for defining and running CFD simulations with modern numerical approaches, including compressible and incompressible flow in fluid dynamics in complex systems.
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