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Introducing the Compressible Navier-Stokes Equation

Key Takeaways

  • What the Navier-Stokes Equations are.

  • The difference between incompressible versus compressible fluid flow.

  • The best way to solve the compressible Navier-Stokes Equation.

Navier-Stokes equations for fluid flow analysis

Navier-Stokes equations

While Galileo passed down many mathematical and scientific gifts to us, the assertion that “Mathematics is the language with which God has written the universe” is one that stands out. Indeed, we are able to understand and even manipulate our environment through the utilization of mathematics, and equations are the keys that open these doors. 

In the field of fluid dynamics, there is probably no more important equation than the Navier-Stokes Equation. In this article, we will explore this equation, with a particular focus on the compressible Navier-Stokes equation.

Why Is the Navier-Stokes Equation Important?

General Navier-Stokes equation

Navier-Stokes equation

The Navier-Stokes equation--shown above--or some form of it is typically at the heart of any analysis of fluid flow, which includes gases and plasma in motion. This equation is employed to analyze both laminar and turbulent flow regimes and can be utilized for 1-D, 2-D, or 3-D evaluations.

Fluid flows may be classified in a number of ways. For example, boundary layer evaluations of airflow across the surface of an aircraft are external flows, while analyzing the fluid changes within an oil pipeline are internal flow studies. Irrespective of these classifications, the Navier-Stokes equation can be employed to determine the pressure and velocity of the fluid. However, an important distinction that affects which Navier-Stokes equations should be used is whether the fluid is compressible or not. 

Incompressible vs. Compressible Fluid Flows

All fluid flow is subject to change due to temperature variation. Although, in some cases, the change may be treated as negligible. With respect to density change, in particular, some fluids do not undergo significant change as a result of pressure variance. When this is the case, the fluid is said to be incompressible and the applicable Navier-Stokes equation becomes the following:
Navier-Stokes equation for incompressible fluids

Incompressible fluid Navier-Stokes equation

We can arrive at the equation above by instituting the assumptions that the fluid is isotropic--or its properties are uniform in all directions--and stress is Galilean invariant or does not depend on velocity directly.

Some fluids, however, may experience a significant change in density in response to pressure variation. These fluids are compressible. Again, it can be assumed that stress is Galilean invariant and the fluid is isotropic. If it is also assumed that the second viscosity, 𝞯, is negligible or zero, the compressible form of the Navier-Stokes equation is as given below.

Compressible flow Navier-Stokes equation

Navier-Stokes equation for compressible flow

In the above equation, the last term on the right, 𝝆g, has been replaced with F for generality. Deconstructing this equation into the different types of forces, we have:

 Equations for different forces                  Inertial forces

 Equations for different forces                                            Pressure forces

 Equations for different forces     Viscous forces   

F                                                   External forces

As shown, the compressible Navier-Stokes equation is a partial differential equation. Typically, a continuity equation, shown below, is simultaneously evaluated, yielding solutions for the momentum and velocity of the fluid flow.      

Accompanying continuity equation for the compressible Navier-Stokes equation

Continuity equation

Solving the Compressible Navier-Stokes Equation

The degree of difficulty in solving the compressible Navier-Stokes equation varies depending upon fluid properties and the dimensionality of the problem space. However, it is advisable to deploy a solver, such as the one shown in the figure below.

Example of fluid flow analysis with Omnis

Mach number determination of compressible fluid flow with Omnis

Whether you are determining the mach number (as shown above), analyzing boundary conditions of a surface, or conducting any other fluid flow analysis that involves the compressible Navier-Stokes equation, the Cadence Omnis solver provides advanced capabilities that will aid your system design.   

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