Introducing the Compressible NavierStokes Equation
Key Takeaways

What the NavierStokes Equations are.

The difference between incompressible versus compressible fluid flow.

The best way to solve the compressible NavierStokes Equation.
NavierStokes 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 NavierStokes Equation. In this article, we will explore this equation, with a particular focus on the compressible NavierStokes equation.
Why Is the NavierStokes Equation Important?
NavierStokes equation
The NavierStokes equationshown aboveor 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 1D, 2D, or 3D 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 NavierStokes equation can be employed to determine the pressure and velocity of the fluid. However, an important distinction that affects which NavierStokes 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 NavierStokes equation becomes the following:
Incompressible fluid NavierStokes equation
We can arrive at the equation above by instituting the assumptions that the fluid is isotropicor its properties are uniform in all directionsand 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 NavierStokes equation is as given below.
NavierStokes 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:
Inertial forces
Pressure forces
Viscous forces
F External forces
As shown, the compressible NavierStokes 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.
Continuity equation
Solving the Compressible NavierStokes Equation
The degree of difficulty in solving the compressible NavierStokes 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.
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 NavierStokes equation, the Cadence Omnis solver provides advanced capabilities that will aid your system design.
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