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The Importance of the Euler Equations and Navier-Stokes Equations in Fluid Dynamics

Key Takeaways

  • The Euler equations apply mostly for inviscid, adiabatic flows with zero viscosity and compressibility.

  • Using the Navier-Stokes equations to analyze fluid takes their viscous properties into account. 

  • The Euler equations can be derived from the Navier-Stokes equations and can provide a more simplified result.

Fluid mechanics equations

The governing set of fluid dynamics equations are derived to simplify the complexity of fluid flow systems. In fluid dynamics, this comes in the form of the Euler equations and the Navier-Stokes equations. For a flow that is inviscid and adiabatic, the governing PDEs (partial differential equations) are the Euler equations. Although these equations can be applied for most flow conditions, they provide the simplest of results when viscosity and compressibility are considered to be negligible. The Navier-Stokes equations, on the other hand, take into account the viscous properties of the fluid to provide a more complete analysis.

In CFD analysis, these continuum equations (the Euler equations and the Navier-Stokes equations) are considered to be especially important given their excellent analytic properties of flow parameters. In this article, we will discuss the Euler equations, how they differ from the Navier-Stokes equations, and their importance in the CFD analysis of a fluid system

Discussing Euler Equations

Euler equations are the set of partial differential equations in fluid dynamics that deal with the flow, which has zero viscosity, and relates the velocity, pressure, and density of the fluid in motion. However, since zero viscosity is not practically possible and the Navier-Stokes equations assume the viscosity of the fluid, the Euler equations can be used for fluid within a limited low-viscous range.

Euler equations are related to Newton’s second law of motion, whereby, they relate the velocity change of the fluid to the presence of force. This is also an applicable theory when discussing the conservation of momentum. We know the Navier-Stokes equation for conservation of momentum is:

 Momentum equation

Note that:

u is the flow velocity

ρ is the fluid density

σ is the Cauchy stress tensor

g is the acceleration

Elaborating the above equation in terms of pressure and stress, we get:  

Cauchy momentum equation

Note that p is the pressure and τ is the deviatoric stress.

For inviscid flow, the above equation assumes zero deviatoric stress, as no external forces apart from gravity are assumed. Thus, we get:

Euler equation

This is the Euler equation as derived from the Navier-Stokes equation in a convective form. This is also called the continuity equation, and can be assumed for the conservation of mass. In fluid dynamics analysis, these equations establish the relationship between fluid movement, pressure, and forces.

It should be noted that for these above-stated equations to be valid, the following assumptions are required:

  • The fluid is inviscid and flows uniformly 

  • The fluid is incompressible, thus, the density is constant 

  • Only pressure force and gravitational forces affect the flow

The above equations can be helpful in elaborating the nature of fluid flow in relation to the internal pressure at different flow points. 

Euler Equations for Compressible Fluids

When the flow of compressible fluids is to be analyzed, the Euler equations can be written as:

Euler equations for compressible fluids

The above set of equations represents the conservation of mass, momentum, and energy for a compressible fluid and facilitates the understanding of the variables, i.e.,  fluid density, flow velocity, and internal energy (e) of the flow system. 

Euler Equations and Navier-Stokes Analysis for CFD Simulation

The Euler equations and Navier-Stokes equations are the basic hydrodynamic equations in a wide range of flow situations. While these equations aim to simplify complex fluid system analysis, deriving the numerical analysis manually for complex fluid systems can be a challenge. Commercial CFD packages offer a solution to design and analysis accuracy.

CFD simulation tools, such as the Fidelity 3D Solver from Cadence, can run high-fidelity CFD simulations while ensuring the Euler equations, Navier-Stokes equations, and other governing equations are properly addressed for compressible or incompressible flows. Numerical simulation abilities can help analyze all associated Eulerian variables while ensuring the fluid system design is done with accuracy and efficiency. 

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