An Overview of Euler's Equations in Fluid Dynamics

November 22, 2021 Cadence CFD

Key Takeaways

  • Euler’s equations in fluid dynamics describe the flow of a fluid without accounting for the fluid’s viscosity.

  • Euler’s equations are derived from the Navier-Stokes equations or from basic equations in continuum mechanics.

  • Although Euler’s equations consider a somewhat impossible physical situation of zero viscosity, they are useful for describing low-viscosity fluids like water or alcohols.

Fluid dynamics graphic

Fluid dynamics is a heavily mathematical discipline that requires careful attention to the physical meanings of differential equations. Among all the equations used in fluid dynamics, Euler’s equations are some of the simplest to work with, particularly within the limits where viscosity and compressibility for a fluid are negligible. Most classes on CFD start with Euler’s equations in fluid dynamics, as they can produce rather intuitive results and are easier to work with than the full Navier-Stokes equations for compressible fluids.

Although inviscid flows, as described by Euler’s equations, are easiest to approach from an analytical standpoint, they are still powerful enough to describe flow phenomena in multiple situations. Compressible and incompressible flows can be treated with Euler’s equations, but turbulence is a theoretical case that cannot be treated with these equations. In this article, we’ll show how Euler’s equations are used, their physical interpretation, and how they produce important results in nearly inviscid flows.

What Are Euler’s Equations in Fluid Dynamics?

Euler’s equations are the equations of motion and continuity that treat a purely theoretical fluid dynamics problem where a fluid has zero viscosity, known as inviscid flow. Real fluids are viscous, although some very low-viscosity incompressible fluids, such as water or alcohols, can be treated in certain flow regimes very accurately with Euler’s equations. Because of this fact, Euler’s equations are describing an ideal case where drag in the fluid and energy dissipation is negligible, something which is unphysical and will never be observed in reality. However, we can approximate the Navier-Stokes equations with very low viscosity to get to Euler’s equations.

The Inviscid Approximation

Although Euler’s equations in fluid dynamics are unphysical, they can be used to describe a situation that would be considered “nearly inviscid,” in which the drag forces are much smaller than any externally applied forces. In this flow regime, Euler’s equations can be quickly derived from the Navier-Stokes equations. First, we can point out the viscous terms in the Navier-Stokes equation of motion:

Navier-Stokes equation of motion momentum

Momentum portion of the Navier-Stokes equations for viscous compressible flows

In this equation of motion, the second and third terms on the RHS of this equation (proportional to μ and λ, respectively) describe the effects of viscosity on shear and energy dissipation. These are the viscous terms to be removed in order to describe inviscid flow. This brings about four cases where such an approximation is valid:

  1. The pressure gradient term (1st term on the RHS) in the system is much larger than the viscous terms.
  2. The body forces or hydrostatic force (4th term on the RHS) are much larger than the viscous terms.
  3. The construction of the system that confines the fluid restricts its motion to vortical flow, where the velocity vector obeys the Laplace equation 𝛻2u = 0 and mimics inviscid flow.
  4. Conditions 1-3 are satisfied.

In any of these four cases, the viscous terms can be ignored in the above equation of motion, and we have Euler’s equation of motion:

Momentum portion of Euler’s equations

Momentum portion of Euler’s equations for inviscid compressible flows

In this equation, we’ve set μ = 0 and λ = 0, although the latter condition is sometimes subject to debate for nearly inviscid flows. If both μ = 0 and λ = 0 were physically realizable, we would have a case where a fluid that was given some momentum would continue flowing forever and would never dissipate its energy and reach equilibrium. In reality, we may have very small values of μ and λ, which are needed to describe an approach to equilibrium, yet they are not critical to describe flow in high pressure gradients.

Continuity and Eulerian Form

In addition to the above equation of motion, we still have the familiar continuity equation describing conservation of mass:

Continuity equation in fluid dynamics

Continuity equation relating fluid density and flow velocity

Finally, we have an additional thermodynamic equation that is related to the dissipation term:

 Fluid enthalpy

Fluid enthalpy equation

Here, we have an equation describing the material derivative of the thermodynamic enthalpy h in terms of the fluid’s thermal conductivity k, the material derivative of the pressure, and dissipation brought on by viscosity. In the case where μ = 0 and λ = 0, the third term on the RHS of the above equation disappears, meaning there is no damping to the motion in the system. 

If we take all of the above equations together, we can write them out in a common form, known as Eulerian form:

Eulerian form of Euler’s equations

Euler’s equations for compressible fluids written in Eulerian form

Here, e is an internal energy per unit mass of fluid. The above set of equations is valid for compressible or incompressible inviscid flows.

The aforementioned unphysical problem with dissipation, and one other problem, means that Euler’s equations are only an approximation and cannot easily treat a specific type of flow: turbulence.

A Note on Turbulence

Recalling again the definition of Reynolds number for a moment, truly inviscid flow would set the Reynolds number for the fluid to infinity. Obviously, such a case would never exist in reality, thus we have a problem where inviscid flow is only laminar if the fluid velocity is zero or its length scale is zero, i.e., flow does not exist!

This is where very lightly viscous approximations must be used to describe the flow, meaning the effects of viscosity are accounted for in some way by carrying specific values for μ and λ in some of the above equations. We can still derive a flow field and, for low-velocity flow situations, we will be able to determine a Reynolds number that might put the flow in the laminar regime. However, we might easily see turbulence in such a fluid, as a higher pressure gradient may be required to ensure the inviscid approximation is maintained in Euler’s equations.

CFD Simulation Packages for Inviscid Flows

While limiting cases of the above equations can be solved by hand in simple systems, 3D systems with complex geometry generally need to be treated with a numerical solver. Commercial CFD simulation packages provide several models and algorithms for calculating solutions to these problems. These applications can help expedite design and evaluation of complex flows, including nearly inviscid flows that might be appropriately described using Euler’s equations.

The complete set of analysis and simulation tools in Omnis 3D Solver from Cadence are ideal for defining and running CFD simulations to solve Euler’s equations in fluid dynamics with modern numerical approaches. The Omnis system can also treat compressible flows with turbulence in complex systems, providing a deeper view into turbulence in nearly inviscid fluids. In Cadence’s simulation suite, physical design data is used for grid generation in CFD simulations; you won’t need to manually construct grids or create generation algorithms.

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