Euler equations vs NS equations
 

Hello,

It has been bothering me for a long time that, for high-speed flow, since NS equations can clearly describe full flow field, why people still want to study Euler equations? I mean without viscous terms, Euler equations can only capture shock waves, but unable to generate turbulence; it's kind of "unphysical".

So, the reason for studying Euler equations, is it because the Euler equations can capture shocks better than NS? Or some other reasons?

:confused:

On the contrary, Euler equations are quite more difficult due to the mathematical singularity in the solution...but often they are used for external flow where the viscosity has almost no relevance.
It is worthwhile to consider that we cannot solve the viscous structure of a shock while considering also problems at large scales, like the flow over an airfoil. Consider that you need to describe a shock layer width that is much smaller than the turbulence Kolmogorov lenght scale. In other words, also using NS equations, the shock is described on a grid as a discontinuity in the Euler equations.

One reason to do so is to understand convective scheme better. Mathematically, Euler equations help to understand the hyperbolic nature of PDE's.

Also, implementation of boundary condition which requires taking into the consideration the direction of flow and the Mach numbers at the boundary.

It is a necessary step before one code for full NS.

Also, do not forget that, for the practical cases where Euler equations are still usually applied (external compressible aerodynamics), using the full Navier-Stokes would either require a full boundary layer resolution (which more often than not is out of question) or the use of wall functions.

In the latter case, even if you have a code with wall functions (which is not necessarily true in general), they are far from reliable in several cases, and also quite wrong in relation to the computed drag on your body.

When you put all these things together, I guess, you start seeing how more attractive Euler equations are in such cases.

I agree, Euler equations are quite suitable for computing pressure distribution and shock location

NS vs Euler equations
 

This question has different answers depending on where you come from. As someone who comes from a *very* pure math background. We look at the compressible Navier-Stokes equations as the next term in the Chapman-Enskog expanion. If you perform the Chapman-Enskog expansion on the Boltzmann equation and retain only first-order terms, you get the compressible Euler equations. If you retain higher order you get the Compressible Navier-Stokes in the hydrodynamic limit. If we retain only first-order terms, taking the hydrodynamic limit we get a purely inviscid system. In fact, its hyperbolic, i.e. its eigenvalues are real and distinct. It is still a valid description where vicous effects are negligible.

In terms of not being able to generate turbulence , I dont know if that is true. If we look at Kelvin-Helmholtz or the Meshkov instablity we do get transitional instabilities.

The inviscid flow assumption is valid when we consider the mechanism of the energy cascade that acts only by means of the non-linear interaction. A real turbulence will be characterized by the fact that a physical dissipation is present at small scales, a fact that terminates the inertial energy cascade.

Thanks for your reply, and it's inspiring. However, I have never worked on solving Euler equation, but one question: if apply Euler to study external flow, due to it's inviscid, would it results in no dray force on, let's say an airfoil (like D'Alembert paradox)?

I can see it's good for shock capturing, but I don't see why it's good for computing pressure field?

Seems you are from math background? I can see Euler can generate instability due to its nonlinearity; but without viscous force, the turbulent energy cascade will go to infinitesimal small, which obvious breaks the continuum assumption.

So, does this mean that Euler equation kind of like a "zoom-in" on certain spatial range so that we can focus on the shocks only?

Euler equations, in general, admit drag in the form of wave drag. The D'Alembert paradox, instead, only applies for potential flows.

This is in theory. In practice, the drag you will see in your computation will depend on the method you use.

When it is possible (e.g., incompressible and inviscid flows), if you solve the potential equation directly, you will see no drag at all. If instead you solve the Euler equations directly, your discretization is likely to introduce numerical viscosity. This, in turn, will make your simulation a viscous like, experiencing separations (thus pressure drag) and entropy production in general (yet, no friction drag).

Note that Euler equations are typically solved directly for this very reason, otherwise you would not see any lift at all as well. For airfoils, that small numerical viscosity will play similarly to a Kutta condition, while still not requiring the resolution of the boundary layers. This also means that, in 3D, you can compute the lift drag.

Note that, especially at high speeds, the pressure-wave drag that you can compute with such a method is much higher than the friction drag you are not computing (and even more so at high Mach numbers). Thus, if the flow is attached, it really is a good approximation.

For Euler equations the shock has zero thickness. For full Navier-Stokes equations the shock has a finite thickness.

In theory you can solve NS equations on a sufficiently fine grid to capture the shock structure (i.e., jumps across the shock become continuous variations over a very small length, function of the mean free path).

In practice, this won't happen and doesn't make much sense, because the length over which this happens is so small that the continuum hypothesis underlying the NS equations is questionable at those scales.

But experts on Boltzmann equations can probably shed more light on this.

In conclusion, yes, if by zoom in you intend that you get a discontinuous shock, no matter at what scale you look at it. That's what you get with Euler equations, but not with NS.

Euler vs NS
 

From a point of view of scales. This is how it is looked at:

Quantum Mechanics -> Kinetic Theory -> Hydrodynamics.
Schrodinger Boltzmann Euler/NS

So to answer the question, yes in the hydrodynamic limit, Euler equations are a zoom in. Something I find interesting is that in very high Mach numbers, Euler's equations become a less adequate description, and in fact the Boltzmann equation becomes more accurate. If you would like to know more about this, I can give you some references. A professor at my university works on multiscale methods (finite volume, finite difference.)

References on limits:
1 .Hydrodynamic Limits of the Boltzmann Equation, Laure Saint-Raymond
2. The Cauchy Problem in Kinetic Theory, Robert Glassey
3. The Boltzmann Equation and its Applications, C. Cercignani (Quite frankly the best book on the Boltzmann equation)
4. Kinetic Equations and Aymptotic Theory, Francois Bouchut, Francois Golse and Mario Pulvirenti

Just some hints:


1) the Prandtl theory about the BL says that the normal derivative of the pressure vanishes as the viscosity goes to zero. That means Euler solution can provide an acceptable pressure field at the wall.


2) Euler equations can generate singularity in the solution but this singularity can be only mathematical and not physical. Jump relation must be satisfied across the singularity. More specifically, the shock must be only compressive.



3) Energy cascade theoretically extends up to infinitesimal lengh scale for invisci flows. Of course, this is a mathematical consequence of the used inviscid approximation. No matter about the continuum description.

Correct me if I am wrong: so you mean, with high-speed flow, the physical drag force will be small, and so lacking of viscous terms (not artificial viscosity) will not cause too much of problem for Euler equations; instead, pressure drag is important, and Euler can compute it well. Am I right?

hm. So you are talking about extremely high-speed flow. I was thinking about in the regime of regular supersonic flight.

Thanks for the recommends ! I'll take a look.

I understand the first two. But for the 3rd, it doesn't seem physical; it appears to me more like a mathematical point of view.

Euler approximation is a mathematical model...the solutions you get can or cannot be an acceptable approximation of the physics, depending on what you want to describe.
The singularity is mathematic, shock Waves are not a discontinuiy in a small scale