# When can I use Euler equations?

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 April 16, 2020, 03:53 When can I use Euler equations? #1 Member   Join Date: Apr 2016 Posts: 90 Rep Power: 10 Hello, I am wondering when I can simplify the Navier-Stokes Equations and just use the Euler Equations in CFD? Euler: Navier-Stokes for inviscid flow Reynolds number= ratio of inertial forces / viscous forces So to my understanding, for really high reynolds numbers, the viscous forces decrease, so we can use the euler equations. On the other hand: when we have high reynolds numbers, the flow is turbulent, where friction/viscous forces play an important role (I assume). I read that Euler can be used for high Mach Numbers, where the shock wave influence plays a more important role than the viscous forces. So for high Mach numbers also the reynolds numbers increase. Could someone bring light into my darkness? To which limits would you use Euler for calculations? Thanks! CellZone

April 16, 2020, 04:40
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Filippo Maria Denaro
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 Originally Posted by CellZone Hello, I am wondering when I can simplify the Navier-Stokes Equations and just use the Euler Equations in CFD? Euler: Navier-Stokes for inviscid flow Reynolds number= ratio of inertial forces / viscous forces So to my understanding, for really high reynolds numbers, the viscous forces decrease, so we can use the euler equations. On the other hand: when we have high reynolds numbers, the flow is turbulent, where friction/viscous forces play an important role (I assume). I read that Euler can be used for high Mach Numbers, where the shock wave influence plays a more important role than the viscous forces. So for high Mach numbers also the reynolds numbers increase. Could someone bring light into my darkness? To which limits would you use Euler for calculations? Thanks! CellZone

Euler equations are a mathematical model that does not allow to set a physical condition about the tangential velocity. Furthermore, the physical energy dissipation is not present and the captare of a real shock wave requires care.
High Re number flows are still governed by the NSE, you can see that as the perturbed form of the Euler equations.
If you need details about the dynamic and thermal BL you cannot use Euler.

 April 16, 2020, 08:05 #3 Senior Member     Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39 Attached, high Re number, external aerodynamic flows is what comes first to my mind. Everything where the inviscid pressure effects are orders of magnitude higher than viscous ones might be a good rule of thumb (maybe blasts, etc.)

April 16, 2020, 10:37
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Lucky
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 Originally Posted by CellZone So to my understanding, for really high reynolds numbers, the viscous forces decrease, so we can use the euler equations. On the other hand: when we have high reynolds numbers, the flow is turbulent, where friction/viscous forces play an important role (I assume).

The viscous forces don't decrease, they're still increasing. They're just increasing slower than the inertial force. Even if you take the limit as viscosity goes to zero, you have non-zero viscous dissipation because the velocity gradients can increase without bound.

 April 16, 2020, 14:58 #5 Senior Member     - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 13 First of all you should be aware that the difference between Euler and Navier-Stokes equations is not only viscosity. In most cases it is useful to define the terms you want to talk about, e.g., what exactly do you mean when your are talking about Euler and Navier-Stokes equations. The classical Navier-Stokes equations consists of at least two different parabolic effects, heat conduction according to Fouriers law and viscous effects according to Stokes law. If you play with more general physics you have to consider more parabolic effects, e.g. Fickian law for multi-species or combustion. __________________ Check out my side project: A multiphysics discontinuous Galerkin framework: Youtube, Gitlab.

April 21, 2020, 15:09
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 Originally Posted by Eifoehn4 First of all you should be aware that the difference between Euler and Navier-Stokes equations is not only viscosity. In most cases it is useful to define the terms you want to talk about, e.g., what exactly do you mean when your are talking about Euler and Navier-Stokes equations. The classical Navier-Stokes equations consists of at least two different parabolic effects, heat conduction according to Fouriers law and viscous effects according to Stokes law. If you play with more general physics you have to consider more parabolic effects, e.g. Fickian law for multi-species or combustion.
Hahaha I don't understand nothing you do not mention.

Thanks to the rest!

April 21, 2020, 21:54
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 Originally Posted by CellZone Hahaha I don't understand nothing you do not mention. Thanks to the rest!
It depends!

No matter how much you talk about Reynolds or Mach number number, if you don't mention your application a advice is meaningless.

Consider fuel injection or multi-phase flows. Here, i would not recommend to run a simulations only with the Euler equations, even if viscous effects are small enough. Other physical effects, e.g. heat conduction, are here far more important.

Last edited by Eifoehn4; April 22, 2020 at 00:50.

April 22, 2020, 08:27
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 Euler and Navier-Stokes equations is not only viscosity
So when I look up at Wikipedia, it says Euler "can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity".

What I learned in school (maybe I did not pay enough attention) is that Euler=Navier Stokes + zero viscosity...

Never heard from conductivity ... makes me confused

April 22, 2020, 08:51
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Filippo Maria Denaro
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Quote:
 Originally Posted by CellZone So when I look up at Wikipedia, it says Euler "can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity". What I learned in school (maybe I did not pay enough attention) is that Euler=Navier Stokes + zero viscosity... Never heard from conductivity ... makes me confused
Think about the total energy equation in terms of the well known thermodynamics law
dE/dt= W - Q

In order to have a reversible transfer of energy you need zero heat flux and the only reversibile part of the mechanical work. That is no viscosity and no conducibility