Is inviscid flow necessarily turbulent?

The Reynolds Number is defined as $Re = \frac{\rho v L}{\mu}$, where $\rho$ is the density, $v$ is the fluid velocity, $L$ is the characteristic length, and $\mu$ is the dynamic viscosity. By definition, inviscid flow implies that $\mu=0$. By this formula, this would make the reynolds number infinite and thus turbulent.

Is it possible to have laminar inviscid flow? Must inviscid flow always (necessarily) be turbulent?

• Could you provide a rough description of the problem you want to solve? – rul30 Oct 8 '15 at 18:04

No, inviscid flows are not necessarily turbulent. If there is nothing to "trip" the turbulence, then the flow will remain laminar. Features which could trip the turbulence include vibration, small temperature fluctuations, any geometric imperfections, velocity field imperfections, and other similar things.

For example, potential flow is a type of inviscid flow. Potential flow solutions are laminar solutions to the Navier-Stokes equations.

As another example, if care is taken to avoid vibration and other flow imperfections, it seems you can generate laminar pipe flows at arbitrarily high Reynolds numbers. Laminar pipe flows have been obtained at Reynolds numbers of about 100,000 under these conditions, which is far higher than the typical transition Reynolds number of about 2,000. From what I understand, there is no indication that 100,000 is any hard limit; you probably could get higher with better experimental setups.

How I think about it is this: Viscosity helps damp out flow imperfections, allowing laminar flows to occur with more imperfections. If you have a truly inviscid flow, it needs to be perfect to not trigger instabilities which lead to turbulence. If you were able to obtain an inviscid flow, you should expect turbulence due to the imperfections inherent in reality.

To answer a question you put in a comment in the other answer, yes, I do believe using a turbulence model for inviscid flow is prudent. For RANS, the Reynolds stress still will exist if the flow is inviscid, and for LES the same is true for the residual stress.

Reynolds number can be more simply defined as the ratio of Inertial forces to Viscous forces.

• viscous >> inertial = Laminar
• viscous << inertial = Turbulent

This is why when speed/flow rate increases fluids go towards/become turbulent as you're increasing the inertia of the fluid.

Based on this logic if you had inviscid (no viscous forces) flow then the inertial forces would have to dominate which should lead to Turbulent flow.

The fringe case

If you're asking if it's possible as a 'fringe case' well... then we're getting to the boundary between engineering and science. Inviscid flow in itself is just an assumption to simplify the NS equations.

AFAIK In real life there's nothing that is truly inviscid, just things that are almost inviscid. Thus as soon as you have some incredibly-close-to-zero viscosity value, then you should have some equally incredibly-close-to-zero flow value that allows laminar flow.

• Indeed, inviscid flow is a simplifying assumption. My concern is whether or not I should use a turbulence model for most practical purposes when simulating inviscid flow (i.e. the velocity is definitely not close to zero). – Paul Oct 8 '15 at 1:59
• While I am definitely not a CFD guru I would assume you should given that in your case viscous<<<inertial forces. Which is what gives rise to turbulence. You might be better off asking on cfd-online – m4p85r Oct 8 '15 at 4:15
• cfd-online.com/Forums/main/10165-inviscid-flow-turbulnces.html It appears to vary based on the application/what is important in your model. – m4p85r Oct 8 '15 at 4:17

As pointed out by @rul30, you should give more details about the problem you have, specifically whether you have an inviscid flow or a high-Reynolds number flow problem.

Anyway, I hope you find this helpful.

What is Reynolds number?

Reynolds number is a dimensionless number obtained from the results of the famous experiments performed by Osborne Reynolds to verify the existence of flow regimes in real-world (viscous) newtonian flow, it was found that the transition from laminar to turbulent flow depends on surface roughness, flow velocity, surface temperature and type of fluid, however, Reynolds found that the transition depends mainly on a ratio between inertial forces $(\rho V_{avg} D)$ and viscous forces $\mu$. With some care using the characteristic length $D$, we can also express the transition from different flow regimes not only in wall-bounded flows but also shear-layer flows, jet flows, wake flows, etc.

For potential flow it's meaningless to find the Reynolds number since $\mu = 0$. Even in some cases when neglecting the effect of viscosity in high speed flows there is a calculable value of Reynolds number, very high indeed but calculable and definitely not infinity.

What causes turbulence?

As the Reynolds number increases, the instabilities within the boundary layer increase (that mainly exist due to viscosity and non-slip condition). Several physical phenomena take place that leads to a disorderly chaotic motion (Tollmien-Schlichting waves, vortex shedding/stretching, etc.).

Now for a pure inviscid flow (potential flow) $\mu = 0$ (flow is governed by Euler equations), the above mentioned instabilities that leads to turbulence would simply not exist as the most important factor in causing turbulence (boundary layer) vanishes.

Conclusion

Is it possible to have laminar inviscid flow? Must inviscid flow always (necessarily) be turbulent?

Laminar and turbulent regimes distinction is not applicable in pure inviscid flow (potential).

My concern is whether or not I should use a turbulence model for most practical purposes when simulating inviscid flow (i.e. the velocity is definitely not close to zero).

If you used CFD codes like FLUENT and STAR-CCM+ you would notice that selecting inviscid model will by default omit the selection of turbulence models (doesn't make sense). So no, you don't use a turbulence model.

However, if your case was a viscous fluid flow that its viscous effect can be neglected due to high inertial forces(High speed flow over an airfoil), that's then a whole other story and turbulence modeling is indeed mandatory in this case, that's why I urge you to expand your question with more details if this is the case.

• Some quibbles. Boundary layers are not needed to develop turbulence. You mentioned shear flows, which are an example of this phenomena. Even an inviscid Kelvin-Heltholtz instability will become turbulent (here is a linear stability analysis. It's also important to note that potential flow is much stricter than inviscid flow (you didn't make this clear). Potential flows are irrotational and steady in addition to being inviscid. The additional two requirements basically prevent anything we might call turbulence from appearing. – Ben Trettel Oct 9 '15 at 22:07