Is turbulence more likely to form with the Euler equation as opposed to Navier-Stokes? The Euler equation models perfectly inviscid fluids. Under this assumption, with $\nu = 0$, the Reynolds number should be infinite. I would guess that this implies the Euler equation is always turbulent, but this is not the case as in practice it is used to model regular (low viscous) fluids. Why does this occur?
And second, does this imply that the Euler equation is always more turbulent than the Navier-Stokes, which accounts for viscosity?
 A: Wiki states on turbulence:

Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration;

So every continuum equation which includes material derivative of flow velocity term :
$$ {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}= {\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} $$
includes time-dependent flow velocity change and convective acceleration term (flow velocity gradient) and is able to describe how flow parcel speed changes along travel trajectory with passage of time. Thus by definition, continuum equations with flow velocity material derivatives can describe flow turbulence, including but not limited to Navier–Stokes equations and Euler equations. So your intuition based on high Reynolds number in Euler equations was right.
A: This is a very good question. What you're talking about in a certain sense is known as the 'dissipative anomaly': in the limit of infinite Reynolds number (or viscosity to zero) the dissipation of energy is finite, but of course the Euler equations are inviscid and its solutions don't dissipate energy, or at least not the solutions in the usual sense. Then, if when $\nu \to 0$ the solutions to Navier-Stokes equations dissipate energy, Euler equations can't describe fully developed turbulence?
This is related to what is known as 'Onsager's conjecture': there are solutions to the Euler equations that dissipate energy but they must have a Hölder exponent $\leq1/3$. What can these 'weak' Euler solutions tell us about fully developed turbulence?. For more information about this you can check the papers by Eyink "Review of the Onsager ideal turbulence theory" and "Onsager and the theory of hydrodinamic turbulence". I would also recommend "Do dissipative weak Euler solutions dream of turbulence?" by Takeshi Matsumoto.
A: What do you mean by "more turbulent"?
Anyway, to qualitatively answer your question, you have to keep in mind that:

*

*viscosity, beside damping smallest scales of turbulence, could trigger instability that can evolve in turbulent flows or regions in the flow. As an example, think as pipe flows, stable in the inviscid limits, unstable for viscous flows (you can have a look at historically noteworthy criteria of stability criteria for oarallel flows, as the Rayleigh, Fjortroft crieteria, and Orr-Sommerfeld equations); or you can think at boundary transition to turbulent regime in real-life flows on solid boundaries at high Reynolds number; in a very extreme case, you can compare the results of a Euler and Navier-Stokes numerical simulation of an airfoil at large angle of incidence: with Euler inviscid equations you have no separation, while with Navier-Stokes taking into account viscosity influence you have large separation regions, turbulent at large Reynolds number;

*Euler equations and Navier-Stokes equations are mathematical models to describe the experience and they have their limits of validity within they give us good results, consistent with experiments and real-life;

*using Euler equations and other inviscid models at high Reynolds number within they're limits of validity, it's like you're lumping all the viscosity and rotational effects in infinitesimally thin regiona of the domain, like vortex sheets or vortex lines. So you can't really appreciate the small-scale turbulence in these thin regions, while it's likely that you can appreciate larger scales coming from the instabilities and interactions of these rotational regions of the fluid

