Transition from laminar to turbulent flow Is there a mathematically rigorous definition of turbulent or laminar flow?
As far as I understand, laminar flow - means, that the velocity fields varies smoothly along the distance and time. Layers of fluid move smoothly along each other, without mixing. It doesn't have to potential, may containt vortices or other topological defects, but is continious in certain sense.
On the contrary, for turbulent flow, velocity is said to vary rapidly and discontiniously
https://www.youtube.com/watch?v=IPExwi2Ar-g&t=839s. Discontiniously means - that it may vary drastically on scales, smaller then some charasteristic size or length in the system?
There is Reynolds number, which is some characteristic value, which measures the ratio of kinetic energy of the flow to the viscous forces, defined by:
$$
\text{Re} = \frac{u L}{\nu}
$$
Where $u$ is a velocity, $L$ - characteistic size, $\nu$ - kinematic viscosity. But it is an approximate quantity, and only the order of its magnitude( whether it is of order $~10$ or $~10^4$) distinguishes what the flow is.
The question is :

Imagine we have flow, and gradually increase the inlet velocity, will there be at some point a discontiniuty? A phase transition. Or the change from one regime to another would be more like a crossover?

 A: There is no mathematically rigorous line.  You are correct that we use the Reynolds number to define the regions, but there is no strict line in the sand between them.
As tough as turbulent flow can be to calculate, its actually easier than calculating behaviors in the transition region between laminar and turbulent flow.  In laminar flow there are things you can handwave away.  For instance, you can assume the flow "sticks" to surfaces.  In turbulent flow there are other things you can handwave away.  For instance, turbulent flow is diffusive, so you can assume that, over a long period of time, flows mix.
In the transition region, both of these sets of simplifications become questionable.  You end up with a really complex system that defies a great deal of simplifications.  And like many transitions between two simplified modes, the line between them is not cut and dry.  It's a murky layer of pain.
A: The change from laminar flow to turbulent flow occurs within a range of flows, not at a particular point. The Moody diagram shows this range, and in that range, it is very difficult to confidently state that the flow is laminar or turbulent. For more info, see en.wikipedia.org/wiki/Moody_chart
A: the transition from laminar to turbulent begins to occur when the magnitude of the forces caused by viscosity (which tend to smooth out the flow and damp out instabilities) become smaller than the magnitude of the forces generated by the motion of a parcel of the fluid, as gauged by that parcel's kinetic energy. The fluid flow is thus stable to perturbations below that threshold (viscosity dominates) and unstable to perturbations above that threshold (viscosity is overwhelmed).
It's hard to pinpoint all the possible sources of perturbations to the flow in a pipe, and thus it is hard to pinpoint exactly at what set of conditions the flow will switch over from laminar to turbulent. This means that the Reynolds number calculation does not furnish a sharply-defined critical value for the transition.
A: There is no rigorous definition of laminar v/s turbulent flow. But here is something that comes close: A turbulent flow is characterised by a wide range of length and time scales, while a laminar flow is not. In practice, you can record a signal from the flow (for e.g. velocity, pressure) and subject it to Fourier transform. If the wavenumber or frequency content of the signal spans many orders of magnitude then you definitely have a turbulent flow.
Usually the entire flow doesn't become turbulent at once (unlike a phase-transition), but isolated patches of turbulence appear which then grow as you increase the Reynolds number and merge until the whole flow becomes turbulent. Also, for certain flows (for e.g. pipe flow), the Reynolds number at which they become fully turbulent depends very much on the strength of external disturbances. By carefully minimising those disturbances, the Reynolds number at which such a flow becomes fully turbulent can be increased by a few orders of magnitude. So I would say that the transition from laminar to turbulent flow is more like a crossover, akin to spreading of an infection, rather than a phase-transition.
