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Water flows in rivers show a wide variety of organized structures due to the interaction of the flow with its boundary. In whitewater, people give these features names like rollers, vortices, holes, hydraulics, and so on, as depicted below: Gauley River

It is my understanding that all of these phenomena should be accessible as solutions to the Navier Stokes equations in the incompressible limit, and while it is certain that a number of direct numerical simulations of flows over rough boundaries have produced these types of features, I never see such features derived from the Navier Stokes equations. The one counterexample I can think of is that the book by Morse and Feshbach does derive a simple laminar vortex as a solution.

I am curious if any works have concentrated on very simplified derivations of the various organized flow structures which appear in nature. For example, one could consider that an underwater ledge produces a horizontal roller, whereas the flow going past a barrier produces a lateral vortex.

Acknowledging that most canonical books on open channel flows (i.e. Nezu and Nakagawa) summarize turbulent velocity profiles in flows over a flat boundary, and do not really go much beyond this, I would love to find a collection of "simple harmonic oscillator" style derivations of the richer phenomena contained in the Navier Stokes equations over more interesting boundaries -- reduced-complexity representations of whitewater, with isolated rollers, solitary lateral vortices, and so on. Do any references exist which provide idealized presentations of these real-world phenomena?

Qmechanic
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kevinkayaks
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1 Answers1

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No, Such a references does not exist.

The reason is simple; The Navier-Stokes existence and Smoothness problem is not solved. Wiki quote;

The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.

Even more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem.

In reality this means that Navier-Stokes equations are mostly solved with Computational Fluid Dynamics (CFD) which always contains a correcting factor which is always originally calibrated in the lab through an real experiment.

This above does not include simple laminar flows, which can be solved correctly.

Here in Stackexchange this question and it's answer is related;
(Horizontal roller = Hydraulic jump)

How should energy loss in a hydraulic jump be calculated?

At the answers there is one derivation presented.

Jokela
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