[marshallward]

We use biharmonic diffusion operators in oceanography as a basic turbulence model, which is effectively a fourth order derivative in space. It is like an accelerated diffusion which aggressively dissipates small scales but preserves the large physical scales more robustly than normal diffusion.

[dhdc]

Kuramoto-Sivashinsky is another fun equation with biharmonic operator.

[boppo1]

Maybe you can help me: what is it about turbulence that remains mathematically 'unsolved'? How is turbulence different than navier-stokes applied to a really fast moving medium with directional randomness?

[btrettel]

Turbulence can be accurately solved by simulating the Navier-Stokes equations on powerful enough computers. [0] The problem is that the computational complexity is enormous. Even relatively simple turbulent flows may exceed the capabilities of a supercomputer 50 years from now. This is pretty easy to show mathematically using the Kolmogorov scales [1] combined with the CFL condition [2] to estimate the computational complexity of unbounded turbulence. (But it's an incomplete estimate. Adding boundaries or other complexities will increase the computational cost.)

The real turbulence problem is figuring out ways to get around the computational complexity with cheaper models. Accurately modeling turbulence without using the full Navier-Stokes equations is really hard.

Also, contrary to what the science media says, the Navier-Stokes existence and uniqueness problem doesn't have anything to do with turbulence being hard aside from that both involve the Navier-Stokes equations. [3, 4] 2D Navier-Stokes, where existence and uniqueness has been proved, still has turbulence.

[0] Given accurate initial and boundary conditions.

[1] https://en.wikipedia.org/wiki/Kolmogorov_microscales

[2] https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%...

[3] https://news.ycombinator.com/item?id=31227133

[4] https://news.ycombinator.com/item?id=31049535

[marshallward]

I think the statement has different meanings to different people, but one "unsolved" problem is the ability to model the dynamics of small scales (the "turbulence") in terms of the large-scale "observed" flow.

The question has very strong analogies to thermodynamics. For example, one can average over microscopic motion to yield something like a diffusion equation, where transport of the (microscopically averaged) density is pushed from high to low concentrations, and all of the microscopic details get wrapped into a single number, the diffusion coefficient.

In fact, averaging over molecular dynamics works in so many contexts that the details end up not being terribly important. You will always end up with something like a diffusion equation.

It's very reasonable to think this ought to work more generally, averaging over the turbulence to produce a similar expression for the dynamics of the large-scale. But if you try to apply similar averaging techniques to the Navier-Stokes equations, the averages never end, no clear solution emerges, and the only hope to terminate the exercise is to insert some kind of "closure approximation".

Some consider a robust theory for such a closure approximation, or any method to resolve the impact of the turbulent flow on large-scale flow, to be an unsolved problem of turbulence.

These questions have been researched for many decades, and a great deal has certainly been learned, but a rigorous closure theory has been elusive. Meanwhile, the computers keep getting bigger and faster, to the point where the turbulence can in many cases be modeled reasonably well. And as the questions around turbulence become relegated to smaller and smaller scales, one starts to wonder if this is a problem that will even need to be solved in the future.

[lutorm]

My understanding is that it's "solved" in the sense you say, we know that the solution obeys the NS equations. The problem is that we can't actually solve them on a sufficient range of scales to actually make that knowledge useful for making predictions, so we have to resort to approximations.

[dhdc]

Turbulence is unsolved because Navier-Stokes is unsolved.