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 much 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, or that if they do exist they have bounded kinetic energy. This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step for understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute offered a US$1,000,000 prize in May 2000, not to whomever constructs a theory of turbulence, but (more modestly) to the first person providing a hint on the phenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem: [snip]
Note: "turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering."
My question is: considering the extensive current use of computational fluid dynamics, what effects would a complete theory of turbulence actually have on engineering?
I hope it would enable some great advances in capability, but I consider it possible that everyone would just say 'oh that's nice - but it doesn't enable us to do anything we can't do with CFD already'.
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In practical terms, a complete theory might enable more efficient computational simulation? If we know that turbulent phenomena always have X, we might be able to use that to shortcut the code instead of integrating NS step-by-step.
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Speed of calculations I should think would be an immediate benefit. The bit design guys at my office will often set up a drill bit model and then leave that model running over night or over the weekend even to see how the drilling mud would flow over and around the bit. Imagine being able to do those same calcs while you walk to refresh your coffee.
The benefits would also spread to things like aircraft design, ship design, automobile design, studies of flow patterns, etc.
As it is, as Surlethe says, the current software integrates and then integrates and then integrates over and over as the flow changes when it moves past whatever is being designed. Which is why the complex models can take hours or days to provide an adequate solution.
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Some phenomenon would be quicker and easier to simulate :
- Simulation of the effects of conventional high-explosive and nuclear weapons under certain conditions.
- Simulation of the dynamic of plasma flow in a tokamak!fusion reactor.
- Weather patterns simulation/prediction (on Earth and on other planets) ==> better weather forecast on earth and understanding of the dynamic of gas giants' atmospheres.
- Stars internal dynamic.
- Simulation of the formation of a stellar system (when it is at the "gas & dust cloud" state).
My intuition (poorly-formed as it is w.r.t. PDEs) is telling me that even a partial theory of turbulence would be enormously helpful if it provides very good qualitative descriptions of the phenomenon. I'm thinking that almost anything would be better than integrating a nonlinear PDE.
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Yeah, pretty much. Obviously, an analytical solution to the NS, even with strings attached, would be ideal, since that allows you to skip the computationally arduous step of numerically integrating the equations. A complete theory of turbulence would potentially allow for better simplifying assumptions to be made, or other shortcuts to be taken. Nonlinear PDEs suck from a modelling standpoint.
Thanks everyone, that's what I wanted to hear. I find it all rather fascinating, even though it's outside my 'proper' field.
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The ideal result would be that we could achieve solution fidelity on par with a DNS simulation (which require computer resources that make use of such simulations untenable within our lifetime for engineering scale problems) but with the computational cost of a zero-equation mixing length type model. If this were to happen, for most applications we'd see an immediate ~10-20%, in terms of turn around time, improvement over current 1,2, or 4 equation turbulence models but solution fidelity (most notably in heat transfer and viscous losses) would be greatly improved. For applications where LES simulations are common, such as combustor modeling, you could see up to 2 orders of magnitude improvement in turn around time.
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A theory of turbulence would presumably imply an ability to predict the onset of turbulence. Even in the low-speed world this would have profound impact: high-energy fluid near a wall implies higher drag but delayed onset of stall, for example (c.f. vortex generators). At higher speeds, turbulent transition represents a multiplicative increase in drag and heat flux by a factor of two, three, five. Quite apart from high-fidelity understanding, an arbitrarily accurate model would allow the calibration of lower-fidelity models that would allow reasonably accurate prediction of a wide variety of engineering flows.
