Is Navier-Stokes a turbulence model? If yes, what is the use of $k$–$\omega$ turbulence model? If no, what does the Navier stokes equation got to do with the turbulence models?
[Physics] Is Navier-Stokes a turbulence model
fluid dynamicsmodelsnavier-stokes;turbulence
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Turbulence is indeed an unsolved problem both in physics and mathematics. Whether it is the "greatest" might be argued but for lack of good metrics probably for a long time.
Why it is an unsolved problem from a mathematical point of view read Terry Tao (Fields medal) here.
Why it is an unsolved problem from a physical point of view, read Ruelle and Takens here.
The difficulty is in the fact that if you take a dissipative fluid system and begin to perturb it for example by injecting energy, its states will qualitatively change. Over some critical value the behaviour will begin to be more and more irregular and unpredictable. What is called turbulence are precisely those states where the flow is irregular. However as this transition to turbulence depends on the constituents and parameters of the system and leads to very different states, there exists sofar no general physical theory of turbulence. Ruelle et Takens attempt to establish a general theory but their proposal is not accepted by everybody.
So in answer on exactly your questions :
yes, solving numerically Navier Stokes leads to irregular solutions that look like turbulence
no, it is not possible to solve numerically Navier Stokes by DNS on a large enough scale with a high enough resolution to be sure that the computed numbers converge to a solution of N-S. A well known example of this inability is weather forecast - the scale is too large, the resolution is too low and the accuracy of the computed solution decays extremely fast.
This doesn't prevent establishing empirical formulas valid for certain fluids in a certain range of parameters at low space scales (e.g meters) - typically air or water at very high Reynolds numbers. These formulas allow f.ex to design water pumping systems but are far from explaining anything about Navier Stokes and chaotic regimes in general.
While it is known that numerical solutions of turbulence will always become inaccurate beyond a certain time, it is unknown whether the future states of a turbulent system obey a computable probability distribution. This is certainly a mystery.
Most importantly, the Navier-Stokes equations are based on a continuum assumption. This means that you should be able to view the fluid as having properties like density and velocity at infinitely small points. If you look at e.g. liquid flows in nanochannels or gas flows in microchannels you could be in a regime where this assumption breaks down. As far as I know there is no hard limit for the continuum assumption, but the Knudsen number is a useful indicator.
Additionally there is, as @ShuchangZhang mentioned, an assumption on the nature of the stress in the fluid. Although I am not sure whether you could really call this an assumption or whether it should be considered a theory (like the NS equations itself).
The strongest assumptions are typically not in the Navier-Stokes equations themselves, but rather in the boundary conditions that should be applied in order to solve them. To give an example, whether the no-slip boundary condition (fluid velocity at the wall equals wall velocity) or the navier slip boundary condition (fluid velocity equals a scaled velocity gradient at the wall) has been a much debated subject, in particular for hydrophobic surfaces (see e.g. Phys. Rev. Lett. 94, 056102 (2005) and references therein and thereto)
Best Answer
The Navier-Stokes Equations are not a 'turbulence model', they are more fundamental than that: they are the fundamental equations that govern all of fluid dynamics (assuming the continuum assumption holds).
The phenomenon of turbulence is believed to be fully captured by the N-S equations, which can be seen from Direct Numerical Simulation of turbulence, which uses the unmodified N-S equations, without a turbulence model.
Turbulence models are separate physical models that are supplementary to the N-S equations, which allow the phenomenon of turbulence to be described more simply. There are two main reasons why they are used:
Firstly, fluid turbulence in reality operates on very small distance and time scales, which means that to simulate (e.g. with CFD) a fully turbulent flow, an impractically huge amount of computational resource would be required. Even with a modern supercomputer, the Direct Numerical Simulation mentioned above can only be performed for a volume of a few cubic millimeters. Turbulence models such as Large Eddy Simulation aim to make turbulent CFD more practical, by 'filtering out' the smaller turbulent eddies.
Secondly, quite often for industrial applications, Engineers don't care about the time-dependent properties of the turbulence. All they really need to know is the time-averaged behavior, in which case trying to simulate the turbulence would be a waste of time. Reynold's Averaged Navier-Stokes Equations (RANS) are models that seek to model and solve only the time-averaged flow properties. The k-$\epsilon$ and k-$\omega$ turbulence models are ways of 'closing' the RANS equations, by modelling average quantities of the turbulence, such as turbulent kinetic energy (k) and turbulent energy dissipation ($\epsilon$).