# Is Navier stokes a turbulence model?

Is Navier stokes a turbulence model? If yes, what is the use of k-omega model.. if no, what does the Navier stokes equation got to do with the turbulence models ..? I am very new to fluid dynamics... Kindly help me understanding this...

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 ($\epsilon$).

• " believed to be fully captured" is correct. The N-S equations describe the behaviour of a continuous medium but of course a real fluid consists of molecules, and even for a simplified model like an ideal gas the behaviour of the individual molecules is not captured at all by N-S. So the physical origin of turbulence is still a mystery, even though N-S "seems to work" modelling it at scales that we can "easily" observe. – alephzero Sep 14 '18 at 20:17
• Thanks. I have another doubt. What do you mean by 'closing' the RANS equation. – Anu Sep 15 '18 at 1:30
• @Anu the Reynolds averaged NS equations include a 'turbulent viscosity' term, which captures the effect of the turbulence on the mean flow. That term has to be closed (i.e. accounted for) in the RANS equations, which is what the turbulence model is doing. – Time4Tea Sep 15 '18 at 17:05
• @Time4Tea . Thank you so much. It helped a lot – Anu Sep 15 '18 at 17:34

Navier-Stokes is the group of three main equations (mass conservation, energy conservation and momentum conservation) which explains the flow of a fluid. Flow of a fluid can be laminar and turbulent both of which NS can explain. However, when mathematicall manipulating the NS equation, one faces some parameters which need be explained more thoroghly for turbulent flows, for instance Reynolds stress. In order to solve numerically the turbulent flows, many "turbulent models" have been developed during the course of the fast few decades. There are one-equation turbulence models such as Spalart-Allmaras, two-equation models such as k-omega and k-epsilon, also there are more advanced, accurate and therefore demanding models such as Large Eddy Simulations (LES), and better than that is Direct Numerical Simulations (DNS).

K-omega is a 2-equation turbulence model which is commonly used for engineering purposes and in the industry. It is commonly used due to its rather acceptable accuracy (for more routine applications and flows) and simple implementation and modest computational power needed. This is also why it is implemented in all commercial CFD solvers along with k-epsilon and k-omega SST (which is a modified K-omega).

There is no correct turbulent solution of the Navier-Stokes equation. There are various approximations. The nonlinear term is approximated linear with dynamic viscosity, which is chosen from the condition of coincidence with the field experiment. Or they introduce a mean of values, but this results in more unknowns than equations. It is necessary to do the approximation of the correlation function. The situation is common for nonlinear partial differential equations. In the real plane, taking into account the nonlinear term of the solution is not. I received a complex solution of the Navier-Stokes equation in the turbulet mode.

Brief summary of scientific direction: Using complex values of velocity and coordinates when solving nonlinear partial differential equations

Just as the square equation has complex roots, the nonlinear partial differential equations have complex solutions. It turns out that the complex solution is probabilistic. The physical meaning of the real part is the average value of the solution, and the imaginary part means the standard deviation. The nonlinear Navier-Stokes equation is reduced to an infinite system of ordinary differential equations of the first order. The complex coordinates of the equilibrium position describe the turbulent solution. Problems arise when recalculating the imaginary part of a complex solution into a real solution. But in the attached articles, for which the abstract describes the solution to these problems. For different types of roughness, the solution to these problems is different.

YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION I. THE GENERAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 60-66. https://world-science.ru/pdf/2016/3/14.pdf

YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION II. THE USE OF LAMINAR SOLUTIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 67-83.https://world-science.ru/pdf/2016/3/15.pdf

YAKUBOVSKIY, E. G. "STUDY OF NAVIER–STOKES EQUATION SOLUTION III. THE PHYSICAL SENSE OF THE COMPLEX VELOCITY AND CONCLUSIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 84-87. https://www.world-science.ru/pdf/2016/3/16.pdf

• Best I can tell online, this journal you've published to is on the "predatory publishers" (cf. this list), so I somewhat doubt the credibility of your papers. – Kyle Kanos Jan 6 '19 at 2:10
• We must look at the content, not on the reputation of the magazine. – Evgeniy Yakubovskiy Jan 6 '19 at 6:18
• I would be very pleased to hear the opinion of a specialist about the shortcomings of my articles. I know some of them, for example, solving a linear problem of the hydrodynamic equation is not a simple thing, and I got it wrong; you need to calculate the derivative along the normal of speed at the surface of body. There may be other shortcomings, but whoever does something new always makes mistakes – Evgeniy Yakubovskiy Jan 6 '19 at 7:45
• And see that's the thing: real papers are peer reviewed (for mistakes & content) prior to being published, not after. If you're still wanting peer review, my doubts are now even stronger. – Kyle Kanos Jan 6 '19 at 11:51
• I honestly describe the situation. If you are against my participation with my files, then take administrative measures. I can not add anything new. – Evgeniy Yakubovskiy Jan 6 '19 at 17:21