# Turbulence Model on Unsteady Navier Stokes

I am asking you if the Unsteady (Time-Dependant) Navier-Stokes Equation is able to predict accurately the Flow Turbulence? I know that the RANS (with different Turbulence Models like Spalart–Allmaras, k–ε and k–ω models...) is the most used method for simulating the Turbulence.

I'd appreciate a constructive response.

Thanks

• DNS of turbulence uses Navier Stokes equation without extra models tagged on. – Deep Oct 6 '17 at 5:58
• Solving Navier--Stokes (DNS) is regarded to give exact solutions for turbulent flows, unlike the RANS methods you mention. We use inaccurate RANS models because exact Navier--Stokes is computationally unfeasible in practice. – Yrogirg Nov 16 '17 at 5:24

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

Turbulence, being a random, chaotic, unsteady and 3D phenomena, is not straightforward to be computed. However, there are some approximations in the form of turbulence models that have made it possible to predict the flow. The models you mentioned Spalart–Allmaras (one-equation model), k–ε and k–ω models (two-equation models) are most commonly used as you said but in the engineering and insustry application. It is the case due to their rather simple implementation, which do not require high computational power. This is also the reason why these models have been implemented in all commercial CFD solvers. On the other hand, there are more advanced models for approximating turbulence which are mostly used in science and for research due to their complexity and hifgh computational demand. Examples are Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS). The title of your question is a bit off since for steady flow, tuebulence modelling is obsolete since turbulence is regarded for unsteady flows. To get a more clear view you can read the first 2, 3 sections of Turbulence chapter of any fluid mechanics books. I also strongly recommend Turbulence for CFD book by Wilcox. It is a greeat route to approach this.