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.


  • $\begingroup$ DNS of turbulence uses Navier Stokes equation without extra models tagged on. $\endgroup$
    – Deep
    Oct 6, 2017 at 5:58
  • 1
    $\begingroup$ 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. $\endgroup$
    – Yrogirg
    Nov 16, 2017 at 5:24

3 Answers 3


First of all the Navier-Stokes equations also hold for turbulent flows without any explicit turbulence modelling. Turbulence models are only introduced due to computational restrictions. Obviously the more physics you cut from the equations the harder it is to find models that reconstruct them.

  • A direct numerical simulation (DNS) of all relevant length scales present in a $dim$-dimensional flow $N^{dim} \sim Re^\frac{3 \cdot dim}{4}$ and time scales $NT \sim \sqrt{Re}$ (according to Kolmogorov's theory, in practice without wall function the requirements are even more restrictive, $Re$ in this context is the Reynolds number $Re := \frac{U L}{\nu}$) present in a realistic technical application (e.g. aerodynamics $Re \gtrsim 10^5$) requires storage and computing capacities that exceed current computing capacities by several orders of magnitude. Whenever available, quality DNS results are though in my opinion far superior to experimental measurements as you can access certain quantities (such as distribution of turbulent kinetic energy) that measurements can't with an accuracy superior to it.

This leaves us with simplified models for highly turbulent flows, which themselves are by nature three-dimensional ($dim = 3$) phenomena. Current methods apply some sort of averaging of the conservation equations in order to reduce computational burden. This leads to unknown fluctuation terms (Reynolds stresses) that have to be modeled accordingly:

  • Three-dimensional Large eddy simulations introduce ensemble averaging of the Navier-Stokes equations and commonly close it with isotropic eddy-viscosity turbulence models on a small sub-grid scale, meaning unresolved scales are simply swallowed by an artificially increased viscosity. While large eddies are resolved, the smallest ones are modeled - often using some variation of the Smagorinsky turbulence model with wall-corrections. This requires resolving most of the energy spectrum present in a flow (as a guideline 80-90% should be resolved and only 10-20% modelled resulting in millions of grid cells with several layers of grid refinement) and if applied correctly has a similar effect to low-pass filtering an image with a Gaussian filter. The results are transient, resolve long term oscillations as well as part of the turbulent fluctuations and are in generally very accurate but still computationally expensive (still millions of grid cells and supercomputers with days/weeks of calculations). They require a fine grid, lot of experience and expertise but in my opinion lead to very accurate simulations that are mostly superior to results obtained by experiments. They are comparably straight forward to extend to moving geometries but the are restricted to moderate turbulent Reynolds numbers ($Re \lesssim 10^5$). Towards the walls the grid has to be significantly refined and/or preferably complemented by a wall function like the Van-Driest damping function in order to capture the viscous effects and reduce the excessive eddy viscosity in these near wall regions.

  • Reynolds-average Navier-Stokes (RANS) methods apply temporal averaging and close this either by introducing again an isotropic viscosity (such as one-equation Smagorinsky model or two-equation $k$-$\epsilon$- or $k$-$\omega$-models but now with a very coarse resolution), solve additional transport equations for the full unknown Reynolds stresses (full RSM or simplified algebraic RSM with 6 additional transport equations) or a mixture between the two like the $ 3 \frac{1}{2}$-equation $k$-$\epsilon$-$\overline{\nu}^2$-$f$-model. The calculations might me performed on a coarser grid with a coarser temporal resolution, as well steady and of lower dimension. In my personal opinion the one- and two-equation models are pretty much worthless for most turbulent applications even though the knowledge behind them is undiscussably remarkable: They overestimate turbulent intensity and struggle with anisotropic flows (they assume isotropy and are not able to resolve the anisotropy) and the wall functions generally struggle with pressure gradients due to curved surfaces where backflow and separation might occur. Reynolds-stress models perform pretty decently but have strict requirements for the meshes and are computationally significantly more expensive and numerically less stable. They can be used to deduct information about technical applications (optimisation) but not about turbulent flows themselves. The three-and-a-half models are a good compromise but due to the isotropic viscosity struggle with anisotropic rotational flows. All of the models belonging to this family are incapable of capturing turbulent transient phenomena even when calculated unsteadily but limited information about average quantities can be deducted if a proper wall treatment (wall function) is applied and the user is fully aware of the limitations of the particular models. Furthermore the models implemented in commercial software are numerically so stable you can achieve results that are far from physical. This often involves eliminating terms for stability reasons.

  • Hybrid combinations of RANS (often in form of a simple Spalart-Allmaras turbulence model for the boundary layer) and LES (for the large eddies) lead to a very consistent compromise for transient technical applications but the quality really depends on the precise model. The interface between the two regions may be handled differently. Some methods apply a zonal two layer model. In this case the interface location depends on the precise mesh. More commonly used are seamless methods like Detached Eddy (DES), Partially Averaged Navier Stokes (PANS) or Scale Adaptive Simulation (SAS) where it is chosen dynamically which method should be chosen depending on a certain threshold. The main advantage of these models is that they are supposed to converge towards the LES solution in the limit of very fine grids, they were designed with the goal to get good results on every possible mesh combining advantages of both modelling paradigms. When applied to a very coarse grid they should outperform LES on the same grid.

  • Additionally some numerical schemes have certain sub-grid properties that allows them to under-resolve while retaining accuracy. The accuracy of such implicit turbulent models can be assess less rigorously.


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.





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.


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