# Numerical cfd methods

Numerical cfd methods

DNS The Direct Numerical Simulation is a method of simulation in Computational Fluid Dynamics that permits to solve directly the Navier-Stokes equations without using any modeling, averaging or approximation other than the numerical discretizations performed on these equations. Consequently, with this method, every instantaneous quantities of the flow have to be known and then, all of the fluid motions contained in the flow are considered to be resolved.

Since the Direct Numerical Solution requires all significant turbulent structures (eddies) to be adequately captured, the mesh size must be smaller than the dissipation scale. That means an important number of mesh elements. That’s why, with this method, you have extremely long calculus time. The more mesh points you have, the longer the calculus will be. And since if the speed of flow increases, the number of time steps increases, then similarly, the faster the flow is, the longer the calculus will be. So the cost of the Direct Numerical Simulation is very high.

Even if the capacity and the power of the tools of calculation keep improving, the use of this method remains limited to simple cases of application with reasonable speeds of flow. Indeed, most of the applications are still too demanding for

Désolé, mais les essais complets ne sont disponibles que pour les utilisateurs enregistrés

the currently available computers. Nevertheless, this method represents a good way of understanding turbulent flow behaviors in practical applications. LES The Large Eddy Simulation is a numerical method of resolution of the partial differential equations governing turbulent fluid flow.

This technique forms a good alternative to Direct Numerical Simulation by solving only the large-scale motions and modeling the small-scale ones. A simulation that treats the large eddies but approximates the small eddies makes perfect sense because the large-scale motions are usually much more energetic than the small ones. In order to define the large-scale field, a filter is used. Kolmogorov’s scale permits to determine which eddies are considered as large eddies, and which eddies are considered as small eddies.

Then, by filtering the Navier-Stokes equations, we obtain resulting equations for the large-scale component of the velocity which contain terms that represent the small-scale (called sub-grid scale) effects on the large scales. These subgrid-scale motions have to be modeled. The mesh size generally corresponds to the filter size. As some of the turbulent scales are not resolved, compensation is made by adding an “eddy viscosity” into the governing equations. Large Eddy Simulation needs less computational effort than the Direct Numerical Simulation but it still remains high.

Resolution is consequently limited, as for the DNS. But, the power of our current computers begins to be sufficient enough to solve advanced problems by LES. In general, DNS is the preferred method because of its accuracy. However, LES is the preferred method for flows in which the Reynolds number is too high or the geometry is too complex. Those two methods are quite promising thanks to the small losses of information they permit. They indeed, contain very detailed information about the flow. However, the cost of a calculation is inversely proportional to its precision.

Therefore, the DNS and the LES method remain inapplicable in an industrial configuration. That’s why the following statistics methods of resolution are commonly used under those conditions. RANS The Reynolds-averaged Navier-Stokes is a statistic method that consists on dealing with the averaged-time equations of motion for fluid flow. It is generally used for turbulent flow. The basic equations are modified with approximations based on the known properties of the turbulent flow in order to obtain averaged solutions to the Navier-Stokes equations.

The assumption (known as the Reynolds decomposition) behind the RANS equations is that the time-dependent turbulent velocity fluctuations can be separated from the mean flow velocity. This introduces a set of unknowns called the Reynolds stresses, which depends on the velocity fluctuations. Those stresses require to use a turbulence model (such as the k-epsilon model) to create a system of solvable equations. Thanks to these averaged-time equations, there is no need to simulate the small-scale motions anymore. Consequently, the mesh size, but also the time step, is increased.

That’s why the RANS method of resolution costs less than the DNS and the LES ones. Therefore, that explains the largely frequent use of this method in industrial applications. Moreover, its field of application is very large (with or without heat transfers). However, this technique of simulation is highly empiric. So under certain conditions, the RANS method can be quite inaccurate. Jiyuan TU, Guan Heng YEOH and Chaoqun LIU. “Computational Fluid Dynamics- A practical approach”. Elsevier, 2008, p. 384-389. Boris GALPERIN and, Steven A. ORSZAG. “Large Eddy Simulation of Complex Engineering and Geographical Flows”.

Cambridge University press, 1993, p. 3-8, p. 37-40, p. 55-58. Yann MARCHESE, « Modelisation de la turbulence ». p. 36-41 CFD Online. 17 December 2008, http://www. cfd-online. com/Wiki/Large_eddy_simulation_(LES). CFD Online. 2 February 2007, http://www. cfd-online. com/Wiki/Direct_numerical_simulation_(DNS). Wikipedia. 28 June 2010, http://en. wikipedia. org/wiki/Direct_numerical_simulation. Wikipedia. 6 July 2010, http://en. wikipedia. org/wiki/Large_eddy_simulation. Wikipedia. 29 May 2010, http://en. wikipedia. org/wiki/Reynolds-averaged_Navier–Stokes_equations.