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| A '''direct numerical simulation (DNS)'''<ref>Here the origin of the term '''direct numerical simulation''' (see {{cite journal|last=Orszag|first=Steven A.|authorlink=Steven A. Orszag|year=1970|title=Analytical Theories of Turbulence|journal=[[Journal of Fluid Mechanics]] (see, e.g., page 385)|volume=41|issue=1970|pages=363–386}}) owes to the fact that, at that time, there were considered to be just two principal ways of getting '''theoretical''' results regarding turbulence, namely via turbulence theories (like the direct interaction approximation) and '''directly''' from solution of the Navier–Stokes equations.</ref> is a [[simulation]] in [[computational fluid dynamics]] in which the [[Navier–Stokes equations]] are numerically solved without any [[turbulence]] model. This means that the whole range of [[Three-dimensional space|spatial]] and [[time|temporal]] scales of the turbulence must be resolved.
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| All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales ([[Kolmogorov microscales]]), up to the integral scale L, associated with the motions containing most of the kinetic energy. The Kolmogorov scale,<math>\eta</math>, is given by
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| :<math>\eta=(\nu^{3}/\varepsilon)^{1/4}</math> | |
| where ν is the kinematic [[viscosity]] and ε is the rate of [[kinetic energy]] dissipation. On the other hand, the integral scale depends usually on the spatial scale of the boundary conditions.
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| To satisfy these resolution requirements, the number ''N'' of points along a given mesh direction with increments ''h'', must be
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| :<math>Nh > L,\,</math>
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| so that the integral scale is contained within the computational domain, and also
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| :<math>h \leq \eta,\,</math>
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| so that the Kolmogorov scale can be resolved.
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| Since
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| :<math>\varepsilon \approx {u'}^3/L,</math>
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| where ''u''' is the [[root mean square]] (RMS) of the [[velocity]], the previous relations imply that a three-dimensional DNS requires a number of mesh points <math>N^{3}</math> satisfying
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| :<math>N^{3}\ge \mathrm{Re}^{9/4} = \mathrm{Re}^{2.25}</math>
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| where Re is the turbulent [[Reynolds number]]:
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| :<math>\mathrm{Re}=\frac{u'L}{\nu}.</math>
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| Hence, the memory storage requirement in a DNS grows very fast with the Reynolds number. In addition, given the very large memory necessary, the integration of the solution in time must be done by an explicit method. This means that in order to be accurate, the integration must be done with a time step, Δ''t'', small enough such that a fluid particle moves only a fraction of the mesh spacing ''h'' in each step. That is,
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| :<math>C = \frac{u'\Delta t}{h} < 1</math>
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| (''C'' is here the [[Courant–Friedrichs–Lewy condition|Courant number]]). The total time interval simulated is generally proportional to the turbulence time scale <math>\tau</math> given by
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| :<math>\tau=\frac{L}{u'}.</math>
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| Combining these relations, and the fact that ''h'' must be of the order of <math>\eta</math>, the number of time-integration steps must be proportional to <math>L/(C\eta)</math>. By other hand, from the definitions for Re, η and ''L'' given above, it follows that
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| :<math>\frac{L}{\eta} \sim \mathrm{Re}^{3/4},</math>
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| and consequently, the number of time steps grows also as a power law of the Reynolds number.
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| One can estimate that the number of floating-point operations required to complete the simulation is proportional to the number of mesh points and the number of time steps, and in conclusion, the number of operations grows as Re<sup>3</sup>.
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| Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the [[supercomputer|most powerful computers currently available]]. However, direct numerical simulation is a useful tool in fundamental research in turbulence. Using DNS it is possible to perform "numerical experiments", and extract from them information difficult or impossible to obtain in the laboratory, allowing a better understanding of the physics of turbulence. Also, direct numerical simulations are useful in the development of turbulence models for practical applications, such as sub-grid scale models for [[Large eddy simulation]] (LES) and models for methods that solve the [[Reynolds-averaged Navier–Stokes equations]] (RANS). This is done by means of "a priori" tests, in which the input data for the model is taken from a DNS simulation, or by "a posteriori" tests, in which the results produced by the model are compared with those obtained by DNS. The biggest DNS in the world, up to this date, used 242 billion degrees of freedom. It was carried out by a team of researchers at [[the University of Texas at Austin]] on Mira, the IBM [[Blue Gene]]/Q supercomputer at [[Argonne National Laboratory]] in 2013.<ref>MK Lee, N. Malaya, and R.D. Moser, "Petascale direct numerical simulation of turbulent channel flow on up to 786K cores", SC '13 Proceedings of SC13: International Conference for High Performance Computing, Networking, Storage and Analysis, Denver, CO (2013) http://dl.acm.org/citation.cfm?doid=2503210.2503298</ref>
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| ==See also==
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| * [[Large eddy simulation]]
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| * [[Reynolds-averaged Navier–Stokes equations]]
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| ==External links==
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| * [http://www.cfd-online.com/Wiki/Direct_numerical_simulation_(DNS) DNS page] at CFD-Wiki
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Direct Numerical Simulation}}
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| [[Category:Fluid dynamics]]
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| [[Category:Turbulence]]
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| [[Category:Turbulence models]]
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| {{Link GA|de}}
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