Editing Computational fluid dynamics (section)

==== Reynolds-averaged Navier–Stokes ====
{{main|Reynolds-averaged Navier–Stokes equations}}
[[File:DrivAer SST-URANS-DDES Comparison.png|thumb|258x258px|External aerodynamics of the [https://www.mw.tum.de/en/aer/research-groups/automotive/drivaer/ DrivAer] model, computed using [[Reynolds-averaged Navier–Stokes equations|URANS]] (top) and [[Detached eddy simulation|DDES]] (bottom)]]

[[File:Verus Engineering Porsche 987.2 Ventus 2 Package.png|thumb|A simulation of aerodynamic package of a [[Porsche Cayman|Porsche Cayman (987.2)]]]]

[[Reynolds-averaged Navier–Stokes equations|Reynolds-averaged Navier–Stokes]] (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new ''apparent stresses'' known as [[Reynolds stresses]]. This adds a second-order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

; [[Turbulence modeling|Boussinesq hypothesis]]: This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε ([[Brian Launder|Launder]] and [[Brian Spalding|Spalding]]),<ref>{{cite journal|last=Launder|first=B.E.|author2=D.B. Spalding|year=1974|title= The Numerical Computation of Turbulent Flows|journal=Computer Methods in Applied Mechanics and Engineering|pages=269–289|doi = 10.1016/0045-7825(74)90029-2|bibcode = 1974CMAME...3..269L|volume=3|issue=2 }}</ref> Mixing Length Model ([[Ludwig Prandtl|Prandtl]]),<ref name=wilcox>{{cite book|last=Wilcox|first=David C.|title=Turbulence Modeling for CFD|year=2006|publisher=DCW Industries, Inc.|isbn=978-1-928729-08-2|edition=3}}</ref> and Zero Equation Model (Cebeci and [[Apollo M. O. Smith|Smith]]).<ref name=wilcox /> The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a "Zero Equation" model because no transport equations are solved; the <math>k-\epsilon</math> is a "Two Equation" model because two transport equations (one for <math>k</math> and one for <math>\epsilon</math>) are solved.
; [[Reynolds stress model]] (RSM): This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.{{Citation needed|date=November 2010}}