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## Reynolds-averaged Navier-Stokes equationsThe ## Additional recommended knowledgeThe left hand side of this equation represents the change in mean momentum of fluid element due to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress due to the mean pressure field, the viscous stresses, and apparent stress due to the fluctuating velocity field, generally referred to as Reynolds stresses. ## Derivation of RANS equationsThe basic tool required for the derivation of the RANS equations from the instantaneous Navier-Stokes equations is the ^{[3]}
where, is the position vector. The following rules will be useful while deriving the RANS. If Now the Navier-Stokes equations of motion Substituting,
, etc. The momentum equation can also be written as, On further manipulations this yields, where, is the mean rate of strain of strain tensor. ## Notes**^**The true time average () of a variable (*x*) is defined by*t*_{0}). This constraint is important for otherwise using time-averaging would be meaning less. This implies that the average value () is independent of time (*t*). Since it is not possible to integrate over an infinte time period, it is necessary to restrict the integration to some finite, yet large time interval. This interval is so selected that the term is independent of the length of the interval (*T*). However, the independence from*t*_{0}can no longer be ensured. Only in case of steady flows will be independent of both*t*_{0}and*T*. Thus,**^**By definition, the mean of the fluctuating quantity is zero().**^**Some authors prefer using*U*instead of for the mean term (since an overbar is used to represent a vector). Also it is common practice to represent the fluctuating term by*u*, even though*u*refers to the instantaneous value. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion we will use to represent the instantaneous, mean and fluctuating term.**^**The equations are expressed in tensor notation, which greatly simplifies the maths.**^****^**This follows from the mass conservation equation which gives,
Categories: Equations of fluid dynamics | Turbulence models |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Reynolds-averaged_Navier-Stokes_equations". A list of authors is available in Wikipedia. |