6.13 Turbulent kinetic energy
The Reynolds stress in the ensemble-averaged momentum conservation Eq. (6.15 ) can be decomposed into viscous and pressure components. The turbulent pressure term, , is commonly subsumed within pressure , to give
Averaging the momentum equation and introducing the eddy viscosity model creates one additional unknown . Additional models are required for to close the system of equations.
By considering from Sec. 6.11 , the model for is typically decomposed into components representing velocity and length scales, and respectively.
The scale of corresponds to turbulent ﬂuctuations , so it is reasonable to assume that . Since the ﬁeld is representative of the component of , it is commonly adopted within turbulence models based on . Also, it is well captured by a suitable conservation equation.
Transport of turbulent kinetic energy
Conservation of turbulent kinetic energy can be written as:
In Eq. (2.56 ), energy from bulk motion is passed to the submicroscopic scale as heat by . In Eq. (6.28), it is converted to turbulent energy by using Boussinesq’s from Eq. (6.20 ). The shear component provides the non-recoverable in Eq. (6.28) and the second () term yields .
While transfers kinetic energy from the bulk ﬂow to , the ﬁnal term transfers on to as dissipated heat. Here, is the turbulent dissipation rate per unit mass from in Sec. 6.6 which, from an ensemble-averaged derivation of Eq. (6.28), is
The equation goes part of the way to closing our system of equations. However, it introduces an additional unknown, , and the model for still requires length scale .