## 6.13Turbulent 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

 (6.25)
The viscous stresses can be combined assuming is modelled as a Newtonian ﬂuid Eq. (2.41 ) and by the eddy viscosity model Eq. (6.20 ) to give
 (6.26)
where the eﬀective kinematic viscosity
 (6.27)
The eﬀective viscosity represents momentum diﬀusion from the combined molecular and turbulent motions. The properties due to molecular motion are often described as laminar, e.g. laminar viscosity .

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:

 (6.28)
where the turbulence generation is
 (6.29)
and is the eﬀective diﬀusivity for . The equation is derived in a manner similar to Eq. (2.56 ) for conservation of speciﬁc internal energy , by (ensemble) averaging the separate energy contributions from and . While includes the kinetic energy of molecular motion, is the equivalent for eddy motion.

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

 (6.30)
Finally, diﬀusion of is represented by , where . This represents diﬀusion by both molecular and turbulent motions and interactions, including an adjustable coeﬃcient which is usually set to 1.

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 .

Notes on CFD: General Principles - 6.13 Turbulent kinetic energy