6.13 Turbulent kinetic energy

The Reynolds stress eqn in the ensemble-averaged momentum conservation Eq. (6.15 ) can be decomposed into viscous and pressure components. The turbulent pressure term, eqn, is commonly subsumed within pressure eqn, to give

@ u ---- + r uu = r ☐☐☐ + ☐☐☐t rp + b: @t \relax \special {t4ht=
The viscous stresses can be combined assuming eqn is modelled as a Newtonian fluid Eq. (2.41 ) and eqn by the eddy viscosity model Eq. (6.20 ) to give
@ u ---- + r uu = 2r effdevD rp + b; @t \relax \special {t4ht=
where the effective kinematic viscosity
eff = + t: \relax \special {t4ht=
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 eqn.

Averaging the momentum equation and introducing the eddy viscosity model creates one additional unknown eqn. Additional models are required for eqn to close the system of equations.

By considering eqn from Sec. 6.11 , the model for eqn is typically decomposed into components representing velocity and length scales, eqn and eqn respectively.

The scale of eqn corresponds to turbulent fluctuations eqn, so it is reasonable to assume that eqn. Since the field eqn is representative of the eqn component of eqn, it is commonly adopted within turbulence models based on eqn. Also, it is well captured by a suitable conservation equation.

Transport of turbulent kinetic energy

Conservation of turbulent kinetic energy eqn can be written as:

|-----------------------------------------| Dk- 2- | Dt =--r-(--Dkrk)--+---G-----3--k(r--u)------
where the turbulence generation eqn is
 G = 2 tdev D ru; \relax \special {t4ht=
and eqn is the effective diffusivity for eqn. The equation is derived in a manner similar to Eq. (2.56 ) for conservation of specific internal energy eqn, by (ensemble) averaging the separate energy contributions from eqn and eqn. While eqn includes the kinetic energy of molecular motion, eqn is the equivalent for eddy motion.

In Eq. (2.56 ), energy from bulk motion is passed to the submicroscopic scale as heat by eqn. In Eq. (6.28), it is converted to turbulent energy by eqn using Boussinesq’s eqn from Eq. (6.20 ). The shear component eqn provides the non-recoverable eqn in Eq. (6.28) and the second (eqn) term yields eqn.

While eqn transfers kinetic energy from the bulk flow to eqn, the final term transfers eqn on to eqn 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

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

The eqn equation goes part of the way to closing our system of equations. However, it introduces an additional unknown, , and the model for eqn still requires length scale eqn.

Notes on CFD: General Principles - 6.13 Turbulent kinetic energy