Notes on Computational Fluid Dynamics: General Principles

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6.13 Turbulent kinetic energy

The Reynolds stress $\text{[math]}$ in the ensemble-averaged momentum conservation Eq. (6.15 ) can be decomposed into viscous and pressure components. The turbulent pressure term, $\text{[math]}$ , is commonly subsumed within pressure $\text{[math]}$ , to give

@ u ---- + r uu = r ☐☐☐ + ☐☐☐t rp + b: @t \relax \special {t4ht=

(6.25)

The viscous stresses can be combined assuming $\text{[math]}$ is modelled as a Newtonian ﬂuid Eq. (2.41 ) and $\text{[math]}$ by the eddy viscosity model Eq. (6.20 ) to give

@ u ---- + r uu = 2r eﬀdevD rp + b; @t \relax \special {t4ht=

(6.26)

where the eﬀective kinematic viscosity

eﬀ = + t: \relax \special {t4ht=

(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 $\text{[math]}$ .

Averaging the momentum equation and introducing the eddy viscosity model creates one additional unknown $\text{[math]}$ . Additional models are required for $\text{[math]}$ to close the system of equations.

By considering $\text{[math]}$ from Sec. 6.11 , the model for $\text{[math]}$ is typically decomposed into components representing velocity and length scales, $\text{[math]}$ and $\text{[math]}$ respectively.

The scale of $\text{[math]}$ corresponds to turbulent ﬂuctuations $\text{[math]}$ , so it is reasonable to assume that $\text{[math]}$ . Since the ﬁeld $\text{[math]}$ is representative of the $\text{[math]}$ component of $\text{[math]}$ , it is commonly adopted within turbulence models based on $\text{[math]}$ . Also, it is well captured by a suitable conservation equation.

Transport of turbulent kinetic energy

Conservation of turbulent kinetic energy $\text{[math]}$ can be written as:

(6.28)

where the turbulence generation $\text{[math]}$ is

G = 2 tdev D ru; \relax \special {t4ht=

(6.29)

and $\text{[math]}$ is the eﬀective diﬀusivity for $\text{[math]}$ . The equation is derived in a manner similar to Eq. (2.56 ) for conservation of speciﬁc internal energy $\text{[math]}$ , by (ensemble) averaging the separate energy contributions from $\text{[math]}$ and $\text{[math]}$ . While $\text{[math]}$ includes the kinetic energy of molecular motion, $\text{[math]}$ is the equivalent for eddy motion.

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

While $\text{[math]}$ transfers kinetic energy from the bulk ﬂow to $\text{[math]}$ , the ﬁnal term $\text{[math]}$ transfers $\text{[math]}$ on to $\text{[math]}$ as dissipated heat. Here, $\text{[math]}$ 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 $\text{[math]}$ is represented by $\text{[math]}$ , where $\text{[math]}$ . This represents diﬀusion by both molecular and turbulent motions and interactions, including an adjustable coeﬃcient $\text{[math]}$ which is usually set to 1.

The $\text{[math]}$ equation goes part of the way to closing our system of equations. However, it introduces an additional unknown, $\text{[math]}$ , and the model for $\text{[math]}$ still requires length scale $\text{[math]}$ .

Notes on CFD: General Principles - 6.13 Turbulent kinetic energy