# Algebraic Models for Eddy Diffusivity

 External Turbulent Flow/Heat Transfer Turbulent Boundary Layer Equations Algebraic Models for Eddy Diffusivity K-ε Model Momentum and Heat Transfer over a Flat Plate

In order to model turbulent flow, one must express the turbulent transports in terms of time-averaged quantities. Such relationships cannot be derived merely from the first principle and therefore must rely on empirical or semi-empirical approaches.[1]

As stated in the preceding subsection, the transport quantities for turbulent flow can be expressed as a sum of molecular and eddy effects. The contributions from the molecular level activities (laminar) are proportional to the gradients of the averaged physical quantities. If it is assumed that the contribution from the eddy level activities is also proportional to the gradients of the averaged quantities, one has

 $-\overline{\rho {u}'{v}'}=\mu ^{t}\frac{\partial \bar{u}}{\partial y}$ (1)
 $\rho c_{p}\overline{{v}'{T}'}=-k^{t}\frac{\partial \bar{T}}{\partial y}$ (2)
 $\rho \overline{{v}'{\omega }'_{1}}=-\rho D^{t}\frac{\partial \bar{\omega }_{1}}{\partial y}$ (3)

where μt,kt, and Dtare turbulent viscosity, conductivity and mass diffusivity, respectively. Substituting eqs. (1) – (3) into the last three equations in turbulent boundary layer equations, we have

 $\bar{\tau }_{yx}=(\mu +\mu ^{t})\frac{\partial \bar{u}}{\partial y}=\rho (\nu +\varepsilon _{M})\frac{\partial \bar{u}}{\partial y}$ (4)
 ${\bar{q}}''_{y}=-k\frac{\partial \bar{T}}{\partial y}-k^{t}\frac{\partial \bar{T}}{\partial y}=-\rho c_{p}\left( \frac{\nu }{\Pr }+\frac{\varepsilon _{M}}{\Pr ^{t}} \right)\frac{\partial \bar{T}}{\partial y}$ (5)
 $\bar{{\dot{m}}''}_{y}=-\rho (D+D^{t})\frac{\partial \bar{\omega }_{1}}{\partial y}=-\rho \left( \frac{\nu }{\text{Sc}}+\frac{\varepsilon _{M}}{\text{Sc}^{t}} \right)\frac{\partial \bar{\omega }_{1}}{\partial y}$ (6)

where εM is the eddy diffusivity for momentum, and Prt and Sct are turbulent Prandtl and Schmidt numbers, which are defined as

 $\Pr ^{t}=\frac{\varepsilon _{M}}{\varepsilon _{H}}$ (7)
 $\text{Sc}^{t}=\frac{\varepsilon _{M}}{\varepsilon _{D}}$ (8)

where $\varepsilon _{H}\text{ and }\varepsilon _{D}$ are eddy diffusivities for heat and mass, respectively. Substituting eqs. (1) – (3) into simplified turbulent boundary layer equations, the momentum, energy, and species equations for turbulent boundary layer become:

 $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+(\nu +\varepsilon _{M})\frac{\partial ^{2}\bar{u}}{\partial y^{2}}$ (9)
 $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial y}=\left( \frac{\nu }{\Pr }+\frac{\varepsilon _{M}}{\Pr ^{t}} \right)\frac{\partial ^{2}\bar{T}}{\partial y^{2}}$ (10)
 $\bar{u}\frac{\partial \bar{\omega }}{\partial x}+\bar{v}\frac{\partial \bar{\omega }}{\partial y}=\left( \frac{\nu }{\text{Sc}}+\frac{\varepsilon _{M}}{\text{Sc}^{t}} \right)\frac{\partial ^{2}\bar{\omega }}{\partial y^{2}}$ (11)

which – together with continuity equation – are governing equations for turbulent boundary layer. Appropriate models for eddy diffusivity for momentum, and turbulent Prandtl and Schmidt numbers are needed in order to describe the transport phenomena in the turbulent boundary layer. Until a viable expression of eddy diffusivity for momentum becomes available, the mathematical description of turbulent boundary layer is not complete. As for the turbulent Prandtl and Schmidt numbers, they are often assumed to be constants near unity because the mechanisms of turbulent transport of momentum, heat and mass are the same. Therefore, our attention now is turning to the models of eddy diffusivity for momentum.

## References

1. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.