# K-epsilon Model

In the turbulent models presented in the preceding section, the eddy diffusivity, $\varepsilon _{M}$, was obtained by algebraic expressions. While they are very easy to apply, their shortcoming is their lack of universal applicability because all of them work well only near the wall. As evidenced by the figures in Two-Layer Model and Van Driest Model, the agreement between the wall function and the experimental results is not very good for large y+. When the above algebraic models are applied to the internal flow, they will break down near the center of the tube. Thus, it is imperative to develop a turbulent model that is applicable in the region that is sufficiently far away from the wall. An ideal turbulent model should be applicable to any region in the turbulent flow although it may not be able to be represented by algebraic equations. Significant advancement of high-speed computers makes developments and applications of non-algebraic turbulent models possible. Among different non-algebraic turbulent models, the K-ε model is one of the most widely used models and it will be discussed below.[1]

The K-ε model is based on the analogy between the motion of the fluid packet in the turbulent flow and the random motion of an ideal gas. The kinetic theory of gases stated that the kinematic viscosity of gas can be obtained by ν = cλ / 3 where c is the mean speed of the molecules and λ is the mean free path. For turbulent flow, the kinetic energy due to the fluid packet random motion is

 $K=\frac{1}{2}(\overline{{u}'^{2}}+\overline{{v}'^{2}}+\overline{{w}'^{2}})$ (1)

and the mean speed of the fluid packet is K1/2. The eddy diffusivity for the momentum can be defined in a similar way to the kinematic viscosity of gas [2], i.e.,

 $\varepsilon _{M}=C_{\mu }K^{1/2}L$ (2)

where Cμ is a dimensionless empirical constant and L plays the same role as the mean free path in the kinematic viscosity of the gas. Therefore, both K and L must be determined in order to use eq. (2) to evaluate the eddy diffusivity.

If we consider a control volume in a horizontal slender flow region (turbulent boundary layer), balance of the kinetic energy for velocity fluctuation is

 $\begin{matrix}{}\\\end{matrix}\text{Convection of }K\text{ = Eddy diffusion of }K$ $\begin{matrix}{}\\\end{matrix}\text{ + Rate of }K\text{ generation }-\text{ Rate of }K\text{ destruction}$ (3)

For a two-dimensional steady state flow, the convection of K is

 $\text{Convection of }K\text{ = }\bar{u}\frac{\partial K}{\partial x}+\bar{v}\frac{\partial K}{\partial y}$ (4)

Since eddy diffusion in the x-direction is negligible for a boundary layer type flow, only diffusion in the y-direction needs to be considered:

 $\text{ Eddy diffusion of }K=\frac{\partial }{\partial y}\left( \frac{\varepsilon _{M}}{\sigma _{k}}\frac{\partial K}{\partial y} \right)$ (5)

where σK is a dimensionless empirical constant. The rate of K production can be obtained by the eddy shear stress and the time averaged velocity gradient

 $\text{Rate of }K\text{ generation }=\left( \varepsilon _{M}\frac{\partial \bar{u}}{\partial y} \right)\cdot \left( \frac{\partial \bar{u}}{\partial y} \right)=C_{\mu }K^{1/2}L\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}$ (6)

where eq. (2) was used to evaluate the eddy diffusivity. The rate of K destruction, $\varepsilon$, can be evaluated by analyzing a fluid packet with diameter L, oscillating in the turbulent flow field. If the oscillating velocity is K1/2, the drag force acting on the fluid packet will be CDρL2(K1 / 2)2, where CD is the drag coefficient that is approximately equal to 1. The mechanical power dissipated per unit mass is

 $\varepsilon =\frac{[C_{D}\rho L^{2}(K^{1/2})^{2}]K^{1/2}}{\rho L^{3}}=C_{D}\frac{K^{3/2}}{L}$ (7)

The K-equation for a boundary layer type flow can be obtained by substituting eqs. (4) – (7) into eq. (2)

 $\text{ }\bar{u}\frac{\partial K}{\partial x}+\bar{v}\frac{\partial K}{\partial y}\text{ }=\frac{\partial }{\partial y}\left( \frac{\varepsilon _{M}}{\sigma _{k}}\frac{\partial K}{\partial y} \right)+\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}-\varepsilon$ (8)

By following a similar procedure, the dissipation equation can be obtained as the following:

 $\text{ }\bar{u}\frac{\partial \varepsilon }{\partial x}+\bar{v}\frac{\partial \varepsilon }{\partial y}\text{ }=\frac{\partial }{\partial y}\left( \frac{\varepsilon _{M}}{\sigma _{\varepsilon }}\frac{\partial \varepsilon }{\partial y} \right)+C_{1}\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}\frac{\varepsilon }{K}-C_{2}\frac{\varepsilon ^{2}}{K}$ (9)

where $\sigma _{\varepsilon },\text{ }C_{1},\text{ and }C_{2}$are additional dimensional empirical constants. After K and ε are obtained, the eddy diffusivity can be obtained by eliminating L between eqs. (2) and (7), i.e.

 $\varepsilon _{M}=C_{\mu }\frac{K^{2}}{\varepsilon }$ (10)

where CD in eq. (7) has been set to 1. Equations (8) – (10) become the equations for the K-ε model. Both eqs. (8) and (9) are parabolic because diffusion in the x-direction is neglected. Jones and Launder [3] suggested the following values for the K-ε model:

$C_{\mu }=0.09,\text{ }C_{1}=1.44,\text{ }C_{2}=1.92,\text{ }\sigma _{K}=1,\text{ and }\sigma _{\varepsilon }=1.3$

Since the effect of molecular viscosity, ν, was neglected in eqs. (8) and (9), the K-ε model cannot be applied in the viscous sublayer where molecular viscosity dominates the velocity profiles. In the sublayer region, Prandtl’s mixing length theory is valid, thus

 $\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{\varepsilon _{M}^{2}}{l^{4}}$ (11)

The generation of K represented by the second term on the right-hand side of eq. (8) becomes $\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{\varepsilon _{M}^{3}}{l^{4}}$

Substituting eq. (2) into the right-hand side of the above equation, one obtains:

$\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{C_{\mu }^{3}K^{3/2}L^{3}}{l^{4}}$

Considering eq. (7), the above equation can be rewritten as

 $\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{C_{\mu }^{3}}{C_{D}}\left( \frac{L}{l} \right)^{4}\varepsilon$ (12)

If the length scale is taken as$L=l\left( \frac{C_{D}}{C_{\mu }^{3}} \right)^{1/4}$

eq. (12) becomes

 $\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\varepsilon$ (13)

where the left and right hand sides, respectively, represent the rates of production and dissipation of K, i.e. generation of K is balanced by dissipation of K in the viscous sublayer. Therefore, the mixing length theory can be used in the sublayer region and the fully turbulent region can be described using the K-ε model.

## References

1. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
2. Kolmogorov, A.N., 1942, “Equations of Turbulent Motion of an Incompressible Turbulent Fluid,” Izv. Akad. Nauk SSSR, Ser. Fiz., Vol. VI, pp. 56-58.
3. Jones, W.P., and Launder, B.E., 1972, “The Prediction of Laminarization with a Two-Equation Model of Turbulence,” Int. J. Heat Mass Transfer, Vol. 15, pp. 301-314.