# Description of turbulence

Turbulent flow is the most commonly encountered type of viscous flow but its theoretical treatment is not as well developed as that for the laminar flow. In this section, turbulent flow will be examined with particular attention paid to mechanism of the momentum and energy transfer. Hinze (1975) was the first one who presented the structure of the turbulent flow in detail. The constituencies of a turbulent flow are eddies and vortices at different sizes. Although the effect of the molecular viscosity are important within these eddies and vortices, the interactions of these eddies and vortices are dominant on the transport phenomena in the turbulent flow. For turbulent air flow at a speed of 100 m/s, the typical size of eddies are on the order of 1 mm; this is much larger than the mean free path of the air molecules, which is on the order of 66 nm standard condition (1 atm pressure and 25 °C). In a cube with 1 mm on a side, there are $2.46\times {{10}^{16}}$ air molecules. Therefore, eddies in the turbulent flow are considered to be “large” from the microscale point of view. On the other hand, they are too small from the macroscopic point of view. When the velocity of the turbulent air flow is measured using a hot wire anemometer, the local instantaneous velocity varies about 10% of its average velocity due to the random eddies and vortices. In other words, for an air flow with mean velocity of 100 m/s, a velocity fluctuation on the order of 10 m/s can be expected. Compared with the average random molecular velocity of 465 m/s, this fluctuation is relatively small. The frequency of the velocity fluctuation is about 1 to 104 Hz, whereas the frequency of molecular collision is about $7\times {{10}^{9}}$ Hz. It appears that the interactions between eddies in a turbulent flow are very different from the molecular collision which is the mechanism of the viscous effect for laminar flow. Since the number of eddies in a turbulent flow is enormous, it is very difficult to model or measure the flow structure within each eddy. Therefore, time averaging technique is usually used to obtain the governing equations that are manageable.

Time dependence of velocity component in the x-direction in a turbulent flow

In turbulent flow, the transport phenomena variables (i.e., u,v,w,T,p, etc.) always vary with time. For example, the magnitude and direction of the instantaneous velocity vector are different from those of the time-averaged velocity. While the instantaneous velocity in a turbulent flow is always time-dependent, the time averaged velocity can be either time-independent or time dependent [for the velocity component in the x-direction, see figure on the right]. For any given location (x,y,z) and time t, the local instantaneous velocity can be expressed as a summation of its mean value and its fluctuation, i.e.,

$u=\bar{u}(x,y,z,t)+{u}'(x,y,z,t)\qquad \qquad(1)$

where

$\bar{u}(x,y,z,t)=\frac{1}{\Delta t}\int_{t}^{t+\Delta t}{u(x,y,z,t)dt}\qquad \qquad(2)$

is the time-averaged velocity at point (x,y,z). The time interval for the time-average, Δt, must be very long compared with the duration of fluctuation. The mean value of the fluctuation must be zero, i.e.,

$\overline{{{u}'}}=\frac{1}{\Delta t}\int_{t}^{t+\Delta t}{{u}'(x,y,z,t)dt}=0\qquad \qquad(3)$

Similarly, the velocity components in the y- and z- directions can be expressed as

$v=\bar{v}(x,y,z,t)+{v}'(x,y,z,t)\qquad \qquad(4)$
$w=\bar{w}(x,y,z,t)+{w}'(x,y,z,t)\qquad \qquad(5)$

In addition, the local instantaneous pressure, temperature and mass fraction can also be expressed as sums of their mean values and fluctuations:

$p=\bar{p}(x,y,z,t)+{p}'(x,y,z,t)\qquad \qquad(6)$
$T=\bar{T}(x,y,z,t)+{T}'(x,y,z,t)\qquad \qquad(7)$
$\omega =\bar{\omega }(x,y,z,t)+{\omega }'(x,y,z,t)\qquad \qquad(8)$

Similar to eq. (3), the volume averages of the fluctuations for these variables are zero, i.e.,

$\overline{{{v}'}}=\overline{{{w}'}}=\overline{{{p}'}}=\overline{{{T}'}}=\overline{{{\omega }'}}=0\qquad \qquad(9)$

For any variables φ and ψ, we have the following properties about the averaging of their fluctuations (Oosthuizen and Naylor, 1999):

$\overline{\frac{\partial \bar{\phi }}{\partial n}}=\frac{\partial \bar{\phi }}{\partial n},\text{ }\overline{\frac{\partial {\phi }'}{\partial n}}=\frac{\partial \overline{{{\phi }'}}}{\partial n}=0$
$\overline{\bar{\phi }\bar{\psi }}=\bar{\phi }\bar{\psi },\text{ }\overline{\bar{\phi }{\phi }'}=\bar{\phi }\overline{{{\phi }'}}=0,\text{ }\overline{\bar{\phi }+\bar{\psi }}=\bar{\phi }+\bar{\psi }\qquad \qquad(10)$

where n can be any of the special variable (x,y,orz).

Although the mean values of the fluctuations are zero, these mean values do contribute to the mean value of some physical quantities. The kinetic energy per unit volume is

\begin{align} & \overline{ke}=\frac{1}{2}\overline{[{{(\bar{u}+{u}')}^{2}}+{{(\bar{v}+{v}')}^{2}}+{{(\bar{w}+{w}')}^{2}}]} \\ & \text{ }=\frac{1}{2}\overline{[({{{\bar{u}}}^{2}}+2\bar{u}{u}'+{{{{u}'}}^{2}})+({{{\bar{v}}}^{2}}+2\bar{v}{v}'+{{{{v}'}}^{2}})+({{{\bar{w}}}^{2}}+2\bar{w}{w}'+{{{{w}'}}^{2}})]} \\ \end{align}

Since $\overline{\bar{u}{u}'}=\bar{u}\overline{{{u}'}}=0,\text{ }\overline{\bar{v}{v}'}=\bar{v}\overline{{{v}'}}=0,\text{ and }\overline{\bar{w}{w}'}=\bar{w}\overline{{{w}'}}=0$, the above expression becomes

$\overline{ke}=\frac{1}{2}[({{\bar{u}}^{2}}+{{\bar{v}}^{2}}+{{\bar{w}}^{2}}+\overline{{{{{u}'}}^{2}}}+\overline{{{{{v}'}}^{2}}}+\overline{{{{{w}'}}^{2}}})]\qquad \qquad(11)$

which indicates that the time average of the squares of the velocity fluctuations are not zero. Similarly, the products of the fluctuations of different variables are generally not zero: $\overline{{u}'{T}'}\ne 0,\text{ }\overline{{u}'{v}'{T}'}\ne 0$, etc. Equation (2.501) shows that both mean velocity and velocity fluctuation contribute to the total kinetic energy in a turbulent flow. The level or intensity of turbulence is defined as (Welty et al., 2000)

$\overline{ke}=\frac{\sqrt{(\overline{{{{{u}'}}^{2}}}+\overline{{{{{v}'}}^{2}}}+\overline{{{{{w}'}}^{2}}})/3}}{{{U}_{\infty }}}\qquad \qquad(12)$

where the numerator is the root mean square value of fluctuation and the denominator, ${U_{\infty}}$, is the mean velocity of the flow. The intensity of turbulence will have very important effect on transition and separation of boundary layer, and the rate of heat and mass transfer. The intensity of turbulence is another quantity in addition to Reynolds number that we will need to describe the turbulent flow.

## References

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

Hinze, J. O. 1975, Turbulence. 2nd ed., McGraw Hill, New York:

Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York.

Welty, J.R., Wicks, C.E., Wilson, R. E., 2000, Fundamentals of Momentum Heat and Mass Transfer, 4th ed., John Wiley & Sons, New York, NY.