Mixing Length Model

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Mixing length model
Figure 1: Mixing length model.

The mixing length model proposed by Prandtl is the simplest turbulent model. The distinctive feature of turbulent flow is the existence of eddies and vortices so that the transport in the turbulent flow is dominated by packets of the molecules, instead of the behavior of individual molecules. Considering a turbulent flow near a flat plat as shown in Fig. 1, the mixing length can be defined as the maximum length that a packet can travel vertically while maintaining its time averaged velocity unchanged. The concept of mixing length for turbulent flow is similar to the mean free path for random molecular motion. When a fluid packet located at point A travels to point B by moving upward a distance that equals the mixing length, l, its time-averaged velocity should be kept at \bar{u} according to the definition of mixing length. On the other hand, the time-averaged velocity at point B is \bar{u}+(\partial \bar{u}/\partial y)l according to the profile of the time-averaged velocity (see Fig. 4.33). Therefore, the packet must have a negative velocity fluctuation equal to -(\partial \bar{u}/\partial y)l in order to keep its time averaged velocity unchanged. Thus, the fluctuation of the velocity component in the x-direction is:

{u}'=-l\left( \frac{\partial \bar{u}}{\partial y} \right)

(1)

When the velocity component in the x-direction has the above negative fluctuation, the velocity component in the y-direction must have a positive fluctuation, v', with the same scale, i.e.,

{v}'=Cl\left( \frac{\partial \bar{u}}{\partial y} \right)

(2)

where C is a local constant. Thus, the time-average of the product of the velocity fluctuations, \overline{{u}'{v}'}, must be negative for this case. Similarly, we can also analyze motion of the fluid packet from point B to point A, in which case u' will be positive and v' will be negative. Therefore, \overline{{u}'{v}'} must be negative for any cases. Combining eqs. (4.410) and (4.411) yields

-\overline{{u}'{v}'}=Cl^{2}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}

Since l is still undetermined, it will be beneficial to absorb C into l and yield

-\overline{{u}'{v}'}=l^{2}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}

(3)

It follows from the definition of the eddy diffusivity [see eqs. (4.399) and (4.402)] that

\varepsilon _{M}=l^{2}\left| \frac{\partial \bar{u}}{\partial y} \right|

(4)

where the absolute value is to ensure a positive eddy diffusivity. While the general rule for determining the mixing length, l, is lacking, the mixing length for turbulent boundary layer cannot exceed the distance to the wall. Therefore, we can assume:

l = κy

(5)

where κ is an empirical constant with order of 1, and is referred to as von Kármán’s constant. Equation (4.414) is valid only if κ is really a constant. Substituting eq. (4.414) into eq. (4.413), the eddy diffusivity of momentum becomes

\varepsilon _{M}=\kappa ^{2}y^{2}\left| \frac{\partial \bar{u}}{\partial y} \right|

(6)

Substituting eq. (4.415) into eq. (4.402), the shear stress in the two-dimensional turbulent flow becomes

\bar{\tau }_{yx}=\rho \left( \nu +\kappa ^{2}y^{2}\left| \frac{\partial \bar{u}}{\partial y} \right| \right)\frac{\partial \bar{u}}{\partial y}

(7)