Prandtl's Mixing Length Hypothesis

• Consider a fully developed turbulent boundary layer . The stream wise mean velocity varies only from streamline to streamline. The main flow direction is assumed parallel to the x-axis (Fig. 33.4). 
• The time average components of velocity are given by  . The fluctuating component of transverse velocity  transports mass and momentum across a plane at y₁ from the wall. The shear stress due to the fluctuation is given by 

(33.14)

• Fluid, which comes to the layer y₁ from a layer (y₁- l) has a positive value of . If the lump of fluid retains its original momentum then its velocity at its current location y₁ is smaller than the velocity prevailing there. The difference in velocities is then

(33.15)

Fig. 33.4   One-dimensional parallel flow and Prandtl's mixing length hypothesis

The above expression is obtained by expanding the function  in a Taylor series and neglecting all higher order terms and higher order derivatives. l is a small length scale known as Prandtl's mixing length . Prandtl proposed that the transverse displacement of any fluid particle is, on an average, 'l' .

Consider another lump of fluid with a negative value of . This is arriving at  from . If this lump retains its original momentum, its mean velocity at the current lamina will be somewhat more than the original mean velocity of . This difference is given by
	(33.16)
The velocity differences caused by the transverse motion can be regarded as the turbulent velocity components at .  We calculate the time average of the absolute value of this fluctuation as
	(33.17)
Suppose these two lumps of fluid meet at a layer  The lumps will collide with a velocity  and diverge. This proposes the possible existence of transverse velocity component in both directions with respect to the layer at . Now, suppose that the two lumps move away in a reverse order from the layer  with a velocity . The empty space will be filled from the surrounding fluid creating transverse velocity components which will again collide at . Keeping in mind this argument and the physical explanation accompanying Eqs (33.4), we may state that
or,
along with the condition that the moment at which  is positive,  is more likely to be negative and conversely when  is negative. Possibly, we can write at this stage
	(33.18)
where C1 and C2 are different proportionality constants. However, the constant C2 can now be included in still unknown mixing length and Eg. (33.18) may be rewritten as
For the expression of turbulent shearing stress  we may write
	(33.19)
After comparing this expression with the eddy viscosity Eg. (33.14), we may arrive at a more precise definition,
	(33.20a)
where the apparent viscosity may be expressed as
	(33.20b)
and the apparent kinematic viscosity is given by
	(33.20c)
The decision of expressing one of the velocity gradients of Eq. (33.19) in terms of its modulus as  was made in order to assign a sign to  according to the sign of  .  Note that the apparent viscosity and consequently,the mixing length are not properties of fluid. They are dependent on turbulent fluctuation.  But how to determine the value of the mixing length? Several correlations, using experimental results for  have been proposed to determine . However, so far the most widely used value of mixing length in the regime of isotropic turbulence is given by
	(33.21)
where  is the distance from the wall and  is known as von Karman constant .