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

The time average components of velocity are given by $\bar{u}=\bar{u}(y),\bar{v}=0,\bar{w}=0$ . The fluctuating component of transverse velocity transports mass and momentum across a plane at y1from the wall. The shear stress due to the fluctuation is given by

$\tau_{t}=-\rho \bar{u^{\prime}v^{\prime}}=\mu_{t}\frac{d\bar{u}}{dy}$

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

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

The above expression is obtained by expanding the function $\bar{u}(y_{1}-l)$ 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 $v^{\prime}$ . This is arriving at $y_{1}$ from $(y_{1}+1)$ . If this lump retains its original momentum, its mean velocity at the current lamina $y_{1}$ will be somewhat more than the original mean velocity of $y_{1}$. This difference is given by

$\Delta u_{2}=\bar{u_{2}}(y_{1}+l)-\bar{u}(y_{1})\approx l(\frac{d\bar{u}}{dy})_{y1}$

• The velocity differences caused by the transverse motion can be regarded as the turbulent velocity components at $y_{1}$ .

• We calculate the time average of the absolute value of this fluctuation as

$|\bar{u^{\prime}}|=\frac{1}{2}(|\Delta u_{1}|+|\Delta u_{2}|=l|(\frac{d\bar{u}}{dy})|_{y1}$

• Suppose these two lumps of fluid meet at a layer $y_{1}$ The lumps will collide with a velocity $2u^{\prime}$ and diverge. This proposes the possible existence of transverse velocity component in both directions with respect to the layer at $y_{1}$ . Now, suppose that the two lumps move away in a reverse order from the layer $y_{1}$ with a velocity $2u^{\prime}$ . The empty space will be filled from the surrounding fluid creating transverse velocity components which will again collide at $y_{1}$ . Keeping in mind this argument and the physical explanation accompanying Eqs , we may state that

$|\bar{v}^{\prime}|\approx |\bar{u}^{\prime}|$

$|\bar{v}^{\prime}|=(const)l|(\frac{d\bar{u}}{dy})|$

or,

along with the condition that the moment at which $u^{\prime}$ is positive, $v^{\prime}$ is more likely to be negative and conversely when $u^{\prime}$ is negative. Possibly, we can write at this stage

$\bar{u^{\prime}v^{\prime}}=-C_{1}|\bar{u^{\prime}}||\bar{v^{\prime}}|$

$\bar{u^{\prime}v^{\prime}}=-C_{2}l^{2}|(\frac{d\bar{u}}{dy}^{2})|$

where $C_1$ and $C_2$ are different proportionality constants. However, the constant C2 can now be included in still unknown mixing length and Eg. may be rewritten as

$\bar{u^{\prime}v^{\prime}}=-l^{2}\frac{d\bar{u}}{dy}^{2}$

• For the expression of turbulent shearing stress $\tau_{t}$ we may write

$\mu_{t}=\rho l^{2}|\frac{d\bar{u}}{dy}|$

• After comparing this expression with the eddy viscosity Eg., we may arrive at a more precise definition,

$\tau_{t}\rho l^{2}|\frac{d\bar{u}}{dy}|\mu_{=t}(\frac{d\bar{u}}{dy})$

where the apparent viscosity may be expressed as

$\mu_{t}=\rho l^{2}|\frac{d\bar{u}}{dy}|$

and the apparent kinematic viscosity is given by

$\mu_{t}= l^{2}|\frac{d\bar{u}}{dy}|$

• The decision of expressing one of the velocity gradients of Eq. in terms of its modulus as $|\frac{d\bar{u}}{dy}|$ was made in order to assign a sign to $\tau_{t}$ according to the sign of $|\frac{d\bar{u}}{dy}|$ .

• 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 "l" the mixing length? Several correlations, using experimental results for $\tau_{t}$ have been proposed to determine $l$ . However, so far the most widely used value of mixing length in the regime of isotropic turbulence is given by $l=x y$ where y is the distance from the wall and $\xi$ is known as von Karman constant .$(\approx 0.4)$