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Prandtl Mixing length theory

According to Prandtl in 1925, the length or mixing length(l), is the distance between two layers in transverse direction.

i.e., the lumps of fluid particles from one layer can reach the other layer and the particles are mixed in the other layer in such a way that the momentum of particles is same in ‘x’ direction. He assumed that the velocity fluctuation in ‘x’ direction is also related to the mixing length l as follows,

$u’=l\dfrac{du}{dy}$

and v’ the fluctuation component of velocity in ‘y’ direction is of same order of magnitude u’ and hence,

$v’=l\dfrac{du}{dy}$

Now, $\overline{u’\times v’}$ becomes as,

$\overline{u’\times v’}= l\dfrac{du}{dy}\times l\dfrac{du}{dy}$

$\overline{u’\times v’}= l^2\left(\dfrac{du}{dy}\right)^2$

Substituting the value of $\overline{u’\times v’}$ from turbulent shear stress equation, i.e.,

$\overline{\tau}=\rho l^2\left(\dfrac{du}{dy}\right)^2$

Thus, the total shear stress at any point in turbulent flow is the sum of shear stress due to laminar shear or viscous shear and turbulent shear and can be written as,

$\overline{\tau}=\mu\dfrac{du}{dy}+\rho l^2\left(\dfrac{du}{dy}\right)^2$

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