In case of turbulent flow, the total shear stress at any point is the sum of viscous shear stress and turbulent shear stress.

In this the viscous shear stress is negligible except near the boundary. Hence it is assumed that shear stress in turbulent flow is given by the equation,

$\overline{\tau}=\rho l^2\left(\dfrac{du}{dy}\right)^2…………..(1)$

Using this equation we can find out the velocity distribution if relation between ‘l’ and ‘y’ is known.

Prandtl assumed the mixing length (y), (l) is a linear function of the distance (y) from pipe wall.

i.e., l=Ky, where K=0.4 (By Karman Prandtl)

Substituting the value of l in equation (1), we get

$\overline{\tau} \text{ or }\tau=\rho \times(Ky)^2\left(\dfrac{du}{dy}\right)^2$

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

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

$\dfrac{du}{dy}=\sqrt{\dfrac{\tau}{\rho K^2y^2}}$

$\dfrac{du}{dy}=\dfrac1{Ky}\sqrt{\dfrac{\tau}{\rho}}…………….(2)$

Some assumptions made by Karman-Prandtl:-

1) For small values of ‘y’, Prandtl assumed

$\tau$ = to is equal to constant and approximately equal to $\tau_0$

Substituting this value of $\tau$ in equation (2),

$\dfrac{du}{dy}=\dfrac1{Ky}\sqrt{\dfrac{\tau_0}{\rho}}………………(3)$

In the above equation $\sqrt{\dfrac{\tau_0}{\rho}}$ has the dimension

$\sqrt{\dfrac{ML^{-1}T^{-2}}{ML^{-3}}}=\sqrt{\dfrac{L^2}{T^2}}=\dfrac LT$

But $\dfrac LT$ is velocity and hence

$\sqrt{\dfrac{\tau_0}{\rho}}$ has the dimension of velocity, which is known as shear velocity and is denoted by $u_*$

$\therefore\sqrt{\dfrac{\tau_0}{\rho}}=u_*$

Therefore equation (3) becomes,

$\dfrac{du}{dy}=\dfrac1{Ky}u_*$

For turbulent flow, ($u_*$)is constant. Hence integrating above equation, we get,

$u=\dfrac{u_*}K\log_ey+C……………..(4)$

The above equation shows that the velocity distribution in turbulent flow is logarithmic in nature.

Now to determine the constant of integration (C) the boundary condition that at y=R and $u=u_{max}$ is substituted in equation (4),

$u_{max}=\dfrac{u_*}K\log_eR+C$

$\therefore C=u_{max}-\dfrac{u_*}K\log_eR$

Substituting the value of ‘C’ in equation (4), we get

$u=\dfrac{u_*}K\log_ey+ u_{max}-\dfrac{u_*}K\log_eR$

$u= u_{max}+\dfrac{u_*}K(\log_ey-\log_eR)$

$u= u_{max}+\dfrac{u_*}{0.4}\log_e\left(\dfrac yR\right)$

[$\because$ Karman constant, K=0.4]

$u= u_{max}+2.5u_*\log_e\left(\dfrac yR\right)…………..(5)$

This equation is known as ‘Prandtl’s’ universal velocity distribution equation for turbulent flow in pipes.

This equation is applicable for both smooth as well as rough pipe boundaries.