The velocity distribution  can be obtained if the relation between mixing length 'ι' and y is known.

Prandtl assumed that near the boundary.

ι = Ky

Where K =  Karman constant = 0.4
Shear stress at boundary,

τ₀ = ρ ι² ( du / dy )²

τ₀ = ρ (Ky)² (du / dy)²

(du / dy)² = τ₀ / ρ. 1/ (Ky)²

du / dy = 1 / (Ky) √τ₀ / ρ

By  √τ₀ / ρ  = √ML^-1 T^-2 / ML^-3 = ( L² / T² )^1/2 = L / T i. e. m/s

√τ₀ / ρ = shear velocity and denoted by u

du/dy  = u / Ky

du  = u./ K dy / y

u  = u./ K .log_e y + C        Put K = 0.4

u  = 2.5 u. log_e y + C

Where C = constant of integration

The equation that in case of turbulent flow, the velocity varies directly with the logarithmic in nature.

Apply boundary condition,

When y = R,      u = u_max put in equation

umax  = 2.5 u. long_e R+C

C  = u_max - 2.5u. log_e R

Substituting the value C in equation

u = 2.5 u. long_e y + u_max - 2.5 u. log_e R

u = u_max + 2.5 u. (log_e y - long_e R)

u  = ( u_max + 2.5 u. log_e v / R )

This equation is called as Prandtl universal velocity distribution for turbulent flow in smooth and rough pipes.

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