Derive Prandtl’s universal velocity distribution equation for turbulent flow in pipes. What do you understand by velocity defect?

(10 Marks) May-2018

Subject Fluid Mechanics 2

Topic Turbulent Flow

Difficulty Medium

1) $\tau = \rho l^2 (\frac{du}{dy})^2$ from this equation, the velocity distribution can be obtained if the relation between L, the mixing length and y is known.

2) Pranditl assumed that mixing length, L is linear function of the distance y from the pipe wall. i.e., L = Ky where k = 0.4 known as karma n constant.

3) Now, $\tau = \tau = \rho l^2 (\frac{du}{dy})^2$

$\tau = \rho \times (ky)^2 (\frac{du}{dy})^2$

i.e. on rearranging above equation in $\frac{du}{dy}$

we get, $\frac{du}{dy} = \sqrt \frac{\tau}{\rho k^2 y^2} = \frac{1}{ky} \sqrt \frac{\tau}{\rho }$ ---- (1)

For small values of y that is very close to the boundary of the pipe, prandtl assumed shear stress $\tau$ to be constant and approximately equal ?o

$\therefore$ putting $\tau = \tau_o \ in \ (1)$

$\therefore \frac{du}{dy} = \frac{1}{ky} \sqrt \frac{\tau \ o}{\rho }$ ---- (2)

now, $\sqrt \frac{\tau_o}{\rho }$ has dimension $\sqrt \frac{ml^{-1} T^{-2}}{ML^{-3}} = \sqrt \frac{L^2}{T^2} = \frac{L}{T}$ which is velocity, known as shear velocity (u).

Thus $\sqrt \frac{?}{\rho } = u , \ then \ \frac{du}{dy} = \frac{1}{ky} \ u$

on integrating above equation, we get,

$u = \frac{u}{k} log_e \ y + c$ ------ (3) (where c = constant of integration)

Above equation (3) shows that in turbulent flow, velocity $\alpha$ $log_e \ Y $ i.e. velocity distribution in turbulent flow is logarithmic in nature.

To find value of C, putting y = R and u = $u_{max}$ (boundary condition) in (3)

Hence,

$u_{max} = \frac{u*}{k} log_e$ R + C ( R is radius of pipe)

$\therefore$ Putting (4) in (3) we get,

$u = \frac{u*}{k} \ log_e \ y + u_{max} - \frac{u*}{k} \ log_e$ R

$= u_{max} + \frac{u*}{k} \times log_e \ y - log_e$ R

$= u_{max} + \frac{u*}{0.4} log_e \ (Y/R)$ ( $because$ k = 0.4 )

$\therefore u = u max + 2.5 \ u * log_e$ (Y/R)

Above equation is called as 'prandtl's universal velocity distribution equation for turbulent flow in pipes. This equation applicable to smooth and rough pipe boundaries.

Velocity defect:

$u = u_{max} + 2.5 \ u* log_e$ (Y/R)

i.e. $u_{max} - u* + 2.5 \ u* log_e$ (R/Y)

Dividing by u*

$\therefore \frac{u_{max} = u}{u*} = 2.5 log_e$ (R/Y)

$\frac{u_{max} - u}{u*} = 2.5 \times 2.3 log_{10} (\frac{R}{Y})$

$( \because log_e \ (R/y) = 2.3 log_{10} (R/Y) )$

$\frac{u_{max} - u}{u*} = 5.75 log_{10} (R/Y)$

The difference between the maximum velocity (u max) and local velocity (u) at any point i.e. (u max - u ) is velocity defect.