A pipeline carrying water has average height of irregularities projecting from the surface of the boundary of the pipe as 0.15 mm. What type of boundary is it? The shear stress developed is 4.9N/m24.9N/m2. The kinematics viscosity of water is 0.01 Stokes.

 A pipeline carrying water has average height of irregularities projecting from the surface of the boundary of the pipe as 0.15 mm. What type of boundary is it? The shear stress developed is 4.9N/m2$4.9N/m^2$. The kinematics viscosity of water is 0.01 Stokes.

Solution:- 

Given:-

K=0.15mm=0.15×10−3$K=0.15mm=0.15\times {10}^{-3}$

τ0=4.9N/m2${\tau }_0=4.9N/m^2$

v=0.01 stokes

v=0.01×10−4m2/s$v=0.01\times {10}^{-4}{m}^2/s$

Density of water, 

ρ=1000kg/m3$\rho =1000kg/m^3$

Shear velocity,

u∗=τ0ρ−−−√$u_{\ast }=\sqrt{{\displaystyle \frac{{\tau }_0}{\rho }}}$

u∗=4.91000−−−−−√=0.07m/s$u_{\ast }=\sqrt{{\displaystyle \frac{4.9}{1000}}}=0.07m/s$

Roughness Reynolds number,

Re=u∗×Kv$R_e={\displaystyle \frac{u_{\ast }\times K}{v}}$

Re=0.07×0.15×10−30.01×10−4$R_e={\displaystyle \frac{0.07\times 0.15\times {10}^{-3}}{0.01\times {10}^{-4}}}$

u∗×Kv=10.5${\displaystyle \frac{u_{\ast }\times K}{v}}=10.5$

Since (u∗×Kv)$\left({\displaystyle \frac{u_{\ast }\times K}{v}}\right)$ lies between 4 and 100, so the pipe behaves as in transition.