Flow in Partly filled Circular Channel

In fluid mechanics, flows in closed circular conduits are usually encountered in places such as drains and sewers.Hence,it becomes very essential for engineers to calculate before hand flow variables such as discharge rate (Q),velocity(V) of the flow for different values of depth of liquid(H) flowing in the circular channel.These calculations can be accomplished either through empirical relations available or through formula derived from the governing principles of the flow.Closed channel flows are generally governed by the principles of channel flow as the liquid flowing possesses free surface inside the conduit.^[1] In the mathematical analysis of uniform flow through a channel,Manning's Equation,Continuity Equation and geometrical relations of the channel's cross-section are employed.

Mathematical Analysis for Flow in Circular Channel

Consider a closed circular conduit of diameter D, partly full with liquid flowing inside it. Let 2θ be the angle subtended by the free surface at the centre of the conduit as shown in figure (a).

The area of the cross-section (A) of the liquid flowing through the conduit is calculated as :

${\begin{aligned}A=&{{D^{2}\theta } \over {4}}-2{\biggl (}\left({\frac {1}{2}}\right){\frac {D}{2}}sin\theta {\frac {D}{2}}cos\theta {\Biggr )}\\&={\frac {D^{2}}{4}}{\biggl (}\theta -{\frac {1}{2}}sin2\theta {\biggr )}\\\end{aligned}}$ (Equation 1)

Now, the wetted perimeter (P) is given by :

$P=D\theta$

Therefore, the hydraulic radius (R_h) is calculated using cross-sectional area (A) and wetted perimeter (P) using the relation:

$R_{h}={\frac {A}{P}}={\frac {D}{4}}{\biggl (}1-{\frac {1}{2}}{\frac {sin2\theta }{\theta }}{\biggr )}$ ^[1] (Equation 2)

The rate of discharge may be calculated from Manning’s equation :

$Q={\frac {D^{2}}{4}}{\biggl (}\theta -{\frac {1}{2}}sin2\theta {\biggr )}{\biggl (}{\frac {1}{n}}{\biggr )}S^{\frac {1}{2}}{{{\biggl (}{\frac {D}{4}}{\biggl (}1-{\frac {1}{2}}{\frac {sin2\theta }{\theta }}{\biggr )}{\biggr )}}^{\frac {2}{3}}}$ ^[1]

$=K{\biggl (}\theta -{\frac {sin2\theta }{2}}{\biggr )}{{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\Biggr )}}^{\frac {2}{3}}}$ (Equation 3)

where the constant $K={\frac {{D}^{\frac {8}{3}}}{{4}^{\frac {5}{3}}}}{\frac {{S}^{\frac {1}{2}}}{n}}$

Now putting $\theta =\pi$ in the above equation yields us the rate of discharge for conduit flowing full (Q_full))

$Q_{(full)}=K\pi$ (Equation 4)

Final Dimensionless Quantities

In Dimensionless form ,the rate of discharge Q is usually expressed in a dimensionless form as :

${\frac {Q}{{Q}_{full}}}={\frac {1}{\pi }}{\biggl (}\theta -{\frac {sin2\theta }{2}}{\biggr )}{{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\biggr )}}^{\frac {2}{3}}}$ ^[1] (Equation 5)

Similarly for velocity (V) we can write :

${\frac {V}{{V}_{(full)}}}={{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\biggr )}}^{\frac {2}{3}}}$ ^[1] (Equation 6)

The depth of flow (H) is expressed in a dimensionless form as :

${\frac {H}{D}}={\frac {1}{2}}-{\frac {1}{2}}cos\theta$ ^[1] (Equation 7)

Flow Characteristics

The variations of Q/Q_(full) and V/V_(full) with H/D ratio is shown in figure(b).From the equation 5 ,maximum value of Q/Q_(full) is found to be equal to 1.08 at H/D =0.94 which implies that maximum rate of discharge through a conduit is observed for a conduit partly full. Similarly the maximum value of V/V_(full) (which is equal to 1.14) is also observed at conduit partly full with H/D = 0.81.The physical explanation for these results are generally attributed to the typical variation of Chezy’s coefficient with hydraulic radius R_h in Manning’s formula.^[1] However, an important assumption is taken that Manning’s Roughness coefficient ‘n’ is independent to the depth of flow while calculating these values.Also, the dimensional curve of Q/Q(full) shows that when the depth is greater than about 0.82D,then there are two possible different depths for the same discharge,one above and below the value of 0.938D.^[2]

In practice,it is common to restrict the flow below the value of 0.82D to avoid the region of two normal depths due to the fact that if the depth exceeds the depth of 0.82D then any small disturbance in water surface may lead the water surface to seek alternate normal depths thus leading to surface instablity.^[3]

References

^ ^a ^b ^c ^d ^e ^f ^g Suman Chakraborty, S K Som (2004). Introduction to Fluid Mechanics and Fluid Machines. New Delhi: McGraw Hill Education. pp. 599, 600. ISBN 978-0-07-132919-4.
^ CHOW, VEN TE (1959). OPEN CHANNEL HYDRAULICS. NEW YORK: McGraw Hill Publications. p. 134. ISBN 07-010776-9. {{cite book}}: ISBN / Date incompatibility (help)
^ SUBRAMANYAM, K. (2009). FLOW IN OPEN CHANNELS. NEW DELHI: McGRAW HILL PUBLICATIONS. pp. 106, 107, 113. ISBN 978-0-07-008695-1.

[:0-1] ^ ^a ^b ^c ^d ^e ^f ^g Suman Chakraborty, S K Som (2004). Introduction to Fluid Mechanics and Fluid Machines. New Delhi: McGraw Hill Education. pp. 599, 600. ISBN 978-0-07-132919-4.

[2] CHOW, VEN TE (1959). OPEN CHANNEL HYDRAULICS. NEW YORK: McGraw Hill Publications. p. 134. ISBN 07-010776-9. {{cite book}}: ISBN / Date incompatibility (help)

[:1-3] SUBRAMANYAM, K. (2009). FLOW IN OPEN CHANNELS. NEW DELHI: McGRAW HILL PUBLICATIONS. pp. 106, 107, 113. ISBN 978-0-07-008695-1.

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