How is the mean velocity found at a shallow vertical with a single reading at 0.6y?

It is taken directly as the mean velocity

It is averaged with the neighbouring vertical

2077 Chaitra · 8 marks

River discharge by mid-section method

Short answer

Using the mid-section method, each vertical strip is 1.2 m wide. Mean velocity is V0.6 for depth ≤0.7 m, else (V0.2+V0.8)/2. Summing width × depth × mean velocity gives discharge = 4.80 m³/s.

Compute the river discharge by mid-section with the help of following data (y is the depth of flow):

Distance from the left bank (m)	Depth (m)	Velocity at 0.6y (m/s)	Velocity at 0.2y (m/s)	Velocity at 0.8y (m/s)
0	0	-	-	-
1.2	0.7	0.4	-	-
2.4	1.7	-	0.7	0.5
3.6	2.5	-	0.9	0.6
4.8	1.3	-	0.6	0.4
6.0	0.5	0.35	-	-
7.2	0	-	-	-

In the mid-section method, each vertical represents a strip extending halfway to its neighbours; here all verticals are 1.2 m apart, so each strip is 1.2 m wide. Mean velocity in a vertical:

Shallow verticals (one reading): $\bar{V} = V_{0.6y}$
Deeper verticals (two readings): $\bar{V} = (V_{0.2y} + V_{0.8y})/2$

Computation

Distance (m)	Width (m)	Depth (m)	$\bar{V}$ (m/s)	Area (m²)	ΔQ (m³/s)
1.2	1.2	0.7	0.40	0.84	0.336
2.4	1.2	1.7	(0.7+0.5)/2 = 0.60	2.04	1.224
3.6	1.2	2.5	(0.9+0.6)/2 = 0.75	3.00	2.250
4.8	1.2	1.3	(0.6+0.4)/2 = 0.50	1.56	0.780
6.0	1.2	0.5	0.35	0.60	0.210
Total				8.04	4.800

(The bank points at 0 and 7.2 m have zero depth and give no flow.)

Q = \sum w_i\,y_i\,\bar{V}_i = 4.80\ \text{m}^3/\text{s}

Answer: River discharge ≈ 4.80 m³/s.

✓ Answer checked by Claude Opus 5.5 (high effort)

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