Applications of the Bernoulli equation to open channel flows
3.5 EXERCISES
In a 3.5 m wide rectangular channel, the flow rate is 14 m^/s. Compute the flow properties in the three following cases:
Flow depth Cross-sectional area Wetted perimeter Mean flow velocity Froude number Specific energy
Case 1 Case 2 Case 3 Units
0.8 1.15 3.9
Considering a two-dimensional flow in a rectangular channel, the velocity distribution follows a power law:
y_
J.
where d is the flow depth and F^ax is a characteristic velocity.
(a) Develop the relationship between the maximum velocity V^^^ and the mean flow velocity QldB.
(b) Develop the expression of the Coriolis coefficient and the momentum correction coefficient as functions of the exponent A^ only.
Explain in words the meaning and significance of the momentum correction coefficient.
Develop a simple flow situation for which you write the fundamental equation(s) using the
3.5 Exercises 47
momentum correction coefficient. Using this example, discuss in words the correction resulting from taking into account the momentum correction coefficient.
Considering water discharging in a wide inclined channel of rectangular cross-section, the bed slope is 30° and the flow is a fully developed turbulent shear flow The velocity distribution follows a l/6th power law:
/ A1/6
i.e. F(y) is proportional to ^
where y is the distance from the channel bed measured perpendicular to the bottom, v(y) is the velocity at a distance y normal from the channel bottom and d is the flow depth. The discharge per unit width in the channel is 2 m^/s. At a gauging station (in the gradually varied flow region), the observed flow depth equals 0.9 m. At that location, compute and give the values (and units) of the specified quantities in the following list: (a) velocity at j = 0.1 m, (b) velocity at the free surface, (c) momentum correction coefficient, (d) Coriolis coefficient, (e) bottom pressure, (f) specific energy.
Considering a smooth transition in a rectangular open channel, compute the downstream flow properties as functions of the upstream conditions in the following cases:
d V B Q
Zo
E d V B
Q
Zo
E d V B Q
Zo
E
Upstream conditions 5 0.5 20 6 0.1 300 12 0 1 2 0.55 3
DIS conditions
18 6
300 12 - 1
0.65 2.2
Units m m/s m m m m mVs m m m/s m m^/s m
Considering a horizontal channel of rectangular cross-section, develop the relationship between E, Q and d. Demonstrate that, for a fixed specific energy E, the maximum discharge (in a hori- zontal channel) is obtained if and only if the flow conditions are critical.
Considering an un-gated spillway crest, the reservoir free-surface elevation above the crest is 0.17 m. The spillway crest is rectangular and 5 m long. Assuming a smooth spillway crest, calculate the water discharge into the spillway.
Considering a broad-crested weir, draw a sketch of the weir. What is the main purpose of a broad-crest weir? A broad-crested weir is installed in a horizontal and smooth channel of rect- angular cross-section. The channel width is 10 m. The bottom of the weir is 1.5 m above the channel bed. The water discharge is 11 m^/s. Compute the depth of flow upstream of the weir, above the sill of the weir and downstream of the weir (in absence of downstream control), assuming that critical flow conditions take place at the weir crest.
A prototype channel is 1000 m long and 12 m wide. The channel is rectangular and the flow con- ditions are t/ = 3 m and Q = 15 m^/s. Calculate the size (length, width and flow depth) and dis- charge of a 1/25 scale model using a Froude similitude.
Considering a rectangular and horizontal channel (z^ constant and H fixed), investigate the effects of change in channel width. Using continuity and Bernoulli equations, deduce the rela- tionship between the longitudinal variation of flow depth (AJ/A^), the variation in channel width (A6/A5) and the Froude number.
Application: The upstream flow conditions being: di = 0.05 m,Bi = lm,Q= 101/s and the downstream channel width being B2 = 0.8 m, compute the downstream flow depth and down- stream Froude number. Is the downstream flow sub-, super- or critical?
A rectangular channel is 23 m wide and the water flows at 1.2 m/s. The channel contracts smoothly to 17.5 m width without energy loss.
• If the flow rate is 41 m^/s, what is the depth and flow velocity in the channel contraction?
• If the flow rate is 0.16 m^/s, what is the depth and flow velocity in the channel contraction?
A broad-crested weir is a flat-crested structure with a crest length large compared to the flow thickness. The ratio of crest length to upstream head over crest must be typically greater than
1.5-3 (Chapter 19). Critical flow conditions occur at the weir crest. If the crest is 'broad' enough for the flow streamlines to be parallel to the crest, the pressure distribution above the crest is hydrostatic and critical depth is recorded on the weir.
Considering a horizontal rectangular channel (B = 15 m), the crest of the weir is 1.2 m above the channel bed. Investigate the two upstream flow conditions (assume supercritical d/s flow) and com- plete the following table:
Case 1 Case 2
Upstream flow depth 1.45m 2.1m Upstream Froude number
Flow depth above the weir crest Upstream specific energy Velocity above the crest Specific energy above the crest Downstream flow depth Downstream Froude number Downstream specific energy
Volume discharge 3.195 m^/s
Notes: For a horizontal rectangular channel, the Froude number is defined as: Fr = V/Jgd. The solution of Case 2 requires the solution of a cubic equation (Appendix A1.4).
3.5 Exercises 49
Top view Fig. E.3.1 Sketch of the measuring flume.
Solution
Case 2: di = 2.1m, Az 1.2 m, 2 = 22.8 mVs, d^ = 0.25 m (d/s depth).
A measuring flume is a short chamiel contraction designed to induce critical flow conditions and to measure (at the throat) the discharge. Considering the measuring flume sketched in Fig. E.3.1, the inlet, throat and outlet sections are rectangular. The channel bed is smooth and it is horizontal at sections 1,2 and 3. The channel width equals 5^ = B^ = 15m,J52 = 8.5 m. The weir height is:
AZQ = 1.1m. The total flow rate is 88m^/s. We shall assume that critical flow conditions (and hydrostatic pressure distribution) occur at the weir crest (i.e. section 2). We shall investigate the channel flow conditions (assuming a supercritical downstream flow).
(a) At section 2, compute: flow depth, specific energy.
(b) In Fig. E.3.1, five water surface profiles are labelled a, b, c, d and e. Which is the correct one?
(c) At section 1, compute the following properties: specific energy, flow depth and flow velocity.
(d) At section 3, compute the specific energy, flow depth and flow velocity.