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EXAMPLE SHEET 1 AUTUMN 2008 Revision Exercises Q1. A long bridge with piers 1.5 m wide, spaced 8 m between centres, crosses a river. The depth of water upstream is 1.6 m and between the piers is 1.45 m. Calculate the volume flow rate under one arch, assuming that the river bed is horizontal, the banks are parallel and frictional effects are negligible. Find the maximum height to which water rises at the front of the piers. Q2. A pipe carrying oil with specific gravity 0.75 tapers from diameter 0.5 m at A to 0.3 m at B, which is 5 m above A. The velocity may be assumed uniform at both points and friction may be neglected. If the velocity and gauge pressure at A are 2 m s–1 and 120 kPa respectively, calculate the gauge pressure at B. Q3. (White) 1 For the reducing section shown, diameters 2 D1 = 80 mm and D2 = 50 mm, whilst p2 is water atmospheric pressure. If u1 = 5 m s–1 and the manometer reading is h = 580 mm, estimate the total force resisted by the flange bolts. h (Take the density of mercury as 13600 kg m–3.) mercury 1 Q4. Water flows through the horizontal elbow shown at a weight flow rate of 150 N s–1. The inlet pipe diameter D1 = 100 mm and the nozzle exit has internal diameter 30 mm. The absolute pressure at inlet is 330 kPa. Neglecting the weight of water and elbow, estimate the force on the pipe bend. 40 2 (Take atmospheric pressure as 101 kPa.) Q5. V(R) The velocity profile V(R) between coaxial cylinders, where the inner cylinder (of radius R1) is fixed and the outer cylinder (of radius R2) is rotating with angular velocity ω, is given by 1 − R12 /R 2 V = ωR 1 − R 2 /R 2 2 R1 R2 1 If the cylinders have length 0.5 m, the inner and outer radii are 20 mm and 40 mm respectively, the outer cylinder is rotating at a steady speed of 120 rpm and the intervening fluid has viscosity 5×10–2 kg m–1 s–1, find: (a) the shear stress on the outer cylinder; (b) the power required to maintain steady rotation. Hydraulics 2 E1-1 David Apsley Q6. (Massey) The air supply to an oil engine is measured by being taken directly from the atmosphere into a large reservoir through a sharp-edged orifice 50 mm diameter. The pressure difference across the orifice is measured by an alcohol manometer set at an angle of arcsin (0.1) to the horizontal. Calculate the volume flow rate of air if the manometer reading is 271 mm, the relative density of alcohol is 0.80, the coefficient of discharge for the orifice is 0.602 and atmospheric pressure and temperature are, respectively, 775 mm Hg and 15.8 °C. Continuity and Momentum Principle For Non-Uniform Velocity Profiles Q7. (Examination, January 2006) A gate is used to control the flow of water in a square-section duct of side h = 0.3 m (Figure below). The upstream flow is uniform, with velocity u0. A short distance downstream of the gate the velocity profile can be adequately approximated by 2, if y ≥ h / 2 u ( y) = 3 + cos(2πy / h), if y ≤ h / 2 –1 where u is the velocity in m s and y is the distance from the floor of the duct in m. gate u0 y u(y) (a) Find the upstream velocity, u0. (b) If the hydrodynamic force on the gate is 400 N, find the pressure drop between the upstream and downstream sections shown (neglecting drag on the duct walls). (c) Hence deduce a pressure loss coefficient for the gate in this position. Q8. (Examination, January 2001) Water flows through a contraction at the end of a horizontal pipe of initial diameter 1.8 m. The diameter at the exit is 1.0 m. The flow exhausts into atmosphere. The velocity distribution across a radius of the pipe immediately upstream of the contraction is as follows: Radius (mm) 0 250 500 675 800 875 900 Velocity (m s–1) 3.0 2.92 2.67 2.34 1.89 1.14 0.0 The velocity at the exit is uniform. (a) What is the velocity at the exit? (b) If the pressure upstream is 30 kPa what is the force exerted on the contraction? Hydraulics 2 E1-2 David Apsley Q9. (Examination, January 2001) Velocities were measured in the wake of a cylinder spanning a 400 mm square test section of a wind tunnel. The cylinder was placed in the middle of the test section and the flow was two-dimensional. The velocity distribution at a downstream cross-section was found to be symmetrical about the centre plane and given by πy u = 20.0 1 − cos 0 .2 where u is the velocity (in m s–1) at a distance y (m) from the axis of the cylinder. (Take the density of air as 1.2 kg m–3.) (a) Find the upstream velocity, which is uniform over the test section. (b) If the force on the body is 3.0 N, determine the pressure drop between upstream and downstream sections. Q10. An axisymmetric jet carrying 20 L s–1 of water impinges normally on a plane wall. The velocity profile in the jet may be approximated by U0 U = 1 + (r / r0 ) 2 , if r < 3r0 0, otherwise where r is the distance from the axis and r0 = 0.025 m. Find: (a) the maximum velocity, U0; (b) the force on the wall; (c) the maximum pressure on the wall. Q11. A hydraulic jump occurs in an open channel of width 1.0 m (see figure). Upstream of the jump the depth is 0.1 m and the velocity is uA (uniform). The velocity profile just downstream of the jump is of the form u πy u = B [1 + cos ] 2 D where u is the velocity at a distance y from the bed of the channel, uB is the velocity near the bed and D (= 0.8 m) is the depth downstream of the jump. 0.8 m uA 0.1 m uB (a) Determine uB, leaving your answer as a function of uA. (b) Calculate the difference between the hydrostatic pressure forces on the fluid cross- sections upstream and downstream of the jump. (c) Neglecting viscous stresses on the channel bed or the free surface, use the momentum principle to find the upstream velocity uA. Hydraulics 2 E1-3 David Apsley Non-Uniform Force Distributions Q12. (Examination, January 2007) The drag and lift on a long-span aerofoil (see figure) are to be found by a wake traverse and by measurement of the surface pressure, respectively. The free-stream velocity is U∞ = 50 m s–1 and the aerofoil has chord c = 0.5 m. y pupper Uoo x U(y) plower The approach velocity is uniform (U∞), whilst the velocity deficit in the wake may be approximated by ∆U [1 + cos( πy / d )] , U∞ −U = if y < d 0 otherwise where ∆U = 5.0 m s , d = 0.1 m and y is the distance from the chord line. The flow may be –1 regarded as two-dimensional, and both the free-stream-velocity and static-pressure differences between upstream and downstream sections may be neglected. (a) Find the drag force per unit span on the aerofoil. (b) Define a suitable drag coefficient for the aerofoil and calculate its value. The pressure distributions on the upper and lower surfaces of the aerofoil are given by plower = 1500 − 3000 x − 48000 x(0.5 − x) 2 pupper = 1500 − 3000 x − 192000 x(0.5 − x) 2 where p is the pressure in Pascals and x is the distance from the leading edge in metres. (c) Find the lift force per unit span on the aerofoil. (d) Define a suitable lift coefficient and calculate its value. Hydraulics 2 E1-4 David Apsley Tank Filling and Emptying Q13. A conical hopper of semi-vertex angle 30º contains water to a depth of 0.8 m. If a small hole of diameter 20 mm is suddenly 0.8 m opened at its point, estimate (assuming a discharge coefficient 30o cd = 0.8): (a) the initial discharge (quantity of flow); (b) the time taken to reduce the depth of water to 0.4 m. Q14. A steep-sided reservoir has a constant surface area 0.3 km2 and discharges over a weir of length b = 4 m. The discharge relationship (discharge Q against height over the sill, h) for the weir is Q = 1.8bh 3 / 2 where Q is in m3 s–1 and b and h are in m. The initial depth of water over the sill of the weir is 0.4 m. If there is no inflow to the reservoir, find the time (in hours) to reduce this depth to 0.2 m. Hydraulics 2 E1-5 David Apsley ANSWERS 1. 23.9 m3 s–1 1.78 m 2. 73.1 kPa gauge 3. 164 N 4. Fx = 2082 N, Fy = 213 N 5. (a) 1.05 N m–2 (b) 0.066 W 6. 21.8 L s–1 7. (a) 2.5 m s–1 (b) 4.9 kPa (c) 1.6 8. (a) 7.3 m s–1 (b) 48 kN 9. (a) 20 m s–1 (b) 259 Pa 10. (a) 4.42 m s–1 (b) 34.6 N (c) 9790 Pa 11. (a) u B = 1 u A 4 (b) 3090 N (c) 6.2 m s–1 12. (a) 51 N (b) 0.068 (c) 750 N (d) 1.0 13. (a) 0.996 L s–1 (b) 177 s 14. 15.2 hours Hydraulics 2 E1-6 David Apsley

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