Civil Engineering Hydraulics Open Channel Flow. Adult: Where s your costume? What are you supposed to be?

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1 Civil Engineering Hydraulics Calvin: Trick or treat! Adult: Where s your costume? What are you supposed to be? Calvin: I m yet another resource-consuming kid in an overpopulated planet, raised to an alarming extent by Madison Avenue and Hollywood, poised with my cynical and alienated peers to take over the world when you re old and weak. Am I scary, or what? Channel Terminology No matter what the shape of the open channel, we use the same terminology in working with the characteristics of the channel We will start with a rectangular channel 1

2 Channel Terminology 3 Channel Terminology Hydraulic Radius (Rh) The ratio of the cross sectional area to the wetted perimeter Mean depth (zm) The ratio of the cross sectional area to the top width (bt) 4

3 Channel Terminology Section factor (Z) Product of the cross sectional area and the square root of the mean depth Z = A zm 5 Channel Terminology Centroid Depth (zc) Depth from the water surface to the centroid of the cross sectional area 6 3

4 Channel Terminology For the section shown, calculate the hydraulic radius, the centroid depth, the section factor, and the mean depth. The rightmost dimension should be b. 7 Froude Number In full pipe flow, the dimensionless ratio of the Reynolds number was used to describe the flow characteristics. 8 4

5 Froude Number In open channel flow, another dimensionless ratio known as the Froude number is used. The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. 9 Froude Number It may also be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to consider the resistance of an object moving through water or of water moving through an object. 10 5

6 Froude Number For rectangular cross-sections with uniform depth d, the Froude number can be simplified to: Fr = 11 v gz m = v gd Froude Number For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterized as supercritical flow. When Fr 1 the flow is denoted as critical flow. 1 6

7 Sluice Gate Sluice gates are used to control flow typically out of an impounded body of water The depth of water upstream of the gate is typically much deeper than the water downstream. 13 Sluice Gate 14 7

8 Sluice Gate The cross sectional area upstream is Aupstream=(bt)(z1) The cross sectional area downstream is Adownstream=(bt)(z) For conservation of mass (continuity) the flow upstream must be equal to the flow downstream so Qupstream = Qdownstream 15 Sluice Gate From this continuity Qupstream = Qdownstream vupstream Aupstream = vdownstream Adownstream vupstreambt z1 = vdownstreambt z 16 8

9 Sluice Gate We can also consider the energy on both sides of the gate neglecting any losses due to friction v1 p v + +zg= + + z g ρ 1 ρ I have adopted the subscripts of the text p1 where 1 denotes upstream and denotes downstream. 17 Sluice Gate Our reference plane is the bottom of the channel and the points are on the surface of the flows v1 p v + +zg= + + z g ρ 1 ρ I have adopted the subscripts of the text p1 where 1 denotes upstream and denotes downstream. 18 9

10 Sluice Gate This means that the pressure at points 1 and are both atmospheric and can be removed from the expression v1 v + z1 g = + z g 19 Sluice Gate If we make the assumption that the downstream velocity is significantly greater than the upstream velocity in the direction of the flow, we can neglect the upstream velocity and the expression reduces to 0 v z1 g = + z g v z1 = + z g 10

11 Sluice Gate v + z g Q v = bt z z1 = Using the downstream area and the volumetric flow rate to calculate the velocity downstream Notice that z has a power of. This would indicate that there are two solutions for z. 1 Q bt z z1 = + z g Q z1 = + z gbt z Problem 7. A rectangular channel 8 ft wide conveys water at 300 ft3/s. A partially raised sluice gate in the channel restricts the flow. Determine the downstream height of the flow if the upstream height is 6ft. Q z1 = + z gbt z 11

12 Problem 7. 3 Problem

13 Problem 7.4 A rectangular channel.3m wide conveys water at a volumetric flow rate of 30 m/s. A partially raised sluice gate in the channel restricts the flow so that the downstream water height is 0.68m. Determine the upstream water height. Q z1 = + z gbt z 5 Problem 7.4 A rectangular channel.3m wide conveys water at a volumetric flow rate of 30 m/s. A partially raised sluice gate in the channel restricts the flow so that the downstream water height is 0.68m. Determine the upstream water height. z1 = m 6 13

14 Venturi Flume 7 Venturi Flume Usually a section manufactured to a specification that can be inserted into an open channel allowing the measurement of volumetric flow rate by measuring the depth of flow at two points in the flume. 8 14

15 Venturi Flume Consider the Bernoulli expression at points 1 and on the surface of the sections. Notice that the channel is narrowed at so it will have a different top width than the section at 1 v1 p v + +zg= + + z g ρ 1 ρ p1 9 Venturi Flume Once again, our reference plane is the base of the channel and the points we are considering are at the top of the water So again, we can eliminate pressure from the expression v1 v + z1 g = + z g 30 15

16 Venturi Flume And substituting for the velocities v1 v + z1 g = + z g Q Q A 1 + z g = A + z g 1 Q z1bt1 31 Q + z g = zbt 1 +z g Venturi Flume Isolating and solving for Q +z g =z g z g 1 Q z1bt1 Q + z g = zbt 1 Q z1bt1 Q zbt Q z1bt1 Q zbt 1 Q z1bt1 Q= 3 = ( z g z1 g ) 1 zbt = ( z g z1 g ) ( z g z1 g ) 1 z1bt1 1 zbt 16

17 Problem 7.10 A venturi flume is placed in a rectangular channel that conveys water at 50 cfs. The upstream channel width is 1 ft, and the minimum width in the flume is 10 ft. The upstream water depth is 8 ft. What is the expected water depth at the minimum width? 33 Homework -1 A rectangular channel 0.5 m wide conveys water. A partially raised sluice gate in the channel restricts the flow such that the upstream height is 1 m and the downstream height is 0.3 m. Determine the volume flow rate under the gate

18 Homework - A venturi flume is placed in a rectangular channel. The upstream channel width is 3 m, and the minimum area section is of width m. The upstream water height is m; the height at the throat is 1 m. Determine the volume flow rate through the channel. 35 Homework -3 An open-channel-flow apparatus in a hydraulics laboratory is rectangular in cross section and 16 in. wide. A venturi flume is to be placed in the channel to give the volume flow rate. The water height upstream of the flume is 18 in., and at the minimum width it is desired to make the depth about 14 in. Determine the minimum width of the venturi flume for a volume flow rate of 7 ft3/s

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