Hydraulic Jumps. CIVE 401: Fall Team 10: Yalin Mao, Natalie Pace, Kyle Nickless. November 19, 2014

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1 Hydraulic Jumps CIVE 401: Fall 014 Team 10: Yalin Mao, Natalie Pace, Kyle Nickless November 19, 014

2 1 INTRODUCTION The fields of fluid mechanics and hydraulics contain a wide range of phenomena. Some concepts are well understood and can be easily modeled (laminar flow) while others are more chaotic and often difficult to represent numerically. One important area of interest within fluid mechanics is the phenomenon of the hydraulic jump. A hydraulic jump occurs in open-channel flow when the height of water above a fixed surface increases from a low depth to a high depth. In practice, these levels are referred to as supercritical flow depth and subcritical flow depth, respectively. A basic visual demonstration of a hydraulic jump is shown in Figure 1. Figure 1. A Weak Hydraulic Jump In this example, the flow upstream of the jump has a high velocity (V 1 ) and shallow depth (y 1 ). Downstream, the flow velocity has decreased (V ) and the depth has increased (y ). When this occurs, the flow almost instantly becomes turbulent and the rough, rapidly changing flow condition can cause significant energy dissipation. The practical application of this energy loss is discussed later in this report. Hydraulic jumps have been studied for over two and a half centuries and include contributions from a range of historical pioneers. Although the description of a hydraulic jump was first documented by Leonardo da Vinci, the first experimental investigations are credited to Italian engineer Bidone in 1818 (Subramanya 48). Many of the occurrences of hydraulic jumps in horizontal channels have since been numerically modeled and are well understood. However,

if the bed slope is relatively steep, the jump becomes harder to analyze (Cruise 400). For this reason, hydraulic jumps are still of research interest today.

3 if the bed slope is relatively steep, the jump becomes harder to analyze (Cruise 400). For this reason, hydraulic jumps are still of research interest today. For practical purposes, the effect of a water-weight component that results from a steep channel is often neglected. Perhaps the most important parameter used in the analysis of hydraulic jumps is the Froude number. The Froude number (Fr) is a function of flow velocity and depth, as well as standard gravity. The value of Fr within the open channel dictates the classification of a particular hydraulic jump. If Fr is less than 1, a hydraulic jump is impossible because it violates the second law of thermodynamics (White 7). For Fr greater than or equal to 1, there are several ways a jump can occur. For example, an Fr of 6.0 would indicate a steady jump, where the downstream flow becomes steady and energy dissipation is between 45 and 70 percent. The jump demonstrated in Figure 1 (a weak jump) has an Fr between 1.7 and.5. The different Froude numbers and their associated classifications are shown in Figure. Figure. Hydraulic Jump Classification

4 3 These Froude numbers represent the parameters of the upstream condition and are often labeled as Fr 1. After a jump occurs, the downstream Fr is less than 1.0. Therefore, a jump only occurs when the Fr changes from 1.0 to <1.0. The position in an open channel where a jump occurs is referred to as the critical point. At this location, the water height changes from being less than the critical depth to greater than critical depth. This is also where the Froude number is reduced to less than 1.0. CALCULATIONS Hydraulic jumps seem complex and almost impossible to predict at first glance. While the flow at the jump is turbulent and complex, the flow upstream (1) and downstream () of the jump can be considered steady, uniform, and one dimensional. Because of these assumptions, a momentum equation can be derived (Eq. 1). F 1 F = ρq(v V 1 ) = ρv 1 y 1 b(v V 1 ) (Eq. 1) Where: F = hydrostatic pressure force ρ = density of water Q = discharge V = velocity y = depth of water from the free surface b = channel width It is important to note that the discharge, Q, is the same upstream and downstream. This is important because then the principle of conservation of mass can be applied to relate the velocity and water depth as seen in Equation.

5 4 Q 1 = Q V 1 y 1 b = V y b V 1 y 1 = V y (Eq. ) It is also important to note that these equations are derived for rectangular channels in which the width remains constant. The hydrostatic pressure force,f, is the pressure of water over the cross-sectional area of the channel. For rectangular channels, this force is determined as such: F = pa = γh A = γ y yb. Note that γ is the specific weight of water, and can be written as γ = ρg, where g is the acceleration of gravity. Then by using these equations, the momentum equation can be simplified to become Equation 3. ρg y 1 b ρg y b = ρv 1y 1 b(v V 1 ) y 1 y = V 1y 1 g (V V 1 ) (Eq. 3) Bernoulli s equation is another useful principle that can be applied in this situation to develop an energy equation for hydraulic jumps (Eq. 4). Where: h L = head loss y 1 + V 1 = y g + V + h g L (Eq. 4) The head loss in this system is due to the jump itself and the energy that is dissipated from it, and minor losses such as shear stress from the walls can be ignored. The energy equation is useful for finding the power that is dissipated by the jump. The power is the force multiplied by distance over time, and can be equated as such: P = FV = γh l AV = γh l Q. The Froude number cannot be ignored when evaluating hydraulic jumps. By combining Equations and 3, and remembering that the Froude number equation is Fr = V 1 gy, a new

6 5 equation can be derived in which the depths upstream and downstream depend only on the Froude number of the upstream flow (Eq. 5). y 1 y = V 1y 1 g (V 1y 1 V y 1 ) y (y 1 y ) y 1 (y 1 y ) = V 1 gy 1 y y 1 + y y 1 Fr = 0 y 1 ± Fr = y 1 ( Fr y 1 ) = y 1 (Eq. 5) With the derived equations, many different problems can be solved for hydraulic jumps. For example, if the upstream velocity and water depth are known, the downstream water depth can be found by using the upstream Froude number and Equation 5. Then by applying the conservation of mass, the downstream velocity can now be found, as well as the downstream Froude number. While the velocities and water depths can be found with the derived equations, determining the length of the jump is a much more complex task and cannot be fit to theoretical equations. The length of the jump can only be studied by experimentation. As discussed earlier, the Froude number is a key factor to the type of jump that will be seen. The hydraulic jump can have a variety of forms, simply due to the different conditions of the velocity and water depth. Please refer to Table 1 for typical classifications of hydraulic jumps based on the Froude number. Hydraulic jumps do not only occur in flat, rectangular channels. Hydraulic jumps can occur in channels with different shapes, the equations above are specifically for a rectangular

does not occur on the surface (Figure 4). Figure 3. Hydraulic jump with a sloped bed Figure 4.

7 6 geometry and would need to be altered slightly for the different cross-sectional area. Jumps also occur in channels with a sloped bed (Figure 3) and with sluice gates, which actually produces a submerged hydraulic jump that does not occur on the surface (Figure 4). Figure 3. Hydraulic jump with a sloped bed Figure 4. Submerged hydraulic jump from a sluice gate EFFECTS AND APPLICATIONS Uncontrolled hydraulic jumps can cause serious consequences such as the failure of the Taum Sauk Pumped Storage Hydroelectric Power Plant's upper reservoir.

7 Taum Sauk has two reservoirs. The lower reservoir is located downstream of Johnson's Shutins along the East Fork in Missouri, and the upper reservoir is located 800 feet higher.

8 7 Taum Sauk has two reservoirs. The lower reservoir is located downstream of Johnson's Shutins along the East Fork in Missouri, and the upper reservoir is located 800 feet higher. The upper reservoir holds 1.5 billion gallons of water when it is full (Figure 5) (Watkins 006). Figure 5. The Taum Sauk Pumped Storage Hydroelectric Power Plant's Upper Reservoir On December 14, 005, the reservoir experienced a failure in which the water overtopped the dike. During 1 minutes, 1.5 billion gallons of water roared down a tributary of the East Fork and straight toward the upstream portion of Johnson's Shut-ins State Park. The average flow was 80,000 cfs, which is larger than the average flow of the Mississippi River. As a result of hydraulic jump, a large scour hole was formed at the transition between the upper portion and the lower portion of Proffit Mountain. The hole was more than 0 feet deep, but it was likely larger before the diminishing flow containing near the end since many debris and stones filled the hole at the end of flow (Figure 6).

8 Figure 6. A Large Scour Hole Formed by Hydraulic Jump Hydraulic jumps can cause serious corrosion. However, after accurate calculation and rational design, hydraulic jumps can be helpful.

9 8 Figure 6. A Large Scour Hole Formed by Hydraulic Jump Hydraulic jumps can cause serious corrosion. However, after accurate calculation and rational design, hydraulic jumps can be helpful. Hydraulic jumps are mainly used: to dissipate energy and thus prevent scouring action on the downstream side of hydraulic structures to maintain high water level on the downstream side of channel for irrigation to reduce uplift pressure under the foundations of hydraulic structures to mix coagulant chemicals in water treatment plants to remove air from water supply and sewage lines to prevent air locking

10 9 A hydraulic jump primarily serves as an energy dissipater to dissipate the excess energy of flowing water on the downstream portion of hydraulic structures. The downstream portion of hydraulic structures needs energy dissipation in order to prevent damage to the downstream channel. That energy dissipation portion is called a stilling basin. Stilling basins are usually provided with special appurtenances including chute blocks, baffles piers and end sills. Chute blocks split and aerate incoming flow; baffle piers dissipate energy mostly by resistance; and end sills reduce further the length of the jump and control scouring action by lifting up the outgoing stream (Cruise et al. 007). This kind of stilling basin is not only highly efficient in dissipating energy, but also very stable.

11 10 REFERENCES Cruise, James F., Sherif, Mohsen M., Singh, Vijay P. (007). Elementary Hydraulics. Cengage Learning, Ohio. "Hydraulic Jump and Its Practical Applications." 8 Apr Web. < Munson, Bruce R., Young, Donald F., Okiishi, Theodore H., Huebsch, Wade W. (013). Fundamentals of Fluid Mechanics, 7th Ed., John Wiley & Sons, Inc., New Jersey. Chapter 10, pp Subramanya, K. (009). Flow in Open Channels, 3rd Ed. Tata McGraw-Hill Publishing. New Delhi, India. Watkins, C. (006). "The St. Francois Mountains -Missouri's Hard Rock Core." RollaNet, < (Oct. 7, 014). White, Frank M. (011). Fluid Mechanics, 7th Ed., McGraw-Hill, New York. Chapter 10, pp 7-73.

Open Channel Flow. Ch 10 Young, Handouts

Open Channel Flow. Ch 10 Young, Handouts Open Channel Flow Ch 10 Young, Handouts Introduction Many Civil & Environmental engineering flows have a free surface open to the atmosphere Rivers, streams and reservoirs Flow in partially filled pipes