MECHANICS of FLUIDS LABORATORY - Mechanical Engineering

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EXPERIMENT 14 OPEN CHANNEL FLOW—HYDRAULIC JUMP When spillways or other similar open channels are opened by the lifting of a gate, liquid passing below the gate has a high velocity and an associated high kinetic energy. Due to the erosive properties of a high velocity fluid, it may be desirable to convert the high kinetic energy (e.g. high velocity) to a high potential energy (e.g., a deeper stream). The problem then becomes one of rapidly varying the liquid depth over a short channel length. Rapidly varied flow of this type produces what is known as a hydraulic jump. Consider a horizontal, rectangular open channel of width b, in which a hydraulic jump has developed. Figure 14.1 shows a side view of a hydraulic jump. Figure 14.1 also shows the depth of liquid upstream of the jump to be h 1 , and a downstream depth of h 2 . Pressure distributions upstream and downstream of the jump are drawn in as well. Because the jump occurs over a very short distance, frictional effects can be neglected. A force balance would therefore include only pressure forces. Applying the momentum equation in the flow direction gives: p 1 A 1 - p 2 A 2 = ρQ(V 2 - V 1 ) Pressure in the above equation represents the pressure that exists at the centroid of the cross section. Thus p = ρg(h/2). With a rectangular cross section of width b (A = bh), the above equation becomes h 1 g 2 (h 1 b) - h 2 g 2 (h 2 b) = Q(V 2 - V 1 ) From continuity, A 1 V 1 = A 2 V 2 = Q. Combining and rearranging, h 2 1 - h 2 2 = Q 2 2 gb 2 ⎝ ⎛ 1 - 1 h ⎠ ⎞ 2 h 1 Simplifying, h 2 2 + h 2 h 1 - 2 Q 2 gb 2 h 1 = 0 Solving for the downstream height yields one physically (nonnegative) possible solution: h 2 = - h 1 2 √⎺⎺⎺⎺ + 2Q 2 4 gb 2 h 1 + h 1 2 from which the downstream height can be found. By applying Bernoulli’s Equation along the free surface, the energy lost irreversibly can be calculated as Lost Energy = E = g(h 2 - h 1 ) 3 4h 2 h 1 and the rate of energy loss is dW dt = ρQE The above equations are adequate to properly describe a hydraulic jump. Hydraulic Jump Measurements Equipment Open Channel Flow Apparatus (Figure 12.1) The channel can be used in either a horizontal or a sloping configuration. The device contains two pumps which discharge water through calibrated orifice meters connected to manometers. The device also contains on the channel bottom two forward facing brass tubes. Each tube is connected to a vertical Plexiglas tube. The height of the water in either of these tubes represents the energy level at the respective tube location. The difference in height is the actual lost energy (E) for the jump of interest. FIGURE 14.1. Schematic of a hydraulic jump in an open channel. h 1 V 2 V 1 p 1 p 2 h 2 34
Develop a hydraulic jump in the channel; record upstream and downstream heights, manometer readings (from which the actual volume flow rate is obtained) and the lost energy E. By varying the flow rate, upstream height, downstream height and/or the channel slope, record measurements on different jumps. Derive the applicable equations in detail and substitute appropriate values to verify the predicted downstream height and lost energy. In other words, the downstream height of each jump is to be measured and compared to the downstream height calculated with Equation 14.1. The same is to be done for the rate of energy loss (Equation 14.2). Analysis Data on a hydraulic jump is usually specified in two ways both of which will be required for the report. Select any of the jumps you have measurements for and construct a momentum diagram . A momentum diagram is a graph of liquid depth on the vertical axis vs momentum on the horizontal axis. The momentum of the flow is given by: M = 2Q2 gb 2 h + h2 4 Another significant graph of hydraulic jump data is of depth ratio h 2 /h 1 (vertical axis) as a function of the upstream Froude number, Fr 1 (= Q 2 /gb 2 h 1 3 ). Construct such a graph for any of the jumps for which you have taken measurements. 35
Page 1 and 2: A Manual for the MECHANICS of FLUID
Page 3 and 4: TABLE OF CONTENTS Item Page Report
Page 5 and 6: as its strong points. Suggest exten
Page 7 and 8: EXPERIMENT 1 FLUID PROPERTIES: DENS
Page 9 and 10: EXPERIMENT 2 FLUID PROPERTIES: VISC
Page 11 and 12: L level R i y zeroing weight torus
Page 13 and 14: 5. Is a low value of the standard d
Page 15 and 16: pivot balancing weight lever arm wi
Page 17 and 18: Questions 1. Can a similar procedur
Page 19 and 20: actual flow rate (measured in the l
Page 21 and 22: manometer orifice meter venturi met
Page 23 and 24: otameter tank valve motor static pr
Page 25 and 26: Pressure Measurement Equipment A Wi
Page 27 and 28: EXPERIMENT 10 DRAG FORCE DETERMINAT
Page 29 and 30: Experiment II For a number of wings
Page 31 and 32: axis) versus (Q 2 /b 2 h 0 3 g). De
Page 33: Therefore, Q = ∫ 0 Integration gi
Page 37 and 38: At any upstream (of the hump) locat
Page 39 and 40: EXPERIMENT 16 MEASUREMENT OF VELOCI
Page 41 and 42: Incompressible Flow Through a Meter
Page 43 and 44: TABLE 16.1. Summary of equations fo
Page 45 and 46: placed on the torque arm to reposit
Page 47 and 48: ∆H = H 2 - H 1 = p 2 ρg + V 2 2
Page 49 and 50: Specific Speed A dimensionless grou
Page 51 and 52: 1 large orifice volume flow rate in

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