Introduction
The transition from a rapid super-critical flow to a slow sub-critical flow is characterised by a strong dissipative mechanism: Hydraulic Jump. If properly designed and implemented, this phenomenon can result in substantial energy dissipation, limiting the damage to hydraulic structures through cavitation and abrasion. Hydraulic Jumps are the most common mechanism deployed by engineers for energy dissipation in hydraulic systems.
Aims & Objectives
This report aims to verify ‘rapidly varied flow (RVF)’ in an open channel through analysing the behaviour of a real fluid through experimentation in laboratory conditions. The objectives of this experiment are to confirm the validity of the Specific Energy and Flow Force equations, confirming the accuracy of the Head Loss and Depth Ratio equations for a hydraulic jump, to analyse and represent the data and subsequent calculations in tabular and graphical form, and to comment upon the accuracy of these findings.
Theory
The phenomenon is dependent upon the initial fluid speed. If the initial speed of the fluid is below the critical speed, then no jump is possible. For initial flow speeds which are not significantly above the critical speed, the transition appears as an undulating wave. As the initial flow speed increases further, the transition becomes more abrupt, until at high enough speeds, the transition front will break and curl back upon itself. When this happens, the jump can be accompanied by violent turbulence and surface undulations.
Specific Energy:
Critical Depth:
Flow Force:
Free Hydraulic Jump:
Head Loss :
Through experimentation and comparison between experimental and theoretical data, the validity of the Specific Energy, Flow Force, Head Loss, and Depth Ratio equations should be confirmed. I expect my experimental and theoretical data to show strong correlation when exhibited graphically, with similar lines of best fit, and R2 values around 1.
Experimental Method
Equipment used: ‘Armfield C4MKII Multi-Purpose Teaching Flume (Depth: 215mm, Width: 75mm, Length: 5000mm) ’, ‘Volumetric Bench (3:1 Ratio)’, ‘Vernier Point Gauges’, ‘Meter Rule’, and ‘Stopwatch’.
Set the level of the sluice gate a suitable distance about the channel bed.
Adjust the discharge control and the tail gate position to obtain a stable hydraulic jumps some distance downstream of the sluice gate.
Record the discharge, Q, using the calibration equipment.
Measure the channel width, B.
Measure the alternative depths y1 and y2 immediately upstream of the sluice gate.
Measure the conjugate depths y3 and y4 immediately upstream and downstream of the hydraulic jump, and estimate the length of the jump, L/mm.
Repeat the experiment for another eight different value of discharge, adjusting the sluice gate and tail gate positions to achieve a free hydraulic jump.
Experimental Results & Calculations
The results table from the experiment does not reflect the true values of the alternative and conjugate depths, as these values must be derived from the experimental data through subtracting the obtained values from the height of the tank. The refined results table below more accurately reflects the results of the experiment.
The flow rate (Q/m3s-1) and discharge per unit width (q/m2s-1) must be calculated for each test.
These calculations are repeated for each test and recorded in results table three.
Using the same value of discharge per unit width (q), specific energy (E) can be calculated for a range of theoretical values of depth (y).
This calculation is used to find E for a range of theoretical values of y from 0.0020m to 0.5000m (intervals of 0.0005m) and recorded in results table 4.
From this data, y/yc and E/yc can be calculated and used to generate a specific energy diagram.
The values of E1 and E2 for each test must be plotted on the specific energy curve using the values of q and yc that have been previously calculated. The following equations are used:
These calculations are repeated for each test and recorded in results table five.
The values of E1 and E2 for each test are plotted on the specific energy diagram:
A theoretical value for y4/y3 is calculated using the following equations:
These calculations are repeated for each test and recorded in the results table below. An experimental value for y4/y3 is calculated by dividing the experimental result for y4 by y3 and recorded in the same table. A theoretical value of y4 is derived by multiplying the theoretical value for y4/y3 by the experimental value for y3.
Using this data, a graph of theoretical values of y4/y3 and experimental values of y4/y3 against Fr3 can be produced.
Using the values previously calculated, a comparison can be made between the experimental and theoretical head loss using the following equations:
These calculations are repeated for each test and recorded in the results table below.
Using this data, a graph of theoretical head loss and experimental head loss against Fr3 can be produced:
The depth ratio must be calculated for both the theoretical value of y4 and the experimental value of y4.
These calculations are repeated for each test and recorded in the results table below.
Result Analysis & Discussion
In a hydraulic jump, the upstream flow conditions must be super-critical. If Fr<1, a hydraulic jump is impossible as it would violate the second law of thermodynamics. These results would indicate that a hydraulic jump did not occur during the experiment as the calculated value of Fr3 for each test is is less than one.
Through analysing the experimental data and subsequent calculations, one can only ascertain that a systematic error has occurred for the observed experimental values of y3 and/or y4. This is likely due to the Vernier Point Gauges not being correctly positioned when recording the data. An example of this is the difference between the conjugate depths on Test III: 0.002m – not an observable a Hydraulic Jump.
This data has resulted in experimental data and theoretical data that does not correlate in any meaningful way. An example of this is the graph of Fr3 against hLexp and hLtheo is an example of a graphical data set that was rendered useless through poor experimental data. The results of the experimental and theoretical head loss equations are so different that there is no way to accurately compare them. The experimental and theoretical depth ratios should be within 1m, however the theoretical depth ratio is nearly double the experimental depth ratio.
Conclusion
These results have not verified the validity of the Specific Energy, Flow Force, Head Loss, and Depth Ratio equations. My experimental and theoretical data did not show strong correlation when exhibited graphically.
References
Lamb, H. (1895). Hydrodynamics. Cambridge: University Press. 6. Batchelor, G. (1967). An Introduction to Fluid Dynamics. Cambridge: University Press. Herschel, C (1895). Measuring water. Providence, R.I. : Builders Iron Foundry Chanson, H. (2004). Environmental Hydraulics of Open Channel Flows: Butterworth-Heinemann