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The following graph represents the flow profiles of the upstream and downstream area using the water depth data obtained from a hydraulic jump laboratory experiment. The channel used along with its characteristics can be found in appendix A.
Figure 1: Measured water depth against distance from the origin
The data used to plot the graph, along with the manometer readings (all lab data can be found in appendix A) and the channel dimensions where used to calculated the required flow characteristics in order to ultimately determine the flow profiles and the location of the hydraulic jump.
Analysis of laboratory results
Calculation of discharge, critical depth and critical slope
The discharge of the channel was found by using equation 1 below, where Hd is the differential manometer reading in mm, i.e. the difference between the top and bottom readings of the manometer.
Hence, the discharge was calculated as follows:
Therefore, the calculated discharge enabled the calculation of the critical depth of the flow using equation 2 below:
(SI units) (2)
Where Q is the discharge in m3/s, g is the gravitational acceleration in m/s2 and b the width if the channel in m.
Hence, the critical depth yc was determined to be:
Using the critical equation calculated above, the critical slope could be determined using the equation below:
However, the Manning's n accounting for the roughness of the channel's surface had to be found in order to proceed to the calculation of the critical slope. Rather than using the empirical equation proposed by Manning in 1889, since the channel was a laboratory channel made of glass and not a material with more irregular surface, the value a standard value of an for glass was used. According to Hamill (2011) would be around 0.009 and 0.010 s/m1/3. Furthermore, the critical hydraulic radius Rc of the channel had to be determined as follows:
Where Ac is the critical area (in m2) and Pc is the critical wetted perimeter (in meters) of the channel. Hence, using a value of n as 0.010 s/m1/3, the critical slope was found to be:
or 34.3mm in 10m distance.
The channel during the experiment had an actual slope of 19mm in 10m which is translated to a slope of 0.0019. The graph plotted in figure 2 was recreated depicted this time all the known information up to now. The location of the jump is also depicted in the graph below for demonstration purposes. However, its calculation will be explained as the report furthers the analysis.
Figure 2: Summarized measured and calculated data in the water depth vs distance graph
Classification of flow profiles
The next step of the analysis is to determine the type of flow at all the three sections of the channel i.e. the upstream, downstream and transition area. Hence, the dimensionless parameter known as Froude number must be calculated for these areas, as this will show whether the flow at a section is subcritical, supercritical or critical. According to Hamill (2011), it is generally known that Froude number determines the type of flow as follows:
F < 1 subcritical flow (relatively deep, slow flow)
F = 1 critical flow (transitional flow)
F > 1 supercritical flow (relatively shallow, fast flow)
The Froude number, for a given discharge, is a function of depth and it is given by the following equation:
Where, Q is the channel's discharge (in m3/s), B the width of the channel (in m) and A the area of the flow (in m2). Equation 4 was applied to the flow in every section of the channel and the results were summarized in the table below:
Table 1: Froude numbers
Hence, having got the actual channel slope, the critical slope and the Froude number for each section, the type of flow profiles can be determined. Since, the actual slope is less than the critical the channel is considered to be a mild one. By making reference to figure 8 in appendix B, the profile curves were determined as follows:
So < Sc and F < 1. Hence, the profile curve is a M1 curve.
So < Sc and F > 1. Hence, the profile curve is a M3 curve.
So < Sc and F < 1. Hence, the profile curve is a M2 curve.
Gradually varied flow profile curves
In a gradually varying flow, the depth varies longitudinally along the channel. What that means is that the energy line and the surface of the water are no longer parallel and hence, it is essential that the slope of the energy line is determined and used throughout the calculations. The energy line slope represents the losses due to friction as the water flows in the channel. For this reason, this slope is also referred to as friction slope, Sf.
Following the direct step method procedure, using Sf, the gradually varying flow equation gives the change in depth D with distance L along the channel:
As it can be understood by the above equation, the rate of change of depth depends on the actual bed slope, the friction slope and the Froude number. Equation 5 was rewritten in the form of finite differences method as shown below:
Where the (1 - F2) and (So - Sf) values are the mean values at the end of the reach. Also, this equation was solved for regions 1 and 3 in order to obtain the M1 and M3 curves respectively. It should be noted that there were insufficient data to produce the M2 curve i.e. the profile curve at region 2.
To be able to do the required calculation using equation 6, the final depth of the curve in region 1 had to be determined. It was assumed that the minimum flow level would be the normal depth of flow which had to be calculated by iterating the Manning's equation below:
The iterative procedure (see appendix B) yielded a normal depth of 73.73mm (final depth). Hence, knowing the final depth and the initial depth (depth at the start of the curve i.e. 202mm) the step length could be calculated by using 20 steps to reduce the percentage error.
Then using equation 6 at small intervals (20 steps), Î”y, Fr, So and Sf were evaluated at each intermediate depth and a solution for Î”x was found. A tabular solution of the equation can be found in appendix B and the M1 profile curve produced is depicted in figure 3.
Figure 3: M1 curve
As the graph illustrates, the water depth curve asymptotically approaches the normal depth line and would need about 90 meters to reach it.
To be able to obtain the M3 curve, a similar procedure as before was followed. This time, the initial depth was known to be 25mm and the final depth was the critical depth of the flow. Hence, using again 20 steps, the step length was calculated to be:
The iterative procedure for equation 6 was again followed and can be founded in a tabulated form in appendix B and the M3 curve was plotted and shown in the following figure.
Figure 4: M3 curve
From the graph above it is shown that the M3 profile curve is approaching the critical depth line at a right angle and would take about 7m for the flow to reach the critical depth.
Location of hydraulic Jump - Conjugate flow curve
An accurate method of determining the location at which the hydraulic jump occurs is the use of a conjugate curve. A conjugate curve is a plot of the depth sequent to the depth of flow of a gradually varying flow profile curve. The conjugate curve was produced by solving the hydraulic equation below.
Where y2 is the conjugate depth of the flow, corresponding to a depth y1 of the M3 curve. The solution to this equation can be found in a tabulated format in appendix B. Two cases were considered. The first case was the location at which the jump would be without the existence of any weir at the end of the channel. The second case was the location of the jump of the actual experiment. For the former, the point at which the normal depth intersects with the conjugate curve is used to interpolate the solutions and find the height of the jump without the weir. For the latter case which is the actual problem, the point at which the mean depth downstream of the M3 curve (section 3) intersects with the conjugate curve is where the jump in the experiment occurs and was used to interpolate the solutions as before to determine the depth of the jump. It should be noted that for better accuracy, the interpolation was done by using the FORECAST excel function. The following graph was produced:
Figure 5: Conjugate depth curve
Change in downstream water level
The water level in the downstream side of the channel can be controlled by the height of the tail gate (weir) at the end of the channel. When the height of the weir increases, the amount of energy in region 2 (M2 curve) is also increased, as more water is withhold between the sluice gate and the weir i.e. increased water level downstream. This means that more energy is required for the flow to "jump" to the new higher water level, as demonstrated by the specific energy curve in figure 6. More energy is translated into higher velocity. The flow has its highest velocity just where water exits the sluice gate, as later it gradually decreases due to the effect of friction. Therefore, as more energy and hence greater velocity is required for the flow to jump to the higher water level, the hydraulic jump moves towards the sluice gate to meet the point at which the velocity is just high enough to provide the extra energy.
Figure 6: Specific energy curve
In opposition to the increase of the water level downstream, a decrease in the weir's height and hence in the water level downstream would have the adverse effect from what was previously discussed. As the water level lowers, the energy is decreased in the M2 profile as well which means that lower energy (hence lower velocity) is need for the water to jump to the new height of the downstream side. Hence, the hydraulic jump moves away of the sluice gate towards the weir at the point where the velocity will be just enough to cause the jump. Note that, this happens due to the fact that velocity decreases as water flows due to friction as mentioned previously.
Moreover, from the conjugated curve in figure 5 it can be seen that if a weir was not existing at all in the downstream side, the jump would occur at 3-4 times greater distance than with the weir as even less energy would be required for the jump to occur. It should also be noted that the water level on the upstream side remains the same as it is totally unaffected by the downstream side, provided that the jump has not reached the gate (Hamill, 2011).
Change in sluice gate opening
The sluice gate is there to control the amount of water to be transferred in the downstream side of the channel. Hence, by altering its height more or less amount of water is passing thought the opening.
When reducing the height of the gate, less amount of water would be allowed to pass through; however, the same discharge should be maintained. For that to happen, the continuity equation demands that the velocity of the water will be increased. Hence, with an increased velocity the hydraulic jump would be pushed away of the sluice gate towards the weir forming a longer gradually varied flow profile (M3). The water level upstream would be increased as less water would escape and at every location downstream of the gate, the depth would decrease. It is important to say that according to Hamill (2011), with a significant reduction of the sluice gate in combination with the absence of a weir would result the velocity of the flow under the gate to get larger and larger, and eventually for a relatively short, smooth laboratory channel with a reasonable slope, supercritical flow would occur downstream up to the end of the channel.
On the contrary to lowering the gate, an increase in the height of the gate would result in more water passing through the gate and hence an increase in the water level at every point after the sluice gate and of course a reduction in the depth upstream of the gate. Moreover, as the same discharge would have to be maintained, the increase of the amount of water would be counterbalanced by a reduction in the flow velocity and hence to the Froude number. Therefore, seeking for higher energy (i.e. velocity) the jump would move towards the upstream side (closer to the sluice gate). With a big enough increase of the sluice gate height, the actual depth would exceed the critical depth and the velocity would be such that the Froude number would be less than one at every point and the entire flow in the channel could be subcritical.
Change in discharge
By increasing the discharge while having the sluice gate and the weir in a stable condition, the flow has to move faster in order to satisfy the continuity equation since the velocity is proportional to the discharge and the area below the sluice gate remains the same. Since more water is introduced into the system, the upstream depth will increase, the transition supercritical flow will retain the same depth (with and increased velocity) and the rest of the downstream are will have an increase in the water level as well. Since the velocity in the transition area will be greater, it will push the jump towards the weir as it would happen by reducing the height of the sluice gate.
On the other hand, when decreasing the discharge, the upstream depth would decrease; the velocity in the transition area would also decrease, along with the depth of the downstream area. Since the velocity at the sluice gate would be decreased, it is obvious that the hydraulic jump would move closer to the gate. Moreover, since the discharge controls the velocity and the Froude number depends on the velocity, it is possible that by decreasing the discharge up to a certain amount to reach a Froude number of 1 (critical flow). Then since the flow would be critical, there would not be needed any alternation in depth according to the specific energy curve and hence, no hydraulic jump would occur. Also, it is important to understand that the critical depth depends in the discharge. Hence, the less the discharge the smaller the critical depth would be.
By increasing the slope of the channel the gravity becomes more and more important as it forces the water to increase its velocity and drags the jump downstream. In contrast to this, decreasing the slope decreases the velocity and hence the jump moves closer to the sluice gate.
Furthermore, the change in slope has a significant affect to the classification of the flow profiles. When for example the slope of the channel So is less that the critical slope Sc the channel is considered as mild with M1, M2 and M3 profile curves for the flow which is the case for the channel examined. For changes in slope relatively to the critical slope, a channel could be of mild, steep, horizontal and adverse slope having an effect on the profile curves.
The jump was found to be located at 10.17m from the origin at a depth of 0.03m as it can be seen in figure 2. The main conclusion is that the theoretical analysis can accurately simulate the actual conditions of the flow. However, it may be subjected to errors depending on the steps used for the direct step method or the choice of Manning's n to account for the roughness of the channel. Moreover, it was found that without a weir the jump would occur at a further distance from the sluice gate.