The undermentioned graph represents the flow profiles of the upstream and downstream country utilizing the H2O deepness informations obtained from a hydraulic leap research lab experiment. The channel used along with its features can be found in appendix A.
Figure 1: Mensural H2O deepness against distance from the beginning
The informations used to plot the graph, along with the manometer readings ( all lab informations can be found in appendix A ) and the channel dimensions where used to cipher the needed flow features in order to finally find the flow profiles and the location of the hydraulic leap.
Analysis of research lab consequences
Calculation of discharge, critical deepness and critical incline
The discharge of the channel was found by utilizing equation 1 below, where Hd is the differential manometer reading in millimeter, i.e. the difference between the top and bottom readings of the manometer.
( l/s ) ( 1 )
Hence, the discharge was calculated as follows:
Therefore, the deliberate discharge enabled the computation of the critical deepness of the flow utilizing equation 2 below:
( SI units ) ( 2 )
Where Q is the discharge in m3/s, g is the gravitative acceleration in m/s2 and b the breadth if the channel in m.
Hence, the critical deepness yc was determined to be:
Using the critical equation calculated above, the critical incline could be determined utilizing the equation below:
( 3 )
However, the Manning ‘s n accounting for the raggedness of the channel ‘s surface had to be found in order to continue to the computation of the critical incline. Rather than utilizing the empirical equation proposed by Maning in 1889, since the channel was a research lab channel made of glass and non a stuff with more irregular surface, the value a standard value of an for glass was used. Harmonizing 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 country ( in M2 ) and Pc is the critical wetted margin ( in metres ) of the channel. Hence, utilizing a value of N as 0.010 s/m1/3, the critical incline was found to be:
or 34.3mm in 10m distance.
The channel during the experiment had an existent incline of 19mm in 10m which is translated to a incline of 0.0019. The graph plotted in figure 2 was recreated depicted this clip all the known information up to now. The location of the leap is besides depicted in the graph below for presentation intents. However, its computation will be explained as the study furthers the analysis.
Figure 2: Summarized measured and calculated informations in the H2O deepness V distance graph
Categorization of flow profiles
The following measure of the analysis is to find the type of flow at all the three subdivisions of the channel i.e. the upstream, downstream and passage country. Hence, the dimensionless parametric quantity known as Froude figure must be calculated for these countries, as this will demo whether the flow at a subdivision is subcritical, supercritical or critical. Harmonizing to Hamill ( 2011 ) , it is by and large known that Froude figure determines the type of flow as follows:
F & lt ; 1 subcritical flow ( comparatively deep, slow flow )
F = 1 critical flow ( transitional flow )
F & gt ; 1 supercritical flow ( comparatively shallow, fast flow )
The Froude figure, for a given discharge, is a map of deepness and it is given by the undermentioned equation:
( 4 )
Where, Q is the channel ‘s discharge ( in m3/s ) , B the breadth of the channel ( in m ) and A the country of the flow ( in M2 ) . Equation 4 was applied to the flow in every subdivision of the channel and the consequences were summarized in the tabular array below:
Table 1: Froude Numberss
Hence, holding got the existent channel incline, the critical incline and the Froude figure for each subdivision, the type of flow profiles can be determined. Since, the existent incline is less than the critical the channel is considered to be a mild one. By doing mention to calculate 8 in appendix B, the profile curves were determined as follows:
So & lt ; Sc and F & lt ; 1. Hence, the profile curve is a M1 curve.
So & lt ; Sc and F & gt ; 1. Hence, the profile curve is a M3 curve.
So & lt ; Sc and F & lt ; 1. Hence, the profile curve is a M2 curve.
Gradually varied flow profile curves
In a bit by bit changing flow, the deepness varies longitudinally along the channel. What that means is that the energy line and the surface of the H2O are no longer parallel and therefore, it is indispensable that the incline of the energy line is determined and used throughout the computations. The energy line incline represents the losingss due to clash as the H2O flows in the channel. For this ground, this incline is besides referred to as clash incline, Sf.
Following the direct measure method process, utilizing Sf, the bit by bit changing flow equation gives the alteration in deepness D with distance L along the channel:
( 5 )
As it can be understood by the above equation, the rate of alteration of deepness depends on the existent bed incline, the clash incline and the Froude figure. Equation 5 was rewritten in the signifier of finite differences method as shown below:
( 6 )
Where the ( 1 – F2 ) and ( So – Sf ) values are the average values at the terminal of the range. Besides, this equation was solved for parts 1 and 3 in order to obtain the M1 and M3 curves severally. It should be noted that there were deficient informations to bring forth the M2 curve i.e. the profile curve at part 2.
To be able to make the needed computation utilizing equation 6, the concluding deepness of the curve in part 1 had to be determined. It was assumed that the minimal flow degree would be the normal deepness of flow which had to be calculated by repeating the Manning ‘s equation below:
( 7 )
The iterative process ( see appendix B ) yielded a normal deepness of 73.73mm ( concluding deepness ) . Hence, cognizing the concluding deepness and the initial deepness ( deepness at the start of the curve i.e. 202mm ) the measure length could be calculated by utilizing 20 stairss to cut down the per centum mistake.
Then utilizing equation 6 at little intervals ( 20 stairss ) , I”y, Fr, So and Sf were evaluated at each intermediate deepness and a solution for I”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 H2O deepness curve asymptotically approaches the normal deepness line and would necessitate about 90 metres to make it.
To be able to obtain the M3 curve, a similar process as before was followed. This clip, the initial deepness was known to be 25mm and the concluding deepness was the critical deepness of the flow. Hence, utilizing once more 20 stairss, the measure length was calculated to be:
The iterative process for equation 6 was once more followed and can be founded in a tabulated signifier in appendix B and the M3 curve was plotted and shown in the undermentioned figure.
Figure 4: M3 curve
From the graph above it is shown that the M3 profile curve is nearing the critical deepness line at a right angle and would take about 7m for the flow to make the critical deepness.
Location of hydraulic Jump – Conjugate flow curve
An accurate method of finding the location at which the hydraulic leap occurs is the usage of a conjugate curve. A conjugate curve is a secret plan of the deepness sequent to the deepness of flow of a bit by bit changing flow profile curve. The conjugate curve was produced by work outing the hydraulic equation below.
( 8 )
Where y2 is the coupled deepness of the flow, matching to a depth y1 of the M3 curve. The solution to this equation can be found in a tabulated format in appendix B. Two instances were considered. The first instance was the location at which the leap would be without the being of any weir at the terminal of the channel. The 2nd instance was the location of the leap of the existent experiment. For the former, the point at which the normal deepness intersects with the conjugate curve is used to extrapolate the solutions and happen the tallness of the leap without the weir. For the latter instance which is the existent job, the point at which the mean deepness downstream of the M3 curve ( subdivision 3 ) intersects with the conjugate curve is where the leap in the experiment occurs and was used to extrapolate the solutions as before to find the deepness of the leap. It should be noted that for better truth, the insertion was done by utilizing the FORECAST excel map. The undermentioned graph was produced:
Figure 5: Conjugate deepness curve
Change in downstream H2O degree
The H2O degree in the downstream side of the channel can be controlled by the tallness of the tail gate ( weir ) at the terminal of the channel. When the tallness of the weir additions, the sum of energy in part 2 ( M2 curve ) is besides increased, as more H2O is withhold between the penstock gate and the weir i.e. increased H2O degree downstream. This means that more energy is required for the flow to “ leap ” to the new higher H2O degree, as demonstrated by the specific energy curve in figure 6. More energy is translated into higher speed. The flow has its highest speed merely where H2O exits the penstock gate, as later it bit by bit decreases due to the consequence of clash. Therefore, as more energy and therefore greater speed is required for the flow to leap to the higher H2O degree, the hydraulic leap moves towards the penstock gate to run into the point at which the speed is merely high plenty to supply the excess energy.
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Figure 6: Specific energy curve
In resistance to the addition of the H2O degree downstream, a lessening in the weir ‘s tallness and hence in the H2O degree downstream would hold the inauspicious consequence from what was antecedently discussed. As the H2O degree lowers, the energy is decreased in the M2 profile every bit good which means that lower energy ( hence lower speed ) is need for the H2O to leap to the new tallness of the downstream side. Hence, the hydraulic leap moves off of the penstock gate towards the weir at the point where the speed will be merely plenty to do the leap. Note that, this happens due to the fact that speed decreases as H2O flows due to clash as mentioned antecedently.
Furthermore, from the conjugated curve in figure 5 it can be seen that if a weir was non bing at all in the downstream side, the leap would happen at 3-4 times greater distance than with the weir as even less energy would be required for the leap to happen. It should besides be noted that the H2O degree on the upstream side remains the same as it is wholly unaffected by the downstream side, provided that the leap has non reached the gate ( Hamill, 2011 ) .
Change in penstock gate gap
The penstock gate is at that place to command the sum of H2O to be transferred in the downstream side of the channel. Hence, by changing its tallness more or less sum of H2O is go throughing thought the gap.
When cut downing the tallness of the gate, less sum of H2O would be allowed to go through through ; nevertheless, the same discharge should be maintained. For that to go on, the continuity equation demands that the speed of the H2O will be increased. Hence, with an increased speed the hydraulic leap would be pushed off of the penstock gate towards the weir organizing a longer bit by bit varied flow profile ( M3 ) . The H2O degree upstream would be increased as less H2O would get away and at every location downstream of the gate, the deepness would diminish. It is of import to state that harmonizing to Hamill ( 2011 ) , with a important decrease of the penstock gate in combination with the absence of a weir would ensue the speed of the flow under the gate to acquire larger and larger, and finally for a comparatively short, smooth research lab channel with a sensible incline, supercritical flow would happen downstream up to the terminal of the channel.
On the contrary to take downing the gate, an addition in the tallness of the gate would ensue in more H2O go throughing through the gate and hence an addition in the H2O degree at every point after the penstock gate and of class a decrease in the deepness upstream of the gate. Furthermore, as the same discharge would hold to be maintained, the addition of the sum of H2O would be counterbalanced by a decrease in the flow speed and hence to the Froude figure. Therefore, seeking for higher energy ( i.e. speed ) the leap would travel towards the upstream side ( closer to the penstock gate ) . With a large adequate addition of the penstock gate tallness, the existent deepness would transcend the critical deepness and the speed would be such that the Froude figure would be less than one at every point and the full flow in the channel could be subcritical.
Change in discharge
By increasing the discharge while holding the penstock gate and the weir in a stable status, the flow has to travel faster in order to fulfill the continuity equation since the speed is relative to the discharge and the country below the penstock gate remains the same. Since more H2O is introduced into the system, the upstream deepness will increase, the passage supercritical flow will retain the same deepness ( with and increased speed ) and the remainder of the downstream are will hold an addition in the H2O degree every bit good. Since the speed in the passage country will be greater, it will force the leap towards the weir as it would go on by cut downing the tallness of the penstock gate.
On the other manus, when diminishing the discharge, the upstream deepness would diminish ; the speed in the passage country would besides diminish, along with the deepness of the downstream country. Since the speed at the penstock gate would be decreased, it is obvious that the hydraulic leap would travel closer to the gate. Furthermore, since the discharge controls the speed and the Froude figure depends on the speed, it is possible that by diminishing the discharge up to a certain sum to make a Froude figure of 1 ( critical flow ) . Then since the flow would be critical, there would non be needed any alternation in deepness harmonizing to the specific energy curve and hence, no hydraulic leap would happen. Besides, it is of import to understand that the critical deepness depends in the discharge. Hence, the less the discharge the smaller the critical deepness would be.
By increasing the incline of the channel the gravitation becomes more and more of import as it forces the H2O to increase its speed and drags the leap downstream. In contrast to this, diminishing the incline decreases the speed and therefore the leap moves closer to the penstock gate.
Furthermore, the alteration in incline has a important affect to the categorization of the flow profiles. When for illustration the incline of the channel So is less that the critical incline Sc the channel is considered as mild with M1, M2 and M3 profile curves for the flow which is the instance for the channel examined. For alterations in incline comparatively to the critical incline, a channel could be of mild, steep, horizontal and inauspicious incline holding an consequence on the profile curves.
The leap was found to be located at 10.17m from the beginning at a deepness of 0.03m as it can be seen in figure 2. The chief decision is that the theoretical analysis can accurately imitate the existent conditions of the flow. However, it may be subjected to mistakes depending on the stairss used for the direct measure method or the pick of Manning ‘s N to account for the raggedness of the channel. Furthermore, it was found that without a weir the leap would happen at a farther distance from the penstock gate.