Fluid Dynamics – Free Surface Profiles in an Open Channel Essay Sample

Fluid Dynamics – Free Surface Profiles in an Open Channel Pages
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The objectives of this laboratory experiment are to analyse the flow of water past a Sluice gate. Flow through a horizontal rectangular channel partially obstructed by a gate has been used to establish a hydraulic jump. Experimental measurements for upstream and downstream depths determining the free surface profile is compared to that predicted by theory for free surface profiles for the horizontal bed and Chezy’s roughness coefficient for the channel. This experiment enables design implications for many real civil engineering problems to be solved, or examined critically.

Introduction

Open channels are frequently encountered, “Natural Streams and rivers, artificial canals, irrigation ditches and flumes are obvious examples; but pipelines or tunnels which are not completely full of liquid also have essential features of open channels.”1 Predicting the free surface profile and the ability to control fluid levels especially the control of water levels and regulation of water discharge is necessary for purposes such as “irrigation, water conservation, food alleviation and inland navigation”2.

This lab examines the rapid increase of depth from super-critical flow to sub-critical flow in a hydraulic jump. A hydraulic jump can occur downstream of a sluice gate (as will be the case in this lab), after a decrease in channel slope, due to an increase in roughness or channel width, or upstream of an obstacle located in a channel. It is an important energy-dissipating phenomenon; practical applications include the dissipation of energy below a spillway for the prevention of scouring farther downstream in the channel. The laboratory experiment not only has allowed visualisation but also made possible quantitative measurements of water depths, location of the jump and discharge rates.

The free surface profile shall be initially plotted using experimental measurements of the initial depths and the sequent depths. This shall then be compared to theoretical values.

Experimental Apparatus

The experiment has been carried out in the flumes of the Imperial College Fluid dynamics laboratory. The flow through a channel in which the sluice gate partially obstructs the flow has been used, (diagram I). The fluid flow is from left to right, with the supply to the flume being gravity driven. A weir at the downstream end of the flume controls the flow. The sluice gate is provided with stagnation tubes, facing directly upstream, these are filled with a colour dye such that the height of a column of water supported by pressure can be quantitatively measured (Diagram II, Appendix A).

The fluid in the upstream section builds up against the gate to a level Y1 and flows under the sluice gate at a height of Z. The fluid gains a higher velocity V2, and a shallower surface height Y2 downstream.

* Once the flume had steady running water in the channel, it was adjusted such that the inflow rate Q was a low rate, and the upstream water depth was above the top stagnation tube on the gate.

* Before any measurements were taken, a small amount of dye was placed upstream of the gate, allowing for a clear visualisation of streamlines.

* The width of the flume and the height of the Sluice gate submerged were measured using a mm ruler.

* Using a pointer gauge the upstream Y1 and downstream Y2 (see Diagram 1) water depths were measured.

* The Discharge rate, Q (m3s-1) was measured. The flow was channelled into a tank of known cross sectional area, the time taken for a specific amount of fluid to discharge was measured, using a meter rule and a stopwatch.

Results and Discussion

The discharge rate calculated experimentally is 8.744 x 10-3 m3s-1. The calculations have been illustrated in Appendix A. The theoretical value of discharge has also been calculated using the energy equation, (Appendix A, Part II) and is 9.235 x 10-3 m3s-1.

Although the two values are different, the discrepancy is 5.31%. The Experimental discharge is smaller than the theoretical discharge. A small error is expected, as the theoretical calculation does not take into account the friction between the water and the rough surface of the boundaries, which reduces the velocity of the flow. This justifies the discrepancy between the values and the percentage difference is small enough for the experimentally calculated discharge to be valid.

The height of water level experimentally measured at intervals of 10m is shown below. The first measurement was taken 20 away from the end of the gate, as this is where the depth y1 is a minimum, and plots have been taken from a constant super critical flow level.

X (mtrs)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Y (mtrs)

0.041

0.0415

0.042

0.0438

0.0441

0.0443

0.0446

0.07

0.0792

0.1

X (mtrs)

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Y (mtrs)

0.1413

0.1454

0.152

0.1577

0.1604

0.161

0.161

0.160

0.159

Measurements of y1 for 0.80.m<x<1.30m have been included, although the average of three measurements they are not accurate as this is where the hydraulic jump takes place and as a result the surface profile is constantly moving.

The values of y for 0<x<0.80 m are called the initial depths and the values of y for 1.30<x<2.0m are known as the sequent depths. The initial depths are super-critical flow, which after the jump will become will become sub-critical.

These figures have been plotted on the graph below. This illustrates the experimentally measured free surface profile.

In order to locate the exact location of the hydraulic jump and to compare the results obtained above, a theoretical free surface profile must be plotted. Theoretically Y, (the height of water, either initial or sequent) is related to X, (distance from the Sluice gate) in an open-channel flow by the equation,

Where: C is a constant called the Chezy coefficient

yc is the critical depth of flow

K is a constant

This equation is derived by integrating the Chezy form of the Bresse equation, (illustrated in Appendix B). The values obtained from this equation enable the theoretical free surface profile to be plotted. However, in order to use this equation the Critical depth yc must be found and a value of C the Chezy coefficient must be chosen.

The Critical depth for this flow is 0.096m (9.6×10-2m). (Calculated in Appendix C)

All Open channel flows are affected by friction losses. The channel has a roughness coefficient the Chezy coefficient, which is used to determine losses. Each channel has a unique value for the Chezy coefficient, which is predetermined. In this laboratory, the Chezy coefficient is an unknown constant which has to be estimated. This can be determined by plotting the super-critical experimental flow and matching it with plots determined using different values of the Chezy coefficient. As the constant K is also dependant upon the variables, a constant has also to been chosen. This has been achieved by substituting the value of Y when X is at 0.2 for Super-critical flow.

Appendix D illustrates the effect of changing the Chezy Coefficient As can be seen in Appendix D Graph II, the line with coefficient of 36 has a very similar gradient and closely matches the experimental super-critical profile. Using this value for the Chezy, the constant K1 is – 5.11103 m.

The profile of the Super-Critical flow can be given mathematically as,

The Theoretical profile of the Supercritical flow is represented by the graph below. X is considered the independent variable and Y to be the dependent Variable.

The theoretical profile is visibly different from the experimental data plot. The theoretical profile does not consider all real variables, whilst the experimental data is real engineering and the result of multiple data measurements. However, the experimental profile is a representation of the flow at that moment in time, whilst the theoretical profile may be considered as a general profile of the flow and as such is a valid generalisation of the flow.

The sub critical part of the flow is located after the hydraulic jump and is where the flow reaches a steady height (Sequent Depth) and velocity. The sequent depths are related to the distance X, by the equation

The Chezy coefficient has been determined as 36 and therefore the value of K is 6.718003 m, therefore the mathematical equation for sub-critical flow is,

Using this equation, a plot of the theoretical and experimental profile of the sub-critical part of the flow can be determined.

Initially the theoretical graph looks a very poor fit for the experimental plot. The Chezy coefficient seems too high as the gradient is too steep and the corresponding K constant is too high; however when 1.5<x<2.0 the graph is a good approximation for the laboratory data.

The fluid flow reached true sub-critical flow once X>1.5. Prior to this, the flow is still in the hydraulic jump region and thus the theoretical profile, which assumes sub-critical flow, is incorrect. During the hydraulic jump region the fluid level fluctuates and errors of measurement are incurred. Downstream of the jump, experimental flow reaches sub critical levels and therefore the theoretical would be a good approximation; this can be seen from the end region of the graph. Also, note that the graph has a Y-axis with increments of 0.5mm, the maximum difference between the theoretical and experimental plot is therefore 2.5mm.

The sequent depths are the depths of the fluid at Y2, once the hydraulic jump has occurred. These are important in deducing the location of the hydraulic jump as where they intersect the sub critical profile is the location of the jump.

The values of the sequent depths expected from the theoretical super-critical free surface profile have been calculated in Appendix E and are listed below.

X Theoretical (m)

0.200

0.2713

0.3425

0.5966

0.6387

0.6667

0.7086

YIntial (m)

0.0410

0.0415

0.0420

0.0438

0.0441

0.0443

0.0446

YSequent (m)

0.1755

0.1741

0.1727

0.1680

0.1672

0.1667

0.1659

These values have been plotted below,

The location of the hydraulic jump may be found by plotting the Theoretical Free surface profile and the Theoretical Sequent depths together. The intersection is the location of the hydraulic jump.

From the graph, it can be seen that the point of intersection falls at the beginning of the region of the fluctuation on the experimental free surface profile. The graph indicates that the Hydraulic Jump occurs 0.94 meters away from the sluice gate, and has an initial height of 0.073m and a sequent height of 0.1604m. The height of the jump is 0.0874m.

The hydraulic jump is an important mechanism of dissipating energy. The head loss caused by the hydraulic jump theoretically is greater than the head loss due to friction. Appendix E, Part I, details the head loss due to the hydraulic jump and is 1.425cm. Part II details the head loss due to friction and is 0.5547cm. The head loss due to the hydraulic jump is 257% greater than the head loss due to friction. This proves the effectiveness of the hydraulic jump at dissipating energy and highlights why civil engineers need to be familiar with them.

Conclusion

The laboratory experiment has illustrated the theoretical and practical behaviour of water in an open channel; in particular, it has illustrated the free surface profiles of open channel flow and illustrated the significance of the hydraulic jump.

Experimental data was very close to theoretical values and as such can be readily applied to many civil engineering problems. Although there was a degree of inaccuracy in the quantities measured, percentage inaccuracies were minimum; the discharge measured was 5.31% different to that theory predicted and the theoretical free surface profile was similar to the experimental free-surface profile. Discrepancies were caused by apparatus inaccuracies, human error, or most significantly due to the fact that the Chezy coefficient chosen was an estimate for the cannel.

This laboratory experiment has enabled the theoretical location of the hydraulic jump, which coincides with that measured experimentally; however, perhaps the most significant inference is that the hydraulic jump is a very effective means of dissipating energy. This coupled with assured theory to predict the location of the hydraulic jump enables real life situations to be accurately modelled in the laboratory, such as damn spillways.

Appendix A

Part I: Experimental Value of the Discharge Rate

Flume width B = 0.1m

Water depth upstream Y1 = 0.283m

Water depth Downstream Y2 = 0.042m

The cross sectional area of the measuring tank was 0.940m2

Water Depth (m)

Initial Final

Difference (m)

Time taken to Discharge (sec)

Volume of Discharged water (m3)

0.12

0.42

0.30

32.15

0.282

0.11

0.41

0.30

32.06

0.282

0.09

0.39

0.30

32.53

0.282

The average time for discharge = 32.25 sec

Average volume of water displaced = 0.282 m3

= 8.744×10-3 m3s-1

Part II: Theoretical Value of the Discharge Rate

The energy Equation states

Where y is the depth at a given position, v is the velocity at this position and g is the acceleration due to gravity.

However, and therefore the equation becomes

by rearranging the formula to make q the subject, and substituting the values of y1, y2 and the value of B, the theoretical flow rate q may be calculated.

q = 9.235×10-3 m3s-1

Appendix B

The Chezy form of the Bresse Equation is,

Note

The equation therefore becomes;

Integrating,

The equation becomes,

Appendix C

The critical depth yc, of a flow in an open channel is given by the formula;

Were q is the discharge per unit width. Having calculated the experimental value for the discharge rate Q, q can easily be calculated using the formula;

Substituting values, q = 0.08744 m2s-1.

The experimental critical depth of this flow is 9.6×10-2m or 9.6cm.

Appendix D

The table below illustrates the effect of changing the Chezy coefficient on the values of the variable X from the measured fluid height Y.

Y (Experimental)

X (Experimental)

X when C=10

X when

C=30

X when

C=32

X when

C=34

X when

C=36

X when

C=38

X when

C=40

0.0410

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.0415

0.3

0.2046

0.2422

0.2480

0.2542

0.2608

0.2677

0.2750

0.0420

0.4

0.2093

0.2843

0.2959

0.3083

0.3214

0.3353

0.3499

0.0438

0.5

0.2261

0.4347

0.4670

0.5014

0.5380

0.5766

0.6172

0.0441

0.6

0.2288

0.4596

0.4953

0.5334

0.5738

0.6165

0.6615

0.0443

0.7

0.2306

0.4761

0.5142

0.5547

0.5976

0.6430

0.6909

0.0446

0.8

0.2334

0.5009

0.5424

0.5865

0.6333

0.6828

0.7350

In order to compare which Chezy coefficient best fits the laboratory conditions, the graph below has been produced.

As can be seen the Chezy Coefficients alters the gradient of the graph. The line with a Chezy Coefficient of 36 matches the experimental plots the best. Using this value for the Chezy, the constant K1 can be calculated as – 5.11103 m.

The profile of the Super-Critical flow can now be given mathematically as

Appendix E

The initial depths of fluid (Yinitial) are related to the sequent depths by the equation,

Where, is called the Froude number, determined by

The initial Equation therefore becomes,

Appendix F

Part I: Head loss due to the Hydraulic jump

The head loss caused by the hydraulic may be calculated by the formula

Where the values y1 and y2 are the depths of water just before and after the hydraulic jump.

Substituting values, y1 = 0.073m and y2 = 0.1604m,

= 0.01425m (1.425cm)

Part II: Head loss due to Friction in the Channel

The formula for calculating head loss due to friction is equivalent to calculating the specific energy head at a specific location and subtracting this from a value obtained upstream. The head loss due to friction is therefore,

Where,

The equation therefore becomes,

Substituting values, y1 = 0.152m, y2 = 0.159m and q = 0.08744m2s-1.

=5.547 x 10-3 m (0.5547cm)

1 Mechanics for fluids B.S. Massey

2 Fluid mechanics for Civil Engineers, N.B. Webber 1971

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