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Scale effects of freeing ports Williamson, Peter Bruce 1991

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Scale Effects of Freeing Ports by Peter Bruce Williamson B.A.Sc. (Mechanical) Queen's University, 1989  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F Master of Applied Science  in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A July 1991 (c)Peter Bruce Williamson, 1991  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  M&zl\cx^7caJ  The University of British Columbia Vancouver, Canada Date j T ^ / y S~ /ff/  DE-6 (2/88)  ^nyne-er^j  Abstract  Freeing ports are found on almost all ships and their ability to drain water from the deck of a ship in rough weather can be critical to both the ship and crews' survival. At present, tests of freeing port performance on models may not be performed due to the effects of viscosity and surface tension that arise due to the decreased scale. In this thesis, the effect of scaling down freeing ports to a model size is examined, and a method of testing freeing ports on models is proposed that would incorporate the monitoring of water on the deck of the model and a control algorithm that would control the discharge from the model. To examine the effect of scale on freeing port discharge, discharge through twelve different sizes of long narrow slots ranging from 1 to 15 millimetres high are tested. Comparisons are made between the cases of steady state discharge, with a constant head, unsteady discharge, with a declining head, and unsteady discharge with the test vessel rocking to generate the sloshing motion of the water that one would expect to find on the deck of a ship in rough seas. It is found that while larger holes drain water off at the rate that one would expect at full scale, the smaller holes can let off either too much water or too little water, depending on the relative water depth.  For this reason, it appears that an active system  is needed to be able to test freeing port performance successfully on models.  ii  Table of Contents Abstract  11  Table of Contents  iii  List of Figures  v  List of Tables  ix  Nomenclature  x  Acknowledgement  x  Chapter L Introduction 1.1  2.3  7  Bernoulli's Equation Dimensional Analysis of Problem 2.2.1 The Important Variables 2.2.2 Formation of Dimensionless Groups 2.2.3 Analysis of Dimensionless Groups Discharge Equation and Discharge Coefficient 2.3.1 The Vena Contracta 2.3.2 Equation for Discharge Coefficient  Chapter 3 Experimentation 3.1 Experimental Apparatus 3.1.1 The Test Vessel 3.1.2 Freeing Port Slots 3.1.3 Apparatus for Steady State Tests  3.2  3.1.4 3.1.5 Testing 3.2.1  1 3 3 4 6  Other Literature  Chapter 2 Background and Governing Equations 2.1 2.2  '  1  Problem Description 1.2 Previous Work 1.2.1 The Capsizing of M/S H E L L A N D - H A N S E N 1.2.2 Experimental Studies on the Shipping Water onto Small Fishing Boats 1.2.3  '  7 9 9 10 11 13 14 17  19 19 19 20 23  Apparatus for Unsteady State Tests Apparatus for Sloshing Tests Procedures Steady State Test Procedure  iii  25 26 2S 29  Table of Concents  3.2.2 3.2.3  Unsteady State Test Procedures Sloshing Test Procedures  30 30  Chapter 4 Experimental Results and Discussion 4.1 Steady State Discharge Test Results 4.1.1 Curve Fitting of Results 4.1.2 Slots 7 lo 15 Millimetres High 4.1.3 The Five Millimetre Slot 4.1.4 4.1.5 4.1.6  4.2  4.3  4.4  4.5  32 32 34 34 35  The Four Millimetre Slot The Three Millimetre Slot The 2.5 Millimetre Slot  37 37 39  4.1.7 The Two Millimetre Slot 4.1.8 The 1.5 Millimetre Slot 4.1.9 The One Millimetre Slot Discussion of Steady State Results 4.2.1 Lowering of Discharge Coefficient at Low Heads 4.2.1.1 Reduced Discharge due to Surface Tension 4.2.1.2 Reduced Discharge due to Viscosity 4.2.2 Raising of Discharge Coefficient at Higher Heads 4.2.3 Relating Discharge Coefficient to Dimensionless Numbers 4.2.3.1 Plotting Discharge Coefficient versus Reynolds Number 4.2.3.2 Plotting Discharge Coefficient versus Weber Number 4.2.3.3 Plotting Discharge Coefficient versus Reynolds and Weber Number 4.2.4 Comparison of Results to Published Values Unsteady Discharge Tests Without Sloshing 4.3.1 Fresh Water Tests 4.3.1.1 Unsteady Discharge Through Large Slots 4.3.1.2 Unsteady Discharge Through Small Slots 4.3.2 Soapy Water Tests 4.3.2.1 Scale Effects from Soapy Water at High Heads 4.3.2.2 Scale Effects from Soapy Water at Low Heads 4.3.3 Video Analysis of Exiting Jet Unsteady Discharge With Sloshing Induced 4.4.1 Measurement of Head al Freeing Port 4.4.2 Measurement of Discharge From the Test Vessel  40 42 43 44 44 45 47 50 53 53 55 57 57 60 60 62 63 64 65 66 66 67 67 71  4.4.3  73 74 77 79 80 80 83  Results of Sloshing Tests 4.4.3.1 Sloshing Tests with 1.5 Millimetre Slot 4.4.3.2 Sloshing Tests with 3 Millimetre Slot 4.4.3.3 Sloshing Tests with 11 Millimetre Slot Discussion of Experimental Error 4.5.J Analysis of Experimental Error 4.5.2 Relative Error in Sloshing Discharge Measurement  Chapter 5 Conclusions and Recommendations  85  IV  Table of Concents  5.1 5.2 5.3 5.4  Summary of Tests Performed Implications of Test Results Control Strategy Recommended Future Tests 5.4.1 Steady State Tests 5.4.1.1 Testing The Effect of Viscosity 5.4.1.2 Testing the Effect of Surface Tension 5.4.2 5.4.3  85 86 87 89 89 90 91  5.4.1.3 Other Test Improvements Unsteady Discharge without Sloshing  92 92  Sloshing Tests 5.4.3.1 Water Motion in the Tank 5.4.3.2 Prediction of Sloshing Discharge  93 93 94  Bibliography  ^6  Appendix A Video Analysis using Frame Grabber A.1 Uses of the Frame Grabber A.2 Hardware and Software A.3 T A N K _ V O L . C Program A.3.1 Use of the Program A.3.2 Calculations Performed in Tank_Vol.C A.3.2.1 Converting to Cartesian Coordinate System A.3.2.2 Converting to Tank Axes A.3.2.3 Calculation o! Tank Volume A.4 DIGITIZE.C Program A. 5 Program Code A.5.1 Tank_Vol.C A . 5.2 Digitize.C  97 97 98 99 99 100 101 101 103 103 105 105 109  Appendix B Data Acquisition System B. l Wave Probes  112 112  B.2 B.3 B.4  B. l .1 Capacitance Wave Probe B . l .2 Kenek Servo Wave Probe The Load Cell Signal Conditioning Hardware and Software  112 113 114 114 115  Appendix C Additional Tables and Figures  118  V  List of Figures Schematic of Flow Through Freeing Port  7  Jet Shape and Dimensions  14  Test Vessel Dimensions  20  Side View of the Tank and Slot Configuration  21  Configuration for 1 to 3 Millimetre Slots  22  Setup For Steady Stale Discharge Tests  24  Unsteady State Discharge Test Apparatus  25  Sloshing Discharge Test Apparatus  27  Measured Discharge Coefficients  32  Computed Curves for Discharge Coefficients  33  Discharge Coefficients for Larger Slots  35  Discharge Coefficients for 5 Millimetre Slot  36  Discharge Coefficients for 4 Millimetre Slot  38  Discharge Coefficients for 3 Millimetre Slot  39  Discharge Coefficients for 2.5 Millimetre Slot  40  Discharge Coefficients for 2 Millimetre Slot  41  Discharge Coefficients for 1.5 Millimetre Slot  42  Discharge Coefficients for 1 Millimetre Slot  43  Orientation of Surface Tension Pressures and Forces  46  Horizontal Forceson an Element in the Test Vessel  '.  48  Forces on the Jet at Low Heads  51  Plot of Discharge Coefficient versus Reynolds Number  54  vi  List of Figures  Plot of Discharge Coefficient versus Weber Number  55  Plot of Discharge Coefficient versus Reynolds Number versus Weber Number  56  Comparison to Published Discharge Rates  58  Comparisons ot Tests With Freeing Ports Submerged  59  Vessel Drainage Through the 2 Millimetre Slot  61  Vessel Drainage Through the 9 Millimetre Slot  63  Vessel Drainage Through the 3 Millimetre Slot  65  Depth Measurement Comparison for the 3 Millimetre Slot  68  Depth Measurement Comparison for the 1.5 Millimetre Slot  69  Depth Measurement Comparison for the II Millimetre Slot  70  Surface Plot From Video Analysis  73  Plot of Volume Flow Rate for Sloshing With 2 Millimetre Slot  75  Remaining Volume in Tank for 1.5 Millimetre Slot  76  Remaining Volume in Tank for 3 Millimetre Slot  78  Remaining Volume in Tank for 1 I Millimetre Slot  80  Plot of Required Area Changes for 3mm Freeing Port  87  Alternate Hole Configurations for Boundary Layer Testing  91  Axes Conversion and Notation  102  Discharge Through the 15 Millimetre Slot  120  Discharge Through the 13 Millimetre Slot  120  Discharge Through the 11 Millimetre Slot  121  Discharge Through the 7 Millimetre Slot  121  Discharge Through the 5 Millimetre Slot  122  Discharge Through the 4 Millimetre Slot  122  vii  List of Figures  Discharge Through the J Millimetre Slot  123  Sloshing Frames for the 1.5 Millimetre Slot  124  Sloshing Frames for the 3 Millimetre Slot  129  Sloshing Frames for the 11 Millimetre Slot  134  viii  List of Tables Important Variables in Discharge  9  Comparison of Discharge Equations  17  Expected Experimental Error  83  Slot Dimensions  •  Coefficient Values for Cd Equations  118 119  ix  Nomenclature b  Freeing Port Slot Width  Cv  Velocity Coefficient  Cc  Contraction Coefficient  Cd  Discharge Coefficient  d  Freeing Port Slot Height  g  Gravitational Acceleration  H  Hydrostatic Head  Hnd  Non Dimensionalized Hydrostatic Head (H/d)  Q  Discharge from Freeing Port  q  T w o Dimensional Discharge Rate  p  Density  u  Dynamic Viscosity  a  Surface Tension Coefficient  V  Discharge Velocity  A  Freeing Port Area  t  Thickness of Wall at Freeing Port  Re  Reynolds Number  We  Weber Number  Fr  Froude Number  F  Interfacial Attraction at top of Freeing Port Slot  l o p  ^bottom  Interfacial Attraction at Bottom of Freeing Port Slot  P  o p p  Opposing Pressure in the Slot Due to Surface Tension  P  H  Hydrostatic Pressure at the Freeing Port Slot  Q Q  Ratio of P  AP  o p p  to P  H  Dimensional Ratio of P  1  H  o p p  to P  H  Difference in Hydrostatic Pressure  x  Acknowledgement Over the course of my research, I have been assisted by many people, too numerous to mention here. There are a few people that I must pay special thanks to. Firstly, I must thank Dr. Sander Calisal for his constant input and direction throughout this project, and for understanding or at least putting up with my hedonistic lifestyle. I must also thank M r . Dan McGreer for his assistance with my experimental apparatus, particularly the sloshing tank, and for answering numerous questions on almost every subject. Thanks must also be extended to Gerry Stensgaard, George Roddan and the rest of the staff at the B C Research Ocean Engineering Centre for their support for my " G R E A T " award, as well as for the use of their facilities and knowledge, particularly in the area of data acquisition. I must thank D R E A , who offerred the initial interest and funding for this project, as well as N S E R C and the B C Science Council for their support. Thanks must also be extended to everyone in the Naval Architecture Lab at U B C , who offerred assistance one way or another and made working here fun. Finally, I must thank my family. A s the fourth person in my immediate family to receive a post graduate degree in fluid related engineering, they have obviously had some influence, whether intended or not, on my decision to pursue this degree.  xi  Chapter 1 Introduction  1.1  Problem Description Almost all ships have bulwarks. They are needed to protect the crew working on  the deck of the ship from being hit by waves, as well as to keep equipment, fish and crew on the ship. Unfortunately, these bulwarks also trap water on the deck of the ship if the seas are large enough to wash over the bulwarks. Ships such as fishing boats and supply ships that have large deck areas close to the water level are particularly prone to this. The water on deck poses a problem because large volumes of water trapped on the deck of the ship raises the centre of gravity and may cause capsizing. Alternatively, this water may slosh from side to side or pocket on one side due to the motion of the ship in the waves. This can lead to severe listing and slow recovery from a large roll, in conditions that are already perilous. Naval Architects who design the bulwarks and freeing ports on ships currently use general formulas to determine the area of the freeing ports that are required for a given length of bulwark and deck area. While this will work for the general case, it does not allow optimization of size and location of freeing ports so that the water may be drained off efficiently with a minimal open area that could allow the loss of the deck contents. The most common method of testing ships is with scaled down models. Model tests are performed to determine the resistance and manoeuvering characteristics with great success, but stability tests are generally performed with a smooth deck and no bulwarks due to the lack of correlation between the drainage of water off of the model 1  Chapter 1: Introduction  and the drainage of the water off of a full scale ship. Without these bulwarks, the performance of the model in simulated rough seas does not accurately model the dynamic performance of the ship due to the lack of complications of the motion and mass of the water that would be on the deck of the ship. It is suspected that the effects of viscosity and surface tension cause this lack of correlation between the full scale and model scale discharge. A n analogy of this problem may be taken from the method of resistance testing model ships. It is well known in this field that it is impossible to keep the Froude Number of the model and the ship the same while maintaining the same Reynolds Number. The nature of the equations and the available liquid do not allow it. This problem is overcome by matching the Froude Numbers and hence wave making drag of the model and ship, and correcting for the difference in Reynolds Numbers to estimate the frictional resistance. Freeing ports are affected by the same parameters as hull drag, meaning that the Reynolds Number and Froude Number are important. In addition, freeing ports deal with the creation of a free surface and as a result, the surface tension, or Weber Number is important. It is proposed that the way that one might avoid this problem of a lack of correlation between ships and models is to follow the example set in resistance testing. If the Froude Number of the freeing port is maintained, one may apply a procedure to account for the differences in surface tension and viscosity. It is suspected that these effects vary with the depth of water at the freeing port, however, so one may have to correct for these effects by a differing amount as the water sloshes back and forth and drains from the deck of the ship. It is proposed that this could be accomplished by  Chapter 1: Introduction  designing an apparatus that monitors the amount of water on the deck of the model and subsequently adjusts the size of the freeing port to take into account this scale effect. Of course, none of this will be possible until a clear understanding of the nature of the flow from the model is achieved. That is the goal of this research - to develop an understanding of the flow characteristics of water through small, model scale freeing ports, and to propose a method of using these findings to enable the tester to test ship stability more accurately.  1.2 Previous Work  A literature review on the subject was performed , covering several databases using a CD ROM catalog. In addition, the data base used by the Institute for Marine Dynamics in St. John's, Newfoundland was also searched using many different keywords including freeing ports, scuppers, scale effects, water on deck, green water, deck drainage, and many more even less likely subjects. It was found that according to the numerous data bases and a number of researchers and professionals in the industry, there appears to have been nothing published specifically on the scale effects of freeing ports. A few papers did surface, however, that are somewhat related to the subject being addressed here and warrant further review.  1.2.1  The Capsizing of M/S HELLAND-HANSEN This paper was written in 1979 by Emil Aall Dahle and Olav Kjaerland of  Norway. The M/S HELLAND HANSEN was a Norwegian research vessel that capsized and sank in September 1976 after being hit broadside by a breaking wave. An  3  Chapter 1: Introduction  investigation was performed that incorporated the modelling of the seas the ship encountered as well as the ship itself. The model ship was built with considerable detail, including fully functioning freeing ports, ln their case, it was noted that there appeared to be no scale effects on the scaled down freeing ports. Examination of Figure 4 in this paper shows us, however, that the freeing ports on the full scale ship were 450 millimetres high and photographs indicate that the width of the freeing port would be approximately 1.5 times the height of the freeing port. The scaled freeing ports would have been 18 millimetres high. The boats that were surveyed around the Vancouver area, which were mainly fishing vessels, were found to have freeing ports that were much lower and longer than those tested by Dahle et al. While they did not notice scale effects on their model, this would not necessarily apply to models of local fishing vessels due to the different shape and size of the freeing ports. Dahle et al. also did tests on full scale freeing ports, testing the different ways of placing hinged covers over the freeing ports and enhancing the performance of the freeing ports with plates above the hole perpendicular to the wall, to aid the flow by reducing the contraction at the exit. These latter tests, however are beyond the scope of this thesis.  1.2.2 Experimental Studies on the Shipping Water onto Small Fishing Boats... "Experimental Studies on the Shipping Water onto Small Fishing Boats and Effectiveness of the Freeing Port on Drawing the Water Off" is a paper by Taihei Yoshino and Tokutaro Yamamoto of Japan. In this paper the authors look at many different aspects of shipping water and freeing port performance. Model tests are  4  Chapter 1: Introduction  performed that explore the actual process of shipping water in different types of seas, ballast conditions and ship speeds. Tests are also performed on model scale freeing ports. Two freeing ports are tested, one that is 15 x 35 millimetres, and one that is round with a 15 millimetre diameter. A scooper is also added to the freeing ports, and tests are performed to compare the performance of the freeing ports with and without the scooper. Tests on these two freeing ports are performed both with the freeing ports in air as well as submerged in water that is flowing past them. While the round freeing port is not addressed here, nor are the scoopers or the case of submerged freeing ports, the 15 x 35 millimetre hole is within the range tested in this research. The hole is quite narrow, and one would expect that it may not behave as a two dimensional hole, but a comparison between the results presented by Yoshino et al and the research performed during this thesis may be used as a measure of the effect of the third dimension of the flow on the overall discharge. It should be noted that the graph that presents the results of the freeing port in air, Figure 44 in Yoshino's paper, shows a final volume of approximately -0.5 litres of water in the tank. It must be emphasized that the comparisons cannot, therefore, be too exact. The results of the tests performed with the freeing ports submerged may be compared to values that are predicted by the findings in this research. Again, it must be emphasized that because the freeing port is submerged, fairly narrow and not at the bottom of the tank, the comparisons should not be expected to yield exact correlations.  5  Chapter 1: Introduction  1.2.3  Other Literature There were other papers that were called upon during this research, but none of  them are directly related to the subject of scale effects of freeing ports. Rules governing the size and location of freeing ports were found in The American Bureau of Shipping's "Rules for Building and Classing Steel Vessels", but these requirements are general and are not related to model testing whatsoever. It is hoped that this research w i l l ultimately aid in the making of such rules in the future. Some other areas of research were uncovered that may be related to discharge through small holes. Experimentation in the area of flow through cracks can be related to the area of this thesis, however the material that was found was not within the scope of the thesis and subsequently are not referred to in this thesis.  6  Chapter 2 Background and Governing Equations  2.1  Bernoulli's  Equation  Bernoulli's Equation is an equation that relates pressure, velocity and hydrostatic head. While it can not be applied to arrive at exact values due to the viscosity of the water and the effect of the vena contracta, it may be used to generate a general relationship between the velocity of the discharged water, the height of the water on the deck and any pressures that may be acting on the flow. The general form of the equation is  p, -  V , - V., 2  P l  +  2  + g (z, - Z j )  =  0  where p, and p are the initial and final pressures, V, and V are the initial and final fluid 2  2  velocities and z, and z, are the initial and final fluid elevations as shown in Figure 2.1.  Figure 2.1  Schematic of Flow Through Freeing Port  7  Chapter 2:  Background and Governing Equations  In the case of water that is trapped on the deck of a ship, this equation may be simplified greatly. Point 1 is on the surface of the water that is trapped on the deck. Point 2 is at the opening of the freeing port, where the water is exiting. The first terms in the equation are the pressure terms. While theoretically there is a difference in atmospheric pressure at the two different heights, the difference is so small that it can be neglected. This means that the pressure terms drop out entirely. The velocity terms are the next terms in the equation, and we are concerned with the value of V , so it must 2  remain in the equation. Because of the nature of the problem, with a large amount of water draining through a relatively small hole,the term V , is going to be much smaller than V , although not zero. While one might doubt that this term V[ can be neglected, it 2  appears that it may be, since the velocity terms are squared, making the difference between V , and V 2  2 2  greater still. For this reason, the velocity term V , may be  neglected. This leaves only the height terms of the equation to be addressed. The difference between z and z, is the hydrostatic head, and we are most certainly interested 2  in this value and not able to neglect it. The final result of this analysis is a greatly simplified equation for the flow through the freeing port. If we let the hydrostatic head which is the difference between the two z values be denoted by the variable H , the equation  + g (z - z,) = 0  -y  2  V = V2TH  can be simplified into its final form  (2.1)  2  8  Chapter 2: Background and Governing Equations  While this equation is not valid for predicting the true discharge from the freeing port due to the losses that were explained earlier, it does give us a relationship between the discharge velocity and the hydrostatic head that w i l l be useful in the future analysis and shall be referred to throughout this chapter.  2.2  Dimensional Analysis of Problem This problem may be broken down into a relationship of significant variables that  may affect this problem. Through an analysis of the dimensions of all of the different variables that are expected to govern the flow, dimensionless groups may be formed that w i l l be significant to the performance of the freeing ports.  2.2.1 The Important Variables The variables that could potentially affect the flow through the model freeing port are as shown in Table 2.1  Symbol  Dimension  Gravity  g  [L / T ]  Freeing Port Height  d  Freeing Port Width  b  [L]  Fluid Density  P  [ M / V]  Variable  2  [M/LT]  Absolute Viscosity Surface Tension  a  [M / T j  Hydrostatic Head  H  [L]  Freeing Port Wall Thickness  t  [LI  2  Table 2.1 Important Variables in Discharge  9  Chapter 2:  2.2.2  Background and Governing Equations  Formation of Dimensionless Groups In forming dimensionless groups, we must consider the number of fundamental  dimensions and variables. The Buckingham Pi method of dimensional analysis for three fundamental variables, length, mass and time, and eight variables that are expected to play a role in the freeing port discharge give five dimensionless groups formed. The hole height, d , the fluid density, p and gravity, g can be selected as the three independent variables as they together contain all three of the fundamental dimensions. Group 1 contains d, p , g and b, the hole width. These can be grouped such that  fjal  ga2  pa3 b  =  a 4  q,  Groups 2, through 5 can be similarly formed such that  d  b l  d  c l  d  di  d  e l  g  b 2  g  c 2  g  d2  g  e 2  p  p  p  b  c  e  3  = q  b 4  o^ = q  2  4  3  d3  p  u  3  H  i4  t  e4  =  3  q and  = q  4  5  A n analysis of the matrices that these equations form gives the results a l = -1, a2 = 0, a3 = 0, a4 = 1 b l = 1 . 5 , b 2 = 0.5,b3 = l , b 4 = -l c l = 2, c2 = 1, c3 = 1, c4 = -1 d l = - 1 , d2 = 0, d3 = 0, d4 = 1 and  10  Chapter 2: Background and Governing Equations  el = -1, e2 = 0, e3 = 0, e4 = 1  These in turn can be used to form the five dimensionless groups that must be considered to be significant to this problem. The groups can be simplified to the forms shown below. d  d'- pVg 5  q, = u  q =  Pgd  2  3  ^  =  *  =  H d t  2.2.3  d  Analysis of Dimensionless Groups  The first group, q, is the ratio of the hole height to the hole width. This term is important since it will affect the shape of the jet if one were looking down on it from above. Due to the narrowing of the jet, or vena contracta, flow could be reduced through the freeing port, particularly on narrow ones. It is hoped that if the width of the freeing port is significantly larger than the height of the freeing port, these effects should not come into play. 11  Chapter 2:  Background and Governing Equations  The second group, q may not be immediately recognizable, but if we call upon 2  the relationship that was derived in section 2.1, we will recognize that q may be written 2  q =  p  A/g~d d  ,— Recognizing A / g d as a velocity term, we may  2  decide to replace the d under the radical with H since they are both length terms and we know that the head, H drives the velocity. This will give us the much more familiar form of the Reynolds Number equation.  iu - Re =  Vd  p  The third term, q , can be rearranged much as the second term was to get it into 3  the more familiar form of the Weber Number. The gravitational term g and the length term d can be rearranged to the form  n j  o  i f the length term d under the radical is replaced by H as was done with the group q „ the entire term -\/g d can be replaced by a velocity squared term, or V . This w i l l 2  2  2  put the equation into the more familiar form of the Weber Number.  p  q = We = ~  V d 2  12  Chapter 2: Background and Governing Equations  The fourth group, q appears to be a simple ratio of hydrostatic head to hole 4  height. Using a similar method of rearrangement that was performed on groups 2 and 3, this may be arranged into the more familiar equation for Froude Number. Multiplying the top and the bottom of the equation by gravity, g, the top of the fraction becomes the square of the familiar velocity term -^g H. The square root of the top and bottom of the fraction leave us with the recognizable form  The final group, q may be left as it is. It is the ratio of the thickness of the wall 3  that the freeing port passes through to the height of the freeing port. It is suspected that this would play a role in the boundary layer development within the freeing port. To avoid complication, this term will be kept as a constant throughout all of the experimental work.  2.3  Discharge Equation and Discharge Coefficient Now that an understanding of the dimensionless terms affecting the flow has been  established, along with a general relationship between head and velocity, it is time to look at the actual equation that is used to determine the discharge through a freeing port.  13  Chapter 2:  2.3.1  Background and Governing Equations  The Vena Contracta When water flows through any hole in a flat plane surface the water that is  approaching the hole will not be flowing perpendicular to the plane. The water is observed to flow somewhat radially towards the hole. That is to say that the water w i l l be flowing generally towards the hole from any point. A s a result, the water that is approaching the centre of the hole will be flowing essentially perpendicular to the hole, while water near the edges w i l l have a component of flow parallel to the plane. Due to the momentum of the water particles, they cannot change direction instantaneously at the hole to flow straight through it. The component of the velocity parallel to the plane is not entirely lost, and as a result, the area of the resulting jet just outside the hole is not the same as the area of the hole itself as can be clearly seen in Figure 2.3.  Figure 2.3 Jet Shape and Dimensions  14  Chapter 2: Background and Governing Equations  Given a hole of dimensions d high and b wide, the resulting jet from the hole will have the dimensions d and b' where each is less than the maximum possible values of d 1  and b. This contraction will result in a reduced area of the jet. A contraction coefficient, Cc may be used to express the ratio of the jet area to the hole area. Due to the decreased hole dimensions, the head will range from H'-d' at the top to FT at the bottom, as opposed to H-d to H as would be expected in the ideal case. As was established earlier, the general formula for velocity is V = ^2 g H as seen in equation 2.1. If a velocity coefficient, G> is introduced as a coefficient of the actual velocity over the ideal velocity through any part of the hole, we may write for any horizontal slice of thickness 6y through the hole, (2.2)  Q = Cv b' Sy ^2gy  If this expression is integrated over the entire area of the jet, we must integrate over the y range of H'-d to H'. This will give us the flow through the jet. 1  H'  Q = Cb«>/2g  JVydy  (2.3)  H'-d'  (2-4)  = | C, bS/fg" { (H') - (H' - d ' ) 3 / 3/2  2  }  (2.5)  Since the terms H', b' and d' vary and are difficult to measure, the best solution, to this problem would be to write an equation for a term Cd, the discharge coefficient,  Chapter 2: Background and Governing Equations  which would incorporate Cv and G allow us to relate the 'prime' values to the 'non prime' values. We may write  b' ( H' ' - (H'-d') ' ) 3  Cd  =  G —) b( H  2  3 2  (2.6)  1 3 / 2  - (H-d) ' ) 3 2  If we now substitute (2.6) into (2.5) we are left with  Q =\  Cd  b yjlg  ( H ' - (H - d) ' ) 3  2  3  2  (2.7)  An alternative and perhaps simpler formula could incorporate the Cd value and rather than integrate over the entire area of the hole, we could consider the head over the entire hole as being equal to that at the hole's centroid. Our simple relationship that was originally found to be V = A/2~g~H Can be simply adjusted to the form  \ d" Q = G / b d * v /2g 2g H-2  (2.8)  This equation (2.8) is much simpler in terms of one trying to solve for any of the variables in the equation. To determine the accuracy of this assumption, one may consider a ratio of equation (2.7) over (2.8). Table 2.2 shows the relationship between the two different equations. It shows that while neither of these equations is valid if the 16  Chapter 2: Background and Governing Equations  water is not higher than the hole, at a water depth of 1.5 times the hole height the simpler equation (2.8) is within 1.10 percent of the expected value, and at five times the hole height, the error is within l/20th of one percent, or for all intents and purposes, they can be considered equal.  Water Depth  Eqn (3) / Cd b  Eqn (4) / Cd b  % Error  0.5d 1.0d 1.5d 2.0d 3.0d 4.0d 5.0d lO.Od  Undefined 2.9530 4.3809 5.3993 6.2477 6.99.18 9.3914 13.6509  0.0000 3.1321 4.4294 5.4249 6.2642 7.0036 9.3963 13.6525  5.72 1.10 0.47 0.26 0.1.7 0.05 0.01  Table 2.2 Comparison of Discharge Equations  2.3.2  Equation tor Discharge Coefficient  N o w that two equations for the discharge from a hole in a vertical plane have been developed, one equation must be selected and rearranged to give us an equation that may be used for Cd from here on in. l f one were looking at the general case, it would be wise to consider the first of the two equations generated, equation (2.7). Since, however, we are going to be looking at water depths that are always greater than the hole height and up to twelve times the hole height, equation (2.8) may be rearranged to give us an equation for discharge coefficient. We may write:  17  Chapter 2: Background and Governing Equations  Q  Cd = bd  2g  (2.9) H  "2.  This equation will provide an accurate measure of Cd, and is i n a simpler form than the other equation, where it is much harder to solve for any one of the variables.  18  Chapter 3 Experimentation  A l l of the experimentation was performed at British Columbia Research's Ocean Engineering Centre. The Ocean Engineering Centre has a towing tank, a ship manoeuvering basin and a shallow water towing tank which is under construction. The towing tank was not used for any model tests, but rather water was drawn off of it and returned to it as it was the most readily available large source of water.  3.1  3.1.1  Experimental Apparatus  The Test Vessel For all of the tests that were performed, a rectangular test vessel made of  Plexiglass was used. The vessel is 90 centimetres long, 25 centimetres high and 30 centimetres wide. The water was allowed to drain from one end of the tank only, through slots in interchangeable aluminum plates to allow for different sizes of slots as seen in Figure 3.1. On most boats, the freeing ports are located such that the bottom of the slot is flush with the deck. This was achieved by having the bottom edge of the slots in the interchangeable plate a fixed distance from the bottom of the tank. Once the aluminum plate was sealed in place, a false bottom was set into the tank so that the bottom edge of the slot was aligned with the surface of the false bottom. T w o false bottoms were used, one that extended the length of the tank and one that extended half of the length of the tank. The wave probe and type of test being performed dictated which bottom would have to be used.  •9  Chapter 3: Experimentation  Figure 3.1 Test Vessel Dimensions  For all of the tests that were performed, it was necessary to measure the water depth accurately so that the water could be kept at a desired depth, or for calibration of the wave probe. This was achieved with a hook gauge with a vernier scale that could be set up on top of the tank at any point. This typically meant that the hook gauge would be set up as close as possible to the end of the test vessel with the freeing port.  3.1.2  Freeing Port Slots The freeing port slots were incorporated into the test vessel by means of  interchangeable aluminum plates that had different sizes of slots in them. Slots of 1, 1.5, 2, 2.5, 3, 4, 5, 7, 9, 11, 13, and 15 millimetres were machined. In each case, the slots were made long, so that the length dimension, b was typically 240 to 250 millimetres, such that the slot covered most of the width of the test vessel. By making the slots long,  20  Chapter 3: Experimentation  the b/d ratio was kept large enough that the flow could be considered to be two dimensional through the freeing port slots.  1/8" A l Water  " Plate  Level H  Yl L  d  t False  Bottom  Tank  Bottom  Figure 3.2  Seal  Side V i e w of the Tank and Slot Configuration  The different slots were made of 1/8 th inch aluminum plate. Figure 3.2 illustrates how the plates were incorporated into the tank. So as not to affect possible boundary layer development within the slot, however, it was necessary to machine the slots such that at the exit the wall thickness was one tenth of the height of the slot, or t/d equaled 1/10. O n the slots ranging from 15 to 5 millimetres, this was achieved by machining the slot first, and then milling the outside of the slot at an oblique angle as seen in Figure 3.2 such that the water did not encounter this machined surface upon exit. The 4 millimetre slot was too small for this method, so instead an initial pass from the outside of the plate with a wide end mill was employed to reduce the thickness of a  21  Chapter 3: Experimentation  section of the plate to 0.4 millimetres. A 4 millimetre slot was then machined through this thinned section.  l/8th" A l  Plate  [  Brass / S h i m - t  V f  H  Stock  \  1 '1.  V,  Figure 3.3 Configuration for 1 to 3 Millimetre Slots  An  attempt was made to use the same method of thinning a section of the plate  before milling the slot for the slots ranging from 1 to 3 millimetres, but it was found that at the thickness required to keep the t/d ratio at 1/10, the material was too soft and flexible to be machined cleanly. A different approach to the construction of these slots was found. A n oversized slot was machined through the plate at its full thickness of 1/8 th of an inch. The bottom side of the long slot was left square and smooth, to essentially be an extension of the false bottom of the tank. Since a certain wall thickness was desired, brass shim stock of the required wall thickness was glued to the outside of the 22  Chapter 3:  Experimentation  plate to reduce the oversized slot down to the required height. The shim stock was also glued at the edges of the slots so as to avoid edge effects from the 1/Sth inch plate at the edges. The configuration of this slot is shown in Figure 3.3. It would appear that the geometry of the slot has been changed for the smaller slot and it has. It was assumed that these effects would be minimal, and tests were performed to check the validity of this assumption.  3.1.3  Apparatus for Steady State Tests The steady state tests with a constant head utilized the test vessel that was  described in section 3.1.1 with a few modifications. A t the end of the tank opposite where the freeing port slot was, a sluice gate arrangement was set up to control the water depth in the tank. While water was draining through the freeing port slot, aluminum strips could be added or taken from the sluice gate at the other end until the desired head was maintained. The discharge through the freeing port end of the tank could then be collected over a period of time and weighed to determine the discharge rate. The water that was being supplied to the tank came from one of two possible sources and was fed in at the end of the tank opposite the freeing port. Baffles were set up such that the water being fed in would be completely calm when it was in the region of the freeing port. The smaller slots did not require a large volume of water, and had the water supplied to them from a simple garden hose whose discharge could be adjusted. Because of the wide range of flow rates experienced, with the largest discharge rate being over 420 times that of the smallest discharge rate, a pump with a capacity of approximately 5 litres per second was employed for the testing of the larger slots. This  23  Chapter 3: Experimentation  discharge was not adjusted, but rather a partial dam was set up between the baffles and the area where the water flowed in, to limit the waterflowingthrough the baffles. Figure 3.4 shows the general arrangement for the steady state flow. The pump would be replaced by a garden hose for smaller freeing port discharges. Pump Outlet.  Test Vessel  Baffles  Sluice Gate  Calm Water  D i s c h a r g e of  E x c e s s Water  _2_  Freeing Port Discharge  ExceSS Water R e t u r n  I Pump  Figure 3.4 Setup For Steady State Discharge Tests  The discharge from the freeing port was allowed to drain back into the towing tank, which is where it was drawn off of. When discharge was measured, water that was collected and weighed was collected in either a 2 gallon bucket or a 30 gallon plastic container. A chute was set in place for a desired period of time and then removed so as to allow for the diversion of the water flow to the bucket in which the measured discharge would be collected.  24  Chapter 3: Experimentation  3.1.4 Apparatus for Unsteady State Tests The unsteady state tests with a declining head were performed in the same test vessel as all of the other tests, with a few modifications specific to the tests performed. The back end of the tank was sealed off and the sluice gate was removed. Because there was no water being added to the tank, the baffles were removed to allow for free flow from all parts of the tank to the freeing port slot. A capacitance wave probe was set up near the centre of the tank to provide a measure of the volume of water within the tank at any given time. Because of the capacitance probe's inability to measure depths in shallow water, the false bottom in the tank was ended near the centre of the tank, such that the probe would be sitting in about 1.2 centimetres of water when the water at the slot was at zero. The depth measured was used as a measure of both head and volume within the tank. The general layout of the apparatus is shown in Figure 3.5.  Capacitance Wave Probe  Figure 3.5 Unsteady State Discharge Test Apparatus  Chapter 3: Experimentation  3.1.5 Apparatus for Sloshing Tests The sloshing tests were performed much as the unsteady state tests were, except that motion of the test vessel was induced and different data acquisition was used. The test vessel was mounted on a platform that had an axis of rotation at the centre, perpendicular to the longest tank dimension. The platform was mounted on a frame, with an actuator driven by a stepper motor attached 25 centimetres from the axis of rotation used to rock the platform. The stepper motor gets its signal from a translator which in turn gets its signal from a computer controlled control module. The discharge and head measurements for the sloshing tests came from two duplicate sources. Each test was recorded on video tape, while also being monitored by the data acquisition system. The capacitance wave probe was found to be inadequate for a number of reasons. The capacitance wave probe does not function in very shallow water and the false bottom covering only half of the bottom of the tank would disrupt the sloshing motion. Additionally, a capacitance probe would obstruct the flow if set up close to the freeing port. A Kennek wave probe which utilizes an electrode in the water and a probe that follows the water surface to complete the electrical circuit was set up at the mouth of the freeing port to measure the head present at the slot. The signal from this was sent to a DT 2801 data acquisition board in one of the computers. The data acquisition board was also sent a signal from a load cell that was used to measure the discharge. Water was caught and channelled away from the control apparatus and allowed to flow into a bucket suspended from the load cell. The discharge weight was sampled at the same rate as the head height, although it was expected that with the water taking a varying amount of time to reach the bucket, and with its impact momentum,  26  Chapter 3: Experimentation  instantaneous discharge rates would be somewhat inaccurate. Figure 3.6 illustrates the general setup of the apparatus used in the sloshing tests.  Depth Probe Test Vessel  Conditioner  n  Discharge  |  IBM PC  IBM PC  o o o o oo Translator  Figure 3.6  J  Controller  Sloshing Discharge Test Apparatus  While one of the computers was being used for the data acquisition and the other to control the sloshing tank, the water draining from the tank was recorded on video for later analysis. The camera was set up with a side view of the tank so that the motion of the water sloshing could be recorded. The water had dye added to it, and because of the two dimensional nature of the sloshing, the water volume visible on the side of the tank could be interpreted to give both head height at the freeing port and volume remaining in the tank at any instant. The video had time recorded on it, and by utilizing a frame grabber card in the computer and a C program that was written to be used with the frame grabber, it was possible to use the mouse to digitize the picture to give values for the angle of the tank and the coordinates of the water surface for any point in time. The  Chapter 3:  Expert in en ta tion  resulting values were used as a check of the values collected by the data acquisition system, and they were also used to allow plots of the motion of the water surface within the tank.  3.2  Testing  Procedures  As was mentioned earlier, there were three different types of general tests that were done. The first tests were the steady state tests, and these incorporated discharge with a constant head achieved through having the water resupplied to the tank. While later tests were performed with both fresh water and soapy water to observe the effect of changing the surface tension, the steady state tests were not possible with the soapy water. The main reason for this was the pump system used. While the pump put out too much flow for the low discharge tests and a garden hose had to be used, running soapy water through the pump for the higher discharge tests would have resulted in excessive suds possibly flooding the work area generated by the high velocity of the incoming flow. The unsteady state tests with a declining head were performed with both fresh and soapy water. Because of the lack of data from the previous tests for the soapy water, the results of the soapy water were used mostly in a qualitative sense. Finally, unsteady tests with sloshing were performed with fresh water to observe the effect that sloshing has on discharge from the test vessel.  28  Chapter 3: Experimentation  3.2.1 Steady State Test Procedure The steady state tests use the apparatus that was described in section 3.1.3. Because of the wide range of discharges that were encountered, the water supply and collection and sampling time had to be adjusted depending on the discharge rate from the vessel. For slots in the 1 to 3 millimetre range, the water was supplied from a garden hose into the far end of the tank. Water from the freeing port could be collected in a two gallon bucket held under the slot for anywhere from 30 to 1.20 seconds depending on the discharge rate. The weight of the water was then weighed on a scale accurate to within less than one gram. The large slots typically had a much larger discharge rate, and the discharge over a reasonable time could not be collected in a small bucket. A much larger 30 gallon bucket was set up on a large scale for these measurements. A chute was used to divert the flow from the drain to the bucket for a given length of time. Again, the discharge could be determined by the weight collected and the time over which it was collected. To check the repeatability of the measurements, tests were performed at least three times each. While two tests would be performed while the head was at a given level, the discharge at other head heights would also be measured before returning to the original head height. This was done to eliminate any bias that could have been created by having the head set inaccurately. Discharge values that were taken were compared to previous runs, and if any discrepancies were noted, additional tests were performed.  29  Chapter 3: Experimentation  3.2.2  Unsteady State Test Procedures Section 3.1.4 describes the apparatus that was used for the unsteady state  declining head tests. In these tests, the vessel was filled with water to a predetermined level with the freeing port closed and left until the water was calm. In the initial tests that were performed, the slot was closed off with duct tape, although later, with the introduction of the soapy water, it became necessary to build a door hinged from the top with a gasket that could be sealed by cantilevering the door closed with weights. In either case, the data acquisition that was used to measure the water depth in the tank was started, and then the freeing port was opened as quickly as possible, to essentially give an instantaneous opening. Before each test was run, the capacitance wave probe was calibrated as the test vessel was filled in increments of two slot heights. Each test was vidoetaped for later analysis. Particular attention was paid to the jet of water exiting the test vessel. The camera was set up such that a side view of the tank would give an accurate measure of head as well as a view of the shape of the jet.  3.2.3  Sloshing Test Procedures The apparatus used in the sloshing tests is described in section 3.1.5. Two initial  water depths were considered, 3 and 6 times the slot heights. Because of the number of possible variables in these tests, it was decided that rather than testing every slot, three slot sizes would be concentrated on. The 1.5 and 3 millimetre slots were selected as this is the region where one expects to find the greatest scale effects, and the 11 millimetre slot was chosen as being in a region that would not be under the same influences as the two much smaller slots. For each slot and each depth, at least two runs were performed  30  Chapter 3:  Experimentation  for each of three different rolling motions. These motions were limited by the capabilities of the sloshing tank, and the three that were decided on were a period of 2 seconds and a peak to peak roll amplitude of 2 degrees, a period of 9 seconds and a peakto-peak roll amplitude 2 degrees, and a period of 9 seconds and a peak-to-peak roll amplitude of 10 degrees. The two periods were chosen such that the larger one would not be an integer multiple of the smaller one to ensure different wave motions within the tank. The first part of each test involved calibration of the data acquisition system. Once the tank had been returned to horizontal after the previous run, the tank was filled up incrementally. Depths were measured with the vernier scale depth gauge to be used in calibration of the Kennek wave probe. The load cell also had to be calibrated and this was done using lead weights covering the expected weight range. With the calibration completed, three things had to be done simultaneously. The freeing port was opened by lifting the weights holding the door shut using a rope and pulley system. At the same time, the data acquisition system was started. These two functions both were performed approximately one half of a second after the button was pushed to start the tank motion. This lag was the time that it took for the motion of the tank to begin. It was found that this time lag could be estimated and accounted for in the timing of the start of the run. A later analysis of the video confirmed this.  31  Chapter 4 Experimental Results and Discussion  4.1  Steady State Discharge Test Results The steady state discharge tests were done to determine the discharge coefficients  for the various slot sizes and heads that were expected to be encountered during model testing. The discharge coefficient was calculated according to the equation in section 2.3.2. Discharge coefficients ranging between 0.32 and 0.98 were recorded for the different slot sizes and heads and are plotted in Figure 4.1. For the purpose of comparing the measured values and ease of analysis, curves were fitted to the data points and are  Measured Steady State Discharge Coefficients  Cd 0.4  A  0.2 ' 0  J  '  0  2  1  1  4  '  1  6  8  10  Water Depth / Slot Height ( H „ ) d  Slot Heighis (d)  " 15mm 0 4mm  ^ 13mm X 3mm  A  11 nun  " 9mm Q 2.5mm A 2mm  * 7mm ^ 1.5mm  >K 5mm ^ I mm  Figure 4.1 Measured Discharge Coefficients  32  1  12  Chapter 4:  Experimental Results and Discussion  shown in Figure 4.2. The curves that were fitted are explained in sections 4.1.2 to 4.1.8. The curves of Cd versus head height that were plotted showed a general trend from a Cd value of 0.81 at a head height of two slot heights to a Cd value of 0.70 at a depth of twelve slot heights for the slots 7 millimetres high and larger. The slots below 7mm high got progressively further from this general curve as the slot size got smaller.  Steady State Discharge Coefficient Curves  0.2 0.1 0  \  1  0  2  1  1  1  1  1  4  6  8  10  12  Water Depth / Slot Height  Figure 4.2  (H„ ) d  Computed curves for Discharge Coefficients  For the smaller slot sizes, the low heads give reduced Cd values while higher heads yield higher Cd values than expected. The smallest slot tested, the 1 millimetre slot, shows this effect the greatest, while the 5 millimetre slot's deviation from the general curve is small and only at the lowest heads.  33  Chapter 4:Experimental Results and Discussion  4.1.1  Curve Fitting of Results If one is going lo be able to utilize the results of the steady state results in a  control algorithm, it would be useful to express the discharge coefficient in terms of head by means of an equation that is valid over the region that was considered here. Most of the curves either are increasing or decreasing with head, but they appear to be reaching an asymptote as the head gets higher. As a result, the expressions for Cd that were calculated generally were an exponential relationship that had been shifted in the y direction to align the asymptote with the measured G/s at the higher heads. A few exceptions were encountered such as the curve for the 4 and 5 millimetre slots. These curves were noted to have a raised hump at the lower heads. This was modelled with either an inverted parabola for the first few points or with an exponential relationship different from the rest of the curve, then the rest of the curve was modelled with the familiar exponential relationship. It must be emphasized that these curves cannot be used to predict the value of Cd outside of the region of 2 to 12 slot heights with any expectation of valid results. At very low heads, Cd was difficult to measure, and this is not a region where water on deck is a concern. Heads greater than 12 slot heights should not occur on models due to the limited height of the bulwark, and therefore should not be a concern to the researcher.  4.1.2  Slots 7 to 15 Millimetres High The slot sizes tested that were above 5 millimetres in height all followed the same  general curve. The lowest head tested was 2 slot heights, and this revealed the highest  34  Chapter 4:  Experimental Results and Discussion  discharge coefficient on the curve at 0.81. The curve is fairly sharp at the beginning, with the curve dropping down to around 0.75 at 3 slot heights. Beyond that the curve becomes much more gradual, appearing to reach an asymptote value of 0.70 by the time the head has reached 12 slot heights. There was obviously some scatter at each of the points due to experimental errors. Typically, the actual measured values for specific slot sizes were always within 0.025 or 3 percent. Figure 4.3 shows the measured values of Cd for slots 7 to 15 millimetres high. The curve of best fit is also shown.  Average Discharge Coefficients for 7 to 15mm Slots 1  T  0 -I 0  1  1  2  4  1  1  6  8  Water Depth / Slot Height — Best Fit Curve ^  Figure 4.3  11 m m  |  1  10  12  (H„ ) d  15min " 9mm  1  13mm •  7mm  Discharge Coefficients for Larger Slots  4.1.3 The Five Millimetre Slot The plot of discharge coefficients versus non dimensionalized head for the 5 millimetre slot is the largest slot whose curve shows a diversion from the general curve  Chapter 4: Experimental Results and Discussion  of the larger slots. From 4 slot heights to 12, the curve fits the general curve that was found from the larger slots. In actual fact, as seen in Figure 4.2, the curve is slightly higher than the curve for the larger slots but this falls within the expected experimental error. A t a head of a little more than three slot heights, the curve starts to divert from the general curve. Just as the Cd values are starting to increase more markedly, they begin to decrease due to the first detectable scale effects.  D i s c h a r g e C o e f f i c i e n t s f o r 5mm S l o t 1• Function Changes 0.8 •  •  0.6  •  .  »  j  •  •  \  Cd 0.4 -  \  1  Cd = -0.334 x H  Cd =-.008 x (H -3.1) + .75  n d  -  | 5 3 6  + .69  2  0.2 "  o0  D1|  | . ,.. ,  2  |  4  .  .  .  .  .  j  |  |  |  6  8  10  12  Water Depth / Slot Height  (H  n i i  )  Figure 4.4 Discharge Coefficients for 5 Millimetre Slot  The drop in the value of Cd at the lowest head height does not allow for the usual shape of the Cd curve to be fitted to the data. The data points and the curve fitted are shown in Figure 4.4. This curve is the standard exponential equation for all of the points  36  Chapter 4: Experimental Results and Discussion  except between the first two points. The first two points were modelled with an inverted parabola. This curve fits the data reasonably well with a residual of 0.81  4.1.4  The Four Millimetre Slot The four millimetre slot is perhaps the largest slot that exhibits scale effects along  its entire Cd versus Head curve. The discharge coefficients curve appears to be parallel the general curve from around 4 slot heights to 12 slot heights with the value of Cd being about 0.08 higher than the general curve. Much as the five millimetre slot does, as head decreases the Cd values on the curve begin to decrease just as the curve is showing signs of increasing more rapidly as head decreases. The reduced Cd values at the low head heights are noticeable, but because of the increased Cd values throughout most of the graph, this curve only drops below the general curve for heads below about 2.5 slot heights. Depths less than 4 slot heights were modelled with a different equation from the rest of the curve. This was necessary due to the increasing and then decreasing nature of the curve. The curve can be seen in Figure 4.5 to be a close fit to the measured data.  4.1.5  The Three Millimetre Slot The three millimetre slot's Cd versus head curve shows characteristics similar to  the four and five millimetre slot except the effects are greater still. Again, for the large heads, the curve is parallel to the previous curves only this time the smoother almost flat line at the higher heads is approximately 0.12 higher. That represents a 17 percent increase in Cd for the heads between 5 and 12 slot heights compared to the values  37  Chapter 4:  Experimental Results and Discussion  without measured scale effects. As the head decreases, the Crf curve increases further still before it begins to decline at 4.5 head heights. At this point, the head declines, but it remains above the curves of the all of the larger slots except for the general curve at all times.  Discharge Coefficients for 4mm Slot Function Changes  1•  /  0.8 •  /  ^sT  4  •  •  •  i  \\  0.6 Cd  \  0.4 -  •  1  Cd=.264 x H "  •  2,057  nd  0.2  Cd  o0  =  1.318  x  H  n d  08'  -  + .778  .65  i  i  i  i  i  i  2  4  6  8  10  12  Water Depth / Slot Height  (H ) nd  Figure 4.5 Discharge Coefficients for 4 Millimetre Slot  The curve of best fit for the 3 millimetre slot shows two different regions and curve shapes. Again, it was necessary to model this curve with two different equations. An inverted parabola and a declining exponential curve were combined to generate an equation for the expected values of Cd. This curve is shown with the data points in Figure 4.6.  38  Chapter 4: Experimental Results and Discussion  Discharge Coefficients for 3mm Slot 1 *  [—*  •  0.8  *  1  0.6 • Cd Function Changes  0.4 -  \ Cd= 13.42 x Hno" -* + .83 3  0.2 -  o 0  Cd = -.019 x (H -4.5) +.863 2  1  1  4  2  1  6  Water Depth /Slot Height  Figure 4.6  4.1.6  1  nd  1  1  8  10  1  12  (H ) nd  Discharge Coefficients for the 3 Millimetre Slot  The 2.5 Millimetre Slot The 2.5 millimetre slot exhibits Cd characteristics very similar to the 3 millimetre  slot for heads from 6 to 12 slot heights. It has slightly higher values in this range but otherwise is very similar. The greatest difference that is seen here is for heads below 6 slot heights. This is the largest slot that does not show the Cd value increasing as the head decreases. The curve does not have the characteristic hump that was shown by the 3, 4 and 5 millimetre slots at depths of around 3 slot heights. This curve is also different from the previously mentioned curves in one other way. A t a head of 2 slot heights all of the other slots that are effected by scale have a Cd value of approximately 0.75. This slot shows a Cd value of 0.7 at a depth of 2 slot heights. The value of Cd is apparently being decreased more significantly by scale effects for slots of this size. 39  Chapter 4:  Experimental Results and Discussion  Figure 4.7 shows the equation that was fitted to the data representing Cd versus water depth for the 2.5 millimetre slot. This curve is the first curve that shows scale effects and may be accurately represented by a single equation in the range of water depths of 2 to 12 slot heights. Discharge Coefficients for 2.5mm Slot 1• •  0.8 •  \ \\  0.6 Cd 0.4 "  Cd=-1.230 x H  3 n  2  2  4  a  +.841  0.2 -  u 0  2  4  6  8  Water Depth / Slot Height  ' 10  12  (H^)  Figure 4.7 Discharge Coefficients for the 2.3 Millimetre Slot  4.1.7  The Two Millimetre Slot The two millimetre slot shows a change from the previous slots in the Cd versus  head curve. This curve is similar to the 2.5 millimetre curve in that this curve is constantly increasing at a decreasing rate. While the general shape of this curve may be compared to the 2.5 millimetre curve, the magnitude may not be. This curve shows a very noticeable increase in Cd values for higher heads, and it shows a drastic decrease in Cd value for heads below 3 slot heights. The highest value of Cd that is attained on this  40  Chapter 4:  Experimental Results and Discussion  curve is 0.89 - higher than any of the curves discussed up to this point, while the value of Cd for the lowest head measured drops from a typical value of 0.75 down to 0.34. That is a value of less than half of what one would expect.  Discharge Coefficients for 2mm Slot 1•  •A. 0.8  / / / /  0.6 Cd 0.4  •  \  •  Cd=-5.253 x H -3245 +.895 nd  0.2  0 0  i  i  i  i  i  2  4  6  8  10  Water Depth / Slot Height  12  (H^,)  Figure 4.8 Discharge Coefficients for the 2 Millimetre Slot  Again, the curve that was fitted to the data points was an exponential equation. A s seen in Figure 4.8, the curve is a very good fit to the data points. A single equation is used to generate this curve. It may be noted that the last two points are both below the curve that was calculated. If this curve was extended to much larger heads it could become a problem, but for the region considered, this curve fit is adequate, considering the other inaccuracies in the testing and the relatively small error in this curve.  41  Chapter 4:  4.1.8  Experimental Results and Discussion  The 1.5 Millimetre Slot The 1.5 millimetre slot is quite similar to the 2 millimetre slot with a continued  progression of the shape of the curve. The actual values of the highest and lowest Cd values are quite similar to those of the 2 millimetre slot. The differences are that the highest Cd value is a little higher, the lowest Cd value is a little lower and that the curve is a little more gradual. At the smaller head heights, the value of Cd does not increase with head at quite as fast a rate. It would appear that the effects of scale, both those at the high and low heads have been drawn to the right on the graph. There is likely a little more scatter in the measured values of Cd due to the increased difficulty in measuring the water depths as the differences become smaller. Despite this difference, the equation that was generated and plotted in Figure 4.9 does show a good fit with the recorded data points. The points do not lend themselves to quite as good a fit as the previous curves, but again, the curve fit is acceptable. Discharge Coefficients for 1.5mm Slot 1  T - •  0.8 0.6 -  Cd 0.4 0.2 " 0 0  2  4  6  Water Depth / Slot Height  8  10  (H ) nfl  Figure 4.9 Discharge Coefficients for 1.5 Millimetre Slot 42  12  Chapter 4:  4.1.9  Experimental Results and Discussion  The One Millimetre Slot The one millimetre slot was the smallest slot tested, and as was stated earlier, it  showed the greatest scale effects. The nature of the curve can be compared to the 1.5 and 2 millimetre slot. In this case, however, the curve is even more gradual and the highest value of Cd is not approached until a head height of 8 times the slot height. In addition, the highest value of Cd is 0.98 - this highest value that was recorded.  Discharge Coefficients for 1mm Slot  Cd  0.2 -  Q 4  1  0  2  '  1  1  4  6  Water Depth / Slot Height  r-  8  1  1  10  12  (H ) nd  Figure 4.10 Discharge Coefficients for the 1 Millimetre Slot  This data was perhaps the hardest to fit a good curve or equation to. T w o equations would not give a much better fit due to the nature of the point scatter. It is likely that again, there is increased error due to the difficulty in maintaining a steady head accurate to well under one millimetre. Figure 4.10 shows the data points and the  43  Chapter 4: Experimental Results and Discussion  equation that was fitted to them. The increase in Cd from 6 to 8 slot heights is probably not just experimental error, but the curve fitted is the closest that a curve may come without having a very strange and unlikely shape to it.  4.2  Discussion  of Steady State  Results  If one were to generalize the results of the determination of the steady state discharge coefficients they would be as follows. Slots 7 millimetres high and higher do not exhibit noticeable scale effects. The first detectable scale effects manifest themselves by lowering Cd slightly at the low heads. A s the slot size is decreased further from 5 millimetres, Cd is lowered at the lowest heads and is higher than would be expected at the higher heads. Clearly there must be more than one scale effect acting here to both raise and lower the value of Cd for a given slot size and varying head.  4.2.1  Lowering of Discharge Coefficient at Low Heads At slot heights of 5 millimetres and less, there is a noticeable reduction in Cd  compared to what the full scale curves would predict, but only for small heads anywhere from below 4 to below 2.5 slot heights, depending on the slot. While this effect may not be attributed to any one cause, both viscous and surface tension could potentially play roles in this discharge reduction.  4.2.1.1  Reduced  Discharge  due to Surface  Tension  Surface tension, or Weber Number is likely the greatest cause of the reduced Cd values at the low heads. This may be explained in terms of attracting forces between the  44  Chapter 4:  Experimental Results and Discussion  water and the material of the slot through which it is passing. The attractive force between the water and the material of the slot is dependant on the material used. Two materials were used in this experimentation, and examination of the contact angles of a droplet of water on the surface of the material indicated that the attractive forces between the brass and aluminum used and the water are quite similar. The units of the surface tension are kg/s , or [M/T ]. If one is to consider a long 2  2  slot of length b and dimensions [L] with water issuing from it, the multiplication of the surface tension coefficient, o by the length b will yield a term of dimensions [ML/T ]. 2  These dimensions are recognizable as a force term. What this means is that there will be a force attracting the water to the entire length of the slot at both the top and bottom surface of the jet coming from the slot. If the attracting force per unit length is considered, the magnitude of this force will be a function of the materials and the geometry of the slot and the liquid flowing through the slot. The force per unit length will not, however, be affected by the water depth and the height of the slot. This force does create a pressure opposing the flow, and this is affected by the slot size. If one considers a long, essentially two dimensional slot, it is going to have an attracting force between the edge of the slot and both the top and bottom of exiting jet. If the two forces are denoted F, the flow, P  opp  and F  bottom  , as shown in Figure 4.11, a pressure opposing  may be calculated according to the equation  F r  +F  _  top T  opp"  b  45  hottom d  , {  . }  Chapter 4: Experimental Results and Discussion  The magnitude of the opposing pressure, P interest is the magnitude of P  opp  is not significant alone. What is of  relative to the pressure from the hydrostatic pressure, P  H  which is determined by the relationship  Figure 4.11 Orientation of Surface Tension Pressures and Forces  P  H  = P g H  or  P  H  = pgHndd  (4.2)  Let this ratio of the two pressures be represented by Q, which shall be calculated by  0=7*  (4.3)  It is clear that as the height of the slot gets smaller, P  is going to increase  proportionally to the decrease in d. Because we are looking at water depths which are non dimensionalized according to d, a decrease in d will also give us a decrease in H such that Hnd remains the same. The net result of this is that for a decrease in the height  46  Chapter 4:  of the slot by a factor of 6d, P  opp  Experimental Results and Discussion  increases by a factor of 6d and P decreases by a factor H  of 6d giving us an increase in Q by a factor of 6d . 2  If this opposing surface tension force is the only force acting here, then one would expect that perhaps this decrease in Cd at low heads may first occur at a certain value of Q. If the constant terms are removed from the equation for Q, we are left with the equation  This equation supports the earlier derivation which indicated that a decrease in slot size by a factor of 6d will increase Q by a factor Sd . 2  The first decrease in Cd that was detected was with the 5 millimetre slot at a head of about 3 slot heights. This translates into an Q' value of approximately 13,000 nr . 2  The 1 millimetre slot shows a decrease in Cd at around 8 slot heights, giving an Q value 1  of 125,000, while examination of an intermediate line such as the 3 millimetre slot gives us an Q' value of 24,700. Clearly, these values do not suggest a constant value of Q' for the region where Cd begins to drop below the expected value. This suggests that this opposing force due to the surface tension is not the only effect acting here.  4.2.1.2 Reduced Discharge clue to Viscosity Reynolds Number is a measure of viscous forces relative to inertial forces. An examination of the graph of Cd versus Reynolds Number indicates that the values of Cd that are less than expected occur at lower values of Re. This can perhaps be attributed to  47  /  Chapter 4:  Experimental Results and Discussion  boundary layer development on the bottom of the tank and on the vertical wall above the slot. The boundary layer results since the water contacting the tank must remain stationary, giving a velocity gradient as the distance from the tank edge increases. This effectively adds a frictional force F, as shown in Figure 4.12 that is acting in the direction away from the slot. Other horizontal forces are also shown on the figure. Boundary Layer development on the wall above the slot would be much less due to the shorter distance for the boundary layer development. The water motion is being driven by the difference in hydrostatic head on each side of the element. It should be noted that the magnitude of the forces are not intended to be to scale and inclination of the surface has been intentionally exaggerated.  Incjjn^^  s | o f  APH  Flow  V  \ \  \  \  Figure 4.12 Horizontal Forces on an Element in the Test Vessel  48  Chapter 4: Experimental Results and Discussion  At low Reynolds Numbers, as are encountered with the smaller slots or low heads, the boundary layer would be laminar, which is a thinner boundary layer than a turbulent boundary layer. Additionally, the fact that the velocity is low gives us a thicker boundary layer than a higher velocity would with the boundary layer in the same regime. With a thick boundary layer on the bottom of the tank approaching a slot that is very low, there would be potential for the boundary layer to restrict the flow through the slot significantly. The action of the water within the tank may be considered as a flow over a flat plate. Recalling that there is a false bottom extending about 40 centimetres ahead of the slot, the volume flow rate divided by the vertical cross sectional area of water in the tank will give an average velocity (neglecting boundary layer effects) that one would expect the flow to be travelling at over the plate. The Blasius solution for steady two dimensional flow over a flat plate may be used to get an idea of the boundary layer growth that we may expect on the bottom of the tank. The results show that indeed the boundary layer is expected to grow to a thickness that is greater than the depth of the water in the tank. This is obviously not possible, and the assumption that there is no flow in the vertical direction will not hold near the slot. This result may be used, however, to illustrate the fact that there could be interference of the discharge through the slots from the boundary layer growth on the bottom of the tank.  49  Experimental Results and Discussion Chapter 4:  4.2.2  Raising of Discharge Coefficient at Higher Heads The smaller slots showed effects at the high heads that were more surprising and  perhaps of greater interest to the modeler than those results observed at the low heads. The discharge coefficient at the high heads was observed to raise the curve on the plot of Cd versus Hnd as the slots got smaller. The increase in Cd that the 5 millimetre slot showed can essentially be absorbed by the experimental error, but the other increases are progressively larger and cannot be ignored. These increases must be of interest to the modeler, because they suggest that the water drained from the deck of a model would be larger than the scaled amount that would be drained off of an actual boat. This could result in model tests that are optimistic if the scale effects are not considered. The increased Cd values are not as easy to explain as the decreased Cd values that were noted at the lower heads. It was observed that during the tests that showed these trends, the jet of water exiting the slot failed to separate cleanly from the bottom edge but rather adhered to the side of the tank and ran down. This may help explain the increase in the measured Cd values, since this means that there was no opposing surface tension force acting on the bottom of the slot. Recalling equation 4.1 from section 4.2.1.1, F  lop  D  "PP  =  + F  hoilom  b d  the opposing pressure is reduced by approximately half, since the force F  bottom  is not  present and would otherwise have been approximately equal to F, . This certainly  50  Chapter 4:  Experimental Results and Discussion  explains why there would be a decrease in the pressure opposing the flow, but it does not explain why the increase in measured Cd values is as great as was experienced. It was proposed in 4.2.1.1 that the ratio of the P to the value of Cd. If P  to P , or Q might be important H  is reduced by a factor of two, then one would expect that  perhaps the reduction in Cd that this opposing force induces would be halved. This would not, however be expected to raise the value of Cd significantly above the value that we expect for discharge from slots that are not affected by scale effects. If P  o p p  was  reduced to a negligible amount, and all other things being equal, the best we would expect would be that the graph of Cd versus Hnd would show that the measured Cd values would be the same as the curve for the slots from 7 to 15 millimetres. It would appear that there must be some other effect acting here.  Figure 4.13  Forces on Jet at L o w Heads  51  Chapter 4: Experimental Results and Discussion  While it was observed that there was no bottom surface on the jet that came out of small slots, it was also observed that there was a continuous free surface that extended down the outside of the tank. As established earlier, there is an attracting force between the top water surface and the top edge of the slot. There is an attraction between the water molecules on the free surface, and there is also another force that is acting on this water - gravity. Because of surface tension, there is a continuous stream of water flowing down the side with an unbroken surface as seen in Figure 4.13. Gravity is pulling this water down, and there is the possibility that this continuous vertical water surface is acting somewhat like a small siphon. The water is pulled down the vertical side by gravity, and rather than break the thin film of water which is maintained by surface tension, more water is drawn through the slot to maintain a continuous surface of water. The surface tension coefficient, o, of water is approximately 0.072 N/m. This means that over a slot one metre long, the greatest force that could be applied due to gravity via surface tension is 0.072 Newtons. Considering the problem in two dimensions only, a 1 millimetre slot with a head of 6 millimetres (H = 6) would have a nd  P of approximately 60 Pascals. On a slot 1 millimetre high, this pressure can be H  translated into a force of 0.06 N/m, which is less than the maximum possible surface tension force that the water could exert through the 'siphoning' effect. What this is suggesting is that the siphoning effect due to surface tension could be greater than the effect from the hydrostatic head! In reality, we may not experience this full tension force of 0.072 N/m, but even a portion of this could have an effect on the flow through a small slot.  52  Chapter 4: Experimental Results and Discussion  The 4 millimetre slot shows increased discharge coefficients at a head of twelve slot heights. This is the largest water depth at which this effect is evident. At the slot, a hydrostatic force of approximately 1.9 N/m is experienced. The surface tension force could only amount to about 4 percent of the hydrostatic force, but for other smaller slots and water depths, as was illustrated by the first example, it can amount to a much greater role.  4.2.3  Relating Discharge Coefficient to Dimensionless Numbers If the scale effects that were observed can be directly attributed to boundary layer  growth alone, then if one were to plot Cd versus Reynolds Number one would expect to see a trend to support this. Alternatively, a plot of Cd versus Weber Number would show a strong trend if surface tension was the cause of the scale effects.  4.2.3.1 Plotting Discharge Coefficient versus Reynolds Number  Figure 4.14 is a graph of Cd versus Reynolds Number. For this case, the equation for Reynolds Number is Re =  t  —  (4.5)  ~  The significant dimension has been chosen as the slot height, d, and the velocity of the water through the slot is expressed by Cd -\]2 g H. It can be seen that the data points from the larger holes tend to fall within a well defined region or envelope of Cd values. As the Reynolds Number decreases, the top of the 'envelope' extends to a Cd value of almost one in the region tested. At the same time, the bottom of the envelope drops out at the low Reynolds Numbers. If general curves 53  Chapter 4:  Experimental Results and Discussion  were fitted within this envelope, they would likely start at low values of Cd and rapidly increase to high values before slowly decreasing to a near constant value as the envelope gets narrower. For smaller holes and heads, not tested here, it is even possible that the top of the envelope may extend beyond the Cd value of one, which would normally be expected to be the limit.  Discharge Coefficient vs. Reynold's Number  Cd  10  100  1000  10000  100000  Reynold's Number Slot Heights  Figure 4.1.4  •  15mm  o  •  4mm  X 3mm  13mm  A  11 mm  •  2.5 mm  ™ 9mm  A 2 mm  ^ 7mm  A  1.5mm  3mm  <3>  Plot of Discharge Coefficient versus Reynolds Number  While this curve could extend beyond a Cd of one, if it did it would be because of the surface tension or siphoning effect that was described in section 4.2.2, and not Reynolds Number related forces. It is impossible to predict, however, whether at the smaller slots and heads, the opposing surface tension force and viscosity effects would be 54  Chapter 4:  Experimental Results and Discussion  stronger than the surface tension siphoning effect. The relative magnitude would dictate the position of the upper limit of the envelope, potentially raising Cd greater than one.  4.2.3.2 Plotting Discharge Coefficient versus Weber Number The plot of Cd versus the dimensional Weber Number, Figure 4.15, shows the same trend as the plot of Cd versus Reynolds Number except it has been scaled differently along the x axis. Because of the lack of a known exact value for the surface tension coefficient a, for the contact between this finish of aluminum and water, o was neglected from the calculation as it would be constant and as a result, this 'Weber Number' is not dimensionless.  Discharge Coefficient vs. Weber Number  Cd  Dimensional Weber Number, pV d 2  Slot Heights  •  15mm  O  13mm  *  s  4inm  X  3irun  t-l 2.5mm  11 mm  " 9mni  *  7mm  A 2inm  A  1.5mm  5mm ^  1mm  Figure 4.15 Plot of Discharge Coefficient versus Weber Number 55  Chapter 4:  Experimental Results and Discussion  A closer examination of the equations for Re and We reveals why the curves are similar. In both cases, the fluid has not been varied throughout the tests, and values such as p, f.i, and a are simply constants that do not affect the general shape of the graph. Reynolds Number is a function of velocity, and Weber Number is a function of velocity squared. That is the fundamental difference between the two curves and explains the variation in the x scaling between otherwise similar graphs.  3D Plot of Cd v$ Re vs We  Figure 4.16 Three Dimensional Plot of Measured Values of C d , Reynolds Number and Weber Number. The lower set of lines are the two dimensional shadow of the upper lines, projected onto the horizontal plane.  The envelope that has been approximated on this graph is much the same as the one that is shown in Figure 4.14. Again, for the same reasons outlined in the previous  56  Chapter 4: Experimental Results and Discussion  section, there is no reason to limit the upper limit of the envelope to a Cd of 1, given the potential surface tension effects at low Weber Numbers.  4.2.3.3 Plotting Discharge Coefficient versus Reynolds and Weber Number It appears that both viscosity and surface tension play a role in the scale effects of the freeing ports. Perhaps the best way to represent this is with a three dimensional plot of Cd versus Reynolds Number and Weber Number. As was stated in the previous section, however, the range tested here is not complete enough to be able to generate a complete three dimensional surface to allow one to determine a value of Cd. Extensive tests with other liquids with differing properties would be required to generate a three dimensional surface plot, but the values that were found in these tests may be plotted as lines in space as shown in Figure 4.1.6. The line that is furthest left is for the 1 millimetre slot, with each movement to the right on the graph representing the next size slot that was tested. The line that is the furthest to the right, therefore is the 15 millimetre slot. The lower set of lines are the shadow of the three dimensional lines on the Reynolds Number Weber Number plane.  4.2.4  Comparison of Results to Published Values The paper by Yoshino et al. presents measurements of discharge through a freeing  port. The only rectangular freeing port they tested is 15mm x 35.3mm. This is within the range of freeing port heights tested although it is much shorter and not of the long two dimensional nature that were tested in this research. Despite these differences and the fact that the hole was not set up so that the bottom edge of the freeing port was flush  57  Chapter 4:  Experimental Results and Discussion  with the deck, reasonable correlation exists between the measured curve presented by Yoshino et al in their "Figure 44" and the predicted curve using the measured Cd values determined in the steady state tests. Figure 4.17 shows a comparison between the measured and calculated curves. The y axis, "Remaining Volume" is based on the tank from which Yoshino's tests were performed. The tank has parallel walls, and Yoshino's initial volume of 22.5 litres corresponds to a depth of 30 centimetres, or a non dimensionalized head of 20. Results from the steady state testing that were used to try to match a predicted curve to this measured curve are only valid for water depths of 2 to 12 slot heights. For this reason, the predicted curve is started part way down at the highest head at which it would be valid. The results of this comparison of curves indicates that the edge effects or third dimension of flow may not have a strong effect on the discharge characteristics.  Comparison of Measured Discharge Rates 25 T  Initial Head 30cm (Yoshino)  20 Remaining  15  Initial Head 13.5cm (Williamson)  Volume (litres)  10  0 0  10  20  30  40  50  Time, seconds Yoshino et al: Measured, 15x35.3mm Hole Williamsont prediction from results of 2D tests, 15mm Slot  Figure 4.17 Comparison to Published Discharge Rates 58  60  Chapter 4:  Experimental Results and Discussion  Further test results presented by Yoshino et al. in their "Figure 47" and "Figure 48" present results of the same freeing port being tested while it is submerged in a cross flow flowing past the hole, parallel to its plane. Figure 4.18 shows a comparison between the predicted discharge rates and the discharge rates measured by Yoshino. It must be emphasized that the predicted curve is one that is for a hole in air, not submerged. This explains why at the lower heads there is a divergence between the measured and predicted curve. The majority of the curve, where the head is above 4 hole heights, again shows a very encouraging correlation between measured and predicted values.  Comparisons to Tests with Freeing Port Submerged 16 T  Yoshino's Tests Freeing Port Submerged in Crossflow by 10cm Freeing Port Size 15x35.3mm Yoshino, 0.8 m/s Crossflow .Yoshino, 1.6 m/s Crossflow .Williamson, Predicted from 2D Tests In Air, No Crossflow  0 0  10  20  30  40  50  Time, seconds  Figure 4.18 Comparison to Tests with Freeing Port Submerged  59  60  Chapter 4: Experimental Results and Discussion  4.3  Unsteady Discharge Tests Without Sloshing  The next phase of the testing involved measuring the decline in head as water was allowed to drain out of the test vessel without resupply. It was assumed that there is no sloshing taking place in the tank as the water was allowed to drain from the tank. Observations showed that in actual fact, there was sloshing occurring due to the rapid opening of the slot at one end of the tank. When the freeing port was suddenly opened, the water level dropped at the end of the test vessel with the hole first. The reduced water level causes the water adjacent to it to drop too, and as a result a wave was sent down the length of the test vessel as the water surface dropped over the length of the tank and it reflected off of the other end. Although the wave was observed to move back and forth in the tank, the water was considered to be not sloshing. While this is not an ideal assumption, it was necessary to differentiate between the case where there is minimal natural sloshing and the case where sloshing is induced through the rolling of the test vessel. Use of the capacitance wave probe dictated that the probe had to be set up beyond the false bottom of the tank. That meant that while the measured water depth was not exactly the same as the water depth at the hole, it did however, provide a useful measure of the remaining water volume within the test vessel.  4.3.1  Fresh Water Tests The water depth was monitored in the vessel as the water was allowed to drain  through the slot. The discharge coefficients determined in the steady state tests were then used to try to predict the discharge in the unsteady case. Starting at the initial depth of  60  Chapter 4: Experimental Results and Discussion  12 times the freeing port height, the discharge was calculated for a small time period At, and after that time period, a new depth and hence new discharge coefficient could be calculated. Plots of the predicted discharge rates compared to the actual measured discharge rates are shown for each of the slot sizes on Figures 4.19 through 4.21 and i n Appendix C , Figures C.3 to C.9. Results of tests with soapy water to reduce surface tension are also presented on some of these graphs. The soapy water tests are addressed in section 4.3.2. It should be noted that the measured experimental curve shows the effects of the wave that travels back and forth in the tank while the predicted values are calculated without considering the wave.  Unsteady Discharge Without Sloshing  Time (seconds)  Figure 4.19  Vessel Drainage Through the 2 Millimetre Slot  To examine the effect of the scale effects, each graph is also shown with the discharge curve that we would have predicted if the scale effects were not considered. 61  Chapter 4: Experimental Results and Discussion  For slots larger than 5 millimetres high, these two curves would be almost the same, and therefore are presented as one curve. For the small slot sizes, the differences between the curves predicted with and without scale effects are quite noticeable. The 5 millimetre slot has the two predicted curves getting very close to each other, and the larger slots cannot have the two predicted curves shown separately. On a graph such as Figure 4.19, for the 2 millimetre slot, there is a great difference between the curves that were predicted with and without the scale effects considered. The lines do intersect when the water depth gets very low due to the fact that for a 2 millimetre slot Cd can be either higher or lower than a full scale value, depending on the water depth. What this means is that if one were to merely measure the time for a vessel to-drain completely, the total times could be closely predicted by the full scale values of Cd, while the intermediate water depths could be significantly different, in this case by up to 18 percent. If one were to discount the scale effects, it could result in tests that suggest that there is less water on the deck of the ship than there actually is.  4.3.1.1 Unsteady Discharge Through Large Slots The plots of discharge versus time for the unsteady state flow without sloshing show very good correlation between the measured curves and the curves that were calculated using the equations that were developed for Cd, as seen in Figure 4.20, the plot for the 9 millimetre slot. Using time increments of one second, the discharge through the slot was calculated for that time period, a new water depth was calculated, and then a new discharge rate was calculated for the new water depth. This process was repeated until a water depth of two slot heights was reached. It was found that for these cases, the  62  Chapter 4:  Experimental Results and Discussion  time step of one second was adequate to produce a predicted line very close to the measured curve.  4.3.1.2 Unsteady Discharge Through Small Slots The slots that were 3 millimetres high and smaller showed a good correlation between the measured and predicted discharge curves for higher heads. A s the head declined, however, the measured discharge curves tended to level off at low heads while the predicted values dropped a little lower. It is likely that this can be attributed to the motion of the water in the tank, but examination of the characteristics of the flow rate are warranted to determine exactly what is happening here.  Unsteady Flow, 9mm Slot  Slot Height 9 mm Slot Length 255 mm Tank Length 895 mm Tank Width 300 mm No Tank Motion  0 0  5  10  15  20  25  30  35  Time (seconds) - Runs I to 4  Predicted from Steady State  Figure 4.20 Vessel Drainage Through the 9 Millimetre Slot  63  40  Chapter 4: Experimental Results and Discussion  It appears that the steady state discharge values may be a little high relative to the unsteady discharge values. When the freeing port is opened, there is an acceleration in the system that is required that was not required in the steady state tests. This acceleration could be the cause of the discrepancy, but there are other accelerations acting. As the head decreases, the flow slows down. A slowing down flow would not have the chance for the boundary layer to thicken to the same extent that it would if the slow flow was maintained for a longer period of time. This reasoning suggests that the unsteady discharge should be greater than the predicted discharge if it is to be different at all. From this analysis, it would appear that if the changing flow is significant, the greatest effect is from the initial acceleration as opposed to a long slow deceleration. It should be noted that when the slot is opened, the water level drops at the hole immediately, while the wave probe at the centre does not detect this change in head height until the wave reaches the centre of the tank. This would result in a predicted discharge that is higher than the actual discharge. The slope of the curve, however, would not make this difference visible at the.high heads as much as it would at the lower heads.  4.3.2  Soapy Water Tests Once the unsteady flow tests were completed with the fresh water, they were  repeated with soap in the water to observe the effect of changing surface tension. As was stated previously, it was not possible to perform the steady state tests with the soapy water, so the same comparisons could not be made between steady state and unsteady  64  Chapter 4:  Experimental Results and Discussion  flow could not be made with the soapy water. The soapy water does, however, give a qualitative view of the effect of different surface tension.  4.3.2.1 Scale Effects from Soapy Water at High Heads A comparison of the unsteady discharge curves for the tests with and without soap in the water show that there is no noticeable difference between discharge rates at high heads. This tends to suggest that if there is a 'siphoning' effect taking place in the smaller slots at high heads, it is not eliminated or reduced significantly by the change in surface tension caused by the addition of the soap. If the effect has been significantly reduced, then the magnitude of this siphoning effect must be quite small.  Unsteady Disharge Without Sloshing 14  T  Slot Height 3mm Slot Length 240mm Tank Length 895mm Tank Width 300mm No Tank Motion  Hnd Fresh Water Test Soapy Water Test  Predicted With / Scale Effects Considered  0 0  5  10  15  20  25  30  35  40  45  Time, seconds  Figure 4.21  Vessel Drainage Through the 3 Millimetre Slot  65  50  Chapter 4:  Experimental Results and Discussion  4.3.2.2 Scale Effects from Soapy Water at Low Heads  A comparison between the discharge curves for fresh and soapy water supports the theory proposed in 4.2.1.1 regarding the attractive force at the exit. The.soap reduces the surface tension, and as a result, the water level in the vessel is allowed to drain to a lower value after enough time has passed that the flow is not showing considerable changes in water depth. With a decreased P , the soapy water flow is not restricted to opp  the same extent as the fresh water is. This is most noticeable with the smallest slots, Figures 4.19, 4.21 and C.8 and C.9, where the relative surface tension effects would be expected to be greater. The 3 millimetre slot, Figure 4.21, shows slight differences in the level that the water is at after a significant period of time, less so than the 2 millimetre slot, Figure 4.19, and the slots larger than that show no noticeable changes in the flow characteristics with the decreased surface tension.  4.3.3  Video Analysis of Exiting Jet The original plan was to video all of the tests with the declining head such that  they could be analysed at a later date. The camera was set up such that the viewing angle would allow a two dimensional view of the jet, as well as a view of the current instantaneous head. A vertical plane of light was shined upon the jet perpendicularly such that the free surface of the jet would be visible, illuminating the shape of the jet. It was observed that the edge effects at the edge of the slot caused the otherwise two dimensional jet to narrow and wrap the edges of the jet over the top of the flatter two dimensional centre of the jet. Upon analysis it was found that the jet shape could not be seen due to distortions of the light through the curved free surface of the edge of the jet  66  Chapter 4: Experimental Results and Discussion  of water. Regardless of the orientation of the light, this problem cannot be corrected, since light changing mediums will be refracted if it does not encounter a perpendicular water surface.  4.4  Unsteady Discharge With Sloshing Induced Following the completion of the tests without induced sloshing, the tank was set  up on a rocking platform to make the test vessel rock to simulate the roll motion that one would expect of a ship rolling in rough seas. Due to limitations of the apparatus, the maximum angle of roll that was tested was 10 degrees from one peak to the other, or an amplitude of 5 degrees. While this may not appear to be a very large angle, it was observed to be enough to cause significant sloshing, such that the water was observed to pocket, or pool entirely on one side when the tank was inclined at its maximum angle.  4.4.1  Measurement of Head at Freeing Port The Kenek servo wave probe was set up such that the probe was near the  centreline of the tank and less than one centimetre from the hole. The electrode used with the probe was also set near the end of the tank, on the back wall. This allowed the electrical circuit to be completed when there was only water directly at the hole and the rest of the tank was 'dry'. The probe was observed to require a minimum water depth of approximately 1 to 2 millimetres for it to function properly. As a result, some plots of water depth versus time show the bottom of the tank as being 'dry' when in actual fact, there could have been a thin film, draining a small amount of water.  67  Chapter 4: Experimental Results and Discussion  This error is small, however, since the probe would mistakenly detect a dry deck when most of the water is at the other end of the tank and the tank is inclined away from the hole, or when the last small amount of water remaining in the tank is draining from the tank and the tank is inclined towards the hole. Neither case is expected to play a major role in the determination of a ship's stability characteristics.  Comparison of Depth Measurements 12 " to  Run  A,  ml A  -  A  Slot Length 240mm Roll Period 9 sec  A  "  8  H„  Roll Angle 5 deg  A  6 ^  Z?  7  Slot Height 3mm  A  Initial  6  A  A A  4 2  A  0  to  t:  14  16  20  Time, seconds Video Analysis  A Wave Probe  Figure 4.22 Depth Measurement Comparison for the 3 Millimetre Slot  An alternate method of measurement of head was that of measuring the image that was captured on video tape using the frame grabber card in one of the computers. This method was performed manually, indicating the tank location and water surface with the mouse, and yielded a good correlation with the wave probe. Comparisons may be seen in Figures 4.22 to 4.24 for the three different slot sizes considered. All of the 68  Chapter 4:  Experimental Results and Discussion  figures show a good correlation between the two different methods of depth measurement, with Figure 4.24, the 11 millimetre slot, showing perhaps the worst correlation. While one would perhaps expect that the larger slot would yield the best correlation between the two measurement methods, this is not the case for two reasons. Comparison of Depth Measurements Run 20 Slot Height 1.5mm  18  Slot Length 211mm  16 14  • • A  12  H„  10 8  A  Period 9 s  •  Roll Angle 5 deg  • .  Initial H „ 6 d  A •  A  A  6  A  A.  •  4 . A  2 -  A  • A  A  0 — 0  4  *  6  8  12  10  14  16  20  Time, seconds A Video Analysis  Figure 4.23  * Wave Probe  Depth Measurement Comparison for the 1.5 Millimetre Slot  While a large slot has a more easily measured head, when the head is rapidly changing the wave probe is required to move over a greater distance in a given time when it encounters a wave peak. A comparison of the peaks at 8.5 seconds in Figure 4.23, the 1.5 millimetre slot, and at 6.5 second on Figure 4.24, the 11 millimetre slot show the lag associated with measurement of large changes of head. Difference in the wave probe and optical measurement of the 11 millimetre slot is also seen in Figure 4.24 as the last thin film of water flowed down the slope of the tank and out of the freeing 69  Chapter 4:  Experimental Results and Discussion  port, at a time of about 12.5 seconds after the hole opening. This difference is because of the fact that the thin flow at the edge of the tank encounters an end wall and the velocity head is transformed into a hydrostatic pressure head. Because of the relatively large slot height adjacent to this raised head, the water was observed to flow out the edges of the slot, and not raise the water depth at the middle of the slot where the wave probe was located. In this case, both measurement techniques are inaccurate, with the best value being somewhere between the two extremes measured.  Comparison of Depth Measurements  4 •  Run 38 Slot Height Slot Lenght Period 9 s Roll Angle Initial Hnd  5  Hnd 4  11mm 257.5mm 5 deg 6  3 2 t 1  0 0  •4  A*  A  O-A  6  8  10  14  18  Time, seconds A Video Analysis  Figure 4.24  * Wave Probe  Depth Measurement Comparison for the 11 Millimetre Slot  Aside from the circumstance of the raised hydrostatic head at the edge of the tank, it is likely that the optical measurement method is more accurate than the wave probe. The movement of the wave probe was observed to have a small lag when it encountered a sudden change in water depth, and this would not occur on the video. The  70  Chapter 4: Experimental Results and Discussion  optical measurements also detect very brief peaks and troughs that can be missed by the wave probe entirely. The wave probe is however, a much more practical and instantaneous method of measuring water depth on a model, as it can be used in real time to control freeing port, without relying on a certain position and angle of viewing that is not practical in model testing.  4.4.2  Measurement of Discharge From the Test Vessel The measurement of the discharge or volume flow rate from the test vessel during  sloshing provided perhaps the greatest challenge of the work performed, and unfortunately also yielded the most inexact results. Again, both the video analysis and the data acquisition were used to try to determine instantaneous discharge rates. The bucket that the discharge was collected in was suspended from a load cell and readings from the load cell were taken by the computer on a regular basis. There are two fundamental problems with the load cell method. The first problem is that the water is dropping from a height of the depth of the bucket, and as a result, an impact force is measured along with the weight of the water. If this were the only problem, then one could account for this in the reading, but this is complicated by the time delay of the falling water and the unsteadiness of the flow. The water that exits the test vessel had to fall approximately 20 centimetres before it encountered a ramp or chute that carried the water away from the sloshing tank's frame and hardware. The water had to flow about 30 centimetres down this chute before falling to the bottom of the collection bucket that was suspended from the load cell. While the load cell did provide an accurate measure of the total water that had flowed out  71  Chapter 4: Experimental Results and Discussion  during the test, the instantaneous discharge was given too much time to reach the bucket and encountered too many obstacles that changed the apparent flow rate. The instantaneous discharge rates were not recorded accurately once the water had travelled to the bottom of the collection bucket. For those tests where the test vessel was 'dry' near the freeing port for a time during one cycle the discharge during the single cycle could be determined with reasonable accuracy. Discharge during one cycle, however, proved to be very difficult to measure. During the same video analysis process in which the tank angle and the head at the freeing port were measured, the entire water surface was digitized, and the instantaneous volume of water remaining in the tank was calculated. Frames at one second increments were digitized, and are presented in Appendix C. An example of this process is shown in Figure 4.25. If the volume in the tank was measured at two different times, the difference in measured volumes divided by the change in time would yield the discharge rate over the time period considered. While volume calculations showed some reasonable results, two measurements one second apart were sometimes observed to show a negative discharge of a significant amount when no water had been observed to be draining from the test vessel. A negative discharge is measured because the second volume calculation gave a value that was larger than the first, when one would expect it to be either the same or less. While this may be expected when there is very little water in the tank, such as at the end of the run or for small holes with small heads, it was found throughout some of the tests. The greatest detectable error in the volume measurement was typically when the tank was inclined away from the hole and a thin film of water was observed to flow  72  Chapter 4: Experimental Results and Discussion  down the slope and pool at the end of the tank where there was no hole. There was invariably a greater water volume measured when the water was pooled at the end of the tank than when it was in a thin film. There could be either one or two reasons for this. The error could be as simple as the fact that a thin film of water is not as visible and appears to be thinner than it is. The success of the previously mentioned head measurements would suggest that this is not the only problem here. Another possible explanation for the error is that when the water flow is arrested at the end of the tank, there is a hydraulic jump. Air may be entrained into the water, making the volume that is measured be not only water but water with air entrained in it. There was not a significant amount of air observed to be drawn into the flow, however, therefore it is likely that it is a combination of these two effects that is causing this error.  Run 7 Time 9 s  X Figure 4.25 Sample Water Surface Plot  4.4.3  Results of Sloshing Tests Despite the problems that were outlined in the previous two sections, there were  some useful results that came from the sloshing tests. Tests were performed on the 1.5, 3  73  Chapter 4: Experimental Results and Discussion  and J1 millimetre slots examining at least two runs for each of three different roll amplitude and period combinations for two different initial water depths.  4.4.3.1 Sloshing Tests with 1.5 Millimetre Slot Sloshing Tests were performed with the 1.5 millimetre slot with initial water depths of 3 and 6 slot heights. It was found that at 3 slot heights the water was not deep enough to be accurately measured either optically or with the wave probe once sloshing was occurring and the tank was draining. Results of these tests are too inaccurate to draw any conclusions from regarding discharge within a single cycle. The tests with an initial water depth of 6 slot heights do, however, provide some reasonable results, although perhaps they are not as accurate as tests with deeper water. Run 20 of the sloshing tests has a 1.5 millimetre slot, an initial depth of 6 slot heights, a period of 9 seconds and an amplitude of 1.0 degrees. It is found that with such a small discharge, measuring instantaneous discharge with the load cell is out of the question due to the high losses as the water travels to the bucket. Using the video analysis to measure volume is also far from ideal, due to the minimal changes in volume over a given time period. Volume measurements at one second intervals from the video for this particular test have a greater error than the typical change in volume. As a result, the plot of discharge versus time, Figure 4.26, has a large amount of scatter on it. If a two second volume sampling increment is considered, the shape of the curve gets to be closer to what we would expect at the beginning of the run, but again, the error is still large. This is especially true as the volume decreases and becomes more difficult to measure as seen by  74  Chapter 4: Experimental Results and Discussion  the increase in scatter as the run progresses. A three second time increment is still not free of negative discharge values, and at this point, the overall shape of the curve is being lost.  Discharge Measurements From Video Analysis Slot Slot Roll Roll  0.0012 0.001  t  0.0008  Height 1.5 mm Length 211 mm Period 9 sec Angle 5 deg  0.0006 Q m  3/  0.0004 0  0  0  0  2  s  Time, seconds O Calculated for 1 second intervals  Figure 4.26  A Calculated for 2 second intervals  — Predicted from Depth Measurements  Plot of Volume Flow Rate for Sloshing With 2 Millimetre Slot  Perhaps the most useful comparison is to try to predict the shape of the discharge curve using the head measured by both the video and the wave probe, and compare it to the measured volume points from the tank. As was seen in Figure 4.23, the measurements of head at the freeing port are both quite similar with the wave probe and the video. Using the steady state predicted equation for Cd, a predicted volume versus time curve may be plotted and compared to the measured values, as shown in Figure  75  Chapter 4: Experimental Results and Discussion  4.27. These differences between the measured and predicted values are quite noticeable, and may be attributed to two causes.  Remaining Tank Volume Run 20  Measured from Video  Slot Height 1.5 mm Slot Length 211 mm Roll Period 9 sec Roll Angle 5 deg Initial H„d 6 Predicted by Wave Probe  Measured from Load Cell 10  15  20  25  30  Time, seconds  Figure 4.27  Remaining Volume in Tank for 1.5 Millimetre Slot  The first source of error has already been outlined, and it is merely error in volume calculations. The second error is to be expected, however, given the nature of the flow. When the tank rolls such that a thin film of water accelerates down the slope of the tank towards the slot, the water can have a considerable approach velocity as it nears the hole. If the freeing port were a single narrow hole, then most of the water would be stopped at the end of the tank, and the head would increase to reflect the energy that the moving water had. Because of the very wide slot, however, a thin film of water that is  76  Chapter 4: Experimental Results and Discussion  approaching the slot will pass through the slot, resulting in a much larger discharge than the measured head would suggest due to the effects of gravity and the resulting approach velocity. This discharge would likely be increased further still, since the value of Cd that was determined for the hole would be too low since there would be increased inertia that would increase the Weber number and hence reduce the surface tension effect. In other words, Cd is a function of the total dynamic head as opposed to the hydrostatic head for this case where the approach velocity is high. For this reason, we would expect to see a greater discharge rate than would be predicted by the steady state results for the case where the tank has a thin layer of water in it and it is being rolled such that the end of the tank with the freeing port is down. This difference would be seen at the end of the run in particular, although what is seen on the graph cannot be separated entirely from the effects of experimental error. This is mostly a problem for very wide holes where the velocity head does not get converted into a true hydrostatic head. It is assumed, however, that for the case of model testing, the slot will not cover the entire length of the bulwark, and this problem will be reduced. Positioning of the wave probe on a model in between two freeing ports could alleviate this problem.  4.4.3.2 Sloshing Tests with 3 Millimetre Slot  The three millimetre slot has twice the water depth of the 1.5 millimetre slot due to the doubling of the height of the slot. As a result, the volume and depth measurements are more accurate on a percentage error basis, although there are still a few anomalies such as a measured increased volume in the tank over the one second increments of  77  Chapter 4: Experimental Results and Discussion  measurement. A comparison of the measured depths using the wave probe and image analysis, however, has shown a good correlation between measurement methods.  Remaining Tank Volume Predicted from Video  Tank  Run 7 Slot Height 3mm Slot Lenght 240mm Roll Period 9 sec Roll Angle 5 deg Initial H 6 n d  Predicted from Wave Probe  Volume 0.003  6  8  10  12  Time, seconds  Figure 4.28 Remaining Volume in Tank for 3 Millimetre Slot  Figure 4.28 shows the discharge curve that is predicted by the measured heads and the steady state discharge results. As in the 1.5 millimetre slot tests, the load cell shows some scatter, with there being areas where the volume appears to have increased Generally speaking, however, the curve is a more logical shape with no scatter such as was observed for the video measurements of the 1.5 millimetre slot. Perhaps the most noticeable difference in the curves on the graph is the increased inaccuracy at the low heads from the head measured from the wave probe. The water depth is too small to measure, the wave probe detects no water, and hence no discharge is predicted. 78  Chapter 4: Experimental Results and Discussion  4.4.3.3 Sloshing Tests with 11 Millimetre Slot  The 11 millimetre slot is a slot which did not show any detectable scale effects. The much larger water depths also allow for a much more accurate measurement of relatively small values of both discharge and water depth at the freeing port. Figure 4.29 shows a much more satisfactory plot of the measured discharge versus time using the video analysis for Run 38. Because of the large volume change, over a one second increment the volume is changing by a far greater magnitude than the expected error, and as a result, the measured curve does not show the scatter from the video analysis that was shown by the plot of the discharge from the 1.5 millimetre slot. The load cell still is not giving acceptable instantaneous discharge readings, in this case it is most likely due to the unsteady flow and relatively large impact velocity causing sloshing in the collection bucket. Using the load cell to determine the general curve of discharge from the tank, however is quite good. The difference between plotting volume and discharge is that to determine the discharge rate, the slope of the line between each point must be considered, and that is much more susceptible to being affected by the scatter of the data. The graph showing predicted tank volume and measured volume versus time show a closer correlation than the smaller slots. The differences in the measured and predicted discharge rates would be expected to be less, not only because of the reduced error, but for another reason as well. It was noted that with the 1.5 and 3 millimetre slot that the approach velocity of the water at the hole when the tank was inclined could increase the discharge considerably. This phenomena would be expected, but not to the same extent, since Cd for the 11 millimetre slot at low heads is higher than it is for small  79  Chapter 4: Experimental Results and Discussion  slots, and could not be increased in the same way that Cd would be increased for a small slot with a low head and high approach velocity.  Remaining Tank Volume  Time, seconds  Figure 4.29  4.5  4.5.1  Remaining Volume in Tank for the 11 Millimetre Slot  Discussion of Experimental Error  Analysis of Experimental Error A first order analysis of the experimental error is possible by means of  differential calculus. If a value W that is found in experimentation is observed to be a function of x, y, and z, we may write; W=/(x,y,z)  (4.6)  80  Chapter 4: Experimental Results and Discussion  The magnitude of the error may be expressed as the total differential of the function, lf the error is then denoted dW, we may then write 6/ 6/ 6/ dW = 7 dx + r d y + 7 d z ox Sy oz L  L  i  J  (4.7) y  ln these tests, the discharge through a slot was measured and the results were non dimensionalized. The discharge is a function of the collected water mass, M, the sampling time, T, the hydrostatic head, H, the slot height, d, the slot width b, and gravitational acceleration, g. The range of values that were encountered for the different parameters are as follows M = 1.4 to 45 kg T = 8 to 100 seconds H = 2 to 180 millimetres d = 1 to 15 millimetres b = 240 to 257 millimetres g = 9.81. m/s  2  The mass of water collected was weighed on one of two different scales, depending on the amount of water to be weighed. The worst possible error was in the scale that was used to weigh the larger volumes of water. The error was approximately ±0.2 kg with the smallest amounts weighed on this scale being approximately 20 kg, or ±1 percent. The collection time of the water varied greatly depending on the rate of discharge from the freeing port. Because of the human element, time was accurate to perhaps 0.25s, giving a worst case scenario error of ±3.125 percent. Hydrostatic head  Chapter 4: Experimental Results and Discussion  was measured with a hook gauge during the steady state tests, and accuracy of that is approximately ±0.2mm, which for the smallest head tested translates into ±10 percent error. The height and width of the freeing port slots can be measured to within ±0.02 millimetres. This error translates into ±2.0 percent error for the slot height, and ±8 x 10° percent error for the width. To compute the error for the tests based on these calculations would give a very poor and inaccurate picture of the accuracy of the tests performed in this research. This is the case since the highest error in time measurement would occur in the tests with the shortest sampling time. The short sampling time corresponds to the largest slot size, water mass and hydrostatic head, all of which give the lowest of their possible error. Conversely, when the error in measuring the slot height and hydrostatic head is greater, this is made up for by the increased accuracy in the sampling time and the volume measurement (due to the more accurate scale available for measuring small masses of water). If one considers the discharge from the 1 millimetre slot, at the lowest head tested which is expected to have the highest experimental error, and the 1.5 millimetre slot at the highest head tested, which has its greatest error from other sources, using equation 2.8 rearranged as M  (4.8)  the total sources of error may be added up as follows  82  Chapter 4: Experimental Results and Discussion  Parameters  Exponent in Equation  Percent 1mm Slot  Error 15mm Slot  Slot Height, d  -1.5  2  0.1  Slot Width, b  -1  .01  .01  Sampling Time, T  -1  0.25  3.125  Discharge Mass, M  1  .25  1  Hydrostatic Head, H  -0.5  10  .1  Gravity  -0.5  -  -  8.51  4.33  Worst Possible Total Table 5.1  Error  Expected Experimental Error  It appears from these results that there may be considerable error associated with the determined values of Cd for the 1 millimeter slot. It should be pointed out, however, that as the hydrostatic head increases from 2 to 4 slot heights, error in measuring head is halved, resulting in a reduction of 2.5 %, or a new total of 6.0 % error. This error will continue to decline as the head is increased.  4.5.2  Relative Error in Sloshing Discharge Measurement In the measurement of the water depth at the freeing port and the discharge of the  sloshing water from the test vessel, there were four methods of measurement employed, all of which had different types of errors. The two methods of water depth measurement were using video and using the wave probe. Measurement using the wave probe was good except when the water became very shallow, when the thin film of water would not register. The response to sudden changes was good, although not as good as the video method of measurement. Depth measurement using the video was perhaps more  Chapter 4: Experimental Results and Discussion  accurate, although it is not a practical method of measurement for real time meaasurements on a model. In measuring the discharge from the sloshing tank, both the video and load cell were not without flaws, for differing reasons. The video method, while again being impractical, experienced amplification of errors, due to the number of depth measurements accross the tank that were required to compute a volume. Measurement from the load cell experienced three sources of error. The first and simplest form of error was that error due to the time lag for the water to reach the collection bucket. This was complicated by the fact that a surge of water leaving the slot would be more spread out by the time that it had reached the collection bucket. The second source of error here was due to the impact velocity of the falling water. It is very difficult to correct for this when it is combined with the error that was described previously. The final source of error in the water mass measurement was due to the motion of the water in the bucket itself. The sloshing of the water resulted in a varying reading.  84  Chapter 5 Conclusions and Recommendations  5.1  Summary of Tests Performed Tests of water discharge through wide small slots were performed to determine  the action of water as it flows through freeing ports that are the size that one may encounter with freeing ports on a model ship. Wide slots were used to minimize the effects of the contraction at the ends of the slots. Twelve different sizes of slots ranging from 1 millimetre high to 15 millimetres high were tested, to cover the expected range of freeing port heights that could be encountered on a model. The first phase of testing was to measure steady state discharge rates for each of the slots over a range of depths from 2 to 12 slot heights. The slots were fitted to one end of a rectangular test vessel, and water was supplied to the vessel to maintain a constant head. Discharge collected over a recorded time was weighed to give the volume flow rate through the slot. Results were nondimensionalized for comparative purposes and are plotted in Figure 4.1. It is believed that this is unique data on discharge from slots such as these. The second phase of testing involved allowing the water in the test vessel to drain without adding water. As a result the head was allowed to drop, and the effect of a decreasing head with changing scale effects was observed. The water depth was monitored over time, and comparisons were made between the measured water depth versus time curve and the depth versus time curve that was predicted using results of the previous steady state tests.  85"  Chapter 5: Conclusions and Recommendations  The final phase of testing involved mounting the test vessel on a rocking platform to induce sloshing in the tank to simulate the motion of the water on a rolling ship. Water depth at the freeing port hole inside the tank was monitored, as was the motion of the tank and water, and the discharge from the tank. An effort was made to predict the discharge from the measured head values, and this was compared to the measured values, despite some detected inaccuracies in the measurement procedure.  5.2  Implications of Test Results The results of these tests confirm that it is not possible to have a fixed size of  freeing port below approximately 5 millimetres high on a model and have the freeing port draining the correct amount of water from the model regardless of the head height. While a freeing port that is 4 millimetres high would drain at a varying yet always too large discharge rate regardless of the head, smaller freeing ports will drain too much water at large heads and will not drain enough water at low heads. Small freeing ports such as these could be observed to drain the expected amount of water in the right time without considering scale effects, but the instantaneous discharge rates from a model do not reflect the true instantaneous discharge rates from a ship. The result of this is that a model's motion may not be representative of a full scale ship's motion. As a result, if one wishes to perform tests on models with freeing ports that are below 5 millimetres in height, the head must be monitored and the discharge must be adjusted accordingly.  86  Chapter 5: Conclusions and Recommendations  5.3  Control Strategy The goal of this research was to study the flow characteristics through small  freeing ports so that a method of model testing with properly functioning freeing ports could be proposed using a control system. The original plan was to use a wave probe to monitor the water depth at the freeing port, and to control the size of the freeing port to account for the differences between the actual value of Cd and the value of Cd that we would expect on the full scale ship. Within the range of heads tested in this research, discharge from freeing ports that are greater than 5 millimetres high does not appear to be affected by scale, yet smaller slots must have their different Cd values accounted for.  Required Area Changes, 3mm Freeing Port  1  2  3  4  5  6  7  8  9  10  11  12  13  14  Time, seconds — W i t t e r Depth  Figure 5.1  —  Area Ratio  Plot of Required Area Change for 3 mm Freeing Port  One may account for the differences in Cd by varying the size of the freeing port through which the water is flowing. For a given model, the tester will know the height of the freeing ports, and subsequently may select an equation that best fits the relationship 87  Chapter 5: Conclusions and Recommendations  between water depth and Cd. For a full scale freeing port, there is also an equation that may be used to generate the value of Cd for any given water depth. A ratio of the two values for Cd may be used to express what area the freeing port should be for a given water depth in relation to its nominal size. For example, a three millimetre high freeing port would have a Cd of approximately 0.86 for a water depth of 12 millimetres. Without scale effects considered, with the same nondimensionalized water depth of four hole heights, we would expect a Cd of 0.73 on the actual ship. The ratio of these two Cd values is 0.73 divided by 0.86, or 0.85. This tells us that on this model at this head height, the freeing port area must be reduced to 85 percent of its nominal size or else too much water will be drained from the model. Figure 5.1 shows a hypothetical water depth measurement versus time plot for a model with a three millimetre high freeing port. The area ratio, which is the ratio of required area of the model freeing port over the theoretical freeing port size, can be seen to be fluctuating to greater and less than one as the water depth varies. The easiest way to vary the size of the freeing port would be to vary the length, since the length is proportional to area. If one were to attempt to vary the height of the slot, the nondimensionalized head would change, changing the very nature of the scale effects that are expected to be acting. Varying the length of the freeing port, however, is not without its flaws. If one were to try to test a 1 millimetre high freeing port, then as the head dropped down to 2 hole heights, the freeing port would have to be more than doubled in length to accommodate the correct discharge, and for smaller heads than that, the hole would likely have to be increased in size still further. Depending on the spacing and length of the freeing ports on the model, testing such small holes and heads may not  88  Chapter 5: Conclusions and Recommendations  be possible. This is a consideration that the tester must remember, so that if a model is to have freeing ports close together, the required range of sizes of the freeing ports must be considered. It is possible that the required variation in freeing port size may dictate the minimum possible model scale. At this point it should be emphasized that the equations that were generated here for Cd versus head for different sizes of slots are for a certain hole geometry. Comparisons with results of tests by Yoshino et al. suggest that the results of the 5 millimetre slot tests are reasonably accurate, despite exact knowledge of Yoshino's different hole configuration. It is likely, however, that as slot size decreases and scale effects become more important, hole geometry could also become more important. Unless there is confidence in the flow characteristics of the water through the freeing ports on the model, the prudent model tester should perform some steady state tests on the actual model to be certain of the flow characteristics for that particular model and freeing port size and configuration. Once the flow characteristics are known, one may then utilize an algorithm to control the freeing port area, as described earlier in this section.  5.4  5.4.1  RecommendedFuntie Tests  Steady State Tests The steady state discharge tests provided perhaps the most useful and  comprehensive results of the tests that were perfomied here. The effects of surface tension and viscosity were observed to be great for the smaller holes.  89  Chapter 5: Conclusions and Recommendations  It appears that the scale effects cannot be attributed to any one effect, but rather a combination of both viscosity and surface tension. If one were wanting to further test the steady state discharge, there are a number of modifications that could potentially be made to determine the relative magnitudes of the viscosity and surface tension.  5.4.1.1 Testing The Effect of Viscosity It was observed that for the small slots, the situation would arise such that the Blasius solution for a laminar boundary layer along the bottom of the tank would tell us that the boundary layer thickness would extend to the water surface as the water approached the slot. While if this is the case it would be hard to avoid this effect on the actual deck of the model, further tests could be performed to determine the magnitude of this effect. If the slot was set up such that it was not flush with the bottom of the tank but instead well above it, we would not see the discharge affected by boundary layer growth on the bottom of the tank, since whatever small boundary layer there would be would be well below the slot, and the water velocity in the tank, or approach velocity, would be reduced to almost zero. Of course, with a slot that is not flush with the bottom of the tank, one would expect to see a contraction in the exiting jet at the bottom of the slot where it was not seen before. This would potentially reduce the flow, and there would be a new effect acting that was not acting in the original tests. If one were able to reduce this contraction, however, the tests would be more accurate. If one were to place a very short plate perpendicular to and flush with the bottom of the slot, perhaps this contraction could be eliminated without causing boundary layer growth of the same magnitude that  90  Chapter 5: Conclusions and Recommendations  was experienced in the experiments that were performed in this research. Another approach to this problem could be to replace the proposed flat plate with a curved plate in the form o f a streamline at the bottom of the hole. Figure 5.2 shows the current hole geometry, along with the proposed changes that could be made to determine the effect of the boundary layer growth.  Water Surface H Cuorant Freeing Pont Configuration  Water Surface  Water Surface H  H |  1  ,  Possible Geometoy Modifications to (Phimgft Boundary Layer  Figure 5.2 Alternate Hole Configurations for Boundary Layer Testing  5.4.1.2 Testing the Effect of Surface Tension A s was mentioned in the description of the apparatus, in the tests performed in this research, it was not possible to perform steady state discharge tests with fluids with reduced surface tension such as soapy water. This constraint came from the accurate flow control that was needed over a very wide range of flow rates, without having the resupplied water entering the tank with a high velocity, since excessive water velocity 91  Chapter 5: Conclusions and Recommendations  and air entrainment in the water translates into excessive suds. Given a good pump, perhaps of variable speed such that the exit velocity is no higher than is required for the freeing port discharge, steady state discharge tests with soapy water could be quite manageable.  5.4.1.3 Other Test Improvements For a complete understanding of the flow through the slots, it would be useful to be able to test the discharge accurately for depths below 2 slot heights. This was not possible with the flow control that was available here, but again, given the ideal pump, or perhaps a cascading system of tanks to steady out the water flow before it entered the test vessel, more steady and accurate flow into the vessel could be achieved. Another improvement to the tests could be analysis of the velocity field in proximity to the slot. Use of laser Doppler anemometry and perhaps some methods of flow visualization could be used to determine the general shape and thickness of the boundary layer approaching the slot.  5.4.2  Unsteady Discharge without Sloshing Many of the suggestions that were made for improving the understanding of the  flow characteristics of the steady state flow may be applied to the tests with the dynamic and variable flow rate. This is to be expected, since for all intents and purposes, the discharge from the vessel that is not being resupplied with water and is not sloshing can be predicted by the results of the steady state flow. Of course, it would be much easier to perform an analysis of flow visualization tests with steady state discharge than with a  92  Chapter 5: Conclusions and Recommendations  declining head , since the flow characteristics for a given head would be visible for an extended period of time, while with the head declining, instantaneous image acquisition or analysis would be required for a given head.  5.4.3  Sloshing Tests A number of conclusions may be drawn from the results of the sloshing tests,  related to both the discharge of the water and the motion of the water in the tank.  5.4.3.1 Water Motion in the Tank  While the scope of the research was not to examine the motion of the water in the vessel during sloshing, some general observations were made regarding the water motion in the tank. Since slot heights down to 1.5 millimetres were tested with the sloshing apparatus, there was opportunity to observe the sloshing motion of some small amounts of water. As the water depth dropped down to 3 slot heights and less with the 1.5 millimetre slot, the sloshing was observed to not resemble the motion of the larger water depths much at all. Wave theory tells us that one should expect there to be some non linear effects on the waves due to the shallow water. While no attempt will be made to set a certain limit as to the water depth at which the scale effects of the water motion in the tank becomes unacceptable, it should be emphasized that the tester must be aware of other scale effects besides those effects that were detected relating discharge to head for small freeing ports.  93  Chapter 5: Conclusions and Recommendations  5.4.3.2 Prediction of Sloshing Discharge While the prediction of the discharge from the stationary tank was quite successful, the measurement and prediction of discharge from the tank with the water sloshing in it were both found to be more difficult. There were some differences that were detected, however, between the actual measured discharge and the discharge that one would expect for a given head. The differences in discharge that were detected are a result of the differences between the measured hydrostatic head and the total head, which includes the velocity head. While the approach velocity in the steady state tests is relatively small, when the tank is inclined there can be a large discharge without a large head due to the approach velocity of the water. There are two ways that one may account for this. Perhaps the most obvious method would be to measure the approach velocity, but this is not as simple as it sounds due to the difficulties in instrumentation and measurement. A simpler method of determining the total hydrostatic and velocity head is to stop the flow. Obviously the flow cannot be stopped across the area of the complete hole, but assuming that the freeing ports are the conventional form of a series of long rectangles with sections of the bulwark in between them, the wave probe that is being used to detect the head may be set up between two freeing ports. This will cause the probe to measure the total head, as was experienced with the video analysis at the edge of the test vessel with the 11 millimetre slot. The flow that is directly approaching the freeing port will pass through without showing signs of a raised head, while the flow that approaches the bulwark between the freeing ports will be stopped, and its head will reflect the total hydrostatic and velocity head.  94  Chapter 5: Conclusions and Recommendations  A n alternate method would be to utilize a pitot tube that extends into the freeing port from the outside. If this were small enough to not affect the flow significantly, and instantaneous pressure readings were available from a pressure transducer, this could be utilized in a control mechanism.  95  Bibliography  American Bureau of Shipping, Rules for Building and Classing Steel Vessels 1974, P120, 121, New York, 1974. Chan, Philip, Sloshing Tank for Fishing Vessel Stability, Volume II - Operations, University of British Columbia Department of Mechanical Engineering, Fourth Year Project Report, July 1988. Dahle, A.E. and Kjaerland, O, 'The Capsizing Institute of Naval Architects, 1979.  of M/S HELLAND - HANSEN', The R  Drysdale, C V . et al, The Mechanical Properties of Fluids, P. 17 - 27, Blackie and Son Limited, Glasgow, 1923. Gibson, A.H. Hydraulics and its Applications, Constable & Company Ltd., London.  'Marcoscopic Wetting Behaviour and a Meth Measuring Contact Angles' ASME Conference Proceedings, June 1990.  Katoh, K., Fujita,H. and Sasaki, H.,  Lamb, Sir H. Hydrodynamics 6th Ed., P 455 - 457, Dover Publications, New York, 1945. Newman, J.N., Marine Hydrodynamics, The MIT Press, Cambridge, Massachusetts, 1986.  'Experimental Studies on the Shipping Water Fishing Boats and the Effectiveness of the Freeing Port on Dra Water'(in Japaneese), Senpaku Gitjutsu Kenkyusho Hokoku V. 22 N. 3, May  Yoshino, T. and Yamamoto, T.,  1985, P 189-217.  96  Appendix A Video Analysis using Frame Grabber  A.l  Uses of the Frame Grabber All of the runs in the second and third stage of testing - the unsteady flow and the  sloshing tests - were recorded on an 8mm videotape. Using a titling attachment on the camera, titles for each run could be recorded, and a stopwatch feature was used to record the time of each run. The stopwatch counts in increments of on tenth of a second, so that there were only three frames for every recorded time step. The video was set up such that it gave a side view of the tank. The camera was set up as far away from the tank as possible with the camera zoomed in as much as possible such that the picture filled the screen appropriately. This gave a view that was as close to two dimensional as possible. With the sloshing tests, the camera was set up so that the entire tank could be seen. The water was dyed so that it would be photogenic, and the side view of the tank allowed for measurement of the head height at the freeing port, measurement of the angle of the tank, and measurement of the area of the water as seen from the side. This area measurement could then be converted to a volume within the tank. Additional frame analysis was performed on videotape of the second phase of the experimentation, where water was allowed to drain out of the tank without an induced rolling motion. For these tests, more general measurements were made, typically of head height at the hole.  97  Appendix A :  A.2  Video Analysis Using Frame Grabber  Hardware and Software One of the computers in U B C ' s Naval Architecture Lab was outfitted with an  Imaging Technology P C V I S I O N p l u s 512 Frame Grabber, with the supporting Imaging Technology I T E X / P C Software Library. This software was used with the Software Development Package sold by Infrascan Inc. of Richmond, B . C . Using the software and hardware from Imaging Technology, it was possible to link the video camera to the computer and also to an additional Multisync monitor. Images recorded on the videotape could be played back through to the computer's frame grabber card, and the picture could be seen on the second monitor. Once the video was paused on the frame that was desired, the image could be captured and stored in the frame grabber's memory and displayed on the screen without the picture coming from the camera. The Software Development Package is a package sold by a local firm that uses the I T E X command library and its own C code to allow the user to use a menu driven mouse compatible program to perform many image capturing and enhancement procedures quickly and easily. Unfortunately, there were no features that were offered that would allow the user to measure dimensions from the screen or digitize points. T w o additional C programs were written and linked with the Software Development Package (SDP) to allow the user to perform these functions.  98  Appendix A :  A. 3  Video Analysis Using Frame Grabber  TANK_VOL. C Program  A.3.1 Use of the Program The TankVol.C program utilizes many of the subroutines defined within the SDP and uses the menus and frame grabbing that are possible within the SDP. The Digitize program can be accessed from the program SDPPC.EXE through the menu selection "Measure Tank Volume" on the main menu. Once this has been chosen, the user is given four options: Change Value of Scale Length Change Aspect Ratio of Coordinates Define Scale on Screen Trace Area with Mouse This program was used for digitizing the coordinates of the water surface and for measuring the tank volume and angle. The first selection that would normally be made is "Define Scale on Screen". The two bottom corners of the tank would be indicated with the mouse, and the computer was set up so that it had the correct default length for the bottom of the tank. From this the tank angle was computed, and the rotation of the tank's axes would be set. If one were not measuring the tank's bottom length, this default length could be changed. The first menu item, "Change Value of Scale Length" allows the user to indicate the actual length of any object that is going to be used to define the scale. The other default value that was set was the aspect ratio of the coordinates. The menu selection "Change Aspect Ratio of Coordinates" allows these to be changed. This is required because it was noticed that while the tank's height over length was physically measured as 0.262, when the tank was measured on the screen by pixels, the height over  99  Appendix A : Video Analysis Using Frame Grabber  length ratio was found to be 0.321. It appears that this ratio may be a function of the video camera. To make the aspect ratio of the pixels the same as the physical tank, the ratio of the two previous ratios was determined and found to be 0.816. All y pixel values are multiplied by this factor. A different camera or set up would require that the pixel to physical measurement ratios be checked before this ratio could be assumed to be correct at 0.84. The "Measure Tank Volume" program is set up specifically to be used with the video of the sloshing tests. Once the bottom of the tank has been defined on the screen using the previous menu selection, the user is asked to indicate the water surface on the screen with the mouse. The coordinate system is shifted such that the bottom left comer of the tank is taken to be the origin of the standard form Cartesian coordinate system, with the x axis aligned with the bottom of the tank. It is assumed that between each of the points, there is a straight line. Once the side area of the water remaining in the tank is determined, the program computes the volume of water within the tank. The user is asked if he would like to save the data and if 'yes,' then he is asked for a filename and for the time on the frame that was analysed. The results are written to a file. The first row in the file contains the tank length in pixels and the tank angle, the second row contains the frame time and the volume, and the following rows contain the x and y coordinates of the points that were indicated with the mouse.  A.3.2 Calculations Performed in Tank_Vol.C The calculations performed in Tank_Vol.C can be broken down into three basic groups. These are Conversion of pixel values to Cartesian coordinate system Converting points to axes of tank  100  Appendix A :  Video Analysis Using Frame Grabber  Tank volume calculations  A.3.2.1 Converting to Cartesian Coordinate System The standard screen that was being used has its coordinate system defined such that the origin is at the top left comer of the screen. The screen has 512 pixels in the horizontal or x direction, and 480 pixels in the vertical or y direction. While the x coordinates were according to convention, two operations had to be performed on the y values. The first operation that was performed was setting the y axis with the zero at the bottom as one would expect with the Cartesian coordinate system. This was achieved by simply letting y = 480 - y for all of the y values. Because of the aspect ratio problem that was described earlier, it was also necessary to multiply all of the y values by the aspect ratio variable, "ar", which was set to default at 0.816, the measured value. The conversion to the Cartesian coordinate system could therefore be performed by the equation y = ar (480 - y)  A.3.2.2 Converting to Tank Axes The angle of the tank is determined after the user defines the two bottom corners of the tank. This angle is determined using the converted Cartesian coordinates according to the equation  6 = tan  1  Yt2 Xt2  - Y.i - X.i  where t denotes tank coordinates as  is shown in Figure A . l  101  Appendix A : Video Analysis Using Frame Grabber  The origin for the new coordinate system is now set at the bottom left corner of the tank, or (Xo,Yti). Any point that is defined by the mouse will have coordinates (X ,Yp) on the Cartesian coordinate system that is oriented to the screen. A line drawn P  from the new tank origin to this point (X ,Y ) may be converted into polar coordinates P  P  such that we may define  (3 = tan  1  Y - Yt P  X  P  r = V(X -X,i) + (Yp- Yn)  and  - Xti  2  Converted. $creen Coordinate (System Figure A.'l  2  P  512,0  Axes Conversion and Notation  The angle y, between the tank axis and the vector V, can be determined by the equation Y = p-e  102  Appendix A : Video Analysis Using Frame Grabber  The new coordinate system that is oriented to the angle of the tank will be called the i-j coordinate system. The i and j values of the points on the new coordinate system may now be determined by the equations  i = r cos (y)  and  P  j = r sin (y) P  Care must be taken when performing these equations and taking an arctangent. It is possible to get an angle from the arctangent that is in the wrong quadrant. An IF statement comparing the sign of the x and y values was used to determine the quadrant that the angle would be in.  A.3.2.3 Calculation of Tank Volume  The volume of water in the tank is determined by multiplying the area of the visible side surface by the width of the tank. The area of the side view of the water in the tank is determined by calculating the area of a series of polygons and keeping a running total. Using the data points on the new i-j axis, starting with the second point and ending with the last point, the area of each polygon that the two points define is computed. Given two points  (inijn-i)  and  (in,jn)  the area may be defined by  Arean = (in - in-1) X (jn-l + jn)/2 A running total of these areas will yield the total area of the side view of the tank.  AA  DIGITIZE.C Program  The Digitize.C program is accessed through the program SDPPC.EXE and is found in the main menu under the heading "Digitize Points on Screen". This program is  103  Appendix A :  Video Analysis Using Frame Grabber  a much simpler version of the previously discussed Tank_Vol.C. The user is given three options this time. Change Aspect Ratio of Coordinates Define Scale on Screen Digitize points The "Change Aspect Ratio of Coordinates" is exactly as described in the previous section B.3. Again, a default value has been set to 0.816 for the apparatus the was used in these tests. This value may be changed if necessary. The second selection "Define Scale on Screen" asks the user to indicate the start and end points of a known length and to input the length of this. Unlike in Tank_Vol.C, there has been no default set for the length. Any object of a known length that is being seen at its true length may be used. The final selection, "Digitize Points" asks the user to indicate with the mouse any points that he may want coordinate values for. The coordinate system has been converted to the standard Cartesian Coordinate system, with the axis at the lower left comer of the screen. Coordinate values are given out in the units that the length of the scale was given in. If no scale was defined, the coordinate values are given out in pixels. As is the case in Tank_Vol.C, the user is asked if he would like the data points saved, and if so, he is asked for a filename and for the time that corresponds to the frame that was analysed, if applicable.  104  Appendix A : Video Analysis Using Frame Grabber  A.5  A.5.1  Program  Code  Tank_Vol.C  j * * * * :|; * * * * * * * * * * * :|: * * * * * ****** * * * * * * * * * * * * * * * * * *************** * * * * * * *j I* Module: T A N K V O L . C  */  /* */ /* This module has been written to compute the area of the side view */ /* of a rectangular tank of liquid. Module C U R S O R . C must be changed */ /* so that get_one_point has the while with buttonQ = L E F T or RIGHT. */  /*  */  /* Copyright (c) 1991: Peter Williamson /*  */ */  j * * * * * * * * * * * :|: * * * * ;|: * :|: * * * * * * * * * * * :|: * * * * * * * * * * * * * * * * * * * * * * * * % * * #include <c:\msc\include\stdio.h> #include <c:\msc\include\math.h> #include <c:\msc\include\float.h> #include "stddefs.h" #include "itexpfg.h" #include "ms_mouse.h" #include "menu.h" #include "cursor.h" #include "tank vol.h"  /  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  void measure_area() { extern short x,y; extern short dx,dy; extern char system_name['|; short itemselected; static char *menu_list[| =  { "Change Value of Scale Length", "Change Aspect Ratio of Coordinates", "Define Scale on Screen", "Trace Area With Mouse", "EXIT to Main Menu" }; float length,scale,area,tang,tang2 ar,volume,time; !  105  H; * * *  * /  Appendix A :  Video Analysis Using Frame Grabber  char ch,string[ 10j,filen[15|; short sxl,sx2,syl,sy2,xp,yp,i,j; float xi[30|,yj[30|; FILE *fp; scale =0; length = 89.5; /* Default value for sloshing tank */ ar = 0.816; /* Default ratio of x/y from screen, maybe */ item_selected = N O N E _ S E L E C T E D ; /* a function of video camera ? */ do  { display_menu(.system_name,"Measure Tank Volume",menujist); /* M E N U . C */ item_selected = menu_selection(menu_list,item_selected); / * M E N U . C */ switch (item_selected) { case 0: tput(21,10,"What is Scale Length? "); / * U T I L . C */ gets(string); length = atof(string); break;  case 1: tput(20,.10,"What is the ratio of measured height over length to"); tput(21,10,"measured pixel height over pixel length? <0.816 if unsure>"); gets(string); ar = atof( string); break;  case 2: tput (19,10,"Bottom Left Corner?"); /* U T I L . C */ get_one_point(&sxl,&syl); /* C U R S O R . C */ syl=ar*(480-syl); tput (19,5,"first coordinates on Cartesian Coord, system are "); printf("%5d,%5d",sxl,syl); delay(30000); tput (20,.10,"Bottom Right Corner?"); /*" U T I L . C */ get_one_point(&sx2,&sy2); / * C U R S O R . C */ sy2 = ar*(480-sy2); tput (20,5,"second coordinates on Cartesian Coord, system are "); printf("%5d,%5d",sx2,sy2); scale = length/sqrt(pow((sxl-sx2),2)+pow((sy l-sy2),2)); tput(22,1.0,"Scale (units per pixel) is "); printf("%8.4f",scale); gets(string); break;  1.06  A p p e n d i x A : V i d e o A n a l y s i s U s i n g Frame Grabber  case 3: if (scale==0) { tput(19,L(),"Define Corners First!"); gets(stTing);  } else { area = 0; i=i; tang = atan(((float)sy2-(float)syl)/((float)sx2-(float)sxl)); tang2 =atan(tan(tang)*ar)* 180/3.1415926; tput(16,10,"Tank A n g l e (degrees) is "); printf("%4.2f",tang2); tput(17,10,"Click on T o p Surface from Left to Right"); tput(18,10,"Hold Right Button on Last Point to End");  get_one_point(&xp,&yp); /* C U R S O R . C */ delay(30000); /* U T I L . C */ yp = ar*(480 - yp); tput (19,.10,"raw coords after aspect scaling are ");  / * * * * Convert data points to coordinate system of tank. * * * * * * * * * / / * * * * There are two sets of equations depending on w h i c h * * * * * * * * * / / * * * * quadrant the acrtangent is being taken in. *********/ printf("%5d,%5d",xp,yp); if (xp ==sxl)  {  yjl'l = yp-syi; x i i i ] = 0;  } else { if ((float)xp > (float)sxl) { xi[ij = sqrt(pow(((float)xp-(float)sxl),2)+pow(((float)yp(float)syl),2))*cos(atan(((noat)yp-(float)syl)/ ((float)xp-(float)sx 1 ))-tang); y j [ i | =sqrt(pow(((float)xp-(float)sxl),2)+pow(((float)yp(float)syl),2))*sin(atan((( float )yp-(float)syl)/ } ((noat)xp-(noat)sxl))-tang);  107  Appendix A :  Video Analysis Using Frame Grabber  else { xi[i | = sqrt(pow(((float)xp-(float)sx l),2)+pow(((rloat)yp(float)syl),2))*cos(atan2(((noat)yp-(float)syl), ((float)xp-(float)sx]))-tang); yj[i|  =sqrt(pow(((float)xp-(float)sxl),2)+pow(((float)yp(float)syl),2))*sin(atan2(((noat)yp-(float)syl), ((float)xp-(float)sxl))-tang);  }  }  tput (20,10,"converted cordinates are"); printf("%5f,%5t",xi[i|,yjli|); if(i!=l) {area = area + (xi|i|-xili-l|)*((yJli]+yj[i-J])/2);} i=i+J;  I while ((button() != RIGHT)&&(i <= 31)); area = area *scale*scale; volume=area/10000.0*0.3; tput (21,10,"The Computed Tank Volume (m"3) is "); printf("%12.5f",volume); tput (22,10,"Would you like this data saved? (y/n)"); ch = getchQ; if (ch == 'y')  { tput (23,1 ('Venter Ihe output filename "); fflush(stdin); gets(filen); tput (25, l(),"enter time of frame "); gets(string); time = atof(string); if ((fp=fopen(filen,"u")) != N U L L )  { fprintf(fp,"\n%8.5At%8.4f\n",time,volume); fprintf(fp,"%8.4t\t%8.4f\n",tang2,length/scale); for (j=l; j < i; j=j+l) fprintf(fp,"%8.4f\t%8.4An" xi|j|,yj|Jl); )  close(filen);  } } } break;  }  } while (slrncmp(menu_lisllitem_selected|,"EXIT",4));  }  ^ ************** * * * * * * * ******************************** * **************** j  108  Appendix A :  A.5.2  Video Analysis Using Frame Grabber  Digitize.C  /*  */  /* Module: DIGITIZE.C  /*  */  */  /* This module has been written to allow the user to define a scale */ /* on the screen and digitize points. Module C U R S O R . C must be changed */ /* so that get_one_point has the while with buttonQ = L E F T or RIGHT. */  j*  *j  I* Copyright (c) 1991: Peter Williamson  I*  */  *j  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + * * * * * * * * * * *^  #include <c:\msc\include\stdio.h> #include <c:\msc\include\math.h> #include <c:\msc\include\float.h> #include "stddefs.h" #include "itexpfg.h" #include "ms_mouse.h" #include "menu.h" #include "cursor.h" #include "digitize.h" ************************************************* void digitizeQ { extern short x,y; extern short dx,dy; extern char system_name[|; short item_selected; static char *menu_list[| = { "Change Aspect Ratio of Coordinates", "Define Scale on Screen", "Digitize Points", "EXIT to Main Menu" I;  float length,scale,ar,time; char ch,string[1.0],filen[1.5]; short sxl,sx2,syl,sy2,xp,yp,i,j; float xi[50],yj[50]; F I L E *fp; scale =0; length =1;  109  Appendix A :  Video Analysis Using Frame Grabber  ar = 0.816; /* Default ratio of x/y from screen */ itemselected = N O N E S E L E C T E D ; do  { dispIay_menu(system_name,"Digitize Points on Screen",menulist); /* M E N U . C */ item_selected = menu_selection(menu_list,item_selected); /* M E N U . C */ switch (item_selecled) { case 0: tput(20,10,"What is the ratio of measured height over length to"); tput(2.1,10, "measured pixel height over pixel length? <0.816 if unsure>"); gets(string); ar = atof( string); break;  case 1: tput(:l9,.10,"First Point of Scale?"); /* U T I L . C */ get_one_point(&sx 1 ,&sy 1); /* C U R S O R . C */ syl=ar*(480-syl); tput (19,5,"first coordinates on Cartesian Coord, system are "); printf("%5d,%5d",sx 1 ,sy 1); delay(30000); tput (20,10,"Second Point of Scale?"); /* U T I L . C */ get_one_point(&sx2,&sy2); /* C U R S O R . C */ sy2 = ar*(480-sy2); tput (20,5,"second coordinates on Cartesian Coord, system are "); printf("%5d,%5d",sx2,sy2); tput (21,10," What is length of scale?"); gets(string); length = atof(string); scale = length/sqrt(pow((sxl-sx2),2)+pow((sy tput(22,10,"Scale (units per pixel) is "); printf("%8.4f",scale); gets(string); break;  l-sy2),2));  case 2: if (scale==l) { tput(19,10,"Scale is in Pixels"); gets(string);  } i = 1;  tput(l 7,10,"Click on Points to Digitize"); tput(.18,10,"Hold Right Button on Last Point to End");  110  Appendix A :  Video Analysis Using Frame Grabber  do  get_one_point(&xp,&yp); delay(30000);  /* C U R S O R . C */ /* U T I L . C */  yp = ar*(480 - yp); tput (19,.10,"raw coords after aspect scaling are "); printf("%5d,%5d",xp,yp); xi[i|=xp*scale; yj[i]=yp*scaie; i=i+l;  } while ((button() != RIGHT)&&(i <= 51)); tput (22,10,"Would you like this data saved? (y/n)"); ch = getch(); if (ch == 'y')  { tput (23,10,"enter the output filename "); fflush(stdin); gets(filen); tput (25,10,"enter time of frame "); gets(string); time = atof(string); if ((fp=fopen(filen,"a")) != N U L L )  { fprintf(fp,"\n%8.5f/n",time); for (j=l; j < i; j=j+l) {fprintf(fp,"%8.4t\t%8.4l.\n",xi|j|,yjlj|);} close(filen);  } break;  } \ while (strncmp(menu_list[item_selected|,"EXlT",4));  } y * *  * * * * * * * *  : H * * * H = * * * * *  * * * * * * * * * * * * * * * * * *  111  Appendix B Data Acquisition System  The data acquisition system can be broken into several groups. They are Wave Probes Weighing Apparatus Signal Conditioning Software and Hardware  B.l  Wave Probes There were two different types of wave probes used in this research. One was a  capacitance wave probe, and the other was the Kenek servo wave probe, an active probe that follows the surface of the water utilizing an electrode within the water.  B.l.l  Capacitance Wave Probe The capacitance wave probe is a probe that was developed by NRC which utilizes  a rod with two horizontal arms that have two wires strung between them. The wires are approximately 6 millimetres apart and while the length of these probes can vary, the wire length of the probe used in this experimentation was 35 centimetres. The probe is oriented vertically in the water, and was supported by a tripod that stood in the test vessel. An excitation voltage of 10 volts dc is provided to the probe, and a signal voltage is returned, with both input and output sharing a common ground terminal. The capacitance of the wires themselves is approximately 1000 pF. This capacitance is changed by the water depth, and through the circuitry of the wave probe's electronics, a voltage is returned from the wave probe which varies linearly with the water depth. HZ  Appendix B:  Data Acquisition System  B.1.2 Kenek Servo Wave Probe The "Kenek" servo wave probe was model SW-301.. It utilizes an electrode within the water, and the probe is the other electrode. The probe follows the water surface such that the circuit is always completed, yet except for very sudden depth increases, the probe remains at the surface, and not below it. The probe's specifications are as follows: Measuring Range  300 mm  Response  500 msec / F.S.  Resolution  ±0.2 mm  Linearity  ±0.1% / F.S.  Drift  ±0.1% / H  Output Voltage  6V/F.S. (0.2V/cm)  Output Current  Max 1 mA  Size  79mm x 60mm x 400mm  Weight Power Source  1.8 kg 1.15 Vac @ 60 Hz  A frame was built that would hold the wave probe above the end of the tank with the freeing port. The frame was mounted on the tank with C clamps such that when the tank was moving, the wave probe was moving too. The end of the probe is very delicate and there was concern that when the tank was dry at the slot, the probe would be hitting the bottom and could potentially be damaged. The frame that was built to hold the probe was designed such that it could hold the probe high enough that when the tank went dry the probe would reach the end of its adjustment just before it hit the bottom. Unfortunately, it was found that the response was slowed considerably when the probe was at its maximum extension. To avoid the problem of a slow response, a new tip for  113  Appendix B:  Data Acquisition System  the probe was machined that was more rugged, and this was allowed to contact the bottom of the tank.  B.2  The Load Cell Water that was exiting the test vessel in the third and final stage of testing was  collected in a bucket that was suspended from a load cell. The load cell that was used was an Omega LCC - 200 load cell. The specifications of the load cell that was used are as follows:  B.3  Rated Capacity  200 lbs  Deflection at Full Load  0.005"  Rated Output  2.0 mV/V±0.1%  Nonlinearity  0.03%  Hysteresis  0.02%  Nonrepeatability  0.01 %  Creep in 20 minutes  0.03%  Zero Balance  1.0%  Compensated Temp Range  -10 to 45 C  Terminal Resistance  Input 350 ± 1 Q  Excitation Voltage  10 V nominal, 18 V max  Insulation Resistance  5000 at 50 V dc  Max Load  Safe 150%, Ultimate 250%  Dimensions  2.25" x 1.25" x 2.38"  U  Signal Conditioning While the signal from the Kenek wave probe was conditioned by its own  conditioner that is part of the system, the capacitance wave probe and the load cell's signal both required amplification. A Terrascience Systems Ltd. ST41B Analog Signal 114  Appendix B: Data Acquisition System  Conditioner was used for this purpose. This signal conditioner has a MB41 card cage to house the signal conditioner cards. The specifications of the signal conditioner are as follows: Channel Excitation  2 to 10 Volts, independently variable  Switchable Gain  1 to 1000  Bridge Completion  1/4 Bridge 120 and 350 W, or 1/2 bridge inputs to each channel  B.4  Line Regulation  less than 0.02% for ± 10% input change  Load Regulation  less than 0.05% for 5 mA to 60 mA  Ripple  Less than ImV peak to peak dc lOOHz  Gain Accuracy  0.05%  Gain Temp Coefficient  0.01% / Deg C  Hardware and Software  The data acquisition computer was an IBM PC equipped with a DT2801 data acquisition board. The specifications of the data acquisition board are as follows: ANALOG INPUTS Number of Analog Inputs  16SE, 8DI  Input Ranges  Jumper Selectable, unipolar or bipolar  Unipolar Input Ranges  0 to +10V, 0 to +5V, 0 to +2.5V, 0 to 1.25V full scale, depending on programmable gain setting  Bipolar Input Ranges  ±10 V, ±5V, ±2.5V, ±1.25V full scale depending  on  programmable gain settings Output Data Codes Software Programmable Gain Range Input Impedance  Straight Binary (unipolar) Offset Binary (bipolar) 1, 2, 4, or 8 Off Channel 100 megohms, lOpF On Channel 100 megohms, lOOpF 115  Appendix B: Data Acquisition System  Bias Current  ±20 nA  Common Mode Input Voltage, Maximum  ±11V  Common Mode Rejection Ratio (CMRR), Gain = 1  80dB at 60 Hz, 1 kilohm unbalanced  Amplifier Input Noise  10 uV, rms  Channel to Channel Input Voltage Error  ±10 uV  ACCURACY Resolution  12 Bits  Differential Nonlinearity  less than ±1/2 LSB  Inherent Quantizing Error  less than ±1/2 LSB  System Accuracy  to within ±.05% FSR  Channel Crosstalk  -80dB at 1 kHz  Sample & Hold Droop Rate .ImV/ms Gain error  Adjustable to Zero  Zero Error  Adjustable to Zero  DYNAMIC PERFORMANCE Channel Acquisition Time to within 1/2 LSB  15 us  A/D Conversion Time 25 us A/D Throughput to System Memory Sample & Hold Aperture Uncertainty Sample & Hold Aperture Delay  13,700 samples per second (using READ A/D with DMA command) 10 ns 50 ns  Sample & Hold Feedthrough Attenuation 80 dB at 1kHz THERMAL CHARACTERISTICS A/D Zero Drift  ±20 uV / deg C (unipolar) 116  Appendix B: Data Acquisition System  ±20 ppm of FSR / deg C (bipolar) Amplifier Zero Drift  ±25 pV / deg C (±(3 pV / Deg C) x Gain)  Gain Drift  ±35 ppm of FSR / deg C  Differential linearity Drift  ±3 ppm of FSR / deg C  Monotonicity  Monotonic, 0 to +50 degrees C  The software that was used for the data acquisition was "DA", a program that was written in the Department for use with the DT2801 data acquisition board. The user is prompted for the start and end channels that should be sampled, the sampling time and frequency, and the filtering frequency. Data is written to a user specified file, with the first number being the sampling frequency, followed by as many columns of data as there were channels selected. The filtering frequency was always selected to be larger than the sampling frequency, thus the filtering in the program did not affect the data.  117  Appendix C Additional Tables and Figures  Slot Size  Slot Height  Slot  Nominal  Actual  Length  (mm)  (mm)  (mm)  15  15.0  253.0  13  13.0  254.0  11  11.0  257.5  9  9.0  255.0  7  7.0  255.0  5  5.0  257.0  4  3.7  256.5  3  3.0  240.0  2.5  2.5  211.0  2  2.0  211.5  1.5  1.5  211.0  1  1.0  211.0  Table C l Slot Dimensions  II?  Appendix C: Additional Tables and Figures  The equations that were found for the various values of Cd may be expressed in the general format Cd = Cl x (Hnd +C2) + C4 C3  The values of the coefficients that were found are as listed below in Table C.2  Slot Size  Cl  C2  C3  C4  Region  1  -0.0886  -1  -0.584  1.2  2 to 12  1.5  -3.563  0  -2.536  0.92  2 to 12  2  -5.253  0  -3.245  0.895  2 to 12  2.5  -1.23  0  -3.224  0.841  2 to 12  3  -0.01.9  -4.5  2  0.863  2 to 5  3  13.42  0  -3.87  0.83  5 to 12  4  -0.07  -3  2  0.727  2 to 3  4  0.246  0  -2.057  0.698  3 to 12  5  -0.008  -3.1  2  0.75  2 to 4  5  0.301  0  -1.732  0.71  4 to 12  7  0.428  0  -2.028  0.715  2 to 12  9  0.955  0  -2.988  0.703  2 to 12  11  0.246  0  -0.591  0.63  2 to!2  13  0.655  0  -2.529  0.69  2 to 12  15  0.306  0  -1.909  0.705  2 to 12  (mm)  Table C.2 Coefficient Values for Cd Equations  119  Appendix C: Additional Tables and Figures  Unsteady Discharge Without Sloshing 14 ;  Slot Height 15 mm Slot Length 253 nun Tank Length 895 mm Tank Width 300 mm No Tank Motion  12 10 •• 8 H„  6 4 " 2 0 — 0  10  15  20  25  30  Time, seconds Run 1, 4 & 5  — Predicted from Steady State  Figure C.l Discharge Through the 15 Millimetre Slot  Unsteady Discharge Without Sloshing 14 12  Slot Height 13 mm Slot Length 254 mm Tank Length 895 trim Tank Width 300 mm No Tank Motion  10 8 H nd  6  4 2 0 10  15  20  25  Time (seconds) Runs 1 to 5  — Predicted from Steady State  Figure C.2 Discharge Through the 13 Millimetre Slot  120  30  Appendix C: Additional Tables and Figures  Unsteady  Discharge Without  Sloshing  14 J Slot Height 11mm Slot Length 257.5mm Tank Length 895mm Tank Width 300mm No Tank Motion  12 10 • 8" t  6  4  0 i 0  1  1  1  1  1  1  5  10  15  20  25  30  Time (seconds) -  Runs 1 to 4  — Steady State Predicted  Figure C.3 Discharge Through the 11 Millimetre Slot  Unsteady  Discharge Without  Sloshing  16 14  Slot Height 7 m m  12 •  S l o t length 2 5 5 m m T a n k L e n g t h 895 m m  10 -  Tank Width 300 m m  8 -  N o Tank  Motion  6 4 2 0 0  10  15  20  25  30  Time (seconds)  Runs I to 4  — Predicted from Steady State  Figure C.4 Discharge Through the 7 Millimetre Slot  35  40  Appendix C: Additional Tables and Figures  Unsteady Discharge Without Sloshing 12  Slot Height 5 mm Slot Lengih Tank Length  257mm 895inm  lank Width 300mm No Tank Motion  H .t n  t>  x Predicted WLfli Scale Effects Considered Hresh Water Test Predicted Without Cosmdemig Scale Effects Soapy ¥/ater Test  0 15  20  25  30  Time, seconds  Figure C.5 Discharge Through the 5 Millimetre Slot  Figure C.6 Discharge Through the 4 Millimetre Slot  35  40  Appendix C: Additional Tables and Figures  Unsteady Discharge Without Sloshing  0  10  20  30  40  50  T i m e , seconds  Figure C.7 Discharge Through the 1 Millimetre Slot  60  Appendix C: Additional Tables and Figures  Run Time  2 0  0 s  Run 20 Time 1 s  Run 20 Time 3 s  Figure C.8 Sloshing Frames for 1.5 Millimetre Slot The slot is at the bottom right corner of the tank. 124  Appendix- C: Additional Tables and Figures  R u n 20 Time 4 s  1  R u n 20 Time 5 s  R u n 20 Time 6 s  R u n 20 Time 7 s  Figure C.8 Sloshing Frames for 1.5 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  125  Appendix C: Additional Tables and Figures  Run  1  Time  20 8 s  Run  20 Time 9 s  Run  20 T i m e 10 s  Run  20 T i m e 11 @  Figure C.8 Sloshing Frames for 1.5 Millimetre Slot, continued The slot is at the bottom right corner of the tank. 126  1  Appendix C: Additional Tables and Figures  Run 20 Time 12 s  Run  Time  13 13  Run 20 Time 15  Figure C.8 Sloshing Frames for 1.5 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  1.27  Appendix C: Additional Tables and Figures  R u n 20 T i m e 16 s  ========—.—1  =  R u n 20 1  Time  17 s  ===  ^—'x,  R u n 20 T i m e IB s  ~ — ' S  R u n 20 T i m e 19 s  Figure C.8 Sloshing Frames for 1.5 Millimetre Slot, continued The slot is at the bottom right corner of the tank. 128  Appendix C: Additional Tables and Figures  Run 7 Time 0 s  Run 7 Time 1 s  Run 7 Time 3 s  ~= z  ' \  Figure C.9 Sloshing Frames for 3 Millimetre Slot The slot is at the bottom right corner of the tank.  129  Appendix C: Additional Tables and Figures  Run 7 Time 4 s  Run 7 Time 5 s  Run 7 Time 6 s  Run 7 Time 7 s  Figure C,9 Sloshing Frames for 3 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  130  Appendix C: Additional Tables and Figures  Run 7 Time 8 s  Run 7 Time 9 s  Run 7 Time  10  s  ~  ~  •  Run 7 T i m e 11  Figure C.9 Sloshing Frames for 3 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  131  _  Appendix C: Additional Tables and Figures  Run 7 T i m e 14 s  Run 7 T i m e 15 s  ~  ' \  Figure C.9 Sloshing Frames for 3 Millimetre Slot, continued The slot is at the bottom right comer of the tank. 132  Appendix' C: Additional Tables and Figures  r  Run 7 Time 16 s  Run 7 Time 18 s  j  Run 7 Time 19 s  Figure C.9 Sloshing Frames for 3 Millimetre Slot, continued The slot is at the bottom right corner of the tank. 133  Appendix C: Additional Tables and Figures  Run 38 Time 0 s  R u n 38 Time 1 s  R u n 38 Time 2 s  Run 38 Time 3 s  Figure C.1.0 Sloshing Frames for 11 Millimetre Slot The slot is at the bottom right corner of the tank.  134  Appendix: C: Additional Tables and Figures  Run 38  Time 4 s  Run 38  Time 5 s  Run 38  Time 6 s  Run 38 Time 7  Figure CIO Sloshing Frames for 11 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  135  Appendix: C: Additional Tables and Figures  Run 38 Time B s  Run 38 Time 9 s  ~  —  -  1  Run 38 Time 10 s  —  _J  Run 38 Time 11 s  Figure C.10  Sloshing Frames for 1.1 Millimetre Slot, continued  The slot is at the bottom right corner of the tank.  136  Appendix C: Additional Tables and Figures  Run 38  Time 12 s  —  ' X  Figure CIO Sloshing Frames for 11 Millimetre Slot, continued The slot is at the bottom right corner of the tank. 137  Appendix C: Additional Tables and Figures  Run 38 Time 16 g  Figure CIO Sloshing Frames for 11 Millimetre Slot, continued The slot is at the bottom right corner of the tank.  138  

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