Tài liệu Cross-sectional stability of a two-inlet bay system

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Cross-sectional Stability of a Two Inlet Bay System Master Thesis C.S. Borsje March 2003 Faculty of Civil Engineering and Geosciences Section of Hydraulic Engineering Cross-sectional Stability of a Two Inlet Bay System Master Thesis Chris Borsje Supervisors: Prof.dr.ir. M.J.F. Stive Dr.Ir. A. van Mazijk Ir. H.J. Verhagen Prof.dr.ir. J. van de Kreeke (TU Delft) (TU Delft) (TU Delft) (University of Miami) TU Delft, Faculty of Civil Engineering and Geosciences, Section of Hydraulic Engineering Delft, March 2003 Preface This master thesis deals with a study on cross-sectional stability of estuaries with two inlets. This has been carried out within the framework of the Faculty of Civil Engineering at Delft University of Technology. Special thanks go to the examination committee, for their supervision during my work: prof. Stive, ir. Verhagen, dr. van Mazijk and prof. van de Kreeke for their devotion and much valuable information and assistance. Also, I would like to thank ir. Tghiem Tien Lam for helping me out with his report. Chris Borsje III IV Abstract Tidal inlets are connections between small, shallow bays and the sea. In a tide-dominated environment, the status of this inlet is a balance between local given conditions such as channel geometry, and the hydraulic environment. According to Escoffier, when the equilibrium is disrupted, it should generally return to its original state. In tidal lagoon systems with more inlets, it is more uncertain to predict a stable condition for each inlet. There are cases where some channels will close and others stay open, while other systems remain overall stable, i.e. each inlet has an equilibrium state. An analytical method to determine this stability is yet no available. Therefore, the focus attends to research a model bay system with two ocean inlets, where there could be an unconditionally stable environment. For this, the following important assumptions and starting angles are set up:    The water level in the bay area moves uniformly; Variations in ocean tide and phase levels may give new insights; A third inlet, imaginary or real, with resulting small water level differences in the bay, is added. For this case, a double inlet lagoon system is modelled: this consists of a bay area with no significant shape, two channels with relevant friction and inertia parameters and an idealized ocean tide, with a tidal range and phase at each inlet throat. Further, a variation of this is done by (virtually) splitting up the bay. On each model, a series of tests has been done with calculation and graphic software. Every inlet has an equilibrium flow curve, i.e. a cross-sectional area where the maximum currents are just enough to flush sediment and keeping the channel open. If this is the case for both cross-sections, the whole lagoon system is referred to as stable (E-type flow condition). In the graphic images, this is shown by the intersections of the equilibrium flow curves. The normal model bay system showed that there is a stable condition when the tidal range at the seaside is different at each inlet, provided that this is not too large. In addition, the channels should be sufficiently short, with friction not too high. This is also valid for higher tidal ranges, where the differences are even smaller. The bay system with the partition had the same stability conditions. Higher tide levels at both inlets could also ensure stability, as the third inlet becomes relatively small. Next, the E-type flow condition is more likely if the third inlet is smaller or longer. Further, the ratio in bay area on either side of the partition channel has some negative influence on this, but in combination with tide difference, a balanced system can remain. The models are further implemented on a present-day situation, a lagoon inlet system located in the Hue province in Vietnam. Here, there are two inlets, located far apart, letting the system be relatively stable. In this particular case, repeated flooding and breaching of the sand barrier resulted into another inlet next to the main inlet in the northern part of the lagoon. The appearance of this third channel has raised the question whether the two close inlets could be stable, although the authorities would rather have this channel closed. The partition model indicated that this can be the case. Although the lagoon is also largely influenced by wave action, frequent storm surges and river runoff, the tidal inlet model is a useful tool in order to understand the basic behaviour of this lagoon better. It should be understood that the models that were used could never fully represent real hydraulic environments, as there are always other influencing factors present. For tide-dominated estuaries, they can however be a valuable starting point. Nevertheless, the results shown by both models are satisfactory. A deeper investigation could be made on parts of this subject. V VI Table of Contents PREFACE ..................................................................................................................... III ABSTRACT ................................................................................................................... V TABLE OF CONTENTS.................................................................................................. VII LIST OF FIGURES ......................................................................................................... XI LIST OF TABLES ........................................................................................................ XIII CHAPTER 1 INTRODUCTION ....................................................................................... 1 CHAPTER 2 PROBLEM ANALYSIS ............................................................................... 3 2.1 Introduction .................................................................................................................................. 3 2.2 One - inlet bay system ................................................................................................................. 3 2.3 Two inlet bay system ................................................................................................................... 4 2.4 Goal of this study ......................................................................................................................... 8 CHAPTER 3 3.1 MODEL FOR CROSS–SECTIONAL STABILITY ............................................... 9 Principles of stability analysis ...................................................................................................... 9 3.2 Classification of equilibrium flow curves. ................................................................................... 11 3.2.1 Type A................................................................................................................................ 11 3.2.2 Type B................................................................................................................................ 12 3.2.3 Type C................................................................................................................................ 12 3.2.4 Type D................................................................................................................................ 13 3.2.5 Type E................................................................................................................................ 15 3.3 Inlet-bay schematization ............................................................................................................ 17 3.3.1 Two inlet bay system ......................................................................................................... 17 3.3.2 Two inlet bay system with partition .................................................................................... 18 VII CHAPTER 4 INLET HYDRODYNAMICS ......................................................................... 19 4.1 Introduction ................................................................................................................................ 19 4.2 One-inlet bay system ................................................................................................................. 19 4.3 Two-inlet bay system ................................................................................................................. 22 4.4 Two-inlet bay system with partition ............................................................................................ 23 4.5 Verification of the solutions ........................................................................................................ 24 CHAPTER 5 5.1 CALCULATIONS .................................................................................... 27 Introduction ................................................................................................................................ 27 5.2 Reference situations .................................................................................................................. 27 5.2.1 Two inlet bay system ......................................................................................................... 27 5.2.2 Two inlet bay system with partition .................................................................................... 28 5.2.3 Discussion.......................................................................................................................... 29 5.3 Symmetrical ocean tide conditions for both inlets...................................................................... 30 5.3.1 Variations with one parameter ........................................................................................... 30 5.3.2 Variations with two parameters.......................................................................................... 31 5.3.3 Discussion.......................................................................................................................... 31 5.4 Different ocean tide conditions for both inlets............................................................................ 32 5.4.1 Variations with one parameter ........................................................................................... 32 5.4.2 Variations with two parameters.......................................................................................... 39 5.4.3 Discussion.......................................................................................................................... 39 5.5 Phase differences inside the bay ............................................................................................... 40 5.5.1 Variations with one parameter ........................................................................................... 40 5.5.2 Variations with two parameters.......................................................................................... 48 5.5.3 Discussion.......................................................................................................................... 49 CHAPTER 6 APPLICATION TO THE TAM GIANG – CAU HAI LAGOON, VIETNAM ............ 51 6.1 Introduction ................................................................................................................................ 51 6.2 Description of the system........................................................................................................... 52 6.3 Objectives .................................................................................................................................. 53 6.3.1 Simulation excluding the Hoa Duan inlet ........................................................................... 54 6.3.2 Simulation excluding the Thuan An inlet............................................................................ 56 6.3.3 Simulations excluding the Tu Hien inlet ............................................................................. 57 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS ................................................ 59 REFERENCES .............................................................................................................. 61 APPENDICES ................................................................................................................. 1 VIII APPENDIX A SOLUTION FOR A TWO INLET BAY SYSTEM ................................................. 2 APPENDIX B SOLUTION FOR A TWO INLET BAY SYSTEM WITH PARTITION ......................... 6 APPENDIX C CALCULATIONS ON THE TWO INLET BAY SYSTEM ................................... 10 C.1. Reference situation .................................................................................................................... 10 C.2. Variations in tidal wave .............................................................................................................. 11 C.2.1. Tidal Period ........................................................................................................................ 11 C.2.2. Tidal amplitude................................................................................................................... 12 C.2.3. Short lengths combined with tidal amplitude ..................................................................... 13 C.2.4. Tidal amplitude difference.................................................................................................. 14 C.2.5. Short lengths combined with an amplitude difference ....................................................... 15 C.2.6. Tide imbalance at higher tidal ranges ................................................................................ 16 C.2.7. Phase lag ........................................................................................................................... 17 C.2.8. Tidal difference in both amplitude and phase .................................................................... 18 C.3. Inlet characteristics .................................................................................................................... 19 C.3.1. Short Inlets ......................................................................................................................... 19 C.3.2. Longer inlets....................................................................................................................... 20 C.3.3. Differences in inlet length................................................................................................... 21 C.3.4. Friction ............................................................................................................................... 22 APPENDIX D D.1. CALCULATIONS ON THE TWO INLET BAY SYSTEM WITH PARTITION ........... 24 Reference situation .................................................................................................................... 24 D.2. Variations in tide and inlet parameters ...................................................................................... 25 D.2.1. Tidal amplitude................................................................................................................... 25 D.2.2. Short lengths combined with tidal amplitude ..................................................................... 26 D.2.3. Amplitude difference .......................................................................................................... 27 D.2.4. Short lengths combined with amplitude difference ............................................................ 28 D.2.5. Stability conditions for increasing tidal difference .............................................................. 29 D.2.6. Stability conditions for increasing tidal difference (big partition)........................................ 30 D.2.7. Short Inlets ......................................................................................................................... 31 D.3. Variations with partition inlet ...................................................................................................... 32 D.3.1. Large cross-sectional area................................................................................................. 32 D.3.2. Larger cross-sectional area ............................................................................................... 33 D.3.3. Small cross-sectional area................................................................................................. 34 D.3.4. Long length ........................................................................................................................ 35 D.3.5. Sensitivity of partition inlet ................................................................................................. 36 D.4. Variations with bay ratio............................................................................................................. 41 D.4.1. Bay Ratio............................................................................................................................ 41 D.4.2. Bay Ratio with small partition inlets ................................................................................... 42 D.4.3. Bay Ratio with tidal amplitude difference........................................................................... 49 APPENDIX E VIETNAM INLET AND BAY PARAMETERS OF THE TAM GIANG - CAU HAI LAGOON, ........................................................................................................... 54 IX APPENDIX F SIMULATIONS ON THE TAM GIANG - CAU HAI LAGOON, VIETNAM ............ 58 F.1. Hoa Duan inlet closed, with partition ......................................................................................... 58 F.2. Thuan An inlet closed, with partition .......................................................................................... 61 F.3. Tu Hien inlet closed ................................................................................................................... 64 F.4. Tu Hien inlet closed with partition .............................................................................................. 67 X List of Figures Figure 2.1. Figure 2.2. Figure 2.3. Figure 2.4. Figure 2.5. Figure 2.6. Figure 3.1. Figure 3.2. Figure 3.3. Figure 3.4. Figure 3.5. Figure 3.6. Figure 3.7. Figure 3.8. Figure 3.9. Figure 3.10. Figure 4.1. Figure 4.2. Figure 4.3. Figure 4.4. Figure 4.5. Figure 5.1. Figure 5.2. Figure 5.3. Figure 5.4. Figure 5.5. Figure 5.6. Figure 5.7. Figure 5.8. Figure 5.9. Figure 5.10. Figure 5.11. Figure 5.12. Figure 5.13. Figure 5.14. Figure 5.15. Figure 5.16. Figure 6.1. Figure 6.2. Figure 6.3. Figure 6.4. Figure 6.5. Figure 6.6. Closure curve of a stable tidal inlet. ................................................................................. 4 Closure surface of Inlet 1 plotted against Inlet 2.............................................................. 5 Equilibrium flow curve of Inlet 1, plotted against Inlet 2................................................... 5 Closure surface of Inlet 2 plotted against Inlet 1.............................................................. 6 Equilibrium flow curve of Inlet 2, plotted against Inlet 1................................................... 6 Possible configurations of equilibrium flow areas. ........................................................... 7 Cross-sectional area – Tidal Prism Relationship for inlets in the U.S.A. ....................... 10 Type A flow curve........................................................................................................... 11 Type B flow curve........................................................................................................... 12 Type C flow curve. ......................................................................................................... 13 Type D flow curve. ......................................................................................................... 13 Examples of instability in type D. ................................................................................... 14 Type E flow curve........................................................................................................... 15 Examples of stability in Type E. ..................................................................................... 16 Two - inlet bay system. .................................................................................................. 17 Two inlet bay system with partition. ............................................................................... 18 Triangular cross section................................................................................................. 20 Closure scenario. ........................................................................................................... 20 Top view of Matagorda Bay. In the centre below is Pass Cavallo; a little to the north, splitting the barrier, is Matagorda Inlet........................................................................... 24 Equilibrium flow curves for Pass Cavallo and Matagorda Inlet (van de Kreeke, 1990). 25 Equilibrium flow curves for Pass Cavallo and Matagorda inlet, calculated by the two inlet bay system model. ................................................................................................. 26 Equilibrium flow curve for the two inlet bay system. ...................................................... 28 Equilibrium flow curve for the two inlet bay system with the partition............................ 29 Changes in the equilibrium flow curves around ĥ01 = 0.59 m. ...................................... 34 Equilibrium flow curves at higher tide differences.......................................................... 35 Closure surfaces of Inlets 1 and 2 at increasing tide differences. ................................. 36 Equilibrium flow curves at increasing phase differences. .............................................. 38 E-type flow curve as a result of tide difference. ............................................................. 40 E-type flow curve, caused by a smaller partition inlet.................................................... 41 Limit E-type condition at A3 = 2900 m2 and a main tide of 0.50 m. .............................. 42 Limit E-type condition at A3 = 6500 m2 and a main tide of 1.00 m. .............................. 43 Limit E-type condition at A3 = 10400 m2 and a main tide of 1.50 m. ............................ 43 Limit E-type condition at A3 = 14000 m2 and a main tide of 2.00 m. ............................ 44 Limit E-type condition at A3 = 18500 m2 and a main tide of 2.50 m. ............................ 44 Water level amplitude differences inside the bay at a bay ratio of 1 (a), 2 (b) and 10 (c). ....................................................................................................................................... 46 Phase level differences inside the bay at a bay ratio of 1 (a), 2 (b) and 10 (c). ............ 47 Equilibrium flow curve at ĥ01 = 0.56 m. .......................................................................... 48 Map of the Tam Giang – Cau Hai lagoon system.......................................................... 51 Overview of lagoon areas and inlets with cross-sectional profiles. ............................... 52 Model of the lagoon system. .......................................................................................... 53 3D image of the closure curve of Thuan An inlet (cross section A1), plotted against Tu Hien inlet (cross section A2). .......................................................................................... 54 3D image of the closure curve of Tu Hien inlet (cross section A2), plotted against Thuan An inlet (cross section A1). ............................................................................................. 54 Equilibrium flow curves for Thuan An inlet and Tu Hien inlet, at ∆φ = 0.18π rad .......... 55 XI Figure 6.7. Figure 6.8. Figure 6.9. Figure 6.10. Figure D.1. Figure D.2. Figure D.3. Figure D.4. Figure D.5. Figure D.6. Figure D.7. Figure E.1. Figure F.1. Figure F.2. Figure F.3. Figure F.4. Figure F.5. Figure F.6. Figure F.7. Figure F.8. Figure F.9. Figure F.10. Figure F.11. Figure F.12. Equilibrium flow curves for Hoa Duan inlet and Tu Hien inlet, at ∆φ = 0.16π rad.......... 56 Equilibrium flow curves for Thuan an and Hoa Duan inlet, at ∆φ = 0.02π rad............... 57 Equilibrium flow curves for Thuan An and Hoa Duan inlet, at ∆φ = 0.02π rad .............. 58 Closure surfaces for Thuan An inlet (left) and Hoa Duan inlet (right)............................ 58 Flow curves for amplitudes at inlet 1 of 0.50 m (green), 0.53 m (blue), 0.56 m (red) and 0.59 m (magenta)........................................................................................................... 29 Flow curves for amplitudes at Inlet 1 of 0.50 m (green), 0.53 m (blue), 0.56 m (red) and 0.59 m (magenta)........................................................................................................... 30 Equilibrium flow curves for Ab1/Ab2 values of 0.1 (red), 1 (green) and 10 (blue)............ 41 Equilibrium flow curves for bay area ratios of 0.10 (a), 1 (b) and 10 (c)........................ 49 Equilibrium flow curves for bay area ratios of 0.10 (a), 1 (b) and 10 (c)........................ 50 Equilibrium flow curves for bay area ratios of 0.10 (a), 1 (b) and 10 (c)........................ 51 Equilibrium flow curves for bay area ratios of 0.10 (a), 1 (b) and 10 (c)........................ 52 Map of the lagoon system with depth profiles................................................................ 54 Simulation with φ2 = 0.18π rad....................................................................................... 58 Simulation with φ2 = 0.0 ................................................................................................. 59 Simulation with φ2 = -0.18π rad...................................................................................... 60 Simulation with φ2 = 0.16π rad....................................................................................... 61 Simulation with φ2 = 0.0 ................................................................................................. 62 Simulation with φ2 = -0.16π rad...................................................................................... 63 Simulation with φ2 = 0.02π rad....................................................................................... 64 Simulation with φ2 = 0.0 ................................................................................................. 65 Simulation with φ2 = -0.02π rad...................................................................................... 66 Simulation with φ2 = 0.02π rad....................................................................................... 67 Simulation with φ2 = 0.0 ................................................................................................. 68 Simulation with φ2 = -0.02π rad...................................................................................... 69 XII List of Tables Table 4.1. Table 5.1. Table 5.2. Table 5.3. Table 5.4. Table 5.5. Table 5.6. Table 5.7. Table 5.8. Table 5.9. Table 5.10. Table 5.11. Table 5.12. Table D.1. Table D.2. Table D.3. Table D.4. Table D.5. Table D.6. Table D.7. Table D.8. Table D.9. Table D.10 Table E.1. Table E.2. Values of parameters used in Matagorda Bay............................................................... 25 Reference values. .......................................................................................................... 28 Reference values. .......................................................................................................... 29 Summary of simulation results with symmetrical tide conditions................................... 31 Observed variations in tide, limit flow areas and flow types. ......................................... 35 E-type conditions at different tide levels. ....................................................................... 37 Instability at higher phase differences. .......................................................................... 37 Summary of simulation results....................................................................................... 39 Summation of flow types at different partition cross sections on a tide of 0.50 m......... 42 Summation of flow types at different partition cross sections on a tide of 1.00 m......... 43 Summation of flow types at different partition cross sections on a tide of 1.50 m......... 44 Ratio between tidal amplitudes and partition cross-sections......................................... 45 Summary of flow types at different simulations. ............................................................ 49 Equilibrium cross-sectional values for different tidal amplitudes at inlet 1. ................... 29 Equilibrium cross-sectional values for different tidal amplitudes at inlet 1. ................... 30 Equilibrium cross-sectional values for different bay area ratios. ................................... 41 Equilibrium cross-sectional values for different bay area ratios. ................................... 42 Equilibrium cross-sectional values for different bay area ratios. ................................... 45 Equilibrium cross-sectional values for different bay area ratios. ................................... 47 Equilibrium cross-sectional values for different bay area ratios. ................................... 49 Equilibrium cross-sectional values for different bay area ratios. ................................... 50 Equilibrium cross-sectional values for different bay area ratios. ................................... 51 Equilibrium cross-sectional values for different bay area ratios. ................................... 52 Average inlet characteristics of the Tam Giang-Cau Hai lagoon system (Lam, 2002).. 55 Inlet parameters of the Tam Giang-Cau Hai lagoon system. ........................................ 56 XIII XIV Cross sectional stability of a two inlet bay system Chapter 1 Introduction Over the years, a lot of study has been done on coastal inlets and tidal basins. Estuaries or lagoons, connected by the outer ocean by one or more channels, are generally subjected to a various dynamic forces such as tides, wind waves, storm surges and river runoff. Next, long-shore and cross-shore currents continuously influence the state of this coastal environment by the sediment balance. In the case of strong long shore transport, the accretion of barrier spits and sand bar migration results into changing positions of these inlets. As far as location is concerned, these systems are unstable. Other inlets remain in relative position because of equilibrium in sediment bypassing, and have a more or less balanced cross section. After a disruption of this equilibrium, some of these estuaries can restore to their original situation over a course of time, while others require constant attention to avoid intolerable damage. There can be a number of reasons for the need to have stable tidal areas: safety for the hinterland, navigational demands for shipping through inlets, environmental and economical aspects are examples of this. In estuaries with more inlets, in some cases an overall stable environment can exist. Previous studies were undertaken to explain this condition analytically. With the approximations that were made, a workable solution to indicate this has not yet been found. This study will put further research in this problem by widening the approach. The scope will be on a tide-dominated bay system, with two ocean inlets. In the following chapter, a problem analysis is given, with a status of the situation as it is right now. Chapter 3 sets some boundaries and starting points on this project. Next, possible different configurations are defined, followed by to main models that will be used. In Chapter 4, the hydrodynamics of the tidal inlet systems modelled into a series of systems of differential equations. These is followed by analytical solutions and finally verified. The next chapter deals with simulations on both models. These are run in a calculation program and further graphically explained with MATLAB. From this, some conclusions will be drawn. In Chapter 6, the two basin models are used to find an explanation on the stability of a lagoon system in Vietnam that is frequently exposed to typhoons and floods. It thus might serve as a basis to understand this coastal area better. Finally, some conclusions are drawn, followed by a few recommendations for further study. 1 Cross sectional stability of a two inlet bay system 2 Cross sectional stability of a two inlet bay system Chapter 2 Problem analysis 2.1 Introduction Tidal inlets are connections between a bay or shallow water environment and the ocean. Each tidal period, water enters and leaves the bay. During flood, sediment is carried into the inlet. The ebb current flushes this sediment out again. In the inlet channel, large ebb and flood velocities can occur due to water level differences on each side of the channel. This results in a maximum value of bottom shear stress through the channel: τ = cf ⋅ ρ ⋅u2 (1) in which cf = friction factor; ρ = density of water u = vertically averaged velocity When dealing with inlet stability, a main distinction can be made between location stability and crosssectional stability. The littoral material transported to a tidal inlet can cause constriction of the inlet throat. The decrease in cross-sectional area will result in greater scouring capacities, which will cause erosion of the downdrift beaches: the inlet tends to migrate, depending on the rate of sediment supply, wave energy and tidal currents. A severe storm can open a new inlet, which makes the old one obsolete, and repeats the process of downdrift migration. This is an example of an unstable inlet as far as location in combination with cross section is concerned. Tidal inlets with a stable throat position have a non-migrating ebb channel. Sand bypassing at these inlets occur through the formation of bars which migrate and attach to the downdrift coast. New bars will constantly develop due to a more or less continuous sand delivery by the ebb tidal channel. Thus, sediment bypassing in this case depends on sand deliveries through the channel system. Subsequently, a stable cross section is largely the result of large ebb currents, which are capable of carrying enough sediment out of the inlet. From here on, the location of an inlet is considered stable; the focus will lie on cross-sectional stability. 2.2 One - inlet bay system Stability, or more precisely the cross-sectional stability, deals with the equilibrium between the inlet’s cross-sectional area and inlet hydrodynamics. The pertinent parameters are the actual tidal maximum of the bottom shear stress τ and the equilibrium shear stress τeq. The equilibrium shear stress is defined as the bottom shear stress induced by the tidal current that is required to flush the sediments carried into the inlet by the long shore currents. The corresponding tidal maximum in the vertically averaged velocity is the equilibrium velocity. Among other things, the value of the equilibrium shear stress is a function of the littoral drift; the larger the littoral drift, the larger the equilibrium shear stress. Escoffier (1940) showed that when the actual current velocity equals the equilibrium velocity, the inlet is in equilibrium with the hydraulic environment. When the actual shear stress is larger than the equilibrium shear stress, the inlet is in scouring mode; when the actual shear stress is smaller than the equilibrium shear stress, the inlet is in shoaling mode. This relationship is shown in figure 2.1. 3 Cross sectional stability of a two inlet bay system uc ueq u = ueq SP1 SP2 Ac Figure 2.1. Closure curve of a stable tidal inlet. Here, the curve represents the inlet’s current velocity as a function of its cross-sectional area Ac. An inlet is referred to as stable when the closure curve has two points of intersection with the equilibrium velocity curve. The first intersection (SP1) is an unstable equilibrium: if Ac is located left of this point, the maximum current is too small to flush sediment through the channel, so it will eventually close. When Ac is located right of SP1, the maximum velocity will increase, the inlet will scour until it reaches the stable intersection point SP2. If Ac is larger than SP2, the tidal currents are too small. Then the inlet will gradually shoal until SP2 is reached. The equilibrium is therefore called stable if after a small change, for example a severe storm, when the value of Ac is moved, the inlet’s cross section returns to its original equilibrium value. When the closure curve lies entirely below the equilibrium velocity there are no intersections, and therefore no equilibrium cross-sectional areas. The inlet will close. 2.3 Two inlet bay system For two inlets connected to the same bay, it is interesting to study the closure curve of one inlet in comparison to the other. When the cross section of Inlet 2 (A2) is 0, Inlet 1 will show a closure curve as shown in figure 2.1. If A2 increases, more flow is transported through Inlet 2, so the maximum velocities in Inlet 1 will decrease, causing the closure curve to ‘flatten’ out more and more. Eventually, there will be a value of A2 where Inlet 1 cannot reach its equilibrium velocity, making it likely to close. This process is illustrated in figure 2.2. A1 is on the x-axis, A2 on the y-axis and û1 (the maximum current in Inlet 1) on the z-axis. For every value of A2, a closure curve for A1 can be drawn. Together they form the closure surface for û1 (A1, A2). The physically interesting part of each closure curve is the stability interval, starting at SP1. Further, to obtain a possible stable cross section for each inlet, the maximum current should be larger than the equilibrium current. In Figure 1, this is the interval of SP1 – SP2. For increasing values of A2, this interval will narrow. It is therefore convenient to portray this interval for all values of Inlet 1 where û1 = ûeq. This is shown in Figure 2.3. In other words, this is a contour plot from figure 2.2 of all combinations of A1 and A2 where û1 = û eq, projected in the (A1, A2) plane. This curve is further referred to as the equilibrium flow curve. The thick part of the curve is the stable section: this runs from (A1, min; A2, max), a 4
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