EXPERIMENTAL AND NUMERICAL STUDY OF SPOILER EFFECT ...

Mansoura University Faculty of Engineering Mechanical Power Eng. Dept.

EXPERIMENTAL AND NUMERICAL STUDY OF SPOILER EFFECT ON SHIP BALANCE Thesis Submitted in Partial Fulfillment of Requirements for the Master of Science in Mechanical Power Engineering

By Eng. Emad Mohammed Saad El-Said B.Sc. of Mechanical Power Engineering Faculty of Engineering Mansoura University Under Supervision of

Prof. Dr. Magdy Abou Rayan

Prof. Dr. Nabil H. Mostafa

Mechanical Power Engineering Dept. Faculty of Engineering Mansoura University

Mechanical Power Engineering Dept. Faculty of Engineering Zagazig University

Assoc. Prof. Dr. Berge Djebedjian Mechanical Power Engineering Dept. Faculty of Engineering Mansoura University 2009

‫ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ‬

‫ﺃﹶﻟﹶ ‪‬ﻢ ﺗ‪‬ﺮ‪ ‬ﺃﹶﻥﱠ ﺍﻟﹾ ﹸﻔﻠﹾﻚ‪ ‬ﺗ‪ ‬ﺠﺮِﻱ ﻓِﻲ ﺍ ﹾﻟﺒ‪ ‬ﺤﺮِ‬ ‫ﻦ ﺁﻳ‪‬ﺎﺗِﻪِ ﺇِﻥﱠ ﻓِﻲ‬ ‫ﻜ ‪‬ﻢ ﻣِ ‪‬‬ ‫ﺑِﻨِ ‪‬ﻌﻤ‪‬ﺖِ ﺍﻟﻠﱠﻪِ ﻟِ‪‬ﻴﺮِﻳ‪ ‬ﹸ‬ ‫ﻜﻞﱢ ﺻ‪‬ﺒ‪‬ﺎﺭٍ ﺷ‪‬ﻜﹸﻮﺭٍ‬ ‫ﺫﹶﻟِﻚ‪ ‬ﻟﹶﺂﻳﺎﺕٍ ﻟِ ﹸ‬ ‫ﺻﺪﻕ ﺍﷲ ﺍﻟﻌﻈﻴﻢ‬ ‫اﻵﻳﺔ )‪ (31‬ﺳﻮرة ﻟﻘﻤﺎن‬

Dedicated to my beloved parents and my wife whose constant prayers, sacrifice and inspiration led to this work

Approval Sheet

Experimental and Numerical Study of Spoiler Effect on Ship Balance Researcher Name: Eng. Emad Mohammed Saad El-Said Supervisors Name Prof. Dr. Magdy Mohamed Abou Rayan Prof. Dr. Nabil Hassan Mostafa Assoc. Prof. Berge Djebedjian

Position Mechanical Power Engineering Department, Mansoura University Mechanical Power Engineering Department, Zagazig University Mechanical Power Engineering Department, Mansoura University

Signature

Head of Department

Vice Dean

Dean

Prof. Dr. Lotfy Rabie

Prof. Dr. Mohamed El-Shabrawy Mohamed Ali

Prof. Dr. Mohamed El-Shabrawy Mohamed Ali

Experimental and Numerical Study of Spoiler Effect on Ship Balance

Researcher Name: Eng. Emad Mohammed Saad El-Said Supervisors Name Prof. Dr. Magdy Mohamed Abou Rayan Prof. Dr. Nabil Hassan Mostafa Assoc. Prof. Berge Djebedjian

Position Mechanical Power Engineering Department, Mansoura University Mechanical Power Engineering Department, Zagazig University Mechanical Power Engineering Department, Mansoura University

Signature

Examining Committee Name Prof. Dr. Sadek Zakaria Kassab Prof. Dr. Mohamed Safwat Saad El-Din Prof. Dr. Magdy Mohamed Abou Rayan Prof. Dr. Nabil Hassan Mostafa

Position Mechanical Engineering Department, Alexandria University Mechanical Power Engineering Department, Mansoura University Mechanical Power Engineering Department, Mansoura University Mechanical Power Engineering Department, Zagazig University

Signature

Head of Department

Vice Dean

Dean

Prof. Dr. Lotfy Rabie

Prof. Dr. Mohamed ElShabrawy Mohamed Ali

Prof. Dr. Mohamed ElShabrawy Mohamed Ali

ACKNOWLEDGMENTS

First of all, I thank Allah the merciful, for helping me to complete this work. I would like to express my deep gratitude and sincere appreciation to Prof. Dr. Magdy M. Abou Rayan, Prof. Dr. Nabil Hassan Mostafa and Assoc. Prof. Dr. Berge Ohanness Djebedjian for their assistance, offering all possible facilities, continuous guidance and encouragement at all stages of this work. Finally, I want to express my great appreciation to all the staff of Mechanical Power Engineering Department for their encouragement.

ABSTRACT This thesis discusses the effect of hull-mounted cavitating spoilers system on ship balance. Three-dimensional flow field around spoiler was simulated experimentally and computationally with injection of air or exhaust gas behind spoiler. The bow spoilers consist of an even number of sections arranged port and starboard forward of the center of mass of the ship. The stern spoilers consist of an even number of sections arranged port and starboard attached to the ship transom or transom plate. Injecting exhaust gas stabilizes cavities behind spoilers. The forces and moments acting on the ship bottom depend on spoiler inclination angle, ship floor angle and air injection positions. The two-phase flow field around a ship spoiler with the free surface simulation in Piecewise Linear Interface Construction method is modeled numerically using a three-dimensional Navier-Stokes code, CFDRC. The governing equations are discretized on a structured grid using an Upwind Difference scheme. For different conditions, the bubbles shape, the three-dimensional flow field around the spoiler body and the pressure variation on the wake of the spoiler body are computed. Furthermore, the spoiler system was applied experimentally to understand the parameters affecting the flow field and bubble formation around ship spoiler and their influences on the hydrodynamic forces which are acting on the spoiler

I

body. These parameters are the spoiler inclination angle, ship floor angle and injected air position. The images of bubble formation and flow filed variation are recorded with scientific video camera and compared with the computed flow field at different conditions and time sequence. The experimental results are compared with the numerical ones at different conditions and time sequence. This comparison display a good agreement between the experimental and numerical simulation of bubble formation in the cases of spoiler with 60° and 90° inclination angles except the computational bubble shape has small dissimilarity and not flatted at bubble tail, and the computational bubble shape has slowly splitting and not attached to ship body in the case of spoiler with 30° inclination angle. This may be contributed to the laminar flow computation without including turbulence and buoyancy effects. The spoiler inclination angle had clear effect on the moment value around the spoiler fixation line. The effect of spoiler inclination angle has the more prominent effect on ship balance than the other two parameters of floor angle and injection position. The high moment values were produced in case of 90° spoiler inclination angle and the small values were produced in case of 30° after 1 second. Therefore, changing spoiler inclination angle, ship floor angle and position relative to injection holes produces different bubble shapes and consequently different forces are introduced to control the roll, pitch motion and speed of the ship leading to ship balance.

II

CONTENTS Subject

Page

LIST OF FIGURES .......................................................................................... I LIST OF TABLES ......................................................................................... IX NOMENCLATURE ......................................................................................... X ABBREVIATIONS ..................................................................................... XIII

CHAPTER 1. INTRODUCTION ................................................................... 1

CHAPTER 2. LITERATURE REVIEW ....................................................... 7 2.1 Introduction ....................................................................................... 7 2.2 Ship Stability Techniques ................................................................. 7 2.3 Studies Related to Flow Field around Spoiler ................................ 10 2.4 Research Related to Free Surface ................................................... 15 2.5 Original Contributions and Main Objectives .................................. 16

CHAPTER 3.

THEORETICAL ANALYSIS AND EXPERIMENTAL TECHNIQUES .................................................................... 19

3.1 Introduction ..................................................................................... 19 3.2 Theoretical Analysis ........................................................................ 19 3.2.1

Similarity for the Spoiler and the Air Injection Holes ....... 19

3.2.2

Prototype’s Spoiler and Injection Hole Calculations ......... 23

3.3 Experimental Test Rig ..................................................................... 24 3.3.1

Air Injection System ........................................................... 25

3.3.2

Lighting System ................................................................. 26

3.3.3

Imaging System .................................................................. 27

3.3.4

Water Tunnel Test Section System .................................... 28 3.3.4.1

Ship Bottom Simulator ....................................... 28

3.3.4.2

Water Channel .................................................... 29

3.4 Measurement Devices ..................................................................... 30 3.5 Test Cases ........................................................................................ 30 3.6 Test Procedure ................................................................................. 31

CHAPTER 4. NUMERICAL PROCEDURE .............................................. 33 4.1 Introduction ..................................................................................... 33 4.2 Computational Fluid Dynamics Tools ............................................ 33 4.3 Governing Equations ....................................................................... 41 4.4 Code Characteristics ........................................................................ 44 4.4.1

Geometry and Grid Generation Tools ................................ 45

4.4.2

Solver Setup Interface and Advanced Polyhedral Solver .. 46

4.4.3

Post Processor ..................................................................... 47 4.4.3.1

4.4.4

Visualization Process .......................................... 47

Discretization ...................................................................... 48

4.4.5

4.4.4.1

Finite Volume Method ........................................ 48

4.4.4.2

Transient Term .................................................... 50

4.4.4.3

Convective Term ................................................. 50

4.4.4.4

Diffusion Term ................................................... 55

4.4.4.5

Source Term Linearization ................................. 56

Velocity-Pressure Coupling ............................................... 57 4.4.5.1

Continuity and Mass Evaluation ......................... 57

4.4.5.2

Pressure Correction and “SIMPLEC” Algorithm ............................................................ 58

4.4.6

Boundary Conditions .......................................................... 63

4.4.7

Solution Methods ............................................................... 65 4.4.7.1

Overall Solution Procedure ................................. 65

4.4.7.2

Under-Relaxation ................................................ 68

4.4.7.3

Linear Equation Solvers ..................................... 69

4.5 Free Surface Simulation .................................................................. 71 4.5.1

Introduction ........................................................................ 71

4.5.2

Theory Background ............................................................ 72

4.5.3

Surface Reconstruction ...................................................... 74

4.6 Numerical Model ............................................................................. 77 4.6.1

Problem Statement ............................................................. 77

4.6.2

Grid Structure ..................................................................... 78

4.6.3

Boundary Conditions .......................................................... 78

4.6.4

Choice of Physical Time-Step ............................................ 80

4.6.5

The Run Time for a Simulation ......................................... 80

4.6.6

Numerical Solution Basic Assumptions ............................ 80

4.6.7

Convergence Criterion and Solver Control Settings .......... 82

4.7 Post-Processing Calculations .......................................................... 83 4.7.1

Forces and Moments Calculations ..................................... 83

4.7.2

Drag, Lift and Heel Moment Coefficients Calculations .... 84

CHAPTER 5. RESULTS AND DISCUSSION ............................................ 87 5.1 Introduction ..................................................................................... 87 5.2 The Spoiler Inclination Angle Effect .............................................. 91 5.2.1

Set (1) ................................................................................. 91 5.2.1.1

5.2.2

Set (2) ............................................................................... 105 5.2.2.1

5.2.3

Results and Discussion ....................................... 91

Results and Discussion ..................................... 105

Drag, Heel Moment and Lift Coefficients for All Cases ..................................................................... 109

5.3 The Injection Position Effect ........................................................ 119 5.3.1

Set (1) ……… ................................................................... 119 5.3.1.1

5.3.2

Set (2) ............................................................................... 132 5.3.2.1

5.3.3



Drag, Heel Moment and Lift Coefficients

for All Cases ..................................................................... 135 5.4 The Floor Angle Effect ................................................................. 145 5.4.1

Set (1) ……… ................................................................... 145 5.4.1.1

5.4.2

Set (2) ............................................................................... 154 5.4.2.1

5.4.3


Set (5) ............................................................................... 172 5.4.5.1

5.4.6


Set (4) ............................................................................... 166 5.4.4.1

5.4.5


Set (3) ............................................................................... 160 5.4.3.1

5.4.4



Drag, Heel Moment and Lift Coefficients for All Cases ..................................................................... 177

5.5 Ship Control Optimization ............................................................ 177

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS ............ 181 6.1 Conclusions ................................................................................... 181 6.2 Recommendations ......................................................................... 183 REFERENCES .............................................................................................. 185 APPENDIX A: Ship Glossary ...................................................................... 191 APPENDIX B: Experimental Error Analysis ............................................ 199

LIST OF FIGURES Figure 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1

Page Ship degrees of freedom, Clark (2002) ................................................ 2 Surging motion ..................................................................................... 2 Swaying motion ................................................................................... 2 Heaving motion .................................................................................... 3 Rolling motion ..................................................................................... 3 Rolling moment, Clark (2002) ............................................................. 3 Pitching motion .................................................................................... 3 Yawing motion ..................................................................................... 3 Example of ship spoiler locations, Soper et al. (1998) ........................ 5 Air controlled tank stabilizer, Muckle (1996) ...................................... 8 Stabilizer fin arrangements, Muckle (1996) ........................................ 9 Bilge keel, Tupper (2004) .................................................................... 9 Method of roll control at boundary layer, Brasher et al. (2001) ……. 12 Experimental rig: side view, Klaka et al. (2005) ............................... 13 Geometric similarity in model testing ................................................ 19 Experimental set-up ........................................................................... 24 Air injection diagram ......................................................................... 26 Simulator plate dimensions ................................................................ 28 Simulator plate construction .............................................................. 28 Water channel ..................................................................................... 29 Computational cell (C.V) ................................................................... 49 Computational boundary cell ............................................................. 64 CFDRC solution flowchart ................................................................ 67 Structured grid of flow field around spoiler ...................................... 78 Two grids over lapping ...................................................................... 81 Two faces with same grid .................................................................. 81 Forces acting on spoiler face at X-Y plane ........................................ 84 Forces acting on spoiler face at Y-Z plane ........................................ 84 Total pressure variation during transient time period 1 second for θ = 30° .......................................................................................... 88

I

5.2 5.3 5.4 5.5 5.6 5.7

5.8 5.9 5.10 5.11 5.12 5.13

5.14 5.15 5.16 5.17

Total pressure variation during transient time period 1 second for θ = 60° .......................................................................................... 89 Total pressure variation during transient time period 1 second for θ = 90° .......................................................................................... 90 Cavities formation due to single injection from Hole (1) with θ = 30° and β = 10° (at Z = -5L) ......................................................... 97 Cavities formation due to single injection from Hole (1) with θ = 60° and β = 10° (at Z = -5L) ......................................................... 98 Cavities formation due to single injection from Hole (1) with θ = 90° and β = 10° (at Z = -5L) ......................................................... 99 Velocity vectors and density around spoiler at different θ and t = 1 sec. with single injection from Hole (1) and β = 10° (at Z = -5L) ....................................................................................... 100 Static pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (1) and β = 10° (at Z = -5L) .................. 101 Total pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (1) and β = 10° (at Z = -5L) .................. 102 Total pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (1) and β = 10° (at X-Z) ........................ 103 Total pressure on spoiler faces at different θ and t = 1 sec. with single injection from Hole (1) and β = 10° ...................................... 104 Drag coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (1) and β = 10° .................... 105 Heel moment coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (1) and β = 10° .............................................................................................. 105 Lift coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (1) and β = 10° .................... 105 Cavities formation due to single injection from Hole (2) with θ = 30° and β = 10° (at Z = -5L) ....................................................... 110 Cavities formation due to single injection from Hole (2) with θ = 60° and β = 10° (at Z = -5L) ....................................................... 111 Cavities formation due to single injection from Hole (2) with θ = 90° and β = 10° (at Z = -5L) ....................................................... 112

II

5.18

5.19 5.20 5.21 5.22 5.23 5.24

5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32

Velocity vectors and density around spoiler at different θ and t = 1 sec. with single injection from Hole (2) and β = 10° (at Z = -5L) ....................................................................................... 113 Static pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (2) and β = 10° (at Z = -5L) .................. 114 Total pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (2) and β = 10° (at Z = -5L) .................. 115 Total pressure around spoiler at different θ and t = 1 sec. with single injection from Hole (2) and β = 10° (at X-Z) ........................ 116 Total pressure on spoiler faces at different θ and t = 1 sec. with single injection from Hole (2) and β = 10° ...................................... 117 Drag coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (2) and β = 10° .................... 118 Heel moment coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (2) and β = 10° .............................................................................................. 118 Lift coefficient with different spoiler inclination angles after 1 sec. with single injection from Hole (2) and β = 10° .................... 118 Drag moment coefficient for all studied cases with different spoiler inclination angles after 1 sec. ............................................... 118 Heel moment coefficient for all studied cases with different spoiler inclination angles after 1 sec. ............................................... 118 Lift coefficient for all studied cases with different spoiler inclination angles after 1 sec. ........................................................... 118 Cavities formation due to double injection from Holes (1) and (2) with θ = 30° and β = 5° (at Z = -5L) ........................................................... 124 Cavities formation due to single injection from Hole (1) with θ = 30° and β = 5° (at Z = -5L) ........................................................ 125 Cavities formation due to single injection from Hole (2) with θ = 30° and β = 5° (at Z = -5L) ......................................................... 126 Velocity vectors and density around spoiler at different injection positions and t = 1 sec. with θ = 30° and β = 5° (at Z = -5L) ....................................................................................... 127

III

5.33 5.34 5.35 5.36 5.37 5.38 5.39 5.40 5.41 5.42 5.43

5.44 5.45 5.46 5.47 5.48

Static pressure around spoiler at different injection positions and t = 1 sec. with θ = 30° and β = 5° (at Z = -5L) .......................... 128 Total pressure around spoiler at different injection positions and t = 1 sec. with θ = 30° and β = 5° (at Z = -5L) .......................... 129 Total pressure around spoiler at different injection positions and t = 1 sec. with θ = 30° and β = 5° (at X-Z) ................................ 130 Total pressure on spoiler faces at different injection positions and t = 1 sec. with θ = 30° and β = 5° .............................................. 131 Drag coefficient with different injection positions after 1 sec. with θ = 30° and β = 5° .................................................................... 132 Heel moment coefficient with different injection positions after 1 sec. with θ = 30° and β = 5° .................................................. 132 Lift coefficient with different injection positions after 1 sec. with θ = 30° and β = 5° .................................................................... 132 Cavities formation due to double injection from Holes (1) and (2) with θ = 90° and β = 5° (at Z = -5L) ........................................... 136 Cavities formation due to single injection from Hole (1) with θ = 90° and β = 5° (at Z = -5L) ........................................................ 137 Cavities formation due to single injection from Hole (2) with θ = 90° and β = 5° (at Z = -5L) ......................................................... 138 Velocity vectors and density around spoiler at different injection positions and t = 1 sec. with θ = 90° and β = 5° (at Z = -5L) ....................................................................................... 139 Static pressure around spoiler at different injection positions and t = 1 sec. with θ = 90° and β = 5° (at Z = -5L) .......................... 140 Total pressure around spoiler at different injection positions and t = 1 sec. with θ = 90° and β = 5° (at Z = -5L) .......................... 141 Total pressure around spoiler at different injection positions and t = 1 sec. with θ = 90° and β = 5° (at X-Z) ................................ 142 Total pressure on spoiler faces at different injection positions and t = 1 sec. with θ = 90° and β = 5° .............................................. 143 Drag coefficient with different injection positions after 1 sec. with θ = 90° and β = 5° .................................................................... 144

IV

5.49 5.50 5.51 5.52 5.53 5.54 5.55 5.56

5.57

5.58

5.59

5.60

5.61 5.62

Heel moment coefficient with different injection positions after 1 sec. with θ = 90° and β = 5° .................................................. 144 Lift coefficient with different injection positions after 1 sec. with θ = 90° and β = 5° .................................................................... 144 Drag coefficient for all studied cases with different injection positions after 1 sec. ......................................................................... 144 Heel moment coefficient for all studied cases with different injection positions after 1 sec. .......................................................... 144 Lift coefficient for all studied cases with different injection positions after 1 sec. ......................................................................... 144 Cavities formation due to double injection from Holes (1) and (2) with θ = 60° and β = 5° (at Z = -5L) ........................................... 149 Cavities formation due to double injection from Holes (1) and (2) with θ = 60° and β = 10° (at Z = -5L) ......................................... 150 Velocity vectors and density around spoiler at different floor angles and t = 1 sec. with θ = 60° and double injection from Holes (1) and (2) with (at Z = -5L) .................................................. 151 Static pressure around spoiler at different floor angles and t = 1 sec. with θ = 60° and double injection from Holes (1) and (2) with (at Z = -5L) ......................................................................... 151 Total pressure around spoiler at different floor angles and t = 1 sec. with θ = 60° and double injection from Holes (1) and (2) with (at Z = -5L) ......................................................................... 152 Total pressure around spoiler at different floor angles and t = 1 sec. with θ = 60° and double injection from Holes (1) and (2) with (at X-Z) ............................................................................... 152 Total pressure on spoiler faces at different floor angles and t = 1 sec. with θ = 60° and double injection from Holes (1) and (2) ..................................................................................................... 153 Drag coefficient with different floor angles after 1 sec. with θ = 60° and double injection from Holes (1) and (2) ....................... 153 Heel moment coefficient with different floor angles after 1 sec. with θ = 60° and double injection from Holes (1) and (2) ..................................................................................................... 153

V

5.63 5.64 5.65

5.66

5.67

5.68

5.69 5.70

5.71 5.72 5.73

5.74

5.75

Lift coefficient with different floor angles after 1 sec. with θ = 60° and double injection from Holes (1) and (2) ....................... 153 Cavities formation due to double injection from Holes (1) and (2) with θ = 90° and β = 10° (at Z = -5L) ......................................... 157 Velocity vectors and density around spoiler at floor angle β = 10° and t = 1 sec. with θ = 90° and double injection from Holes (1) and (2) (at Z = -5L) ........................................................... 158 Static and total pressures around spoiler at floor angle β = 10° and t = 1 sec. with θ = 90° and double injection from Holes (1) and (2) (at Z = -5L) .......................................................................... 158 Total pressure around spoiler at floor angle β = 10° and t = 1 sec. with θ = 90° and double injection from Holes (1) and (2) with (at X-Z) ............................................................................... 159 Total pressure on spoiler faces at floor angle β = 10° and t = 1 sec. with θ = 90° and double injection from Holes (1) and (2) ..................................................................................................... 159 Drag coefficient with different floor angles after 1 sec. with θ = 90° and double injection from Holes (1) and (2) ....................... 159 Heel moment coefficient with different floor angles after 1 sec. with θ = 90° and double injection from Holes (1) and (2) ..................................................................................................... 159 Lift coefficient with different floor angles after 1 sec. with θ = 90° and double injection from Holes (1) and (2) ....................... 159 Cavities formation due to double injection from Holes (1) and (2) with θ = 30° and β = 10° (at Z = -5L) ......................................... 163 Velocity vectors and density around spoiler at floor angle β = 10° and t = 1 sec. with θ = 30° and double injection from Holes (1) and (2) (at Z = -5L) ........................................................... 164 Static and total pressures around spoiler at floor angle β = 10° and t = 1 sec. with θ = 30° and double injection from Holes (1) and (2) (at Z = -5L) ........................................................................... 164 Total pressure around spoiler at floor angle β = 10° and t = 1 sec. with θ = 30° and double injection from Holes (1) and (2) (at X-Z) ........................................................................................ 165

VI

5.76

5.77 5.78

5.79 5.80 5.81

5.82

5.83

5.84 5.85 5.86 5.87 5.88 5.89

Total pressure on spoiler faces at floor angle β = 10° and t = 1 sec. with θ = 30° and double injection from Holes (1) and (2) ..................................................................................................... 165 Drag coefficient with different floor angles after 1 sec. with θ = 30° and double injection from Holes (1) and (2) ....................... 165 Heel moment coefficient with different floor angles after 1 sec. with θ = 30° and double injection from Holes (1) and (2) ..................................................................................................... 165 Lift coefficient with different floor angles after 1 sec. with θ = 30° and double injection from Holes (1) and (2) ....................... 165 Cavities formation due to single injection from Hole (1) with θ = 60° and β = 5° (at Z = -5L) ......................................................... 169 Velocity vectors and density around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (1) (at Z = -5L) ......................................................................... 170 Static and total pressures around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (1) (at Z = -5L) ....................................................................................... 170 Total pressure around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (1) (at X-Z) ............................................................................................. 171 Total pressure on spoiler faces at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (1) .............. 171 Drag coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (1) ...................................... 171 Heel moment coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (1) .................... 171 Lift coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (1) ...................................... 171 Cavities formation due to single injection from Hole (2) with θ = 60° and β = 5° (at Z = -5L) ......................................................... 174 Velocity vectors and density around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (2) (at Z = -5L) ......................................................................... 175

VII

5.90

5.91

5.92 5.93 5.94 5.95 5.96 5.97 5.98 A.1 A.2

Static and total pressures around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (2) (at Z = -5L) ....................................................................................... 175 Total pressure around spoiler at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (2) (at X-Z) ............................................................................................. 176 Total pressure on spoiler faces at floor angle β = 5° and t = 1 sec. with θ = 60° and single injection from Hole (2) .............. 176 Drag coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (2) ...................................... 176 Heel moment coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (2) .................... 176 Lift coefficient with different floor angles after 1 sec. with θ = 60° and single injection from Hole (2) ...................................... 176 Drag moment coefficient for all studied cases with different floor angles after 1 sec. .................................................................... 177 Heel moment coefficient for all studied cases with different floor angles after 1 sec. .................................................................... 177 Lift coefficient for all studied cases with different floor angles after 1 sec. ........................................................................................ 177 Ship starboard profile ....................................................................... 193 Ship glossary schematic ................................................................... 193

VIII

LIST OF TABLES Table 3.1 4.1 5.1 5.2 5.3 5.4 A.1

Page Parameters of experimental test cases..................................................30 Solver control settings ........................................................................ 82 Cases of spoiler inclination angle effect ............................................ 91 Cases of injection position effect ..................................................... 119 Cases of floor angle effect ............................................................... 145 Forces, moments and coefficients on ship spoiler ........................... 178 Ship types and speed ranges, Muckle (1996) ................................... 191

IX

NOMENCLATURE Latin Symbols Af

area of air feeding tube, m2

Ao

area of air orifice, m2

Cd

discharge coefficient, dimensionless

CD

drag coefficient, dimensionless

CL

lift coefficient, dimensionless

Cϕ

heel moment coefficient, dimensionless

dc

local cell “dimension” (or length-scale), m

F

liquid volume fraction, dimensionless

FD

drag force, N

FL

lift force, N

Fr

Froude number, dimensionless

FX, FY, FZ

force component in X, Y, Z direction respectively, N

g

gravitational acceleration, m/s2

H

injected air Pressure head, mm water

L

spoiler length, m

Mϕ

heel moment, N.m

P

fluid static pressure, N/m2

PJ

injected air pressure, N/m2

X

Pt

total pressure, N/m2

Sφ

source term of property for fluid

t

time, second

U

free water flow velocity, m/s

u, v, w

velocity in x, y, and z directions respectively, m/s

UJ

injected air velocity, m/s

Vc

volume of the whole cell, m3

Vcut

volume of the cell truncated by the cutting plane, m3

vc

local velocity vector, m/s

W

spoiler width, m

Greek Symbols α

blending factor, dimensionless

β

rise of floor angle, degree

Γ

Diffusion coefficient, dimensionless

∆t

physical time step, second

∆t c

maximum time step that can be taken in cell c, second

∆x , ∆y , ∆z

control volume sizes in x, y and z directions, m

ϑ

volume, m3

θ

spoiler inclination angle, degree

µ

dynamic viscosity, kg/m.s

XI

ρ

density, kg/m3

σ

surface tension between two fluids, N/m

τ

viscous stress tensor, N/m2

φ ~ φ

value of the property for fluid volume-averaged quantity of the property for fluid

Subscripts 1

value of the fluid property (air)

2

value of the fluid property (water)

D

drag

f

feeding

J

injected air

L

lift

m

scale model

mix

mixture

o

orifice

P

prototype

Τ

total

φ

heel

XII

ABBREVIATION AMG

Algebraic Multigrid Solver

CAD

Computer Aided Design

CFD

Computational Fluid Dynamics

CFD-ACE

Advanced Polyhedral Solver

CFD-GEOM

Geometry

CFD-GUI

Graphical User Interface

CFD-VIEW

Post Processor

CFL

Courant-Friedrichs and Lewy Number

CGS

Conjugate-Gradient-Squared

CIF

Caltech Intermediate Form

CSF

Continuum Surface Force

DTF

Data Transfer Facility Format

GGD

Geometry and Grid Definition

IGES

Initial Graphics Exchange Specification

LAMP

Large Amplitude-Motion Program

MEMS

Micro-Electro Mechanical Systems

MURI

Multi Universities Research Interactive Projects

NITER

Number of Iterations

NURBS

Non-Uniform Rational B-Splines

XIII

PLIC

Piecewise Linear Interface Construction Method

SIMPLE

Semi-Implicit Method for Pressure-Linked Equations

SIMPLEC

Semi-Implicit Method for Pressure-Linked Equations Consistent

SIMPLER

Semi-Implicit Method for Pressure-Linked Equations Revised

SLIC

Single Line Interface Construction Method

TFI

Transfinite Interpolation Methodology

VOF

Volume-Of-Fluid

XIV

1

INTRODUCTION

CHAPTER 1

Introduction

1.1 Introduction The motion of a ship in six degrees of freedom is considered as a translation motion (position) in three directions: surging, swaying, and heaving; and as a rotation motion (orientation) about three axes: rolling, pitching and yawing, Figure (1.1). Surging, Figure (1.2), being the fore and aft motion, whilst swaying, Figure (1.3), and heaving, Figure (1.4), are respectively the athwart ship motion and the vertical motion of the ship. Rolling, Figures (1.5, 1.6), is rotation about a longitudinal axis, while pitching, Figure (1.7), is rotation above a transverse axis and yawing, Figure (1.8), is rotation about a vertical axis. Pitching and rolling are true oscillatory motions. The control and reduction of ship motion is important as it is related to the structural loading and comfort of a ship. In realistic sea state all six motions can be generated, but here attention will be focused mainly on pitching and rolling as these are true oscillatory motions, that is to say if the ship is displaced from its equilibrium position by some force, when that force is removed the ship will oscillate until the motion is damped out, Muckle (1996).

1

CHAPTER 1

Introduction

On the other hand, resistance of a ship is decomposed into three components; the wave-making resistance, the frictional drag, and the viscous form drag. The reduction of the wave-making resistance is achieved through modification of hull forms. The reduction of ship frictional drag has been investigated using many techniques, e.g., microbubbles (Madavan et al., 1984), air films (Murai et al., 2007), riblets (Walsh and Lindemann, 1984), super-water-repellent surface and air injection (Fukuda et al., 2000) etc.

Surging Yawing Swaying

Pitching Rolling Heaving

Fig. (1.1) Ship degrees of freedom, Clark (2002) Z

Z Water line

×× X

× × Y

Fig. (1.2) Surging motion

Fig. (1.3) Swaying motion

2

CHAPTER 1

Introduction

Z

Water line ×

Y

×

Fig. (1.4) Heaving motion

Z

Water line

Y

×

Fig. (1.5) Rolling motion

Fig. (1.6) Rolling moment, Clark (2002)

Y Z

X ×

× X

Fig. (1.7) Pitching motion

Fig. (1.8) Yawing motion

3

CHAPTER 1

Introduction

1.2 Spoiler Basic Idea The idea of using a system of spoilers as a device to control and reduce ship motions is very simple to understanding, the spoiler which mounted at the bottom of the ship when water flow around it, two flow fields with different pressure are created, one of them in upstream region and the other one in downstream region. Hence, the pressure difference is carried out in low speed flow field, so the pressure drop in downstream region can be increased by air injection. This creates two-phase flow; air and water; with high density ratio that plays an important role in dynamic pressure and total pressure values. The pressure difference between the two regions generates forces acting on the spoiler; these forces are used to control and reduce ship motions. A system of spoilers is mounted at the bottom of the ship. They can be classified into bow spoilers and stern spoilers as shown in Figure (1.9). The results quoted by Soper et al. (1998) show that by appropriately choosing the number, extension, and distribution of the spoilers, one can affect an optimal trim of the ship over the whole speed range and a reduction in the ship motion when moving in waves.

4

CHAPTER 1

Introduction CG Rudder ×

Stern spoiler

Bow spoiler

Starboard

Port

×

Fig. (1.9) Example of ship spoiler locations, Soper et al. (1998)

1.3 Organization of the Thesis Chapter 2 reviews the previous researches related to reduction and control the ship rolling and pitching motion. Chapter 3 discusses the details of the testrig used to simulate the flow field around the spoiler and the recording of the bubble formation under different conditions. Chapter 4 explains the mathematical model, the numerical technique procedures and the solving commercial code CFDRC. Chapter 5 presents the experimental and numerical results of the eighteen test cases and the estimation of forces and moments for each case. The results demonstrate the advantages spoiler system as a ship motion control. Finally, Chapter 6 gives the conclusions and discusses the future works associated with this thesis.

5

CHAPTER 1

Introduction

6

2

LITERATURE REVIEW

CHAPTER 2

Literature Review

2.1 Introduction This chapter reviews the background and the available previous literatures regarding the ship stability techniques, ship spoiler flow field and flow filed simulation. The first section covers coherent techniques and devices which have been suggested for the reduction of rolling in ships. The second section covers the studies related to flow field around spoiler, and the viewpoints about the flow phenomena. The third section covers theories, models and simulations relating to free surface structures. The fourth section discusses the objectives of this thesis.

2.2 Ship Stability Techniques Many devices have been suggested for the reduction of rolling in ships. The most successful being the passive water tank or channel, activated fins, rudder, moving weights and bilge keel. The former consists of a channel or tank placed athwart ships in which water is free to oscillate in the transverse direction as shown in Figure (2.1). By suitably

7

CHAPTER 2

Literature Review

adjusting the period of the water in the tank to be approximately equal to the ship period, it is possible to get the water moving in a direction which applies a moment to the ship of opposite sign to that applied by the waves, thus reducing the rolling motion. The device has proved to be very suitable for ships which operate for long periods at zero or near zero speeds.

Fig. (2.1) Air controlled tank stabilizer, Muckle (1996)

The activated fin system consists in having one or sometimes two fins projecting from each side of the ship as shown in Figure (2.2), the fins being inclined at angles of incidence to the flow so that lift forces are generated as the ship moves through the water. The angles of incidence on the two sides of the ship are of opposite sign so that a couple is generated. The attitudes of the fins are

8

CHAPTER 2

Literature Review

controlled by a mechanism which in turn is controlled by the motion of the ship, and the mechanism can be so arranged that the couple generated by the fins opposes the motion of the ship. A bilge keel is a permanent flat plate projection from the ship's bilge region which extends for the midship section. It is fitted at right angles to the bilge radiused plating and projects a sufficient distance to penetrate the boundary layer of water along the hull, see Figure (2.3). The bilge keel also creates a damping moment resulting from the pressure resistance of the projecting area and the viscous-eddy flows around it. The system has been widely adopted in passenger ships and naval vessels from the point of view of human comfort and cargo damage. It is more effective in fast ships, since the forces generated on the fins are proportional to the square of the speed through the water. Flat bar

Shell plating

δ

Doubling plate

Lift developed by angle of attack (a) All movable surface δ1

Offset bulb plate

Flap

(a)

δ Shell plating

Lift developed by angle of attack + variable camber (b) All movable surface with tail flap

Flat bar

Movable portion Fixed portion

Bilge keel ( Offset bulb plate )

Lift developed by variable camber (c) Surface behind fixed structure

(b)

Fig (2.2) Stabilizer fin arrangements, Muckle (1996)

Fig. (2.3) Bilge keel, Tupper (2004)

9

CHAPTER 2 2.3

Literature Review

Studies Related to Flow Field around Spoiler Hull-mounted cavitating spoilers have been shown by the Krylov

Shipbuilding Research Institute (Russia) to be an effective means for motion reduction in high-speed displacement-hull vessels. The specific design details of the Russian system are not available for propriety reasons. In Multi Universities Research Interactive projects (MURI), Virginia Technology, USA, a theoretical investigation of the effectiveness and design of such a system has been undertaken by Soper et al. (1998) to choose the number, extension, and distribution of the spoilers, one can affect an optimal trim of the ship over the whole speed ranges and a reduction in the ship motion when moving in waves. They used the panel method by developing a two-dimensional quasi-steady numerical local model for the fixed-cavitation region. The computational-fluid-dynamics

boundary-element

model

is

based

on

a

distributed-vorticity potential-flow form. The steady-flow streamlines are determined via an iterative approach that converges to a pre-specified level of accuracy. The cavity shape and quasi-steady hydrodynamic forces have been generated over a wide range of forward-speed/cavity-pressure and spoilerprojection-angle parameter values. Depending on the sea state, roll reduction by a factor of 2 to 5 and pitch reduction by a factor of 1.2 to 1.5 can be achieved. Consequently, an increase in the efficiency and hence the speed are by a factor of 10% to 20%.

10

CHAPTER 2

Literature Review

Owis et al. (2000) developed a code of Navier-Stokes equations with cavitation procedure function of local pressure and density to solve the same problem. Mostafa (2001a) illustrated that there is no unique definition of the interface between two phases flow. And also, the cavitation phenomenon depends on the local pressure and density. Mostafa (2001b) showed that the fundamental of ship spoiler problem is different; it depends on air injection into water and it is not a very high speed phenomenon. Mostafa (2003) analyzed the transient two-dimensional flow around ship spoiler numerically and experimentally, by choosing spoiler inclination angles at 45° and 90° with different air injection positions. The moment around the 45° spoiler fixation point is seven times that around the 90° spoiler. In double injection condition beside and at a distance equal to the horizontal spoiler length downstream the spoiler, the bubbles are more stable and the moment is higher by 7.9 and 17.8 times after 1.2 and 2 seconds, respectively, than that around the 90° spoiler. The matching between the numerical and experimental results was not good in all conditions due to two-dimensional computational model consideration. Brasher et al. (2001) investigated an alternative method to control rolling motion of wing of the Wide Area Surveillance Projectile (WASP), using pressurized air

11

CHAPTER 2

Literature Review

in a wing venting through a jet spoiler to induce flow separation over a wing, Figure (2.4).

Fig. (2.4) Method of roll control at boundary layer, Brasher et al. (2001)

Klaka et al. (2005) investigated a series of experiments to measure the hydrodynamic forces and moments acting on sailing yacht keels. A vertical flat plate in open flow running normal to the surface of the plate has been the subject of investigation for several decades, Figure (2.5). However, the forces and moments acting on surface-piercing plates, plates slightly inclined to the vertical and plates extending close to a lower boundary have not received such attention. It was found that drag force and heel moment varied approximately as flow speed squared for all plates investigated; the influence of plate angle on drag coefficient was small, with no clear trend; the magnitude of the drag coefficients for the surface-piercing plates was greater than for published data on fully submerged plates; and the centre of pressure was at approximately 50% of the span.

12

CHAPTER 2

Literature Review channel rim hinge support water line flow plate

channel bed

Fig. (2.5) Experimental rig: side view, Klaka et al. (2005)

Djebedjian et al. (2008a, 2008b); extracted from this thesis; investigated experimentally and numerically the characteristics of the flow field around the ship spoiler to study the effect of spoiler inclination angle on the ship stability. They represented the numerical results of the two-phase flow field around the ship spoiler using three-dimensional Navier-Stokes equations with the Free Surface simulation in PLIC method. The moment around the 90° inclined spoiler fixation line is 1.651 times that around the 30° inclined spoiler and 1.1 times that around the 60° inclined after 1 second. The spoiler inclination angle of 90° has the highest value of drag coefficient on spoiler upstream face and the smallest value in case of 30°. The effect of spoiler inclination angle θ has the prominent effect on ship stability.

13

CHAPTER 2

Literature Review

Djebedjian et al. (2008c); extracted from this thesis; presented the effect of ship floor angle on the ship stability by studying the three-dimensional two-phase flow field around the spoiler body and the pressure variation on the wake of the spoiler body. They recorded experimentally the flow field and bubble formation around the spoiler at different conditions. The Froude number was about 3.76 based on spoiler length. The computational code performed relatively well, with the free surface simulation more closely matching the experimental results. The moment around the 5° floor angle spoiler fixation line was 1.034 times that around the 10° floor angle after 1 second. Therefore, the effect of floor angle of the ship has a small influence on ship stability. Djebedjian et al. (2008d); extracted from this thesis; studied the effect of air injection position on the ship stability. The three-dimensional flow field around spoiler was simulated experimentally and computationally with injection of air behind spoiler to evaluate the effect of air injection position on the flow field and bubble formation around the ship spoiler and their influences on the hydrodynamic forces which act on the spoiler body. The moment around the spoiler fixation line in case of double injection in two holes was 1.025 times that in case of single injection in a hole beside the spoiler fixation, whereas equal about 0.886 times that in case of single injection in a hole 0.7L from the spoiler fixation, after 1 second (L is the spoiler length). For the inclined spoiler and air injection, the injection position has a slight influence on ship stability.

14

CHAPTER 2 2.4

Literature Review

Research Related to Free Surface The free surface simulation computes the mixture of two incompressible,

immiscible fluids, including the effects of surface tensions. Thus, the strength of the free surface simulation can model the injection of one fluid into the second fluid with arbitrary immiscible fluid-fluid interfaces, which includes two fluids with very high density ratio, such as air and water. Noh and Woodward (1976) developed the Single Line Interface Construction (SLIC) method with upwind scheme, by assuming the fluid surface to be parallel to the currently selected cell face, with the relative position of the second fluid depends on the flow direction and the upstream or downstream value of liquid volume fraction. Hirt and Nichols (1981) published early form of the free surface simulation called Volume-Of-Fluid (VOF) method which was developed later by Rider et al. (1995). Kothe et al. (1996) developed the Piecewise Linear Interface Construction (PLIC) method with upwind scheme, by assuming the liquid-gas interface to be planar and allowed to take any orientation within the cell, and will therefore generally have the shape of an arbitrary polygonal shape. So, PLIC is more accurate method. Fukuda et al. (2000) presented a new technique for reducing ship bottom frictional drag using a super-water-repellent surface and air injection, the

15

CHAPTER 2

Literature Review

frictional resistance on the super-water-repellent (SWR) surface was reduced by 80% at a speed of 4 m/s and 55% at 8 m/s. Liovic et al. (2002) studied splash as result of top-submerged air injection into water. They simulated a 2D numerical model of free surface flows using a Piecewise Linear volume tracking scheme to track interfaces, maintain sharp interfaces and capture fine-scale flow phenomena such as fragmentation and coalescence. Nagayosh et al. (2003) studied experimentally and numerically the gas-liquid two-phase behavior in systems of actual nuclear reactor, by simulating a 2D model of air injection into water pool from the center of the flat tube bottom. The interface-tracking of the free surface and bubbles is calculated by using the VOF (Volume-Of-Fluid) technique.

2.5 Original Contributions and Main Objectives The objectives of this thesis are to investigate experimentally and numerically the characteristics of the flow field around the ship spoiler. The main objective is studying the effects of spoiler inclination angle, ship floor angle, and air injection position on the ship balance. To achieve this objective three targets are studied as follows: * The two-phase flow field around a ship spoiler was recorded experimentally by trigging the bubble formation images at three different spoiler angles (θ),

16

CHAPTER 2

Literature Review

two different rise of floor angles (β) and three different injected air conditions; including the variation with time. * The three-dimensional two-phase flow field around a ship spoiler model using the three-dimensional Navier-Stokes equations with the Free Surface simulation in PLIC method; including the variation with time was studied computationally. * The hydrodynamic forces acting on the spoiler faces and the moment around the spoiler fixation line, according to the pressure variation on the wake which affects the roll, pitch and speed of the ship were calculated.

17

CHAPTER 2

Literature Review

18

3

THEORETICAL ANALYSIS AND EXPERIMENTAL TECHNIQUES

CHAPTER 3

Theoretical Analysis and Experimental Techniques

3.1 Introduction This chapter is concerned with describing the details of the theoretical analysis and experimental test-rig used to simulate the flow field around the spoiler and the record of the bubble formation with the different parameters (spoiler inclination angle, rise of ship floor angle and air injection position).

3.2

Theoretical Analysis

3.2.1 Similarity for the Spoiler and the Air Injection Holes The general cargo ships are taken as studying ship type. Figure (3.1) illustrates the geometric similarity between the prototype and model.

θ

Up

Um

Lp β

θ

Lm

Wp

( a ) Prototype

(b) Scale model

Fig. (3.1) Geometric similarity in model testing

19

β

Wm

CHAPTER 3


The lift, drag, and heel moment coefficients are given as function of the geometrical and physical parameters of the spoiler and the injection hole. For example, the drag, D, on a thin rectangular plate (L x W in size) placed normal to a fluid with velocity U can be assumed as: D = f ( L, W , µ , ρ , g , U , θ , β , θ J , N J , U J , D J , µ J , ρ J )

(3.1)

where θ J is the angle of injection hole, N J is the number of injection holes, U J is the velocity in the injection hole, and D J is the air injection hole diameter. Taking the similarity based on the spoiler length L, the application of the pi theorem yields: ⎛ L ρU L U DJ ρ J U J DJ ⎞ D ⎟ ⎜ , φ θ β θ N , , , , , , , = J J ⎟ ⎜W µ µ L L2 ρ U 2 g L J ⎠ ⎝

(3.2)

For the model (denoted by subscript “m”) and prototype (denoted by subscript ‘’p”), Equation (3.2) gives respectively:

Dm L2m ρ m U m2

⎛L ρ U L Um D ρ U D =φ⎜ m , m m m , , θ m , β m , θ Jm , N Jm , Jm , Jm Jm Jm ⎜W Lm µm µ Jm g Lm ⎝ m

⎞ ⎟ ⎟ ⎠

(3.3)

20

CHAPTER 3

Dp L2p ρ p U 2p


⎛ Lp ρ p U p Lp U p D Jp ρ Jp U Jp D Jp , , , θ p , β p , θ Jp , N Jp , , =φ⎜ ⎜Wp L µ µ Jp g L p p p ⎝

⎞ ⎟ ⎟ ⎠ (3.4)

where:

U m : Scale model velocity, U p : Prototype velocity,

Lm : Spoiler of scale model, Lp :

Spoiler of prototype.

The similarity requirements for the model spoiler and prototype spoiler are the length scale ratio, Reynolds number, Re , Froude number, Fr , spoiler inclination angle, floor angle, number of injection holes, hole diameter to spoiler length ratio, Reynolds numbers for the injected air and drag coefficient are given respectively as:

Lp Lm = Wm W p Re m = Re p =

(3.5)

ρ m U m Lm ρ p U p L p = µm µp

(3.6)

2

2 Up Um Frm = Frp = = gLm gL p

(3.7)

21

CHAPTER 3


θm = θ p

(3.8)

βm = β p

(3.9)

θ Jm = θ Jp

(3.10)

N Jm = N Jp

(3.11)

D Jm D Jp = Lm Lp

(3.12)

ρ Jm U Jm D Jm ρ Jp U Jp D Jp = µ Jm µ Jp

(3.13)

C Dm = C D p =

Dm L2m ρ m U m2

=

Dp L2p ρ p U p2

Since the model and prototype have the same length scale ratio, same time scale ratio,

(3.14)

Lm Wm , and = Lp Wp

Tm Lm / U m = , the result is that the velocity-scale ratio Tp Lp /U p

will be the same for both. The motions of two systems are kinematically similar if homologous particles lie at homologous points at homologous time, White (1998). Dynamic similarity exists, simultaneous with kinematic similarity, if model and prototype forces are in a constant ratio. For incompressible flow with a free surface, the model and prototype Reynolds numbers and Froude numbers, are correspondingly equal, Equations (3.6) and (3.7) respectively.

22

CHAPTER 3


For the model and prototype, the spoiler inclination angle, floor angle, angle of injection hole, and the number of injection holes are equal, Equations (3.8) to (3.11) respectively. For the injected air hole, the hole diameter to spoiler length ratio and the Reynolds numbers for the injected air through the hole for the model and prototype are equal, Equations (3.12) and (3.13) respectively. The drag coefficient, Equation (3.14), and also the lift and heel moment coefficients are similar in both the model and prototype.

3.2.2 Prototype’s Spoiler and Injection Hole Calculations

Assuming the prototype spoiler dimensions are 1 x 1 m, i.e. the spoiler length of prototype L p is 1 m, and for the model, the scale model velocity

U m = 6 m/s and the spoiler length of model Lm = 0.01 m, then by substituting in Equation (3.7), the Froude number is Frm = 3.67 and the prototype velocity ( U p ) is equal to 0.6 m/s. Similar calculations can be done for the air injection hole diameter for the prototype using Equation (3.12).

23

CHAPTER 3


3.3 Experimental Test-Rig The experimental test-rig as shown in Figure (3.2) consists of four main systems: air injection system, lighting system, imaging system and water channel test section system. The measurements were carried out in a water channel facility at the Fluid Mechanics Laboratory, Faculty of Engineering, Mansoura University. Details of the experimental setup are briefly mentioned in the following sections.

Lighting Box

Test Section y Water Inlet

Water Outlet

x

Power Supply

Video camera

Valve

Computer Air Compressor

Fig. (3.2) Experimental set-up

24

Printer

CHAPTER 3


3.3.1 Air Injection System

The air injection system as shown in Figure (3.3) consists of the following components: 1-

Compressor to supply air at certain pressure.

2-

Pressure gauge to measure air source pressure.

3-

Air regulator to control the flow.

4-

Water U-tube manometer to measure air pressure before entering hole (orifice).

5-

Three valves (1), (2) and (3) to control air bleeding from the air line, feeding air to water U-tube manometer, and feeding air to test section, respectively.

25

CHAPTER 3


Pressure Gauge

Compressed Air Source

Regulator Valve

Valve (1) Bleeding Air

Valve (2)

Valve (3)

Water U-Tube Manometer Water Tunnel Test Section

Water Surface Water Flow U

Spoiler

Injected Air Orifices

Fig. (3.3) Air injection diagram

3.3.2 Lighting System

A special lighting system is designed to obtain a thin light sheet. The lighting system consists of a 30 cm x 20 cm x 25 cm black wooden box. The inside box walls were coated with aluminum papers and a 1000 Watt light bulb was mounted inside the box. The lighting system was placed over the test

26

CHAPTER 3


section, and the produced light was passed from a window 20 cm in length, and 6 mm in width, fixed in the bottom of the black wooden box, to pass through the test section.

3.3.3 Imaging System 1- Video Camera

The bubbles growth was recorded with a video camera (Panasonic M9500), which allowed the collection of 25 frames per second. 2- Video Recorder

The scientific video recorder (Panasonic- NV-HD100) has 6 heads, which allow the ability for displaying each frame without image noise. 3- Image Capture Software

Commercial software is used to capture the experimental results images from movie clips.

27

CHAPTER 3


3.3.4 Water Channel Test Section System 3.3.4.1 Ship Bottom Simulator

A Plexiglas plate allowing the light rays to pass through has 15.5 cm width, 80 cm length and 4 mm thickness is used as a model to simulate the ship bottom surface, Figure (3.4). The spoiler is manufactured steel sheet 1 cm x 1cm wide and 1 mm thick. Two holes are located on the longitudinal center line of plate; hole (1) touches the spoiler down face and hole (2) is at a distance 7 mm from the spoiler face, Figure (3.5).

Two feeding tubes I.D. 5 mm

Two injection holes I.D. 1 mm

15.5 cm

4 mm 80 cm

Fig. (3.4) Simulator plate dimensions Air injection holes (1)

0.7 L

(2)

θ Flow

Spoiler downstream edge L

β

Spoiler

Spoiler upstream edge (a) Spoiler inclination angle

W (b) Floor angle

Fig. (3.5) Simulator plate construction

28

CHAPTER 3


3.3.4.2 Water Channel

The water channel test section shown in Figure (3.6) is 2.43 m long, 15 cm wide and 30 cm deep. The pump has a discharge of 360 lit/min at 0.75 bar head and power of 0.75 hp. The width and depth of water in the water channel were 15 and 10 times the spoiler height, respectively. The two sides of the water channel test section are from Plexiglas which allows for visualization of the injected air, their interaction with the free surface, and bubble entrainment. 3

5 7

1

9

12

2 11

8 6

10

4

1. Slope Indicator 2. Control Panel 3. Head Gate, Manual 4. Valve 5. Manometer 6. Slope Mechanism

7. Plexiglas Channel 8. Pump 9. Aluminum Rail 10. Pivot 11. Fiber Glass Reservoir 12. Tail Gate, Manual

Fig. (3.6) Water channel

29

CHAPTER 3


3.4 Measurement Devices This section is concerned with representing the measurement devices used in this study.

Water U-tube Manometer For the small measured pressure value (60 N/m2), low density liquid for the Utube manometer must be used to give sensible and visible differences. Therefore, the water U-tube manometer is suitable as pressure measuring device for this study.

3.5 Test Cases Eighteen test cases were performed which include three angles of θ and for each angle, two floor angles β were used. For each β, there were three injection conditions as given in Table (3.1). Table (3.1) Parameters of experimental test cases Parameter Spoiler angle, Rise of floor angle, Spoiler width, Injected air pressure, Water flow velocity, Injection Condition

θ (°) β (°) W (cm) PJ (N/m2) U (m/s)

Test Case Set (1) (2) 30 60 5, 10 5, 10 1 1 60 60 0.6 0.6 - Single injection in Hole (1) - Single injection in Hole (2) - Double injection in Holes (1) and (2)

30

(3) 90 5, 10 1 60 0.6

CHAPTER 3


3.6 Test Procedure In order to obtain robust data on the stability boundaries in flow field, special steps must be established. There are four basic parameters that can be controlled by the operator: injection position, air flow pressure, spoiler angle and plate angle. At each operating condition, the test section and experimental set-up are prepared; air source tube is fitted with injection tubes of the simulator plate and fitted with a water channel. The lighting box is positioned at a distance about 20 cm from a water surface, so that, the thin light sheet is parallel at test section center line. A video camera is positioned at a distance of 30 cm from the test section side, as shown in Figure (3.2). The whole apparatus is set at a dark room to prevent any other light source. After experiment preparations, the lighting system is powered on and the air pressure is adjusted at 6 mm water on U-tube manometer for single injection conditions and at 24 mm water for double injection conditions by using the regulating valve (1) and bleeding valve (2), the video camera is powered on. Finally, valve (3) is opened to allow air passing through injected holes. The flow condition image is recorded and extracted by using the imaging system during test time needed. The experimental error analysis is mentioned in Appendix B.

31

CHAPTER 3


32

4

NUMERICAL PROCEDURE

CHAPTER 4

Numerical Procedure

4.1 Introduction A two-phase flow field around a ship spoiler with the free surface simulation in Piecewise Linear Interface Construction method is modeled numerically using a three-dimensional Navier-Stokes code. In this chapter, the governing equations discretization on a structured grid using an Upwind Difference scheme, boundary initial conditions and the method of solution of the model will be presented. For different conditions, the bubbles shape, the threedimensional flow field around the spoiler body and the pressure variation on the wake of the spoiler body are computed.

4.2 Computational Fluid Dynamics Tools From the 1960s onwards the aerospace industry has integrated Computational Fluid Dynamics (CFD) techniques into the design, R&D and manufacture of aircraft and jet engines. More recently the methods have been applied to the design of internal combustion engines, combustion chambers of gas turbines and furnaces. Furthermore, motor vehicle manufactures now routinely, predict drag forces, under-bonnet air flows and the in-car environment

33

CHAPTER 4

Numerical Procedure

with CFD. Increasingly CFD is becoming a vital component in the design of industrial products and processes. The ultimate aim of developments in the CFD field is to provide a capability comparable to other CAE (Computer-Aided Engineering) tools such as stress analysis codes. The main reason why CFD has lagged behind is the tremendous complexity of the underlying behavior, which precludes a description of fluid flows that is at the same time economical and sufficiently complete. The availability of affordable high performance computing hardware and the introduction of user-friendly interfaces have led to a recent upsurge of interest and CFD is poised to make an entry into the wider industrial community in the 1990s. Moreover, there are several unique advantages of CFD over experiment-based approaches to fluid systems design: 1. Substantial reduction of lead times and costs of new designs. 2. Ability to study systems where controlled experiments are difficult or impossible to perform (e.g. very large systems). 3. Ability to study systems under hazardous conditions at and beyond their normal performance limits (e.g. safety studies and accident scenarios). 4. Practically unlimited level of detail of results.

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CFD codes are structured around the numerical algorithms that can tackle fluid flow problems. In order to provide easy access to their solving power, all commercial CFD packages include sophisticated user interfaces to input problem parameters and to examine the results. Hence all codes contain three main elements: (i) A pre-processor, (ii) A solver, and (iii) A post-processor. We briefly examine the function of each of these elements within the context of a CFD code. (i) Pre-processor Pre-processing consists of the input of a flow problem to a CFD program by means of an operator-friendly interface and the subsequent transformation of this input into a form suitable for use by the solver. The user activities at the preprocessing stage involve: 1. Definition of the geometry of the region of interest: the computational domain. 2. Grid generation: the sub-division of the domain into a number of smaller, non-overlapping sub-domains: a grid (or mesh) of cells (or control volumes or elements).

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3. Selection of the physical and chemical phenomena that need to be modeled. 4. Definition of fluid properties. 5. Specification of appropriate boundary conditions at cells which coincide with or touch the domain boundary. The solution to a flow problem (velocity, pressure, temperature etc.) is defined at nodes inside each cell. The accuracy of a CFD solution is governed by the number of cells in the grid. In general, the larger the number of cells is, the better the solution accuracy. Both the accuracy of a solution and its cost in terms of necessary computer hardware and calculation time are dependent on the fineness of the grid. Optimal meshes are often non-uniform: finer in areas where large variations occur from point to point and coarser in regions with relatively little change. Efforts are under way to develop CFD codes with a self-adaptive meshing capability. Ultimately such programs will automatically refine the grid in areas of rapid variations. A substantial amount of basic development work still needs to be done before these techniques are robust enough to be incorporated into commercial CFD codes. At present, it is still up to the skills of the CFD user to design a grid that is a suitable compromise between desired accuracy and solution cost.

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(ii) Solver There are three distinct streams of numerical solution techniques: finite difference, finite element and spectral methods. In outline, the numerical methods that form the basis of the solver perform the following steps: •

Approximation of the unknown flow variables by means of simple functions.

•

Discretization by substitution of the approximations into the governing flow equations and subsequent mathematical manipulations.

•

Solution of the algebraic equations.

The main differences between the three separate streams are associated with the way in which the flow variables are approximated, and with the discretization processes. Finite Difference Methods Finite difference methods describe the unknowns φ of the flow problem by means of point samples at the node points of a grid of co-ordinate lines. Truncated Taylor series expansions are often used to generate finite difference approximations of derivatives of φ in terms of point samples of φ at each grid point and its immediate neighbors. Those derivatives appearing in the governing equations are replaced by finite differences yielding an algebraic equation for the values of φ at each grid point.

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Finite Element Methods Finite element methods use simple piecewise functions (e.g. linear or quadratic) valid on elements to describe the local variations of unknown flow variables φ . The governing equation is precisely satisfied by the exact solution φ . If the piecewise approximating functions for φ are substituted into the equation it will not hold exactly and a residual is defined to measure the errors. Next the residuals (and hence the errors) are minimized in some sense by multiplying them by a set of weighting functions and integrating. As a result we obtain a set of algebraic equations for the unknown coefficients of the approximating functions. The theory of finite elements has been developed initially for structural stress analysis. Spectral Methods Spectral methods approximate the unknowns by means of truncated Fourier series or series of Chebyshev polynomials. Unlike the finite difference or finite element approach the approximations are not local but valid throughout the entire computational domain. Again we replace the unknowns in the governing equation by the truncated series. The constraint that leads to the algebraic equations for the coefficients of the Fourier or Chebyshev series is provided by a weighted residuals concept similar to the finite element method or by making the approximate function coincide with the exact solution at a number of grid points.

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Finite Volume Method The finite volume method was originally developed as a special finite difference formulation. The solving commercial code (CFDRC), which is used in this thesis, uses a finite volume solution approach. So, the finite volume discretization method will be mentioned in the next sections. The numerical algorithm consists of the following steps: •

Formal integration of the governing equations of fluid flow over all the (finite) control volumes of the solution domain.

•

Discretization involves the substitution of a variety of finite-differencetype approximations for the terms in the integrated equation representing flow processes such as convection, diffusion and sources. This converts the integral equations into a system of algebraic equations.

•

Solution of the algebraic equations by an iterative method.

The first step, the control volume integration, distinguishes the finite volume method from all other CFD techniques. The resulting statements express the (exact) conservation of relevant properties for each finite size cell. This clear relationship between the numerical algorithm and the underlying physical conservation principle forms one of the main attractions of the finite volume method and makes its concepts much more simple to understand by engineers than finite element and spectral methods.

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CFD codes contain discretization techniques suitable for the treatment of the key transport phenomena, convection (transport due to fluid flow) and diffusion (transport due to variations of φ from point to point) as well as for the source terms (associated with the creation or destruction of φ ) and the rate of change with respect to time. The underlying physical phenomena are complex and nonlinear so an iterative solution approach is required. Many popular solution procedures are used to solve the algebraic equations and the algorithms used to ensure correct linkage between pressure and velocity. Commercial codes may also give the user a selection of further, more recent, techniques such as Stone's algorithm and conjugate gradient methods. (iii) Post-Processor As in pre-processing a huge amount of development work has recently taken place in the post-processing field. Owing to the increased popularity of engineering workstations, many of which have outstanding graphics capabilities, the leading CFD packages are now equipped with versatile data visualization tools. These include: 1. Domain geometry and grid display, 2. Vector plots, 3. Line and shaded contour plots, 4. 2D and 3D surface plots, 5. Particle tracking, 6. View manipulation (translation, rotation, calling etc.),

40

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7. Color postscript output. More recently, these facilities may also include animation for dynamic result display and in addition to graphics all codes produce trusty alphanumeric output and have data export facilities for further manipulation external to the code. As in many other branches of CAE the graphics output capabilities of CFD codes have revolutionized the communication of ideas to the non-specialist, Versteeg and Malalasekera (1995).

4.3 Governing Equations 4.3.1 Momentum Equations Treatment Theory The governing equations for the flow module represent mathematical statements of the conservation laws of flow: *

The mass of a fluid is conserved.

*

The time rate of change of momentum equals the sum of the forces on the fluid (Newton’s second law).

These two laws can be used to develop a set of equations (known as the NavierStokes equations) for CFD-ACE to solve numerically.

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• Mass Conservation Conservation of mass requires that the time rate of change of mass in a control volume be balanced by the net mass flow into the same control volume (outflow - inflow). This can be expressed as:

v ∂ρ + ∇ • ( ρV ) = 0 ∂t

(4.1)

The first term on the left hand side is the time rate of change of the density (mass per unit volume). The second term describes the net mass flow across the control volumes boundaries and is called the convective term. •

Momentum Conservation

This section describes the mathematical equations used by the flow module. Newton's second law states that the time rate of change of the momentum of a fluid element is equal to the sum of the forces on the element. We distinguish two types of forces on the fluid element: - Surface forces: • pressure forces • viscous forces - Body forces: • gravity force • centrifugal force

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• electromagnetic force • etc. The x-component of the momentum equation is found by setting the rate of change of x-momentum of the fluid particle equal to the total force in the xdirection on the element due to surface stresses plus the rate of increase of xmomentum due to sources:

v ∂ (− p + τ xx ) ∂τ yx ∂τ zx ∂ ( ρu ) + ∇ • ( ρVu ) = + + + S Mx ∂t ∂x ∂y ∂z

(4.2)

Similar equations can be written for the y- and z-components of the momentum equation: ∂ (− p + τ yy ) ∂τ xy ∂τ zy v ∂ ( ρv ) + ∇ • ( ρVv ) = + + + S My ∂t ∂y ∂x ∂z

(4.3)

v ∂ (− p + τ zz ) ∂τ yz ∂τ xz ∂ ( ρw ) + ∇ • ( ρVw) = + + + S Mz ∂t ∂z ∂y ∂x

(4.4)

Navier-Stokes Equations

The momentum equations, given above, contain as unknowns the viscous stress components τij. A model must be provided to define the viscous stresses. In Newtonian flows the viscous stresses are proportional to the deformation rates. The nine viscous stress components (of which six are independent for

43

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isotropic fluids) can be related to velocity gradients to produce the shear stress terms and by substitution of these terms into the momentum equations yields the Navier-Stokes equations:

v v⎤ ∂ ( ρu ) ∂p ∂ ⎡ ∂u 2 ( )⎥ + ∇ • ( ρVu ) = − + − ∇ • V 2 µ µ ∂t ∂x ∂x ⎢⎣ ∂x 3 ⎦ ∂ ⎡ ⎛ ∂u ∂v ⎞⎤ ∂ ⎡ ⎛ ∂u ∂w ⎞⎤ + µ⎜ + ⎟ + S Mx ⎢ µ ⎜ + ⎟⎥ + ∂y ⎣ ⎜⎝ ∂y ∂x ⎟⎠⎦ ∂z ⎢⎣ ⎝ ∂z ∂x ⎠⎥⎦

(4.5)

v v⎤ ∂p ∂ ⎡ ∂v 2 ∂( ρv ) + ∇ • ( ρVv ) = − + ⎢2 µ − µ (∇ • V )⎥ ∂y ∂y ⎣ ∂y 3 ∂t ⎦ +

∂ ∂z

⎡ ⎛ ∂v ∂w ⎞⎤ ∂ ⎡ ∂u ∂v ⎤ + ⎥ + S My ⎟⎟⎥ + ⎢ µ ⎢ µ ⎜⎜ + ∂ z ∂ y ∂ x ∂ y ∂x ⎦ ⎠⎦ ⎣ ⎣ ⎝

v v⎤ ∂p ∂ ⎡ ∂w 2 ∂( ρw) − µ (∇ • V )⎥ + ∇ • ( ρVw) = − + ⎢2µ ∂z ∂z ⎣ ∂z 3 ∂t ⎦ ∂ ⎡ ⎛ ∂v ∂w ⎞⎤ ∂ ⎡ ∂u ∂w ⎤ + ⎢µ ⎜ + µ + + S Mz ⎟ + ∂y ⎣ ⎝ ∂z ∂x ⎠⎥⎦ ∂x ⎢⎣ ∂z ∂x ⎥⎦

(4.6)

(4.7)

4.4 Code Characteristics CFDRC code provides a variety of tools for the simulation and analysis of fluid flow and associated physics in a wide variety of industrial applications. The goal of CFDRC is to provide Software and Services for Multi-Disciplinary Engineering Simulations so that the average engineering professionals can use these exciting technologies in their day-to-day work.

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CFD-ACE+ is a set of computer programs for multi-physics computational analysis. The programs provide an integrated geometry and grid generation module, a graphical user interface for preparation of the model, a computational solver for performing the simulation, and an interactive visualization program for examination and analysis of the simulation results. The standard CFD-ACE+ package includes the following applications: • CFD-GEOM - geometry and grid generation • CFD-GUI – solver setup interface • CFD-ACE(U) - advanced polyhedral solver • CFD-VIEW - post processor 4.4.1 Geometry and Grid Generation Tools

CFDRC has three applications for geometry and grid generation: general purpose CFD-GEOM, CFD-Micromesh which is optimized for Micro-Electro Mechanical Systems (MEMS) geometries, and CFD-VisCART which builds Cartesian grid systems which can be projected with viscous layering, but here attention will be focused mainly on CFD-GEOM geometry and grid generation application. CFD-GEOM

CFD-GEOM provides interactive geometry modeling and grid generation capabilities. Similar to computer aided design (CAD), geometry modeling is the process of creating a computer model of the geometry that makes up a problem

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domain. CFD-GEOM offers a Non-Uniform Rational B-Splines (NURBS) based geometry engine with a variety of geometric construction tools. It also has the ability to import and export Initial Graphics Exchange Specification (IGES) data from most major industry CAD programs. Grid generation is the process of discretizing the problem domain with individual cells over which the flow equations are integrated. CFD-GEOM provides two classes of cells: structured and unstructured. 4.4.2 Solver Setup Interface and the Advanced Polyhedral Solver

CFD-ACE(U) is an unstructured, polyhedral cell flow solver. It is also integrated with a wide variety of other physics modules making it the core of a multi-disciplinary analysis environment. Inputs are specified for CFD-ACE(U) using CFD-GUI, an advanced graphical user interface that allows complete specification of the multi-physics problem. 4.4.2.1 Solution Methodology

CFD-ACE(U) employs a cell-centered control volume solution approach. This approach implies that the discrete equations are formulated by evaluating and integrating the fluxes across the faces that surround each control volume. In addition, CFD-ACE(U) uses a pressure-based methodology in which pressure becomes one of the dependent equations evaluated at each cell. This method allows the solver to be used for incompressible problems where pressure is loosely coupled to density.

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4.4.2.2 Phenomenon Simulated

The CFD-ACE(U) unstructured flow solver simulates a wide variety of flow regimes and phenomenon: • Laminar Flow • Turbulent Flow • Incompressible or Compressible Flow • Heat Transfer • Mixing and Reaction • Droplet/ Particle Tracking • Steady-state or Transient • Free-Surface Liquid Tracking

4.4.3 Post Processor

One of the challenges in computational modeling is that each simulation generates a large volume of data which must be reduced to extract useful information that can be applied to practical problems in science and engineering. To aid in the data reduction process, the CFD-ACE+ suite includes a 3-D graphical post processor, CFD-VIEW. 4.4.3.1 Visualization Process

Simple mouse operations select and control the visualization models displayed in the GUI. The visualization process is truly interactive; that is, you

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select a computational plane, Cartesian cutting plane, or shaded iso value surface and interactively move that entity throughout its entire volumetric domain. A turbo option is provided to ensure acceptable interactive response times for fairly large applications (over 500,000 nodes) or on relatively slow computers.

4.4.4 Discretization

To start the numerical solution process, discretization of the differential equations is introduced to produce a set of algebraic equations. In CFDACE(U), the finite-volume approach is adopted due to its attractive capability of conserving solution quantities. The details of the finite volume approach and the discretization methods implemented in CFD-ACE(U) are presented in the following sections. 4.4.4.1 Finite Volume Method

The solution domain is divided into a number of cells known as control volumes. In the finite volume approach of CFD-ACE(U), the governing equations are numerically integrated over each of these computational cells or control volumes. An example of one such control volume is shown in Fig. (4.1).

48

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n B1 Ae

●

z y

τ n

● P

●

e

e C1

●

E

EE

C3

P

B2 ●

B3

●

x

(a) 3D Computational cell (C.V)

(b) 2D Computational cell (C.V)

Fig. (4.1) Computational cell (C.V)

The geometric center of the control volume, which is denoted by P, is also often referred to as the cell center. CFD-ACE(U) employs a collocated cell centered variable arrangement, i.e. all dependent variables and material properties are stored at the cell center P. In other words, the average value of any quantity within a control volume is given by its value at the cell center. Most of the governing equations can be expressed in the form of a generalized transport equation, ∂ρ φ ∂t {

transient

r + ∇ • ( ρ V φ ) = ∇ • ( Γ ∇ φ ) + Sφ 14243 14243 { convection diffusion source

(4.8)

This equation is also known as the generic conservation equation for a quantity

φ . Integrating this equation over a control-volume cell, we have,

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v ∂( ρ φ ) d ϑ + ∇ • ( ρ V φ ) dϑ = ∫ ∇ • ( Γ ∇ φ ) dϑ + ∫ S φ dϑ ∫ ∂t ∫ ϑ ϑ ϑ ϑ

(4.9)

4.4.4.2 Transient Term

The transient term in Equation (4.9) is integrated as follows, ∂( ρ φ ) ρ φ ϑ − ρ o φ oϑ o ∫ ∂ t dϑ = ∆t ϑ

(4.10)

where the superscript “o” denotes an older time, while no superscript denotes the current or the new time. The cell volume, represented by ϑ , may change with time (in particular when moving grids are used).

4.4.4.3 Convective Term

The convection term is discretized as follows:

v v v ∇ • ( ρ V φ ) d ϑ = ρ φ V • n dA = ∑ ( ρ eφeVen ) Ae = ∑ C eφe ∫ ∫

ϑ

A

e

(4.11)

e

where subscript e denotes one of the faces of the cell in question, Ae is the area of face e, Ven represents the velocity component in the direction that is normal to the face, and C e is the mass flux across the face. Evaluation of the mass flux,

C e , will be described later when the issue of velocity-pressure coupling is addressed.

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The evaluation of φ at control volume faces is described next. For ease of illustration, let us consider a 2-D control volume as shown in Figure (4.1b). Because the solution variable φ is available only at the cell-centers, the cell-face values of φ need to be interpolated. Interpolation Schemes

Various interpolation schemes with varying levels of numerical accuracy and stability are in use today. In CFD-ACE(U) the user has a choice of several popular schemes, each of which is illustrated below in the evaluation of φ e , the value at face e. - First-Order Upwind Scheme

In this scheme, φ e is taken to be the value of φ at the upstream grid point, i.e.

φ e equals either φ P or φ E depending on the flow direction at cell face e. Mathematically, φ at this face can be expressed as:

φe UP = φ P φe UP = φ E

if if

Ven > 0 ⎫ ⎪ ⎬ ⎪ Ven < 0⎭

(4.12)

As its name implies, this scheme has first-order accuracy and is one of the most stable schemes.

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- Central Difference Scheme

In the conventional second-order accurate central difference scheme, φ e is evaluated by arithmetically averaging the values at cell centers P and E, assuming that φ varies linearly between the two centers, namely,

φe CD = Ω e φ P + (1 − Ω e ) φ E

(4.13)

where Ωe is the geometrical weighting function at face e. It is well known that conventional central difference scheme may give rise to non-physical oscillations in the numerical solution. Moreover, most iterative methods tend to be unstable when central differencing is used to evaluate convection terms. For most problems, some damping (or artificial/ numerical viscosity) is needed for stability. In CFD-ACE(U), central difference scheme with damping is constructed,

φ e = α φ e UP + (1 − α ) φ e CD

(4.14)

where α is a blending factor that combines the central scheme with the firstorder upwind scheme to produce a stable scheme. The scheme represented by the preceding equation has an order of accuracy between 1 and 2. α = 0 yields the conventional second-order accurate central difference scheme, while α = 1 results in the first-order accurate upwind scheme.

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- Second-Order Upwind Scheme

The original idea of this scheme is to evaluate the cell face value by using linear interpolation of values at two upstream cells, instead of one. In order to implement this scheme in unstructured grid system, quadrangle cells are treated differently from triangle cells in CFD-ACE(U). With the index notation in Figure (4.1b), the second-order upwind scheme gives,

φe SUD = f (φC 3 , φ P ) φe SUD = f (φ EE , φ P )

if if

Ven > 0 ⎫ ⎪ ⎬ ⎪ n Ve < 0⎭

(4.15)

Again, a blending factor is used with this scheme,

φ e = α φ e UP + (1 − α )φ e SUD

(4.16)

- Second-Order Upwind Scheme with Limiter

In order to achieve boundedness of the interpolation given by the second-order upwind scheme, the face value may be further limited to within the range of the surrounding cell center values. Such a scheme is also implemented in CFDACE(U).

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- Smart Scheme

For the central difference scheme implemented in CFD-ACE(U), the damping coefficient, i.e. the blending factor α , which is specified by the user, is used throughout the flow domain. However, experience has shown that damping is needed only in certain limited regions with steep spatial variation of flow variables. In CFD-ACE(U), a unique damping scheme, which will be called the ‘Smart Scheme’, is designed to adaptively calculate the damping coefficient depending on the local variation of φ . A new function known as the Minimode limiter is defined as follows: Minimode(λ , β ) = sign (αλ ) max[0, min ( λ , β )]

(4.17)

The smart scheme with adaptive damping can also be represented by defining damping factor as:

α e = 1 − Minimode (1, re )

(4.18)

where re is a parameter defined to measure the variation of φ. It is convenient to introduce numerical subscripts for φ to represent the functional form of re. Let i−1, i, and i+1 denote the grid points B1, P, and E respectively (see Figure (4.1b)),

re =

φi +1− u − φi − u φi +1 − φi

with u = sign (G e )

54

(4.19)

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Thus u is +1 or −1. The smart scheme reduces to the first-order upwind scheme when re < 0, to central difference scheme when re > 1, and to the second-order upwind scheme for 0 < re < 1.

- Third-Order Scheme

A third-order scheme with limiters is available in CFD-ACE(U) for accurate shock capturing in high-speed flows. This scheme is also represented by Equation (4.14) but with different expressions for α ,

αe = 1 −

η=

where

1 3

1+η 1 −η Minimode (1, β re ) Minimode ( β , re ) + 2 2

and

β=

(4.20)

3 −η 1 −η

This scheme yields third-order accuracy.

4.4.4.4 Diffusion Term

The diffusion term is discretized as follows, ⎛ ∂φ ⎞

v

∫ ∇ • ( Γ∇φ ) dϑ = ∫ Γ∇φ • ndA = ∑e Γe ⎜⎝ ∂n ⎟⎠ e Ae

ϑ

(4.21)

A

With the three unit vectors defined in Fig. (4.1b), we have, 1 ⎛ ∂φ v v ∂φ ⎞ ∂φ = v v⎜ − e •τ ⎟ ∂τ ⎠ ∂n n • e ⎝ ∂e

(4.22)

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the diffusion term becomes:

v v τ • e Γe ⎛ ∂φ ⎞ Γe ⎛ ∂φ ⎞ ∫ ∇ • ( Γ∇φ )dϑ = ∑e nv • ev ⎜⎝ ∂e ⎟⎠ e Ae − ∑e nv • ev ⎜⎝ ∂τ ⎟⎠ e Ae ϑ

(4.23)

where,

φE − φP ⎛ ∂φ ⎞ ⎜ ⎟ = δ P,E ⎝ ∂e ⎠ e

φC 2 − φC1 ⎛ ∂φ ⎞ ⎜ ⎟ = δ C 2,C1 ⎝ ∂τ ⎠ e

and

where δP,E and δC2,C1 represent the distance between E and P, and C2 and C1, respectively.

4.4.4.5 Source Term Linearization

If the source term is a function of φ itself, it is linearized as: Sφ = S U + S Pφ

(4.24)

such that SP is negative. In general, both SP and SU can be functions of φ . They are evaluated using the latest available value of φ , which is generally taken to be the value of φ ` at the end of previous iteration. The linearized source term is integrated over the control volume which results in:

∫ Sφ dϑ = SU + S P φP

(4.25)

ϑ

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where S P = S Pϑ and

SU = S U ϑ .

4.4.5 Velocity-Pressure Coupling

In this section the discretization of the continuity equation will be described. The continuity equation, which governs mass conservation, requires special attention because it cannot be written in the form of the general convection-diffusion equation. Moreover, it is used to determine the pressure field in the pressure-based method as employed in CFD-ACE(U). 4.4.5.1 Continuity and Mass Evaluation

The mass conservation equation can be written as: v ∂ρ + ∇ • ( ρ V ) = m& ∂t

(4.26)

Integrating the above equation over the cell in Figure (4.1b), we have:

ρ ϑ − ρ oϑ o ∆t

+ ∑ ρ eVen Ae = m& ϑ

(4.27)

e

where Ven is the face-normal component of the velocity at face e, which is obtained by the inner product of the velocity vector (u, v, w) and the face-normal unit vector (nx, ny, nz), Ven = u e n x + v e n y + we n z

(4.28)

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Because the fluid density and velocities are available only at cell centers, their face values need to be interpolated from cell-center values. Linear interpolation de-couples the velocity and pressure fields giving rise to the well known checkerboard instability, in particular, for incompressible flows. In this method, the cell-face mass flux is evaluated by averaging the momentum equation to the cell faces and relating the cell face velocity directly to the local pressure gradient. Without going into the details of the derivations, the three Cartesian components at the center of face e are obtained as follows, v v u e = γ e u P + (1 − γ e )u E + D ue (PE − PP ) − D ue (∇P )e • (rE − rP )

(4.29)

v v v e = γ e v P + (1 − γ e )v E + D ve (PE − PP ) − D ve (∇P )e • (rE − rP )

(4.30)

v v we = γ e w P + (1 − γ e )w E + D we (PE − PP ) − D we (∇P )e • (rE − rP )

(4.31)

where γ e is again the geometrical weighting function at face e and Ds are evaluated from link coefficients of momentum equations. The density at the face may be estimated with any one of the differencing schemes discussed earlier in this chapter. With the face values of both velocity and density obtained, the mass flux evaluation is completed.

4.4.5.2 Pressure Correction and “SIMPLEC” Algorithm

Solutions of the three momentum equations yield the three Cartesian components of velocity. Even though pressure is an important flow variable, no

58

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governing partial differential equation for pressure is presented. Pressure-based methods utilize the continuity equation to formulate an equation for pressure. In CFD-ACE(U), the SIMPLEC scheme has been adopted. The basic framework of this method is briefly described below. SIMPLEC stands for “Semi-Implicit Method for Pressure-Linked Equations Consistent”, and is an enhancement to the well known SIMPLE algorithm. In SIMPLEC, an equation for pressure-correction is derived from the continuity equation. It is an inherently iterative method. The finite-difference form of the x-momentum equation can be rewritten as: ⎛ ⎞ ⎛ ⎞ a P u P = ⎜⎜ ∑ a nb u nb + SU ⎟⎟ − ⎜⎜ ∑ Pe Ae n xe ⎟⎟ ⎝ nb ⎠P ⎝ e ⎠P

(4.32)

with the subscript P again indicating that the equation is written for cell center P. The pressure field should be provided to solve Equation (4.32) for u.

However, the pressure field is not known a priori. If the preceding equation is solved with a guessed pressure (or the latest available pressure in an iterative procedure) P * , it will yield velocity u * which satisfies the following equation: ⎛ ⎞ ⎛ ⎞ * + SU ⎟⎟ − ⎜⎜ ∑ Pe* Ae n xe ⎟⎟ a P u *P = ⎜⎜ ∑ a nb u nb ⎝ nb ⎠P ⎝ e ⎠P

(4.33)

In general, u * will not satisfy continuity. The strategy is to find corrections to u * and P * so that an improved solution can be obtained.

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Let u ′ and P ′ stand for the corrections. u = u* + u′

(4.34)

P = P* + P′

(4.35)

An expression for u ′P can be obtained by subtracting Equation (4.33) from Equation (4.32), ⎛ ⎞ ⎛ ⎞ ′ ⎟⎟ − ⎜⎜ ∑ Pe′ Ae n xe ⎟⎟ a P u ′P = ⎜⎜ ∑ a nb u nb ⎝ nb ⎠P ⎝ e ⎠P

(4.36)

Because our aim is to find an incrementally better solution, we will approximate

′ by u ′P giving us an expression for u ′P : u nb ⎛ ⎞ u ′P = ⎜⎜ ∑ Pe′ Ae n xe ⎟⎟ d P ⎝ e ⎠P

(4.37)

−1 a P − ∑ a nb

(4.38)

where

dP =

Moreover, an expression for u e′ , the velocity correction at the face, is obtained by averaging from the cell-center values: u e′ = γ e u ′P + (1 − γ e ) u E′

(4.39)

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Let us now define a mass flux as, C e = ρ eV en Ae

(4.40)

The mass conservation equation (4.27) is rewritten as, ρϑ − ρ o ϑ o ∆t

+ ∑ C e = m& ϑ

(4.41)

e

If u * and ρ * is used to evaluate C e and the resulting C e* will not satisfy the preceding equation. Let C ′ represent the correction to C, i.e.

C e = C e* + C e′

(4.42)

Equation (4.41) can be recast as,

ρ P′ ϑ ∆t

+ ∑ C e′ = S m

(4.43)

e

S m represents the mass correction or mass source in the control volume, ⎛ ρϑ − ρ o ϑ o S m = ⎜⎜ ∆t ⎝

⎞ ⎟ + m& ϑ − ∑ C e* ⎟ e ⎠

(4.44)

From Equation (4.42), we have,

(

) (

C e′ = ρ * V n ' A e + ρ ′V n* A

)

(4.45)

e

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Numerical Procedure

It should be noted that in deriving the preceding equation, terms containing products of primes have been neglected. For incompressible flows, ρ ′ is zero. For compressible flows, ρ ′ is estimated as: *

⎛ ∂ρ ⎞ ρ ′ = ⎜⎜ ⎟⎟ p ′ ⎝ ∂p ⎠

An equation of state is used to estimate

(4.46) ∂ρ . If the fluid is a perfect gas, ∂p

∂ρ 1 = ∂p RT

(4.47)

The density correction at the cell face ρ e′ , is estimated from ρ P′ and using ρ E′ the same scheme used to estimate ρ e . The correction to face-normal velocity is obtained as,

Ven ′ = ue′ n x + ve′ n y + we′ n z

(4.48)

With the face-normal velocity correction and the density correction thus expressed in terms of pressure correction, a pressure correction equation can be obtained from Equation (4.43), a P PP′ = ∑ a nb Pnb′ + S m

(4.49)

nb

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The SIMPLEC procedure can be summarized as follows: 1. Guess a pressure field P * . 2. Obtain u * , v * , and w * by solving discretized momentum Equation (4.33). 3. Evaluate C * from ρ * , u * , using the procedure outlined earlier in this section. 4. Evaluate mass source from Equation (4.44). 5. Obtain P ′ by solving Equation (4.49). 6. Use P ′ to correct the pressure and velocity fields using Equation (4.35) and Equation (4.34). 7. Solve the discretized equations for other flow variables, scalars etc. Go to step 2 and repeat the procedure until convergence is obtained. 4.4.6 Boundary Conditions

The general treatment of boundary conditions in the CFD-ACE(U) finitevolume equations is discussed in this section. A control cell adjacent to the west boundary of the calculation domain is shown in Figure (4.2).

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Numerical Procedure N

P

E

B

S

Fig. (4.2) Computational boundary cell

A fictitious boundary node labeled B is shown. The finite-volume equation for node P will be constructed as: a Pφ P = a E φ E + a N φ N + a S φS + S

(4.50)

The coefficient aW is set to zero after the links to the boundary node are incorporated into the source term S in its linearized form,

S = SU + S P φ P

(4.51)

All CFD-ACE(U) boundary conditions are implemented in this way.

Fixed Value Boundary Condition

If the boundary value is fixed as φ B , the source term is modified as,

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SU = SU + aW φ B

(4.52)

S P = S P − aW

(4.53)

Zero-Flux Boundary Conditions

At zero-flux boundaries, such as adiabatic walls for heat and symmetric boundaries for any scalar variables, the boundary link coefficients are simply set to zero without modifying source terms. 4.4.7 Solution Methods

CFD-ACE(U) uses an iterative, segregated solution method wherein the equation sets for each variable are solved sequentially and repeatedly until a converged solution is obtained. In this chapter, the overall solution method is presented in flowchart form for the solution algorithm. Additional details of the solution procedure are also presented, namely under-relaxation, relevant details of the linear equation solvers.

4.4.7.1 Overall Solution Procedure

The overall solution procedure is shown in Fig. (4.3). All the parameters that dictate how many times a procedure is repeated can be specified by the user. This is the number of iterations (NITER), and, in the case of a transient simulation is simply repeated at each time step of a transient simulation. The number of iterations to be performed is dictated by the overall residual reduction desired. At each iteration, the program will calculate a residual for each variable,

65

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which is the sum of the absolute value of the residual for that variable at each computational cell. In general, the algorithm aims to see a high order of magnitude reduction in the residual before declaring that convergence has been obtained.

66

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Numerical Procedure Start

At t = tf prescribe initial flow t = t + ∆t

Evaluate Link Coefficients Repeat NSTEP times

Solve u, v, w

Evaluate Mass Imbalances

Solve p’

Repeat NITER times

Correct p, u, v, w

Solve other equations

No

Convergence ?

Yes

STOP

Fig. (4.3) CFDRC solution flowchart

67

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4.4.7.2 Under-Relaxation

Under-relaxation of dependent and auxiliary variables is used to constrain the change in the variable from iteration to iteration in order to prevent divergence of the solution procedure. If the numerically integrated transient, convection, diffusion and source terms are assembled together, it results in the following linear equation. (a P − S P ) φ P =

∑ a nb φnb + SU

(4.54)

nb

where the subscripts “nb” denote values at neighboring cells, a nb are known as the link coefficients. For all dependent variables, this is done by modifying Equation (4.54) in the following way,

a P (1 + I ) φ P = ∑ a nb φnb + SU + a P I φ P*

(4.55)

nb

where φ P* is the current value of φ P . At convergence, when there is no change in

φ P from one iteration to the next, the equation is not modified at all by the addition of these terms. Prior to convergence, however, they provide a link between the new value of φ P and the current iteration value φ P* . The larger is this link or the value of I, the stronger the under-relaxation. Reasonable values for I can be deduced if we think of it as being analogous to the transient term link coefficient:

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CHAPTER 4 I=

Numerical Procedure

ρϑ

(4.56)

∆t f aP

where ∆ t f is a ‘false’ time step. If I is equal to 1.0, then ∆t f is comparable to the maximum time step allowed in an explicit time-marching solution scheme. Since CFD-ACE(U) uses an implicit solution scheme it is possible to use larger false time steps and, hence, smaller values of I. Values of I in the range of 0.2 to 0.8 are common. In very difficult problems it may be necessary to use values of I in excess of 1.0. The auxiliary variables ρ, P, T and µ can be under-relaxed by specifying a linear under-relaxation factor, ψ , which is applied in the following way,

δ new = ψ δ + (1 − ψ )δ *

(4.57)

where δ new is the updated value of the auxiliary variable, δ is the value of that auxiliary variable that would be calculated with no under-relaxation and δ * is the current iteration value of the auxiliary variable.

4.4.7.3 Linear Equation Solvers

Once the equation set for a variable has been assembled it must be solved. Iterative equation solvers are preferred for this task because they are more economical in terms of memory requirements than are direct solvers.

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Three types of linear equation solvers are available in CFD-ACE(U). They are briefly discussed in the next three subsections.

4.4.7.3.1 Conjugate Gradient Squared (CGS) + Preconditioning Solver

Conjugate gradient type solvers have many advantages over classic iterative methods such as suitability for vectorization and the lack of userspecified parameters. The CGS algorithm has been incorporated into the CFDACE(U) program. The CGS algorithm applied to the system Aφ = S . The convergence rate of the above CGS solver depends on the spectral radius of the coefficient matrix and can be effectively accelerated by preconditioning the system. This preconditioning is accomplished by transforming the system Aφ = S to P −1 Aφ = P −1 S , where P is a preconditioning matrix which

approximates A. Incomplete Cholesky decomposition is used as a preconditioner in CFD-ACE(U), Saad (2000).

4.4.7.3.2 Algebraic MultiGrid (AMG) Solver

Multigrid approach offers two major advantages over other iterative methods including the CGS solvers discussed above: (a) the CPU time only increase in proportion to the number of unknowns in the equations; (b) faster convergence particularly for fully-unstructured meshes.

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The basic idea of a multigrid solution is to use a hierarchy of grids, from fine to coarse, to solve a set of equations, with each grid being particularly effective for removing errors of wavelength characteristic of the mesh spacing on that grid. The solution method can be best illustrated for a two-grid system: • On the fine grid (original grid) obtain residual after performing few iterations, • Perform iterations on the coarse grid to obtain corrections, with the finegrid residual being imposed as a source term, • Interpolate the corrections to the fine grid and update the fine grid solution, • Repeat the entire procedure until the residual is reduced to the desired level. There are basically two types of multigrid methods, geometric and algebraic multigrid, respectively. In CFD-ACE(U) the latter method is adopted because it uses only the equation matrix itself (not the geometry) to generate the coarse grids.

4.5 Free Surface Simulation 4.5.1 Introduction

The Free Surface capability in CFD-ACE(U) allows the simulation of a mixture of two incompressible, immiscible fluids, including the effects of surface tensions. Common examples of such problems include the sloshing of

71

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Numerical Procedure

water in a tank, injection molding, and capillary flows. The relative mixture of the two fluids within the problem domain is tracked in terms of a secondary fluid volume fraction, F, which, by definition, ranges between 0 and 1. The approximate interface shape between the two fluids can be reconstructed based on an interpolation to determine the iso-surface corresponding to a volume fraction of 0.5. The basis of the method used in the CFD-ACE(U) Free Surface Module is the Volume-Of-Fluid (VOF) method, as published in a nearly form by Hirt and Nichols (1981), and as extended by Rider et al. (1995). The model can currently accommodate any two fluids that are incompressible and immiscible and where localized slip between the two is negligible. Water and air (when compressibility is not significant) are a good example, and will be used below for the purpose of illustration.

4.5.2 Theory Background

The characterizing feature of the VOF methodology is that the distribution of the second fluid (e.g. water) in the computational grid is accounted for using a single scalar field variable, F, which specifies the fraction of the volume of each computational cell in the grid occupied by fluid two (water). Thus, F takes the value 1 in cells which contain only fluid two (water) and the value 0 in cells which contain only fluid one (air). A cell that contains an

72

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Numerical Procedure

interface would have a value of F between 0 and 1. By definition, any values outside the range of zero to one are physically unrealistic. Given a flow field and an initial distribution of F on a grid, the manner in which the volume fraction distribution F (and hence the distribution of fluid 2) evolves is determined by solving the passive transport equation: ∂F ∂F ∂F ∂F r + V• v F = +u +v =0 ∂t ∂y ∂t ∂x

(4.58)

This equation must be solved together with the fundamental equations of conservation of mass and momentum in order to achieve computational coupling between the velocity field solution and the liquid distribution. This requires three related actions: - Compute mixture properties, - Reconstruct the fluid-fluid interface in each cell, - Determine the contribution of the secondary fluid flux to the equations for conservation of mass, momentum, energy, and volume fraction. The proportions of fluid one and fluid two in the mixture, together with the properties of these fluids, determines the average density, viscosity, thermal conductivity, and specific heat capacity, etc. in a given computational cell. In particular, the average value of any volume specific quantity, φ , in a computational cell can be computed from the value of F in accordance with:

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Numerical Procedure

~

φ = Fφ 2 + (1 − F )φ1

(4.59)

~ where φ is the volume-averaged quantity, for an intensive quantity, and Equation (4.61) can be extended to include the effect of density, ρ:

~

φ = [Fρ 2 φ2 + (1 − F )ρ 1φ1 ] ρ mix

(4.60)

4.5.3 Surface Reconstruction

The classification of the VOF method as a volume-tracking method follows directly from the use of the single scalar variable F to describe the liquid distribution and to solve for the liquid volume evolution. One consequence of the purely-volumetric representation of the phase distributions is that there is no unique definition of the interface between the two phases. As such, if the location of the interface is required for any modeling, computational, or visualization purpose, then it must be dynamically reconstructed from the F distribution. In CFD-ACE(U), surface reconstruction is a prerequisite for determining the flux of fluid two from one cell to the next and for determining surface curvatures when the surface tension model is activated. The following three methods of surface reconstruction are currently available: - A "homogeneous" upwind scheme; - An upwind scheme with the Single Line Interface Construction (SLIC) Method (Noh and Woodward, 1976); and

74

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Numerical Procedure

- An upwind scheme with the Piecewise Linear Interface Construction (PLIC) Method (Kothe et al., 1996). In the "homogenous" scheme, no attempt is made to determine the position or orientation of the free surface in a cell, such that a mixed cell is treated as homogenous foam. In the SLIC approach, the fluid surface is assumed to be parallel to the currently selected cell face, with the relative position of fluid two dependent on the flow direction and the upstream or downstream value of F. The third reconstruction method, PLIC is the most accurate of the three. In the PLIC scheme, the liquid-gas interface is assumed to be planar and allowed to take any orientation within the cell, and will therefore generally have the shape of an arbitrary polygonal facet. The facet in a cell is fully defined by specifying: (i) the spatial orientation of the infinite plane that contains the facet; and, (ii) the location of a point within the cell through which the infinite plane passes. The orientation is specified by specifying the outward-pointing unit normal of the infinite plane, and the sense of the normal is here arbitrarily chosen so that it points out of the liquid phase and into the gas phase. The unit normal of the plane is determined by assuming that it is parallel to the gradient vector of F. The gradient of F is determined from the local distribution of F in a set of cells which includes the target cell and all its immediate neighbors. The location of the "anchor point" is determined by finding the infinite cutting plane

75

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Numerical Procedure

perpendicular to the unit normal of the infinite plane that truncates the correct liquid volume from the cell, i.e., that satisfies the condition:

Vcut = F Vc

(4.61)

In the PLIC scheme, each cell has a unique surface normal that can be used to compute the surface curvature from cell to cell. This enables the calculation and addition of surface tension forces for the free surfaces. Within each computational cell, the stability limit is given by the so-called Courant Condition: ∆t c =

dc vc

(4.62)

The net normal force acting on the surface is equal to the summation of all tangential forces due to surface tension:

∫ ∆ pd s = ∫τ d x

(4.63)

Since the tangential force is equal to:

r r dx τ =σn × r dx

(4.64)

this leads to:

r r ∆ p ds = σ n × d x , ∫ ∫

r n = ∇F

(4.65)

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Numerical Procedure

The computing process can be implemented to find the fluid-fluid interface at each cell, find the unit normal of the interface, then apply the above integral to determine an effective cell volume force, CFDRC (2004).

4.6 Numerical Model 4.6.1 Problem Statement

The problem considered is depicted schematically in Figure (4.4) and refers to the three-dimensional flow under horizontal flat plate with length 40L, width 11L and the flow depth 10L and take angle β around x-axis. The flow is restricted by vertical flat plate with 1L x 1L and 0.1L thickness (spoiler) fixed with horizontal flat plate centerline at distance 14L from the inlet and inclined by angle θ with the horizontal flat plate. The outlet of the problem domain is open to the atmosphere. One or two squared holes 0.1L x 0.1L, according to test case are located just after the vertical flat plate and after 0.7L at centerline. The Cartesian coordinate is chosen and the origin. The present study considers the unsteady-state, laminar, two-phase, threedimensional, and air injection from holes. All the thermo-physical properties are assumed to be constants. The multi block calculation system approach is applied. The structured grids are divided into fourteen 3-D blocks.

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Numerical Procedure

4.6.2 Grid Structure

The structured grids are divided into fourteen 3-D blocks. Twelve of them are under water: eleven blocks beside the spoiler have grids 6 x 2 x 3, 2 x 2 x 7, 6 x 10 x 6, 6 x 8 x 36, 6 x 20 x 6, 6 x 8 x 36, 6 x 2 x 3, 6 x 2 x 3, 6 x 5 x 6, 6 x 3 x 2, and 2 x 2 x 7 grid points and the twelfth one have 7 x 20 x 36 grid points. The grids are clustered near the spoiler to solve the fluid interaction. The other two 3-D blocks are in air and represent the injection pipes. They have 2 x 2 x 6 and 2 x 2 x 6 grid points. The total number of nodes is 10,004.

y

y

x

x z

Inlet Wall

z Symmetry

Symmetry

Outlet

Inlet Symmetry

(a) front view

(b) side view

Fig. (4.4) Structured grid of flow field around spoiler

4.6.3 Boundary Conditions

On each of the computational domain boundaries, conditions are required to solve the differential equations. Generally there are six types of boundaries in this problem, namely, the inlet section, the outlet section, the solid walls, inter faces and the axis of symmetry.

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Numerical Procedure

Boundary Condition at the Walls

A no slip boundary condition is imposed on the velocity components at the walls. Inlet Condition of Main Flow

At the inlet, the pressure is set equal to atmospheric pressure, the axial velocity is set equal to constant value 0.6 m/s and the fluid volume fraction F equal to 1 (water). Inlet Condition of Injection Flow

At the inlet, the pressure is set equal to 60 N/m2, the axial velocity is set equal to 10 m/s and the fluid volume fraction F equal to 0 (air). Outlet Condition

At the outlet, the pressure is set equal to atmospheric pressure and all velocities are set equal to zero and the fluid volume fraction F equal to 1 (water). Axis of Symmetry

Here the derivative of the variables is set equal to zero. Interface Boundary Condition

Interface boundary condition is used to allow separate grid blocks or computational regions to communicate information.

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Numerical Procedure

All blocks are starting with water except the two injection pipes are completely full of air at constant temperature of 300 K. 4.6.4 Choice of Physical Time-Step

The relative mixture of the two fluids within the computational cell changes with time during the air injection. Since, the stability time limit of this changing, ∆t c , must be larger than the physical time-step. So, the physical time step is taken as 1 x 10-4 second for the unsteady flow computations in order to resolve accurately the transients of the cavity formation and according to Equation (4.62) is suitable.

4.6.5 The Run Time for a Simulation

The run time for a simulation to calculate the development of fluid motion during one second takes from 5 to 10 days on Intel(R) Pentium(R) M Processor (CENTRINO): 1.81 GHZ, RAM: 1 GB.

4.6.6 Numerical Solution Basic Assumptions

There are five basic assumptions that underlie the solution procedure: 1) The flow field around the spoiler will be solved as laminar flow. 2) The air injection holes are square with 1x1 mm and not circular with diameter 1 mm. 3) The air injection pressure is constant.

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Numerical Procedure

4) The downstream boundary conditions are atmospheric, pressure gradient is negligible. 5) The kinematic viscosity of water and air is constant. For the first and second assumptions, the free surface module in numerical code CFDRC, which used in this thesis, has the following limitations: • Not compatible with the turbulent flow module. • Use with Arbitrary Interface. For PLIC method, arbitrary interface (over

lapping grids cases) has to be submerged in one of the fluids. The second limitation will be verified when air injected through circular hole with circular grid structure of air, which make over lapping with rectangular grid structure of water, as shown in Fig. (4.5), so that we had to treat the interface of air and water as one face as shown in Fig. (4.6).

Fig. (4.5) Two grids over lapping

Fig. (4.6) Two faces with same grid

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4.6.7 Convergence Criterion and Solver Control Settings

The use of an iterative solution method necessitates the definition of a convergence and stopping criteria to terminate the iteration process. And also, the solver control settings. The measure of convergence is a norm on the change in the solution vector between successive iterations. The iterative algorithm is terminated after a fixed number of iterations if the convergence has not been achieved. This criterion is used to prevent slowly convergent or divergent problems from wasting computation time. Convergence in this thesis is defined to have been obtained after all the criteria and solver control settings in Table (4.1).

Table (4.1) Solver control settings Spoiler Angle θ (°) Rise of Floor Angle β (°) Injection Condition Max. No. of Iterations Convergence Criterion Pressure Relaxation Factor Velocities Relaxation Factor Time Step (sec.)

30

60

5

10

90

5

10

5

10

Single from hole (1), single from hole (2) and double from holes (1&2) 100

100

100

100

100

100

100

100

100

100

100

100

10-9

10-9

10-9

10-9

10-9

10-9

10-9

10-9

10-9

10-9

10-9

10-9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.5

0.5

0.9

0.9

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.1

0.1

10-4

10-4

10-4

10-4

10-4

10-4

10-4

10-4

10-4

10-4

10-4

10-4

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4.7 Post-Processing Calculations 4.7.1 Forces and Moments Calculations

The CFDRC code does not calculate the force and moment around the spoiler fixation line but the pressure distribution values only can be calculated. So, from Equation (4.66) by pressure integration over the spoiler faces (upstream and downstream faces) the normal force Fn is determined: A

Fn = ∫ PdA

(4.66)

0

Since the spoiler is inclined by angle (θ) and ship floor inclined also by angle (β), as shown in Figures (4.7, 4.8). So, by analyzing the normal force Fn in X, Y, and Z directions, the three components of force are: FX = Fn sin θ

(4.67)

FY = Fn cos θ

(4.68)

FZ = 0

(4.69)

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Numerical Procedure Y X θ FX

U

θ

Fn

FY

Fig. (4.7) Forces acting on spoiler face at X-Y plane Y

Z

Fn

β

Fig. (4.8) Forces acting on spoiler face at Y-Z plane

4.7.2 Drag, Lift and Heel Moment Coefficients Calculations

Drag and lift coefficients on the spoiler face (upstream face) and also heel moment coefficient are calculated by the following equations, Klaka et al. (2005): CD =

CL =

FD

(4.70)

1 ρ AU 2 2 FL

(4.71)

1 ρ AU 2 2

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CHAPTER 4 Cϕ =

Numerical Procedure Mϕ

(4.72)

1 ρ A LU 2 2

where: Drag force FD = F X , Lift force FL = FY , Heel moment M ϕ = moment around spoiler fixation line.

85

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Numerical Procedure

86

5

RESULTS AND DISCUSSION

CHAPTER 5

Results and Discussion

5.1 Introduction The experimental and numerical results obtained for flow filed around ship spoiler are analyzed as well as the discussion of the bubbles formation, the three-dimensional flow field pressures around the spoiler body, forces and moments. The transient flow around a ship spoiler is affected by air injection, spoiler inclination and ship floor angle. For these different conditions, eighteen experimental test cases are presented and compared to the corresponding numerical results at the same time sequence. The eighteen test cases are divided into three sections according to the three parameters effects (inclination angle θ, floor angle β and injection position). To obtain a suitable time limit for result analysis, Figures (5.1-5.3) show the comparison curves between the static pressure and total pressure variations with time on downstream spoiler face edge midpoint and downstream spoiler face center of β = 10o and β = 5o at different injection conditions. These variations become approximately constant after 0.2 second. So, all the next numerical result analysis is carried out during test time equal 1 sec.

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CHAPTER 5


Spoiler face center

β = 5o with double injection from holes (1&2) β = 5o with single injection from hole (1) β = 5o with single injection from hole (2) β = 10o with double injection from holes (1&2) β = 10o with single injection from hole (1) β = 10o with single injection from hole (2)

Spoiler face edge Y

z

Total pressure, N/m2


x

Time, sec.

Time, sec.

(b) Downstream spoiler face center



(a) Upstream spoiler face center

Time, sec.

Time, sec.

(c) Upstream spoiler face edge

(d) Downstream spoiler face edge

Fig. (5.1) Total pressure variation during transient time period 1 second for θ = 30°

88

CHAPTER 5


Spoiler face center Spoiler face edge Y

z

x




Time, sec.

Time, sec.





Time, sec.

Time, sec.




89

CHAPTER 5


Spoiler face center Spoiler face edge Y

z

x




Time, sec.

Time, sec.





Time, sec.

Time, sec.




90

CHAPTER 5


5.2 The Spoiler Inclination Angle Effect The effect of the spoiler inclination angle is presented in this section using two sets; each constitutes of 3 cases which are computed numerically and tested experimentally, Table (5.1). In these cases, the floor angle β = 10° and the injection is in Hole (1) for set (1), whereas for set (2), single injection in Hole (2) is used.

Table (5.1) Cases of spoiler inclination angle effect Parameters

Set (1) Case (1)

Spoiler angle, θ (°) Floor angle, β (°) Injection position

Case (2)

Set (2) Case (3)

30 60 90 10 10 10 Single injection in Hole (1)

Case (4)

Case (5)

Case (6)

30 60 90 10 10 10 Single injection in Hole (2)

5.2.1 Set (1): Case (1): θ = 30°, β = 10° and single injection in hole (1) Case (2): θ = 60°, β = 10° and single injection in hole (1) Case (3): θ = 90°, β = 10° and single injection in hole (1) 5.2.1.1 Results and Discussion Cavities Formation Figures (5.4-5.6) display the iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different spoiler inclination angles, θ.

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Figure (5.4) illustrates the cavities formation for spoiler inclination angle θ = 30° for a total time of 1 second starting from the air injection. The numerical results demonstrate that the cavity formation has mainly three stages. First, a cavity starts to grow at the wake of the spoiler body as shown in Figure (5.4a-d). At the second stage, the cavity starts to split, Figures (5.4e-f). Finally, the split cavity runs away at the third stage. The computational and experimental results show that there is a good matching of bubble formation except the computational bubble has slowly splitting and not attached to ship body, so, a layer of water was isolated between the bubble and the ship body, Figures. (5.4d-f). This may be due to that the flow computed as a laminar flow. The free surface module in the numerical code CFDRC is not compatible with the turbulent flow module and the buoyancy effect is not considered. For spoiler inclination angle θ = 60°, Figure (5.5) shows the cavities formation. Similar to the previous case, the cavity formation passes through three stages. The growth of cavity at the wake of the spoiler body, Figures (5.5ab), the circulation of the flow inside the bubble, Figures (5.5c-d), and finally the growth of cavity downstream and taking the final form at the third stage, Figures (5.5e-g). Both the experimental and numerical results show that bubble formation is attached to the ship surface from the beginning and it is more stable than that for inclination angle θ = 30°. Good matching between numerical and experimental results except the computational bubble shape has small dissimilarity and not flatted at bubble tail.

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Figure (5.6) displays the cavities formation for spoiler inclination angle θ = 90°. The main features observed from the numerical results are the attachment of cavity to the ship surface is from the beginning of injection, the cavity takes a rectangular shape due to the perpendicular spoiler on the ship surface and gradual increase in the length of the cavity. However, the comparison with the experimental results reveals that the cavity height decreases from the free point of spoiler and flatten to ship surface downstream. Velocity Vectors Figure (5.7) represents the velocity vectors and density contours of different cases for different spoiler inclination angles. For θ = 30°, it is demonstrated from Figure (5.7a) that the reverse flow occurs when the cavity bubble reaches to quasi-steady. For θ = 60°, Figure (5.7b) displays clearly that the more bubble stability make the velocity more uniform. Figure (5.7c) indicates that for θ = 90°, the velocity uniformity is according to the increasing in spoiler inclination angle and bubble stability formation. Similar to the previous cases, it is evident from the last three figures that the maximum velocities are concentrated around the injection hole and spoiler edges. Static Pressure Figures (5.8) shows the static pressure contours around the spoiler, upstream and downstream spoiler faces for three inclination angles at time t = 1 second. For the spoiler inclination angle θ = 30°, the static pressure contours downstream the

93

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spoiler shows the existence of different pressure regions due to the small angle of inclination. However, increasing this angle to θ = 60° one pressure region can be observed with extension beyond 7L downstream. Finally, the pressure contours for θ = 90° illustrate the mostly uniform static pressure downstream the spoiler. On the other hand, the static pressure distribution on the upstream and downstream faces of the spoiler illustrated in Figure (5.8) shows an increase in static pressure on the upstream spoiler face related to the increase in inclination angle of spoiler. Also, less pressure is observed on the three free edges of the spoiler. Meanwhile, the mean negative static pressure on the downstream face of spoiler increases. Total Pressure Figures (5.9, 5.10, 5.11) represent the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for three inclination angles at time t = 1 second. It is demonstrated that the iso-total pressure contours at the spoiler downstream are approximately similar to the isodensity contours which indicated that modeling the two phase flow field around a ship spoiler using three-dimensional Navier-Stokes with the Free Surface simulation in Piecewise Linear Interface Construction (PLIC) method is successful. The total pressure distribution on the upstream face for spoiler inclination angle θ = 30° is more uniform more than cases of θ = 60° and θ = 90°. It illustrated that the decrease in total pressure on the downstream faces related to the increase in inclination angle of spoiler.

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Moment The major objective of this study is the development of a controlling system of ship motions and speed. This system can not be verified without the determination of the moment around the fixation line. It is observed from the numerical results that the line of maximum pressure drop of spoiler face is its longitudinal centerline. The maximum of moment values for θ = 30° equals 4.124 x 10-4 N.m. For θ = 60°, the maximum value of moment is 6.177 x 10-4 N.m while for θ = 90°, the maximum value of moment is 6.81 x 10-4 N.m. Therefore, the moment around the 90° inclined spoiler fixation line is 1.651 times that around the 30° inclined spoiler and 1.1 times that around the 60° inclined spoiler after 1 second. Drag, Heel Moment and Lift Coefficients Figure (5.12) shows the variation of the drag coefficient as a function of spoiler inclination angle for β = 10° and single injection in hole (1). The effect of spoiler inclination angle on drag coefficient is high because of the produced total pressure on spoiler faces. Increasing of angle θ increases the total pressure on upstream face of spoiler and decreases it on downstream face. So, the drag coefficient is gradually increased with increasing of spoiler inclination angle. Figure (5.13) illustrates the variation of the heel moment coefficient with different inclination angles θ = 30°, 60°, and 90°. The heel moment variation follows generally the trends of drag coefficient variation with inclination angle.

95

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Figure (5.14) shows the lift coefficient at different spoiler inclination angles. Conversely, the lift coefficient decreases with increasing of spoiler inclination angle. The lift coefficient is zero for spoiler with angle θ = 90°.

96

CHAPTER 5

Density kg/m3

-3

-3

y/L -2 -1

Air injection

y/L -2 -1

Spoiler


y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0

(a) Computational results of density contours

1

2

3

4

5

6

7

8

(b) Experimental results

Fig. (5.4) Cavities formation due to single injection from Hole (1) with θ = 30° and β = 10° (at Z = -5L).

97

9 X/L

CHAPTER 5

Density kg/m3

-3

-3

y/L -2 -1

Air injection

y/L -2 -1

Spoiler


y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



98

9 X/L

CHAPTER 5


-3

-3

y/L -2 -1

Density kg/m3

y/L -2 -1

Spoiler Air injection

y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



99

9 X/L


-2

y/L

-1

0

CHAPTER 5

-2

y/L

-1

0

(a) Spoiler inclination angle θ = 30°

-2

y/L

-1

0

(b) Spoiler inclination angle θ = 60°

0

1

2

3

4

5

6

7

X/L

(c) Spoiler inclination angle θ = 90°

Fig. (5.7) Velocity vectors and density around spoiler at different θ and t = 1 sec. (at Z = -5L). β = 10° and single injection from Hole (1)

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CHAPTER 5

Results and Discussion (1) Front view

(2) Upstream face

(3) Downstream face

0 y/L 0

-1

-1

-3

y/L

y/L -2 -1

0

Static pressure N/m2

1 Z/L

0

1 Z/L

0 y/L 0

-1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

1 Z/L

0

1 Z/L

0

1

2

3

4

5

6

7

8

y/L 0

-1

-1

-3

y/L

y/L -2 -1

0

0


1 Z/L

9 X/L

(c) Spoiler inclination angle θ = 90° Fig. (5.8) Static pressure around spoiler at different θ and t = 1 sec. (at Z = -5L) β = 10° and single injection from Hole (1)

101

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(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m2

0

1 Z/L

0

1 Z/L

0 -1

-1

-3

y/L

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

0

1 Z/L

0

1

2

3

4

5

6

7

8

9 X/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L


Fig. (5.9) Total pressure around spoiler at different θ and t = 1 sec. (at Z = -5L) β = 10° and single injection from Hole (1)

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CHAPTER 5

Results and Discussion Spoiler Injection hole

-4

-3

z/L -2

-1

0

Total pressure N/m2

-4

-3

z/L -2

-1

0


-4

-3

z/L -2

-1

0


0

1

2

3

4

5

6

7

8

9

X/L


Fig. (5.10) Total pressure around spoiler at different θ and t = 1 sec. (at X-Z). β = 10° and single injection from Hole (1)

103

CHAPTER 5 (1) Upstream face

Results and Discussion (2) Downstream face

(3) Dimensions of spoiler

Spoiler

0.0 0.1 0.2 0.3 1.0 0.4 0.5 0.8 0.6 0.6 0.7 0.4 0.8 0.2 0.9 1.0 0.0




Fig. (5.11) Total pressure on spoiler faces at different θ and t = 1 sec. β = 10° and single injection from Hole (1)

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CHAPTER 5


8

5

5 β = 10o, Hole 1

β = 10o, Hole 1

β = 10o, Hole 1 4

4

6

3

3

Cφ

CD 4

2

CL 2

2

1

1

0

0 0

30

60

90

120

0

30

60

90

0

120

0

θ( )

θ( )

Fig. (5.12) Drag coefficient with different spoiler inclination angles after 1 sec.

Fig. (5.13) Heel moment coefficient with different spoiler inclination angles after 1 sec.

30

60

90

θ ( o)

o

o

Fig. (5.14) Lift coefficient with different spoiler inclination angles after 1 sec.

β = 10° and single injection from Hole (1)

5.2.2 Set (2): Case (4): θ = 30°, β = 10° and single injection in hole (2) Case (5): θ = 60°, β = 10° and single injection in hole (2) Case (6): θ = 90°, β = 10° and single injection in hole (2) 5.2.2.1 Results and Discussion Cavities Formation The iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different spoiler inclination angles are displayed on Figures (5.155.17).

105

120

CHAPTER 5


As mentioned in previous set for inclination angle θ = 30°, the cavities formation could be observed on Figure (5.15) over total time of 1 second starting from the injection of air. Numerically, the cavity formation could be resulted by three main stages which are started by growing of cavity at spoiler body wake as shown in Figures (5.4a-d). Then, the cavity starts to split, Figures (5.4e-f). Finally, the split cavity runs away. The computational and experimental results show that there is a good matching of bubble formation except the computational bubble has slowly splitting and not attached to ship body for the same reasons which are mentioned before in the previous set. For spoiler inclination angle θ = 60°, Figure (5.16) shows the cavities formation. Like the previous set, the cavity formation passes through the same three stages. The growth of cavity at the wake of the spoiler body, Figures (5.5ab), the circulation of the flow inside the bubble, Figures (5.5c-d), and finally the growth of cavity downstream and taking the final form at the third stage, Figures (5.5f-g) and also the matching condition between computational and experimental results. Figure (5.17) displays the cavities formation for spoiler inclination angle θ = 90°. The main features observed from the numerical results are the attachment of cavity to the ship surface from the starting of injection. Similar to the previous set, the cavity takes the same shaping for the previously mentioned reasons.

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Velocity Vectors Figure (5.18) shows the velocity vectors and density contours of different cases. It is demonstrated that the velocity vectors has the same flow behavior and the maximum velocities are concentrated around the injection hole and spoiler edges. Static Pressure The static pressure contours around the spoiler, upstream and downstream spoiler faces for different inclination angles at time t = 1 second are indicated in Figure (5.19). Resembling the previous set, the static pressure distribution on the upstream and downstream faces of the spoiler is related to the spoiler inclination angle. An increase in the static pressure on the upstream face of spoiler is observed with increasing inclination angle θ. Also, it is noted a significant decrease in pressure at the free spoiler edges. Total Pressure The total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for different inclination angles at time t = 1 second are represented in Figures (5.20-5.22). Like the previous set, it is evident from Figure (5.20) that the iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours. It is also noted that increasing the inclination angle of spoiler decreases the value of total

107

CHAPTER 5


pressure on the downstream face and increases the uniformity of the total pressure distribution on the upstream face. Moment As mentioned in the previous set, the numerical results indicate that the maximum pressure drop of spoiler face on the spoiler longitudinal centerline is affected. The maximum of moment values around the spoiler fixation line for θ = 30° equals 4.406 x 10-4 N.m. For θ = 60°, the maximum value of moment is 6.592 x 10-4 N.m whereas for θ = 90°, the maximum value of moment is 6.992 x 10-4 N.m. Therefore, the moment around the 90° inclined spoiler fixation line is 1.587 times that around the 30° inclined spoiler and 1.061 times that around the 60° inclined spoiler after 1 second. Drag, Heel Moment and Lift Coefficients The variation of the drag coefficient with spoiler inclination angle is represented in Figure (5.23). As mentioned in the previous set, the value of angle θ has a significant effect on the drag coefficient. The increase in inclination angle θ offsets an increase in total pressure on upstream face of spoiler and a decrease on downstream face. Therefore, the drag coefficient increases with the increase in the spoiler inclination angle. Figure (5.24) illustrates the variation of the heel moment coefficient with three different inclination angles θ = 30°, 60°, and 90°. The heel moment

108

CHAPTER 5


variation follows generally the trends of drag coefficient variation with inclination angle. Figure (5.25) shows the lift coefficient at different spoiler inclination angles. Similar to the previous set, the lift coefficient decreases with increasing of spoiler inclination angle. The lift coefficient is zero for spoiler with angle θ = 90°.

5.2.3

Drag, Heel Moment and Lift Coefficients for All Cases The variation of drag, heel moment and lift coefficients for the 18 cases

are shown in Figures (5.26-5.28) respectively. Clearly, the effect of spoiler inclination angle θ has the more prominent effect on ship balance.

109

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Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



110

9 X/L

CHAPTER 5 Air injection

-3

-3

y/L -2 -1

Density kg/m3

y/L -2 -1

Spoiler


y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



111

9 X/L


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



112

9 X/L


-2

y/L

-1

0

CHAPTER 5

-2

y/L

-1

0


-2

y/L

-1

0


0

1

2

3

4

5

6

7

X/L


Fig. (5.18) Velocity vectors and density around spoiler at different θ and t = 1 sec. (at Z = -5L). β = 10° and single injection from Hole (2)

113

CHAPTER 5


(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

0

1 Z/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

0

1 Z/L

0

1

2

3

4

5

6

7

8

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

9 X/L

(c) Spoiler inclination angle θ = 90° Fig. (5.19) Static pressure around spoiler at different θ and t = 1 sec. (at Z = -5L) β = 10° and single injection from Hole (2)

114

1 Z/L

CHAPTER 5


(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m2

1 Z/L

0

1 Z/L

0

y/L -1

-1

-3

y/L

y/L -2

-1

0

0


0

1 Z/L

0

1 Z/L

0

1

2

3

4

5

6

7

8

-1

-1

-3

y/L

y/L

y/L -2 -1

0

0


0

1 Z/L

0

9 X/L


Fig. (5.20) Total pressure around spoiler at different θ and t = 1 sec. (at Z = -5L) β = 10° and single injection from Hole (2)

115

1 Z/L

CHAPTER 5

Results and Discussion Spoiler Injection hole

-4

-3

z/L -2

-1

0

Total pressure N/m2

-4

-3

z/L -2

-1

0


-4

-3

z/L -2

-1

0


0

1

2

3

4

5

6

7

8

9

X/L


Fig. (5.21) Total pressure around spoiler at different θ and t = 1 sec (at X-Z). β = 10° and single injection from Hole (2)

116

CHAPTER 5

Results and Discussion (1) Upstream face

(2) Downstream face




Fig. (5.22) Total pressure on spoiler faces at different θ and t = 1 sec. β = 10° and single injection from Hole (2)

117

CHAPTER 5


5

5

8 β = 10o, Hole 2

β = 10o, Hole 2 4

β = 10o, Hole 2

4

6

3

3

Cφ

CD 4

CL 2

2

1

1

2

0 0

30

60

90

0

120

0

θ (o)

30

60

90

0

120

0

30

θ( ) o

Fig. (5.23) Drag coefficient with different spoiler inclination angles after 1 sec.

60

90

120

θ (o)

Fig. (5.24) Heel moment coefficient with different spoiler inclination angles after 1 sec.

Fig. (5.25) Lift coefficient with different spoiler inclination angles after 1 sec.

β = 10° and single injection from Hole (2)

8

5

5

4

4

6

3

CD 4

β = 5o, Hole 1 β = 10o, Hole 1 β = 5o, Hole 2 β = 10o, Hole 2 β = 5o, Holes 1&2 β = 10o, Holes 1&2

2

30

60

θ( ) o

90

1

120

CL


2

0 0

3

Cφ

1

0 0

30

60

90

120

θ( ) o

Fig. (5.26) Drag coefficient for Fig. (5.27) Heel moment all studied cases with different coefficient for all studied cases spoiler inclination angles with different spoiler inclination after 1 sec. angles after 1 sec.

118


2

0 0

30

60

90

120

θ (o)

Fig. (5.28) Lift coefficient for all studied cases with different spoiler inclination angles after 1 sec.

CHAPTER 5


5.3 The Injection Position Effect The effect of injection position is studied numerically for two sets; each has three test cases which are similar to cases of experimental work, Table (5.2). The effect of injection position is presented for spoiler inclination angle θ = 30° and floor angle β = 5° for the first set whereas the second set is for spoiler inclination angle θ = 90° and floor angle β = 5°.

Table (5.2) Cases of injection position effect Parameters Spoiler angle, θ (°) Floor angle, β (°) Injection position

Set (1)

Set (2)

Case (1)

Case (2)

Case (3)

Case (4)

Case (5)

Case (6)

30 5 Single injection in Hole (1)


30 5 Double injection in Holes (1) and (2)



90 5 Double injection in Holes (1) and (2)

5.3.1 Set (1): Case (1): θ = 30°, β = 5° and single injection in hole (1) Case (2): θ = 30°, β = 5° and single injection in hole (2) Case (3): θ = 30°, β = 5° and double injection in holes (1) and (2)

119

CHAPTER 5 5.3.1.1



Cavities Formation The iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different three injection positions are displayed in Figures (5.295.31) for a total time of 1 second starting from the injection of air. The numerical results demonstrate that the cavity formation has mainly three stages. First, a cavity starts to grow at the wake of the spoiler body as shown in Figures (5.29a-c), (5.30a-d) and (5.31a-c). At the second stage, the cavity starts to split, Figures (5.29d), (5.30e) and (5.31d). Finally, the split cavity runs away at the third stage. The computational and experimental results display a good agreement between the experimental and numerical simulation of bubble formation except the computational bubble has slowly splitting and not attached to ship body, so, a layer of water was isolated between the bubble and the ship body, Figures (5.29d-g), (5.30d-g) and (5.31d-g). This may be due to that the flow computed as a laminar flow. The free surface module in the numerical code CFDRC is not compatible with the turbulent flow module and the buoyancy effect. Velocity Vectors The velocity vectors and density for three injection positions at time t = 1 second are indicated in Figure (5.32). The maximum velocities are concentrated around the injection hole and spoiler edges. There is big difference in the general

120

CHAPTER 5


behavior of velocity vectors paths according to injection position. The double injection at holes (1) and (2) makes the cavity more extended and long path for reverse velocity vectors, and non-uniform movement at flow contact with spoiler surface. This differs from the cases of single injection where the extent of cavity and the path of velocity vectors are less; therefore the vectors are more uniform beside the spoiler. Static Pressure The static pressure contours around the spoiler and on the upstream and downstream spoiler faces at time t = 1 second are shown in Figure (5.33). The static pressure contours downstream the spoiler shows the existence of different pressure regions due to the small angle of inclination. The static pressure distribution on the upstream face is more uniform than in the downstream face. This may be due to the phase variation and flow interruptions do not occur like the downstream region. Also, less pressure is observed on the free edges of the spoiler. Meanwhile, the mean negative static pressure on the downstream face of spoiler increases. Total Pressure The total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces at time t = 1 second are displayed in Figures (5.34-5.36). The iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours. The total pressure distribution

121

CHAPTER 5


on the upstream face for the three cases is not uniform and the maximum total pressure is near the spoiler edge. On the downstream face, there is an increase in total pressure at air injection holes for the cases of double injection and injection at hole (1) only and this does not appear when injection at hole (2) as this hole is far from the spoiler surface which is the effective surface. In this region, the increase in total pressure is 50 N/m2 which is related to the increase in static pressure in the same region. Moment It is observed from the numerical results the moments for studied cases (θ = 30° and β = 5°). The moment around the spoiler fixation line in case of double injection in holes (1) and (2) is 3.533 x 10-4 N.m. For case of single injection in hole (1), it is 3.692 x 10-4 N.m, while for case of single injection in hole (2), the value of moment is 4.204 x 10-4 N.m. Therefore, the moment around the spoiler fixation line in first case is 0.957 times that in second case and equal about 0.84 times that in third case after 1 second. Drag, Heel Moment and Lift Coefficients Figure (5.37) shows the effect of injection position on the drag coefficient for the studied cases (θ = 30° and β = 5°). It is noted that the single injection from hole (2) has a higher drag coefficient than either double injection from holes (1) and (2) or single injection from hole (1). This may be attributed to the high total pressure on the downstream face of spoiler with small inclination angle for the

122

CHAPTER 5


cases of double injection and injection at hole (1) only. This high total pressure decreases the total force applying on the spoiler face and consequently the drag coefficient decreases in these two cases. The heel moment and lift coefficients variation with injection position has generally similar trends to that for drag coefficient as illustrated in Figures (5.38, 5.39) due to the effect of total pressure distribution on the spoiler faces.

123

CHAPTER 5


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L


g) 0

1

2

3

4

5

6

7

8

9 X/L


Fig. (5.29) Cavities formation due to double injection from Holes (1) and (2) with θ = 30° and β = 5° (at Z = -5L).

124


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L


g) 0

1

2

3

4

5

6

7

8



125

9 X/L


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



126

9 X/L


-2

y/L -1

0

CHAPTER 5

-2

-1

y/L

0

(a) Double injection in Holes (1) and (2)

-2

y/L -1

0

(b) Single injection in Hole (1)

0

1

2

3

4

5

6

7

X/L

(c) Single injection in Hole (2)

Fig. (5.32) Velocity vectors and density around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 30° and β = 5°

127

CHAPTER 5


(1) Front view

(2) Upstream face

(3) Downstream face

0 y/L 0

-1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

1 Z/L

0

1 Z/L

0

1 Z/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1

2

3

4

5

6

7

8

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

9 X/L


Fig. (5.33) Static pressure around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 30° and β = 5°

128

CHAPTER 5


(2) Upstream face

(3) Downstream face

y/L -1

-1

-3

y/L

y/L -2 -1

0

0

Total pressure N/m2

0

1 Z/L

0

1 Z/L

0

1 Z/L

0

1 Z/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

y/L -1

-1

-3

y/L

y/L -2 -1

0

0


0

0

1

2

3

4

5

6

7

8

1 Z/L

9 X/L


Fig. (5.34) Total pressure around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 30° and β = 5°

129

CHAPTER 5

Results and Discussion Spoiler

Total pressure N/m2

-4

-3

z/L -2

-1

0

Injection holes

-4

-3

z/L -2

-1

0


-4

-3

z/L -2

-1

0


0

1

2

3

4

5

6

7

8

9

X/L


Fig. (5.35) Total pressure around spoiler at different injection positions and t = 1 sec. (at X-Z) θ = 30° and β = 5°

130

CHAPTER 5


(2) Downstream face




Fig. (5.36) Total pressure on spoiler faces at different injection positions and t = 1 sec. θ = 30° and β = 5°

131

CHAPTER 5


8

5

5 θ = 30 , β = 5 o

θ = 30 , β = 5

o

o

4

6

θ = 30o, β = 5o

o

4 3

3

CD 4

CL

Cφ 2

2

1

1

2

0 0

30(1&2) Holes

60(1) Hole

90 (2) Hole

120

0

0 0

Holes 30(1&2)

Injection Position

Fig. (5.37) Drag coefficient variation with different injection positions after 1 sec.

60 (1) Hole

Hole90 (2)

120

0

Holes 30(1&2)

Fig. (5.38) Heel moment coefficient variation with different injection positions after 1 sec.

Hole 60(1)

90(2) Hole

Injection Position

Injection Position

Fig. (5.39) Lift coefficient variation with different injection positions after 1 sec.

θ = 30° and β = 5°

5.3.2 Set (2): Case (4): θ = 90°, β = 10° and single injection in hole (1) Case (5): θ = 90°, β = 10° and single injection in hole (2) Case (6): θ = 90°, β = 10° and double injection in holes (1) and (2) 5.3.2.1 Results and Discussion Cavities Formation Figures (5.40-5.42) represent the iso-density computational contours for cavities formation in a time sequence beside the ship spoiler and the experimental images for the bubble at the same time for different three injection positions. They illustrate for all injection positions, the cavities formation for a total time

132

120

CHAPTER 5


of 1 second starting from the injection of air. Similar to the previous section, it is observed from the numerical results for θ = 90° that the attachment of cavity to the ship surface is from the starting of injection, the cavity takes a rectangular shape due to the perpendicular spoiler on the ship surface and gradual increase in the length of the cavity. The comparison with the experimental results reveals that the cavity height decreases from the free point of spoiler and flatten to ship surface downstream. Velocity Vectors The velocity vectors and density for different injection positions at time t = 1 second are displayed in Figure (5.43). As mentioned in the previous case, it is indicated significantly that the air injection from different positions makes the velocity vectors paths change. The maximum velocities are concentrated around the injection hole and spoiler edges. The vectors are more uniform at contacting with the spoiler. Static Pressure Figure (5.44) shows the static pressure contours around the spoiler, upstream and downstream spoiler faces at time t = 1 second. The static pressure distribution on the upstream face is more uniform than in the downstream face. This may be due to that the phase variation and flow interruptions do not occur like the downstream region. It is observed less pressure on the free edges of the spoiler.

133

CHAPTER 5


Total Pressure Figures (5.45-5.47) display the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces at time t = 1 second. Similar to the previous set, the iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours. The total pressure distribution on the upstream face for the three cases is not uniform and the maximum total pressure is near the spoiler edge. Moment It is observed from the numerical results that the moment around the spoiler fixation line in case of double injection in holes (1) and (2) is 6.732 x 10-4 N.m. For case of single injection in hole (1), it is 7.116 x 10-4 N.m, while for case of single injection in hole (2), the value of moment is 7.065 x 10-4 N.m. Therefore, the moment around the spoiler fixation line in first case is 0.946 times that in second case and equal about 0.953 times that in third case after 1 second. Drag, Heel Moment and Lift Coefficients Figure (5.48) shows the effect of injection position on the drag coefficient for the studied cases (θ = 90° and β = 5°). It is noted that the single injection from hole (1) has a higher drag coefficient than either double injection from holes (1) and (2) or single injection from hole (2). This is in contrary to the first set and

134

CHAPTER 5


may be attributed to the big inclination angle of 90° and the single injection from hole (1) which is very near to the downstream surface of the spoiler. The heel moment coefficient variation with injection position has generally similar trends to that for drag coefficient as illustrated in Figure (5.49) due to the effect of total pressure distribution on the spoiler faces. The lift coefficient is zero as the spoiler inclination angle is 90°, Figure (5.50).

5.3.3

Drag, Heel Moment and Lift Coefficients for All Cases The drag, heel moment and lift coefficients for the 18 tested cases are

shown in Figures (5.51-5.53), respectively. For different injection holes, the drag coefficient has a magnitude in the range of 2.0–7.8, the magnitude of the lift coefficient was in the range of 0.0–4.3 and the heel moment coefficient was in the range of 1.96–3.95. For each spoiler inclination angle, the maximum drag, heel moment and lift coefficients are obtained with injection in hole (2). Noting that the flow rate was constant for the single injection whether to hole (1) or hole (2), the injection to hole (2) is superior to injection in hole (1). These high values of coefficients are attributed to the existence of hole (2) far from the spoiler fixation line compared to hole (1). The air injected from hole (2) makes a complete isolation of the spoiler downstream area from the incoming water stream by creating a stable cavity.

135


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8

9 X/L



136


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

y/L

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



137

9 X/L

CHAPTER 5


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



138

9 X/L


-2

y/L

-1

0

CHAPTER 5

-2

y/L

-1

0


-2

y/L

-1

0


0

1

2

3

4

5

6

7

X/L


Fig. (5.43) Velocity vectors and density around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 90° and β = 5°

139

CHAPTER 5


(1) Front view

(2) Upstream face

(3) Downstream face

y/L -1

-1

-3

y/L

y/L -2 -1

0

0


0

1 Z/L

1 Z/L

0

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

0

1 Z/L

0

1

2

3

4

5

6

7

8

0 y/L 0

1 Z/L

-1

-1

-3

y/L

y/L -2 -1

0


9 X/L


Fig. (5.44) Static pressure around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 90° and β = 5°

140

CHAPTER 5


(2) Upstream face

(3) Downstream face

y/L -1

-1

-3

y/L

y/L -2 -1

0

0

Total pressure N/m2

0

1 Z/L

0

1 Z/L

0

1 Z/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0 y/L

-1

-1

-3

y/L

y/L -2 -1

0


0

0

1

2

3

4

5

6

7

8

1 Z/L

0

9 X/L


Fig. (5.45) Total pressure around spoiler at different injection positions and t = 1 sec. (at Z = -5L). θ = 90° and β = 5°

141

1 Z/L

CHAPTER 5


Total pressure N/m2

-4

-3

z/L -2

-1

0

Injection holes

-4

-3

z/L -2

-1

0


-4

-3

z/L -2

-1

0


0

1

2

3

4

5

6

7

8

9

X/L


Fig. (5.46) Total pressure around spoiler at different injection positions and t = 1 sec. (at X-Z) θ = 90° and β = 5°

142

CHAPTER 5


(2) Downstream face




Fig. (5.47) Total pressure on spoiler faces at different injection positions and t = 1 sec. θ = 90° and β = 5°

143

CHAPTER 5

Results and Discussion 5

5

8

θ = 90o, β = 5o

θ = 90o, β = 5o

4

θ = 90o, β = 5o

4

6

3

3

Cφ

CD 4

2

CL 2

2

1

1

0

0 0

30(1&2) Holes

60 1 Hole

90 2 Hole

Holes30 (1&2)

0

120

Hole 60 1

Hole 90 2

120

0 0

30(1&2) Holes

Injection Position

Injection Position

Fig. (5.48) Drag coefficient variation with different injection positions after 1 sec.

60 1 Hole

90 2 Hole

120

Injection Position

Fig. (5.49) Heel moment coefficient variation with different injection positions after 1 sec.

Fig. (5.50) Lift coefficient variation with different injection positions after 1 sec.

θ = 90° and β = 5°

θ = 30o, β = 5o θ = 30o, β = 10o θ = 60o, β = 5o θ = 60o, β = 10o θ = 90o, β = 5o θ = 90o, β = 10o

θ = 30o, β = 5o θ = 30o, β = 10o θ = 60o, β = 5o θ = 60o, β = 10o θ = 90o, β = 5o θ = 90o, β = 10o 8

6

θ = 30o, β = 5o θ = 30o, β = 10o θ = 60o, β = 5o θ = 60o, β = 10o θ = 90o, β = 5o θ = 90o, β = 10o

5

5

4

4

3

3

Cφ

CD 4

2

CL 2

2

1

1

0

0 0

30(1&2) Holes

60 1 Hole

90 2 Hole

120

Injection Position

Fig. (5.51) Drag coefficient for all studied cases with different injection positions after 1 sec.

0

0

Holes30 (1&2)

60 1 Hole

90 2 Hole

120

Injection Position

Fig. (5.52) Heel moment coefficient for all studied cases with different injection positions after 1 sec.

144

0

30(1&2) Holes

60 1 Hole

90 2 Hole

120

Injection Position

Fig. (5.53) Lift coefficient for all studied cases with different injection positions after 1 sec.

CHAPTER 5


5.4 The Floor Angle Effect Floor angle effect is presented in this section with illustration of 5 sets; each consisted of 2 cases, Table (5.3). The four highlighted cases are mentioned before in the previous sections. In these cases, the floor angles are 5° and 10°.

Table (5.3) Cases of floor angle effect Set (1) Set (2) Set (3) Set (4) Set (5) Case Case Case Case Case Case Case Case Case Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Spoiler angle, θ (°) 60 60 90 90 30 30 60 60 60 60 Floor angle, β (°) 5 10 5 10 5 10 5 10 5 10 Single Single Injection position Double injection in Holes (1) and (2) injection in injection in Hole (1) Hole (2) Parameters

5.4.1 Set (1): Case (1): θ = 60°, β = 5° and double injection in holes (1) and (2) Case (2): θ = 60°, β = 10° and double injection in holes (1) and (2) 5.4.1.1 Results and Discussion Cavities Formation Figures (5.54-5.55) illustrate the iso-density computational contours for cavities formation in a time sequence beside the ship spoiler and the experimental images for the bubble at the same time for different floor raise angles, β. It is

145

CHAPTER 5


illustrated for two floor angles β = 5° and 10° for a total time of 1 second starting from the injection of air. The numerical results demonstrate that the cavity formation has mainly three stages. First, the growth of cavity at the wake of the spoiler body as shown in Fig. (5.54a-b) and Fig. (5.55a-b). At the second stage, the circulation of the flow inside the bubble and also the internal splitting of bubble, Fig. (5.54c) and Fig. (5.55c). At the third stage, the growth of the cavity and the final form shaping. The computational and experimental results show that there is a good matching of bubble formation except the computational bubble shape has small dissimilarity and is not flatted at bubble tail, Fig. (5.54d-f) and Fig. (5.55d-f). This may be due to the flow is computed as a laminar flow. The free surface module in the numerical code CFDRC is not compatible with the turbulent flow module and the buoyancy effect. Velocity Vectors Figure (5.56) shows the velocity vectors and density for two floor angles at time t = 1 second. The behavior and paths of the velocity vectors are not affected by the ship floor angle. It is displayed that the more bubble stability makes the velocity more uniform. The maximum velocities are concentrated around the injection hole and spoiler tips. Static Pressure Figure (5.57) shows the static pressure contours around the spoiler, upstream and downstream spoiler faces for two floor angles at time t = 1 second. For the

146

CHAPTER 5


raise angle β = 5°, the static pressure contours downstream the spoiler shows the existence of different pressure regions. The static pressure contours downstream has mostly uniform pressure distribution. No variation in pressure region with increasing in angle to β = 10° can be observed. On the other hand, the static pressure distribution on the upstream and downstream faces of the spoiler, Fig. (5.57), shows that the static pressure on the front surface is not affected by the raise angle of ship. Also, less static pressure is observed on the two free edges of the spoiler. Total Pressure Figures (5.58, 5.59, 5.60) displays the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for two floor angles at time t = 1 second. It is demonstrated that the iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours which indicate that modeling the two-phase flow field around a ship spoiler using three-dimensional Navier-Stokes with the Free Surface simulation in Piecewise Linear Interface Construction (PLIC) method is successful. Small variation in total pressure region with increasing in angle β can be observed. Moment It is observed from the numerical results that the moments for case of β = 5° equals 6.006 x 10-4 N.m, while for β = 10°, the value of moment is 5.81 x 10-4

147

CHAPTER 5


N.m. Therefore, the moment around the 5° floor angle spoiler fixation line is 1.034 times that around the 10° floor angle after 1 second. Drag, Heel Moment and Lift Coefficients Figures (5.61) through (5.63) show the drag, heel moment, and lift coefficients corresponding to two floor angles β = 5° and 10°. In spite of the insufficiency in floor angles data, the figures illustrate small variation in coefficients for different floor angles. This may be attributed to the edge clearance from the infinite bed that eliminates the spoiler effect on the flow restriction, and the closely total pressure values on spoiler faces for different floor angles.

148


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8

9 X/L



149

CHAPTER 5

Density kg/m3

-3

-3

y/L -2 -1

Air injection

y/L -2 -1

Spoiler


y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8

9 X/L



150


y/L

(b) Floor angle β = 10°

-2

-1

0

-2

(a) Floor angle β = 5°

y/L

-1

0

CHAPTER 5

0

1

2

3

4

5

6

7

X/L

Fig. (5.56) Velocity vectors and density around spoiler at different floor angles and t = 1 sec. (at Z = -5L). (1) Front view

(2) Upstream face

(3) Downstream face 0 y/L

0

1 Z/L

-1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

0

1

2

3

4

5

6

7

8

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

9 X/L


Fig. (5.57) Static pressure around spoiler at different floor angles and t = 1 sec. (at Z = -5L). θ = 60° and double injection from Holes (1) and (2)

151

CHAPTER 5


(2) Upstream face

(3) Downstream face

y/L -1

-1

-3

y/L

y/L -2 -1

0

0

Total pressure N/m2

0

1 Z/L

0

1 Z/L

0

1 Z/L

0

1 Z/L

0

1

2

3

4

5

6

7

8

0 y/L

9 X/L

-1

-1

-3

y/L

y/L -2 -1

0



Fig. (5.58) Total pressure around spoiler at different floor angles and t = 1 sec. (at Z = -5L). Spoiler

Injection holes

-3 -4


z/L -2

-1

0

-4

-3


z/L -2

-1

0

Total pressure N/m2

0

1

2

3

4

5

6

7

8

9

X/L

Fig. (5.59) Total pressure around spoiler at different floor angles and t = 1 sec. (at X-Z). θ = 60° and double injection from Holes (1) and (2)

152

CHAPTER 5


(1) Upstream face

(2) Downstream face



Fig. (5.60) Total pressure on spoiler faces at different floor angles and t = 1 sec. 8

5

5 θ = 60 , Double injection

θ = 60 , Double injection

θ = 60o, Double injection

o

o

4

6

4

3

CD 4

3

Cφ

CL 2

2

1

1

2

0 0

5

10

15

β (o)

0 0

5

10

15

0 0

β( )

Fig. (5.61) Drag coefficient with different floor angles after 1 sec.

Fig. (5.62) Heel moment coefficient with different floor angles after 1 sec.

10

Fig. (5.63) Lift coefficient with different floor angles after 1 sec.

θ = 60° and double injection from Holes (1) and (2)

153

5

β (o )

o

15

CHAPTER 5


5.4.2 Set (2): Case (3): θ = 90°, β = 5° and double injection in holes (1) and (2) Case (4): θ = 90°, β = 10° and double injection in holes (1) and (2) 5.4.2.1 Results and Discussion Cavities Formation Figures (5.64) and (5.40) illustrate the iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different floor raise angles, β. It is illustrated for two angles β = 5° and 10° for a total time of 1 second starting from the injection of air. As observed in section (5.3.2) for the case (θ = 90°, β = 5° and double air injection in holes (1&2)), the numerical results show an attachment of cavity is to the ship surface from the starting of injection beginning, the cavity takes a rectangular shape. The comparison with the experimental results reveals that the cavity height decreases from the free point of spoiler and flatten to ship surface downstream. Velocity Vectors Figures (5.65) and (5.43a) show the velocity vectors and density for two floor angles at time t = 1 second. The behavior and paths of the velocity vectors are not affected by the ship floor angle. It is displayed that the more bubble stability makes the velocity more uniform. The maximum velocities are concentrated around the injection hole and spoiler tips as observed in section (5.3.2).

154

CHAPTER 5


Static Pressure Figures (5.66a) and (5.44a) shows the static pressure contours around the spoiler, upstream and downstream spoiler faces for two floor angles β = 10° and 5°, respectively, at time t = 1 second. For the floor angle β = 5°, the static pressure contours downstream the spoiler shows the existence of different pressure regions. No variation in pressure region with increasing in angle to β = 10° can be observed, except that the static pressure contours downstream is more uniform pressure distribution than in floor angle β = 5°. On the other hand, the static pressure is attribution on the upstream and downstream faces of the spoiler, Figures (5.66a) and (5.44a), shows that the static pressure on the front surface is not affected by the raise angle of ship. Also, less static pressure is observed on the two free edges of the spoiler. Total Pressure Figures (5.66b), (5.45a); (5.67), (5.46a); (5.68) and (5.47a) displays the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for two floor angles β = 10° and 5°, respectively, at time t = 1 second. As remarked in section (5.3.2), the iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours which indicate that modeling the two phase flow field around a ship spoiler using three-dimensional Navier-Stokes with the Free Surface simulation in

155

CHAPTER 5


Piecewise Linear Interface Construction (PLIC) method is successful. Small variation in total pressure region with increasing in angle β can be observed. Moment It is observed from the numerical results that the moments for studied cases (θ = 90° and double air injection in holes (1) and (2)) with β = 5° equals 6.723 x 10-4 N.m, while for β = 10°, the value of moment is 7.065 x 10-4 N.m. Therefore, the moment around the 5° floor angle spoiler fixation line is 0.952 times that around the 10° floor angle after 1 second. Drag, Heel Moment and Lift Coefficients Figures (5.69) and (6.70) show the drag and heel moment coefficients corresponding to two floor angles β = 5° and 10°. The figures illustrate small variation in coefficients for two floor angles. Figure (5.71) shows the lift coefficient which is zero as the spoiler inclination angle is 90°.

156


Density kg/m3

a)

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8

9 X/L



157


-2

y/L -1

0

CHAPTER 5

0

1

2

3

4

5

6

7

X/L

Fig. (5.65) Velocity vectors and density around spoiler at floor angle β = 10° and t = 1 sec. (at Z = -5L). (1) Front view

(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

0

1 Z/L

(a) Static pressure

0

1

2

3

4

5

6

7

8

9 X/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m 2

0

(b) Total pressure

Fig. (5.66) Static and total pressures around spoiler at floor angle β = 10° and t = 1 sec. (at Z = -5L). θ = 90° and double injection from Holes (1) and (2)

158

1 Z/L

CHAPTER 5


Injection holes

-4

-3

z/L -2

-1

0

Total pressure N/m2

0

1

2

3

4

5

6

7

8

9

X/L

Fig. (5.67) Total pressure around spoiler at floor angle β = 10° and t = 1 sec. (at X-Z).

(1) Upstream face

(2) Downstream face

Fig. (5.68) Total pressure on spoiler faces at floor angle β = 10° and t = 1 sec. 8

5

5 θ = 90o, Double injection

6



4

4

3

3

CD 4

CL

Cφ

2

2 2

1

1 0 0

5

10

15

β( )

0

0 0

5

10

15

0

o



10



159

5

β (o )

β( )

o

15

CHAPTER 5


5.4.3 Set (3): Case (5): θ = 30°, β = 5° and double injection in holes (1) and (2) Case (6): θ = 30°, β = 10° and double injection in holes (1) and (2) 5.4.3.1 Results and Discussion Cavities Formation Figures (5.72) and (5.29) illustrate the iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different floor raise angles, β. It is illustrated for two angles β = 5° and 10° for a total time of 1 second starting from the injection of air. As remarked in section (5.3.1) for the case (θ = 30°, β = 5° and double air injection in holes (1) and (2)), the numerical results demonstrate that the cavity formation has mainly three stages. First, a cavity starts to grow at the wake of the spoiler body. At the second stage, the cavity starts to split. Finally, the split cavity runs away at the third stage. The computational and experimental results display a good agreement between the experimental and numerical simulation of bubble formation except the computational bubble has slowly splitting and not attached to ship body. Velocity Vectors Figure (5.73) and (5.32a) show the velocity vectors and density for two floor angles β = 10° and 5°, respectively, at time t = 1 second. Similar to the case

160

CHAPTER 5


mentioned in section (5.3.1), the cavity is more extended and long path for reverse velocity vectors, and non-uniform movement at flow contact with spoiler surface. Also, the behavior and paths of the velocity vectors are not affected by the ship floor angle. Static Pressure Figures (5.74a) and (5.33a) show the static pressure contours around the spoiler, upstream and downstream spoiler faces for two floor angles β = 10° and 5°, respectively, at time t = 1 second. As mentioned before in section (5.3.1), the static pressure contours downstream the spoiler shows the existence of different pressure regions due to the small angle of inclination. The static pressure distribution on the upstream face is more uniform than in the downstream face for the two floor angles. No variation in pressure region with increasing in angle to β = 10° can be observed. Total Pressure Figures (5.74b), (5.34a), (5.75), (5.35a), (5.76) and (5.36a) display the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for two floor angles β = 10° and 5°, respectively, at time t = 1 second. The iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours as observed in section (5.3.1). It is noted that the total pressure distribution on the upstream face for β = 5° is more uniform than β = 10°.

161

CHAPTER 5


Moment It is observed from the numerical results that the moments for studied cases (θ = 30° and double air injection in holes (1) and (2)) and for β = 5° equals 3.533 x 10-4 N.m, while for β = 10°, the value of moment is 3.633 x 10-4 N.m. Therefore, the moment around the 5° floor angle spoiler fixation line is 0.973 times that around the 10° floor angle after 1 second. Drag, Heel Moment and Lift Coefficients Figures (5.77) through (6.79) show the drag, heel moment and lift coefficients corresponding to two floor angles β = 5° and 10°. The figures illustrate small variation in coefficients for different floor angles.

162


Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8

9 X/L



163


-2

y/L

-1

0

CHAPTER 5

0

1

2

3

4

5

6

7

X/L


(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

0

1 Z/L

(a) Static pressure

0

1

2

3

4

5

6

7

8

9 X/L

0 y/L -1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m2

0

(b) Total pressure

Fig. (5.74) Static and total pressures around spoiler at floor angle β = 10° and t = 1 sec. (at Z = -5L). θ = 30° and double injection from Holes (1) and (2)

164

1 Z/L

CHAPTER 5


Injection holes

-4

-3

z/L -2

-1

0

Total pressure N/m2

0

1

2

3

4

5

6

7

8

X/L

9


(1) Upstream face

(2) Downstream face


5

5 θ = 30 , Double injection

θ = 30 , Double injection


o

o

4

6

4

3

CD 4

3

Cφ

CL 2

2

1

1

2

0 0

5

10

15

β (o)

0 0

5

10

15

0 0

β( )



10



165

5

β (o )

o

15

CHAPTER 5


5.4.4 Set (4): Case (7): θ = 60°, β = 5° and single injection in hole (1) Case (8): θ = 60°, β = 10° and single injection in hole (1) 5.4.4.1 Results and Discussion Cavities Formation Figures (5.80) and (5.5) illustrate the iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for floor raise angles, β = 5° and 10°, respectively. As for the case (θ = 60°, β = 10° and single air injection in hole (1)); section (5.2.1); the cavity formation passes through three stages. The growth of cavity at the wake of the spoiler body, the circulation of the flow inside the bubble, and finally the growth of cavity downstream and taking the final form at the third stage. Both the experimental and numerical results show that bubble formation is attached to the ship surface from the beginning. Good matching between numerical and experimental results except the computational bubble shape has small dissimilarity and not flatted at bubble tail. Velocity Vectors Figure (5.81) and (5.7b) represent the velocity vectors and density for two floor angles at time t = 1 second. Alike the case of section (5.2.1), it is clear that the more bubble stability make the velocity more uniform. The maximum velocities are concentrated around the injection hole and spoiler edges. Similar to the

166

CHAPTER 5


previous set, the behavior and paths of the velocity vectors are not affected by the ship floor angle. Static Pressure Figures (5.82a) and (5.8b) shows the static pressure contours around the spoiler, upstream and downstream spoiler faces for two floor angles at time t = 1 second. It is observed that there no variation in static pressure region with increasing in floor angle to β = 10°. Total Pressure Figures (5.82b), (5.9b); (5.83), (5.10b); (5.84) and (5.11b) displays the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for two floor angles at time t = 1 second. Identical to the case mentioned in section (5.2.1), the iso-total pressure contours at the spoiler downstream are approximately similar to the iso-density contours. No variation in total pressure region with increasing in floor angle to β = 10° is observed. Moment The numerical results estimate the moments for studied cases (θ = 60° and single air injection in hole (1)) and for β = 5° as 6.274 x 10-4 N.m, while for β = 10°, the value of moment is 6.177 x 10-4 N.m. Therefore, the moment around

167

CHAPTER 5


the 5° floor angle spoiler fixation line is 1.016 times that around the 10° floor angle after 1 second. Drag, Heel Moment and Lift Coefficients Figures (5.85), (5.86) and (5.87) show the drag, heel moment and lift coefficients corresponding to two floor angles β = 5° and 10°. The figures illustrate small variation in coefficients for two floor angles.

168

CHAPTER 5


-3

-3

y/L -2 -1

Density kg/m3

y/L -2 -1


y/L -2 -1

a)

-3

-3

y/L -2 -1

a)

y/L -2 -1

b)

-3

-3

y/L -2 -1

b)

y/L -2 -1

c)

-3

-3

y/L -2 -1

c)

y/L -2 -1

d)

-3

-3

y/L -2 -1

d)

y/L -2 -1

e)

-3

-3

y/L -2 -1

e)

y/L -2 -1

f)

-3

-3

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8



169

9 X/L


-2

y/L

-1

0

CHAPTER 5

0

1

2

3

4

5

6

7

X/L


(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


0

1 Z/L

0

1 Z/L

(a) Static pressure

0 y/L

0

1

2

3

4

5

6

7

8

-1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m2

0

1 Z/L

0

9 X/L

(b) Total pressure

Fig. (5.82) Static and total pressures around spoiler at floor angle β = 5° and t = 1 sec. (at Z = -5L). θ = 60° and single injection from Hole (1)

170

1 Z/L

CHAPTER 5


Injection holes

-4

-3

z/L -2

-1

0

Total pressure N/m2

0

1

2

3

4

5

6

7

8

X/L

9


(1) Upstream face

(2) Downstream face


5

5 θ = 60o, Single injection (1)

θ = 60o, Single injection (1)


4

6

4

3

CD 4

3

Cφ

CL 2

2

1

1

2

0 0

5

10

15

β( )

0 0

5

10

0

15

0

β (o)

o



10

β (o )


θ = 60° and single injection from Hole (1)

171

5

15

CHAPTER 5


5.4.5 Set (5): Case (9): θ = 60°, β = 5° and single injection in hole (2) Case (10): θ = 60°, β = 10° and single injection in hole (2) 5.4.5.1 Results and Discussion Cavities Formation Figures (5.88) and (5.16) illustrate the iso-density computational contours for cavities formation in a time sequence beside ship spoiler and the experimental images for the bubble at the same time for different floor angles, β. It is illustrated for two angles β = 5° and 10° for a total time of 1 second starting from the injection of air. As mentioned before in section (5.2.2) for case (θ = 60°, β = 10° and single air injection in hole (2)), the cavity formation passes through the same three stages and same matching condition between the computational and experimental results for the same reasons. Velocity Vectors Figure (5.89) and (5.18b) illustrate the velocity vectors and density for two floor angles at time t = 1 second. Similar to the previous set, the behavior and paths of the velocity vectors are not affected by the ship floor angle. Static Pressure Figures (5.90a) and (5.19b) show the static pressure contours around the spoiler, upstream and downstream spoiler faces for two floor angles at time t = 1 second.

172

CHAPTER 5


It is observed that the static pressure distribution on the downstream face for β = 5° is more uniform than that for β = 10°. Total Pressure Figures (5.90b), (5.20b); (5.91), (5.21b); (5.92) and (5.22b) display the total pressure contours around the spoiler in X-Y and X-Z planes, upstream and downstream spoiler faces for two floor angles at time t = 1 second. The iso-total pressure contours at the spoiler downstream are approximately similar to the isodensity contours which are remarked also in the previous set. It is observed that there no variation in pressure region with increasing floor angle β from 5° to 10°. Moment The moments calculated from the numerical results for Case (9) is 6.044 x 10-4 N.m, whereas for Case (10) is 6.592 x 10-4 N.m, i.e., the moment around the 5° floor angle spoiler fixation line is 0.917 times that around the 10° floor angle after 1 second. Drag, Heel Moment and Lift Coefficients The drag, heel moment and lift coefficients corresponding to two floor angles β = 5° and β = 10° are shown in Figures (5.93), (5.94) and (5.95), respectively. The variation in these coefficients for the two floor angles is small.

173

CHAPTER 5

Air injection

Density kg/m3

-3

-3

y/L -2 -1

y/L -2 -1

Spoiler


a)

-3

-3

y/L -2 -1

y/L -2 -1

a)

b)

-3

-3

y/L -2 -1

y/L -2 -1

b)

c)

-3

-3

y/L -2 -1

y/L -2 -1

c)

d)

-3

-3

y/L -2 -1

y/L -2 -1

d)

e)

-3

-3

y/L -2 -1

y/L -2 -1

e)

f)

-3

-3

y/L -2 -1

y/L -2 -1

f)

g) 0

1

2

3

4

5

6

7

8

9 X/L

g) 0


1

2

3

4

5

6

7

8


Fig. (5.88) Cavities formation due to single injection from Hole (2)

174

9 X/L

CHAPTER 5


-2

y/L

-1

0

with θ = 60° and β = 5° (at Z = -5L).

0

1

2

3

4

5

6

7

X/L


(2) Upstream face

(3) Downstream face

0 y/L -1

-1

-3

y/L

y/L -2 -1

0


1 Z/L

0

1 Z/L

0

(a) Static pressure

0 y/L

0

1

2

3

4

5

6

7

8

-1

-1

-3

y/L

y/L -2 -1

0

Total pressure N/m 2

0

1 Z/L

0

9 X/L

(b) Total pressure

Fig. (5.90) Static and total pressures around spoiler at floor angle β = 5° and t = 1 sec. (at Z = -5L). θ = 60° and single injection from Hole (2)

175

1 Z/L

CHAPTER 5


Injection holes

-4

-3

z/L -2

-1

0

Total pressure N/m2

00

1

2

3

4

5

6

7

8

9

10 X/L


(1) Upstream face

(2) Downstream face


5

5 θ = 60o, Single injection (2)



4

4

6

3

3

CD 4

CL

Cφ

2

2 2

1

1 0 0

5

10

15

β( )

0

0 0

5

10

15

β (o)

o



θ = 60° and single injection from Hole (2)

176

0

5

10

β (o )


15

CHAPTER 5 5.4.6


Drag, Heel Moment and Lift Coefficients for All Cases For the 18 cases studied, the effect of floor angle on drag, heel moment

and lift coefficients are shown in Figures (5.96-5.98) respectively. It is evident that the floor angle β is not effective on ship balance.

θ = 30o, Single injection (1) θ = 30o, Single injection (2) θ = 30o, Double injection θ = 60o, Single injection (1) θ = 60o, Single injection (2) θ = 60o, Double injection θ = 90o, Single injection (1) θ = 90o, Single injection (2) θ = 90o, Double injection



8

6

5

5

4

4 3

3

CD 4

CL

Cφ 2

2

1

1

2

0 0

5

10

15

0

0 0

5

10

β( )

Fig. (5.96) Drag coefficient for all studied cases with different floor angles after 1 sec.

15

β (o)

o

Fig. (5.97) Heel moment coefficient for all studied cases with different floor angles after 1 sec.

0

5

10

Fig. (5.98) Lift coefficient for all studied cases with different floor angles after 1 sec.

5.5 Ship Control Optimization According to the obtained results from CFDRC code and post processing calculations by Equations (4.63-4.65), Table (5.4) summarizes the forces acting

177

15

β (o)

CHAPTER 5


on spoiler face, moments around its fixation line, Figures (4.7, 4.8) and the drag, lift and heel moment coefficients.

Table (5.4) Forces, moments and coefficients on ship spoiler

Case

θ

β

(°)

(°)

1

Normal Injection Force (N) Position Fn

Location (m) ∆y

∆z

Force Moment Component Component Coefficients (N) (N.m) x 104 Fx Fy Mz CD CL Cφ

Holes (1&2) 0.0726 0.00490 0.00470 0.036 0.063

3.533

2.02 3.49 1.96

Hole (1)

0.0737 0.00590 0.00450 0.037 0.064

3.692

2.05 3.55 2.05

Hole (2)

0.0840 0.00590 0.00475 0.042 0.073

4.204

2.33 4.04 2.34

Holes (1&2) 0.0722 0.00590 0.00446 0.036 0.062

3.633

2.00 3.47 2.02

Hole (1)

0.0808 0.00519 0.00448 0.040 0.070

4.124

2.24 3.89 2.29

6

Hole (2)

0.0890 0.00496 0.00478 0.044 0.077

4.406

2.47 4.28 2.45

7

Holes (1&2) 0.1205 0.00499 0.00489 0.104 0.060

6.006

5.80 3.35 3.34

Hole (1)

0.1252 0.00502 0.00488 0.108 0.063

6.274

6.02 3.48 3.49

Hole (2)

0.1276 0.00474 0.00479 0.110 0.064

6.044

6.14 3.54 3.57

Holes (1&2) 0.1182 0.00492 0.00480 0.102 0.059

5.810

5.69 3.28 3.23

Hole (1)

0.1234 0.00500 0.00485 0.107 0.062

6.177

5.94 3.43 3.43

12

Hole (2)

0.1322 0.00498 0.00489 0.115 0.066

6.592

6.36 3.67 3.66

13

Holes (1&2) 0.1342 0.00500 0.00500 0.134

0

6.723

7.46

0

3.73

Hole (1)

0.1421 0.00500 0.00510 0.142

0

7.116

7.89

0

3.95

Hole (2)

0.1411 0.00500 0.00510 0.141

0

7.065

7.84

0

3.92

Holes (1&2) 0.1337 0.00502 0.00497 0.134

0

6.717

7.43

0

3.73

Hole (1)

0.1360 0.00500 0.00497 0.136

0

6.809

7.55

0

3.78

Hole (2)

0.1411 0.00496 0.00512 0.141

0

6.992

7.84

0

3.88

5

2 3 4

30 10

5

5

8 9 10

60 10

11

5

14 15 16 17 18

90 10

For all cases: Fz = 0 and Mx = My = 0

178

CHAPTER 5


It can be observed that in all cases with inclination angle θ = 90°, the force component Fx and moment component Mx are maximum compared to the other cases. According to this result, it can be concluded that when the two spoilers that exist at the left and right of ship are inclined by θ = 90° and β = 30° and air injection at hole (1) and making the other two spoilers with zero inclination angle gives the optimum case of decreasing the rolling of ship. Similarly, the pitching of ship can be decreased by similar choices for the two spoilers existing at the front and aft of ship.

179

CHAPTER 5


180

6

CONCLUSIONS AND RECOMMENDATIONS

CHAPTER 6

Conclusions and Recommendations

6.1 Conclusions The study of the transient flow around ship spoiler with injection of exhaust gases or air reveals that the three main parameters that have influence on the flow characteristics are spoiler inclination angle, floor angle and air injection positions. The main conclusions are summarized: 1-

The ship spoiler has a strong wake effect with gas injection, which can control the ship oscillation.

2-

The comparison between the numerical and experimental results shows a good matching of bubble formation in the cases of spoiler with 60° and 90° inclination angles except the computational bubble shape has small dissimilarity and not flatted at bubble tail, and the computational bubble shape has slowly splitting and not attached to ship body in the case of spoiler with 30° inclination angle. This may be contributed to the laminar flow computation without including turbulence effects.

181

CHAPTER 6 3-


The cavity has the same formation stages for different floor angles values and injection positions, i.e. the cavity formation does not depend on ship floor angle and position of air injection holes.

4-

The iso-total pressure contours at the spoiler’s downstream are approximately similar to the iso-density contours which indicated that modeling the two-phase flow field around a ship spoiler using threedimensional Navier-Stokes with the Free Surface simulation in Piecewise Linear Interface Construction (PLIC) method is successful.

5-

There is a reverse flow in the horizontal velocity component at the cavities region near to the ship body.

6-

The maximum vertical velocity component is concentrated around the injection hole.

7-

The moment around the spoiler fixation line of spoiler inclination angle of 90° is recorded the highest values in all different injection cases. On the other hand, the lowest values are recorded in spoiler inclination angle of 30°.

8-

Any variation in the position of air injection holes affected the moment value around the spoiler fixation line.

9-

The moment around the spoiler fixation line value depends on the ship

182

CHAPTER 6


floor angle. 10- The heel moment coefficient variation followed generally similar trends to that for drag coefficient. 11- The effect of spoiler inclination angle θ has the more prominent effect on ship balance than the other two parameters of floor angle and injection position. 12- The acting point of pressure was approximately at the centre of the spoiler face.

6.2 Recommendations Some proposed studies for future work could be useful, and they are recommended as follows: 1-

The present work deals only with a laminar flow model. It needs to be explored how the results would present with a turbulent flow model when all turbulence effects are included. Wherever, the free surface module in numerical code CFDRC, which used in this thesis, has not compatible with the turbulent flow.

2-

Measuring of stress affecting on spoiler body.

183

CHAPTER 6 3-


Studying the effect of using a free water and air injected at different velocities.

4-

Studying the effect of ship draught on bubble formation and also the moment value acting on the spoiler.

5-

It is observed from the results that the maximum moment value is at the spoiler center line. So, it is useful to study the effect of air injection positioned on a line parallel to the spoiler fixation line.

184

REFERENCES

References

References [1]

Brasher, M., Cesnik, C., and Smith, J., 2001, "Alternative Methods of Roll Control", Final Report 16.621, Fall 2001, VA., USA.

[2]

CFDRC, 2004, "CFD-ACE+ Theory and Users' Manuals Ver.6.4," CFD Research Corporation, Huntisville, AL, USA.

[3]

Clark, I.C., 2002, The Management of Merchant Ship Stability, Trim & Strength, The Nautical Institute.

[4]

Djebedjian, B., Mostafa, N.H., El-Said, E.M.S., and Rayan, M.A., 2008a, "Experimental and Numerical Study of Spoiler Effect on Ship Stability: Effect of Spoiler Inclination Angle," Twelfth International Water Technology Conference, IWTC 2008, Alexandria, Egypt, 27-30 March, pp. 729-746.

[5]

Djebedjian, B., Mostafa, N.H., El-Said, E.M.S., and Rayan, M.A., 2008b, "Experimental and Numerical Study of Spoiler Effect on Ship Stability: Effect of Spoiler Inclination Angle", Proceedings of ICFDP9, Ninth International Congress of Fluid Dynamics & Propulsion, 18-21 December, Alexandria, Egypt, ICFDP-EG-289.

[6]

Djebedjian, B., Mostafa, N.H., El-Said, E.M.S., and Rayan, M.A., 2008c, "Experimental and Numerical Study of Spoiler Effect on Ship

185

References Stability: Effect of Ship Floor Angle", Proceedings of ICFDP9, Ninth International Congress of Fluid Dynamics & Propulsion, 18-21 December, Alexandria, Egypt, ICFDP-EG-238. [7]

Djebedjian, B., Mostafa, N.H., El-Said, E.M.S., and Rayan, M.A., 2008d, "Experimental and Numerical Study of Spoiler Effect on Ship Stability: Effect of Air Injection Position", Proceedings of ICFDP9, Ninth International Congress of Fluid Dynamics & Propulsion, 18-21 December, Alexandria, Egypt, ICFDP-EG-237.

[8]

Fukuda, K., Tokunaga, J., Nobunaga, T., Nakatani, T., Iwasaki, T., and Kunitake, Y., 2000, "Frictional Drag Reduction with Air Lubricant over a Super-Water-Repellent Surface," Journal of Marine Science and Technology, Vol. 5, pp. 123-130.

[9]

Hirt, C.W., and Nichols, B.D., 1981, "Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries," Journal of Computational Physics, Vol. 39, pp. 201-225.

[10]

Holman, J.P., and Gajda, Jr. W.J., 1978, Experimental Methods for Engineers, McGraw-Hill, New York.

[11]

Klaka, K., Penrose, J.D., Horsley, R.R., and Renilson, M.R., 2005, "Hydrodynamics Test on a Fixed Plate in Uniform Flow," Experimental Thermal and Fluid Science, Vol. 30, pp. 131-139.

186

References [12]

Kothe, D.B., Rider, W.J., Mosso, S.J., and Brock, J.S., 1996, "Volume Tracking of Interfaces having Surface Tension in Two and Three Dimensions" AIAA Paper 96-0859.

[13]

Liovic, P., Rudman, M., and Liow, J.-L., 2002, "Numerical Modeling of Free Surface Flows in Metallurgical Vessels," Applied Mathematical Modelling, Vol. 26, pp. 113-140.

[14]

Madavan, N.K., Deutsch, S., and Merkle, C.L., 1984, "Measurements of Local Skin Friction in a Microbubble-Modified Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 27, pp. 356-363.

[15]

Mostafa, N.H., 2001a, "Computed Transient Supercavitating Flow Over a Projectile," Mansoura Engineering Journal, MEJ; Vol. 26, No. 2, pp. M79-M91.

[16]

Mostafa, N.H., 2001b, "Flow Around Ship Spoiler," Mansoura Engineering Journal, MEJ; Vol. 26, No. 3, pp. M1-M14.

[17]

Mostafa, N.H., 2003, "Numerical and Experimental Analysis of Transient 2D Flow around Ship Spoiler," Mansoura Engineering Journal, MEJ; Vol. 28, No. 1, pp. M1-M19.

[18]

Muckle, W., 1996, Muckle’s Naval Architecture, ButterworthHeinemann Ltd; Third Edition.

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References [19]

Murai, Y., Fukuda, H., Oishi, Y., Kodama, Y., and Yamamoto, F., 2007, "Skin Friction Reduction by Large Air Bubbles in a Horizontal Channel Flow," International Journal of Multiphase Flow, Vol. 33, pp. 147-163.

[20]

Nagayosh, T., Minato, A., Misawa, M., Suzuki, A., and Kuroda, M., 2003, "Simulation of Multi-dimensional Heterogenous and Intermittent Two-Phase Flow by Using an Extended Two-Fluid Model," Journal of Nuclear Science and Technology, Vol. 40, No. 10, pp. 827-833.

[21]

Noh, W.F., and Woodward, P.R., 1976, "SLIC (Simple Line Interface Method)," In A.I. van de Vooren and P.J. Zandbergen, editors, Lecture Notes in Physics, Vol. 59, pp. 330-340.

[22]

Owis, F., Nayfeh, A.H., and Mook, D.T., 2000, "Control of Roll Motion Using a System of Spoilers," Seventh Semi-Annual MeetingMURI- Nonlinear Active Control of Dynamical Systems, Blacksburg, VA, USA.

[23]

Rider, W.J., and Kothe, D.B., 1995, "Stretching and Tearing Interface Tracking Methods", AIAA Paper 95-1717.

[24]

Rider, W.J., Kothe, D.B., Mosso, S.J., Cerrutti, J.H., and Hochstein, J.I.,

1995,

"Accurate

Solution

Algorithms

Multiphase Fluid Flows," AIAA Paper 95-0699.

188

for

Incompressible

References [25]

Saad, Y., 2000, Iterative Methods for Spare Linear Systems, 2nd Edition, PWS, Boston.

[26]

Soper, R.R., Nayfeh, A.H., and Mook, D.T., 1998, "Control of Rolling Ships by Hull-Mounted Spoilers," Department of Engineering Science and Mechanics, MC 0219, Virginia Tech., Blacksburg, VA 24061.

[27]

Tupper, E.C., 2004, Introduction to Naval Architecture, 4th Edition, Elsevier Ltd.

[28]

Versteeg, H.K., and Malalasekera, W., 1995, An Introduction to Computational Fluid Dynamics - the Finite Volume Method, Longman Scientific Technical, England.

[29]

Walsh, M.J., and Lindemann, A.M., 1984, "Optimization and Application of Riblets for Turbulent Drag Reduction," AIAA Paper No. 84-0347.

[30]

White, F.M., 1998, Fluid Mechanics, Fourth Edition, McGraw-Hill.

189

References

190

SHIP GLOSSARY

Appendix A

A.1 Ship Types and Speed Range Table (A.1) Ship types and each speed range, Muckle (1996) Ship Type General Cargo Ships Container Ships Refrigerated Cargo Ships Roll-on Roll-off Ships Barge Carriers Oil Tankers Bulk Carriers Passenger Ships

Speed Range (m/s) 6–9 Up to 15 11 9 – 11 9 6–8 6–8 11

Speed Range (Knots) 12 – 18 Up to 30 22 18 – 22 18 12 – 16 12 – 16 22

A.2 Glossary It is imperative in the study of this subject to have a clear understanding of the meaning of the various terms abbreviations and symbols used, Muckle (1996), as shown in Figures (A.1, A.2). After Body

The immersed body aft of the midship section.

Breadth Moulded

Measured at amidships and is the maximum breadth over the frames.

Center of Gravity

The point through which the total weight of the ship may be assumed to act.

Depth Moulded

The vertical distance at amidships from top of keel to top of the deck beam at side or underside of deck plating at

191

Appendix A the ship side; the deck to which the depth is measured should be stated. Draught Moulded

The distance of the top of the keel below the water line.

Draught

The distance of the lowest point of the keel below the water line.

Flat of Keel

The small portion of the bottom which is horizontal.

Fore Body

The immersed body forward of the midship section.

Free Board

The vertical distance between the actual or permissible waterline and upper surface at side of the deck to which is measured.

Heel

The angle of inclination of the ship in the transverse direction; measured in degrees.

Rise of Floor

The rise of the bottom shell plating measured transversely amidships at the moulded breadth line.

Water Plans

Plans at right angles to the middle line plan; they are symmetrical about the middle line plane. Water plans looked at edge on in the profile are called water lines.

Yaw

The movement from the mean course of a ship in the horizontal plane; measured in degrees.

192

Appendix A

Depth Moulded

Fig. (A.1) Ship starboard profile

Breadth Moulded

Rise of Floor

Flat of Keel Fig. (A.2) Ship glossary schematic

193

Appendix A

A.3 Ship Types and Specific Details Cable Ship TELIRI (Owner Electra)

Length

111,5 mt.

Breadth

19 mt.

Building height

12,5 mt.

Draft

6,51 mt.

GRT

8.345 T/Tons

NRT

2.503 T/Tons

DWT

3.400 T/Tons

Full load

2.000 c. mt.

Full load max speed 16,5 Knots Chemical Tanker MONTE BELLO (Owner Marittima Fluviale di Navigazione)

Length

135 mt.

Breadth

20,4 mt.

Building height

11 mt.

Draft

7,55 mt.

DWT

14.000 T/Tons

Full load

16.800 c. mt.

Full load max speed

13,5 Knots

Chemical Tanker MARTINA (Owner Navigazione di Cabotaggio srl, for Cantieri Apuania)

Length

117 mt.

Breadth

20 mt.

Building height

10 mt.

Full load

12.000 c. mt.

194

Appendix A Chemical Tanker GIOVANNI FAGIOLI (Owner Finaval)

Length

125 mt.

Breadth

19,7 mt.

Building height

9,55 mt.

Draft

7,55 mt.

DWT

9.550 T/Tons

Full load

10.657 c. mt.

Full load max speed

15,6 Knots

Chemical Tanker MIMMO IEVOLI (Owner Marnavi)

Length

125,2 mt.

Breadth

19,6 mt.

Building height

9,55 mt.

Draft

7,55 mt.

DWT

9.550 T/Tons

Full load

10.657 c. mt.

Full load max speed

15,6 Knots

Chemical Tanker IEVOLI SHINE (Owner Marnavi)

Length

126,76 mt.

Breadth

19,7 mt.

Building height

9,8 mt.

Draft

7,7 mt.

DWT

10.000 T/Tons

Full load

10.910 c. mt.

Full load max speed

15,4 Knots

195

Appendix A Chemical Tanker ISOLA AMARANTO (Owner Finaval)

Length

126,76 mt.

Breadth

19,70 mt.

Building height

9,8 mt.

Draft

7,7 mt.

DWT

10.250 T/Tons

Full load

11.060 c. mt.

Full load max speed

15,4 Knots

Chemical Tanker MONTALLEGRO (Owner Eurochem Italia)

Length

135 mt.

Breadth

20,4 mt.

Building height

11 mt.

Draft

8,6 mt.

DWT

15.100 T/Tons

Full load

17.000 c. mt.

Full load max speed

13,6 Knots

Chemical Tanker ENRICO IEVOLI (Owner Marnavi)

Length

138,31 mt.

Breadth

20,40 mt.

Building height

12,15 mt.

Draft

9,2 mt.

DWT

16.400 T/Tons

Full load

17.000 c. mt.

Full load max speed

14,4 Knots

196

Appendix A Chemical Tanker ISOLA ATLANTICA (Owner Finaval)

Length

138,31 mt.

Breadth

20,40 mt.

Building height

12,15 mt.

Draft

9,2 mt.

DWT

16.400 T/Tons

Full load

17.000 c. mt.

Full load max speed

14,4 Knots

197

Appendix A

198

EXPERIMENTAL ERROR ANALYSIS

Appendix B

B.1 Experimental Error Analysis It is possible to summarize the sources of errors which can be recorded as three main sources: 1. Error from reading pressure difference in U-tube manometer - Pressure loss (PL)

(± 1.5 %)

2. Error in measuring dimensions - Width of model plate

(± 1 mm)

- Diameter of injection air orifice

(± 0.01 mm)

- Length of the spoiler

(± 0.1 mm)

- Width of the spoiler

(± 0.1 mm)

- Injection distance "L" from the spoiler

(± 0.1 mm)

3. Error in measuring angles due to parallax - Model plate inclination angle

(± 0.5 %)

- Spoiler inclination angle

(± 0.5 %)

4. Systematic errors a- Light beam reflection Light beam reflection effect can be prevented by using the light sheet in dark room.

199

Appendix B b- Water lens magnification factor Water lens magnification factor effect is not important, so it can be dissipated in images capturing. c- Plexiglas flexibility The flexibility property of Plexiglas slabs with limited different thicknesses cased deflection. In the following sections, uncertainty due to both reading pressure drop and measuring dimensions will be calculated, in addition to the systematic errors.

B.2 Uncertainty in Experimental Results The uncertainty in the result on the basis of the uncertainties in the laboratory measurements is calculated from the following formula, Holman and Gajda (1978): 2 2 2 ⎡⎛ ∂R ⎛ ∂R ⎞ ⎞ ⎛ ∂R ⎞ W R = ⎢⎜⎜ ⋅ Wn ⎟⎟ ⋅ W2 ⎟⎟ + ............. + ⎜⎜ ⋅ W1 ⎟⎟ + ⎜⎜ ⎢⎣⎝ ∂X 1 ⎠ ⎝ ∂X 2 ⎠ ⎝ ∂X n ⎠

⎤ ⎥ ⎥⎦

1/ 2

(B.1)

for a result given as:

R = R ( X 1 , X 2 ,..., X n )

(B.2)

where: WR :

Uncertainty in the result,

200

Appendix B

X 1 , X 2 ,..., X n :

Independent variables, and

W1 , W2 ,..., Wn :

Uncertainty in the corresponding variables.

The uncertainty analysis in this work comprises uncertainty in calculating the velocity of the injected air. It is function of several variables in the form of: U J = f (H , C d , Ao , A f

)

(B.3)

This velocity is given as, Holman and Gajda (1978):

U J = Cd A f

2 gH A 2f − Ao2

(B.4)

where: H : Injected air head, (m water) A f : Feeding tube cross-sectional area, (m2)

Ao : Orifice cross-sectional area, (m2) C d : Discharge coefficient. The uncertainty in injected air velocity is given by:

WU J

⎡⎛ ∂ U J = ⎢⎜⎜ ⋅WH ⎢⎝ ∂ H ⎣

2

⎛ ∂U J ⎞ ⎟⎟ + ⎜⎜ ⋅ WC d C ∂ ⎠ d ⎝

2

⎞ ⎛ ∂U J ⎞ ⎟⎟ + ⎜⎜ ⋅ W Ao ⎟⎟ ⎝ ∂ Ao ⎠ ⎠

2

⎛ ∂U J +⎜ ⋅W Af ⎜∂A f ⎝

⎞ ⎟ ⎟ ⎠

2 1/ 2

⎤ ⎥ ⎥ ⎦

(B.5)

201

Appendix B

The following partial derivatives can be obtained from Equation (B.4): ∂ U J Cd A f = 2 ∂H ∂U J = Af ∂ Cd

2g H ( A 2f − Ao2 )

(B.6)

2gH A 2f − Ao2

(B.7)

∂ U J Cd A f Ao 2 g H = ∂ Ao ( A2f − Ao2 ) 3/2

(B.8)

Cd Ao2 2 g H ∂U J =− ∂ Af ( A2f − Ao2 ) 3/2

(B.9)

From Equations (B.6) to (B.9), substituting in Equation (B.5) and using Equation (B.4), the injected air velocity uncertainty (WU J ) can be determined as:

WU J UJ

⎡⎛ W = ⎢⎜⎜ H ⎢⎝ 2 H ⎣

2

⎞ ⎛ WC d ⎞ ⎛⎜ Ao W Ao ⎞⎟ ⎛⎜ ⎟⎟ + + ⎟⎟ + ⎜⎜ 2 2 ⎟ ⎜ ⎜ C − A A ⎠ ⎝ d ⎠ ⎝ f o ⎠ ⎝ 2

2

− Ao2 A2f −

⎞ ⎟ ⋅ W A f ⎟ Ao2 ⎠ Af

2

⎤ ⎥ ⎥ ⎦

1/ 2

(B.10)

The measured values of the variables affecting the uncertainty in the mean air pressure head are: H = 0.006 m water

( WH = ± 0.001 m)

C d = 0.98

( WC d = ± 0.005)

202

Appendix B

Ao = 7.854 x 10-7 m2

( WAo = ± 1.6 x 10-6 m2)

A f = 1.963 x 10-5 m2

( WA f = ± 1.6 x 10-6 m2)

Therefore, from Equation (B.4) the injected air velocity U J = 10 m/s, and its uncertainty, Equation (B.10), WU J = 0.8353 m/s.

203

‫ﻣﻠﺨﺺ اﻟﺮﺳﺎﻟﺔ‬

‫ﺗﺘﻨﺎﻭﻝ ﺍﻟﺮﺳﺎﻟﺔ ﺩﺭﺍﺳﺔ ﺣﺴﺎﺑﻴﺔ ﻭﲡﺮﻳﺒﻴﺔ ﻟﺘﺄﺛﲑ ﺍﻟﺰﻭﺍﺋﺪ ﺍﳉﻨﻴﺤﻴﺔ ﻋﻠﻰ ﺍﺗﺰﺍﻥ ﺍﻟﺴﻔﻦ ﻭﺫﻟﻚ ﻣﻦ‬ ‫ﺧﻼﻝ ﳕﺬﺟﺔ ﳎﺎﻝ ﺍﻟﺴﺮﻳﺎﻥ ﺛﻨﺎﺋﻲ ﺍﻟﻄﻮﺭ ﺛﻼﺛﻲ ﺍﻷﺑﻌﺎﺩ ﺣﻮﻝ ﺍﻟﺰﺍﺋﺪﺓ ﻭﻋﻨﺪ ﺣﻘﻦ ﻫﻮﺍﺀ ﺧﻠﻒ ﻫﺬﻩ‬ ‫ﺍﻟﺰﺍﺋﺪﺓ ﺍﳉﻨﻴﺤﻴﻪ ﺍﻟﱵ ﰎ ﺇﻟﺼﺎﻗﻬﺎ ﻋﻠﻰ ﺍﻟﺴﻄﺢ ﺍﻟﺴﻔﻠﻰ ﻟﻐﺎﻃﺲ ﺍﻟﺴﻔﻴﻨﺔ ﻣﻊ ﺗﻐﲑ ﻣﻜﺎﻥ ﺍﳊﻘﻦ‬ ‫ﺳﻮﺍﺀ ﻣﻼﺻﻖ ﻟﻠﺬﺍﺋﺪﺓ ﺃﻭ ﻋﻠﻰ ﻣﺴﺎﻓﺔ ﻣﻨﻬﺎ‪ .‬ﺗﺘﻜﻮﻥ ﺍﻟﺰﻭﺍﺋﺪ ﺍﳉﻨﻴﺤﻴﺔ ﲟﻘﺪﻣﺔ ﺍﻟﺴﻔﻴﻨﺔ ﻛﺄﻗﺴﺎﻡ‬ ‫ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻌﺪﺩ ﻭﻣﺮﺗﺒﺔ ﻋﻠﻰ ﺍﳌﻴﻤﻨﺔ ﻭﺍﳌﻴﺴﺮﺓ ﻟﻠﺴﻔﻴﻨﺔ ﰲ ﻭﺿﻊ ﺃﻣﺎﻣﻲ ﳌﺮﻛﺰ ﺍﻟﺜﻘﻞ ﻭﺑﺎﳌﺜﻞ ﺗﺘﻜﻮﻥ‬ ‫ﺃﻳﻀﹰﺎ ﺑﺎﳌﺆﺧﺮﺓ ﻛﺄﻗﺴﺎﻡ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻌﺪﺩ ﻭﻣﺮﺗﺒﺔ ﻋﻠﻰ ﺍﳌﻴﻤﻨﺔ ﻭﺍﳌﻴﺴﺮﺓ ﻭﻣﺜﺒﺘﺔ ﺑﺎﻟﺮﺍﻓﺪﺓ ﺍﳌﺴﺘﻌﺮﺿﺔ‬ ‫ﻟﻠﺴﻔﻴﻨﺔ‪ .‬ﺇﻥ ﺣﻘﻦ ﺍﳍﻮﺍﺀ ﺧﻠﻒ ﺍﻟﺰﺍﺋﺪﺓ ﳚﻌﻞ ﺍﻟﺘﻜﻬﻒ ﺍﻟﻨﺎﺷﺊ ﺃﻛﺜﺮ ﺍﺳﺘﻘﺮﺍﺭﹰﺍ ﻭﻛﺬﻟﻚ ﺍﻟﻘﻮﻯ‬ ‫ﻭﺍﻟﻌﺰﻭﻡ ﺍﻟﱵ ﺗﻨﺸﺄ ﻋﻠﻰ ﺍﻟﺴﻄﺢ ﺍﻟﺴﻔﻠﻰ ﺍﻟﻐﺎﻃﺲ ﻟﻠﺴﻔﻴﻨﺔ ﻭﺍﻟﱵ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﺩﺭﺟﺔ ﻣﻴﻞ ﻫﺬﻩ‬ ‫ﺍﻟﺰﻭﺍﺋﺪ ﻭﻃﻮﳍﺎ ﻭﻭﺿﻌﻬﺎ ﺑﺎﻟﻨﺴﺒﺔ ﳌﺮﻛﺰ ﻛﺘﻠﺔ ﺍﻟﺴﻔﻴﻨﺔ ﻭﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﻘﻌﺮ ﺍﳌﺴﻄﺢ ﻟﻠﺴﻔﻴﻨﺔ‪.‬‬ ‫ﰎ ﳕﺬﺟﺔ ﺣﺴﺎﺑﻴﺔ ‪‬ﺎﻝ ﺍﻟﺴﺮﻳﺎﻥ ﺛﻨﺎﺋﻲ ﺍﻟﻄﻮﺭ ﺣﻮﻝ ﻫﺬﻩ ﺍﻟﺰﻭﺍﺋﺪ ﺍﳉﻨﻴﺤﻴﻪ ﻟﺪﺭﺍﺳﺔ ﺗﺄﺛﲑ ﺯﺍﻭﻳﺎ‬ ‫ﻣﻴﻠﻬﺎ ﻭﻣﻮﺍﺿﻊ ﺣﻘﻦ ﺍﳍﻮﺍﺀ ﺍﳌﺨﺘﻠﻔﺔ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻪ ﻭﺯﺍﻭﻳﺎ ﻣﻴﻞ ﺍﻟﻘﻌﺮ ﺍﳌﺴﻄﺢ ﻟﻠﺴﻔﻴﻨﺔ ﺑﻮﺍﺳﻄﺔ‬

‫‪1‬‬

‫ﺍﺳﺘﺨﺪﺍﻡ ﺑﺮﻧﺎﻣﺞ ﺣﺴﺎﰊ ﳊﻞ ﻣﻌﺎﺩﻻﺕ ﻧﺎﻓﲑ‪-‬ﺍﺳﺘﻮﻛﺲ ﰲ ﺍﻟﺜﻼﺙ ﺍﲡﺎﻫﺎﺕ ﻣﻊ ﺍﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ‬ ‫ﺍﶈﺎﻛﺎﺓ ﻟﻠﺴﻄﺢ ﺍﳊﺮ ﺑﻄﺮﻳﻘﻪ ﺍﻟﺒﻨﺎﺀ ﺍﻟﻘﻄﻌﻲ ﺍﳋﻄﻰ ﺍﻟﺬﻛﻲ ﻟﻠﺴﻄﺢ ﺍﻟﺒﻴﲏ ﺣﻴﺚ ﰎ ﲤﺜﻴﻞ‬ ‫ﺍﳌﻌﺎﺩﻻﺕ ﺍﳊﺎﻛﻤﺔ ﻋﻠﻰ ﺷﺒﻜﺔ ﻣﻨﺸﺄﺓ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻧﻈﺎﻡ ﻓﺮﻕ ﺍﻻﺧﺘﻼﻑ ﺍﳌﻀﺎﺩ ﻟﻼﲡﺎﻩ‪ .‬ﰎ ﺣﺴﺎﺏ‬ ‫ﺷﻜﻞ ﺍﻟﻔﻘﺎﻋﺔ ﻭﳎﺎﻝ ﺍﻟﺴﺮﻳﺎﻥ ﺛﻼﺛﻲ ﺍﻷﺑﻌﺎﺩ ﺣﻮﻝ ﺟﺴﻢ ﺍﻟﺰﺍﺋﺪﺓ ﺍﳉﻨﻴﺤﻴﻪ ﻭﺍﺧﺘﻼﻓﺎﺕ ﺍﻟﻀﻐﻂ‬ ‫ﻋﻠﻰ ﺍﻣﺘﺪﺍﺩ ﻣﺴﺎﺭ ﺍﳌﺎﺋﻊ ﳍﺬﻩ ﺍﻟﺰﺍﺋﺪﺓ ﺍﳉﻨﻴﺤﻴﺔ ﰲ ﺍﻟﻈﺮﻭﻑ ﺍﳌﺨﺘﻠﻔﺔ‪.‬‬ ‫ﻭﺑﺎﻹﺿﺎﻓﺔ ﻟﺬﻟﻚ ﰎ ﻋﻤﻠﻴﹰﺎ ﺗﺼﻮﻳﺮ ﳎﺎﻝ ﺍﻟﺴﺮﻳﺎﻥ ﺣﻮﻝ ﺍﻟﺰﺍﺋﺪﺓ ﺍﻟﺴﺮﻳﺎﻥ ﰲ ﺍﳊﺎﻻﺕ ﻭﺍﻷﺯﻣﻨﺔ‬ ‫ﺍﳌﺨﺘﻠﻔﺔ ﺑﻨﻔﺲ ﻣﻘﻴﺎﺱ ﺍﻟﻨﻤﻮﺫﺝ ﺍﳊﺴﺎﰊ‪ ،‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﻛﺎﻣﲑﺍ ﻓﻴﺪﻳﻮ‪ .‬ﻭﻗﺪ ﺃﻇﻬﺮﺕ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﻌﻤﻠﻴﺔ‬ ‫ﻭﺍﳊﺴﺎﺑﻴﺔ ﺗﻄﺎﺑﻖ ﺟﻴﺪ ﰲ ﺣﺎﻻﺕ ﻣﻴﻞ ﺍﻟﺰﺍﺋﺪﺓ ﺑﺰﻭﺍﻳﺎ ‪ ٦٠‬ﺩﺭﺟﺔ ﻭ‪ ٩٠‬ﺩﺭﺟﺔ ﻓﻴﻤﺎ ﻋﺪﺍ ﺍﺧﺘﻼﻑ‬ ‫ﺍﻟﻨﺘﺎﺋﺞ ﺍﳊﺴﺎﺑﻴﺔ ﻋﻦ ﺍﻟﻌﻤﻠﻴﺔ ﰲ ﺃﻥ ﺷﻜﻞ ﺍﻟﻔﻘﺎﻋﺔ ﻏﲑ ﻣﻨﺒﺴﻂ ﺑﺎﻟﻨﻬﺎﻳﺔ‪ ،‬ﺃﻣﺎ ﰲ ﺣﺎﻟﺔ ﺍﻟﺰﺍﺋﺪﺓ‬ ‫ﺍﳉﻨﻴﺤﻴﺔ ﺫﺍﺕ ﻣﻴﻞ ‪ ٣٠‬ﺩﺭﺟﺔ ﻓﺈﻥ ﺍﻟﻔﻘﺎﻋﺔ ﻻ ﺗﻠﺘﺼﻖ ﺑﺎﻟﺴﻄﺢ ﺍﻟﺴﻔﻠﻰ ﻟﻠﺴﻔﻴﻨﺔ ﻭﺗﻨﻔﺼﻞ ﺑﺸﻜﻞ‬ ‫ﺑﻄﻲﺀ‪ .‬ﻭﻗﺪ ﺗﺒﲔ ﺃﻧﻪ ﰲ ﺣﺎﻟﺔ ﻣﻴﻞ ﺍﻟﺰﺍﺋﺪﺓ ﺍﳉﻨﻴﺤﻴﺔ ﺑﺰﻭﺍﻳﺎ ‪ ٦٠‬ﺩﺭﺟﺔ ﻭ‪ ٩٠‬ﺩﺭﺟﺔ ﲢﺪﺙ ﺩﻭﺍﻣﻪ‬ ‫ﺩﺍﺧﻞ ﺍﻟﻔﻘﺎﻋﺔ ﻭﺗﻜﻮﻥ ﺍﻟﻔﻘﺎﻋﺔ ﺃﻃﻮﻝ ﳑﺎ ﻳﻌﻄﻰ ﺿﻐﻄﹰﺎ ﻣﻨﺨﻔﻀﹰﺎ ﻋﻠﻰ ﻣﺴﺎﺣﻪ ﺃﻛﱪ ‪ ،‬ﻭﻗﺪ ﻳﻜﻮﻥ‬

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‫ﺳﺒﺐ ﻫﺬﻩ ﺍﻻﺧﺘﻼﻓﺎﺕ ﺃﻥ ﺍﳊﺴﺎﺑﺎﺕ ﲤﺖ ﻋﻠﻰ ﺍﻋﺘﺒﺎﺭ ﺃﻥ ﺍﻟﺴﺮﻳﺎﻥ ﺭﻗﺎﺋﻘﻲ ﺑﺪﻭﻥ ﺍﻟﺘﻌﺮﺽ‬ ‫ﻟﺘﺄﺛﲑﺍﺕ ﺍﻟﺴﺮﻳﺎﻥ ﺍﳌﻀﻄﺮﺏ‪.‬‬ ‫ﻭﻗﺪ ﺗﺒﲔ ﺃﻥ ﻟﺰﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﺰﺍﺋﺪﺓ ﺗﺄﺛﲑ ﻛﺒﲑ ﻋﻠﻰ ﺗﻐﲑ ﻗﻴﻤﺔ ﺍﻟﻌﺰﻭﻡ ﻭﻣﻦ ﰒ ﻗﻴﻤﺔ ﺃﻱ ﻣﻦ‬ ‫ﺍﳌﻌﺎﻣﻼﺕ ﺫﺍﺕ ﺍﻟﻌﻼﻗﺔ ﺑﻘﻴﻤﺔ ﺍﻟﻘﻮﻯ ﻭﺍﻟﻌﺰﻭﻡ ﻭﻋﻠﻰ ﺫﻟﻚ ﻓﺈﻥ ﲨﻴﻊ ﺣﺎﻻﺕ ﺍﻟﺰﺍﺋﺪﺓ ﺫﺍﺕ ﻣﻴﻞ‬ ‫‪ ٩٠‬ﺩﺭﺟﺔ ﺳﺠﻠﺖ ﺃﻋﻠﻰ ﻗﻴﻢ ﻟﻠﻌﺰﻭﻡ ﻭﰲ ﺍﳌﻘﺎﺑﻞ ﻗﺪ ﺗﺒﲔ ﺃﻥ ﲨﻴﻊ ﺣﺎﻻﺕ ﺍﻟﺰﺍﺋﺪﺓ ﺫﺍﺕ ﻣﻴﻞ‬ ‫‪ ٣٠‬ﺩﺭﺟﺔ ﺳﺠﻠﺖ ﺃﻗﻞ ﻗﻴﻢ ﻟﻠﻌﺰﻭﻡ ﻭﺫﻟﻚ ﺑﻌﺪ ﻣﺮﻭﺭ ﺯﻣﻦ ﻗﺪﺭﻩ ‪ ١‬ﺛﺎﻧﻴﺔ ﻣﻦ ﺑﺪﺍﻳﺔ ﺍﳊﻘﻦ‪.‬‬ ‫ﻭﺑﻨﺎﺀﹰﺍ ﻋﻠﻴﻪ ﻓﺈﻥ ﺃﻱ ﺗﻐﻴﲑ ﰲ ﻭﺿﻊ ﺍﻟﺰﻭﺍﺋﺪ ﻭﺯﺍﻭﻳﺎ ﻣﻴﻠﻬﺎ ﻳﻨﺘﺞ ﻋﻨﻪ ﺗﻐﲑ ﰲ ﺷﻜﻞ ﺍﻟﻔﻘﺎﻋﺔ‬ ‫ﻭﺍﻟﻀﻐﻮﻁ ﺣﻮﳍﺎ ﳑﺎ ﻳﺴﺎﻋﺪ ﻋﻠﻰ ﺍﻟﺘﺤﻜﻢ ﰲ ﺍﳊﺮﻛﺔ ﺍﻻﻫﺘﺰﺍﺯﻳﺔ ﺍﻟﺮﺃﺳﻴﺔ ﻭﺍﻟﺪﻭﺍﺭﺓ ﻭﺳﺮﻋﺔ‬ ‫ﺍﻟﺴﻔﻴﻨﺔ ﳑﺎ ﻳﺆﺩﻱ ﺇﱃ ﺟﻌﻞ ﺍﻟﺴﻔﻴﻨﺔ ﺃﻛﺜﺮ ﺍﺗﺰﺍﻧﺎﹰ‪.‬‬

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‫ﺍﻟﺴﺎﺩﺓ ﺍﻟﻤﺸﺭﻓﻴﻥ‬ ‫ﻋﻨﻭﺍﻥ ﺍﻟﺭﺴﺎﻟﺔ‪ :‬ﺩﺭﺍﺴﺔ ﺘﺠﺭﻴﺒﻴﺔ ﻭﺤﺴﺎﺒﻴﺔ ﻟﺘﺄﺜﻴﺭ ﺍﻟﺯﻭﺍﺌﺩ ﺍﻟﺠﻨﻴﺤﻴﺔ ﻋﻠﻰ ﺍﺘﺯﺍﻥ ﺍﻟﺴﻔﻥ‬ ‫ﺍﺴﻡ ﺍﻟﺒﺎﺤﺙ‪ :‬ﻡ‪ /‬ﻋﻤﺎﺩ ﻤﺤﻤﺩ ﺴﻌﺩ ﺍﻟﺴﻌﻴﺩ‬ ‫ﻟﺠﻨﺔ ﺍﻹﺸﺭﺍﻑ‪:‬‬ ‫ﺍﻻﺴﻡ‬

‫ﻡ‬

‫ﺍﻟﻭﻅﻴﻔﺔ‬

‫‪١‬‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻤﺠﺩﻱ ﻤﺤﻤﺩ ﺃﺒﻭ ﺭﻴﺎﻥ‬

‫‪٢‬‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻨﺒﻴل ﺤﺴﻥ ﻤﺼﻁﻔﻲ‬

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‫ﺃ‪.‬ﻡ‪.‬ﺩ‪ /.‬ﺒﻴﺭﺝ ﺃﻭﻫﺎﻨﺱ ﺠﺒﻪ ﺠﻴﺎﻥ‬

‫ﺍﻟﺘﻭﻗﻴﻊ‬

‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬

‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬ ‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﺯﻗﺎﺯﻴﻕ‬

‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬

‫ﺭﺌﻴﺱ ﻤﺠﻠﺱ ﺍﻟﻘﺴﻡ‬

‫ﻭﻜﻴل ﺍﻟﻜﻠﻴﺔ ﻟﻠﺩﺭﺍﺴﺎﺕ ﺍﻟﻌﻠﻴﺎ ﻭﺍﻟﺒﺤﻭﺙ‬

‫ﻋﻤﻴﺩ ﺍﻟﻜﻠﻴﺔ‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻟﻁﻔﻲ ﺤﺴﻥ ﺭﺒﻴﻊ‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻤﺤﻤﺩ ﺍﻟﺸﺒﺭﺍﻭﻯ ﻤﺤﻤﺩ ﻋﻠﻲ‬


‫ﺍﻟﺴﺎﺩﺓ ﺃﻋﻀﺎﺀ ﻟﺠﻨﺔ ﺍﻟﻤﻨﺎﻗﺸﺔ ﻭﺍﻟﺤﻜﻡ‬ ‫ﻋﻨﻭﺍﻥ ﺍﻟﺭﺴﺎﻟﺔ‪ :‬ﺩﺭﺍﺴﺔ ﺘﺠﺭﻴﺒﻴﺔ ﻭﺤﺴﺎﺒﻴﺔ ﻟﺘﺄﺜﻴﺭ ﺍﻟﺯﻭﺍﺌﺩ ﺍﻟﺠﻨﻴﺤﻴﺔ ﻋﻠﻰ ﺍﺘﺯﺍﻥ ﺍﻟﺴﻔﻥ‬

‫ﺍﺴﻡ ﺍﻟﺒﺎﺤﺙ‪ :‬ﻡ‪ /‬ﻋﻤﺎﺩ ﻤﺤﻤﺩ ﺴﻌﺩ ﺍﻟﺴﻌﻴﺩ‬ ‫ﻟﺠﻨﺔ ﺍﻹﺸﺭﺍﻑ‪:‬‬ ‫ﻡ‬

‫ﺍﻻﺴﻡ‬


‫‪١‬‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻤﺠﺩﻱ ﻤﺤﻤﺩ ﺃﺒﻭ ﺭﻴﺎﻥ‬

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‫‪٣‬‬

‫ﺃ‪.‬ﻡ‪.‬ﺩ‪ /.‬ﺒﻴﺭﺝ ﺃﻭﻫﺎﻨﺱ ﺠﺒﻪ ﺠﻴﺎﻥ‬


‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬ ‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﺯﻗﺎﺯﻴﻕ‬ ‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬

‫ﻟﺠﻨﺔ ﺍﻟﻤﻨﺎﻗﺸﺔ ﻭﺍﻟﺤﻜﻡ‪:‬‬ ‫ﻡ‬

‫ﺍﻻﺴﻡ‬


‫‪ ١‬ﺃ‪.‬ﺩ‪ /.‬ﺼﺎﺩﻕ ﺯﻜﺭﻴﺎ ﻜﺴﺎﺏ‬ ‫‪ ٢‬ﺃ‪.‬ﺩ‪ /.‬ﻤﺤﻤﺩ ﺼﻔﻭﺕ ﺴﻌﺩ ﺍﻟﺩﻴﻥ‬


‫ﻗﺴﻡ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻻﺴﻜﻨﺩﺭﻴﺔ‬ ‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬

‫‪ ٣‬ﺃ‪.‬ﺩ‪ /.‬ﻤﺠﺩﻱ ﻤﺤﻤﺩ ﺃﺒﻭ ﺭﻴﺎﻥ‬

‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬

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‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ –‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﺯﻗﺎﺯﻴﻕ‬

‫ﺭﺌﻴﺱ ﻤﺠﻠﺱ ﺍﻟﻘﺴﻡ‬

‫ﻭﻜﻴل ﺍﻟﻜﻠﻴﺔ ﻟﻠﺩﺭﺍﺴﺎﺕ ﺍﻟﻌﻠﻴﺎ ﻭﺍﻟﺒﺤﻭﺙ‬

‫ﻋﻤﻴﺩ ﺍﻟﻜﻠﻴﺔ‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻟﻁﻔﻲ ﺤﺴﻥ ﺭﺒﻴﻊ‬



‫ﺠﺎﻤﻌﺔ ﺍﻟﻤﻨﺼﻭﺭﺓ‬ ‫ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴــــــﺔ‬ ‫ﻗﺴﻡ ﻫﻨﺩﺴﺔ ﺍﻟﻘﻭﻯ ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬

‫دراﺳﺔ ﺗﺠﺮﻳﺒﻴﺔ وﺣﺴﺎﺑﻴﺔ ﻟﺘﺄﺛﻴﺮ اﻟﺰواﺋﺪ اﻟﺠﻨﻴﺤﻴﺔ ﻋﻠﻰ اﺗﺰان اﻟﺴﻔﻦ‬ ‫رﺳـﺎﻟﺔ‬ ‫ﻣﻘﺪﻣﺔ ﻟﻠﺤﺼﻮل ﻋﻠﻰ درﺟﺔ اﻟﻤﺎﺟﺴـﺘﻴﺮ‬ ‫ﻓﻲ هﻨﺪﺳـﺔ اﻟﻘﻮي اﻟﻤﻴﻜﺎﻧﻴﻜﻴﺔ‬

‫ﻣﻦ‬

‫اﻟﻤﻬﻨﺪس‪ /‬ﻋﻤﺎد ﻣﺤﻤﺪ ﺳﻌﺪ اﻟﺴﻌﻴﺪ‬ ‫ﺑﻜﺎﻟﻮﺭﻳﻮﺱ ﻫﻨﺪﺳـﺔ ﺍﻟﻘﻮﻱ ﺍﳌﻴﻜﺎﻧﻴﻜﻴﺔ ‪ -‬ﺟﺎﻣﻌﺔ ﺍﳌﻨﺼﻮﺭﺓ‬ ‫ﲢﺖ ﺇﺷـﺮﺍﻑ‬

‫ﺃ‪.‬ﺩ‪ /.‬ﳎﺪﻯ ﳏﻤﺪ ﺃﺑﻮ ﺭﻳﺎﻥ‬

‫ﺃ‪.‬ﺩ‪ /.‬ﻧﺒﻴﻞ ﺣﺴﻦ ﻣﺼﻄﻔﻲ‬

‫ﺃﺳﺘﺎﺫ ﻣﺘﻔﺮﻍ ﺑﻘﺴﻢ ﻫﻨﺪﺳﺔ ﺍﻟﻘﻮﻯ ﺍﳌﻴﻜﺎﻧﻴﻜﻴﺔ‬

‫ﺃﺳﺘﺎﺫ ﻭﺭﺋﻴﺲ ﻗﺴﻢ ﻫﻨﺪﺳﺔ ﺍﻟﻘﻮﻯ ﺍﳌﻴﻜﺎﻧﻴﻜﻴﺔ‬

‫ﻛﻠﻴﺔ ﺍﳍﻨﺪﺳﺔ – ﺟﺎﻣﻌﺔ ﺍﳌﻨﺼﻮﺭﺓ‬

‫ﻛﻠﻴﺔ ﺍﳍﻨﺪﺳﺔ – ﺟﺎﻣﻌﺔ ﺍﻟﺰﻗﺎﺯﻳﻖ‬

‫ﺃ‪.‬ﻡ‪.‬ﺩ‪ /.‬ﺑﲑﺝ ﺃﻭﻫﺎﻧﺲ ﺟﺒﻪ ﺟﻴﺎﻥ‬ ‫ﺃﺳﺘﺎﺫ ﻣﺴﺎﻋﺪ ﺑﻘﺴﻢ ﻫﻨﺪﺳﺔ ﺍﻟﻘﻮﻯ ﺍﳌﻴﻜﺎﻧﻴﻜﻴﺔ‬ ‫ﻛﻠﻴﺔ ﺍﳍﻨﺪﺳﺔ – ﺟﺎﻣﻌﺔ ﺍﳌﻨﺼﻮﺭﺓ‬ ‫‪٢٠٠٩‬‬