Twenty-Third Symposium on Naval Hydrodynamics

Twenty-Third Symposium on Naval Hydrodynamics

  • Publisher: National Academies Press
  • ISBN: 9780309254670
  • eISBN Pdf: 9780309593571
  • Place of publication:  United States
  • Year of digital publication: 2001
  • Month: December
  • Pages: 1025
  • DDC: 530
  • Language: English

"Vive la Revolution!" was the theme of the Twenty-Third Symposium on Naval Hydrodynamics held in Val de Reuil, France, from September 17-22, 2000 as more than 140 experts in ship design, construction, and operation came together to exchange naval research developments. The forum encouraged both formal and informal discussion of presented papers, and the occasion provides an opportunity for direct communication between international peers.

This book includes sixty-three papers presented at the symposium which was organized jointly by the Office of Naval Research, the National Research Council (Naval Studies Board), and the Bassin d'Essais des Carènes. This book includes the ten topical areas discussed at the symposium: wave-induced motions and loads, hydrodynamics in ship design, propulsor hydrodynamics and hydroacoustics, CFD validation, viscous ship hydrodynamics, cavitation and bubbly flow, wave hydrodynamics, wake dynamics, shallow water hydrodynamics, and fluid dynamics in the naval context.

  • Twenty-Third Symposium on NAVAL HYDRODYNAMICS
  • Copyright
  • Contents
  • FOREWORD
  • OPENING REMARKS—RADM JAY M.COHEN, USN CHIEF OF NAVAL RESEARCH
  • Contents
  • LIST OF ATTENDEES
    • AUSTRALIA
    • AUSTRIA
    • BELGIUM
    • CANADA
    • CHINA
    • DENMARK
    • FINLAND
    • FRANCE
    • GERMANY
    • GREECE
    • INDIA
    • ITALY
    • JAPAN
    • KOREA
    • THE NETHERLANDS
    • NORWAY
    • POLAND
    • PORTUGAL
    • RUSSIA
    • SINGAPORE
    • SPAIN
    • SWEDEN
    • TURKEY
    • UNITED KINGDOM
    • UNITED STATES
  • Modern Seakeeping Computations for Ships
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 BACKGROUND
      • Historical Approaches to Seakeeping
      • Taxonomy of Seakeeping Computations
    • 3 CONTEMPORARY CALCULATION METHODS
      • Unsteady Viscous Flow
      • Potential Flow Formulation
      • Motions by Fully Nonlinear Potential Flow
      • Derived Quantities
        • Green Water on Deck
        • Structural Loads
        • Added Resistance in Waves
    • 4 MAJOR RESEARCH ISSUES
      • Efficient Numerical Methods
      • Large Amplitude Motions and Capsizing
        • Capsizing
        • Roll and Cross-Flow Drag
        • Nonlinear Dynamics and Bifurcation
        • State-of-the-Art in Large Amplitude Motion Predictions
      • Horizontal Plane Motions
        • Rudder Roll Stabilization
        • Broaching
      • Finite Depth Problems in the Littorals
      • Verification and Validation
    • 5 CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • APPENDIX—CORRECTION TO THE ADDED RESISTANCE OF LIN AND REED (1976)
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Forces, Moment and Wave Pattern for Naval Combatant in Regular Head Waves
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. TEST DESIGN
    • 3. MEASUREMENT SYSTEMS
      • 3.1. Forces and moment
      • 3.2. Farfield free surface elevations
      • 3.3. Nearfield free surface elevations
    • 4. UNCERTAINTY ASSESSMENT
      • 4.1. Raw time histories
      • 4.2. FS harmonics
      • 4.3. FS-reconstructed time histories
    • 5. RESULTS AND DISCUSSIONS
      • 5.1. Incident wave
      • 5.2. Median test case for forces and moment
      • 5.3. Linear response for forces and moment
      • 5.4. Non-linear response for forces and moment
      • 5.5. Free surface elevations
    • 6. SUMMARY AND CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • New Green-Function Method to Predict Wave-Induced Ship Motions and Loads
    • ABSTRACT
    • INTRODUCTION
    • BOUNDARY VALUE PROBLEM AND NEW GREEN FUNCTION
    • BI-QUADRATIC PATCH METHOD
    • NUMERICAL RESULTS
      • Hulme's hemisphere
      • Farell's ellipsoid
      • Wu's sphere
      • Journee's Wigley-hull
    • DISCUSSIONS AND CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
  • Validation of Time-Domain Prediction of Motion, Sea Load, and Hull Pressure of a Frigate in Regular Waves
    • ABSTRACT
    • INTRODUCTION
    • THEORETICAL ANALYSIS
      • Equations of Ship Motion
      • External Forces
      • Pressure
      • Sea Loads
    • VALIDATION
      • Model Test
      • Motions, Sea Loads and Pressures
    • CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • ADDITIONAL REFERENCES
  • Ship motions and loads in large waves
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 EXPERIMENT
    • 3 RESULTS OF THE EXPERIMENT
      • 3.1 Motion
      • 3.2 Pressure
      • 3.3 Bending moment
    • 4 COMPUTATION
      • 4.1 Basic idea of the computation
      • 4.2 Computation procedure
      • 4.3 Grid system
      • 4.4 Results and their comparison with experiments
        • 4.4.1 Frequency response characteristics
        • 4.4.2 Nonlinearities with respect to waveheight
        • 4.4.3 Time histories
    • 5 CONCLUSIONS
    • ACKNOWLEDGEMENT:
    • REFERENCES
  • Prediction of Vertical-Plane Wave Loading and Ship Responses in High Seas
    • ABSTRACT
    • INTRODUCTION
    • THE TIME-DOMAIN STRIP THEORY
    • MODELING OF GREEN WATER LOADS
    • THE S175 CONTAINER SHIP IN HEAD WAVES
    • SHIP RESPONSES IN ALL HEADINGS
      • A Panamax Container Ship
      • A Very Large Crude Carrier
    • CONCLUSION
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • REFERENCES:
  • Basic Studies of Water on Deck
    • ABSTRACT
    • INTRODUCTION
    • ASSUMPTIONS AND MODELING
    • PHYSICAL INVESTIGATIONS
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Second Order Waves Generated by Ship Motions
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. MEASUREMENT OF THE SECOND ORDER WAVES
    • 3. ASYMPTOTIC CHARACTERISTCS
    • 4. COMPARISON WITH THEORETICAL PREDICTION
    • 5. CONCLUDING REMARKS
    • ACKNOWLEDGEMENT
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Prediction of Nonlinear Motions of High-speed Vessels in Oblique Waves
    • ABSTRACT
    • INTRODUCTION
    • THEORETICAL FORMULATION
      • Coordinate System
      • Incident Waves
      • Transformation of Hydrodynamic Coefficient Matrix
      • Sectional Force Components
      • Equations of Motions
    • NUMERICAL ALGORITHM
      • Sectional Hydrodynamic Coefficients
      • Time Integration
      • Viscous Roll Damping
      • Artificial Spring
    • COMPARISON OF PREDICTION AND EXPERIMENTAL RESULTS
    • CONCLUSION
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Optimizing Turbulence Generation for Controlling Pressure Recovery in Submarine Launchways
    • ABSTRACT
    • INTRODUCTION
    • GOVERNING EQUATIONS AND SOLUTION METHOD
    • DYNAMIC SUBGRID SCALE MODEL
    • MODEL COEFFICIENT
    • RESULTS AND DISCUSSION
    • CONCLUSIVE REMARKS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • DISCUSSION
    • AUTHOR'S REPLY
  • HULL DESIGN by CAD/CFD SIMULATION
    • ABSTRACT
    • INTRODUCTION
    • HULL DESIGN SYSTEM
      • Design system for America's Cup 2000
      • Advanced design system with wave statistics
    • CFD SIMULATION FOR STEADY PERFORMANCE
      • Performance prediction simulation
      • Accuracy problem
    • CFD SIMULATION FOR MOTION PERFORMANCE IN WAVES
      • Grid system for motion in waves
        • Density function method for free-surface motion
      • Motion simulation method
      • Two-degrees of freedom motion
      • Three-degrees of freedom motion
    • CASE OF HIGH-SPEED FERRY DESIGN TUMMAC-IV METHOD FOR FAST SHIP
      • Optimization of principal particular
      • Hull-form optimization
    • CONCLUSION
    • REFERENCES
  • Steady-State Hydrodynamics of High-Speed Vessels with a Transom Stern
    • ABSTRACT
    • INTRODUCTION
      • Literature Review
      • Current Work
    • THEORY
      • Definition of the Problem
      • Discretization of the Hull
      • Equations for the Potential
      • Potential-Flow Solution
      • Modeling of the Hollow
      • Forces and Moment on the Vessel
      • Equilibrium of the Vessel
      • Iteration of the Hollow
      • Simplistic Resistance
    • LEGO SHIP MODEL SERIES
    • RESULTS
      • Numerical Convergence Tests
      • Resistance Components
      • Comparison with Experiments
    • CONCLUDING REMARKS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Practical CFD Applications to Design of a Wave Cancellation Multihull Ship
    • ABSTRACT
    • INTRODUCTION
    • HAVELOCK AND FOURIER-KOCHIN REPRESENTATION OF WAVE DRAG
    • APPLICATION TO MULTIHULL SHIPS
    • FOURIER-KOCHIN REPRESENTATION OF NEAR-FIELD STEADY SHIP WAVES
    • SLENDER-SHIP APPROXIMATION
    • FOUR METHODS OF ANALYSIS
    • RESULTS OF ANALYSIS
    • CONCLUSION
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • APPENDIX
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Simulation of Ship Maneuvers Using Recursive Neural Networks
    • ABSTRACT
    • INTRODUCTION
    • DESCRIPTION OF DATA
    • RNN ARCHITECTURE
    • FORCE AND MOMENT INPUTS
    • TRAINING PROCEDURE
    • RESULTS
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • REFERENCE
  • Flow- and Wave-Field Optimization of Surface Combatants Using CFD-Based Optimization Methods
    • ABSTRACT
    • NOMENCLATURE
    • INTRODUCTION
    • COMPUTATIONAL METHOD
      • RANS Equation Solver
      • Computational Grids
      • Uncertainty Assessment
      • Nonlinear Optimization Problem
      • Nonlinear Programming Algorithm
      • Hull Form Modification Function
    • RESULTS
      • Model 5415—Stern Optimization
      • Model 5415—Sonar Dome Optimization
      • Model 5415—Bow Optimization
    • SUMMARY AND CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • MARINE PROPULSOR NOISE INVESTIGATIONS IN THE HYDROACOUSTIC WATER TUNNEL “G.T.H.”
    • 1. ABSTRACT
    • 2. INTRODUCTION
    • 3. PRINCIPAL CHARACTERISTICS OF THE G.T.H.
    • 4. MODEL EQUIPMENTS AND INSTRUMENTATION FOR THE G.T.H.
      • 4.1 Model equipment
      • 4.2 Instrumentation
    • 5. HYDROACOUSTIC PERFORMANCES OF THE GTH
      • 5.1 Kinetic performances of the flow:
      • 5.2 Deaeration and cavitation nuclei control performances:
      • 5.3 Hydroacoustic performances:
    • 6. SOME RESULTS OF HYDROACOUSTIC SURVEYS IN GTH
    • 7. CONCLUSIONS
    • REFERENCES
    • NOMENCLATURE
    • APPENDIX I: Tests & equipment in the GTH
    • APPENDIX II: BACKGROUND NOISE OF THE GTH
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Propulsor Design Using Clebsch Formulation
    • ABSTRACT
    • INTRODUCTION
    • THEORETICAL FORMULATION
    • ALGORITHM DESCRIPTIONS
    • DESIGN CONSIDERATIONS
    • DESIGN EXAMPLE 1—DUCTED POD PROPULSOR
    • DESIGN EXAMPLE 2—AXIAL PRESWIRL
    • DESIGN EXAMPLE 3—MIXED FLOW PRESWIRL
    • CONCLUSIONS
    • ACKNOWLEDGEMENT:
    • REFERENCES:
    • DISCUSSION
    • AUTHOR'S REPLY
  • Unsteady Flow Quantities on Two-Dimensional Foils: Experimental and Numerical Results
    • ABSTRACT
    • NOMENCLATURE
    • INTRODUCTION
    • NUMERICAL SIMULATION
    • EXPERIMENTAL SET-UP
    • COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS
      • Numerical flow description:
      • Numerical pressure signal comparison:
      • Numerical and experimental Spectrum
    • EFFECT OF REYNOLDS NUMBER ON CAVITATION
    • CONCLUSION
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Hydrofoil Turbulent Boundary Layer Separation at High Reynolds Numbers
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL SET-UP AND UNCERTAINTY
    • EXPERIMENTAL RESULTS
    • PRELIMINARY COMPARISONS TO CALCULATIONS
    • SUMMARY AND CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Pressure Fluctuation on Finite Flat Plate above Wing in Sinusoidal Gust
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. CALCULATION METHODS FOR PRESSURE FLUCTUATION
    • 3. RESULTS OF 2-D PROBLEM
      • 3.1 Pressure fluctuation due to wing in a gust
      • 3.2 Pressure fluctuation due to thickness-varying wing in a gust
    • 4. RESULTS OF 3-D PROBLEM
    • 5. CONCLUSION
    • ACKNOWLEDGEMENT
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Control of the Turbulent Wake of an Appended Streamlined Body
    • ABSTRACT
    • INTRODUCTION
    • BOUNDARY LAYER AND WAKE CONTROL
    • EXPERIMENTAL SET UP
    • BLOWING SYSTEM EVALUATION
      • Wake fraction
      • Wake fraction without tail planes
      • Wake fraction with tail planes
    • EFFECT OF TAIL PLANES ON THE WAKE DISTRIBUTION
    • EFFECT OF TAIL PLANES ON THE WAKE HARMONIC DISTRIBUTION
    • TURBULENCE
    • POWERING PERFORMANCE
    • CONCLUSIONS
    • REFERENCES
    • DISCUSSION
    • REFERENCES:
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Investigation of Global and Local Flow Details by a Fully Three-dimensional Seakeeping Method
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. THEORY
      • 2.1. Physical model
      • 2.2. Mathematical model
      • 2.3. Added resistance
    • 3. APPLICATIONS
      • 3.1. Local pressures
      • 3.2. Added resistance
    • ACKNOWLEDGMENT
    • REFERENCES
    • DISCUSSION
    • AUTHORS' REPLY
    • DISCUSSION
    • AUTHORS' REPLY
    • DISCUSSION
    • AUTHORS' REPLY
  • Prediction of Wave Pressure and Loads on Actual Ships by the Enhanced Unifed Theory
    • ABSTRACT
    • INTRODUCTION
    • ENHANCED UNIFIED THEORY
      • Mathematical formulation
      • Radiation problem
      • Diffraction problem
      • Hydrodynamic and hydrostatic pressure
      • Hydrodynamic forces
      • Ship motions
      • Wave loads
    • RESULTS AND DISCUSSION
      • Outline of the strip method
      • Wave-induced ship motions
      • Pressure distribution
      • Wave loads
    • CONCLUDING REMARKS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Frequency Domain Numerical and Experimental Investigation of Forward Speed Radiation by Ships
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL STUDY
      • Experimental set-up
      • Models and test conditions
      • Measurements of forces and moments
      • Wave patterns measurements
      • Acquisition system and signal analysis
    • TEST MEASUREMENTS
      • Added-mass and damping coefficients
      • Wave pattern measurements
      • Analysis of flow
    • NUMERICAL STUDY
      • Mathematical model
      • Numerical method
    • NUMERICAL RESULTS AND COMPARISON WITH TEST MEASUREMENTS
      • Series 60 hulls
      • Flat plate in forced sway motion
    • CONCLUSION
    • AKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • REFERENCE:
  • INTERNATIONAL COLLABORATION ON BENCHMARK CFD VALIDATION DATA FOR SURFACE COMBATANT DTMB MODEL 5415
    • ABSTRACT
    • INTRODUCTION
    • OVERLAPPING TEST DESIGN, COMPARISON VARIABLES, AND CONDITIONS
    • FACILITIES, MEASUREMENT SYSTEMS, AND PROCEDURES
    • UNCERTAINTY ASSESSMENT
    • CFD VALIDATION/COMPLEMENTARY CFD
    • COMPARISON OF RESULTS
    • HIGHLIGHTS OF OVERALL TEST PROGRAM
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications
    • ABSTRACT
    • INTRODUCTION
      • Nomenclature
      • Physical Model
    • NUMERICAL METHOD
    • RESULTS
    • CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY:
    • DISCUSSION
    • AUTHOR'S REPLY:
    • DISCUSSION
    • AUTHOR'S REPLY:
    • DISCUSSION
    • AUTHOR'S REPLY:
    • DISCUSSION
    • AUTHOR'S REPLY:
    • DISCUSSION
    • AUTHOR'S REPLY:
    • REFERENCES FOR DISCUSSION:
  • Free Surface Viscous Flow Computation Around A Transom Stern Ship By Chimera Overlapping Scheme
    • ABSTRACT
    • INTRODUCTION
    • NUMERICAL SCHEME
    • SINKAGE AND TRIM COMPUTATION
    • HULLFORM GEOMETRY
    • CHIMERA OVERLAPPING GRID
      • Grid Topology
      • Extent of Domain and Grid Distribution, Size
    • EXPERIMENTAL DATA
    • RESULTS AND ANALYSIS
      • Model 5365
      • Model 5415
    • CONCLUSION
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION:
    • REFERENCES:
    • AUTHOR'S REPLY
  • ANTI-ROLL TANK SIMULATIONS WITH A VOLUME OF FLUID (VOF) BASED NAVIER-STOKES SOLVER
    • ABSTRACT
    • INTRODUCTION
    • COMFLO—MATHEMATICAL MODEL AND NUMERICAL METHOD
    • FREE-SURFACE TANKS IN REGULAR MOTION
    • U-TUBE TANKS IN REGULAR MOTION
    • U-TUBE TANKS IN IRREGULAR MOTION
    • U-TANKS WITH ACTIVE CONTROL
    • COUPLED SHIP AND TANK FLUID MOTION
    • COUPLED SHIP AND TANK FLUID MOTION WITH ACTIVE CONTROL
    • CONCLUSIONS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Validation of Tab Assisted Control Surface Computation
    • ABSTRACT
    • INTRODUCTION
    • GOVERNING EQUATIONS
    • NUMERICAL METHOD
      • Preconditioned Method
      • Multigrid Method
      • Boundary Conditions
      • Two-Equation Turbulence Models
    • DESCRIPTION OF EXPERIMENT
    • DISCUSSION OF RESULTS
      • Convergence and Grid-Independent Solution
      • Forces and Moments
    • CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • REFERENCES:
    • AUTHOR'S REPLY
  • Experimental And Numerical Investigation Of The Flow Around The Appendices Of A Whitbread 60 Sailing Yacht
    • ABSTRACT
    • INTRODUCTION
    • WIND TUNNEL TESTS
      • Description of the facility
      • Description of the two geomletries
      • Flow visualization
      • P.I.V. measurements
      • Drag and lift measurements
    • NUMERICAL SIMULATION
      • Description of the simulations
      • Results—Angle of attack=0°
      • Results—Angle of attack=6°
      • Validation
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
  • Propeller Wake Analysis by Means of PIV
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL SET-UP
    • IMAGE ANALYSIS
    • MEASUREMENT UNCERTAINTY
    • PROPELLER WAKE ANALYSIS
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS.
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • REFERENCES
  • Experimental and Numerical Investigation of the Unsteady Flow around a Propeller
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL ANALYSIS
      • Experimental setup
      • Phase sampling technique
    • THEORETICAL ANALYSIS
      • Governing equations
      • Boundary integral formulation
      • Wake analysis
      • Discretization
      • Solution procedure
    • FLOW FIELD INVESTIGATIONS
      • Hull nominal wake
      • Flow field around the propeller
    • CONCLUDING REMARKS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • 1.1. REFERENCES
  • Simulation of Incompressible Viscous Flow Around a Ducted Propeller Using a RANS Equation Solver
    • ABSTRACT
    • INTRODUCTION
    • NUMERICAL METHOD
      • Governing Equations and Turbulence Closure
      • Pseudo-compressibility
      • Discretization
      • Solution Algorithm
    • NUMERICAL RESULTS
      • Geometry, Mesh and Boundary Conditions
      • Convergence
      • Forces and Pressures
      • Velocities and Hydrodynamic Pitch
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHORS' REPLY
    • DISCUSSION
    • AUTHORS' REPLY
  • On Submerged Stagnation Points and Bow Vortices Generation
    • ABSTRACT
    • INTRODUCTION
    • SSP THEORY AND BOW VORTICES GENERATION
    • RESULTS AND DISCUSSION
      • SSP-Existence and Location
      • Pressure Profile and Flow with SSP
      • Calculation of FSSP
      • Comparison with Experimental Results
      • Inertial-Gravitational Effect
      • Bow Drag
      • Bow Contour Design-two dimensional case
      • SSP for Vertical Step and Bulbons bow
      • Bow Vortices—three dimensional case
    • CONCLUSIONS
    • SCOPE OF THE FUTURE WORK
    • REFERENCES
  • Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 TURBULENCE MODELS
      • 2.1 Algebraic Models
      • 2.2 One-Equation models
        • 2.2.1 Spalart & Allmaras
        • 2.2.2 Menter
        • 2.2.3 Boundary Conditions
      • 2.3 Two-equation Models
        • 2.3.1 Two-layer k-E
        • 2.3.2 Chien's k-E model
        • 2.3.3 Standard K-w model
        • 2.3.4 Menter's K-w model
        • 2.3.5 q-ζ model
        • 2.3.6 Boundary Conditions
    • 3 RESULTS AND DISCUSSION
      • 3.1 General
      • 3.2 Wall Boundary Condition for ω
      • 3.3 Scaling Effects
    • 4 CONCLUSIONS
    • REFERENCES
  • The Experimental and Numerical Study of Flow Structure and Water Noise Caused by Roughness of the Body
    • ABSTRACT
    • INTRODUCTION
    • NUMERICAL METHOD
      • Fast vortex method
      • Mathematical roughness
      • Hydroacoustic theory
    • EXPERIMENT
    • RESULTS
    • CONCLUSION
    • REFERENCE
    • DISCUSSION
    • REFERENCES.
    • AUTHOR'S REPLY
  • Large-Eddy Simulations of Turbulent Wake Flows
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 MATHEMATICAL/NUMERICAL MODELS
      • 2.1 Navier-Stokes Solver
      • 2.2 Random Flow Generation (RFG) methodology
    • 3 APPLICATIONS
    • 4 RESULTS
      • 4.1 Flat Plate Wake
    • 4.2 SHIP WAKE
    • 5 CONCLUSIONS AND FUTURE WORK
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • QUESTIONS
    • DISCUSSION
    • REFERENCES
    • AUTHOR'S REPLY
    • REFERENCE:
  • Instability of Partial Cavitation: A Numerical/Experimental Approach
    • ABSTRACT
    • INTRODUCTION
    • THEORETICAL CONSIDERATIONS
      • Compressibility boundary layer theory
      • A single-phase flow approach for cavitating flows
      • Numerical Approach
    • EXPERIMENTAL METHOD
    • LIFT OSCILLATIONS
    • COMPARISONS WITH SIMULATIONS
      • Bubble/Patch Cavitation
      • Sheet-Cloud Cavitation (partial cavitation)
      • Sheet-Cloud Cavitation (full cavitation)
      • Wake Characteristics
    • NOISE AND SURFACE PRESSURE CHARACTERISTICS
    • COMPARISON WITH NON-CAVITATING SEPARATED FLOWS
    • SUMMARY AND CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • An Unsteady Three-Dimensional Euler Solver Coupled with a Cavitating Propeller Analysis Method
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 PRESENT METHOD
    • 3 FORMULATION
      • 3.1 Steady Euler Solver
      • 3.2 Unsteady Euler Solver
        • 3.2.1 Finite Volume Discretization for Momentum Equations
        • 3.2.2 Pressure Correction Method for Continuity Equation
          • Sequence of Operations
      • 3.3 Boundary Conditions
    • 4 BODY FORCE
      • 4.1 Two-Dimensional Body Force
        • 4.1.1 Surface Distribution Model
        • 4.1.2 Camberline Pressure Model
      • 4.2 Body Force Representation of an Actuator Disk
      • 4.3 Steady Body Force Model
      • 4.4 Unsteady Body Force Model
    • 5 NUMERICAL RESULTS
      • 5.1 Actuator Disk
      • 5.2 Uniform Inflow
      • 5.3 Propeller DTMB 4842 in Uniform Inflow
      • 5.4 Nominal Wake with Only the Third Harmonic
      • 5.5 Cavitation on Propeller DTMB 4148
      • 5.6 Total Velocity in front of Propeller DTMB 4148
    • 6 CONCLUSIONS
    • ACKNOWLEDGMENT
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL SETUP AND PROCEDURE
    • CAVITATION INCEPTION INDICES AND BUBBLE DYNAMICS
    • PIV RESULTS
    • 4. EFFECT OF GAP SIZE ON THE STRENGTH DISTRIBUTION OF TIP VORTICES AND PRESSURE MINIMA
    • SUMMARY
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • REFERENCES
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • An Experimental Investigation of Cavitation Inception and Development of Partial Sheet Cavities on Two—Dimensional…
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL APPARATUS
      • Test facility
    • RESULTS
      • Cavitation inception
        • NACA 66 12% 100 mm
        • NACA66–6% 100 mm and 150 mm.
        • Eppler E817–100 mm.
      • Cavitation development
        • Cavitation types
        • Partial sheet cavity characteristics
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • NOMENCLATURE
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Modeling 3D unsteady sheet cavities using a coupled UnRANS-BEM code
    • ABSTRACT
    • INTRODUCTION
      • Overview of the method
    • MATHEMATICAL FORMULATION
      • Cavity Model
      • Mixed Approach
      • UnRANS Approach for the Liquid Behavior
      • Reynolds Averaging
      • UnRANS Free Surface Conditions
      • Grid Generation
      • Treatment of the Cavity
      • Dual Reciprocity BEM
      • Dual BEM Approach
      • Discretization
      • Time Integration
      • Wake and Tip Vortex Modeling in the BEM
    • RESULTS AND DISCUSSIONS
      • Complex 3D simulations
    • CONCLUSIONS
    • ACKNOWLEDGMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Ship Wake Detectability in the Ocean Turbulent Environment
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. THEORETICAL MODEL
    • 3. NUMERICAL MODEL
    • 4. EXPERIMENTAL STUDY
    • 5. CONCLUSION
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • An Experimental and Computational Study of the Effects of Propulsion on the Free-Surface Flow Astern of Model 5415
    • ABSTRACT
    • INTRODUCTION
    • MODEL DESCRIPTION
    • EXPERIMENTAL PROCEDURES
    • LONGITUDINAL WAVE CUTS
      • Theory of Operation
      • Experimental Setup
      • Calibration
      • Operating Procedures
    • STERN TOPOGRAPHY
      • Theory of operation
      • Experimental Setup
      • Calibration
      • Operating Procedures
    • PRESENTATION OF RESULTS
      • Longitudinal Wave Cut Measurements
      • Stern Topography
    • MEASUREMENT UNCERTAINTY
    • COMPUTATIONS
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Breaking Waves in the Ocean and around Ships
    • ABSTRACT
    • 1 FORWARD
    • 2 OVERVIEW
      • 2.1 1985
      • 2.2 Wave Modulation
      • 2.3 Wave Deformation
      • 2.4 Inception and Mechanism of Breaking
      • 2.5 The Effects of Breaking on Wave Evolution
      • 2.6 On the Modeling of Splashing, and its Consequences
      • 2.7 Surface Tension Effects; Microbreakers
    • 3 WHEN AND WHY DO MODULATED GRAVITY WAVES BREAK?
      • 3.1 The breaking process, and criterion
      • 3.2 Theory of the breaking criterion.
      • 3.3 Experimental verification
      • 3.4 The limits of breaking
    • 4 THE SIMULATION OF SPLASHING
      • 4.1 Basic SPH Equations
      • 4.2 The Breaking Bore
      • 4.3 Breaking Bow Waves
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Numerical and Experimental Study of the Wave Breaking Generated by a Submerged Hydrofoil
    • ABSTRACT
    • INTRODUCTION
    • EXPERIMENTAL INVESTIGATION
      • Experimental system and techniques
      • Wave-pattern and vortex-shedding visualization
    • NUMERICAL MODELING
      • Navier-Stokes solver for the two-phase flow
      • Free surface motion via the Level-Set technique
      • Solid boundaries modeled via body forces
      • Domain decomposition
    • VALIDATION
      • Case study: wavy flow induced by a moving bottom topography
      • Submerged hydrofoil: non breaking regime
        • FNS results
        • DD results
      • Submerged hydrofoil: breaking regime
    • CONCLUDING REMARKS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • The Numerical Simulation of Ship Waves Using Cartesian Grid Methods
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 FIELD EQUATIONS
    • 3 ENFORCEMENT OF BODY BOUNDARY CONDITIONS
      • 3.1 Free-slip conditions
      • 3.2 No-slip conditions
    • 4 INTERFACE TRACKING
      • 4.1 CLS method
        • 4.1.1 CLS Contact angle boundary conditions in general geometries
      • 4.2 Level-set method
    • 5 FLUX INTEGRAL METHODS
    • 6 PRELIMINARY RESULTS
      • 6.1 Ship Wave Results
      • 6.2 Spray Sheet Results
    • 7 CONCLUSION
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • QUESTIONS
    • REFERENCES
    • AUTHOR'S REPLY
  • Radiation Loads on a Cylinder Oscillating in Pycnocline
    • ABSTRACT
    • INTRODUCTION
    • IMPULSE RESPONSE TECHNIQUE
    • EXPERIMENTAL ARRANGEMENT
    • THEORETICAL ANALYSIS
    • NUMERICAL RESULTS
      • Comparison with experimental results
    • CONCLUSION
    • ACKNOWLEDGMENTS:
    • REFERENCES
  • Wave Resistance Computations—A Comparison of Different Approaches
    • ABSTRACT
    • 1. INTRODUCTION
    • 2. POTENTIAL FLOW COMPUTATIONS
      • Description of Method
      • EXAMPLE COMPUTATIONS
    • 3. VOLUME OF FLUID EULER METHOD
      • DESCRIPTION OF METHOD
      • EXAMPLE COMPUTATIONS
    • 4. RANS-SOLVER COMET
      • DESCRIPTION OF METHOD
      • EXAMPLE COMPUTATION
    • 5. COMPARISON OF RESULTS
      • WAVE PATTERNS
      • PRESSURE DISTRIBUTION AND RESISTANCE
    • 6. PRACTICAL CONSIDERATIONS
    • 7. CONCLUSIONS
    • 8. REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Computation of Nonlinear Turbulent Free Surface Flows Using the Parallel Uncle Code
    • ABSTRACT
    • INTRODUCTION
    • GOVERNING EQUATIONS
    • NUMERICAL PROCEDURE
    • PARALLEL ITERATION HIERARCHY
    • MULTIGRID STRATEGY
    • GRID GENERATION
    • RESULTS
    • CONCLUSION
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Submarine manoeuvrability assessment using Computational Fluid Dynamic tools
    • ABSTRACT
    • INTRODUCTION
    • MESH GENERATION
      • Body and deck
      • Appendages
      • Combination
      • Maneuvering
    • NUMERICAL TESTS
      • Different mesh
      • Pure incidence/pure drift
      • Rotation (vertical and horizontal planes)
      • Rudder effectiveness
    • CONVERGENCE
    • EFFECT OF MESH SIZE
      • Horizontal plane
      • Vertical plane
        • Pure incidence:
        • Low rotation rates:
        • Higher rotation rates:
    • COMPARISON WITH MODEL TESTS DATA
      • Pure incidence (without rotation)
      • manoeuvring (vertical plane)
        • Pure drift (without rotation)
      • Rotation/incidence (vertical plane)
      • Rotation/drift (horizontal plane)
      • Rudder effectiveness
    • PREDICTION OF SUBMARINE BEHAVIOUR
    • CONCLUSIONS
    • FUTURE DEVELOPMENT
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Simulation of UUV Recovery Hydrodynamics
    • ABSTRACT:
    • INTRODUCTION:
    • METHODOLOGY:
      • Surface Definition:
      • Satisfaction of the Surface Boundary Condition:
        • Initial Volume Vorticity Distribution:
        • Linearized Tetrahedral Vorticity Elements:
        • Computation of Derivatives:
        • Evolution of the Vorticity Field:
        • Baldwin-Lomax Turbulence Model:
        • Vorticity Boundary Condition:
        • Initial Volume Vorticity Distribution, Euler Layer and Point Creation:
        • Numerical Accuracy:
      • UUV and Cone Definition:
    • RESULTS:
      • UUV Docking Simulation:
    • DISCUSSION:
      • Unsteady Wake Development:
        • Full-scale UUV Recovery:
    • CONCLUSIONS:
    • ACKNOWLEDGMENTS:
    • REFERENCES:
    • DISCUSSION
    • AUTHOR'S REPLY
  • Reynolds-Averaged Modeling of High-Froude-Number Free-Surface Jets
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 GOVERNING EQUATIONS
      • 2.1 Reynolds-Averaged Navier-Stokes Equations
        • The Exact Averaged Equations
        • Approximate Equations for Small Surface Fluctuations
        • Near-Surface Reynolds Stress Modeling
      • 2.2 Surface Fluctuation Model
        • Wave-Action Spectrum Model
        • Turbulent Source Term
    • 3 NUMERICAL IMPLEMENTATION
    • 4 RESULTS
      • 4.1 Modeling of a Low-Froude-Number Jet
      • 4.2 One-Way-Coupled Modeling for Unsteady Waves
      • 4.3 Two-Way-Coupled Modeling of a High-Froude-Number Jet
    • 5 SUMMARY AND CONCLUSIONS
    • ACKNOWLEDGMENT
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • On Roll Hydrodynamics of Cylinders Fitted with Bilge Keels
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 THEORETICAL MODELS
      • 2.1 The Free Surface Random Vortex Method
      • 2.2 The Boundary-Fitted Finite-Difference Method
      • 2.3 Forces and moment
    • 3 EXPERIMENTAL SETUP AND HYDRODYNAMIC COEFFICIENTS
      • 3.1 Measured forces and moment
      • 3.2 Hydrodynamic coefficients
    • 4 RESULTS AND DISCUSSIONS
      • 4.1 Inviscid-fluid results
      • 4.2 Theory versus experiments—a validation
      • 4.3 Vorticity and flow patterns
    • 5 CONCLUSIONS
    • ACKNOWLEDGEMENT
    • REFERENCES
    • DISCUSSION
    • AUTHORS' REPLY
    • DISCUSSION
    • AUTHORS' REPLY
  • Combining Accuracy and Effciency with Robustness in Ship Stern Flow Computation
    • ABSTRACT
    • 1 INTRODUCTION
    • 2 ORIGINAL SOLUTION STRATEGY
    • 3 NEW ELEMENTS IN SOLUTION PROCEDURE
      • 3.1 Larger sub-domains
      • 3.2 The linear system solver
      • 3.3 Preconditioning
      • 3.4 Including ‘negligible' terms
    • 4 NUMERICAL RESULTS
    • 5 CURRENT DEVELOPMENTS: INCORPORATING THE FREE SURFACE
    • 6 CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • An Unstructured Multielement Solution Algorithm for Complex Geometry Hydrodynamic Simulations
    • ABSTRACT
    • INTRODUCTION
    • GOVERNING EQUATIONS
    • NUMERICAL APPROACH
      • Reconstruction
      • Residual Evaluation
        • Spatial Residual
        • Temporal Residual
      • Time Evolution
      • Boundary Conditions
      • Turbulence Modeling
    • PARALLELIZATION
    • APPLICATIONS
      • Prolate Spheroid
      • NOAA FRV-40 Hull
      • SUBOFF Model
      • DTMB Model 5415 Hull
        • Nominal Wake Calculations
        • Fully Appended with Rotating Propulsors
    • CONCLUSIONS
    • ACKNOWLEDGEMENTS
    • REFERENCES
  • Ship Stern Flow Calculations on Overlapping Composite Grids
    • ABSTRACT
    • INTRODUCTION
    • GRID GENERATION
      • The present grid generator
    • SOLVER
      • Governing equations
      • Numerical method
    • VALIDATION
      • HSVA Tanker
      • Ryuko Maru
      • Modern container ship
    • FUTURE WORK
    • ACKNOWLEDGEMENTS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Study on the Prediction of Flow Characteristics Around a Ship Hull
    • ABSTRACT
    • INTRODUCTION
    • NUMERICAL METHODS
      • Viscous flow
      • Potential flow
    • SELECTION OF CODES
    • EXPERIMENTAL METHOD
      • Test description
      • Uncertainty analysis
    • SELECTION OF THE OBJECT SHIPS
    • RESULTS AND DISCUSSIONS
      • Resistance characteristics
      • Wave profile
      • Local resistance
      • Velocity distribution
      • Limiting streamline
      • Pressure distribution
      • Wake (at the propeller plane)
    • CONCLUSIONS
    • REFERENCES
  • Analysis of Turbulence Free-Surface Flow around Hulls in Shallow Water Channel by a Level-set Method
    • ABSTRACT
    • INTRODUCTION
    • MATHEMATICAL FORMULATION
      • Governing Equations
    • LEVEL-SET FORMULATION
      • Local Level-Set Method
      • Local Level-Set Method
    • LEVEL-SET FORM OF NAVIER-STOKES EQUATIONS
    • NUMERICAL PROCEDURE
    • NUMERICAL RESULTS
    • CONCLUSIONS
    • REFERENCE
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • A Design Tool for High Speed Ferries Washes
    • ABSTRACT
    • INTRODUCTION
    • SHIP WAVE FIELD AND AMPLITUDE SPECTRUM
      • Free wave amplitude spectrum
      • Principal of calculation of the free wave amplitude spectrum
      • Particular case of ultra high-speed ships
      • Numerical computation of the wave amplitude spectrum
      • Results
        • Longitudinal cuts
        • Wave amplitude spectrum
        • Conclusion
    • COASTAL WAVE PROPAGATION MODELLING
      • Coupling between amplitude spectrum and variance (or energy) spectrum
      • Highlighting simulations
      • Simulations parameters
      • Simulations results
    • FUTURE WORK AND CONCLUSIONS
    • ACKNOWLEDGMENTS:
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
    • DISCUSSION
    • AUTHOR'S REPLY
  • Flow Around Ships Sailing in Shallow Water—Experimental and Numerical Results
    • ABSTRACT
    • INTRODUCTION
    • MODELS INVESTIGATED
      • Shallow river vessel
      • Inland waterway ship
    • EXPERIMENTS
      • Ship A
      • Ship B
    • CFD-METHODS APPLIED
      • RANSE-Solver CFX-5 from AEA Technology (Methods 1 & 2)
      • RANSE-Solver based on the principle of artificial compressibility (Method 3)
      • Shallow water potential-theoretic transcritical treatment (Method 4)
    • EFD RESULTS
      • Ship A
      • Ship B
    • COMPARISON OF RESULTS
    • CONCLUSION
    • ACKNOWLEDGEMENTS
    • REFERENCES
  • Ship Stability Study in the Coastal Region: New Coastal Wave Model Coupled with a Dynamic Stability Model
    • ABSTRACT
    • I. INTRODUCTION
    • II. A COUPLED SURFACE WAVE MODEL AND SHIP DYNAMIC STABILITY MODEL
      • A. Surface Wave Model:
      • B. FREDYN Ship Dynamic Stability Model
    • III. CAPSIZE PREDICTIONS:
      • A. Case I: Saint Lawrence Gulf
      • B. Case II: Pacific Ocean Seamount
    • IV. CONCLUSIONS:
    • V. ACKNOWLEDGEMENT
    • VI. REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY
  • Waves and Forces Caused by Oscillation of a Floating Body Determined through a Unified Nonlinear Shallow-Water Theory
    • ABSTRACT
    • INTRODUCTION
    • THEORETICAL FORMULATION
      • General Description
      • Approximation of the Outer-Field Flow
      • Approximation of the Inner-Field Flow
      • Unified Shallow-Water Theory
      • Coupling Conditions
    • NUMERICAL IMPLEMENTATION
      • Crank-Nicolson Scheme
      • Staggered Grid
      • Scheme for Coupling Conditions
      • Solution Method
    • RESULTS AND DISCUSSION
    • CONCLUSIONS
    • REFERENCES
    • DISCUSSION
    • AUTHOR'S REPLY

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