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## Twenty-First Symposium on Naval Hydrodynamics (1997) Commission on Physical Sciences, Mathematics, and Applications (CPSMA)

### Citation Manager

. "Self-Propelled Maneuvering Underwater Vehicles." Twenty-First Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press, 1997.

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 Page 481

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Twenty-First Symposium on NAVAL HYDRODYNAMICS

In this expression, is the Jacobian of the numerical flux vector, with the first subscript representing the position of the cell face of the numerical flux vector and the second subscript representing the position of the dependent variable vector that the numerical flux vector is differentiated with respect to. Ia is an identity matrix, except the first diagonal element is zero in order to satisfy the true incompressible continuity equation.

A linear system of equations must be solved at each iteration of Newton's method. For three-dimensional problems a direct solution seems to be impractical [17] and in this work symmetric Gauss-Seidel relaxation is used. Because the flux Jacobian of the flux vector based on Roe's formulation is difficult to obtain analytically in three dimensions, and also in the interest of simplicity, the flux Jacobian is obtained numerically [18]. The solution scheme is referred to as discretized Newton-relaxation [16], or the DNR scheme [17]. Multigrid is used to accelerate the numerical solutions [19]. This multigrid scheme has been extended to multiblock [20] and unsteady flow [21]. The solution process is, therefore, a multigrid scheme for three-dimensional unsteady viscous flow on dynamic relative motion multiblock grids.

##### PROPULSOR TREATMENTS

From the beginning of the program it was determined that there needed to be two methods of handling the propulsor. One was to be a simulation of the propulsor using a relatively simple model that was computationally efficient, and the other was to incorporate the capability of handling the actual rotating propulsor. Both methods were developed, have been included in the code, and are discussed below.

###### Body Force Propulsor

The model selected that satisfies the conditions of simplicity and efficiency was one that has been used for a number of years for similar simulations such as open propellers [22] and ducted fans [23]. The basic approach is explained in [22] and consists of including body forces in Eq. (1) which operate on the fluid in a manner similar to the way an actual propulsor operates on the fluid. All three components of the body force vector were taken into account and consequently thrust, swirl, and their radial distributions can be included in the computations. In any given situation this force data may be obtained from conventional propulsor design tools or data bases. Here this information is based on thrust and torque coefficient data which were obtained from an actual marine propulsor. A description of the experiment and the measured results are reported in [24]. This coefficient information was used to determine the components of the body force vector and then distribute these components to the center of the cells in this cell-centered finite volume scheme in the region where the propulsor was located.

###### Actual Rotating Propulsor

The method used to include an actual rotating propulsor is one that has been continually developed and also used for a number of years, primarily for compressible flows [2531]. The approach is to use relative motion blocks with structured grids, whereby the blocks that include the rotating blades move relative to adjacent blocks with a region of the blocks near the relative motion interface being treated using the localized grid distortion technique introduced by Janus [31]. This method of handling relative motion blocks insures the continuity of grid lines (although they do change partners periodically, or “click”) and restricts the maximum distortion of the grid to be of the order of the distance between grid points which for viscous grids is, of course, small. This approach eliminates the need to interpolate the solution vector from one grid to another. The cell volumes do change in time, however, and the geometric conservation law must be satisfied [31].

##### DYNAMIC MOTION OF THE VEHICLE

The trajectory of the body at any instant of time is described by its linear velocities u, v, w, and by its angular velocities p, q, r, in the body fixed frame of reference and its position and orientation in an inertial frame of reference. The governing six-degree-of-freedom (6-DOF) Equations Of Motion (EOM) can be written as:

Axial Force

Lateral Force

Normal Force

 Page 481
 Front Matter (R1-R16) Opening Remarks (1-4) Progress Toward Understanding How Waves Break (5-28) Radiation and Diffraction Waves of a Ship at Forward Speed (29-44) Nonlinear Ship Motions and Wave-Induced Loads by a Rankine Method (45-63) Nonlinear Water Wave Computations Using a Multipole Accelerated, Desingularized Method (64-74) Computations of Wave Loads Using a B-Spline Panel Method (75-92) Simulation of Strongly Nonlinear Wave Generation and Wave-Body Interactions Using a 3-D Model (93-109) Analysis of Interactions Between Nonlinear Waves and Bodies by Domain Decomposition (110-119) Fourier-Kochin Theory of Free-Surface Flows (120-135) 24-inch Water Tunnel Flow Field Measurements During Propeller Crashback (136-146) Accuracy of Wave Pattern Analysis Methods in Towing Tanks (147-160) Unsteady Three-Dimensional Cross-Flow Separation Measurements on a Prolate Spheroid Undergoing Time-Dependent Maneuvers (161-176) Time-Domain Calculations of First-and Second-Order Forces on a Vessel Sailing in Waves (177-188) Third-Order Volterra Modeling Ship Responses Based on Regular Wave Results (189-204) Nonlinearly Interacting Responses of the Two Rotational Modes of Motion-Roll and Pitch Motions (205-219) Nonlinear Shallow-Water Flow on Deck Coupled with Ship Motion (220-234) Radar Backscatter of a V-like Ship Wake from a Sea Surface Covered by Surfactants (235-248) Turbulent Free-Surface Flows: A Comparison Between Numerical Simulations and Experimental Measurements (249-265) Conductivity Measurements in the Wake of Submerged Bodies in Density-Stratified Media (266-277) Macro Wake Measurements for a Range of Ships (278-290) Time-Marching CFD Simulation for Moving Boundary Problems (291-311) Yaw Effects on Model-Scale Ship Flows (312-327) A Multigrid Velocity-Pressure-Free Surface Elevation Fully Coupled Solver for Calculation of Turbulent Incompressible Flow around a Hull (328-345) The Shoulder Wave and Separation Generated by a Surface-Piercing Strut (346-358) Vorticity Fields due to Rolling Bodies in a Free Surface-Experiment and Theory (359-376) Numerical Calculations of Ship Stern Flows at Full-Scale Reynolds Numbers (377-391) Near-and Far-Field CFD for a Naval Combatant Including Thermal-Stratification and Two-Fluid Modeling (392-407) Water Entry of Arbitrary Two-Dimensional Sections with and Without Flow Separation (408-423) Coupled Hydrodynamic Impact and Elastic Response (424-437) A Practical Prediction of Wave-Induced Structural Responses in Ships with Large Amplitude Motion (438-452) Evaluation of Eddy Viscosity and Second-Moment Turbulence Closures for Steady Flows Around Ships (453-469) On the Modeling of the Flow Past a Free-Surface-Piercing Flat Plate (470-477) Self-Propelled Maneuvering Underwater Vehicles (478-489) Spray Formation at the Free Surface of Turbulent Bow Sheets (490-505) Numerical Simulation of Three-Dimensional Breaking Waves About Ships (506-519) Generation Mechanisms and Sources of Vorticity Within a Spilling Breaking Wave (520-533) The Flow Field in Steady Breaking Waves (534-549) Freak Waves-A Three-Dimensional Wave Simulation (550-560) Bluff Body Hydrodynamics (561-579) Large-Eddy Simulation of the Vortical Motion Resulting from Flow over Bluff Bodies (580-591) The Wake of a Bluff Body Moving Through Waves (592-604) Low-Dimensional Modeling of Flow-Induced Vibrations via Proper Orthogonal Decomposition (605-621) Measurements of Hydrodynamic Damping of Bluff Bodies with Application to the Prediction of Viscous Damping of TLP Hulls (622-634) Hydrodynamics in Advanced Sailing Design (635-660) Divergent Bow Waves (661-679) A Method for the Optimization of Ship Hulls from a Resistance Point of View (680-696) Hydrodynamic Optimization of Fast-Displacement Catamarans (697-714) On Ships at Supercritical Speeds (715-726) The Influence of a Bottom Mud Layer on the Steady-State Hydrodynamics of Marine Vehicles (727-742) A Hybrid Approach to Capture Free-Surface and Viscous Effects for a Ship in a Channel (743-755) Shock Waves in Cloud Cavitation (756-771) Asymptotic Solution of the Flow Problem and Estimate of Delay of Cavitation Inception for a Hydrofoil with a Jet Flap (772-782) Examination of the Flow Near the Leading Edge and Closure of Stable Attached Cavitation (783-793) Numerical Investigation on the Turbulent and Vortical Flows Beneath the Free Surface Around Struts (794-811) Steep and Breaking Faraday Waves (812-826) The Forces Exerted by Internal Waves on a Restrained Body Submerged in a Stratified Fluid (827-838) Influence of the Cavitation Nuclei on the Cavitation Bucket when Predicting the Full-Scale Behavior of a Marine Propeller (839-850) Inception, Development, and Noise of a Tip Vortex Cavitation (851-864) Velocity and Turbulence in the Near-Field Region of Tip Vortices from Elliptical Wings: Its Impact on Cavitation (865-881) Calculations of Pressure Fluctuations on the Ship Hull Induced by Intermittently Cavitating Propellers (882-897) Hydroacoustic Considerations in Marine Propulsor Design (898-912) Prediction of Unsteady Performance of Marine Propellers with Cavitation Using Surface-Panel Method (913-929) A Comparitive Study of Conventional and Tip-Fin Propeller Performance (930-945) A New Way of Stimulating Whale Tail Propulsion (946-958) Effects of Tip-Clearance Flows (959-972) Experiments in the Swirling Wake of a Self-Propelled Axisymmetric Body (973-985) Hydrodynamic Forces on a Surface-Piercing Plate in Steady Maneuvering Motion (986-996) Advances in Panel Methods (997-1006) Effect of Ship Motion on DD-963 Ship Airwake Simulated by Multizone Navier-Stokes Solution (1007-1017) Large-Eddy Simulation of Decaying Free-Surface Turbulence with Dynamic Mixed Subgrid-Scale Models (1018-1032) Fully Nonlinear Hydrodynamic Calculations for Ship Design on Parallel Computing Platforms (1033-1047) Validation of Incompressible Flow Computation of Forces and Moments on Axisymmetric Bodies Undergoing Constant Radius Turning (1048-1060) The Validation of CFD Predictions of Nominal Wake for the SUBOFF Fully Appended Geometry (1061-1076) Appendix-List of Participants (1077-1084)