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APPENDIX D Description of Mathematical Model The navigation channel design model is intended to simulate the reactions of the design vessel ~ s ~ in a given set of conditions. The model is composed of several modules intended to evaluate squat, bank suction, vessel attitude, thrust for a ppecif fed stopping distance, and ship-generated waves ~ Figure D-1 ) . The modular construction allows f or changes to be made with relative ease as advances are made in an area of analysis. As specif fed by the control cards, data are read in and distributed by the main module to the respective modules. After the specified modules have completed their evaluation, control is transferred back to the main module. The input data consist of eight cards. is listed below: The purpose of each card Card #1 specifies the analysis to be performed and the azimuth of the inbound vesse 1. In addition, it specif ies whether an inbound, an outbound, or a combination analysis is to be performed. Card # 2 specif ies title inf ormation which would identify the channel analysis being performed. Card # 3 specif ies the section geometry f or the channel under , . consideration. Card #4 specif ies the magnitude and direction of botch the current and the wind. In addition, the water density and temperature are given. Card #S specifies the navigable channel lion; ts, the thickness of the channel boundary layer, and the bank elevations . Card #6 specif ies the primary vessel characteristics , e. g., draft, beam, length, minim and maximum speeds, vessel boundary layer thickness , position from centerline, maximum rudder angle, wind area, displacement, and stopping distance. Card #7 specifies the secondary vessel characteristics, if the passing condition has been specified. Card #8 specif ies the entrance length of the primary vessel. D1

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D-2 Vessel Attitude 1~ Bank Suction Neutral Steering Line Boundary Layer L Data _ . two_ Main Control Squa t _ l Sa linity l 1 ;hip -Genera~ced Waves Stopping Dis tance Power Required \ ~1 -1 Area Figure D-1 Channel design evaluation model

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D-3 Description of Modules Squat computation module The squat module checks the salinity of the water to determine if vessel draft needs to be adjusted. Squat is computed for the primary vessel at various speeds for the on-centerline or passing case (or both). The basic equation (Garthune et al., 1948; Tothill, 1966) is 22.6 [ A ] where Ad = squat ( in ft) if = velocity (in Its) 2 A = cross-sectional area of flow in channel, absent vessel (in ft ) A' = cross-sectional area of flow with ship in channel section, experiencing squat The computation terminates when the incremental speed reaches one of the following: o maximum vessel speed; o critical velocity; or o vessel touches bottom. In the development of the model, calculated values were compared to field measurements and found to agree satisfactorily (Figure D-2. Figures D-3(a) through D-31 illustrate results for squat in nine channels selected by the panel. Neutral Steering Line and Bank Suction Module If bank suction is required, then the neutral steering line is calculated. This is the sailing line of a vessel along which lateral f orces and turning moments owing to bank suction are balanced. If the channel cross-section is symmetrical, the neutral steering line is the channel centerline; if asymmetrical, the module calculates the distance from centerline to neutral steering line, adding this value algebraically to the distance from the centerline of the ship's position. These values are then used in the bank suction computations, which encompass lateral forces and moments due to bank suction effects for various speeds, using the method proposed by Schoenherr (1960~. In validation of the model, computed and measured values (from model tests) were in satisfactory agreement for symmetric and asymmetric cross-sections of channels. Vessel Attitude Module This module computes the additional forces due to current and wind and adds these values to the forces calculated in the bank suction module. The rudder and drift angles required to neutralize the total forces and moments are then calculated for selected speeds, based on the method described by Bindel (1960), and the addition of an off-centerline coefficient.

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D-4 2.0 - u' 1.0 a) o c' Figure D-2 r It- -- 1-.1( 1 1~-1 1 Or I I 0 1..0 2.0 Measured Squat (ft) Comparison of computed and measured squat

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D-9 Ship-Generated Waves Module The principal purpose of this module is to determine the effects of ship-generated waves on barge trains, but it also gives a general indication of other effects that may be expected in particular channels from ship-generated waves--for example, bank erosion--and may suggest operational or design changes. The module computes ~ f or selected speeds ~ the ship wave height at the vessel, and the wave height at each cusp on lines of propagation. The methods are based on those suggested by Saunders (1957) and Havelock (Wigley, 1963~. There are few measurement data for comparison. Stopping Power Computation Module This module computes the reverse thrusts needed to bring the vessel (sailing at selected speeds) to a standstill in a spec' fled distance. The computational procedure suggested by D' Arcangelo ~ 1957 ~ was adjusted in consultations with p ilots, ship masters, and other sources of inf ormation about the behavior of ships in restricted waters.

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D-10 Ref erences Bindel, S . ~ 1960 ), "Turning Characteristic Coefficients for a Cargo Ship and a Destroyer, " Report 1461, David Taylor Model Basin, U.S. Navy. 'Arcangelo, A. M., gent ed. ( 1957 ), "Guide to the Selection of Backing Power, " Tech . and Res . Bull . No . 3-5, Panel M-9, Society of Natural Architects and Marine Engineers. Garthune, R. S., et al. ~ 1948 ), "The Performance of Model Ships in Relation to the Design of the Ship Canal, " Report 601, David Taylor Model Basin, U. S. Navy. Saunders, H. E . ~ 1957 ), Hydrodynamics in Ship Design, Vol. II ~ New York : Society of Naval Architects and Marine Engineers ~ . Schoenherr , K. E . ~ 196 0 ), "Data f or Estimating Bank Suction Ef f eats in Restricted Waters and on Merchant Ship Hulls, " Report 1461, David Taylor Model Basin, U. S. Navy. Tothill, J. T. (1966), Ships in Restricted Channels, Report MB-264 ~ Ottawa: National Research Council of Canada ~ . Wigley, C., gent ed. (1963), The Collected Papers of Sir Thomas Havelock on Hydrodynamics (Washington, D.C.: Office of Naval Research).