A prediction method for squat in restricted and unrestricted rectangular fairways
Graphical abstract
Highlights
► Model tests are carried out in different rectangular fairways. ► Based upon the conservation of mass and Bernoulli the sinkage is calculated. ► Measurements and calculations are compared. ► An equivalent canal width and forward ship speed is introduced. ► Based upon the equivalent width and speed the sinkage can be calculated.
Introduction
A sailing ship continuously displaces and accelerates a significant amount of water, which, according to the Bernoulli principle, results in a pressure drop around the vessel. The latter yields a vertical displacement characterized by values of the running sinkage at the forward and the aft perpendiculars, or alternatively, a mean running sinkage and trim. This phenomenon is commonly referred to as squat (Briggs et al., 2010). Principally, the ship does not undergo an increase of its draft or volume displacement, but the level of the free water surface around the vessel changes and, on average, drops. As a result, the ship translates vertically and rotates around its lateral axis.
Squat also occurs in deep and open water but is more pronounced in restricted or confined waters. Moreover, in deep and open water this phenomenon has no major consequences, while in more restricted waters the vessel can run aground. A decrease of the net underkeel clearance as a result of squat may also affect the ship’s manoeuvrability dramatically, resulting in loss of control.
An accurate prediction of the squat that is valid in the complete range from deep and open waters up to narrow and shallow waterways (i.e., channels, canals, docks, locks) is therefore important, to avoid not only groundings but also manoeuvrability issues due to insufficient underkeel clearances. The aim of the present paper is to predict the squat (sinkage and trim) for a wide range of water depths and widths of a canal with rectangular cross section. The forward speed, lateral position of the vessel in the canal, propeller action and hull geometry are taken into account.
In June 2010 systematic model tests were carried out in the Towing Tank for Manoeuvres in Shallow Water (cooperation Flanders Hydraulics Research — Ghent University). The sinkage of a scale model of a very large crude carrier was measured, the lateral position and the forward speed of the model were varied systematically resulting in a database of about 400 different model tests.
The model test results are compared with the semi-empirical calculation method as published by Dand and Ferguson (1973). Based upon this theory a new mathematical model is proposed for the calculation of squat in any rectangular cross section, from a wide and deep fairway, up to underkeel clearances of only a few percent of the vessel’s draft and widths of a fraction more than the beam of the ship. The application range of the proposed model is restricted to channels with a rectangular cross section, and to vessels sailing without drift and parallel to the longitudinal channel boundaries.
Section snippets
State of the art
For years the combination of mean running sinkage zVM and running trim tV (or squat) has been investigated at different research institutes. Published scientific research on squat initiated with Constantine (1960) who discussed the different squat behaviour for subcritical, critical and supercritical vessel speeds and its theoretical relation with the blockage factor m (or the ratio between the midship section AM and the cross section of the fairway AC). Constantine’s reflections were
Test facilities
All the test results discussed in the present paper are obtained by captive model tests carried out in the Towing Tank for Manoeuvres in Shallow Water (cooperation Flanders Hydraulics Research — Ghent University). The main dimensions of the empty towing tank are 88×7×0.5 m3 (Van Kerkhove et al., 2009), but vertical walls parallel to the longitudinal tank walls were built into the tank to reduce the channel width over a length of 30 m. The tank is equipped with a planar motion carriage that allows
Principle
Dand and Ferguson (1973) proposed a semi-empirical theory to calculate the squat as a combination of mean sinkage and trim based on the mass conservation law and the Bernoulli principle. Hereafter, Dand and Ferguson (1973) will be referred to as D&F. This theory takes into account the ship’s beam B(x) on the waterline at a longitudinal position x, the cross-sectional area A(x) and the longitudinal distribution of these areas. It supposes that the vessel sails at a constant speed V in a waterway
Running sinkage at the FP zVF
Based upon the measured running sinkage at the FP zVF and the D&F-method for every ship speed, eccentricity y and water depth h, a new width can be calculated which results in the same running sinkage at the FP as the measured running sinkage during the model tests. This results in an equivalent width Weq based upon the measured sinkage at the FP zVF. As discussed in Section 4, this new canal width Weq is smaller than the real canal width when the canal is wider than about 1.7 the ship’s beam
Future research opportunities
The newly proposed mathematical model will be validated against the sinkage of other ship types. The independence of the coefficients relative to the ship’s geometry will also be tested. The cross section of a natural river or a manmade canal is seldom rectangular, therefore other cross sections than rectangular cross sections of the fairway will be investigated. As a consequence a new formulation for the influence of the lateral position of the vessel on the equivalent width will be
Conclusions
Model tests have been carried out with a scale model of a very large crude carrier in canals of different widths. The sinkage of the ship has been measured at FP and AP and the measurements were compared with the empirical model determined by Dand and Ferguson (D&F).
The original D&F method for the sinkage at the FP corresponds to the measurements when . When the section is wider the original D&F method underestimates the sinkage at the FP, while an overestimation is found when the section
Acknowledgments
This research project was funded by the Flemish Government, Department Mobility and Public Works, and the model tests were executed in 2010 by Flanders Hydraulics Research (Antwerp, Belgium). The presented research is executed in the frame of the Knowledge Centre Ship Manoeuvring in Shallow and Confined Water, a cooperation between Flanders Hydraulics Research and the Maritime Technology Division of Ghent University.
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