Shallow water and squat prediction

In 2003 we investigated on behalf of the German Waterways and Shipping Administration (WSV) the dynamic response of very large (mega jumbo) container vessels sailing up the River Elbe. In navigation channels vessel dynamic sinkage and trim, known as squat, is a dynamic function of vessel size and speed, and channel bottom features.

In the past all investigations into this topic were done using model ships and basins constructed exclusively for this purpose, such as those at the Federal Waterways Engineering and Research Institute. This method incurs high costs related to the acquisition of physical models and managing water levels during the tests.

We were able to successfully run and validate simulations of the squat phenomena using simple techniques. The channel bottom was an idealized plane and the computational domain was split into three regions, the air, water, and ground. The computational mesh was then allowed to move vertically and rotate around a transverse axis. The material properties in the three regions are set based on the concentration of the species 1 (air) and 2 (solid) in the background fluid (water). The concentration of air in water is imposed at the boundaries and is transported inside the computational domain by solving an additional equation. The concentration of solid in water below the bottom plane is set at the boundaries and also inside the computational domain in each time step. Control volumes (CVs) which are completely below the bottom plane are considered to be 100% solid. The solid concentration of CVs cut through the bottom plane is carefully interpolated between 0 and 100% as a function of the truncated volume of the CV. Since the position of the CVs relative to the bottom surface constantly changes, this interpolation has to be repeated at each time step. Based on the solid concentration the velocity vector in the CV is modified at each time step to match the computed velocity by the solver or the undisturbed velocity. This means for instance that for CVs below the bottom plane the velocity is set equal to the undisturbed velocity each time step, which is equivalent to zero velocity (solid) as experienced by the vessel.

The computations were carried out at a model scale of 1:40 at three speeds. The main dimensions of the investigated vessel are:

  • length water line = 355.8 m
  • breadth = 55.0 m
  • draft = 16.0 m
  • depth = 30.0 m
  • displacement = 221,191 t

The sailing conditions at full scale are 9.8, 12.3, 14.7 knots vessel speed and 2.0 metres under-keel-clearance at rest.

This very simple approach has shown qualitatively acceptable results as seen in the diagram below. There are many improvements that could be made to extend and improve these studies in the future.

The figure below shows the position of the container vessel at rest (black) and at 3 different speeds. The geometry has been scaled by a factor of 5 in the vertical direction for better presentation of the differences in squat. The figure also shows the undisturbed water plane and the sea bottom. The under-keel-clearance at the stern of the full scale vessel shrinks from 2 m at rest to 0.57 m at the higher speed.