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Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

Preliminary study on the 2D+t method for ship bow waves with Cartesian grid
Yifan Zhang

Shan Ma*

Xuhui Li

Yunxing Zhang

Wenbo Shao

College of Shipbuilding Engineering, Harbin Engineering University
Harbin, Heilongjiang, China

ABSTRACT1Munk (1924) first proposed the 2D+t theory for the calculation oftransverse loads on an airship inclined in a stream. The local loads
are given in terms of the added mass of transverse sections and theirrate of axial change. Von Karman (1929), when investigating
high-speed impact problem, also expressed the vertical loads, likeMunk, in terms of added mass variation in time. Tulin (1957) used
2D+t theory for slender planing surface, which considers thenon-linear effects at the wet chine and a spray jet. Tulin and Wu(1996) successfully used 2D+t non-linear slender body theory
method to predict entire divergent pattern.

Ship bow waves are quite common during ship navigation. Athorough investigation of the bow waves will contribute to a better
understanding of the formation process of wave pattern as well as
its wave-forming resistance. At present, computational cost of direct
simulation of ship bow waves using three-dimensional
Computational Fluid Dynamics (CFD) is still very large, which is
not so practical in industry. To investigate the phenomenon more
efficiently, an incompressible two-dimensional CFD model
combined with two-dimensional (2D) + time (t) theory is developedin this paper. Based on the 2D+t theory, quantities in the flow
direction would change slowly compared with those in thetransverse and vertical directions for slender bodies. In this way, thesteady 3D problem can be simplified to unsteady 2D free-surface
flows generated by a deformable body in the vertical planetransverse to the vessel. To validate the capability of 2D+t method
to simulate wave generation, two cases are provided. The first one is
the wave generated by the forced motion of a rigid body. The otherone is the wave generated by the deformation of a constantly
deforming body. In both cases, water elevation data recorded by thewave probes installed near the body are compared with analyticalsolution obtained using potential flow theory.

In real case, when ship speed is high, breaking phenomenon ofdivergent waves can be frequently observed near the bow. Potential
theory is no longer suitable to simulate the case. To improve theaccuracy of simulation and the efficiency to simulate breaking
waves. 2D+t theory has also been extended to CFD method bypioneers. Many applications of 2D+t in Smoothed Particle
Hydrodynamics (SPH) can be found in literature. Marrone (2011)
first applied 2D+t method to SPH and successfully predict the wave
pattern generated by high-speed slender ships with a sharp stem.
Landrini et al. (2012) combined the 2D+t concept with a moreadvanced Boundary Element Method-Smoothed Particle
Hydrodynamics (BEM–SPH) solver to simulate the bow waves atthe pre-breaking and post-breaking stages, respectively. Zheng(2019) also proposed a combined ISPH (Incompressible Smoothed
Particle Hydrodynamics) with 2D+t modeling approach forsimulating the bow wave generations by a high-speed slender vessel.
The application of 2D+t method to cartesian gird solver can befound in Weymouth (2013). Euler equation is satisfied incomputational domain, near-wall viscosity is enforced by adding
slip model to no-penetration condition. With the model, non-linear
ship bow waves are successfully predicted.

KEY WORDS: Ship bow waves, 2D+t, VOF-PLIC, VOS-IB
INTRODUCTION
Ship-generated waves are significant in the field of surface-shiphydrodynamics because of its intensive relationship with hull
resistance (Inui, 1981). Especially for high-speed vehicle,wave-generated resistance accounts for a large percent of total dragforce. Therefore, an accurate prediction of wave patterns andhydrodynamic force could have reference meanings for reduction ofwave resistance and improvement of ship speed.

To sum up, the achievements accomplished by previous research on
predicting ship bow waves using 2D+t method are numerous.
However, to the best of our knowledge, the combination of 2D+t
method and incompressible two-phase flow CFD solver is rare. In
this article, an approach to directly add 2D+t method into theincompressible two-phase flow CFD solver is proposed.

Nowadays, although with the rapid improvement of computer
computing ability, computational cost of direct simulation of ship
bow waves using three-dimensional CFD is still very large, which is
not so practical in industry. To fulfill the needs of efficientcomputation, the idea of 2D+t is proposed.

CFD could be a good alternative with the development ofcomputing power. Relevant methods can be divided into body-fitted
grid methods and non-body-fitted grid methods. For a body-fittedgrid in which the grid is aligned with the body surface, theenforcement of boundary conditions is simple and accurate.
However, the mesh generation could be complicated especially

The early application of 2D+t is based on potential flow theory.
*Corresponding Author: Shan Ma2535