The Influence of the Steady Flow in Seakeeping Computations
Volker BertramInstitut für Schiffbau, Univ. Hamburg, bertram©schiffbau.uni-hamburg.d400.de
Consider a ship with average speed U in a regular wave of small amplitude h. The task it to compute the motions of this ship. In reality, there are a number of nonlinearities involved in the
phySicS which are usually ignored in practice:
- The noulinearities in the fundamental field equations (conservation of mass and momentum) are neglected, assuming an ideal fluid subject to Laplace's equation (conservation of mass). - Nonlinearities of the geometries of ship and water surface are also neglected.
- Relevant quantities, namely the potential, are developed in Taylor series and all nonlinear terms in these series are neglected.
Neglecting these nonlinearities naturally looses some of the physics, but it is generally felt that at least heave and pitch motions can still be well reproduced following this simplified approach. 'Nonlinear' methods introduce some nonlinear elements, but this may also be in the sense that one of the above noalinearities is approximated by a quadratic expression. Somecare has to be taken in what a 'nonlinear' method really considers and what it still neglects. If a statistical approach :with superposition of reactions to a wave spectrum is pursued, e.g. as for estimating the added
resistance in a seaway for a ship, the above linearisations are deemed necessary.
The methods used in studying the ship motion problem are predominantly boundary element methods, with more recently Green function methods and Rankine panel methods as sophisticated 3-d alternatives to close-fit strip methods. Almost all methods presented so far to solve the ship motion problem introduce a further simplification: The first term in the Taylor expansion ofthe potential, i.e. the steady flow potential, is crudely approximated by uniform flow. Correspondingly, these methods introduce usually the following errors in the physical model:
- neglect of dynamic trim and sinkage of the ship
- neglect of the steady wave profile (average wetted surface) and generally the steady wave elevation on the free surface
- neglect of the local steady flow field especially on the hull
It may be subject to discussion if it is worth the expense of sophisticated 3-d boundary element methods in an approach using both 'wrong' boundary and 'wrong' boundary conditions! As with all simplifications, the fundamental question is how large the associated error is. This issue will be Studied in this paper numerically.
The applications will be limited to head waves as this simplifies the computational effort consid-erably, but still allows to galn insight into the influence of the steady flow respectively the effect of its common simplification in seakeeping computations. The Rankine panel method employed Solves for the source strengths (indirect method) and is described in detail in Bertram (1995). In a first step, the steady wave-resistance problem is solved by a 'fully nonlinear' code employing higher-order panels on the hull to evaluate also second derivatives of the potential there, Hughes arid Bertram (1995). In a second step, the seakeeping problem is solved to determine the ship motions. Laplace's equation is solved subject to the following boundary conditions:
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Water does not penetrate the ship hull.
Water does not penetrate the free water surface. There is atmospheric pressure on the free surface. There is undisturbed flow far away from the ship.
Waves created by the ship propagate away from the ship. For r > 0.25 (this is fulfilled for all relevant wave lengths in head waves), these waves are limited to a sector downstream. Waves created by the ship must leave an artificial boundary of the computational domaj without reflection.
[h] The forces on the ship result in periodic motions. (We assume that the time-averaged added resistance is compensated by increased propulsion forces, i.e. the average speed remains con-stant.)
. Condition [a] and a combination of [b] and [c] result in mathematical conditions for the unknown time-harmonic potential that have been derived in an equivalent form before by Newman (1978). The conditions assume the steady potential as known and linearise only with respect to the wave
height. The conditions are enforced on the average wetted hull (including dynamic trim and sinkage and the steady wave profile) and on the average free surface (the steady wave pattern) in a collocation scheme, employing first-order Rankine singularities on the hull and above the free surface (desingularized). The numerical 'shifting' or 'staggered grid' technique to enforce the radiation condition [e], developed originally for the steady wave-resistance problem, can be adapted without problem to the time-harmonic problem, Bertram (1990), and fulfills also automatically the open-boundary condition [f]. Decay condition [d] and Laplace's equation are automatically fulfilled in a Rankine panel method.
As usual, the seakeeping method solves first for the the diffraction and (unit motion) radiation potentials. This forms for the symmetric case of head waves 4 systems of equations which share the same coefficient matrix with only the r.h.s. being different. All four cases are solved simulta-neouslyusing Gauss elimination. Then the computation of all potentials and their derivatives at all collocation points is straight-forward. But for the total potential, the so-far unknown motion amplitudes still need to be determined. The expressions for this final step are derived in principle from 'force = mass acceleration' yielding a system of linear equations for the motion amplitudes which is quickly solved by Gauss elimination.
The method described so far will be denoted by RPM in the following. If the steady potential is set to uniform flow, the ship is only considered at calm-water draft, and the free surface is taken as flat, the same method is denoted as SRPM (simplified RPM).
The Series-GO with CB = 0.6 was selected as test case, Fig.1. The steady proMem was solved for F = 0.2 with 1422 elements on the free surface. The nonlinear computation converged rapidly reducing the error in the nonlinear free-surface condition to 1% of its initial value in 3 iterative steps. Trim and sinkage agreed well with experiments. All flow details appeared plausible, giving no indication of undue errors in this step.
The same hull grid as for the steady computations was used for the seakeeping computations. For each investigated wave length, a new structured grid was generated on the free surface and the results of the steady computation were interpolated in this grid. The grids had between 1400 and 1700 elements. The RPM results are compared with experiments, SRPM, and a close-fit strip method.
Fig.2. shows the response amplitude operators for heave and pitch. The results can be summarized
as follows:
-- The strip method reproduces tlìe motions satisfactorily. The RPM agrees best with experiments.
- The uniform-flow approximation for the steady flow introduces in the SRPM considerable errors for medium wave lengths. In the investigated case, these errors are sometimes larger
than for the strip method.
g.l: Discretizatiort of Series-60 ship with 594 elements. For uniform flow approximation (SR PM), ly the ]ower part up to CWL (marked by the two short lines) with 495 elements is considered.
p71/pçjh o o o loo o o o 31 -+ o + 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
g.2: Response amplitude operators for Series 60, CB = 0.6, head waves, F,. = 0.2 experiment, + strip method, o RPM, * SRPM
0.5 1.0 1.5 .0 0.5 1.0 1.5 2.0
g.3: Diffraction pressure and pressure due to unit pitch motion at poi lit I n bow region; 0 RPM, * SRPM -
ûsL/h
8 6 + 4- 2-psj/pg o 6 e Q Q Q o oBesides integral values like forces and motions, sometimes local pressures are of interest, espe in the bow region. Therefore, for a strip of panels in the forebody(the third strip from the
b0 Fig.1) the computed pressures were compared between RPM and SRPM, Bertram (1996b) fo wave length. For one point in the bow region (ilL = 0.4753, y/B = 0.00107, z/T = 0.9488) F shows the diffraction pressure (without the contribution of the incident wave) and the racjj. pressure due to unit pitch motion for various wave lengths. For the diffraction pressure differences between RPM and SRPM increase for longer wave lengths. For the radiation pre due to pitch unit motion, significant differences occur only for shorter waves. The diffrac pressure dominates for short waves (motions almost zero), but the radiation pressures for- lo waves. This explains probably why the motions are computed well for long and short waves. large relative errors occur here in unimportant pressure parts. However, for medium wave len - where the maxima of the motions occur - errors apparently superpose to such an extent that influence of the steady flow becomes considerable for the motion prediction.
As far as the comparison with experiments for integral values allows conclusions, capturing steady flow in seakeeping computations apparently increases the accuracy of prediction consjd ably. Substituting the steady flow by uniform flow introduces considerably different motion resul for medium wave lengths. Japanese experiments for a tanker model indicate that for blunt ship the diffraction pressures in the bow region are predicted with errors of about 50%, if they computed using uniform flow as steady flow approximation, Iwashita et al. (1993). Unfortunatel the geometry of the investigated ship is classified. Neither are any other comparable measuremen published for an unclassified ship geometry. Therefore investigations as the one presented here a so far limited to purely numerical studies. I hope that in the future more such investigationsw shed light on the topic and that eventually we will have experiments to validate the computatio also for flow details like pressures.
Acknowledgements
I am indebted to James Filhol of HSVA for supplying the strip method results, MIII R+D Center Nagasaki for supplying the panel grid for the Series-GO, and the Deutsche Forschungsgemeinschaft for financial support.
References
BERTRAM, V. (1990), Fulfilling open-boundary and radiation condition in free-surface problems using Ranlrine sources, Ship Techn. Res. 37/2
BERTRAM, V. (1996a), A 3-d Ranlcine panel method to compute added resistance of ships, IfS-Report 566, Univ. Hamburg
BERTRAM, V. (1996b), Comparison of various 3-d methods to compute seakeeping of ships, Jahrbuch STG, Springer (in German)
HUGHES, M.; BERTRAM, V. (1995), A higher-order panel method for steady 3-d free-surface flows, IfS-Report 558, Univ. Hamburg
IWASHITA, H.; ITO, A.; OKADA, T.; OHKUSU, M.; TAKAKI, M.; MIZUGUCHI, S. (1993), Wave force acting on a blunt ship with forward speed in oblique sea (2nd report), J. Soc. Nay. Arch. Japan 173 (in Japanese)
NEWMAN, J.N. (1978), The theory of ship motions, Advances in Applied Mechanics 18
