Armin W. Troesch

APPLICATION OF GLOBAL METHODS FOR ANALYZING DYNAMICAL SYSTEMS TO SHIP ROLLING MOTION AND CAPSIZING

Ship capsizing is a highly nonlinear dynamic phenomenon where global system behavior is dominant. However the industry standards for analysis are limited to linear dynamics or nonlinear statics. Until recently, most nonlinear dynamic analysis relied upon perturbation methods which are severely restricted both with respect to the relative size of the nonlinearity and the region of consideration in the phase space (i.e., they are usually restricted to a small local region about a single equilibrium), or on numerical studies of idealized system models. In this work, recently developed global analysis techniques (e.g., those found in Guckenheimer and Holmes [1986], and Wiggins [1988, 1990]) are used to study transient rolling motions of a small ship which is subjected to a periodic wave excitation. This analysis is based on determining criteria which can predict the qualitative nature of the invariant manifolds which represent the boundary between safe and unsafe initial conditions, and how these depend on system parameters for a specific ship model. Of particular interest is the transition which this boundary makes from regular to fractal, implying a loss in predictability of the ship’s eventual state. In this paper, actual ship data is used in the development of the model and the effects of various ship and wave parameters on this transition are investigated. Finally, lobe dynamics are used to demonstrate how unpredictable capsizing can occur.

A NONLINEAR PROBABILISTIC METHOD FOR PREDICTING VESSEL CAPSIZING IN RANDOM BEAM SEAS

The capsizing of vessels in random beam seas is investigated using a single degree-of-freedom model which is limited to the roll motion. Several factors, including sea wave spectrum, nonlinear righting moment characteristics, and nonlinear damping, are taken into account in the analysis. A nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics (specifically, the Melnikov function and the phase space area flux) and random vibrations. Conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics (significant wave height and characteristic wave period) and the vessel parameters (damping and stiffness coefficients). Extensive numerical simulations are carried out to demonstrate the validity of the analytical results. It is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing.

Hydrodynamic forces acting on cylinders oscillating at small amplitudes

The hydrodynamic forces resulting from small-amplitude harmonic oscillations of arbitrarily shaped cylinders are considered both experimentally and theoretically. The fluid is assumed to be initially at rest. The theoretical model assumes a laminar, nonseparating flow, where the in-line force has two components, one due to normal pressure stresses and one due to skin friction. In the limit of zero amplitude oscillations, comparisons between theory and experiment demonstrate that the nonseparating theoretical model captures the essential behavior of real fluid hydrodynamics. This is valid for a variety of shapes including sharp-edged bodies such as squares. Through model testing, it is possible to estimate an “effective eddy viscosity” which can then be used in conjunction with the theoretical laminar flow model to give empirical drag coefficients.

A VORTEX LATTICE METHOD FOR HIGH-SPEED PLANNING

SUMMARY A three-dimensional numerical model using vortex lattice methods (VLMs) is developed to solve the steady planing problem. Planing hydrodynamics have similarities to the aerodynamic swept wing problem-the hdamental difference being the existence of a free surface. Details of the solution scheme are discussed, including the special features of the VLM used here in obtaining accurate flows at the leading and side edges. Computational results are presented and compared with existing theories and experiments.

Streaming flows generated by high‐frequency small‐amplitude oscillations of arbitrarily shaped cylinders

Small‐amplitude harmonic oscillations of arbitrarily shaped cylinders are considered both experimentally and theoretically. For the theoretical model, the flow regime is separated into inner and outer regions. In the inner region, the flow is governed by the classical Stokes boundary layer equation. In the outer region, the full Navier–Stokes equation for the steady streaming flow is solved numerically by using a finite difference method coupled with conformal mapping techniques. Numerical results of streaming, a nonlinear response to harmonic motion, show complicated flow schemes. Experimental results confirm the existence of such flows. Streaming flow around a sharp corner of a square cylinder is investigated through numerical calculation and experimental flow visualization. The absence of any vortex shedding on the time scale of the streaming flow is noted. These results suggest that in the limit of a small amplitude of oscillation, or equivalently large Strouhal numbers, sharp‐edged bodies experience attached flow in the mean sense.

Transom-stern Flow for High-speed Craft

Abstract Two series of experiments have been conducted, one at the University of Michigan (U-M), and one by The University of New South Wales (UNSW), with a focus to characterize the flow in the transom region of a high-speed vessel. At U-M, we have tested a destroyer type model, with and without a stern flap, while measuring pressures in the aft region of the hull and on the flap. The model was tested in both the free-to-sinkand-trim condition and the fixed condition. At UNSW, a series of geosimilar models was tested while measuring the free-surface elevation behind the vessel. The non-dimensional free-surface elevation was found to be primarily a function of the calm-water-transom-draft Froude number. To this end, an empirical formula that estimates the unwetting of the transom has been developed. This formula can be employed in a resistance prediction computer program which will provide an accurate calculation of the hydrostatic force on the transom. As a consequence, the total resistance of the vessel can now be computed accurately, even in the low-Froude-number region.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions

It is the aim of this paper to further the use of Lyapunov and local Lyapunov exponent methods for analyzing phenomena involving nonlinear vessel dynamics. Lyapunov exponents represent a means to measure the rate of convergence or divergence of nearby trajectories thus denoting chaos and possibly leading to the onset of conditions that produce capsize. The work developed here makes use of Lyapunov exponent methodologies to study capsize and chaotic behavior in vessels both experimentally and numerically using a multi-degree of freedom computational model. Since, the Lyapunov exponent is defined in the limit as time approaches infinity, one encounters fundamental difficulties using Lyapunov exponents on the capsize problem, which is inherently limited to a finite time. This work also incorporates the use of local Lyapunov exponents, which do not require an infinite time series, to demonstrate their usefulness in analyzing finite time chaotic vessel phenomena. The objective is to demonstrate the value of the Lyapunov exponent and local Lyapunov exponent as a predictive tool with which to indicate regions with crucial sensitivity to initial conditions. Through the intelligent use of Lyapunov exponents in vessel analysis to indicate specific regions of questionable stability, one may significantly reduce the volume of costly simulation and experimentation.

The generation of digital random time histories

Abstract “A method to generate digital random time histories is described. A random number sequence is shaped to give the desired spectral density curve. This finite set of numbers is then Inverse Fast Fourier Transformed (IFFT). The result is a pseudo random time history which has given spectral characteristics. An application of this technique is described.”

Hydrodynamics of Damping Plates at Small KC Numbers

The performance of circular thin plates in enhancing hydrodynamic damping of lightly damped offshore structures such as Spar Platforms and Tension Leg Platforms is studied. These platforms can experience resonant oscillations in heave under first and more likely second-order wave forces. As such, drag-augmenting devices are desired to limit the response amplitude to a safe range. This work includes two parts. The first part focuses on the damping coefficients’ parametric dependence (KC number) and geometric dependence (thickness-to-diameter ratio). The study exploits a series of forced oscillation experiments. The experiment spans a range of KC numbers from 0.01 to 1.1 and a range of thickness-to-diameter ratios from 1/87.5 to 1/25. The second part of this study focuses on the underlying flow physics utilizing flow visualization experiments. The results of KC number dependence indicate three KC regimes where the damping coefficient behaves differently. Further flow visualization experiments demonstrate four unique vortex formation modes in these three KC regimes. A comparison of the slopes of the damping curve indicates that the interaction of vortices generated from two half cycles increases the damping effectiveness. For plates with different thickness-to-diameter ratio, similar characteristics of KC dependence are observed. The transitional KC numbers are thickness-to-diameter ratio dependent with the transitions occurring at larger KC numbers for thicker plates. While the total force experienced by oscillating plates is linear with KC number and essentially independent of the thickness-to-diameter ratio, a significant reduction in damping with an increase in thickness-to-diameter ratio is observed.

NUMERICAL STABILITY ANALYSIS FOR FREE SURFACE FLOWS

SUMMARY A systematic methodology of numerical stability is presented here in the study of numerical properties of mixed Eulerian‐Lagrangian schemes for the numerical simulation of non-linear free surface flows. Two different numerical schemes, i.e. a source‐doublet panel method and a desingularized method, are investigated. The present work provides theoretical foundations and applications for numerical stability analysis theory. The matrix stability method has been developed to obtain the spectral radii and normal modes associated with free surface discretization. Some examples considered illustrate the usefulness of this analysis. 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 893‐912, 1997.