Plane and axisymmetric mixing layers

Christophe Bogey and Christophe Bailly

Velocity-gradient regions between two streams are found in numerous flows such as plane or axisymmetric jets. These simple flow configurations, usually referred to as shear layers or mixing layers, have been extensively investigated as reported in the review of Ho and Huerre (1984). These studies have allowed better understanding of the transition to turbulence occurring in initially laminar or transitional shear layers thanks notably to the development of instability theories (Michalke 1984) and to the observations of coherent structures. Among the experiments supporting the latter observations, the famous one conducted by Brown and Roshko (1974) shows clearly that large-scale coherent structures are intrinsic features of mixing layers even at high Reynolds numbers. These structures, through their interactions such as vortex mergings, have been recognized to be appreciably responsible for the spreading of the shear layers or for noise generation, as stated by Winant and Browand (1974).

From the preceding considerations, the large-eddy simulation (LES) approach appears especially well suited to mixing-layer computations because with LES all the scales larger than the grid size, and consequently a large part of the coherent structures, are calculated. The earlier large-eddy simulations of practical flows have thus often involved mixing layers. They have permitted testing of the LES methodology for simple transitional flows.

Introduction

As described in Chapter 1, the general arrangement and layout of ship-shaped offshore units designed for oil and gas operations may be grouped into several major parts: hull structures including storage tanks, topsides (processing facilities), export facilities, mooring facilities, accommodations, machinery space, subsea systems, and flowlines. All of these various parts are equally important to achieve successful operation, with due consideration of safety, health, environment, and costs versus benefits.

This chapter focuses on topsides, moorings, and export facilities. The material presented herein is aimed at the nonspecialist introductory reader. It is consistent with the content of this book and is included, primarily, to complete the coverage of the various aspects relating to ship-shaped offshore units.

Topsides consist of processing facilities that are typically located as elevated modules that are several meters (say, 3m or more) above the main deck of the vessel hull, but related piping systems may be located on the main deck of the vessel hull. Depending on the vessel size and topsides layout, the topsides modules may have multiple decks that contain the oil-, water-, and gas-processing facilities; utility systems; and similar functions. The preferred configuration, however, may be that to the extent possible, the topsides facilities would be incorporated as single-layer “pancake” units. The single-layer unit arrangement requires a larger main deck area for a given set of needs.

The determination of wind effects, such as forces and heeling moments for the hull, topsides, accommodation areas, and helideck of a ship-shaped offshore unit, can be an essential task for the analysis of intact and damage stabilities and other strength aspects. Wind forces and wind moments should also be predicted for the analyses of mooring and thruster systems. Although theoretical and numerical simulations including computational fluid dynamics (CFD) methods may be employed, wind-tunnel tests are highly desirable to get more reliable estimates in this regard.

Wind-tunnel tests are also usually used to analyze smoke ingress and ventilation problems on board a ship-shaped offshore unit, aspects that are always involved in various environmental and safety risk assessments. Examples include assessment and optimization of the areas over the helideck, which are affected by disturbed flow and by temperature rises due to turbine exhaust emissions. To model emergency gas releases and fire scenarios and to identify the regions of poor ventilation, wind-tunnel tests may be required. The natural ventilation within the process areas of an FPSO can also be assessed by wind-tunnel testing.

For a detailed description of the wind-tunnel testing of ship-shaped offshore units involving test procedures, measurement techniques, and assessment criteria, refer to the UK HSE report titled Review of model-testing requirements for FPSOs, Offshore Technology Report, 2000/123, Health and Safety Executive, UK, 2000.

Introduction

The first two chapters provided the basics of small-rotation dynamics with applications to rigid bodies or near rigid bodies. The purpose of this chapter is to provide an introduction to the finite element method (FEM) of structural analysis to the limited extent necessary for the use of this textbook. That is, the present discussion of the FEM is limited mostly to one-dimensional structural elements.Specifically, all the example problems and exercises deal only with linear, beam finite elements and linear,spring finite elements. Neither of these elements require the finite series sophistication of two- orthree-dimensional elements such as plate or solid finite elements. A very brief introduction to multidimensional finite elements is presented in Endnote (). If the reader is already familiar with the FEM, then this chapter can be skipped, particularly because the next chapter provides ample further review of this topic. Since the finite element method is so extensively used for, and so particularly suited to, structural dynamics analyses, no other method of structural analysis is used for the calculations presented in this textbook. For the sake of instruction, the use of the FEM in this textbook is oriented to hand calculations rather than the use of one of the many available and routinely used commercial software programs that all do essentially the same things and differ only in style. Thus the reader should be able to gain insight into what all such FEM programs need to do.

Introduction

The purpose of this chapter is to introduce damping forces into the structural equations of motion. Simply speaking, damping forces are internal or external friction forces that dissipate the energy of the structural system. Although damping forces are usually much smaller than their companion inertia and elastic forces, they nevertheless can have a significant affect on a vibratory motion, especially after many periods of vibration, or when the system is vibrating at one of certain important frequencies called the system's natural frequencies. This chapter describes various ways of characterizing damping and explains how the damping properties of a vibratory system can be measured. Solutions for the motion of one-DOF systems are presented for force free and certain applied forces to better explain the role that damping plays in structural systems.

Descriptions of Damping Forces

When an actual, force free, structural system is set in motion by means of initial deflections or initial velocities, or both, any point within the system generally vibrates with amplitudes that are very little different over short time intervals; that is, time intervals lasting typically five or fewer periods of the vibration. Figure 5.1(a) shows the calculated amplitude–time trace of such a vibration where the period t of the vibration is 1 sec and the initial displacement has a unit value. As will soon be seen, the sinusoidal expression that describes the force free motion of a one-DOF undamped system, has to be modified, in this case by an exponential multiplier, when one representative form of system damping is present.

Introduction

The focus of this textbook is on the vibrations of engineering structures, not mechanisms. However, this chapter focuses on pendulums as representative of mechanisms. Pendulums are rarely a part of an engineering structure. However, because the motions of pendulums are familiar to everyone, they do provide a comparatively simple means for both visualizing and explaining some basic aspects of more general vibratory systems. Pendulums also provide an opportunity to consolidate the lessons on dynamics set forth in the first chapter without the complication of dealing with flexible structures. As an aside, pendulums also provide a relatively simple introduction to the quite challenging topic of nonlinear vibrations. Thus, despite their limited relevance to engineering structures in general, this introductory study of structural dynamics begins with the study of the back-and-forth motion of pendulums.

The static equilibrium position (SEP) of any dynamical system is the deflected position of that system in response to all the applied static loads and their support reactions, if any. A stable pendulum system is defined as any system that, when displaced from its static equilibrium position, tends to return to that SEP as a result of the presence of a gravitational force field or other force field. An example of body forces other than gravitational forces that stabilize a structural system is the centrifugal force field acting on a rotating helicopter blade.

The vorticity equation

Irrotational flow can be established from a state of rest in an ideal incompressible fluid by the instantaneous transmission throughout the fluid of impulsive pressures from a moving boundary. If the boundary motion is subsequently arrested the motion everywhere ceases immediately. Kelvin's theorem (§2.10), that the kinetic energy of an irrotational flow is always smaller than that of any other flow consistent with the same boundary conditions, is a consequence of the fact that the number of degrees of freedom of irrotational motion is exactly the same as the number of degrees of freedom of the boundary itself. In a real fluid, however, there are typically an unlimited number of degrees of freedom, the flow is rotational, and the motion continues after the boundary stops moving. Kelvin (1867) therefore proposed the following definition of a vortex in a homogeneous incompressible fluid: ‘… a portion of fluid having any motion that it could not acquire by fluid pressure transmitted from its boundary’. Vorticity is actually a derived kinematic quantity, but its introduction greatly increases understanding of a complex flow and a knowledge of its distribution frequently permits the description of the fluid motion to be simplified.

When a small fluid particle is imagined to be suddenly solidified without change in its angular momentum, it continues to translate and rotate as a solid body. Its initial angular velocity of rotation is determined by its moment of inertia tensor, which depends on the particle shape.

No actual structure is rigid. All structures deform under the action of applied loads. When the applied loads vary over time, so, too, do the deflections. The time-varying deflections impart accelerations to the structure. These accelerations result in body forces called inertial loads. Since these inertia loads affect the deflections, there is a feedback loop tying together the deflections and at least the inertial load part of the total loads. When the applied loads result from the action of a surrounding liquid, then the deflections determine all the applied dynamic loads. Therefore, unlike static loads (i.e., slowly applied loads), differential equations based on Newton's laws are required to mathematically describe time-varying load–deflection interactions. Inertial loads can also have the importance of being the largest load set acting on parts of a structure, particularly if the structure is quite flexible.

In order to appreciate how significant time-varying forces can be, consider, for example, the time-varying loads that act on a typical large aircraft. After the aircraft starts its engines, it generally must taxi along taxiways to a runway and then travel along the runway during its takeoff run. Taxiways and runways are not perfectly flat. They have small alternating hills and valleys. As will be examined in a simplified form later in this book, these undulations cause the aircraft to move up and down and rock back and forth on its landing gear, that is, its suspension system.

This textbook is designed to be the basis for a one-semester course in structural dynamics at the graduate level, with some extra material for later self-study. Using this text for senior undergraduates is possible also if those students have had more than one semester of exposure to rigid body dynamics and are well versed in the basics of the linear, stiffness finite element method. This textbook is suitable for structural dynamics courses in aerospace engineering and mechanical engineering. It also can be used in civil engineering at the graduate level when the course focus is on analysis rather than earthquake design. The first two chapters on dynamics should be particularly helpful to civil engineers.

This textbook is a departure from the usual presentation of this material in two important ways. First, from the very beginning, descriptions of system dynamics are based on the simpler-to-use Lagrange equations. To this end, the Lagrange equations are derived from Newton's laws in the first chapter. Second, no organizational distinctions are made between multidegree of freedom systems and single degree of freedom systems. Instead, the textbook is organized on the basis of first writing structural system equations of motion and then solving those equations mostly by means of a modal transformation. Beam and spring stiffness finite elements are used extensively to describe the structural system's linearly elastic forces. If the students are not already confident assemblers of element stiffness matrices, Chapter 3 provides a brief explanation of that material.

Fluid mechanics impinges on practically all areas of human endeavour. But it is not easy to grasp its principles and ramifications in all of its diverse manifestations. Industrial applications usually require the numerical solution of the equations of motion of a fluid on a very large scale, perhaps coupled in a complicated manner to equations describing the response of solid structures in contact with the fluid. There has developed a tendency to regard the subject as defined solely by its governing equations whose treatment by numerical methods can furnish the solution of any problem.

There are actually many practical problems that are not yet amenable to full numerical evaluation in a reasonable time, even on the fastest of present-day computers. It is therefore important to have a proper theoretical understanding that will permit sensible simplifications to be made when formulating a problem. As in most technical subjects such understanding is acquired by detailed study of highly simplified ‘model problems’. Many of these problems fall within the realm of classical fluid mechanics, which is often criticised for its emphasis on ideal fluids and potential flow theory. The criticism is misplaced, however: For example, potential flow methods provide a good first approximation to airfoil theory, and ‘free-streamline’ theory (pioneered in its modern form by Chaplygin) permits the two-dimensional modelling of complex flows involving separation and jet formation.

Introduction

A knowledge of the rudiments of dynamics is essential to understanding structural dynamics. Thus this chapter reviews the basic theorems of dynamics without any consideration of structural behavior. This chapter is preliminary to the study of structural dynamics because these basic theorems cover the dynamics of both rigid bodies and deformable bodies. The scope of this chapter is quite limited in that it develops only those equations of dynamics, summarized in Section 1.10, that are needed in subsequent chapters for the study of the dynamic behavior of (mostly) elastic structures. Therefore it is suggested that this chapter need only be read, skimmed, or consulted as is necessary for the reader to learn, review, or check on (i) the fundamental equations of rigid/flexible body dynamics and, more importantly, (ii) to obtain a familiarity with the Lagrange equations of motion.

The first part of this chapter uses a vector approach to describe the motions of masses. The vector approach arises from the statement of Newton's second and third laws of motion, which are the starting point for all the material in this textbook. These vector equations of motion are used only to prepare the way for the development of the scalar Lagrange equations of motion in the second part of this chapter. The Lagrange equations of motion are essentially a reformulation of Newton's second law in terms of work and energy (stored work).