The Mechanisms

The mechanisms of wake-induced instabilities of pairs and small groups of cylinders, on the one hand, and cylinder arrays involving perhaps hundreds of cylinders, on the other, are fundamentally similar. Therefore, in principle, one could conceive of a common introduction for both Chapters 4 and 5; in practice, however, there are sufficient peculiarities in both, in particular with regard to sign conventions, to make it desirable to have different introductory sections in the two chapters, despite the repetitiousness in terms of the basic ideas involved. Wake-induced flutter is discussed in Section 4.2.

As in the introductory sections for other chapters, what is presented here is highly simplified, leaving most of the complexities and refinements to the sections that follow. Specifically, following Païdoussis & Price (1988), in Section 4.1.1, a modified form of quasi-steady theory is presented, suitable for the subject matter of Chapters 4 and 5. In Section 4.1.2, the damping-controlled instability mechanism is revisited in the context of this modified quasi-steady theory.

In terms of systems to which this work can be related, the main ones are overhead transmission lines subjected to wind and clustered offshore oil and gas production risers subjected to ocean currents.

Modified quasi-steady theory

Consider a small group of cylinders subjected to cross-flow, as shown in Figure 4.1.

Introductory Comments

In this chapter, we use the word prism in a general sense to denote a structure of noncircular, not necessarily polygonal, cross-section. We purposely avoid such expressions as “rectangular cylinders”, preferring instead “prisms of rectangular cross-section” or simply “rectangular prisms”.

Consider a slender flexible beam or cable, cross-sectionally bluff, submitted to flow normal to its long axis; or alternatively, a flexibly supported bluff body, as in Figure 1.1. We define galloping as a velocity-dependent, damping-controlled instability, giving rise to transverse or torsional motions – for the present, considering it as a one-degree-of-freedom (1-dof) instability. Parkinson (1971) finds the name “rather appropriate”, “because of the visual impression given” when it occurs in transmission lines: typically a low-frequency (~1 Hz), high-amplitude (as much as 3 m) oscillation, reminiscent of a galloping horse – in contrast, on both counts, to the vortex-shedding related Aeolian vibration. For the same reasons presumably, in the early days, galloping was also referred to as “dancing vibrations”, e.g. by Davison (1930) who was among the first to describe the phenomenon in detail.

A circular cylinder in cross-flow is immune to galloping. As illustrated in Figure 2.1(a), the flow-related force does not change magnitude and is always in the direction of the flow. Hence, when the body is in motion, the cylinder velocity and the component of force in that direction oppose each other, thus negating the possibility of sustenance of the vibration by the flow.

Description, Background, Repercussions

A cylinder array is an agglomeration of cylinders parallel to one another in a geometrically repeated pattern. Typical examples are tube arrays in heat exchangers, involving hundreds or thousands of tubes; one fluid flows within the tubes, while another flows around them, partly normal to their long axis (cross-flow). Here, by heat exchangers we understand also steam generators and boilers, where the outer fluid boils and produces steam, and condensers, where the reverse process is involved. Figure 5.1 shows sectional views of two different types of steam generators. In both cases, the outer fluid flows axially in some regions and transversely as a cross-flow in others (Païdoussis 1981, 1983, 2004). It is well known that cross-flow promotes heat transfer, but at the expense of higher vibration levels and the possibility of cross-flow-induced instabilities, the subject matter of this chapter. Figure 5.2 shows the winding pattern of the outer flow in another kind of heat exchanger, involving cross-flow nearly everywhere.

What distinguishes heat-exchanger arrays compared with the groups of cylinders discussed in Chapter 4 is that (i) we have many more cylinders here and (ii) they are more closely spaced. The significance of (i) is that, other than the cylinders crosssectionally on the periphery, the flow around any given cylinder is confined by the presence of adjacent cylinders, rather than being unconfined at least on one side.

Here are some closing remarks on the topics covered, some on what is not covered and some on what remains to be done to reach “perfect understanding”, fully realizing, of course, that this is an unattainable goal.

Transverse galloping may be said to be well understood and that it can be well predicted. The same cannot be said of torsional galloping or of mixed transversetorsional galloping/flutter, e.g. of bridge decks, or in situations involving substantial intertwining with vortex shedding. In this regard, the advent of models based on measured force and moment coefficients as in aeroelasticity has resulted in adequate predictive ability. However, this has had a deterious effect on the desire for enhanced physical understanding and on the funding opportunities towards achieving it.

Vortex-induced vibrations under lock-in conditions have received a great deal of attention over the years. Although understanding and modelling such vibrations has come a long way over the years, including the past decade, we still have a lot of ground to cover. Indeed, predictive tools are largely semi-empirical or they depend on CFD (which bypasses the need to understand physically). Thus, a great deal remains to be done.

Motivated in the 1970s and 1980s by wind-induced vibration problems on overhead transmission lines, and more recently by current-induced instabilities on offshore structures, wake-induced flutter of small groups of cylinders has received considerable attention.

A Historical Perspective

Ovalling oscillation, or simply “ovalling”, of chimney stacks refers to the wind-induced shell-mode oscillation of thin metal stacks, involving deformation of the cross-section – strictly in the second circumferential mode (hence, the name) but, by common usage, in higher circumferential modes also. In a cross-section of the chimney the radial component of shell vibration varies proportionately to cos nθ, where n is the circumferential wavenumber (see Figure 6.1). Thus, for ovalling oscillations, n ≥ 2; whereas n = 1 for conventional beam-like lateral, or “swaying”, vibrations of the stack which are discussed only parenthetically here.

Ovalling as a technological problem first arose with the construction of thin-walled, tall and metallic chimney stacks; thin enough to easily deform as shells, tall enough to be unprotected by the earth's boundary layer near the top and with low internal damping in the metal construction. Dickey & Woodruff (1956) and Dockstader et al. (1956) describe some full-scale ovalling experiences: e.g. at Moss Landing Harbor, California, where a tall chimney (L = 68 m; D = 3.44 m, h = 7.9 mm at the top) developed ovalling in a U = 40 km/h wind with a frequency of about 1.47 Hz. Johns & Allwood (1968) describe a case of large-amplitude ovalling, which eventually led to a collapse of the chimney during a typhoon.

A number of experimental studies followed, notably by Heki & Hawara (1965) and Langhaar & Miller (1967).

General Overview

Cross-flow-induced vibration of bluff bodies, i.e. bodies whose aspect is not small compared with the streamwise dimension, are ubiquitous, in nature as well as in man-made constructions. The wind-induced fluttering of leaves and tree branches and the waving motions in wheat fields are examples of the former. The Aeolian harp, going back perhaps 3000 years, is an example of the earliest realization and/or exploitation of the existence of these vibrations made by man.

Perhaps the first documented and surviving realization of the existence of vortex shedding as such goes back to two Renaissance paintings in Bologna and a sketch by Leonardo da Vinci, thus, to the 14th and 15th centuries. The modern study of vortex shedding began in the late 19th century, with Strouhal (1878), Bénard (1908) and von Kármán (1912). Studies on vortex-induced vibrations followed soon after; lock-in, or shedding frequency synchronization, was first documented by Bishop & Hassan (1964).

With such a venerable and long pedigree, it is not surprising that the topic of cross-flow-induced vibrations and instabilities of bluff bodies, notably cylinders or groups of cylinders, is truly vast. To make any headway in this topic, one must first understand the fluid mechanics of the flow around bluff bodies, while stationary or in motion, and the forces generated thereon.

Of all cross-flow induced instabilities, rain-and-wind-induced vibrations, referred to as RWIV, are a special case. They have been identified quite recently and are certainly a challenge to understand. For the purposes of this book, they somehow gather together several of the issues that have been presented: hence, they deserve some treatment, even if the relentless evolution of knowledge on the subject makes any review soon obsolete.

Experimental Evidence

Field cases

This surprising case of cross-flow-induced motion was identified in the 1970s on cables of cable-stayed bridges. First reported by Wianecki (1979), it was described in detail by Hikami & Shiraishi (1988) as follows. Large-amplitude motion of some cables occurred in the presence of wind with rain but disappeared when the rain stopped. This is illustrated in Figure 7.1, where the existence of motion of the cables, as recorded in situ, is clearly correlated to the occurrence of rain. Similar occurrences have been reported in various bridges over the years; see for instance Ruscheweyh & Verwiebe (1995), Matsumoto et al. (1989), Zuo et al. (2008), Ni et al. (2007) and Main & Jones (1999). Although the cables undergoing such vibrations differed from one bridge to another, some common features were identified. First, the motion was specific to the concurrent existence of significant rain and wind, but it stopped if the wind velocity exceeded some value.

Structures in contact with fluid flow, whether natural (e.g. wind and ocean currents) or man-made, are inevitably subject to flow-induced forces and flow-induced vibration: from plant leaves to traffic signs, to more substantial structures, such as bridge decks and heat-exchanger tubes. These vibrations may be of small or large amplitude, and they may be inconsequential, or of mild or even grave concern.

Consider overhead transmission lines, bridges, tall buildings, and chimneys subjected to wind, offshore risers and umbilicals in ocean currents, cylinders and cylindrical tube arrays in power-generating and chemical plants, for example. Such structures vibrate to some extent at any flow velocity, e.g. due to turbulence or vortex shedding. If the vibrations are of small amplitude, they may lead to fatigue or fretting wear in the long term. However, under certain circumstances, the vibration amplitude is large, and damage may occur in the short term, in hours or weeks. Moreover, the vibration may be self-excited. Typically, but not universally, such vibration is associated with a threshold of flow velocity: on one side of the threshold, oscillations due to some perturbation imparted to the system die out; on the other side, oscillations grow. More generally, we may define self-excited vibration simply as one that grows exponentially with time until it settles down to a limit-cycle motion. Clearly, such phenomena, more specifically the conditions under which they arise, are of great importance to designers and operators of the systems concerned, because of the great potential to cause damage in the short term. Such flow-induced instabilities are the subject of this book.

This chapter describes the governing systems of equations that can serve as the basis for atmospheric models used for both operational and research applications. Even though most models employ similar sets of equations, the exact formulation can affect the accuracy of model forecasts and simulations, and can even preclude the existence in the model solution of certain types of atmospheric waves. Because these equations cannot be solved analytically, they must be converted to a form that can be. The numerical methods typically used to accomplish this are described in Chapter 3.

The equations that serve as the basis for most numerical weather and climate prediction models are described in all first-year atmospheric-dynamics courses. The momentum equations for a spherical Earth (Eqs. 2.1–2.3) represent Newton's second law of motion, which states that the rate of change of momentum of a body is proportional to the resultant force acting on the body, and is in the same direction as the force. The thermodynamic energy equation (Eq. 2.4) accounts for various effects, both adiabatic and diabatic, on temperature. The continuity equation for total mass (Eq. 2.5) states that mass is neither gained nor destroyed, and Eq. 2.6 is analogous, but applies only to water vapor. The ideal gas law (Eq. 2.7) relates temperature, pressure, and density. The variables have their standard meteorological meaning.

Background

Sometimes the standard dependent variables of NWP and climate models are all that are required for making decisions. But, frequently these meteorological variables influence some other physical process that also must be simulated before a weather-dependent decision can be made. As we will see, there are myriad examples of such situations. These models that are coupled with the atmospheric model may be referred to as special-applications models or secondary models. Examples include the following.

• Air-quality models

• Infectious-disease models

• Wave-height models

• Agricultural models

• River-discharge, or flood, models

• Wave-propagation models – sound and electromagnetic

• Wildfire-behavior and -prediction models

• Electricity-demand models

• Dust-elevation and -transport models

• Ocean-circulation models

• Ocean-drift models

• Aviation-hazard models – turbulence, icing, visibility

Sometimes the secondary model is embedded within the code of the atmospheric model, and the coupled system is run simultaneously. And, sometimes there are two distinct model codes that are run sequentially. When the code that represents the secondary process is run within the atmospheric model, the secondary process may interact with the atmospheric simulation. Or, the flow of data may be in one direction only, where the atmospheric variables are used in the secondary model without feedback. There are some secondary-model processes that have strong feedbacks to the atmosphere, and for their prediction there is of course a greater need to have a two-way exchange of information between the atmospheric and secondary models.

The surface processes whose numerical simulation is discussed here occur near both the land–atmosphere and the water–atmosphere interfaces. Over land, the movement of heat and water within the plant canopy and the ground beneath it must be represented in both weather- and climate-prediction models. Through this movement of heat and water across the land–atmosphere interface, properties of the land surface such as temperature and wetness are felt by the atmospheric boundary layer and the free atmosphere above. The atmosphere, in turn, affects the substrate and vegetation properties through radiation, precipitation, and controls on evapotranspiration. The effect of the surface on the frictional stress felt by the air moving over it is more the subject of boundary-layer meteorology and parameterizations rather than land-surface physics, so most of the discussion of this topic is found in Chapter 4. Over water, the interaction is complicated by the fact that the wind stress causes currents, waves, and vertical mixing of the water, which affect surface temperature and evaporation.

The skillful numerical prediction of atmospheric processes of many types and scales depends on the proper representation of surface–atmosphere interactions. For example, the prediction of convection relies on the accurate calculation by the model of surface fluxes of heat and water vapor. And, direct thermal circulations on the mesoscale, forced by horizontally differential heating at the surface, can dominate the local weather and climate near coastlines and sloping orography.

Background

What is verification?

Forecast verification involves evaluating the quality of forecasts. Various methods exist to accomplish this. In all cases, the process entails comparing model-predicted variables with observations of those variables. The term validation is sometimes used instead of verification, but the intended meaning is the same. That said, the root word “valid” may imply to some that a forecast can either be valid, or invalid, whereas obviously there is a continuous scale that measures forecast quality. Thus, the term verification is preferable to many, and will be employed here. Special verification measures that are most applicable to ensemble predictions have been discussed in Chapter 7. There is an extensive body of literature on the subject of model verification, and students and researchers should read beyond the summary material in this chapter to ensure that they understand underlying statistical concepts and that they use the verification metrics that are most appropriate for their needs.

Reasons for verifying model simulations and forecasts

There are multiple motivations for evaluating the quality of model forecasts or simulations.

• Most models are under continuous development, and the only way modelers can know if routine system changes, upgrades, or bug fixes improve the forecast or simulation quality is to objectively and quantitatively calculate error statistics.

• For physical-process studies, where the model is used as a surrogate for the real atmosphere, the model solution must be objectively verified using observations, and if the observations and model solution correspond well where the observations are available, there is some confidence that one can believe the model where there are no observations.

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