Dynamic simulation of the fluidelastic instability in a single cylinder of an in-line tube array

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Vibration Mechanisms

The fluid flow through the tube arrays is fairly complex. There exists several excitation mech-anisms leading to flow-induced vibrations in tube arrays. Broadly, these are classified as, Vortex shedding, Turbulent buffeting, Acoustic resonance and Fluidelastic excitations. In ad-dition, there are other mechanisms which may simultaneously exist, such as the parallel flow turbulent pressure fluctuations, hydraulic noise, structural noise transmitted by the external vibrations, fluctuations in the flow etc.

Vortex shedding / Strouhal periodicity

The fluid flow past a bluff body generates the low pressure alternate vortices behind the body, known as the vortex shedding. The flow past a circular cylinder has been extensively studied physics since a long time. The shedding of vortices results in lift forces in addition to drag forces on the cylinder. The flow depends mainly on the Reynolds number (Re), which is a ratio of fluid inertia forces to viscous forces. The frequency of vortex shedding is normalised by using the flow velocity and cylinder diameter. The resulting dimensionless number is called the Strouhal Number (Sh). Generally the relationships between the forces (drag, lift), Strouhal frequency and the Reynolds number are established empirically. The drag and lift forces beating at Strouhal frequency induce cylinder vibrations in the cylinder. Further, if the Strouhal frequency synchronises with the cylinder’s natural frequency, the cylinder resonates and produces large amplitude vibrations, know as vortex-induced vibrations.
The flow through tube arrays forms multiple flow channels passing around the lines (columns) of cylinders. The presence of Strouhal periodicity in tube arrays also results in high amplitude vibrations, when it resonates with the cylinders’ natural frequency. The flow instabilities in the tube arrays are largely disputed for the presence of distinct vortex shedding similar to the classical vortex shedding behind a single cylinder. The topic is well reviewed by (Païdoussis, 1983b) and (Weaver and Fitzpatrick, 1988b). Until the early 70’s, only the excitations due to vortex shedding were held responsible for the vibrations in tube bundles. The dispute over unanimous values of the Strouhal frequency and its existence in tube arrays led to further in-vestigations of the phenomenon. While summarizing the research work on the existence of vortex shedding in tube arrays, (Païdoussis, 1983b) stated that, the Strouhal periodicity com-monly appeared for the first few rows, provided the upstream turbulence had not suppressed it. The appearance of vortex shedding deep within the arrays is found to be dependent on the Reynolds number, array geometry, mechanical properties of the tubes and also the amplitudes of tube vibration. In the later research works, (Weaver et al., 1987), (Fitzpatrick et al., 1988) observed the presence of vortex shedding, if not a flow periodicity even deep within the arrays. In an interesting flow visualization study by (Abd-Rabbo and Weaver, 1986), the development of vortex shedding in arrays is observed similar to the classical vortex shedding behind a single cylinder. The Strouhal periodicity is observed in the closely placed staggered array configurations, which is reported to be absent in the in-line array configurations. Further study on the in-line tube arrays by (Ziada and Oengören, 1992) and (Ziada and Oengören, 1993) has enhanced the understanding of the generation of vortex shedding excitations. The fluid flowing in lanes forms the jet like structures, while the wakes of the cylinders are con-fined. The vortex-shedding excitations are generated in the first row due to the jet instabilities and persist for couple of rows downstream. The tube pitch ratio and the upstream turbulence has a major impact on the vortex-shedding in the front rows as well as deep in the arrays. The vortex shedding in the staggered (normal triangular) arrangement is studied in (Polak and Weaver, 1995) for various pitch ratios and Reynolds numbers. In a comprehensive work by (Ziada, 2006), the vortex-shedding is shown to be generated by either jet, wake or shear layer instabilities, depending on the tube spacing, upstream turbulence, Reynolds number and the array configuration.
(a) Staggered array (b) In-line array
Figure 1.2 Vortex-shedding in tube arrays. Source: (Ziada, 2006)
Figure (1.2) shows vortex shedding patterns in a staggered and in-line tube array arrange-ments. The sub-figures in Figure 1.2(a) show the influence of increasing Reynolds number on the flow vorticity. The closely placed cylinders in-line array shown in Figure 1.2(b) shows the shear layer instabilities as the source of Strouhal periodicities.
A review on the cross flow induced vibrations and theoretical models of the fluidelastic instability in tube arrays

Turbulent buffeting

The flow in heat exchanger tube arrays is turbulent in almost all industrial configurations. In addition to vortex-shedding / Strouhal periodicity, instabilities due to the flow turbulence are present separately in the flow. They are present in a wide range of frequencies of the flow spectra. If the cylinder natural frequency is in the range of turbulence frequencies, the cylinder is fed with more energy, resulting in high amplitudes of cylinder vibration. The historical development on the topic is provided in the review articles by (Païdoussis, 1983a) and (Weaver and Fitzpatrick, 1988b). The very discussed article by (Owen, 1965) questioned the presence of vortex shedding in the confined arrays. The excitations are attributed to the range of turbulence frequencies and a dominant frequency from the range. The development by (Chen, 1968), is based on the similar concept, but the dominant frequency is used as a vortex shedding frequency. The experimental work by (Weaver and Yeung, 1984) showed the presence of both, the vortex periodicity and the range of turbulence frequencies. Figure (1.3) shows separated peaks for the vortex periodicity and the turbulence range of the spectrum. The experiments were performed in a water tunnel to study the influence of mass ratio on vibrations in all four standard tube bundles with a pitch ratio 1.5. The figure shows a frequency response spectrum for a rotated square array of aluminium tubes and the water flow with an upstream Reynolds number ≈ 1500.
Figure 1.3 Distinct spectral peaks of the vortex shedding and turbulence. Source: (Weaver and Yeung, 1984)
In order to predict the cylinder’s vibrational response to the turbulent buffeting, (Pettigrew and Gorman, 1978) estimated the turbulence spanwise correlations. The random correlation coefficients are used to obtain the power spectral density and thereby the cylinder vibration. On a similar ground, (Blevins et al., 1981) derived an equally simple relation to predict the vibrational response of a cylinder, in a non dimensional form. Figures 1.4(a), 1.4(b) show the plots of random correlation coefficients by (Pettigrew and Gorman, 1978) and (Blevins et al., 1981) respectively. The main difference between Figures 1.4(a) and 1.4(b), as also addressed by (Païdoussis, 1983b), is the difference between the stability limits for the first row or upstream cylinders and the downstream cylinders, which is attributed to the upstream flow turbulence.
In the recent studies on the influence of upstream flow turbulence on the tube array insta-bilities, (Romberg and Popp, 1998), (Popp and Romberg, 1998) (Rottmann and Popp, 2003) found that, the increased upstream turbulence has a stabilizing effect on the cylinders response. The increase of upstream turbulence intensity resulted in a shift in the instability boundaries of critical velocities to higher values.

Acoustic resonance

The acoustic resonance occurs when the acoustic natural frequencies of the tube bundle shell resonate with the flow periodicity in the tube array. The heat exchanger shells with water like fluids and with compact structures are less susceptible to acoustic resonance, since the speed of sound is relatively high in liquids. On the other hand, the gaseous flows through the tube arrays can undergo acoustic vibration, which may result in an intense magnitude noise. The phenomenon is briefly reviewed in (Païdoussis, 1983b). Similar to the very vortex shedding and the turbulent buffeting phenomena, the acoustic resonance is also disputed over the source of acoustic excitations, which is discussed in (Weaver and Fitzpatrick, 1988b). An experimen-tal study by (Parker, 1978) showed that the effective speed of the sound reduces because of the presence of tube arrays. Furthermore the acoustic resonance is analysed and compared with the experiments. In (Blevins, 1984), Blevins (1986), it is reported that the vortex shedding acted as a dipole source of the sound. A theoretical development based on Lighthill’s analogy is postulated. The acoustic modes of heat exchangers are identified using the finite element technique and compared with experimental data. The acoustic resonance strongly modified the vortex shedding and increased the strength and correlation of the vorticity. In (Fitzpatrick, 1985), the source of acoustic resonance is assumed to be the vortex shedding as well as the turbulent buffeting, in order to predict the flow induced noise. The various methods for pre-dicting the acoustic resonance are compared and design guidelines are formulated in (Ziada and Oengören, 2000) to avoid the acoustic resonance.
Figure 1.5 Illustration of the vortex shedding under the influence of acoustic resonance. (a) with resonance (b) without resonance. Source: (Ziada and Oengören, 1992) the influence of acoustic resonance. In general, the flow instabilities behind the cylinders are accounted as the excitation source of the acoustic modes in the system. Figure (1.5) shows a schematic illustration of the influence of acoustic resonance (lock-in) on the vortex shedding behind the cylinders. In a recent article, (Ziada, 2006) provided an improved guidelines for the acoustic resonance in tube arrays, detailing its dependence on the tube layout pattern, pitch ratio and Reynolds number. The influence of the acoustic sound field on the lift force in terms of the change in vortex shedding at the onset of acoustic resonance is further studied in (Hanson and Ziada, 2011).

Fluidelastic excitations

The galloping phenomenon of the ice-laden transmission wires due to the wind and the flutter in the aircraft wings are closely linked with the fluidelastic instability in tube arrays. Both the galloping and the flutter instabilities involve either an asymmetry in the associated geometry or/and the interaction with an incident flow with dynamically changing the angle of attack. The fluidelastic instability in tube arrays is also classified under the Movement Induced Vibrations (MIV), since a small movement of the structure (tubes) acts as an origin of the fluidelastic excitations. A small vibration in a tube array excited by the interstitial flow modifies the flow itself and thereby the fluid forces acting on the tubes, which results in the increased vibration, until the array becomes completely unstable. The damage caused by the fluidelastic instability is within a short term and devastating (Weaver and Fitzpatrick, 1988b). Figure (1.6) shows an idealized response of a cylinder from an array under the different excitations mechanisms (Païdoussis, 1983b).
Until the 60’s, the flow-induced vibrations due to the fluidelastic instability were attributed to the vortex shedding resonance. In its PhD work (Roberts, 1962), discovered the self-excitation mechanism in a staggered row of cylinders under cross flow, for the vibration in inflow direction. A theoretical development for predicting the critical flow velocity is pro-posed, which is based on a jet switching phenomenon (Roberts, 1966). The flow jet, which is formed due to the cylinders arrangement, oscillates at a relatively lower frequency than the cylinder natural frequency. Later, in the breakthrough work (Connors, 1970), (Connors, 1978) proposed a relatively simple dependency of the critical flow velocity on the structural parame-ters, i.e. on the mass-damping parameter. The model is developed using a single in-line row of cylinders instead of the staggered arrangement in (Roberts, 1962). (Blevins, 1974) extended the model of (Connors, 1970) for the tube arrays. Although the extended model incorporated the changing damping with respect to the flow velocity, the dependence on the mass-damping parameter remained in the same form. Equation (1.1) is the general form of the fluidelastic instability criteria. upc is the effective critical flow velocity (also called as gap or pitch or intertube velocity). fn, m and δ represent the cylinder natural frequency, mass of cylinder per unit length and the logarithmic decrement of the cylinder free vibration decrease respectively. Where as D stands for the cylinder diameter. The mechanical variables ( fn, m and δ ) can be defined in several ways, but generally they are defined with respect to the fluid medium at rest. K is the constant of proportionality. The exponent a is often taken as 0.5.
upc mδ a = K (1.1) fnD ρD2
A detailed historical development on the topic is reviewed in (Païdoussis, 1983b). The de-pendence of the critical velocity on the mass-damping parameter is not much disputed, while the value exponent a and the proportionality constant (K) varies depending on configurations. Some of the earlier works is concerned about obtaining an appropriate value of the constant K, e.g. (Pettigrew and Gorman, 1973). (Gibert et al., 1977), (Pettigrew et al., 1978), (Blevins, 1979a). The value of Kmin = 3.3 from the work of (Pettigrew et al., 1978) is widely accepted.
In further investigations for the value of the constant K and in general for the validity of the model (Equation (1.1)) for a wide range of mass-damping parameters, experimental results showed a wide scatter instead of a single instability boundary as per the Equation (1.1). The experimental work of (Weaver and Lever, 1977) and (Southworth and Zdravkovich, 1975) showed that a single cylinder from an array can become fluidelastically unstable. In the experiments of (Gorman, 1977) the effect of stiffness on the stability limit is tested. In addition to the study on whether the open tube lanes serve as a trigger for the fluidelastic instability is carried out. The results are contradicting with the formulation of (Connors, 1978) (Equation 1.1). Further, in (Weaver and Grover, 1978) the dependence of critical velocity only on the logarithmic decrement is tested, which resulted in a different value of the exponent a for the δ (a ≈ 0.21). There are many experimental evidences reported against grouping the mass-ratio (m/ρD2) and logarithmic decrement (δ ) together. (Weaver and El-Kashlan, 1981) studied the effect of mass ratio (m∗ = m/ρD2) and damping ratio (ζ ) on the instability limits and found different values (a = 0.29, b = 0.21 respectively for m∗ and δ ) of the exponents, in a single oscillating cylinder from a rotated/parallel triangular array configuration. In similar studies, (Nicolet et al., 1976), (Heilker and Vincent, 1981), (Chen and Jendrzejcyk, 1981), (Price and Païdoussis, 1989), (Price and Kuran, 1991) obtained different set of values for the exponents in a single as well as multiple oscillating cylinders configurations. (Tanaka et al., 2002) suggested that for the mass ratio m∗ = m/ρD2 smaller than 10, mass and damping should be treated separately.
Other set of experiments evidenced the presence of multiple stability limits for a single as well as multiple cylinders from an array, e.g. (Chen and Jendrzejczyk, 1983), (Andjelic and Popp, 1989) and (Austermann and Popp, 1995). The multiple stability boundaries are gener-ally observed for lower values of the mass-damping parameter. This feature of the instability does not reflect in the model of (Connors, 1978) (Equation 1.1). Figure (1.7) shows the mul-tiple stability regions for a single cylinder form a triangular normal array, in the multiple sets of experiments performed by (Andjelic and Popp, 1989).
In attempts to model the fluidelastic instability, several theories have been developed in the past couple of decades, in addition to the pioneering work of (Roberts, 1966) and (Connors, 1978). The parameter space being large, none of the theories holds good in all circumstances. The behaviour of the instability largely changes for different values of the mass-damping pa-rameter. In (Chen, 1983a), the instability is further attributed to the different mechanisms in place, namely, fluidelastic stiffness controlled mechanism and damping controlled mecha-nism. It is recommended to use difference stability criteria for the different parameter range (Chen, 1983b). In the damping controlled mechanism, the fluid forces act in phase with the velocity of cylinder vibration and there exists a phase lag between the displacement of cylin-der and the fluid forces acting on it. The dependence of the instability on phase lag is not fully understood. The finite fluid inertia model by (Lever and Weaver, 1986a) (Lever and Weaver, 1986b), the fluid flow hysteresis effect model by (Price and Païdoussis, 1984), (Granger and Païdoussis, 1996) consider the phase or finite time lag between the fluid forces and the cylin-der displacement. In addition to the very clear presence of the multiple stability limits and the independence of the mass ratio and damping ratio in the formation of the mass-damping pa-rameter, there are other parameters that greatly affect the fluidelastic stability limits, namely, orientation of the array, transverse and longitudinal pitch ratios, Reynolds number, position of the cylinder in an array etc. (Chen and Jendrzejcyk, 1981). Further the authors found that the fluidelastic instability may interact with the other excitation mechanisms (e.g. Strouhal periodicity/ Vortex shedding). The flow through tube arrays in steam generators is normally two-phase flow, an additional parameter that influences the stability limits, which can be a separate topic of research in itself and not considered in this study.

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The fluidelastic instability models

The mathematical models of fluidelastic instability can be categorized based on the physi-cal assumptions and methodology used to derive it, namely, jet switch models, quasi-static models, quasi-steady models, semi-analytical models and unsteady models.

Jet switch model

The very first model developed for the fluidelastic instability by (Roberts, 1962) is for a single and a double rows of cylinders in cross flow. It is derived for the instability in the in-flow direction. A slight staggering in the cylinder rows is necessary in developing the wake dy-namics and the jet switching instability. Figure 1.8(a) shows the schematic of the jet switch mechanism. The sudden expansion of the flow jet below the separation locations makes it highly transient in nature, while the asymmetry in the geometry creates one small and an-other big wakes downstream the cylinders. The in-flow oscillations of cylinders favour the jet switching by extracting the energy form the fluidelastic switching of the jet. The author further suggests that the jet switching is possible only if the frequency of jet switching is lower than the cylinder frequency. Further, the occurrence of jet switching is constrained to u/ωnD ≥ 2 condition. Where u, ωn, D represent the inflow velocity, cylinder angular frequency and its diameter respectively.
The in-flow fluid force as a function of the jet switching mechanism is written as, Fx = 1 ρu2 0.717 1 −Cpb(x, τ) −2 ωnD (1 −Cpb)mean dx (1.2) 2 u dτ Where, Cpb represents the theoretical base pressure for the two adjacent cylinders, which is a function of both the displacement (x) and the dimensionless time τ = (tωn). The solution of the equation of motion by using the in-flow force in Equation (1.2) leads to the stability boundaries as shown in Figure 1.8(b). The simplified form of the solution by neglecting the damping and unsteady terms from the force Fx leads to, uc mδ 0.5 = K (1.3) ωnεD ρD2
Where ε represents the ratio of fluidelastic (jet switching) frequency to the cylinders fre-quency. The symbols, m, ρ and δ stand for the mass per unit length, density of fluid and logarithmic decrement of the cylinder vibration respectively. K is the proportionality con-stant.
The model shows a poor agreement with the experimental results for the cylinder rows in cross flow (Païdoussis et al., 2010, chap. 5). The model is developed for the motion of cylin-ders in the in-flow direction, while as the experiments show the dominance of the fluidelastic instability in the flow normal direction. In the theory, instability can not occur if uc/ωnεD ≤ 2.

Quasi-static, quasi-steady models

Similar to Equation (1.3), (Connors, 1970) developed a quasi-static model for the fluidelas-tic instability in a row of cylinders under cross-flow. The forces (drag and lift) are directly measured by performing experiments. Mainly two types of (whirling) patterns of the adjacent cylinders from a row with pitch ratio p∗ = 1.41 are observed during real experiments, which are either symmetric or anti-symmetric. The force coefficients are obtained by simulating the dominant patterns of the cylinder oscillations using the static displacement of the adjacent cylinders. In the analysis, the instability found to be more dependent on the transverse direc-tion. Also the effect of jet switching on the instability is found to be negligible. The cross-flow and inflow energy balances based on the measurements led to, upc mδ 0.5 = K (1.4) fnD ρD2
Where, upc is the critical pitch velocity, the value of constant K = 9.9. In an extension of the model (Equation (1.4)) to the tube arrays, the value of K = 3.3 is recommended in (Gorman, 1976) and (Pettigrew et al., 1978). Later, in (Connors, 1978), provided a relation for the value of constant K, in terms of the transverse pitch ratio (p∗y = T /D) as, K = (0.37 + 1.76p∗y) for 1.41 < p∗y < 2.12. On the same ground, (Blevins, 1974) derived a relation similar to Equation (1.4). The cylinders from a row as well as from an array are assumed to whirl in interdependent orbits, mainly, in out of phase to the adjacent cylinders. The quasi-state formulation resulted in, upc 2(2π)0.5 mδ 0.5 = (1.5) fnD ( x y)0.25 ρD2 C K
Where, Cx = ∂Cx/∂ y and Ky = ∂ Ky/∂ x are the fluid-stiffness terms, provided in the formula-tion of (Connors, 1970).
In the quasi-steady approach of modeling the interaction between cylinder in motion and fluid around, the instantaneous velocity vector is repositioned for the lift and drag direction, while the effective values of the coefficients are kept the same as in the stationary case. Figure 1.9(a) shows the relative change of the drag and lift directions on a cylinder with a change in the inflow velocity direction. (Blevins, 1979c) used a quasi-steady approach in order to incor-porate the effect of the fluid dependent damping in the relation. The logarithmic decrement of the (Connors, 1978) model is given by, δ = 2π ζxζy (1.6)
where, ζx and ζy are the damping factors in the inflow x and the cross flow y directions respec-tively. A considerable work was done by (Whiston and Thomas, 1982), in order to take into account the phase angle between the adjacent cylinders and the details of the intertube flow physics.
(Gross, 1975) used the quasi-steady analysis for the tube arrays, stating that, the instability occurs due to two distinct mechanisms, namely, negative-damping and stiffness controlled. The fluid force is taken proportional to the variation of the lift coefficient linearly with the relative angle between the incident velocity and cylinder position, as shown in Figure 1.9(a). The cylinder is expected to be unstable when the effective damping becomes zero. The model is given as, upc = mδ (1.7) fnD ρD2(−∂C/∂α)
In contrast to the experimental evidences (e.g. (Chen, 1984), (Weaver and Fitzpatrick, 1988b)) the dependence of the critical dimensionless velocity on the mass-damping parameter is linear, against the low values (less than one) of the exponent (a) in the experiments. In the similar analysis, (Price and Païdoussis, 1983) developed a quasi-steady relation between the force and the displacement of a cylinder from the two rows of cylinders. Further, they improved the model by incorporating the dependency of the fluid coefficients on the displacement of the surrounding cylinders by using a constrained modal analysis (Price and Païdoussis, 1986a) as well as importantly, by adding the flow retardation effect in the model (Price and Païdoussis, 1984), (Price and Païdoussis, 1986b). It is suggested that, the instability for the higher values of mass-damping parameter (mδ /ρD2) is stiffness-controlled, on the other hand it is damping controlled for the smaller values of the mass-damping parameter. In an analysis of a single cylinder from an array, the instability is attributed to the negative effective damping of the cylinder, which resulted due to the flow retardation or the phase lag effect. The equation proposed is, upc = 4mδ /ρD2 (1.8) fnD (−CD − µD ∂CL/∂ y)
where, CD, CL are drag and lift coefficients, while µ is the flow retardation parameter. The resulted instability map for different values of the flow retardation is shown in Figure 1.9(b).

Semi-analytical models

An analytical approach is adopted in (Lever and Weaver, 1982), in order to model the fluide-lastic instability for a single cylinder in an array. It is concluded based on the experimental studies in (Weaver and Grover, 1978) and (Lever and Weaver, 1982) that the instability in a single cylinder is representative for an array since the value of the critical flow velocity remains more of less the same. In addition, the model is developed for the cross flow instabilities only. In the following improvements, (Lever and Weaver, 1986a), (Lever and Weaver, 1986b) incor-porated the cylinder motion in the in-flow direction as well, although treated independently. In the model, the authors considered the interactions of the adjacent flow channels with the cylinder displacement. The flow channels are assumed to be inviscid with a resistance term for the frictional losses. The area of the inviscid stream-tubes is preserved, if it is modified by the movements of the cylinder. The dynamic interaction between the flow tubes and the cylinder displacement resulted in a finite phase lag between the two, due to the fluid inertia. An empirical relation is used to model the phase lag. Figure 1.10(a) shows the idealized flow channels adjacent to the cylinder under consideration.

Table of contents :

1 A review on the cross flow induced vibrations and theoretical models of the fluidelastic instability in tube arrays 
1.1 Introduction
1.2 Vibration Mechanisms
1.2.1 Vortex shedding / Strouhal periodicity
1.2.2 Turbulent buffeting
1.2.3 Acoustic resonance
1.2.4 Fluidelastic excitations
1.3 The fluidelastic instability models
1.3.1 Jet switch model
1.3.2 Quasi-static, quasi-steady models
1.3.3 Semi-analytical models
1.3.4 Unsteady models
1.4 The parameter space and definitions
1.4.1 Array orientation
1.4.2 Natural frequency
1.4.3 Mass of the tube
1.4.4 Damping
1.4.5 The critical flow velocity
1.5 Conclusion
2 Numerical simulation of the flow in tube arrays by using the Unsteady Reynolds Averaged Navier-Stokes turbulence modeling 
2.1 Introduction
2.2 Surface pressure profiles in triangular arrays
2.2.1 Experimental and numerical configurations
2.2.2 Results and discussion
2.3 Dynamic simulation of the fluidelastic instability in a single cylinder of an in-line tube array
2.3.1 Experimental and numerical configurations
2.3.2 Results comparison and discussion
2.4 Conclusion
3 Analysis of the fluidelastic instability by using the Large Eddy Simulations 
3.1 Introduction
3.2 Configuration
3.2.1 Experiments
3.2.2 Large Eddy Simulations (LES)
3.2.3 Fluid-structure coupling
3.3 Results comparison
3.4 Flow analysis
3.5 The onset of fluidelastic instability
3.5.1 Comparison between the static and dynamic case simulations
3.5.2 Dynamics of the fluid forces acting on the cylinder
3.6 Conclusion
4 A theoretical model of the fluidelastic instability in square inline tube arrays 
4.1 Introduction
4.2 Theory
4.2.1 Mathematical Model
4.2.2 Estimation of the critical flow velocity
4.3 Model predictions of experimental results
4.4 Conclusion
5 Introduction to Reduced-Order Modeling 
5.1 Introduction
5.2 Preliminary definitions
5.2.1 Dynamical systems
5.2.2 Transfer functions
5.2.3 Controllability and Observability Gramians
5.2.4 Stability and Passivity
5.2.5 Subspace projections
5.2.6 Hankel singular values
5.3 Model order reduction techniques
5.3.1 Truncated Balanced Realization
5.3.2 Krylov subspaces
5.3.3 Proper Orthogonal Decomposition
5.4 Conclusion
6 A Galerkin-free model reduction approach for the Navier-Stokes equations 
6.1 Introduction
6.2 Mathematical formulation
6.2.1 Method of snapshots POD
6.2.2 Periodicity of POD temporal modes
6.2.3 Linear interpolation
6.2.4 A posteriori error estimate
6.2.5 Stability of the interpolation ROM
6.3 Flow past a cylinder at low Reynolds number – a case study
6.3.1 Governing flow equations and numerical methods
6.3.2 Results and discussion
6.4 Conclusion
7 Model reduction of fluid-structure interactions by using the Galerkin-free POD approach 
7.1 Introduction
7.2 Mathematical formulation
7.2.1 The Snapshots POD
7.2.2 The POD time modes (Chronos)
7.2.3 Linear interpolation
7.2.4 Error estimate
7.3 Vortex induced vibration of a cylinder at Re = 100 for various mass ratios
7.3.1 The flow equations
7.3.2 Fluid-structure coupling
7.3.3 POD analysis
7.3.4 ROM solution states
7.4 Conclusion
8 Conclusions and outlook 
8.1 Conclusions
8.2 Outlook
Appendix A Turbulence modeling
A.1 Unsteady Reynolds Averaged Navier-Stokes (URANS)
A.1.1 Linear eddy viscosity models
A.1.2 Non-linear eddy viscosity models
A.2 Large Eddy Simulations (LES)
A.2.1 Smagorinsky-Lilly Model
Appendix B Modal analysis 163
B.1 Half-Power Bandwidth Method (HBM)
B.2 Time Domain Modal Analysis (TMA)
B.2.1 The characteristic functions
B.2.2 The number of modes and parameters of the characteristics functions
B.2.3 Statistical estimation of the modal parameters
References 

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