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Vortex-induced vibrations of slender structures with circular cross section
A brief introduction
Flow around a smooth circular cylinder and the vortex-shedding from it can be complex. For instance, the location of minimum pressure and the minimum pressure coefficient changes with the Reynolds number. In addition, the separation point and pressure coefficient in the sep-arated region is highly Reynolds number dependent as is the behavior of the shed vortices in the wake behind the cylinder. Add motion and the flow field becomes even more complex by adding nonlinear coupling between the fluid and structural oscillation. This coupling gives rise to vortex-induced vibrations which are hard to model and not fully understood. When adding 3D dimensional effects, the flow and vortex-induced vibration becomes even more complex with correlation, coherence, end-effects and more complex modes of oscillation. This chapter is an introduction to the parameters and factors that affects the flow around cylinder like structures with circular cross-sections and vortex-induced vibrations. It will also be shown that there are important knowledge gaps later chapters will fill.
Stationary cylinders in cross flow
The Reynolds number effect
The Reynolds number is one of the most important parameters in classifying and predicting the flow field near a circular cylinder. This dimensionless parameter, given in equation (2.1), relates the inertial and viscous forces experienced by a fluid and helps determine how flows behave. For a circular cylinder, the inertial forces are ρU2, with ρ and U being the fluid density and speed respectively, and the viscous forces µU/d, with µ and d being the dynamic viscosity and diameter respectively.
Rather than being separated into two flow categories, e.g. laminar or turbuent as for pipe flow, the flow around a cylinder can be categorized into several distinct regions based on the Reynolds number. This is visualized in figure 2.1 which is based on the wake and flow separation visualization and descriptions of Lienhard [9] and Blevins [6]. The difference is that figure 2.1 uses the naming convention of Szechenyi [10] which is more similar and consistent with the naming of Mach number regimes. In addition to showing how flow around a circular cylinder changes, figure 2.1 shows the changes in the mean drag coefficient with the Reynolds number (cf. the textbook of White [11] or paper of Roshko [8] for the evolution of Cd). The flow changes around a cylinder and the mean drag are intrinsically linked as the mean drag around a cylinder is dependent on the shear and pressure.
Figure 2.1: Definition of Reynolds number regions and how mean drag coefficients changes with the Reynolds number. Inspired descriptions by Roshko [8] and Lienhard [9] as cited by Blevins [6].
At very low Reynolds numbers, Re 5, the boundary layer flow around the cylinder and in the wake behind the cylinder is laminar and fully attached. As the Reynolds number is increased, up to Re ≈ 40, the flow stays laminar but separates from the cylinder as symmetric vortex pairs behind the cylinder. Beyond Reynolds numbers of 40 and up to 150, the shed vortices are alternating instead of symmetrically pairwise and are called von Kármán vortices. These shed vortices are strong and gives rise to alternating lift and drag forces. While the flow separation changes from pairwise to periodically alternating vortices, the flow over and behind the cylinder is still laminar at these Reynolds numbers. For the regions mentioned so far, the drag coefficient decreases from a value of O(102) to O(100) [6, 8].
For industrial chimneys (cf. figures 1.1 and 1.2), launch vehicles and other circular civil engineering structures, the Reynolds number will be greater than what’s described so far. Even for a wire with a 1 cm diameter, the Reynolds number will be above 600 when the wind speed is 1 m/s. The first Reynolds number range of interest, is the sub-critical Reynolds number range which is the region between first and second dashed line of figure 2.1 (Re ∈ [150, 3 • 105]). In this Reynolds number range, there’s a large change in the flow behind the cylinder. While the boundary layer flow over the cylinder stays laminar up to separation point (near θs ≈ ±80◦ from the stagnation point1), the shed vortices and wake is increasingly turbulent. The width of the vortex street behind the cylinder and strength of the shed vortices are similar to at lower Reynolds number, i.e. the width is relatively large and the shed vortices are strong and periodic [6, 8, 12]. Canonically, the mean drag converges towards a value of Cd = 1.2 in this region and lasts until the sub-critical Reynolds region ends at Re ≈ 3 • 105 [6, 10, 12–14].
Beyond Reynolds numbers of Re ≈ 3 • 105, the flow around circular cylinders changes dra-matically and enters the critical region. This region comprises of Reynolds numbers between 3 • 105 and 1-3 • 106 for circular 2D cylinders as shown in figure 2.1 (between the second and third dashed lines). In this range, the boundary layer flow starts transitioning from being fully laminar to being turbulent but can have a double separation process with the final separation being in the rear (θs ≈ ±140◦) [10, 13].
At early critical Reynolds numbers, the boundary layer flow can start as laminar, separate from the cylinder before reattach as turbulent boundary layer flow. If asymmetrically formed, these laminar separation bubbles give a mean lift. As the final separation is far in the rear, the wake is relatively narrower and more chaotic with an irregular and weak vortex-shedding signature. In addition to changing the surface flow, the mean drag coefficient has a steep decline at low critical Reynolds numbers before reaching the global minimum drag coefficient and subsequently increase. This drastic change in mean drag coefficient is why the flow change at critical Reynolds number is often referred to as the « drag crisis » [6, 8, 12, 15].
The final Reynolds number region, and the one that’s most relevant for chimneys and launch vehicles, is the super-critical Reynolds number region at Re > 1-3 • 106 (after the last dashed line in figure 2.1) [6, 10, 12, 13]. Experimental data at super-critical Reynolds numbers are scarcer than the previously described regions [16, 17]. A reason for the lack of data is the need for more complex and specialized wind-tunnels2 (e.g. highly pressurized wind tunnels [8, 16, 17, 19–22], different flow mediums [23] or high speed wind tunnels giving high Mach numbers [24]) or modified cylinders triggering an earlier boundary layer flow transition [10, 17, 25–28].
The experimental data available at super-critical Reynolds numbers do identify some general trends. Firstly, the boundary layer flow around the cylinder is fully turbulent just like the wake’s. Secondly, regular vortex-shedding is re-established at super-critical Reynolds numbers with a shedding point that starts near θs ≈ ±140◦ though it’s much more chaotic [6, 8, 15, 17, 21, 24, 29] [15]. As the Reynolds number is increased, the shedding point moves towards the front (θs ≈ ±110◦) but stays in the rear half. This change in the separation point means that the wake width is narrow at the start of the super-critical region but increases with the Reynolds number [6, 8, 12, 16, 17, 21, 24]. Like the boundary layer flow, the drag coefficient stabilizes at a fairly constant value at super-critical Reynolds numbers and this new value is lower than at sub-critical Reynolds numbers (in the range Cd ∈ [0.7, 0.75] [8, 21, 22] though other works cite a lower value cf. [13, 30, 31]).
Steady and unsteady aerodynamic forces
Pressure distribution
At Reynolds numbers of interest (Re > 3 •105), the distribution of unsteady pressure is the main source of aerodynamic forces on a cylinder; skin friction is at most 3% of the total mean drag [11, 19, 20]. To compare the pressure distributions from different experiments, it’s better to use a dimensionless pressure like the pressure coefficient Cp,θi (t) = 2(p(θi, t) − pinf ) , (2.2) with p and pinf being the static pressure on the cylinder and in the free stream respectively. From these time series, statistics such as the mean value and standard deviation (SD or subscript σ) or root-mean-square (rms) can be found. Note that the mean, SD and rms values are related through rms = √ mean2 + SD2. This means that the « rms of the fluctuating components » sometimes used in the literature is identical to the standard deviation [26, 32–40]. The mean and SD pressure coefficients are shown in figures 2.2 and 2.3 respectively. The mean pressure distributions on a smooth cylinder at several Reynolds number regions are based on the data of Achenbach [19] while standard deviation results at the sub-critical are by West and Apelt [41].
The mean pressure distribution is the major contributor to the mean drag and lift forces. Ideally, the mean pressure distribution will have bilateral/reflection symmetry around the cen-terline between 0 and 180◦ (mirror image). This should lead to a zero mean lift force but as seen in figure 2.2, the mean force isn’t always symmetric [19, 20]. Due to flow and surface imperfections, there can be a nonzero mean lift force and the lift can be significant with larger flow instabilities like separation bubbles [19–21, 24].
In potential flow theory, the flow is fully attached around the cylinder and the pressure drag zero. Due to flow separation in the rear, the pressure « flats out » (like in 2.2) and there’s a pressure imbalance in the flow direction which leads to a mean drag force. When the separation occurs further in the rear, the flat pressure value in the rear is increased which decreases the drag. This can be seen when comparing figures 2.1 and 2.2 at sub-critical (Re = 1 .0 • 105), critical (Re = 8.5 • 105) and super-critical Reynolds numbers (Re = 3.6 • 106). At Re = 2.6 • 105, there’s a transition from sub-critical to critical Reynolds numbers which leads to lower drag and later separation than at lower Reynolds numbers. The base pressure in the wake and separation angle are largest for Reynolds numbers in the critical Region and smallest at sub-critical which corroborates the separation angle/drag coefficient relationship [11, 19, 20, 22, 24, 42].
Data on the distribution of standard deviation of pressure coefficient is rarer than the mean distribution. Basu [13] explains the difference for this as « more sophisticated instrumentation and measurement techniques are required for measuring [Cl,σ and Cd,σ] ». This also explains why data on unsteady drag and lift are rarer than mean drag and Strouhal number [22, 24]. It also explains why figure 2.3, showing the distribution of standard deviation of pressure coefficient for a smooth cylinder, only contains Reynolds numbers up to 1.31 • 105. Still, the standard deviation of pressure shows that the pressure oscillations are higher at the top of the cylinder (θ ∈ [70, 100]) which should contribute to unsteady lift and a second higher oscillation region in the rear which should contribute to unsteady drag [41, 43, 44].
Calculating aerodynamic forces
As mentioned, the aerodynamic forces on a cylinder can be found using the surface pressure distribution as the effect of skin friction is relatively low at high Reynolds numbers [11, 19, 20, 26]. These forces are found by spatially integrating the pressure distributions projected in the directions of the drag or lift (aligned and perpendicular with the free stream direction respectively). For discretely measured pressure, the integral is replaced by a summation [26, 42] and the two-dimensional drag coefficients is calculated by Cd(t) = 2 p(t, θi) cos(θi)rφi, (2.3)
where K is the set of measurement locations in a 2D ring and p(t, θi) is the pressure at location i at time t. θi is the angle between location i and the reference location (making for force at the reference purely in drag direction), and rφi is the « integration length » (radius times angle
Unsteady forces
Using equations (2.3) and (2.4), the statistics of the force coefficients can be calculated similarly to the pressure’s. While the mean drag and lift have been discussed previously, but the fluctu-ating force components have not. Relative to the total force, the fluctuating drag components are of lesser importance than the lift’s. Up to 100% of the total unsteady lift (real rms) is from the SD value whereas the total unsteady drag is dominated by the mean [45, 46].
For sub-critical Reynolds numbers, 0.3 has been reported as a « classical » value for the lift coefficient’s oscillation amplitude due to vortex-shedding [6, 14] which is similar to Schewe’s results at low Reynolds numbers [16] (see figure 2.4). This value is lower than the results of Cheung and Melbourne [30], Blackburn and Melbourne [47] and the summarized results of Basu [13], Ribeiro [48] and Ruscheweyh [49]; the latter specified the SD lift coefficient as 0.7 at sub-critical Reynolds numbers.
In the review of Demartino and Ricciardelli [31], a large range of SD lift coefficients were found at sub-critical Reynolds numbers (Cl,σ ∈ [0.09, 0.5]) [8, 16, 36, 50–52]. A possible reason for the scatter in SD lift coefficient, is the method for calculating the forces: If found by inte-grating over the entire length of the cylinder, e.g. using a force balance, the phase difference between the forces (e.g. from cells of vortex-shedding) can change the total force from the 2D pressure values. Another posibility, is the influence of roughness. While the SD lift coefficient is scattered, the « classical » value reported isn’t too far off from the other references. It’s possible that the SD lift coefficient is in the range Cl,σ ∈ [0.3, 0.4] at sub-critical Reynolds numbers and that it increases towards 0.4 when it’s about to transition to critical Reynolds numbers [6, 13, 14, 16, 22].
What’s more consistent among experiments, is how the SD lift coefficient changes qualita-tively at critical and super-critical Reynolds numbers. As the Reynolds number reaches critical Reynolds numbers, ≈ 105, the SD lift coefficient drops before reaching a minimum value at a Reynolds number of 4 • 105. From this point, the SD lift coefficient slowly increases towards a value in the range Cl,σ ∈ [0.08, 0.15]; the exact value depends on the experiment in question [13,
16, 24, 30, 31, 47, 48]. The change in SD lift coefficient is, therefore, similar to how the mean drag coefficient changes with Reynolds number. The biggest difference between the mean drag and SD lift data, is the increased scatter that should be due to the lack of data and difficulty in getting accurate measurements (e.g. force-balance issues and number and precisison of unsteady pressure sensors) [13, 22, 24].
There’s even less information on SD drag coefficient than lift but the available data suggests it’s lower than the SD lift and behaves similarly to it. The exact value varies and some studies give a high SD drag coefficient (≈ 0.35) at sub-critical Reynolds numbers [30, 53] but the majority gives a low value in the range Cd,σ ∈ [0.03, 0.12] [12, 29, 31, 33, 38, 39, 44]. For all studies, the SD drag coefficient behaves like the mean drag and SD lift when increasing the Reynolds number. On a fixed cylinder, the characteristic frequency of the fluctuating drag tends to be twice the oscillation frequency of the fluctuating lift [6].
Characteristic shedding frequency
Like how the forces change with the Reynolds number, so do the characteristic vortex-shedding frequency and forced frequency. To better compare experiments, a characteristic and dimension-less shedding frequency called the Strouhal number is used; this number is defined in equation (2.5). The experimental data on the vortex-shedding frequency is more available than the un-steady forces. This is because the vortex-shedding frequency can be measured both from the applied force (e.g. unsteady pressure [54] or force-balance [22, 24]) and the wake fluctuations which is easier to measure (e.g. using Cobra probes or hot-films and wires to measure the speed fluctuations in the wake) [8, 21, 23, 25].
A representative selection of the Strouhal number for a smooth circular cylinder at sub-critical, critical and super-critical Reynolds numbers is shown in figure 2.5. The presented Strouhal number is measured using two different locations and those of Achenbach and Heinecke and Adachi [25] uses the wake whereas Zan’s [54] uses pressure and wake measurements. As mentioned, the vortex-shedding is strongly periodic at sub-critical Reynolds numbers and the bandwidth of vortex-shedding is narrow making it easy to identify. For these Reynolds numbers, the characteristic shedding frequency is consistent in the literature and is in the range St ∈ [0.18, 0.21] [5, 6, 8, 9, 12–14, 16, 21, 23–25, 31, 36, 49, 54–56].
At critical Reynolds numbers, the vortex-shedding’s bandwidth widens and more chaotic. The Strouhal numbers that are identified are higher and more diverse than at sub-critical Reynolds numbers and can be found in the range St ∈ [0.25, 0.55] as figure 2.5 shows. At super-critical Reynolds numbers, the vortex-shedding becomes easier to identify and the bandwidth becomes narrower than the critical’s. The super-critical Strouhal number has a commonality with sub-critical values in that both converge towards a constant value but the exact super-critical value varies between experiments. As shown in figure 2.5, some experiments give a Strouhal number in the range St ∈ [0 .25, 0.27] [8, 13, 21, 23–25, 36, 56] whereas others give a super-critical Strouhal number close to 0.2 [17, 54]. There is one major difference between the studies giving high and low super-critical Strouhal numbers: The tests giving the higher Strouhal numbers tend to be measured in the wake whereas the lower Strouhal numbers tend to be measured using unsteady pressure measurements.
Modified Reynolds numbers
One of the best practices for determining the loading on large civil engineering structures, like vortex-shedding, is to use boundary layer wind tunnels. As the models need to be scaled down in the wind tunnels, it’s difficult to reach super-critical Reynolds numbers and the load will be incorrect [4]. To fix this, the test conditions are normally changed so that super-critical Reynolds numbers flow on smooth cylinders can be simulated. There are several tactics to achieve this and it includes using uniformly distributed roughness elements (e.g. sandpaper or emery paper) [10, 21, 25, 57, 58], discrete roughness elements (e.g. ribs or wires) [28, 37, 48, 59], turbulence intensity and atmospheric boundary layers [40, 60]. While triggered, the resulting super-critical unsteady forces can be different from that on a smooth cylinder.
Effect of Roughness on aerodynamics
As mentioned, added surface roughness can cause the transition to critical and super-critical Reynolds numbers to occur at lower Reynolds numbers (in absolute value) [11, 13, 40]. Most studies on the effect of surface roughness on flow over a cylinder focused on uniform surface roughness and its effect on the mean drag coefficient and the Strouhal number. The sparse data on SD lift with added uniform roughness suggest that the SD lift coefficient is increased when using a rougher cylinder. This is shown in figure 2.4 using the data of van Hinsberg on a cylinder with uniform roughness of height Rr ≈ 10−3 [22, 48]. Its postulated that this is due to a stronger and stable vortex-shedding process [25, 61].
Surface roughness affects the mean drag in two ways and both can be seen in figure 2.6a. Firstly, the drag crisis and turbulent boundary layer flow (critical Reynolds numbers) is triggered at lower Reynolds numbers. Secondly, the triggered boundary layer flow has a larger mean drag coefficient at super-critical Reynolds numbers. The increase in drag is related to the thickness of the boundary layer flow: Added roughness gives a thicker boundary layer that separates earlier. This in turn decreases the base pressure (in the rear) and increases the minimum pressure on the cylinder [10, 13, 17, 19–23, 26, 27, 31, 40, 58, 62, 63]. Güven et al. [63] attributed the higher mean drag to a decreased difference between the minimum and rear pressures but a lower rear pressure acting over a larger area will increase the drag alone.
The best description of what roughness does to the Strouhal number, is that it « smooths out » the curve as a function of Reynolds number as shown in figure 2.7a. With added roughness, the higher Strouhal number « hump » at critical Reynolds numbers is reduced as is the Strouhal number at super-critical Reynolds numbers [10, 13, 17, 21, 22, 25, 54, 58, 61, 62, 64]. With the largest uniform roughness, the hump is completely smoothed out and there’s only a small increase in the Strouhal number.
In addition to changing the shedding frequency, the added roughness has a second effect: It makes the vortex-shedding more periodic and stronger at super-critical Reynolds numbers and the vortex-shedding’s frequency spectrum becomes a narrowband single peak [10, 13, 17, 21, 22, 25, 58, 61, 62, 64]. This effect is also seen at critical Reynolds numbers which could explain the smaller Strouhal number hump. Essentially, the overall effect of uniformly added roughness is to increase the mean drag coefficient, give clearer and stronger vortex-shedding peaks (in frequency domain), move the transition to and from critical Reynolds number to lower Reynolds numbers and to « smooth out » any instabilities caused during the transition [61, 64].
The most promising results at simulating higher Reynolds number flow is by Ribeiro [28, 37]. Ribeiro’s lengthwise ribs not only best reproduced the mean forces [28], but also the unsteady force spectrum [37]. Still, the ribs trigged earlier flow separation which lowered the minimum pressure coefficient and increased it in the rear; This increased the mean and fluctuating force coefficients [28, 31, 37, 48]. While it’s effective at triggering super-critical Reynolds numbers, the rib height needs to be as small or the flow will be overly distorted when compared to smooth, super-critical data. For instance, the added ribs quickly cause the point of minimum pressure to move forward when compared to a smooth cylinder. An alternative to ribs, is to use either two strategically placed roughness elements to prematurely trigger super-critical flow or to have a surface similar to that of stranded cables [31].
Nigim and Batill [65] and Perry et al. [66] explained how the attached ribs affect the surface flow. They found that the surface flow depends on the ratio of rib height to spacing. For small ratios, the surface vortices are small and becomes trapped between rib ridges while for large ratios, the surface vortices can interact with and become part of the flow over cylinders [65, 66]. The use of ribs have some limitations: Only one inflow direction can be investigated (the unsteady pressure distribution can easily become asymmetric), they need to be placed at specific intervals and there’s no simple formula for determining the required height, width and spacing.
Table of contents :
I Introduction and state of the art
1 Foreword and motivation
2 Vortex-induced vibrations of slender structures with circular cross section 5
2.1 Stationary cylinders in cross flow
2.1.1 The Reynolds number effect
2.1.2 Steady and unsteady aerodynamic forces
2.1.3 Modified Reynolds numbers
2.1.4 Changes in forces with 3D effects
2.1.5 What’s missing?
2.2 Fundamentals of vortex-induced vibrations
2.2.1 A short introduction to vibrations and aeroelasticity
2.2.2 2D model of vortex-induced vibrations
2.2.3 Lock-in of vortex-shedding
2.2.4 Amplification of forces with motion
2.3 Vortex-induced vibrations of slender structures
2.3.1 Incoming wind
2.3.2 Structural parameters
2.3.3 Derived parameters
2.3.4 Previous full-scale tests
2.3.5 What’s missing from literature
2.4 Predicting vortex-induced vibrations
2.4.1 Spectral model
2.4.2 Correlation length model
2.4.3 Simiu and Scanlan’s design approximation
2.4.4 Coupled wake oscillator models
2.4.5 How to improve predictions of vortex-induced vibrations
2.5 Original contributions of this work
II 2D wind tunnel experiments on stationary cylinders
3 Wind tunnel results up to super-critical Reynolds numbers
3.1 Background
3.2 Experimental methodology
3.2.1 Wind tunnel description
3.2.2 The test model and setup
3.2.3 Roughness elements
3.2.4 Instrumentations and measurements
3.2.5 Tested wind speeds
3.2.6 Characteristic dimensions
3.3 Aerodynamic forces
3.3.1 Unsteady lift and drag forces
3.3.2 Correlation and coherence
3.4 Unsteady pressure distributions
3.4.1 Bi-orthogonal decomposition
3.4.2 Smooth cylinder’s topos and Reynolds number effect
3.4.3 The effect of roughness and simulated super-critical Reynolds numbers
3.5 Wake and lift Strouhal number
3.6 Summary of large-scale results
4 Simulation of super-critical Reynolds numbers in a smaller wind tunnel
4.1 Background
4.2 Experimental methodology
4.2.1 Wind tunnel description
4.2.2 The test model and setup
4.2.3 Roughness elements
4.2.4 Instrumentations and measurements
4.2.5 Tested wind speeds
4.3 Aerodynamic forces
4.3.1 Strouhal number
4.3.2 Unsteady lift and drag coefficients
4.3.3 Correlation and coherence
4.4 Simulated super-critical unsteady pressure distributions
4.4.1 Topos and eigenvalue changes with Reynolds number
4.4.2 Spatial energy distribution at simulated super-critical Reynolds numbers
4.4.3 MAC comparison of topos
4.4.4 Eigenvalue comparison
4.4.5 Chronos comparison
4.4.6 Relative forces from BOD pairs
4.5 Best configuration for simulating super-critical Reynolds numbers at small-scale
4.6 Summary of small-scale results
III Aeroelastic experiments
5 Small-scale aeroelastic wind tunnel experiments
5.1 Background
5.2 Experimental methodology
5.2.1 The generated atmospheric wind
5.2.2 Test model and setup
5.2.3 The first mode shape
5.2.4 Model configurations and parameters
5.2.5 Instrumentation and measurements
5.2.6 Analysis of displacement and acceleration
5.3 Wake and response results
5.3.1 Strouhal number
5.3.2 Structural response
5.3.3 Correlation and coherence
5.4 Summary of small-scale aeroelastic results
6 Field tests: The Bouin chimney
6.1 Background
6.2 Chimney design and methodology
6.2.1 Structural characteristics of the chimney
6.2.2 Field-test location and instrumentation
6.2.3 Data analysis process
6.3 Wind characteristics
6.3.1 Distribution of wind speed and direction
6.3.2 Mean incoming wind profiles
6.4 Cross-wind vibrations
6.4.1 Amplitude response
6.4.2 Frequency of motion
6.4.3 Directional response statistics
6.5 Summary of field-test results
IV Predicting the response due to vortex-induced vibrations
7 Mathematical modeling of response
7.1 Background
7.2 A new amplitude approximation
7.2.1 Approximation of amplitudes and phase
7.2.2 Stability of solutions
7.3 Predicting when vortex-induced vibrations occurs
7.3.1 Defining the lock-in regions
7.3.2 Changes in lock-in region with mass-ratio and damping
7.4 Predicted maximum response
7.5 An approximate summary
8 Predictions compared with experimental results
8.1 Background
8.2 Model parameters and versions used
8.2.1 Design code models
8.2.2 Wake oscillator modification
8.3 Predicting the response from the small-scale wind tunnel
8.3.1 Model configurations and parameters
8.3.2 Structural response
8.4 Predicting the field test’s amplitude response
8.4.1 Aerodynamic and structural parameters
8.4.2 Comparison of amplitude response
8.5 Predicting maximum response
8.6 Which model to use?
V Outlooks and conclusion
9 General conclusions
9.1 2D wind tunnel experiments on stationary cylinders
9.2 Aeroelastic experiments
9.3 Predicting the response due to vortex-induced vibrations
9.4 Perspectives
VI Appendix
A Mathematical tools
A.1 Transforming acceleration to response
A.2 Hilbert transform
A.3 Modal assurance criterion
A.4 Bayesian inference
A.5 Analytic approximation of deterministic equations
B Additional 2D results 135
B.1 Ergodicity of unsteady pressure
B.2 Strouhal number
B.2.1 Strouhal number comparison
B.2.2 Strouhal comparison using single pressure tap and vortex-lift’s chronos
B.3 Unsteady forces
B.4 Super-critical Reynolds numbers
B.5 Correlation and coherence
B.5.1 Correlation
B.5.2 Coherence
B.6 Topos comparison using MAC
B.7 Relative forces
B.8 Inference results
C Additional data for the 3D wind tunnel experiment
C.0.1 The first mode shape
D Chimneys compared
E Publications
F Changes suggested by the jury
G Résumé en français
G.1 L’état de l’art
G.1.1 Écoulement 2D et 3D
G.1.2 Réponse aéroélectrique
G.1.3 Objectifs de la thèse
G.2 Expériences en soufflerie 2D sur des cylindres stationnaires
G.3 Expériences aéroélectriques
G.4 Prédiction de la réponse due aux vibrations induites par les vortex
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