Empirical Methods Numerical Analysis Wind Dynamic Assessment Methods for Medium-span Bridges A comprehensive review of empirical and numerical approaches Master’s thesis in Master Programme Structural Engineering and Building Technology LUKAS EHN SVEN LUNDELL DEPARTMENT OF MECHANICS AND MARITIME SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se www.chalmers.se Master’s thesis 2021:52 Wind Dynamic Assessment Methods for Medium-span Bridges A comprehensive review of empirical and numerical approaches LUKAS EHN SVEN LUNDELL Department of Mechanics and Maritime Sciences Division of Dynamics Chalmers University of Technology Gothenburg, Sweden 2021 Wind Dynamic Assessment Methods for Medium-span Bridges A comprehensive review of empirical and numerical approaches LUKAS EHN SVEN LUNDELL © LUKAS EHN, SVEN LUNDELL, 2021. Supervisor: Christoffer Svedholm, Structural Engineer at ELU Supervisor: Per Söderström, Structural Engineer at ELU Examiner: Professor Thomas Abrahamsson, Department of Mechanics and Maritime Sciences Master’s Thesis 2021:52 Department of Mechanics and Maritime Sciences Division of Dynamics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: To the left is a flowchart from the quick reference guide in Appendix A, presented in Chapter 4. To the right is a CFD simulation of the vortex shedding around a rectangle and graphs of the corresponding data from Chapter 7. Typeset in LATEX, template by Magnus Gustaver Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iv Wind Dynamic Assessment Methods for Medium-span Bridges A comprehensive review of empirical and numerical approaches LUKAS EHN SVEN LUNDELL Department of Mechanics and Maritime Sciences Chalmers University of Technology Abstract Generally, bridge engineers are unfamiliar with wind dynamics as it falls in-between the fields of structural engineering and fluid dynamics. Therefore, there is a need to summarize the field in a digestible manner. Procedures for wind dynamic as- sessments of medium-span bridges (e.g. bridges with longest spans of 50 to 200 metres) are investigated by studying both the current norm in Sweden, and an in- ternational alternative. A quick reference guide for wind dynamic assessment is developed, simplifying the procedure for bridge engineers. It offers significant time savings, especially in early stages of design, and it can prevent unexpected issues in later stages. However, to verify its reliability large scale testing on bridges is recommended. Additionally, possibilities of further analysis using computational fluid dynamics is investigated. Simulation data show some promising results and with further development, the methodology could provide better estimations than the norm. Conclusively, two useful tools for wind dynamic assessment of bridges are developed, and with further work, application in practice is possible for both methods. Keywords: Aeroelastic instability, Bridge engineering, Computational fluid dynam- ics, Detached-eddy simulation, Eurocode, OpenFOAM, Strouhal number, Structural Engineering, Vortex induced vibrations, Vortex shedding. v Acknowledgements We wish to acknowledge the effort of all those involved in the making of this thesis. Firstly, we would like to thank ELU Konsult AB, who proposed the subject of this thesis, and for providing guidance and support. We would like to especially extend our gratitude to our supervisors Christoffer Svedholm and Per Söderström, structural engineers at ELU, attending countless meetings and giving invaluable opinions and insights based on their experience as bridge engineers. We would also like to thank our examiner Thomas Abrahamsson, professor in Struc- tural Dynamics at the Department of Mechanics and Maritime Sciences at Chalmers University of Technology, for his encouragement and willingness to aid us through the entire process. Lastly, we would like to extend our gratitude to Håkan Nilsson, professor in Fluid Dynamics at the Department of Mechanics and Maritime Sciences at Chalmers Uni- versity of Technology, for answering questions regarding CFD which our supervisors and examiner were unfamiliar with. Lukas Ehn Sven Lundell Gothenburg, June 2021 vii Contents List of Figures xiii List of Tables xv 1 Introduction 1 1.1 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 I Wind Dynamics of Bridges; Theory and Norms 5 2 Wind Dynamic Phenomena 7 2.1 Aerodynamic Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Buffeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Aeroelastic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Galloping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Torsional Divergence . . . . . . . . . . . . . . . . . . . . . . . 20 3 Current Norms for Wind Dynamic Assessment of Bridges 23 3.1 Buffeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 Empirical Method in Eurocode . . . . . . . . . . . . . . . . . 24 3.2.2 Empirical Method in British Annex . . . . . . . . . . . . . . . 25 3.3 Galloping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Empirical Method in Eurocode . . . . . . . . . . . . . . . . . 26 3.3.2 Empirical Method in British Annex . . . . . . . . . . . . . . . 26 3.4 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.1 Empirical Method in Eurocode . . . . . . . . . . . . . . . . . 28 3.4.2 Empirical Method in British Annex . . . . . . . . . . . . . . . 28 3.5 Torsional Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ix Contents 4 Quick Reference Guide 31 4.1 Development and Example of Derivation . . . . . . . . . . . . . . . . 31 4.2 Example of Application . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Comments and Further Analysis . . . . . . . . . . . . . . . . . . . . . 36 II Numerical Analysis of Vortex Shedding 39 5 Computational Fluid Dynamics 41 5.1 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Solving the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . 42 5.3 Reynolds-Averaged Navier-Stokes . . . . . . . . . . . . . . . . . . . . 43 5.3.1 RANS Based Turbulence Models . . . . . . . . . . . . . . . . 43 5.3.2 Initial Turbulence Conditions . . . . . . . . . . . . . . . . . . 44 5.3.3 Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.4 Hybrid LES/RANS Technique . . . . . . . . . . . . . . . . . . . . . . 45 5.5 OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Methodology for Numerical Analysis of Vortex Shedding 49 6.1 Pre-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2 Solver Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2.1 Choice of Turbulence Model . . . . . . . . . . . . . . . . . . . 50 6.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 50 6.2.3 Solvers in OpenFOAM . . . . . . . . . . . . . . . . . . . . . . 51 6.3 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3.1 Visualization in ParaView . . . . . . . . . . . . . . . . . . . . 52 6.3.2 Data Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 53 7 Results from Simulations of Vortex Shedding 55 7.1 Variation of Width-to-Height Ratio . . . . . . . . . . . . . . . . . . . 55 7.2 Transition from Separated to Reattached Type Vortex Shedding . . . 56 7.3 Influence of Wind Velocity . . . . . . . . . . . . . . . . . . . . . . . . 58 Discussion and Conclusion 61 8 Discussion 63 8.1 Comparison of the Norms . . . . . . . . . . . . . . . . . . . . . . . . 63 8.2 Quick Reference Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.3 Numerical Simulations using OpenFOAM . . . . . . . . . . . . . . . . 65 9 Conclusion and Recommendations on Further Studies 69 Bibliography 71 x Contents Appendices 77 A Quick Reference Guide for Wind Dynamic Assessment of Bridges B Background Document for Quick Reference Guide C Verification of Strouhal Numbers from OpenFOAM on Rectangles D Matlab Code for Data Interpretation E Directory Structure and Settings in OpenFOAM xi List of Figures 1.1 Wind dynamic assessment methods based on level of complexity. . . . 1 2.1 Bluff body aerodynamics analogy of an airfoil and a bridge deck. . . . 7 2.2 Overview of wind dynamics of bridges. . . . . . . . . . . . . . . . . . 8 2.3 One oscillation period for the motion patterns related to each oscil- latory aeroelastic instability of a bridge deck cross-section. . . . . . . 8 2.4 Wind velocity ranges at which the oscillatory phenomena are domi- nant with vertical and or torsional response. Reproduced from illus- tration by Prof. Fujino from the University of Tokyo. . . . . . . . . . 9 2.5 Illustration of vortex shedding around a bridge deck. . . . . . . . . . 11 2.6 Velocity flow fields for visualization of separated type vortex shedding (a) and reattached type vortex shedding (b). Blue and red colour indicate low and high velocity, respectively. . . . . . . . . . . . . . . . 11 2.7 Graph of vortex shedding frequency for an arbitrary bluff body. . . . 12 2.8 Oscillations of the Volgograd Bridge in May 2010. Licensed under CC-BY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.9 One oscillation period of galloping motion of a bridge deck cross- section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.10 General relation between lift coefficient and angle of attack. . . . . . 15 2.11 One oscillation period of the motion pattern for torsional flutter (top) and classical flutter (bottom) of a bridge deck cross-section. . . . . . 17 2.12 Sketch of vortices forming on the Tacoma Narrows bridge, leading to classical flutter ultimately resulting in collapse. Courtesy of Dr. Allan Larsen, chief engineer at COWI DK. . . . . . . . . . . . . . . . 17 2.13 Collapse of the Tacoma Narrows Bridge. (James Bashford / The News Tribune, 1940) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.14 CFD simulated velocity flow fields in m/s for the bridge deck of the Hongkong-Zhuhai-Macao Bridge with a) upward central stabilizer b) sealed side traffic barrier c) partially sealed side and central traffic barrier. Courtesy of Guoji Xu, professor at Southwest Jiaotong Uni- versity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.15 Illustration of an airfoil at a) a stable angle b) the critical angle where flow separation begins c) an angle greater than the critical one with ongoing torsional divergence, i.e., stalling. . . . . . . . . . . . . . . . 21 2.16 Sketch of a bridge deck undergoing torsional divergence. . . . . . . . 21 xiii List of Figures 2.17 Xihoumen bridge by Roulex 45, distributed under a CC-BYSA 3.0 licence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1 Design curves for Ed,V S to be compared with Rd,V S = n1d4. Valid for all bridge types. Linear interpolation is allowed. . . . . . . . . . . . . 32 4.2 Strouhal number for rectangles with sharp corners. Reproduced from Figure A.1 of PD 6688-1-4:2009. . . . . . . . . . . . . . . . . . . . . . 36 4.3 Strouhal number for bridge cross-sections. Reproduced from Figure A.2 of PD 6688-1-4:2009. . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.1 The Navier-Stokes equation, with description of the terms. . . . . . . 41 5.2 The three major numerical CFD techniques for solving the Navier- Stokes equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 Graphical representation of K-Omega SST turbulence model. Cour- tesy of Aidan Wimshurst, Senior Engineer in CFD. . . . . . . . . . . 44 5.4 Overview of OpenFOAM structure, from the OpenFOAM user guide. 46 5.5 2D wind tunnel in grey, with carved out U-shaped geometry. . . . . . 46 6.1 Boundary conditions for a quasi two-dimensional wind tunnel. . . . . 50 6.2 Velocity flow field around a U-shaped geometry during vortex shed- ding, visualized in ParaView. Blue and red colour indicate low and high velocity, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3 Lift coefficient variation over time during vortex shedding and Strouhals number, from an FFT of the lift coefficient. . . . . . . . . . 53 6.4 Example of application of first rule to extract Strouhals number from simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.5 Example of application of second rule to extract Strouhals number from simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.1 Simulation results of rectangle with width-to-height ratio of 1. Top left and right depict lift coefficients determined with RANS and DES, respectively. Bottom depicts Strouhal numbers extracted from re- spective lift coefficients, between times 25 to 100 and 20 to 99 seconds, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.2 Simulation results of rectangle with width-to-height ratio of 2 and wind velocity 10 m/s. Top graphs depict lift coefficients, with RANS to the left and DES to the right. Bottom graph depicts Strouhal numbers from the DES simulation. . . . . . . . . . . . . . . . . . . . 57 7.3 Simulation results of rectangle with width-to-height ratio of 5 with wind velocities 5, 10 and 20 m/s. The graphs at the top depict lift coefficients for the respective velocities. Bottom graph shows the average Strouhal number distribution as well as Strouhal numbers for the respective velocities. . . . . . . . . . . . . . . . . . . . . . . . 58 8.1 Strouhal numbers for simulations of a rectangle with width-to-height ratio of 5, at wind velocities 5, 10 and 20 m/s, and average Strouhal number distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 xiv List of Tables 6.1 Boundary conditions in OpenFOAM. . . . . . . . . . . . . . . . . . . 51 6.2 Inlet turbulence conditions in OpenFOAM for K-Omega SST. . . . . 51 7.1 Simulation results of Strouhal numbers and mean drag and lift coeffi- cients, simulated at a wind velocity of 10 m/s, together with tabulated data, for rectangles of various width-to-height ratios, b/d4. . . . . . . 55 7.2 Mean drag and lift coefficients and Strouhal numbers, together with tabulated norm values, extracted from results of RANS and DES sim- ulations of a rectangle with width-to-height ratio 1 and wind velocity 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3 Mean drag and lift coefficients and Strouhal numbers, together with tabulated norm values, extracted from results of RANS and DES sim- ulations of a rectangle with width-to-height ratio 2 and wind velocity 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.4 Mean drag and lift coefficients and corresponding Strouhal numbers for each respective wind velocity, tabulated norm values, and Strouhal number from average distribution. . . . . . . . . . . . . . . . . . . . . 59 xv 1 Introduction The year is 1940 and the original Tacoma Narrows Bridge, crossing Puget Sound in Washington State, has just collapsed after violently twisting in the wind during an intense autumnal storm (Gaal, 2016). In the aftermath, the field of wind dynamics on bridges was born. Advances in material science and the ever-growing need of bridging larger obstacles means that bridges designed today are longer than ever before. For prestigious long suspension and cable stayed bridges wind dynamic assessments, in the form of wind tunnel testing, are code of practice. However, the wind dynamic phenomena are not only limited to the longest bridges. Therefore, the current Swedish norm for bridge design stipulate that the wind dynamic response for all bridges where the longest span exceeding 50 metres must be analysed. Section 8.2 of Eurocode 1:4, published by the Swedish Institute for Standards [SIS] (2009), treats wind dynamic assessments of medium-span bridges, i.e., bridges with span lengths between 50 and 200 metres. Guidance on how to empirically assess the wind dynamic response for bridges within this range is given in the informa- tive Annex E. However, the annex is ill-suited for bridges and the Swedish Trans- port Agency prohibits the use of several parts, without giving any further guidance (Transportstyrelsen, 2018). Wind tunnel testing is a reliable experimental alter- native to employing the empirical methods in the Eurocode. For long suspension bridges the immense cost and time investment of conducting wind tunnel tests are justified, but for medium-span bridges they are not a viable alternative. Generally, bridge engineers lack knowledge of the complexities of wind dynamics as the field lies in-between structural engineering and fluid dynamics. Therefore, the need to map and present available assessment methods in the field in a digestible manner is identified. The findings indicate that there are three main wind dynamic phenomena relevant for medium-spam bridges, presented in Figure 1.1. Level 1 Level 1.5 Level 2 Empirical Formulae Numerical Analysis Wind Tunnel Test Figure 1.1: Wind dynamic assessment methods based on level of complexity. 1 1. Introduction Assessment can be performed at two levels of complexity, with an intermediate level separating the first and the second. An intricate review of the first level is conducted, comparing two empirical approaches. Based on the first level, an empirical quick reference guide for wind dynamic assessment, aimed at assisting bridge engineers in early design phases, is produced. Furthermore, the intermediate level is investigated, and a method using computational fluid dynamics (CFD) is proposed. 1.1 Aim and Objectives The aim of this thesis is to investigate and summarize procedures for wind dynamic assessment of medium-span bridges. The goal is to present procedures of varying complexity, that bridge engineers unfamiliar to the field can easily apply in practice. To achieve this, the following objectives are identified: • Identify and define relevant wind dynamic phenomena for medium-span bridges. • Investigate the empirical procedure in the current Swedish norm and compare with international alternatives. • Investigate numerical and experimental procedures. • Compile procedures at different levels of complexity. 1.2 Methodology In order to understand the dynamic structural response of bridges subjected to wind flows, a comprehensive literature study was conducted. The physics of relevant phenomena, associated terminology, and their design implications was compiled. Examples of bridges where wind dynamic phenomena caused significant issues or collapse were studied. Additionally, the current research front was reviewed, and findings of relevant articles and papers have been summarized. Next, the current Swedish norm (SIS, 2009) and the national annexes were studied in detail. An international alternative was found in the British Annex to Eurocode (British Standards Institute [BSI], 2009). The methods in the norms were com- pared, and based on the findings a quick reference guide was developed, resting on the methods in the British Annex. It guides bridge engineers to design curves for relevant wind dynamic phenomena through the means of a flowchart, simplifying the assessment procedure. While researching possible numerical alternatives for deeper analysis, it was found that vortex shedding could be simulated with CFD to more accurately determine the Strouhal number. To ensure that educated choices were made, a theoretical back- ground of CFD was gathered and compiled. The open source CFD software Open- FOAM was used for two-dimensional simulations of virtual wind tunnels, analysing the variation of the lift coefficient over time to extract the Strouhal number. The procedure was verified against reference cases, both from literature and tabulated data in the norms. 2 1. Introduction 1.3 Limitations The norm stipulate that the wind dynamic response of medium-span bridges, i.e., bridges with longest spans of 50 to 200 metres, must be investigated. Therefore, the main focus of this thesis is methods for wind dynamic assessments of medium- span bridges. Furthermore, the norm is not directly applicable to arch, suspension, cable-stayed and movable bridges, as well as bridges with strong curves. Hence, these types of bridges are not considered in this thesis. Additionally, the wind dynamic response of individual members of the bridges are not considered. The purpose of the numerical simulation method is to be used by bridge engineers in practice, and their computational resources are often limited. Hence, the method is limited to 2D in order to reduce the computational cost otherwise associated to 3D CFD simulations. Furthermore, the numerical simulations are limited to the phenomena vortex shedding, as it was considered most relevant for medium-span bridges. 1.4 Outline The thesis is divided into two main parts to guide the reader through the contents. Where the first part treats wind dynamic assessment of bridges on a comprehensive level, the second part dives deeper into the analysis of one specific phenomena, highlighting possibilities of applying new techniques to the field. The parts are tied together in a mutual discussion and conclusion. Part I treats the theory and norms of wind dynamics of bridges. In Chapter 2, the aerodynamic phenomena are defined and explained. Also, the respective research fronts are summarized. In Chapter 3, the methods for wind dynamic assessment in the current Swedish norm, Eurocode 1:4, is presented. Furthermore, an alternative method from the British Annex to Eurocode is introduced. In Chapter 4, a quick reference guide for wind dynamic assessment of bridges, based on the method in the British Annex, is presented. The entirety of the quick reference guide, and an accompanying background document, are appended in Appendices A and B. Part II treats numerical analysis of vortex shedding. In Chapter 5, an introduc- tion to the field of computational fluid dynamics relevant for bridge engineers is given. In Chapter 6, an approach to study vortex shedding in bridges is presented. In Chapter 7, simulation results are compared against data from literature, pre- sented in detail in Appendix C, in order to verify the approach. In Appendix D, MATLAB code for data processing is presented. In Appendix E, directory structure and files for simulations in OpenFOAM are presented. The Discussion and Conclusion is mutual for Parts I and II. In Chapter 8, the results and the implications of both parts are discussed in detail. In Chapter 9, the findings of the thesis are summarized in conjunction with possibilities for future research. 3 Part I Wind Dynamics of Bridges; Theory and Norms 5 2 Wind Dynamic Phenomena Wind dynamics, in other industries known as aerodynamics, is the study of air flows that interact with solid bodies (Simiu & Yeo, 2019). The flow around a body will produce lift and drag forces acting on its center of lift. Lift forces are generated by a pressure difference due to the body acting as a divider, where a low pressure zone is developed on one side the body and a high pressure zone on the other. The force acts towards the low pressure zone and its magnitude is described by a lift coefficient. Drag forces arise in the opposite direction of the flow due to its volume disturbing the flow field. The magnitude of the drag force depends on how streamlined the body is, described by a drag coefficient, where a lower value corresponds to a more streamlined body. Bodies with high drag coefficients, i.e., non-streamlined, are called bluff. In structural engineering, bluff body aerodynamics is of interest as few structures are designed to be aerodynamically efficient. Although large differences to the aviation industry exist, more or less the same phenomena affect the wings of airplanes and bridges. This is visualized in Figure 2.1, where an airfoil is compared to a bridge deck. Figure 2.1: Bluff body aerodynamics analogy of an airfoil and a bridge deck. One difference between airfoils and bridges is that the former travels through the air while the latter is stationary. However, physically, it is the relative wind velocity that is of importance when studying the response. Some phenomena are more intuitive and discernible when studying airfoils rather than bridges. Therefore, airfoils can be used to illustrate and explain some of the phenomena of relevance in bridge engineering. The field of wind dynamics of bridges is complex with extensive and sometimes con- fusing and ambiguous terminology. The motion patterns of some phenomena are similar and it can be difficult to distinguish them. In order to establish the termi- nology of this thesis, a comprehensible graphical overview of the dynamic effects on bridges due to wind flow is presented in Figure 2.2. 7 2. Wind Dynamic Phenomena Wind Dynamics of Bridges Aerodynamic Actions Limited Amplitude Oscillations Buffeting Vortex Shedding Aeroelastic Instabilities Divergent Amplitude Oscillations Galloping Flutter Aerostatic Torsional Divergence Figure 2.2: Overview of wind dynamics of bridges. Wind dynamics of bridges can be divided into aerodynamic actions and aeroelas- tic instabilities. The aerodynamic actions are limited amplitude oscillations, and can be further subdivided into buffeting and vortex shedding, leading to vortex in- duced vibrations (VIVs). While limited amplitude oscillations may cause noticeable displacements, they are generally not a direct cause of structural failure. Instead, they may initiate divergent amplitude oscillations. The instabilities and VIVs are of aeroelastic nature, meaning that the influence of the aero-part, i.e., wind flow, is coupled with the elastic motions of the structure. The term aeroelastic instability includes both oscillatory phenomena of diverging amplitude, galloping and flutter, and the static phenomenon torsional divergence. These three phenomena may, by their divergent nature, cause failure if not interrupted. In Figure 2.3, the motion patterns of galloping, torsional flutter and classical flutter are illustrated. Galloping Torsional Flutter Classical Flutter Figure 2.3: One oscillation period for the motion patterns related to each oscilla- tory aeroelastic instability of a bridge deck cross-section. 8 2. Wind Dynamic Phenomena Galloping corresponds to pure longitudinal bending motion which appears vertical in the section view of Figure 2.3. Torsional flutter, also known as stall flutter, is pure torsion and classical flutter is coupled bending and torsion (De Miranda, 2016). As per the terminology of Eurocode, both torsional flutter and classical flutter are denoted as flutter. A schematic diagram depicting the relation of the oscillatory phenomena, with vi- bration amplitude as a function of wind velocity, is illustrated in Figure 2.4. It shows the ranges of wind velocities where the respective phenomena are expected to dominate the vibration amplitude. Figure 2.4: Wind velocity ranges at which the oscillatory phenomena are dominant with vertical and or torsional response. Reproduced from illustration by Prof. Fujino from the University of Tokyo. It is apparent that the ranges of when the phenomena are relevant overlaps. As seen in Figure 2.4, the peak for VIVs occur at low wind velocities. However, as it is limited in amplitude it can in some cases be allowed to occur. For the instabilities flutter and galloping, the amplitude diverges and therefore is never allowed. Note that the vibration amplitude can either be vertical or torsional. This is because, similarly to natural frequencies, aeroelastic phenomena are associated with either vertical or torsional movement. A further note is that torsional divergence is of static nature. Hence, it is not visible in the figure as there is no associated amplitude. In the following sections, the aerodynamic actions and aeroelastic instabilities will be defined and visualized, and their effect on bridges described. Also, the respective research fronts will be summarized. 2.1 Aerodynamic Actions The aerodynamic actions lead to limited amplitude oscillations in the form of either buffeting or VIVs. Both phenomena are external dynamic actions, acting on a body or structure and are of turbulent nature. However, while buffeting stems from the natural turbulence of the wind, VIVs are caused by turbulent vortices in the wake of a bluff body. 9 2. Wind Dynamic Phenomena 2.1.1 Buffeting Structures, similarly to bushes and trees, sway in the wind, albeit with smaller motions. The swaying motion stems from fluctuations in the air pressure acting on the object due to wind gusts, generated by atmospheric turbulence (Larose & Larsen, 2015). Resulting structural vibrations are called buffeting and, as previously mentioned, it is not an aeroelastic phenomena. This means that the structural response does not influence the interaction with the wind. In bridge engineering, most analyses of structural response due to wind assume constant wind velocity, considering the gusty and turbulent nature of wind through partial factors. The statistical modelling approach of buffeting was developed in the 1960s by professor Alan G. Davenport, a renowned contributor to the field of wind engineering, and it is the basis of the current theory of buffeting response (Cheynet et al., 2016). It is based on relating the gust velocity to the mean wind velocity and it divides the structural response into resonant and background parts, where the resonant response accounts for 70-80 % of the response according to Larose & Larsen (2015). Generally, buffeting response is of interest for slender and flexible bridges, such as suspension bridges. The turbulence of the wind flow depends significantly on the topography in the vicinity of the bridge, i.e., terrain or presence of other structures. Buffeting due to turbulence in the wake of an adjacent bluff body is called wake buffeting. Although the buffeting theory was introduced more than half a century ago, few full scale studies to verify the theory have been conducted. According to Cheynet et al. (2016), the conducted studies have all been flawed as the statistical significance of them were insufficient. Specifically, the duration of the tests were too short. One of the studies, conducted by Katsuchi et al. (2002), during six hours of typhoon con- ditions of the Akashi Kaikyō bridge in Japan partially verified the theory. However, Cheynet et al. (2016) deemed it irrelevant for European bridge engineering in part due to the short time duration, but also due to the typhoon conditions which are limited to the western parts of the pacific ocean. In their own research, Cheynet et al. (2016) measured the buffeting response and wind conditions of the Lysefjord Bridge in Norway. During two tests, each with a duration of 24 hours, two main wind directions, approximately 180 degrees apart, were observed. Good correlation between model and measurements were found for one wind direction but less so for the other, in part due to differing turbulence properties between the two directions. Hence, a case-by-case approach was recommended as buffeting is so dependent on local conditions, such as the terrain. 2.1.2 Vortex Shedding Vortex shedding is a phenomena where vortices form in the wake of a bluff body subjected to wind flow (Simiu & Yeo, 2019). As seen in Figure 2.5, the vortices forming in the wake of the bridge deck have alternating rotational direction, and therefore produce fluctuating upwards and downwards lift force on the body. The oscillatory motion of the body generated by the lift forces is what is known as VIVs. 10 2. Wind Dynamic Phenomena Figure 2.5: Illustration of vortex shedding around a bridge deck. Note that vortices originate from sharp edges, where a low pressure area is formed. According to Bruno & Khris (2003), there are two types of vortex shedding; sepa- rated and reattached. The former is typical for compact bodies, such as cylinders and squares, producing orderly von Kármán vortex streets in the wake of the body. The latter is typical for wider bodies, such as most bridge cross-sections, producing vortices that detach and then reattach to the body further downstream. When these vortices interact with those forming behind the body, a more complex and chaotic wake pattern with large variations in vortex size and frequency is formed. In Figure 2.6, the two types of vortex shedding are visualized with separated type in (a) and reattached type in (b). (a) (b) Figure 2.6: Velocity flow fields for visualization of separated type vortex shedding (a) and reattached type vortex shedding (b). Blue and red colour indicate low and high velocity, respectively. The vortex shedding frequency, nvs, i.e., the frequency of fluctuation in lift force direction, is determined by Equation 2.1 nvs = USt D (2.1) where U is the wind velocity, St is the Strouhal number and D is a characteristic body dimension (Simiu & Yeo, 2019). For bridges D is the cross-sectional height. The Strouhal number is a dimensionless number depending on Reynolds number and body geometry. Vortex shedding can lead to large amplitude oscillations when the vortex shedding frequency is in proximity to a natural frequency of the bluff body, where the VIVs are reinforced (Bourguet et al., 2011). This is called lock-in, which is visualized in Figure 2.7 for the first two natural frequencies, n1 and n2. As VIVs can be both vertical and torsional, the degree of freedom (DOF) with the lowest natural frequency is governing, meaning that n1 and n2 can be either vertical or torsional. 11 2. Wind Dynamic Phenomena Figure 2.7: Graph of vortex shedding frequency for an arbitrary bluff body. As vortices form in the wake of all bluff bodies, VIVs can be a significant issue in bridge design, especially for bridge decks. However, the conditions for when lock-in occurs varies dependent on the characteristics of the cross-section and the natural frequencies of the bridge. In design, the aim is to ensure that lock-in does not occur for the wind velocities the bridge is subjected to. The amplitude of displacements caused by VIVs at lock-in may be relatively large, but in general it does not put the structural integrity at risk (Gimsing & Larsen, 1992). It may, however, give rise to physical discomfort for users and, in long-term, wear in bearings and joints. While it is not alone capable of causing structural fail- ure, it may initiate more damaging phenomena, in the form of aeroelastic instability, which was the case in the well known Tacoma Narrows Bridge (Gaal, 2016). Flutter and its role in the catastrophe will be treated in a subsequent chapter. Three other examples of large amplitude oscillations due to VIVs are the Rio-Niterói Bridge in Brazil in episodes between 1980 to 1998, the Trans-Tokyo Bay Bridge in Japan in 1995 and the Volgograd Bridge in Russia in 2010. Notably, none of these three lead to structural failure and, due to precise and effective countermeasures, they are still operational to this day (Corriols, 2015). Figure 2.8 is a photograph of the oscillations, with amplitudes of about 70 cm, of the Volgograd bridge in 2010. 12 2. Wind Dynamic Phenomena Figure 2.8: Oscillations of the Volgograd Bridge in May 2010. Licensed under CC-BY. A common denominator of the three bridges are their steel box girder cross sections with low deck-width-to-span-length ratios, i.e. high slenderness. This gave the structures low natural frequencies, for example in the range of 0.3-0.6 Hz for the Rio-Niterói bridge (Battista & Pfeil, 2000). The vortex shedding frequency matched this range, inducing lock-in effects for wind velocities in the region of 15-17 m/s, a condition that was met several times between 1980 and 1998. A suggested mitigation measure was to alter the cross-sectional shape by installing aerodynamic appendages in order to alter the wake frequency. However, this proved ineffective, and for all three bridges the ultimately decided upon countermeasure was installing mass dampers inside the box-sections (Corriols, 2015). Wind tunnel testing is an effective tool to evaluate bridge response due to VIVs, but there are some limitations. Firstly, wind tunnel testing is an expensive and exten- sive measurement technique requiring both carefully crafted models and advanced equipment. Secondly, the scaled models generate some inaccuracies as the flow of air also must be scaled down (Wu et al., 2019). The Reynolds number is an important parameter that describes how turbulent the wind flow is, where higher number cor- responds to more turbulent flow. Due to the scaling, the wind flow is more turbulent in reality than what is reproduced in wind tunnel testing. The Reynolds number is often assumed to not influence the air flow around a bridge deck, but according to Larsen & Schewe (1998) it is in some cases not negligible. Specifically, bluff bodies with sharp edges, such as bridge box girders, violate the assumption. In these cases, the lift and drag coefficients may be inaccurately estimated. The alternative to wind tunnel testing is to simulate a wind tunnel using Compu- tational Fluid Dynamics (CFD). CFD is proving to be a very useful tool for bridge engineering applications without the financial drawbacks of wind tunnel tests. While the use is widespread in other industries such as the aviation and automotive in- dustries, it has yet been widely implemented in bridge engineering. However, its 13 2. Wind Dynamic Phenomena application is of ever increasing interest. On the other hand, there are still some limitations as the simulations are computationally expensive and turbulent flows are challenging to model (Wu et al., 2019). Due to the shortcomings of wind tunnel testing and CFD simulations, Wu et al. (2019) sought to establish a semi-empirical model for VIVs of bridge decks. This was accomplished by using sinusoidal input describing functions (SIDF) which ap- proximates nonlinear systems as quasi-linear by neglecting higher-order components. For example, as the lock-in effect is the most influential factor for VIVs, components of the system not relevant for lock-in can be neglected. Based on comparisons with experimental results from case studies, conclusions were drawn that the SIDF ap- proach was sufficiently accurate at predicting vertical VIVs. However, modelling of torsional VIVs was not satisfactory and requires additional investigation. 2.2 Aeroelastic Instabilities The aeroelastic instability phenomena are, once certain conditions are met, divergent and sustained by internal self-excited forces. This entails that the elastic structural response increases the response due to the phenomena, leading to divergence. In bridge engineering, the three relevant aeroelastic instability phenomena are gallop- ing, flutter, and torsional divergence. They can, in practice, be distinguished by their respective motion pattern or amplitude and frequency. 2.2.1 Galloping Galloping describes low frequency oscillations with large amplitude, in the order of at least a cross-sectional dimension of the body (Simiu & Yeo, 2019). The motion pattern for galloping of a bridge deck is illustrated in Figure 2.9 where the deck moves upwards and downwards in a bouncing motion perpendicular to the wind direction. Figure 2.9: One oscillation period of galloping motion of a bridge deck cross- section. Wake galloping, also known as interference galloping, is a separate aeroelastic phe- nomenon where oscillations are induced in a cylindrical body by turbulence in the wake of an adjacent but not connected cylindrical body (Dielen & Ruscheweyh, 1995). It is of interest for closely grouped chimneys and power-cables, but it is rarely relevant in bridge engineering. However, assessing the susceptibility for ordi- nary galloping of a bridge deck is an important part of a wind dynamic assessment. 14 2. Wind Dynamic Phenomena The Den Hartog stability criterion can be used to assess the galloping stability of a bridge deck and is presented in Equation 2.2[ dCL(α) dα + CD(α) ] α=0 < 0 (2.2) where a body is unstable if the expression on the left is smaller than zero (Simiu & Yeo, 2019). CL and CD are the lift and drag coefficients, respectively, and α is the angle of attack of the wind flow. In empirical calculations, the factor of galloping instability, aG, is often used as a substituted to the stability criterion. It closely related to Equation 2.2, and it is defined as aG = − [ dCL(α) dα + CD(α) ] α=0 (2.3) Similarly to the parameters of importance for torsional divergence, the parameters of galloping can be determined while the body is at rest. For a bridge deck, an angle of attack of zero degrees corresponds to wind parallel to the deck. As seen in Figure 2.10, when the angle of attack exceeds a certain threshold the slope of the lift coefficient becomes negative. Figure 2.10: General relation between lift coefficient and angle of attack. Galloping was first discovered by den Hartog in the 1930s, when he observed an oscillatory motion in partially ice-covered power lines. With ice build up on one side of the cable, a profile similar to an airfoil is formed altering its aerodynamic properties. Today, it is a well known phenomena in the power line industry and has been studied extensively. In recent years, super-long suspension bridges have been on the rise, especially in China, and the straits and canyons being bridged offer increasingly challenging wind conditions (Chen et al., 2020). Longer spans amount to longer main cables, and problems with galloping has arisen during the construction phase. The safety issues connected to galloping in the construction phase has been studied on the Xihoumen bridge, where it was found that problems with galloping may arise. In a finished state, the main cables are built up of hundreds 15 2. Wind Dynamic Phenomena of strands encased in a circular tube. While circular profiles are not susceptible to galloping, the unenclosed partially built up main cable is. The main cables are built up gradually, from the bottom up, by adding one strand at a time. So, at the early stages, the cross-section of the combined strands resembles either a triangle or a half circle. In other words, the cross-section at some stages resembles that of an airfoil. Moreover, as the main cables are not loaded during the construction phase, the cables are less tensioned and therefore are less stiff. Hence, they are more prone to galloping which can lead to safety issues. An important parameter when studying galloping is the Scruton number, which describes the mass-damping interaction between a body and fluid (Bartoli et al., 2020). Heavy and damped bodies have high Scruton numbers, while light and undamped bodies have low numbers. For example, a cast-in-situ concrete bridge has a higher Scruton number than a steel truss bridge. There are two prevalent types of galloping, quasi-steady and unsteady galloping. The former is evaluated, with high precision, using the den Hartog instability criterion (Equation 2.2), but it requires a certain wind velocity to be valid. According to Wawzonek (1979), the influence of vortex shedding invalidates the quasi-steady theory for wind velocities below 2.5 times the wind velocity at which the lock-in phenomenon of vortex shedding occurs. For bodies that are lightweight and have low stiffness, i.e., a low Scruton number, the galloping instability threshold may be inaccurately modelled by quasi-steady theory (Mannini, 2020). A better suited theory is unsteady galloping, combining the influence of galloping and vortex shedding. However, the theory is underdeveloped due the complex interaction. A study on unsteady galloping, analysing a pedestrian bridge in the UK, found that for bridges with low Scruton numbers, there is a strong tendency of interaction between VIV and galloping (Bagnara et al., 2017). A particular flaw of the studied bridge is the parapets, which form a U-shaped cross-section, enabling generation of wind vortices. One suggested solution was to use porous barriers as parapets which partially ventilates the trapped vortices. The galloping response improved, however, the altered wind flow decreased the critical wind velocity at which flutter arise, as the wind flow is flattened. Another study, by a Croatian research team, found the influence of wind barriers to be negligible for cable-supported bridges with regards to galloping (Buljac et al., 2017). A more recent research study focused on composite bridges, where steel box girders have been found susceptible to galloping during launching, when their Scruton number is low (Bartoli et al., 2020). Their attempt to model unsteady galloping behaviour was largely unsuccessful as the complex behaviour observed in wind tunnel testing was not replicated. 2.2.2 Flutter For either torsional or classical flutter to ensue, a small perturbation that disturbs the equilibrium of the body is required. This perturbation often comes in the form of VIVs, as flutter is always accompanied by vortex shedding (Simiu & Yeo, 2019). The motion pattern of flutter for a bridge deck is illustrated in Figure 2.11. 16 2. Wind Dynamic Phenomena Figure 2.11: One oscillation period of the motion pattern for torsional flutter (top) and classical flutter (bottom) of a bridge deck cross-section. The aeroelastic stability of a body describes how susceptible it is to flutter (Simiu & Yeo, 2019). Small perturbations invoke self-excited forces that returns the body to a state of equilibrium, due to mechanical damping. According to commonly used models, based on the linear model proposed by Scanlan and Tomko (1971), wind velocities exceeding a critical value, denoted as the flutter velocity, causes the self- excited forces to shift the equilibrium state of the body. This corresponds to a negative aerodynamic damping effect, resulting in growing oscillation amplitudes, i.e., divergence. This is also known as hard flutter where constant, or increasing, wind speeds always leads to structural failure. Non-divergent flutter is called soft flutter. In 1940, hard flutter led to the collapse of the original Tacoma Narrows Bridge, illustrated in Figure 2.12. Figure 2.12: Sketch of vortices forming on the Tacoma Narrows bridge, leading to classical flutter ultimately resulting in collapse. Courtesy of Dr. Allan Larsen, chief engineer at COWI DK. 17 2. Wind Dynamic Phenomena The original Tacoma Narrows bridge was given the nickname "Galloping Gertie" by the construction workers as it galloped in the wind during construction (Gaal, 2016). Ultimately, however, it was flutter that caused the structural collapse initiated by severe torsional stiffness degradation. Galloping led to a cable band, the connec- tion between a hanger and the main cable, sliding on the main cable, creating an asymmetric hanger arrangement. This enabled the torsional motion of the deck as vortices formed within the H-shaped section, causing VIVs that lead to divergent flutter. Nowadays, H-shaped cross-sections are rarely used, due to their poor flutter performance and low torsional stiffness, resulting in a low flutter velocity. According to Simiu & Yeo (2019), the flutter velocity for bridge decks, Uc, can be determined with Equation 2.4 Uc = Bn1 Kc (2.4) where B is the width of the deck, n1 is the the fundamental frequency and Kc is the non-dimensional reduced frequency. Kc depends on aeroelastic parameters called flutter derivatives, that can only be accurately estimated with wind tunnel testing or coupled fluid-structure interaction simulations. The flutter derivatives describe the structural response in the vertical, torsional and horizontal DOFs. The torsional flutter derivative of the Tacoma Narrows Bridge, with an H-shaped cross- section having an inherently low torsional stiffness, generated negative damping for a relatively low wind velocity of 20 m/s. The day of the catastrophe, this velocity was exceeded and torsional flutter ensued leading to the dramatic collapse captured in Figure 2.13. Figure 2.13: Collapse of the Tacoma Narrows Bridge. (James Bashford / The News Tribune, 1940) 18 2. Wind Dynamic Phenomena The tools needed to analyze bridges with respect to flutter were not available in the 1940s. Scanlan and Tomko (1971) developed the first widespread methodology to evaluate flutter derivatives, also known as aerodynamic derivatives, for bridge decks in 1971. They are estimated using wind tunnel tests at a range of velocities and motion frequencies (Siedziako & Øiseth, 2018). The standard procedure involves only motion in one DOF at one velocity per test, resulting in a large number of tests required in order to obtain estimations for all derivatives. Developments in the last decade of CFD application for bridge engineering has opened the door to numerically determine the flutter derivatives which is of great interest as wind tunnel testing is generally expensive. Gu & Zhu (2014) achieved good correlation with wind tunnel test results for both a hexagonal plate and a real bridge deck. As the Scanlan based models are only able to describe linear aeroelastic behaviour, due to their linear nature (Gao et al., 2020), the models can predict the onset of flutter but are unable to include the influence of aeroelastic nonlinearity, i.e., complex effects of higher order. This is significant for bridge decks, and especially for intricately engineered cross-sections such as twin box girders. The higher order effects may contribute with additional damping, meaning that divergence can be prevented even after the flutter velocity has been reached, i.e., soft flutter. While available methods for CFD simulations have seen rapid development in the last decade, the accuracy is not satisfactory for the most prestigious bridge projects. Generally, all super-long and most long-span bridges are subject to wind tunnel testing to study their aerodynamic behaviour. The standard identification proce- dure requires several test configurations, which is expensive and time consuming. An improved identification procedure was presented by Siedziako & Øiseth (2018), where all derivatives are estimated from a single test by subjecting the body to a general random motion, activating all DOFs, and a single wind velocity. As only a single test is needed, this is a significant step of the optimization. However, a more advanced forced vibration setup and validation of test results against reference data is required. All bridge decks have some sort of vertical obstructions in the form of traffic barriers, railings, parapets and so on. A recent study, by Bai et al. (2020), investigated the influence of the obstructions on flutter and VIVs as well as the possibility of using them as passive aerodynamic measures. In the design of the bridge deck of the Hong Kong–Zhuhai–Macau Bridge, a central upward stabilizer was used as a mitigation measure. The results of CFD simulations on the bridge deck with alternative mitigation measures are presented in Figure 2.14. 19 2. Wind Dynamic Phenomena Figure 2.14: CFD simulated velocity flow fields in m/s for the bridge deck of the Hongkong-Zhuhai-Macao Bridge with a) upward central stabilizer b) sealed side traffic barrier c) partially sealed side and central traffic barrier. Courtesy of Guoji Xu, professor at Southwest Jiaotong University. The partially sealed traffic barriers ventilates the vortices, decreasing their size sig- nificantly and thus reducing VIVs. Wind tunnel testing was conducted to find the optimal sealing form and it was concluded that improvements on both VIVs and flutter behaviour are possible with good design choices. However, at certain angles of attack the flutter velocity is reduced for partially sealed traffic barriers. 2.2.3 Torsional Divergence Torsional divergence, or aerostatic divergence, is the result of a positive feedback loop where the angle of attack of the wind flow grows as the torsional resistance of a body is exceeded, causing rotation (Andersen et al., 2016). Due to the aeroelastic moment, caused by wind, acting with an eccentricity from its torsional centre, and with a certain angle of attack, the body rotates in order for it to obtain equilibrium. As the body rotates, the angle of attack increases, thereby increasing the aeroelas- tic moment and thus the rotation angle. The torsional divergence of an airfoil is illustrated in Figure 2.15. 20 2. Wind Dynamic Phenomena Figure 2.15: Illustration of an airfoil at a) a stable angle b) the critical angle where flow separation begins c) an angle greater than the critical one with ongoing torsional divergence, i.e., stalling. Note that for airfoils, the divergence results in stalling as the air flows above and below are separated, giving a sudden loss of lift force. A certain relative wind velocity, together with either an angle of attack that is not parallel to the body or an initial rotation of the body, is required for the divergence to initiate. The critical torsional divergence velocity, Udiv, depends on parameters that can be determined while the body is stationary (Simiu & Yeo, 2019). It is determined with Equation 2.5 Udiv = √√√√√√ 2kα ρB2 dCM dα ∣∣∣∣∣ α=0 (2.5) where kα is the torsional stiffness, ρ is the fluid density, B is the width of the body and CM is the aerodynamic moment coefficient about the elastic axis. The phenomena of torsional divergence is in general only found in flat bluff bodies with low torsional stiffness, such as airfoils. Hence, it is also relevant for bridge decks with large width-to-height ratios, found mostly in suspension and cable-stayed bridges. An example of a structural model of a bridge deck with torsional stiffness, kα, subjected to the wind flow, U , is visualized in Figure 2.16. Figure 2.16: Sketch of a bridge deck undergoing torsional divergence. 21 2. Wind Dynamic Phenomena It must be noted that structural collapse due to torsional divergence is only possible if the rotation of the bridge deck is restrained below a critical angle (Andersen et al., 2016). It corresponds to the angle at which the aerodynamic response of the bridge is similar to that of a stalling airfoil, where the lift force is lost. If it is unrestrained, an oscillatory motion can occur. Generally, oscillations in bridges should be minimized as they may cause excessive wear and tear in bearings and reduce the fatigue life of the structure. A model of the Xihoumen Bridge in China was rigorously analysed through wind tunnel testing and nonlinear FEM to determine the effects of torsional divergence for super-long suspension bridges (Ge et al., 2013). Given that a significant portion of the torsional stiffness of a suspension bridge stems from the tension in the main cables, a conclusion was drawn that if the tension was lost, severe stiffness degrada- tion would occur. Hence, if sufficient lift force on the bridge deck is generated by the wind to make the main cables stress-less, torsional divergence may occur. For the Xihoumen bridge, the critical wind velocity was determined as approximately 100 m/s, depending on the angle of attack. While it was determined not to be an issue in this case, Ge et al (2013) proposed measures to ensure tension in the main cables if the critical wind velocity is deemed too low. An elevation view of the Xihoumen bridge is presented in Figure 2.17. Figure 2.17: Xihoumen bridge by Roulex 45, distributed under a CC-BYSA 3.0 licence. The current research regarding torsional divergence is primarily centred around super-long suspension bridges. For example, the previously mentioned and already built Xihoumen Bridge, the East Great Belt bridges and the proposed bridges for fjord-crossings for Coastal Highway Route E39 in Norway and the crossing of the strait of Gibraltar (Andersen et al., 2016; Andersen & Brandt, 2018). For these kinds of bridges, minimizing the mass of the bridge deck is of utmost importance to become economically feasible. However, this increases flexibility and decreases tor- sional stiffness. The aerodynamic instability of a triple-box girder for this purpose was investigated by Andersen & Brandt (2018), where the challenge was to ensure that neither flutter nor torsional divergence occurs. Through the means of nonlinear finite-element analysis and extensive wind tunnel testing, it was shown that satis- factory aeroelastic performance was obtained for low torsional-to-vertical frequency ratios, through the means of nonlinear finite-element analysis and extensive wind tunnel testing. 22 3 Current Norms for Wind Dynamic Assessment of Bridges Eurocode 1:4, published by the Swedish Standards Institute [SIS] (2005), is the current norm for design of bridges with regard to wind actions in Sweden, and the national choices are stipulated by Transportstyrelsen (2018:57) and by Trafikverket (2019:3). Wind actions on bridges is treated in Section 8, and Section 8.2 treats dynamic effects. Section 8 is only applicable to bridges consisting of a single deck of constant depth. However, Transportstyrelsen allows for use of applicable sections of the norm as guidance for other bridge types. Trafikverket (2019:3) states that the dynamic response of bridges with spans longer than 50 metres must be assessed. An upper limit of Eurocode 1:4 is that it is not to be used for spans exceeding 200 metres. Hence, the empirical methods are limited to bridges with longest spans in the range of 50 to 200 metres, i.e, medium-span bridges. Guidance on how to assess the dynamic response for bridges within this range is given in the informative Annex E. Due to Eurocode 1:4 being a general norm applicable to a wide range of structures by design, it is inevitable that some guidelines are less suited to specific structures. This is especially true for Annex E, where, for example, some formulae are de- rived for tall chimneys. Therefore, Highways England, the British counterpart to Transportstyrelsen, still use methods developed before the implementation of the Eurocodes. These methods are published in Annex A of the British Annex (BSI, 2009), and are tailored specifically for wind dynamic assessment of bridges. For bridges not satisfying empirical requirements, the British annex recommends wind tunnel testing on scaled models. This method is a reliable option to ex- perimentally assess the wind dynamic response of bridges, which is the norm for prestigious super-long bridges (Belloli, Diana, & Rocchi, 2014). The bridge models are in various scales, ranging from 1:200 for entire bridges and 1:20 for sections. However, the models require meticulous scaling of the material properties in order to replicate the eigenfrequencies of the real bridge. Furthermore, as the models are to scale, the wind flow and its turbulence is also scaled down. Hence, the influence of wind turbulence cannot be captured, and there may arise discrepancies. Wind tunnel testing is an alternative that is rarely used for medium-span bridges, mostly due to economical reasons as the expense of wind tunnel testing cannot be justified compared to adjusting the design. 23 3. Current Norms for Wind Dynamic Assessment of Bridges In cases where the requirements for either vortex shedding or galloping are not met, recent advancements in the field of CFD provides alternative methods to reliably estimate certain parameters with good accuracy. Therefore CFD is becoming an increasingly viable option to assess dynamic performance as it can be used to justify higher capacity than what the empirical method in the norm predict. A promis- ing alternative currently in development is coupled fluid-structure interaction (FSI) simulations (Braun & Sangalli, 2020). With an increased computational cost to normal CFD, it is the virtual equivalent to experimental wind tunnel testing, where the movement of the structure is simulated in conjunction with the wind flow. This option provides designers a complete tool for dynamic analyses of bridges, without the use of empirical or experimental methods. However, it is not feasible for appli- cations in practice due to immense computational costs, especially for medium-span bridges. 3.1 Buffeting Vibrations of the structure arising due to buffeting is not considered in the wind dynamic assessment in Eurocode 1:4 (SIS, 2005). However, the influence of buffeting is considered in Section 6, either as a structural factor to be applied on calculated static wind loads or as wake buffeting for certain conditions. Wake buffeting is irrelevant for dynamic assessments of bridges with longest spans in range of 50 to 200 metres. 3.2 Vortex Shedding There are, in general, two parts included in the assessment of vortex shedding around bridges. The first part is related to the critical vortex shedding velocity at which lock-in occurs, and the second to assess vibration amplitudes and accelerations due to VIVs. However, the second part is only relevant if requirements in the first part are not met. 3.2.1 Empirical Method in Eurocode Vortex Shedding is treated in the informative Annex E.1 in Eurocode 1:4 (SIS, 2005). Transportstyrelsen (2019:3), providing the Swedish national annex regarding Eurocodes on road and rail infrastructures, states that it is not allowed to use Annex E.1. The approaches described in Annex E.1 were developed for use on chimneys and similar structures, making it ill-suited for bridge design. No further guidance regarding assessment of vortex shedding is given by Transportstyrelsen. In EKS 11, the corresponding Swedish national annex for applications of the Eurocodes on buildings, the use of Annex E.1 is also prohibited. In previous editions, no further guidance was given either, but EKS 11 now refers back to the old norm BSV 97. 24 3. Current Norms for Wind Dynamic Assessment of Bridges 3.2.2 Empirical Method in British Annex The British counterpart of Transportstyrelsen, Highways England Co. LTD, also identifies the flaws of Annex E.1. Therefore, the old national norm preceding the Eurocodes has been kept in use with minor adjustments. Alternative empirical approaches to the Eurocode, based on wind tunnel test data of bridge decks, are presented in the British Annex (BSI, 2009). The critical wind velocity check, de- scribed in Annex E.1.2, and the second approach for physical discomfort check, described in Annex E.1.5.3, are adapted. The steps are: 1. Check span length-to-height ratio. If larger than 6, continue with following steps. Otherwise, vortex shedding need not be investigated. 2. Calculate critical wind velocity for vortex shedding, vcrit. 3. Check that the critical wind velocity is more than 1.25 times larger than the mean wind velocity; vcrit > 1.25vm. If the inequality holds true, no further action is needed. Otherwise, continue with Step 4 and 5. 4. Calculate maximum predicted amplitude, ymax, due to vortex shedding. 5. Calculate dynamic sensitivity parameter, KD, and check against comfort cri- teria. The critical wind velocity for which lock-in occurs is determined with Equation 3.1 vcrit = d4n1 St (3.1) where d4 is the cross-sectional height of the bridge, n1 is the cross-wind fundamental frequency in bending or torsion, whichever is lowest, and St is the Strouhal number defined in Section A.1.3.2. As the Strouhal number for most bridges in this method is set to the value 1/6.5, there is potential to ascertain higher capacity if a lower Strouhal number can be justified. The annex states that the Strouhal may be gathered from an attached diagram, indicating that it may also be gathered by other means. One alternative method to determine the Strouhal number is to simulate the wind flow around a bridge section with CFD and measure the vortex shedding frequency. This frequency can then be converted to a Strouhal number which may be lower than 1/6.5. For the critical wind velocity, an approximation of the maximum vibration ampli- tude, ymax, is determined according to Section A.1.5.4.3 with varying formulae for vertical and torsional vibrations. It is used to estimate the sensitivity parameter, KD, determined with Equation 3.2 KD = ymaxn 2 1 (3.2) where ymax is the maximum predicted deflection and n1 the fundamental frequency. KD is an acceleration in mm/s2 that is compared to criteria for physical comfort of pedestrians. Furthermore, an investigation of the structural response due to effective loading from vortex shedding should be conducted for all bridges whereKD is greater than or equal to 12.5. 25 3. Current Norms for Wind Dynamic Assessment of Bridges 3.3 Galloping Galloping is pure longitudinal bending, corresponding to vertical translation in a section view of a bridge deck. Therefore, the vertical bending frequency of the bridge is a very important parameter when studying the phenomena. This is reflected in the empirical formulae used in both Eurocode 1:4 and the British Annex. The British annex also assesses torsional flutter of bridge decks in conjunction with galloping, unlike Eurocode 1:4 that only assesses galloping. 3.3.1 Empirical Method in Eurocode The simplified method, proposed in Eurocode (SIS, 2005), to evaluate the risk of galloping is given in Section 2 of the informative Annex E. The major steps are: 1. Calculate onset wind velocity of galloping, vCG. 2. Check that the onset wind velocity of galloping is more than 1.25 times larger than the mean wind velocity; vCG > 1.25vm. 3. Check that onset wind velocity of galloping is not close to critical vortex shedding velocity; 0.7 < vCG vcrit < 1.5. The complexity lies in determining the onset wind velocity as it depends on some parameters found in standardized tables and figures, where the correct choice may not be obvious. It is determined with Equation 3.3 vCG = 2Sc aG n1,yb (3.3) where Sc is the Scruton number defined in Annex E.1.3.3, n1,y is the first vertical natural frequency, determined approximately in Section 2 of Annex F or through solution of the eigenvalue problem with FEM. For bridge decks, this corresponds to the fundamental bending frequency. The factor of galloping instability, aG, can be determined using Table E.7 where the width, b, is also defined based on the type of cross-section. If the cross-section shape does not correspond to those listed in the table, aG may be set to 10. Alternatively, by determining the drag and lift coeffi- cients for a range of attack angles using CFD, aG can be determined with Equation 2.3. Then, a more accurate onset wind velocity of galloping can be calculated with Equation 3.3. 3.3.2 Empirical Method in British Annex The British Annex (BSI, 2009) uses the same simplified method as Eurocode 1:4 for individual members. However, an alternative procedure developed specifically for bridge decks is given in Section A.2.4. This is useful as bridge decks are often incompatible with the cross-sections that the factor of galloping instability is given for in Eurocode. Furthermore, the galloping section is expanded by distinguishing vertical and torsional motion, corresponding to galloping, and torsional flutter, i.e., stall flutter. The steps of the procedure are: 26 3. Current Norms for Wind Dynamic Assessment of Bridges 1. Determine bridge type according to Figure A.3. 2. Calculate onset wind velocity, vG, for torsional and, if relevant, vertical motion. 3. Calculate wind storm velocity, vWO. 4. Check that the smallest onset wind velocity is larger than the wind storm velocity; vG > vWO. For vertical motion, only relevant for some bridge types and with certain cross- section width-to-height ratios, the onset velocity is determined using Equation 3.4 vg = vRgn1,bd4 (3.4) where n1,b is the fundamental bending frequency, d4 is the height of the bridge cross-section and vRg is the reduced velocity defined as vRg = Cg(mδs) ρd2 4 = 1 2CgSc (3.5) where Cg is a factor, defined as either 1 or 2, based on bridge type and geometry. Comparing Equation 3.4 to 3.3 from Eurocode 1:4, the similarities are evident. The factor Cg corresponds to 2/aG, and it simplifies the risk assessment of bridge decks as aG does not need to be identified. The onset velocity for torsional motion is relevant for all bridge types and is deter- mined, depending on bridge type, by either Equation 3.6 or 3.7 vg = 3.3n1,tb (3.6) vg = 5.0n1,tb (3.7) where n1,t is the fundamental torsional frequency and b is the width of the bridge. The equations are similar to Equation 3.4, but the constants 3.3 and 5.0 have been empirically determined based on data from wind tunnel tests on a variety of bridge types. The wind storm velocity denotes the wind velocity that the bridge must be stable for with respect to divergent amplitude phenomena. It is defined in Equation 3.8 vWO = K1UK1Avm(z) ( 1 + 2Iv(z) √ B2 ) (3.8) where K1U is an uncertainty factor with a default value of 1.1 and K1A is a factor taking climactic region into consideration, which for locations in the UK, and Swe- den, is set to 1.25. The turbulence intensity factor, Iv, and the background factor, B2, are defined by Transportstyrelsen in the National Annex. The wind storm ve- locity is used for both galloping and flutter checks and if the criteria is not met, stability must be verified through wind tunnel testing. 27 3. Current Norms for Wind Dynamic Assessment of Bridges 3.4 Flutter Flutter, in Eurocode 1:4 referring to classical flutter, is a coupling of bending and torsional motion of the bridge deck, i.e., galloping and torsional flutter. There is a considerable dissimilarity between how flutter is treated in Eurocode 1:4 and the British annex. Eurocode 1:4 states three conditions that indicate a risk of flutter, if all criteria is fulfilled. The British annex, on the other hand, present empirical formulae to assess the phenomena. 3.4.1 Empirical Method in Eurocode In Eurocode 1:4 (SIS, 2005), flutter is treated in Annex E.4, together with torsional divergence. No formulae is given to determine a critical wind velocity for flutter. Instead, a structure is deemed prone to flutter and torsional divergence if three criteria are met. Otherwise, no further check is required. The criteria are: 1. The structure has a flat shape with height-to-width ratio smaller than 0.25. 2. Position of torsional axis fulfils certain geometrical conditions. 3. The fundamental frequency is torsional, or the lowest torsional frequency of the structure is lower than two times the fundamental translational frequency. If all criteria are met there is a risk of flutter, and then Eurocode suggest seeking expert advice. 3.4.2 Empirical Method in British Annex The British Annex (BSI, 2009) uses the same approach as Eurocode 1 for plate-like structures, but, similarly to galloping, they have developed a procedure to assess the flutter response specifically for bridge decks, in Section A.4.4. The onset velocity of flutter for bridge decks is determined in Equation 3.9 vf = vRfn1,tb (3.9) where n1,t is the fundamental torsional frequency and b is the cross-sectional width of the bridge deck. The reduced flutter velocity, vRf , is defined as vRF = 1.8 [ 1− 1.1 (n1,b n1,t )2]1/2(mr ρb3 )1/2 (3.10) where n1,b is the fundamental bending frequency, ρ is the air density, m is the mass per unit length and r is the radius of gyration of the cross-section. Note that vRF can not be less than 2.5. The bridge deck is considered stable with regard to flutter if the onset wind velocity for flutter is larger than the wind storm velocity, determined in Equation 3.8. If the criteria is not met, stability must be verified through wind tunnel testing. 28 3. Current Norms for Wind Dynamic Assessment of Bridges 3.5 Torsional Divergence In Eurocode 1:4 (SIS, 2005), torsional divergence is treated in Annex E.4, in con- junction with flutter with the criteria stated in Section 3.4.1. However, unlike with flutter, a formula to determine the critical wind velocity for torsional divergence is given. It is estimated with Equation 3.11 vdiv = [ 2kθ ρd2 dCM dθ ]1/2 (3.11) where kθ is the torsional stiffness, ρ is the density of air and d is the width of the bridge deck. The factor dCM/dθ is the gradient of the aerodynamic moment coefficient. The critical divergence velocity should be more than two times larger than the mean velocity, vm(z). The method in the British annex (BSI, 2009) does not differ from the method presented in Eurocode 1, where no specific section is dedicated to bridges. Therefore, it is concluded that torsional divergence is not relevant for bridges with longest spans in the range of 50 to 200 metres. 29 4 Quick Reference Guide A quick reference guide for wind dynamic assessment of bridges is developed based on the empirical formulae in the British annex to Eurocode (BSI, 2009), for use by bridge engineers. It consists of a flowchart guiding the user to a number of checks, in the form of design curves, based on input data. The quick reference guide in its entirety with an accompanying background document, describing its development in detail, are presented in Appendices A and B, respectively. 4.1 Development and Example of Derivation Four different checks are required for a complete wind dynamic assessment of bridges according to the British Annex. In total, six design curves are produced. Here, the derivation of the design curve for vortex shedding is presented. The design curves for all checks are derived in a similar manner, starting with a requirement to be fulfilled. For vortex shedding, the requirement reads vcrit > 1.25vm (4.1) where vm is the mean wind velocity and the critical vortex shedding velocity, vcrit, is calculated as vcrit = n1d4 St (4.2) The mean wind velocity depends on several other parameters, that in turn depend on site conditions. Rewriting and rearranging Equation 4.1, and inserting all pa- rameters, yields n1d4 = 1.25krSt · ln ( z z0 ) vb (4.3) The left-hand-side is henceforth defined as the capacity for vortex shedding, Rd,V S, and the right-hand-side as the effect for vortex shedding, Ed,V S. Several choices and conservative assumptions are made to simplify the effect as to only depend on the bridges height above ground, z, and the basic wind velocity, vb. The resulting effect is presented in Equation 4.4. Ed,V S := 0.037 ln ( z 0.05 ) vb (4.4) Using the relationship in Equation 4.4, the design curves in Figure 4.1 are produced in MATLAB. 31 4. Quick Reference Guide Figure 4.1: Design curves for Ed,V S to be compared with Rd,V S = n1d4. Valid for all bridge types. Linear interpolation is allowed. Once an effect, Ed,V S, has been extracted from Figure 4.1 and a capacity has been calculated with Rd,V S = n1d4, the check is performed by confirming that Rd,V S > Ed,V S (4.5) If the inequality is true the check has been passed, otherwise the bridge has unsat- isfactory wind dynamic response. For vortex shedding in particular, investigation of vibration amplitudes must be conducted. However, for the other checks, failure to meet requirements entails redesign of the bridge. Detailed derivation procedures for all checks, where all choices and assumptions are explained, are presented in Appendix B. 4.2 Example of Application An application example of the quick reference guide on a bridge is presented on the following pages. Note that only two checks are required for this specific bridge. Certain bridge types in the British Annex (BSI, 2009) do not require check of gal- loping, and the checks for torsional flutter and classical flutter is combined into one check. Furthermore, only six unique input parameters are required to assess the wind dynamic response of a bridge. 32 4. Quick Reference Guide Wind Dynamic Assessment of Pedestrian Bridge Using Quick Reference Guide Consider the pedestrian steel truss bridge shown to the left. It has a a main span of 60 metres and is located in Falköping. The bridge deck is 4 metres wide and the truss height varies from 3 to 5 metres. However, as solid glass barriers are mounted on both sides of the other- wise permeable trusses, it is the height of the barriers that are of interest for wind flow. The distance from the top of the barriers to the bottom of the bridge is approximately 2.4 metres. The bridge sits 8 metres above the ground and the basic wind velocity in Falköping is 24 m/s, according to Transportstyrelsen (2018:57). From an FE-analysis, the fundamental frequencies for bending and torsion are determined as 2.71 Hz and 4.24 Hz, respectively. The first step of the quick reference guide is to confirm that the span length of the bridge is between 50 and 200 metres, which it is. Next is to determine the bridge type from Figure A.2. As seen on the side, it is identified as Bridge type 5. Also, the required input parameters are gathered and compiled in the table below. Parameter Variable Value Height above ground z 8.0 m Basic wind velocity vb 24 m/s Cross-sectional height d4 2.4 m Cross-sectional width b 4.0 m Fundamental bending frequency n1,b 2.71 Hz Fundamental torsional frequency n1,t 4.24 Hz Lowest fundamental frequency n1 2.71 Hz Torsional-to-bending frequency ratio n1,t/n1,b 1.56 The next step is to evaluate vortex shedding. The resistance is determined as Rd,V S = n1d4 = 2.71 · 2.4 = 6.5 and the effect, Ed,V S, is extracted from the figure below. 33 4. Quick Reference Guide The red lines intersect at approximately Ed,V S = 4.5. Hence, we can conclude that no risk of vortex shedding exists as Rd,V S = 6.5 > 4.5 = Ed,V S. Moving on, we check the torsional-to-bending frequency ratio. As it is larger than 1.45, we move right in the flowchart and check if the bridge is of type 3, 3A, 4 or 4A. As it is of type 5, we evaluate flutter with Figure A.5. The resistance is determined as Rd,F = n1,tb = 4.24 · 4 = 17.0 and the effect, Ed,F , is extracted from the figure below. The red lines intersect at approximately Ed,F = 13.4. Hence, we can conclude that no risk of flutter exists as Rd,F = 17.0 > 13.4 = Ed,F . Following the flowchart, it is apparent that no more checks are necessary and the wind dynamic response of the bridge is OK! 34 4. Quick Reference Guide The path taken through the flowchart is marked in red below. Start No Yes 50m ≤ L ≤ 200m Manual not applicable Evaluate Vortex Shedding Resistance: Rd,VS = n1d4 Effect: Ed,VS Fig. A.3 Yes Rd,VS ≥ Ed,VS Check bridge type: Fig A.2 Gather input data* Wind dynamic response NOT OK! Further investigations recommended Yes No n1,t / n1,b ≤ 1.45 Evaluate Flutter Resistance: Rd,F = n1,tb Effect: Ed,F Fig. A.4 Bridge type: 3, 3A, 4, 4A Yes Rd,F ≥ Ed,F No No Yes Bridge type: 3, 3A, 4, 4A AND b < 4d4 Wind dynamic response OK! Evaluate Galloping Resistance: Rd,G = n1,bd4 Effect: Ed,G Fig. A.8 Yes Rd,G ≥ Ed,G No Evaluate Flutter Resistance: Rd,F = n1,tb Effect: Ed,F Fig. A.5 No Rd,F ≥ Ed,F Yes b < 2.4d4 No Evaluate Flutter Resistance: Rd,F = n1,tb Effect: Ed,F Fig. A.7 Evaluate Flutter Resistance: Rd,F = n1,tb Effect: Ed,F Fig. A.6 No Yes Yes Rd,F ≥ Ed,F Yes b < 4d4 Rd,F ≥ Ed,F No Yes Yes No No List of input data z Height above ground [m] vb Basic wind velocity [m/s] d4 Cross-sectional height [m] b Cross-sectional width [m] n1,b Fundamental bending frequency [Hz] n1,t Fundamental torsional frequency [Hz] n1 Lowest fundamental frequency [Hz] n1,t / n1,b Torsional-to-bending frequency ratio [-] Investigate vibration amplitudes according to Section A.1.5.4 in PD 6688-1-4:2009 No 35 4. Quick Reference Guide 4.3 Comments and Further Analysis While the bridge in this example fulfils all checks, it is important to keep in mind that failure to meet the requirements of the quick reference guide is not always equivalent to an unsatisfactory wind dynamic response, especially if the checks are almost satisfied. If one or more checks are not fulfilled, detailed calculations according to the procedure in the British Annex (BSI, 2009), is recommended. Some guidance for such calculations can be found in the background document in Appendix B. Studying the empirical equations of the British Annex (BSI, 2009), a possibility to justify higher capacity for vortex shedding than the norm estimates is identified. In the norm, the Strouhal number is identified as a conservative parameter, where CFD simulations could be used to determine it more precisely. According to Larsen & Walther (1998), the Strouhal number for bridges is defined as St = v d4 · fcr (4.6) where v is the wind velocity in metres per second, d4 is the bridge height and fcr is the critical vortex shedding frequency. In section A.3 of the British Annex (BSI, 2009), two figures are presented to retrieve empirical Strouhal numbers for rectangles and bridge decks, respectively. In Figures 4.2 and 4.3, slightly reproduced versions of the Strouhal number figures from the British Annex are presented. Figure 4.2: Strouhal number for rectangles with sharp corners. Reproduced from Figure A.1 of PD 6688-1-4:2009. 36 4. Quick Reference Guide Figure 4.3: Strouhal number for bridge cross-sections. Reproduced from Figure A.2 of PD 6688-1-4:2009. Comparing Figures 4.2 and 4.3, it is apparent that the Strouhal number for bridges with ratios b∗/d4 below 5 are governed by the peak of Strouhal number for rectan- gles with ratios b/d4 around 3.5. However, for rectangles of other ratios the Strouhal number is lower. This indicates that many bridges with ratios b∗/d4 below 5 could have Strouhal numbers below 1/6.5. The possible increased capacity due to a de- creased Strouhal number is apparent when studying the formula for critical vortex shedding velocity in Equation 4.7. vcrit = n1d4 St (4.7) As the critical vortex shedding velocity is inversely proportional to Strouhal number, a decrease in Strouhal number of 10 % yields a 10 % increase in critical vortex shedding velocity. 37 Part II Numerical Analysis of Vortex Shedding 39 5 Computational Fluid Dynamics The field of computational fluid dynamics (CFD) uses numerical methods to solve the Navier-Stokes equations in 3D to simulate fluid flow. Most air flows encountered in reality are of turbulent nature (Davidson, 2021). Pronounced examples can be found in the flows around and behind airplanes, and in the wake of high-speed trains. However, wind flow around buildings and other structures is also commonly turbulent. 5.1 Navier-Stokes Equation Fluid dynamics is based on the Navier-Stokes equation, which is derived by applying Newtons law of motion on fluids. The equations describe, together with a continuity equation, the motion of all fluids in space over time. In Figure 5.1, the Navier- Stokes equation in vector form is presented. The terms, their physical significance and correlation to Newtons law of motion are highlighted. Figure 5.1: The Navier-Stokes equation, with description of the terms. While the equation is known, it has only been smoothly solved in two dimensions by Olga Ladyzhenskaya in 1958. In three dimensions, only conditional proofs of smoothness have been presented. The Navier-Stokes equation is one of the millen- nium prize problems with a one million dollar reward for an unconditional solution (Clay Mathematics Institute, 2021). The complexity in the solution stems from the convection term, (v̄ · ∇)v̄, which dominates the solution in turbulent flows. To determine whether a flow is laminar or turbulent, the Reynolds number is used. It is a dimensionless parameter, defined as the ratio between inertial and viscous force. In aerodynamics, air flow transitions from laminar to turbulent at Reynolds numbers of approximately 2300 (Davidson, 2021). In boundary layers, i.e., the thin 41 5. Computational Fluid Dynamics film of air closest to the surface of an object, the transition occurs at approximately 500,000. For low Reynolds numbers, i.e., for cases where the diffusion term, µ ·∇2v̄, is dominant over the convection term, (v̄ ·∇)v̄, the equation can be smoothly solved in 3D. However, for turbulent flows, which is the most common occurrence, the flow is chaotic and currently impossible to smoothly solve. 5.2 Solving the Navier-Stokes Equations In order to approach the solution of the Navier-Stokes equations numerically, dis- cretization is required. The most common method is the finite volume method (FVM), where domains with known boundary conditions are isolated and parti- tioned into a mesh of cells with a finite volume (SimScale, 2016). While other methods are possible, such as the finite element method (FEM), the decisive advan- tage of using FVM is that the solution will always fulfil the continuity condition. Note that, as FVM is three dimensional by nature, true two dimensional simulations are not possible. Rather, a quasi-two dimensional domain, with a thickness of one cell, can be utilized. In turbulent conditions, the range of the spatial and temporal scales of flows are wide (Zhiyin, 2015). Spatial scales represent vortex size and temporal scales represent velocities, dissipation and frequencies of the vortices. In Figure 5.2, the three major numerical techniques to treat the wide range are presented. Figure 5.2: The three major numerical CFD techniques for solving the Navier- Stokes equations. At the top of the pyramid resides direct numerical simulations (DNS), accurately resolving all turbulence scales. However, even for simple laminar cases the com- putational cost is very high and the use of supercomputers is necessary. As such, DNS is limited to the very front of the field and not yet applicable for industry use. Consequently, the need for approximative techniques further down the pyramid is ev- ident. They use turbulence models to a varying extent, sacrificing accuracy in favor of lower computational cost. Large-eddy simulations (LES) filters out the smallest length scales in favour of directly computing the largest scales. Hence, the largest source of turbulence is accurately simulated but the computational cost remains sig- nificant. The third and most widespread alternative, due to its low computational cost, is the Reynolds Averaged Navier-Stokes (RANS) approach. 42 5. Computational Fluid Dynamics 5.3 Reynolds-Averaged Navier-Stokes RANS implements time averaging on the Navier-Stokes equation and utilizes tur- bulence models, essentially providing a steady-state solution which is sufficient for many industrial applications. Due to the time averaging, RANS is the only mod- elling method that is physically suitable for 2D simulations (Davidson, 2021). As the number of cells needed for 2D simulations is approximately the cube root squared of the number of cells in 3D simulations, it is apparent that 2D simulations are preferable as long as the accuracy is sufficient. However, as the eddies are enclosed, i.e., trapping energy that would otherwise dissipate in the unsolved direction, the magnitudes of the drag- and lift coefficients may be overestimated. RANS time averages both mean and turbulent motion. A variation of RANS is unsteady-RANS (URANS), which does not time average the mean motion and is therefore able to capture variations in the mean motion of the flow (Davidson, 2021). Therefore, URANS can be used to simulate von Karman vortex streets, i.e., vortex shedding, for flows where the time scale of the mean flow is much larger than the time scale of the turbulence. For example, laminar vortex shedding in the wake of a cylinder. 5.3.1 RANS Based Turbulence Models The three most frequently used turbulence models in the RANS family are the K- Epsilon (k− ε), K-Omega (k−ω) and K-Omega Shear Stress Transport (k−ω SST) models, with some respective variants. Common for all RANS based models is that all turbulence effects are modelled, i.e., in addition to the conservation equations, partial differential equations are used to capture turbulence history effects in the fluid. The K-Epsilon model is a two-equation model, meaning that two transport equations are used, describing the turbulent kinetic energy, k, and the turbulent dissipation rate, ε (SimScale, 2020a). The K-Epsilon model captures free-stream flows with good accuracy, but struggles in resolving large pressure gradients, flow separations and flows with strong curvatures, occurring, for example, near walls. The K-Omega model, similarly to the K-Epsilon model, also uses two equations to account for the turbulence history effects; one for turbulent kinetic energy, k, and one for specific turbulent dissipation rate, ω (SimScale, 2020b). It is a model appropriate for low Reynolds numbers, meaning that it resolves the field near walls well. However, this model is highly susceptible to turbulence at the inlet. An effective remedy of the respective shortcomings in both models is to combine them, and utilizing them in the regions where they are most accurate. This is the basis of the K-Omega SST model, depicted in Figure 5.3. 43 5. Computational Fluid Dynamics Figure 5.3: Graphical representation of K-Omega SST turbulence model. Courtesy of Aidan Wimshurst, Senior Engineer in CFD. By blending the models their respective disadvantages are mitigated, yielding a robust and versatile turbulence model (SimScale, 2020b). For example, adverse pressure gradients and flow separation are captured with good accuracy. Hence, the turbulence model K-Omega SST is widely used in the industry. 5.3.2 Initial Turbulence Conditions Prior to initializing simulations employing the K-Omega SST turbulence model, the initial inlet turbulence condition parameters, k and ω, need to be defined (Sim- Scale, 2020b). The turbulent dissipation rate, ε, can be determined with k and ω and therefore does not need to be specified. The inlet turbulent kinetic energy is determined according to Equation 5.1 k = 3 2(v · I)2 (5.1) where v is the mean free-stream velocity and I is the inlet turbulence intensity. For highly turbulent flows, such as wind flows, the turbulence intensity lies in the range of 5 to 20 percent. The specific turbulent dissipation rate, ω, is defined according to Equation 5.2 ω = √ k l (5.2) where l is the characteristic turbulent length scale. Physically, the characteristic turbulent length scale corresponds to the size of the largest eddies (CFD Online, 2012). For vortex shedding in 2D, a good estimate is the cross-wind dimension of the geometry, as the size of the eddies formed behind the geometry cannot be larger than this. 5.3.3 Wall Functions In proximity to boundaries inside the domain, denoted as walls, the fluid flow is more turbulent and viscous effects are significant, yielding sharp gradients of the velocity and dissipation profiles (Davidson, 2021). Closest to the wall, viscous effects dominate, and at a certain distance, turbulence effects described by the log-law govern the flow. To capture this behaviour, a very fine mesh can be used, but 44 5. Computational Fluid Dynamics this requires significant computational resources. As these resources are finite and preferably allocated to other regions of the domain, the use of wall functions is an efficient alternative for modelling of complex flows. Wall functions are empirically derived equations that approximate the influence of the viscous sublayer, rather than resolving it explicitly (SimScale, 2018). However, in order to obtain reliable results, certain requirements of a dimensionless distance parameter, y+, must be satisfied. It describes the relation between the viscous stress and turbulence stress, similar to the Reynolds number. For low y+ values, viscous stresses dominate the flow. The viscous stresses decrease with increasing values of y+, and conversely turbulent stresses increase. At values of around 30, the viscous stresses are approximately 1 percent of the turbulence stresses. Hence, it is a lower bound for the y+ value, assuring that the center of the boundary layer cell closest to the wall lies within the log-law region. An upper bound is set to 300, as values exceeding 300 yield poor resolution of the wall. As such, a y+ value between 30 and 300 is a verification that the mesh resolves the wall correctly. 5.4 Hybrid LES/RANS Technique URANS is an insufficient technique for simulations of certain phenomena, such as vortex shedding of a streamlined body. In these cases, the assumption that the time scale of the mean flow is much larger than the time scale of the turbulence is no longer true. Hence, neglecting the influence of the turbulence over time produces steady-state solutions to inherently transient phenomena. Similar to the blending of the two turbulence models K-epsilon and K-omega into the K-omega SST model, hybrid LES/RANS entails blending of LES and RANS to remedy the limitations of RANS while keeping the computational cost down (Chaouat, 2017). One such model is K-omega SST DES, where DES stands for Detached Eddy Simulation. It detects highly refined regions in the mesh where the cells are small and applies LES in them, i.e., directly simulating the large eddies. In the other regions, including boundary layers, RANS is applied. Therefore, the user controls which regions are modelled with RANS and LES, respectively. Although the computational cost of the hybrid model is relatively low, the high mesh resolution required for LES lead to a significant computational cost compared to pure RANS simulations. Researchers Mannini & Schewe (2011) conducted a numerical study of vortex shed- ding around a rectangular cylinder with a width-to-height ratio of 5 using DES. Notably, this ratio is commonly used as a reference case for study of aerodynamics of bridges. A 3D approach, with limited depth, was adopted with a depth-to-width ratio of 1, meaning that some out-of-plane elements are used. The influence of numerical dissipation was studied, as well as different algorithms for discretization of inviscid fluxes. Various levels of correlation to a reference wind tunnel test was achieved, with Strouhal numbers differing by 21.6 % in the worst case, and only 2.7 % in the best case. It is concluded that 3D DES simulations with a limited depth can, with carefully chosen settings, yield reliable results, but it is sensitive to numerical dissipation. 45 5. Computational Fluid Dynamics 5.5 OpenFOAM There are many commercial CFD softwares available on the market, often coupled with hefty license fees. The open source software OpenFOAM is an attractive al- ternative, especially for academic applications. It is a text-based C++ library, with no graphical user interface. OpenFOAM provides a range of solvers based on the Finite Volume Method (FVM), as well as utilities for data manipulation. In Figure 5.4, a visualization of the structure of the OpenFOAM library is presented. Figure 5.4: Overview of OpenFOAM structure, from the OpenFOAM user guide. While OpenFOAM offers tools for meshing of domains, it is not sufficient for mod- elling of complex geometries or structures. For this purpose, CAD-programs are used to model geometries, and a mesh of the geometry is typically generated with a free meshing program, such as GMSH. However, the mesh of the geometry is not used in the simulations, rather it is a required input for creating the mesh of the fluid. This is a key difference between fluid and structural dynamic analyses, i.e., that the structure is not studied, rather the fluid flow around it. An example of this is seen in Figure 5.5, where a geometry with a U-shape is carved out of a 2D virtual wind tunnel. Figure 5.5: 2D wind tunnel in grey, with carved out U-shaped geometry. 46 5. Computational Fluid Dynamics In CFD simulations, there is always a choice between performing steady-state or transient analyses. Steady-state simulations are preferable as transient simulations are vastly more computationally expensive. Notably for a bridge engineer, the defi- nition of steady-state and transient conditions differ between the fields of structural dynamics and fluid dynamics. In structural dynamics, a harmonic oscillation is considered steady-state, while in fluid dynamics a harmonically oscillating flow is considered transient. Hence, vortex shedding is a transient phenomena in fluid dy- namics. There are several different solvers in the OpenFOAM library, suitable for different types of problems. For steady-state problems of incompressible fluid flows with turbulence, such as wind load on a structure, the solver simpleFoam is suitable. For transient problems of incompressible fluid flows with turbulence, such as vortex shedding in wind flows, the solver pimpleFoam can be employed effectively. 47 6 Methodology for Numerical Analysis of Vortex Shedding A methodology to analyze vortex shedding of bridge cross-sections using 2D CFD simulations is presented in this chapter. It is structured with the chronology of a simulation, covering pre-processing, solving and post-processing. In Appendix E, the directory structure and typical settings of a simulation is presented. 6.1 Pre-Processing As meshing of the wind tunnel in OpenFOAM for simulation of vortex shedding around an object entails carving it out, it must first be created in external software. Quasi-2D geometries are modelled in the CAD program Autodesk Inventor Profes- sional with an arbitrary depth, and exported as two separate file formats, one as .stp and one as .stl. Other CAD-programs can be used, as long as these two file types can be exported. The .stp file is loaded into the free meshing program GMSH, where a 2D mesh, with arbitrary refinement is generated. This mesh is then exported as a .msh file. The 2D virtual wind tunnel in OpenFOAM is constructed by making use of the meshing tools and utilities blockMesh, snappyHexMesh and extrudeMesh. A base mesh of the wind tunnel is generated using the blockMesh, where the outer dimen- sions of the tunnel and the size of the coarsest cells is determined. The second utility, snappyHexMesh, is an algorithm that carves out the object by refering to the .stl and .msh files. The refinement level around the object is determined by user input. Further refinement is possible by manually defining regions in which the refinement level is increased. Generally, the refinement is highest near the object. However, the wake behind the object must also be refined in order to capture the effects of the vortices. As snappyHexMesh refines in all dimensions, extrudeMesh is used to reduce the 3D wind tunnel to quasi-2D, by extruding the mesh in the x-y plane to a thickness of one element. 6.2 Solver Settings Computational fluid dynamic simulations are complicated and many choices, with significant influence on simulation results, need to be made. These include choice of tu