Governing equations of the separated method for RWIV of stay cables

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Cable vibrations in cable-stayed bridge system

The trend of building super longer span cable-stayed bridges introduces many new chal-lenges to the bridge engineers and designers. One of the new challenges is related to long and flexible stay cables due to the low internal structural damping and prone to paramet-ric excitation vibration, vortex-induced vibration, wake galloping, and rain-wind-induced vibration.

Parametric excitation vibration

Parametric excitation vibration of stay cable is driven by harmonic oscillator, which are in-duced by varying system parameter variations at different frequencies. Vibrations of cables may be induced by small periodic movements of their cable anchorages either on the deck or the pylons. The vibrations of the decks and the towers, which are excited by wind actions on them or traffic loading on the deck, induce the movement of the cable anchorages on the bridge [75, 102, 109, 124, 151, 174, 123, 32, 24, 157, 176, 150]. Due to these low frequen-cies movement effect, parametric excitation vibration is probable to occur. The unstable range of excitation increases very quickly with excitation amplitude, and large amplitudes of the end point oscillations, almost all the frequency ranges become unstable. Amplitude oscillations causing fatigue and contact between stay cables would bring a great damage to bridges. Kovács pointed out the danger of instability when the girder or mast frequency is closed to twice a cable frequency, an instability condition and a simplified formula for the maximum amplitude were given; Tagata studied the first mode parametric excitation and de-rived a a non-dimensional non-linear Mathieu equation for the movement at mid-span of a weightless string[131]; Labeeuw gave minimal values for cable damping to avoid instability due to a time variable cable tension; Takahashi calculated the instability boundaries of the main instability region of the Strutt diagram based on the eigenvalue method; and Hsu and Szemplinska´-Stupnicka[130] have made two important advances in the knowledge of the mathematics of systems of equations with periodic coefficients.

Vortex-induced vibration

Vortex-induced vibrations(VIV) occurs, when the vortex shedding frequency is synchro-nized to the natural frequency of the structure, i.e., the structure frequency would lock-in the vortex shedding frequency. Other significant characteristics of VIV are that the excitation frequency of the fluctuating drag frequency being twice that of the fluctuating lift; hysteresis behavior, deriving from the fluid due to different wake patterns that form in response to the amplitude of oscillation; and the abrupt phase shift between the lift and the cylinder cross-flow displacement. Vortex-induced vibration (VIV), has been investigated over the last two decades, many of which are related to the push to explore very low mass and damping, and to the unavailability of new computational and experimental techniques. New concepts and phenomena generic to VIV systems, pay special attention to the vortex dynamics and en-ergy transfer that give rise to modes of vibration, the importance of mass and damping, the concept of a critical mass, the relationship between force and vorticity, and the concept of “effective elasticity”, among other points [155, 62, 22, 64, 38, 37, 177, 86, 55, 39, 146, 11].

Wake galloping

Wake galloping has been reported to be caused by the wake interference between two cir-cular cylinders [137, 170, 89, 172, 171, 173, 8, 43, 166, 162, 165]. It has been pointed out that accelerated gap flow around the cylinder in the leeward plays an important role for the violent vibration. The characteristics of the vibration against the cable arrangement, wind speed, Reynolds number, and Scruton number have been clarified to some degree. The characteristics are listed as follows, the most significant vibration was observed when the spacing ratio W /D was 4.3 and the incidence angle of wind axis was 15◦; 1st mode was dominant; and dominant axis of the vibratory displacement was changed by the combined effect of Reynolds number and reduced wind speed.

Rain–wind-induced vibration

Rain–wind-induced vibration is hypothesized to be a new type of instability phenomenon, which often occurs by the combined influences of rain and wind, may cause fatigue dam-age and corrosion to cable, and affect the safety of the entire bridge eventually. The main characteristics of RWIV can be summarized as follows,
(1) The occurrence of rain–wind-induced vibration is significantly related with com-bined influences of rain and wind[58];
(2) The occurrence of rain–wind-induced vibration is significantly limited to the cables which geometrically decline in direction of wind;
(3) Rain–wind-induced vibration occurs within the limited range of wind speed, espe-cially in subcritical and critical Reynolds number range;
(4) Rain–wind-induced vibrations are characterized by a much lower frequency than vortex induced-vibration and by a much higher amplitude with the modes of vibration varied from a fundamental to 4th mode and the dominant frequency of vibration felt into the range of 1 Hz to 3 Hz;
8 Introduction
(5) Rainwater gathers and forms rivulets, which oscillating in circumferential direction of cable surface when RWIV occurs.
The rain–wind-induced vibration has been the subject of significant work in recent years to ensure the serviceability and safety of the bridge, and to reduce maintenance costs. Maximum double amplitudes of rain–wind-induced vibrations have been reported as 600 mm(Brotonne Bridge), 1000 mm(Koehlbrand Bridge), 2000 mm(Faroe Bridge), and 1950 mm(Tenpohzan Bridge). The phenomenon is of great interest in this thesis. Therefore, a more detailed literature review on this topic is provided in the next section.

Literature review: rain–wind-induced vibration

Most have focused on rain–wind-induced vibration(RWIV) from the field observation, wind tunnel tests, analytical model, and the numerical investigation aspects.

Field measurement

Under conditions of rain combined with the wind loads, RWIV phenomena prone to occur in the cable-stayed bridges(Fig. 1.2). RWIV was firstly observed and defined during the construction of Meikonishi bridge by Hikami[58], characterized by a much lower frequency around 1 −3Hz than a vortex induced oscillation combined with a much higher amplitude around 55cm under wind velocity 14 m/s combined with the rain. The measurement was carried out by the 24 cables of south-side plane extending from the east-side pylon, which lengths varied from 65 to 200 m. Particular wind-rain-stay cable configuration, participation of the medium intensity rainfall and the limit range of the wind speed were the necessary conditions of RWIV excitation. Furthermore, it was also observed that the attached rainwa-ter would forming rivulet, and oscillating along the circumference of cables.
Fig. 1.2 RIWV phenomena observed in the cable-stayed bridges: Meikonish Bridge (Japan) [58], Fred Hartman Bridge (Houston, USA) [78, 179, 180, 77], and Donating Lake Bridge (China) [93] from left to right.
During a long-term filed measurement at the Fred Hartman Bridge in Houston, Texas, some characterizations of RWIV were present [78, 77]: root mean square displacement am-plitudes were estimated as large as 50 cm among 15 stay-cables; the number of the dominant vibration mode ranged from 1 to 4; the dominant vibration frequencies were between 1 to 3 Hz. Furthermore, the wind direction relative to the stay cables, the yaw angles and inclina-tions of stay cables had significant effects on RWIV phenomena. The wind velocity ranged 4-14.5 m/s and the rain intensity were also considered during the observation.
Observations were investigated further on the Fred Hartman Bridge [179, 180]. Stay cables can exhibit multiple modes due to three-dimensional nature of the cable-wind en-vironment, similar to vortex induced vibration(VIV). The similarities between RWIV and VIV might be due to vortex-induced type of excitation, different from Karman vortex shed-ding, however, RWIV is induced by Karman vortex shedding enhanced by periodic vortices along the stay cable surface remained unclear. The difference between RWIV and VIV was that RWIV occurred at much larger amplitude and at much higher wind speed than it did in VIV, although the amplitude of both did appear to be self-limiting.
A full-scale stay cable model with 30 m long polyethylene pipe for cable-stayed bridges was exposed to natural wind and rain conditions to investigate the RWIV phenomena [84]. Without precipitation, the vibration was induced by the Karman vortex shedding, due to the dominant frequencies of acceleration PSD 14.6 Hz(7th mode) and 18.6 Hz(8th mode) approached to the natural frequency of the stay cable; with precipitation, the vibration, with the dominant frequencies of acceleration PSD 4.0 Hz(3th mode) and 5.96 Hz(4th mode) at high reduced wind velocity, might be considered as the rain-wind–induced vibration.
The field measurements were conducted on the cable-stayed Donating Lake Bridge for continuous 45 days to investigate the RWIV phenomena [93]. The critical mean wind ve-locity producing RWIV were 6-14 m/s; the critical mean wind direction(relative yaw angle) ranged from 10◦ to 50◦; RWIV occurred in light-to-moderate rain(less than 8 mm/h); the cable in-plane acceleration response amplitudes were approximately two times the out-of-plane acceleration response amplitudes in RWIV, furthermore, dominant mode of RWIV was the third mode in all RWIV events; the majority of the response involves participation of the dominant mode as well as other low-order modes.

Wind tunnel tests

In wind engineering, wind tunnel tests are used to measure the velocity around, and forces or pressures upon structures. Very tall buildings, buildings with unusual or complicated shapes (such as a tall building with a parabolic or a hyperbolic shape), cable suspension bridges or cable stayed bridges are analyzed in specialized atmospheric boundary layer wind tunnels.
There is a long upwind section of the wind tunnel to accurately reproduce the wind speed and turbulence profile acting on the structure. Wind tunnel tests provide the necessary design pressure measurements in use of the dynamic analysis and control of tall buildings. For RWIV phenomena, two ways are divided for investigation: Artificial rain infall wind tunnel tests, which employed the rain infall instrument to simulate the real process of the natural rain; Artificial rivulets wind tunnel tests, which assumed a fixed solid attached on the cable surface to simulate the rivulets effect.
Artificial rainfall wind tunnel tests for RWIV
To simulate the real rainfall process, to control the physical parameters during rainfall, i.e., rain intensity and the diameters of the rain droplets, and to reveal the rain factors influencing the whole process of RWIV, the rainfall instruments were incorporated in wind tunnel test, so called artificial rainfall wind tunnel tests for RWIV. The artificial rainfall wind tunnel tests parameters, including the size of wind tunnel, the size, materials, mass, damper, frequency, the wind inflow velocity, and the maximum displacement of the cable, are listed in Table 1.1 and the scheme for simulating the process of natural rain was employed either by sprinkling water from the showers near the top boundary [147] or form the top plastic tube injecting on the cable [5](Fig. 1.3).
Fig. 1.3 The scheme for simulating the process of natural rain was employed either by sprinkling water from the showers near the top boundary (left-side) [147] or form the top plastic tube injecting on the cable (right-side) [5] in the wind tunnel experiments.
The roles of rain on the occurrence of the vibration were investigated by reproducing the rain condition in the wind tunnel tests through spraying water by Hikami [58]. To investigate the details of RWIV through the cable’s responses with rain and without rain conditions, the RWIV phenomena was validated by the wind tunnel tests. Furthermore, the influence of the cable frequency varying from 1 to 3 Hz was conducted to capture the details of RWIV, i.e., the relationship between wind speed and the cable’s amplitude, the locations of the rivulets adhering to the cable surface, and try to clarify the possible mechanism of RWIV phenomena; the configuration of wind-rain-cable structure, the rivulets oscillating during the RWIV occurred.
A series of wind tunnel tests were conducted, and the fundamental aerodynamic charac-teristics of a yawed and inclined circular cylinder with and without rain were investigated [82, 83, 81, 85]. An intense secondary axial flow and the formation of the upper rivulet af-fects significantly RWIV phenomena. The secondary axial flow resulted in an aerodynamic exciting force acting on the yawed/stay cable formed in the early wake, similar role a splitter plate submerged in the wake; the role of rain is considered as an amplifier of the essential unstable aerodynamic exciting force acting on the yawed/stay cable.
The fundamentally different exciting mechanisms of RWIV of cables or steel bars were explained through wind tunnel tests [147]. The frequency of model up to 8.9 Hz with the wind speed above 8 and up to 30 m/s, RWIV phenomena were observed in the cross-wind direction as well as in the wind direction, depending on the wind-rain-cable configuration, i.e., yaw and inclination angles of the cable and the wind speed. When the wind force act-ing in the wind direction, changing in the rhythm of the natural frequency was caused by a rhythmic shift of the separation lines due to the shift of the rivulets on the cable surface, the wind direction vibration occurred; the lateral force was changing due to the changeable section of the cable with one or two lateral rivulets and by the rhythmic shift of the separa-tion lines, and the cross-wind direction vibration occurred. Furthermore, the cable surface and the permanently changing rivulets’ shape locked-in each other with the same frequency. Additionally, RWIV phenomena were identical to a self-excited oscillation.
Full scale for a section of the stays of the Normandie Bridge were conducted in the wind tunnel [42] under several materials, shapes and surface coatings. RWIV phenomena were re-produced and identified with standard polyethylene cylindrical smooth casing at wind speed from 7 to 13 m/s. The prevalent role of the upper rivulet was verified. Additionally, dirt coating of the cable surface was sensitive to formation and evolution of the upper rivulet, compared with clean or greasy cable surface, consequently easier to excite the RWIV phe-nomena; a spiral disorganizing the movement of the upper rivulet every 0.3 m is efficient enough to stabilize the cable to decrease the vibration.
The stay cable with high-density polyethylene was conducted in the wind tunnel [30]. The pressure distribution around the cable surface was firstly proposed, meanwhile, the thickness of water film was proposed to be captured by means of the resistance wire dis-tributed around the cable surface.
RWIV phenomena were reproduced under conditions of spraying water onto the cable model to simulate the natural rain process [51]. Parametric analysis, i.e., inclination angle, frequency, yaw angle, and damping of the cable, was performed. Additionally, the velocity-and amplitude-restricted vibration, i.e., RWIV, was considered to occur based on the forma-tion of the upper rivulets and its motion around the cable surface.
RWIV were systematically investigated through the measurement performed for the for-mation and behavior of water rivulets [5]. Flow was classified into five categories for dif-ferent Reynolds number with varied stagnation line; the circumferential oscillation of water rivulets influenced on the fluid forces and near the wake of the cylinder more considerably and significantly than steady, without oscillating cases; and when the vortex-shedding fre-quency approached the natural frequency of the circumferentially oscillating rivulets, the rivulet-vortex-induced vibration caused RWIV phenomena.
Initial attempts to capture the upper rivulet, used an ultrasonic transmission thickness measurement system(UTTMS)(Fig. 1.4)–capable of measuring time-dependent spatial distribution of rainwater around the surface of an stay cable [70]. The location, geometry, and oscillation data of rainwater rivulets were recorded and analyzed (Fig.1.5). Additionally, quantitative investigation of the upper rivulet oscillation indicated that the oscillation fre-quency of the upper rivulet, obtained by wind tunnel experiment, approached to the natural frequency of the segment model, and the RWIV phenomena were reproduced.
Fig. 1.4 The ultrasonic transmission thickness measurement system(UTTMS)–capable of measuring time-dependent spatial distribution of rainwater around the surface of an stay cable [70].
Fig. 1.5 The location, geometry, and observed data of rainwater rivulets were recorded by the UTTMS [70].
To capture the details of the rain roles in RWIV, i.e., the thickness, and the positions of the rainwater films etc., artificial rainfall in the wind tunnel experiments were conducted(Fig. 1.6), however, to obtain the aerodynamic force during RWIV, the artificial rivulets were incorporated in wind tunnel test, and the rain effect was considered as a solid adhering to the stay cables.
Fig. 1.6 The artificial rivulets method to capture the details of the RWIV phenomena in the wind tunnel experiments [52].
Artificial rivulet wind tunnel tests for RWIV
Flexural rivulet adhering to the cable surface combined with the cable were conducted at the wind speed around 10 m/s in the wind tunnel [163]. The frequency of oscillation of upper rivulet approached the natural frequency of the cable motion, and thus the RWIV phenomena were reproduced. Two possible reasons for RWIV were presented either by quasi-steady Den Hartog galloping theory or by rivulet oscillation along the circumference of the cable.
The rivulets were simulated by two bars placed at different locations on the cable surface in the wind tunnel tests [21]. Changing the shape and the position of the two bars were employed to reproduce RWIV phenomena. RWIV phenomena were caused by the presence of the two bars; RWIV corresponded to galloping and were independent of bar shapes and dimensions, within the limits of the tested configurations.
To explore the mechanism of RWIV and its mitigation, a section model was adopted in the wind tunnel experiments [175]. Artificial rivulets were used to reproduce and validate the RWIV phenomena. The important parameters, i.e., yaw angle, inclination angle, rain-fall were performed and analyzed. The velocity-restricted and amplitude-restricted RWIV phenomena were observed, with the amplitude more than 180 mm, in addition, the rain-fall intensity, cable damping ratio, inclination angle, and yaw angle of the cable affected significantly RWIV phenomena, and the control schemes were also presented.
A 3-D sectional cable model was employed to investigate RWIV phenomena [53, 52]. The pressure and aerodynamic force acting on the cable model and the upper artificial rivulet were obtained from the wind tunnel tests. Through the variation of the yawed angle and inclination angle factors, violent variations of the wind forces acting on the cable model could be the key aerodynamic factors of RWIV phenomena.

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Theoretical analysis

Three ways were divided to investigate the mechanism of RWIV phenomena from the ana-lytical model aspect. The multi-degree-of-freedom method, which were coupled the equa-tions of stay cables model and rivulets dynamic model based on the aerodynamic force and pressure distribution around the stay cable obtained from the wind tunnel experiments, was employed to clarify the characteristics of the rivulets dynamic and the cable motions; the single-degree-of-freedom method, which simplified the rivulets motion as a series of simple harmonic oscillation mixed to be substituted to the cable dynamic equations, was used to capture the characteristics of the cable motion during RWIV phenomena; the zero-degree-of-freedom method, which ignored the rivulets motion effects, just considering the rivulets affecting the flow structure when the wind flowed past the stay cable, was adopted to clarify the mechanism of the RWIV phenomena.

Table of contents :

1 Introduction 
1.1 Background and motivations
1.2 Research objectives
1.3 Assumptions and limitations
1.4 Cable vibrations in cable-stayed bridge system
1.4.1 Parametric excitation vibration
1.4.2 Vortex-induced vibration
1.4.3 Wake galloping
1.4.4 Rain–wind-induced vibration
1.5 Literature review: rain–wind-induced vibration
1.5.1 Field measurement
1.5.2 Wind tunnel tests
1.5.3 Theoretical analysis
1.5.4 Numerical simulation
1.6 Literature review: multiphase interface tracking algorithm
1.6.1 Advecting the color function
1.6.2 Front tracking method
1.6.3 Level set method
1.6.4 Phase field method
1.6.5 CIP method
1.6.6 Volume of fluid method
1.7 Research contents
2 Numerical simulations for RWIV of stay cables based on the separated method 
2.1 Introduction
2.2 Assumptions and limitations
2.3 Governing equations of the separated method for RWIV of stay cables
2.4 Boundary conditions and parameter selections
2.4.1 Boundary conditions
2.4.2 Parameter selections
2.5 Numerical simulations for RWIV of stay cables based on the separated method 33
2.5.1 Vortex shedding characteristics analysis
2.5.2 Pressure distribution characteristics analysis
2.5.3 Aerodynamic forces characteristics analysis
2.6 Brief summary
3 Numerical simulations of RWIV based on semi-coupled and coupled methods 
3.1 Introduction
3.2 Assumptions, limitations and preparations for numerical model
3.2.1 Assumptions
3.2.2 Limitations
3.2.3 Preparations
3.3 Numerical simulation of RWIV based on semi-coupled method
3.3.1 Governing equation for semi-coupled method
3.3.2 The rainwater morphology evolution on a stay cable
3.4 Numerical simulation of RWIV based on the coupled method
3.4.1 Governing equation for the coupled method
3.4.2 Computational domain decomposition, boundary conditions and parameter selections
3.4.3 Rainwater morphology evolution validation on a stay cable subjected to gravity and surface tension
3.4.4 Rainwater morphology evolution on a stay cable subjected to gravity, surface tension and wind
3.5 Brief summary
4 Multiphase and multiscale model for rain–wind-induced vibrations(RWIV)
4.1 Introduction
4.2 Multiphase and multi-scale model
4.2.1 Governing equations for incompressible background fluid
4.2.2 Method for tracking Lagrangian particles
4.2.3 Two-way coupling term and transformation rules
4.3 Numerical schemes for multiphase and multiscale model
4.3.1 General method
4.3.2 Temporal discretization
4.3.3 Spatial discretization
4.3.4 Smoothness for the source term in LES zone for Lagrangian particle tracking
4.4 Numerical considerations for multiphase and multiscale model
4.4.1 Rain model
4.4.2 Transformation from the physical model to computational model
4.4.3 Boundary conditions
4.4.4 Parameter selections
4.4.5 Parameter definitions
4.4.6 Modeling limitations
4.5 Brief summary
5 Rainwater morphology and aerodynamic characteristics of a cylinder in RWIV
5.1 Introduction
5.2 Macroscopic analysis of the lift/drag force coefficient evolution
5.3 Collision–splashing pattern
5.3.1 Rainwater morphology evolution analysis
5.3.2 Aerodynamic characteristics analysis
5.4 Accumulation–slipping pattern
5.4.1 Rainwater morphology evolution analysis
5.4.2 Aerodynamic characteristics analysis
5.5 Formation–breaking pattern
5.5.1 Rainwater morphology evolution analysis
5.5.2 Aerodynamic characteristics analysis
5.6 Dynamic equilibrium state
5.6.1 Rainwater morphology evolution analysis
5.6.2 Aerodynamic characteristics analysis
5.7 Brief summary
6 Conclusions and Recommendations 
6.1 Conclusions
6.2 Recommendations
References 

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