Evolution laws of distributed vortex-induced pressures and energy of a flat-closed-box girder via numerical simulation

Abstract

This article investigates the vertical vortex-induced vibration of a flat-closed-box girder using numerical simulation method. The accuracy of simulation results is verified at first by comparing the displacement responses and vortex-induced force of vertical vortex-induced vibration with those obtained in a previous wind tunnel test of large-scale sectional model. The precision of extracting the vortex-induced pressures from the surface pressures and decomposing the vortex-induced pressures via the mathematical model is validated later. Subsequently, the vortex-induced pressures and energy distribution, and the evolution laws of vortex-induced pressures and energy are discussed. The results show that the linear aerodynamic negative damping and nonlinear aerodynamic positive damping are key factors of the rapid development of vortex-induced vibration and the self-limiting phenomenon separately. The positive aerodynamic damping is mainly provided by the lower surface and the middle of the upper surface, and the negative aerodynamic damping is primarily provided by the middle and downstream of the upper surface.

Keywords

aerodynamic damping pressure energy distribution evolution law flat-closed-box girder numerical simulation vortex-induced pressure vortex-induced vibration

Introduction

With the increase of bridge span length, the stiffness and stability of modern long-span bridges decrease obviously. Thus, many long-span bridges, such as the Storebelt Bridge (Larsen et al., 2000) and the Xihoumen Bridge (Laima et al., 2013; Li et al., 2011), are often subjected to violent vortex-induced vibration (VIV). Results observed on real bridges and in wind tunnel test show that VIV is a nonlinear wind-induced vibration with limited amplitude and does not cause disastrous consequences, but it can cause discomfort to drivers and even fatigue problems to structural members of bridges. Therefore, VIV has being taken seriously and some classical research about VIV (Scanlan, 1981; Simiu and Scanlan, 1996; Ehsan and Scanlan, 1990; Larsen, 1995) was made.

Computational fluid dynamics (CFD) method has been extensively used in the research fields of bridge aerodynamics and aeroelastics for its advantages of visualization and convenience. Sun et al. (2009) attested that aerodynamic coefficients can be acquired through CFD with k−ω turbulence model, and this turbulence model can well balance the computational efficiency and accuracy. Hallak et al. (2013) discussed the aerodynamic behaviors of a bridge deck with tall vehicle through two-dimensional (2D) numerical simulation, while there are some differences between the simulation results and the test results in the critical wind speed, which may be due to the small number and large size of grids. He and Li (2015) investigated the effects of some aerodynamic measures on VIV of a separate twin-box girder based on both wind tunnel tests and numerical simulations, but the results of the two methods are not compared. In general, numerical simulation is affected by human subjective operation, and reasonable setting of grid size, turbulence model, and other parameters can keep a good balance between calculation accuracy and calculation efficiency. In order to ensure the correctness of CFD simulation, it is necessary to verify the simulation results, and the comparison objects include analytical solutions, measured data, test results, or data in existing research.

Although there have been a great deal of researches on VIV of bridges, most of them have focused on the responses of VIV while only a few researches have involved in the nonlinear mechanisms of the VIV and the vortex-induced force (VIF). Diana et al. (2006) indicated that the VIF on a multi-box deck has nonlinear characteristics significantly. Sarwar and Ishihara (2010) discussed the vibration reduction mechanism of VIV of a bridge deck with some countermeasures. Zhu et al. (2013, 2015, 2017) have discussed the VIF on typical box decks from a macro perspective, and two important conclusions were obtained. First, the negative aerodynamic damping is the major source of power driving the development of the vertical VIV. Second, the positive nonlinear aerodynamic damping is the inherent key factor leading to the self-limited amplitude phenomenon of the vertical VIV. However, the investigation on the nonlinear mechanism of VIV has been rarely carried out from a meso-perspective of the vortex-induced pressure (VIP) distribution. Chen et al. (2018) discussed the contribution of location to VIF for a flat steel box girder. However, due to the phase difference between the force and displacement, the value of force could not represent the real energy contribution. Xu et al. (2017) carried out a wind tunnel test, and the relationship between the hysteresis phenomenon and the saddlenode bifurcation were explored based on the Poincare mapping techniques and state-space reconstruction. In summary, it has been not clear in which zones of the deck surface the VIPs provide the negative linear aerodynamic damping or the positive nonlinear aerodynamic damping, and this is just the aim of this study. To emphasize, it is significant to ascertain the contributions of VIPs in various zones of the deck surface to the VIF and the VIV response, because of its ability to carry out target-oriented inventions of proper and effective aerodynamic countermeasures for VIV mitigations. However, relevant researches have still been very limited till now.

This article is organized as follows. In section “Numerical simulation and verification,” the CFD method and numerical model are to be introduced first, and the numerical simulation results are then verified. Section “Extraction, decomposition, and verification of VIPs” presents the processes to extract the VIPs from the surface pressures and then to decompose the VIPs via mathematical model. Meanwhile, the correctness of the processes and the accuracy of the key components of VIPs are verified. In section “Evolution laws of VIPs and energy,” the distribution of VIPs and the evolution law of distributed VIPs at a cycle of different stages of VIV are discussed first. And then the energy distribution and evolution law of the VIPs and key components of VIPs are focused on. Some conclusions of this study are finally drawn in section “Conclusion.”

Numerical simulation and verification

CFD model

Figure 1(a) shows a schematic view of the flat-closed-box girder considered in this article. The width B is 1.6 m, and the height of the box girder is D, which is 0.26 m. α presents positive angle of attack for the box girder, which is +5° in this study. Moreover, the mass of per unit length of this flat-closed-box girder is 50.61 kg, and the vertical natural frequency and structural damping ratio of this flat-closed-box girder are 2.807 Hz and 0.5%, respectively. The computational domain and boundary conditions are shown in Figure 1(b). The length of computational domain is 17 B in the mean-flow direction and 28 D in the cross-flow direction. To simulate the positive angle of attack, the left-lower directions of the girder are assumed as the sources of the upcoming wind, and the right-upper directions are assumed as the outlet of the wind flow. The boundary conditions are set as follows: the left and lower boundaries are the velocity inlets with a uniform wind speed of 9.1 m/s, a turbulent intensity of 0.02 and a turbulent viscosity ratio of 2; the right and top boundaries are the pressure outlets where the relative pressure is set as zero; the surface of the flat-closed-box girder is defined as no-slip wall boundary. All the aforementioned parameters are assumed as consistent as possible with the wind tunnel tests in Zhu et al. (2013).

Figure 1.

Computational domain and mesh: (a) schematic view of flat-closed-box girder, (b) computational domain, and (c) local computational mesh.

As can be seen from Figure 1, the computational domain is comprised of the following four parts: the rigid zone, the dynamic-mesh zone, the wake zone, and the outer zone. The rigid zone is filled with quadrilateral grids and moves synchronously with the bridge girder to guarantee the accuracy of the simulation results near the girder. The size of the dynamic-mesh zone is 3B × 10D, and it is filled with a dynamic mesh which is triangular grids. To capture the force and response of the box girder under the effect of wake flow, the quadrilateral grids in wake zone are densified. The outer domain far away from the girder is filled with larger quadrilateral girds to improve the computational efficiency. Furthermore, around the surface of the flat-closed-box girder, 15 densified boundary layers are set, and the maximum y⁺ is less than 1 in this study. It should be noted that the detailed independence study of grid and time step of this simulation has been carried out by Chen et al. (2018) and Li et al. (2018). Thus, the computational mesh is generated as shown in Figure 1(c) and the time step is set equal to 0.000445 s. In addition, the detailed information of how the motions of the model are modeled in this simulation was also introduced by Chen et al. (2018).

In this study, the CFD software FLUENT is employed, and the SST−k−ω model is adopted for Reynolds average numerical simulation. The governing equations are discretized using the second-order scheme based on the finite-volume method. The SIMPLEC algorithm is employed for pressure–velocity coupling. The pressure is discretized in the standard format. The momentum, turbulent kinetic energy, and specific dissipation rate are also discretized by utilizing the second-order form. Gradient terms are handled using a least-square cell-based method.

Simulation results and verification

For having a more reasonable comparison, the following linear relationship between the damping ratio and the vibration amplitude fitted by Zhu et al. (2013) based on the measured data in the wind tunnel test was adopted for the motion-dependent mechanical damping ratio of the sectional model system in the numerical simulation of VIV in this sub-section

ξ = 0.24434 a (t) + 0.00158

where ξ is the structural damping ratio; a(t) is the transient amplitude of vibration, and defined as follows

a (t) = \sqrt{y^{2} (t) + {\dot{y}}^{2} / ω_{0} ​^{2}}

where ω ₀ is the dominant circular frequency of vibration.

Both the simulated and tested stable displacement time histories and their amplitude spectrums are shown in Figure 2 for comparison. It can be seen that the simulated displacement agrees fairly well with the measured one for both the time history and spectrum. The discrepancy of the accumulated phases of the simulated and measured time histories is due to the small discrepancy between the simulated and measured frequencies.

Figure 2.

Comparison between simulated and tested displacements and VIFs at stable stage of VIV: (a) time history of displacement, (b) amplitude spectrum of displacement, (c) time history of VIF, and (d) amplitude spectrum of VIF.

Besides the vertical displacements, the simulated and measured VIFs are also compared in this sub-section for verifying the numerical simulation. According to Zhu et al. (2013), VIF can be extracted from the lift through the following formulas

{\tilde{f}}_{​_{V}}^{0} (t) = - m_{0} {\ddot{y}}^{0} (t) - c_{0} {\dot{y}}^{0} (t)

f_{V I} (t) = {\tilde{f}}_{V} (t) + m_{0} \ddot{y} (t) + c_{0} \dot{y} (t)

where the superscript “0” represents the zero wind speed condition; ${\tilde{f}}_{_{V}}^{0}$ is the fluctuating lift force acting on the girder model per unit length during a free decay vibration excited by an initial displacement under the zero wind speed condition; ${\ddot{y}}^{0}$ and ${\dot{y}}^{0}$ are the free decay responses of acceleration and velocity of the girder model under the zero wind speed condition; m ₀ and c ₀ are the non-wind-induced additional mass and damping coefficients of the girder model per unit length, respectively, and can be determined via least-square fitting using the target function as shown in equation (3) and the simulated data of ${\tilde{f}}_{_{V}}^{0}$ , ${\ddot{y}}^{0}$ and ${\dot{y}}^{0}$ ; ${\tilde{f}}_{V}$ and $f_{V I}$ are the total fluctuating lift force and the vortex-induced lift force acting on the deck model per unit length during the VIV at a certain wind speed, respectively; $\ddot{y}$ and $\dot{y}$ are the VIV responses of acceleration and velocity of the girder model, respectively. ${\tilde{f}}_{_{V}}^{0}$ and ${\tilde{f}}_{V}$ can be obtained by integrating the simulated surface dynamic pressures. The fitted values of m ₀ and c ₀ in this simulation case are 6.051 kg/m and 3.098 N s/m² separately.

Figure 2(c) shows the comparison between the simulated and measured time histories of VIF, while Figure 2(d) displays the comparison between the amplitude spectra of the simulated and measured VIFs. It can be seen that the simulated VIF also agrees fairly well with the tested one for both the time history and spectrum. One can also find that the curve pattern of the simulated VIF is stable but that of the tested VIF is time-varying to some extent possibly because of measurement error or unstableness of wind field.

Furthermore, it can be found from Figure 2(c) and (d) that both the curves of the simulated and measured VIF time histories are remarkably distorted from normal sinusoidal curves and there are significant multiple-frequency components in VIF. This indicates out that the vertical VIF on the flat-closed-box girder has stronger nonlinearity than the displacement response of VIV.

Extraction, decomposition, and verification of VIPs

Arrangement of pressure points

In order to discuss the mechanisms of VIV of the flat-closed-box girder, it is needed to acquire the distribution characteristics of pressures on the surface of the girder. To this end, 302 pressure points are set non-uniformly on the surface of the girder cross-section as shown in Figure 3, to pick out the time histories of the fluctuating surface pressures from the simulated results. It can be seen that the pressure points are distributed intensively at the corners and the webs of this girder.

Figure 3.

Schematic diagram of pressure points on the cross-section of girder.

Extraction and verification of VIPs

Similar to the total VIF, which can be determined by equations (3) and (4), the VIPs at the 302 pressure points can also be extracted from the total fluctuating pressures by the following formulas

{\tilde{p}}_{V, i}^{0} (t) s_{i} = {\tilde{p}}_{i}^{0} (t) s_{i} sin β_{i} = - m_{0, i} {\ddot{\tilde{y}}}^{0} (t) - c_{0, i} {\dot{\tilde{y}}}^{0} (t)

f_{V I, i} (t) = p_{V I, i} (t) s_{i} = {\tilde{p}}_{V, i} (t) s_{i} + m_{0, i} \ddot{\tilde{y}} (t) + c_{0, i} \dot{\tilde{y}} (t)

where, i, ranging from 1 to 302, is the order number of the pressure point; ${\tilde{p}}_{i}^{0}$ is the total fluctuating pressure strength at the ith point during a free decay vibration excited by an initial displacement under the zero wind speed condition, and ${\tilde{p}}_{V, i}^{0}$ is the vertical component of the pressure strength in the body axis coordinate system; β_i is the angle from the horizontal axis of the cross-section to the outward normal $({\vec{n}}_{p})$ of the exterior profile of the cross–section at ith pressure point; m _0,i and c _0,i are, respectively, the non-wind-induced additional mass and damping coefficients at the ith pressure point and can be determined by least-square fitting; $f_{V I, i}$ is the VIP at the ith point; ${\tilde{p}}_{V, i}$ and $p_{V I, i}$ are, respectively, the total fluctuating pressure strength and the VIP strength at the ith point during the VIV at a certain wind speed; s_i is the length of the control zone of the ith pressure point, and for each pressure point, the control zone contains two parts: one is the half space between the target point and the previous one and the other one is the half space between the target point and the next one.

Based on equation (6), the VIPs at each pressure point on the box girder can be obtained. As the cross-section is supposed to be rigid, the vertical velocities and accelerations at all pressure points on the deck surface are fully correlated or synchronous. On this account, the total VIF can be calculated via numerical integral of VIPs along the cross-section profile, that is

f_{V I} (t) = \sum_{i = 0}^{302} f_{V I, i} (t) = \sum_{i = 0}^{302} p_{V I, i} (t) s_{i}

The correctness of the above process can be verified by comparing the total VIF obtained by the integral of VIPs with that extracted directly from aerodynamic lift and judging whether they are equal or not, as shown in Figure 4. It can be found that the time history of numerical integral of VIPs along the cross-section profile shows a close agreement with that of total VIF at any stage of VIV, which illustrates that the method of extracting VIPs from fluctuating surface pressures is appropriate and also accurate.

Figure 4.

Comparison between the time history of total VIF and that of numerical integral of VIPs along the cross-section profile: (a) development stage and (b) stable stage.

Decomposition of VIPs

In the previous section, the extraction process of the VIP is described and the reliability of the process is verified. However, in order to further explore the distribution and characteristics of the components of the VIPs, it is necessary to decompose the VIP and obtain the components of the VIP.

According to Chen et al. (2018), for this type of girder, the VIF can be expressed as

f_{V I} = ρ U^{2} D [\begin{array}{l} P_{10} {\dot{y}}_{*} + P_{01} y_{*} + P_{20} {\dot{y}}_{*}^{2} + P_{11} {\dot{y}}_{*} y_{*} + P_{02} y_{*}^{2} + P_{30} {\dot{y}}_{*}^{3} \\ + P_{21} {\dot{y}}_{*}^{2} y_{*} + P_{03} y_{*}^{3} + P_{13} {\dot{y}}_{*} y_{*}^{3} + V_{s} sin (ω_{s} t + ϕ) \end{array}]

where the last item is vortex-shedding force (VSF); ρ is the air density; U is the wind speed; D is the height of the girder; ${\dot{y}}_{*}$ is the dimensionless velocity of VIV $({\dot{y}}_{*} = \dot{y} / U)$ ; $y_{*}$ is the dimensionless displacement of VIV $(y_{*} = y / D)$ ; P_ij , V_s , $ω_{s}$ and ϕ are undetermined constants. It can be seen that the VIF is comprised of several components.

Similarly, the VIPs can be decomposed in the same way, which is

f_{V I, i} = ρ U^{2} D [\begin{array}{l} P_{10, i} {\dot{y}}_{*} + P_{01, i} y_{*} + P_{20, i} {\dot{y}}_{*}^{2} + P_{11, i} {\dot{y}}_{*} y_{*} + P_{02, i} y_{*}^{2} + P_{30, i} {\dot{y}}_{*}^{3} \\ + P_{21, i} {\dot{y}}_{*}^{2} y_{*} + P_{03, i} y_{*}^{3} + P_{13, i} {\dot{y}}_{*} y_{*}^{3} + V_{s, i} sin (ω_{s, i} t + ϕ_{i}) \end{array}]

When the wind angle of attack is 5°, the airflow near streamlined box girder is disturbed by the central guardrail, which may cause a large pressure gradient in the central part of the upper surface. Therefore, the reliability of the VIP decomposition method is verified by taking the 108th pressure point behind the central barrier as an example. The comparison between the time history of the VIP and the reconstituted VIP via Chen’s model at this point is shown in Figure 5, where, the nonlinear least-square fitting method was adopted. From Figure 5, it can be seen that the reconstituted VIP is in conformity with the simulated one at any period of VIV.

Figure 5.

Comparison between the VIP of the 108th pressure point and the reconstituted VIP via Chen’s model: (a) development stage and (b) stable stage.

Although the accuracy of the decomposition of VIPs via Chen’s model was verified earlier, it is yet to be verified whether the numerical integral of each decomposed component of VIPs along the cross-section profile is consistent with the corresponding decomposed component of VIF. As reported by Zhu et al. (2013, 2015), the linear aerodynamic negative damping force (LADF) and VSF provide energy for VIV, and the nonlinear aerodynamic positive damping force (NADF) dissipates energy during VIV. Therefore, just the linear aerodynamic negative damping pressures (LADPs) which are equal to $P_{10} {\dot{y}}_{*}$ , the nonlinear aerodynamic positive damping pressures (NADPs) which correspond to $P_{30} {\dot{y}}_{*}^{3}$ , and the vortex-shedding pressures (VSPs) would be discussed in further analysis. Meanwhile, it needs to be emphasized that the VSF not only caused by the shedding of vortices but also the generation and moving of vortices.

Figure 6 displays the comparison between the time histories of the key components of total VIF and those of integral of VIPs. It can be seen that the two times histories of each key component agree well with each other for the overall trend. Although there are small differences existed because that the numerical integral of 302 pressure points causes the error superposition, the small error is acceptable. Moreover, from Figure 6, it can be found that the fluctuation frequencies of those three key components are similar, and that the LADP has synchronous fluctuation with the VSF, but the fluctuation of NADP is in the opposite with that of LADP and VSF. This phenomenon is also consistent with the conclusions of Zhu et al. (2013, 2015).

Figure 6.

Comparison between the time histories of key components of total VIF and those of numerical integral of VIPs along the cross-section profile: (a) LADF at development stage, (b) LADF at stable stage, (c) NADF at development stage, (d) NADF at stable stage, (e) VSF at development stage, and (f) VSF at stable stage.

It can be found that it is correct and pretty accurate to extract the VIPs from the fluctuating pressure of the 302 pressure points and decompose the VIPs distributed along the cross-section profile via Chen’s model, and that the distributed LADPs and NADPs and VSPs are relatively precise. In summary, it can then be concluded that the methods and processes and results described in this section are applicable and reliable.

Evolution laws of VIPs and energy

Evolution law of VIPs

Figure 7 shows the vertical displacement response of VIV. In order to explore the evolution laws of VIV, the process of VIV is divided into three stages: the first one is the stage of rapid development, in which the amplitude of VIV of box girder increases rapidly and the envelop curve of the amplitude concaves upward; the second stage is the gradual convergence stage, in which the increase rate of the amplitude of VIV gradually slows down and the amplitude tends to be stable, and the inflection point of the contour is the critical state between the first and the second stage, the envelop curve of the amplitude concaves downward in this stage; the third stage is the stable vibration period. Subsequently, one period is taken as example in the following discussion for each stage of the VIV, which is represented by a blue line in Figure 7, and the development process of VIPs distributed along the cross-section profile are shown in Figures 8 to 10. In these figures, the distribution lines of pressure are inside the box profile when the pressures are positive and vice versa. That is, for the upper half of the box girder, the positive VIP is the pressure with downward direction, and the negative VIP is the pressure with upward direction, but the opposite is true for the lower half of this girder.

Figure 7.

Vertical displacement response and stage division of VIV.

Figure 8.

Development process of distributed VIPs during one cycle at the first stage of VIV.

Figure 9.

Development process of distributed VIPs during one cycle at the second stage of VIV.

Figure 10.

Development process of distributed VIPs during one cycle at stable stage of VIV: (a) 0∼π/2, (b) π/2∼π, (c) π∼3π/2, and (d) 3π/2∼2π.

The rapid development stage of VIV

Figure 8 shows the distribution of the VIPs at four typical moments in the period A of the VIV of box girder. When the box girder is moving upward at the equilibrium position, the distribution of the VIP is shown with a red solid line in this figure. It can be seen that most pressure points are subjected to upward pressure, and the VIPs in the middle part of the upper surface of box girder are larger than others.

At the moment with a phase of π/2, the distribution of VIPs is represented by a green dash line in Figure 8. Compared with the distribution of the VIPs at the moment with a phase of 0, it can be seen that the action direction of the VIP in the upper part of the box girder is still upward, while the action direction of the other parts is downward. The range of negative pressure zone on the upper surface increases, but the absolute value of negative pressure decreases.

At the moment with a phase of π of Cycle A, VIPs are represented with blue dot dash line in Figure 8. It can be seen from the figure, the absolute negative pressure on the windward side of the upper surface decreases slightly. The VIPs on the leeward side of the upper surface change from negative to positive, and the absolute value of these pressures increase.

The magenta double-dot-dash line represents the distribution of VIPs at the moment with a phase of 3π/2. In the region of leeward upper surface close to midpoint, the absolute value of VIPs decreases. Except that, the absolute values of the pressures on the upper surface of the box girder increase, whether the pressure is positive or negative. The VIPs on the lower surface change from negative to positive.

Based on the above analysis, it can be obtained that except the windward area on the upper surface, the direction of VIPs in most areas of box girder is consistent with motion direction of the box girder during most of the period of VIV.

The gradual convergence stage of VIV

In the limiting stage, the amplitude of VIV gradually converges, and the distribution and variation of VIPs in period B are quite different from that in the first stage, as shown in Figure 9.

When the box girder moves up to the equilibrium position, it can be seen that the upper surface of the box girder is dominated by negative pressure, and the large negative pressures are concentrated in the middle of the box girder and the leeward side of the upper surface. There are small positive pressure regions and values on the upper surface. The direction of VIPs acting on the oblique web and the bottom plate is upward. From the upwind side to the leeward side, the VIPs gradually decrease, while the oblique web of the leeward side is subjected to negative pressures.

At the π/2 moment of Cycle B, the VIPs on the top of the windward side of the upper surface change to positive pressure, and the absolute negative pressures on the leeward upper surface increase slightly. The VIPs of the oblique web on the windward side of the lower surface decrease greatly and change from positive pressure to negative pressure. At the same time, the pressures on the lower surface also decrease slightly.

As can be seen from this figure, at the moment with a phase of π, the VIPs on the top of the upper surface are almost positive pressure, and the positive pressure in the central region is the largest. The VIPs of the oblique web on the leeward side of the lower surface is very small, and the pressure around the nose of wind fairing is more prominent. The direction of VIPs in the rest of the box girder is downward, and compared with that of π/2 moment, the value of VIPs is obviously larger.

Subsequently, at the moment with a phase of 3π/2, except for the top of the leeward upper surface, the direction of the VIPs in the other areas of the box girder is upward, and the pressures of the lower oblique web are larger. There are positive and negative VIPs on the upper surface of windward side, in which, the front edge of box girder is mainly positive pressure, and the negative pressure in front of the central anti-collision guardrail is more prominent. In addition, the VIPs on the leeward side of the upper surface are positive, while the VIPs on the tail of box girder are very small.

It can also be found from the figure that the action direction of VIPs on the surface of box girder is almost synchronized with the moving direction of the box girder in a period of VIV, except for the upper surface area on the windward side. The areas with large pressure values include the upper central and leeward of the upper surface, the lower oblique web and the bottom plate. In addition, there are significant differences in the distribution and variation of the VIP in the gradual convergence stage compared with those of the rapid development stage of VIV. The fluctuation range of VIPs is more intense, the change of the VIPs on the upper surface of the windward side becomes more disordered, and the area where the positive and negative pressure alternate increases obviously.

The stable vibration stage of VIV

In order to facilitate a more detailed observation of changes in the distribution of VIPs at the stable stage of VIV, one cycle of the stabilization phase of VIV is divided into eight stages, as shown in Figure 10.

As shown in Figure 10, in the process of the girder moving from the equilibrium position to the positive displacement peak, the upstream of the upper surface is subjected to positive pressures, and the range and value of positive pressures increase gradually. The VIPs on the downstream of the upper surface are negative, and the range and value of positive pressures decrease gradually. The positive pressures on the underside of downstream web decrease and turn to negative.

In the process of the girder moving from the positive displacement peak to the negative displacement peak, for the upstream of the upper surface, the positive VIPs with pretty large value slowly decrease and then almost turn to negative pressures in this process. The value of negative VIPs gradually decrease and turn to positive VIPs with large value. The VIPs on the lower surface are upward when the girder reaches the positive displacement peak, then decrease and change to downward pressures quickly, and then the downward pressures turn to upward pressures when the girder reaches the negative displacement peak. The pressures on the upper of upstream web are small and the magnitudes of the changes are not significant. For the downstream webs, the change of VIPs are complex, the positive pressures change to negative pressures quickly, and then rapidly turn to positive pressures again.

By comparing with the distribution and variation law of the VIPs in the first two stages, it can be seen that the variation range of the VIPs on the leeward side of the upper surface and the lower leeward side oblique web increases and the pressure value also increases, while the variation range of the VIPs on the leeward side of the lower surface decreases slightly.

Energy distribution and evolution laws of VIPs and their components

The distribution of VIPs was analyzed from the perspective of force in section “Evolution law of VIPs,” while it is not appropriate to determine the contribution of VIPs at different locations to the VIV simply according to the values of VIPs because of the diverse phase differences between the VIPs and the displacement of VIV. As mentioned previously, the negative aerodynamic damping and the positive nonlinear aerodynamic damping present opposite energy contribution to VIV. They are two items of the components of VIPs, and can be extracted from VIPs via equation (9).

From the perspective of energy, the work of VIPs and their components can be obtained, and then it is possible to quantitatively analyze the energy contribution of each component to VIV during one cycle and to determine the location where the surface of the box girder provides and dissipates energy obviously. Therefore, the energy distribution and evolution laws of VIPs and their components are explored in this section, and the mesoscopic mechanisms of the nonlinear characteristics of VIV of the flat-closed-box girder are revealed in this section.

Energy calculation method

According to equation (9), the components of VIPs can be decomposed, the linear aerodynamic damping component and the third-order nonlinear aerodynamic damping component are shown as follows

f_{L A D, i} = ρ U^{2} D \cdot P_{10, i} {\dot{y}}_{*}

f_{N A D, i} = ρ U^{2} D \cdot P_{30, i} {\dot{y}}_{*}^{3}

Since the time histories of pressures and displacement are known, the work of the VIPs and their linear and nonlinear aerodynamic damping components at point ith in one cycle are performed in the following equations, respectively

E_{V I, i} = \sum_{t = 0}^{T} f_{V I, i} (t) \cdot Δ y

E_{L A D, i} = \sum_{t = 0}^{T} f_{L A D, i} (t) \cdot Δ y

E_{N A D, i} = \sum_{t = 0}^{T} f_{N A D, i} (t) \cdot Δ y

where $Δ y$ is the difference of displacement between the current moment and the previous moment.

Distribution laws of energy

As is known to all, for the contribution of force to the VIV, the discussion about the amplitude of the force is not appropriate because the phase between force and displacement may be different. And the traditional approach is to discuss the work of force to this girder. It means that does positive work provides energy for the movement of structure and does negative work dissipates energy and obstructs the movement of structure. In this section, in which zones of the deck surface the VIPs provide the LADPs or NADPs was found, while the quantities of energy provided by LADPs or dissipated by NADPs were also determined.

Figures 11 to 13 illustrate the average energy distribution in one cycle of LADPs, NADPs, and VIPs at the first stage to the third stage, respectively. In these figures, the distribution line located at the inside of the girder means that these pressures do positive work, and vice versa. Moreover, it needs to be stressed that these contour lines are magnified in different multiple for convenience of observation.

Figure 11.

Energy distribution at the first stage.

Figure 12.

Energy distribution at the second stage.

Figure 13.

Energy distribution at the third stage.

As displayed in Figures 11 to 13, it can be seen that only the absolute value of linear aerodynamic damping work (LADW) increases, but the distribution line of LADW keeps same at each stage. So does the nonlinear aerodynamic damping work (NADW). The reason is as follows. According to equation (9), all the LADPs and the NADPs are merely related to velocity of each point which equals to the velocity of the girder. It means that there is no phase difference for all the aerodynamic damping pressures. In the same way, all the aerodynamic damping work fluctuates synchronously. Therefore, the contribution lines of aerodynamic damping work keep same, except the absolute values increase.

It can also be found that there is a big difference between the energy distribution of VIPs and LADPs at the first stage, and then the distribution line of vortex-induced work (VIW) gradually tends to be close to that of LADW. The reason is that both the LADPs and the VSPs contribute energy for VIV, as introduced in section “Decomposition of VIPs.” Compared with the LADPs, the VSPs cannot be ignored at the initial period, but these VSPs are small enough to be ignored with the significant increase of LADPs and NADPs.

One can also find from these figures that at the middle and downstream of the upper surface, the contour of LADW is significant. It means that the LADPs on the middle and downstream of the upper surface contribute a lot of energy to VIV. Moreover, the energy contributed by LADPs on the upper of upstream web and the lower surface also cannot be ignored. For the NADPs, the pressures on the lower surface and the middle of upper surface dissipate energy significantly. It is because of the existence of central barriers, the distribution of pressures is disorder at the middle of upper surface, thus, the change rule of energy is complex at the middle of upper surface.

Furthermore, it can be inferred that the significant energy supply is due to the backward movement and shedding of large-scale vortices on the upper surface, so some ways to reduce the energy input can be considered, such as prevent the large vortex from moving and falling off on the upper surface, or prevent the large vortex from forming at the leading edge of the box girder. Therefore, some targeted vibration suppression measures may be adopted, including the higher central upper stabilizer to block the movement of vortices, and the wind fairing with smaller angle to reduce the size of leading edge vortex.

Evolution laws of energy

The distributions of energy for each type of pressure at every stage were discussed above, and for a quantitative analysis of evolution law of energy, the quantities of energy for each type of pressure at every stage are determined as shown in Table 1. It can be found that besides the vortex-induced force, there is also a structural damping force exerted on the flat-closed-box girder. Obviously, the LADPs and VSPs provide energy to VIV, and the NADPs and the structural damping force dissipate energy all the time. In addition, Table 1 demonstrates the significance of the VSPs at the first stage and illustrates the reason why the outline of VIW gradually tends to be close to that of LADW at the second and the third stages.

Table 1.

The quantities of energy for each type of pressure at every stage (J).

Type of pressures	First stage	Second stage	Third stage
LADPs	0.086	0.521	0.855
NADPs	–0.002	–0.105	–0.198
Vortex-shedding pressures	0.004	0.004	0.003
VIPs	0.088	0.420	0.661
Structural damping force	–0.019	–0.407	–0.660
Total	0.069	0.013	0.000

LADP: linear aerodynamic negative damping pressure; NADP: nonlinear aerodynamic positive damping pressure; VIP: vortex-induced pressure.

At the first stage, the VIV develops rapidly. As the amplitudes of displacement and velocity are small, the NADPs which are proportional to the cube of velocity are very small, so very little energy is dissipated by the NADPs. In addition, the product of structural damping and velocity is small, thus, the energy dissipated by the structural damping force is also little. Therefore, it can be found that this girder absorbs a lot of energy to promote the rapid development of VIV at this period.

The average amplitude of displacement at the second stage is much higher than that at the first stage, also the average amplitude of velocity increases significantly, as shown in Figure 7. As listed in Table 1, the absolute values of energy for each type of pressure increase rapidly, especially the energy dissipated by NADPs, which cannot be ignored in this period. Although the growth rate of the dissipated energy is larger than that of the absorbed energy, the quantity of the dissipated energy is smaller than that of the absorbed energy. Thus, at the second stage, this girder also absorbs energy to supply the VIV, but the total absorbed energy decreases obviously compared with the previous stage.

At the stable stage of VIV, both the average amplitude of displacement and velocity increase slightly compared with the second stage, so the absolute values of energy for each type of pressure also increase. Similar to the second stage, the growth rate of the dissipated energy is also larger than that of the absorbed energy, and the total absorbed energy continue decreases. As shown in Table 1, the total absorbed energy is almost zero, it means that the energy provided by LADPs and VSPs equals to the energy dissipated by NADPs and structural damping force, so that the VIV is in balance.

In summary, at the rapid development period, the energy provided by LADPs accounts for a large proportion, it can be concluded that the LADPs are the key factor of the rapid development of VIV. Then, the proportion of the energy dissipated by NADPs increases rapidly, it can be considered that NADPs play key roles for the self-limiting phenomenon. At the stable stage of VIV, the absorbed energy and the dissipated energy reach equilibrium, which causes the steady vibration to continue.

Conclusions

Studies on the VIV of a flat-closed-box girder based on distributed pressures were carried out through numerical simulation in this study. The major works and some conclusions are drawn below.

The numerical simulation method is acceptable and accurate, the simulation results agree well with the test results.

It is appropriate and precise to extract the VIPs from the surface fluctuating pressures and to decompose the VIPs by applying the mathematical model of VIF.

The direction of the VIPs at the rest of the girder is synchronized with the direction of motion of the girder, except that at the upstream of the upper surface. Moreover, the area with large pressure value contains the middle and downstream of the upper surface and the underside of upstream web and the lower surface.

At the rapid development period of VIV, the LADPs are the key factor of the rapid development of VIV. And the NADPs play key roles for the self-limiting phenomenon. At the stable stage of VIV, the absorbed energy and the dissipated energy reach equilibrium, which leads to the stable VIV continues.

At any stage of VIV, the zones of the deck surface that the VIPs provide the negative damping or the positive damping are the same. That is, the negative aerodynamic damping is mainly provided by the middle and downstream of the upper surface. At the lower surface and the middle of upper surface, the nonlinear aerodynamic positive damping pressures dissipate energy significantly.

Since the main energy supply is caused by the backward movement and shedding of large vortices on the upper surface, some targeted vibration-suppression measures can be considered, such as the higher central upper stabilizer to block the movement of vortices and the wind fairing with smaller angle to reduce the size of leading edge vortices.

Furthermore, the relationship between the evolution laws of VIPs and energy and the evolution law of flow field, and the investigations into target-oriented invention of proper or effective aerodynamic countermeasures for VIV mitigations are worth further study.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the supports from China Railway Eryuan Engineering Group Co, Ltd, the National Key Research and Development Program (grant no. 2017YFB1201204), the Sichuan Province Youth Science and Technology Innovation Team (grant no. 2015TD0004), and the Open Subject of State Key Laboratory of Disaster Reduction in Civil Engineering (grant no. SLDRCE14-01).

ORCID iDs

Xingyu Chen

Haojun Tang

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