The impact of the wind attack angle on a typical bridge deck’s flutter behavior by the distributed aerodynamic characteristics method

Abstract

To investigate the influence of the wind attack on a bridge deck’s flutter performance, the concept of “pressure flutter derivatives” was introduced as a part of the “distributed aerodynamic characteristics” method, which can be used to visualize the dominant fluctuations of the pressure field. The distributions of the aerodynamic damping and stiffness under different wind attack angles were visualized, instead of using an overall force perspective. The analysis was carried out through two-dimensional computational fluid dynamics simulations based on a typical box deck section. Necessary data were obtained through the forced vibrations of the deck. The dominant pressure fluctuations were demonstrated, around which the wind attack angles’ impact on the Scanlan flutter derivatives was discussed. For the deck section used in this study, the wind attack angle was found to most affect the windward parts of the deck. When wind attack angle ranged from $- 1.5 °$ to $3 °$ , two regions contributed the most: one located behind the windward faring vertex and the other located behind the first vertex of the upper slab. When the wind attack angle decreased to $- 3 °$ , a new dominant region appeared behind the first vertex of the lower slab, which caused a sudden reduction of the flutter critical speed.

Keywords

Bridge aerodynamics wind attack angle aerodynamic damping distributed aerodynamic characteristics method computational fluid dynamics simulation

1. Introduction

Flutter is an aerodynamic instability phenomenon. Scanlan and Tomko (1971) developed a formula that contained eight parameters named flutter derivatives to describe the wind force acting on the bridge deck. Miyata and Yamada (1990) investigated the coupled flutter oscillations of a very-long-span bridge by using a complex eigenvalue analysis. Iwamoto and Fujino (1995) proposed a method based on the extended Kalman filter (EK-WGI method) to distinguish the eight parameters, which simplified the procedure significantly. Based on flutter derivatives, various researchers have studied the flutter mechanism of the bridge. Agar (1989) proposed a method to calculate the flutter’s critical wind speed, considering multiple degrees of freedom (DoFs). Ge and Tanaka (2000) introduced a multi-mode approach and a full-mode approach to calculate the flutter speed and frequency. Yang et al. (2007) introduced a two-dimensional three-degree-of-freedom method, and they investigated the flutter mechanism through comparisons of the aerodynamic damping and stiffness. Chen and Kareem (2006) derived closed-form solutions of the bimodal flutter, which could improve the understanding of a bimodal coupled flutter. Based on the closed-form solutions of the bimodal flutter, Li et al. (2019) introduced a concept of “surface flutter derivatives” as a part of the “distributed aerodynamic characteristics” method, to analyze the contribution from the girder’s surface to the modal characteristics. Based on this method, the control mechanism of a central stabilizer was studied.

As for the research on the wind attack angle effect, Ouyang and Chen (2015) studied the change of the flutter’s critical wind speed with varying wind attack angles. Tang et al. (2017) experimentally investigated the flutter characteristics of a bridge deck at large wind attack angles, pointing out that the flutter was mainly controlled by the torsional mode. Li et al. (2018) discovered that the flutter of a flat box girder is dominated by the branch of the torsional mode when the angle is in the range of 0°–5°. Tang et al. (2019) discovered that vortices formed at the windward edge become larger with increasing wind speed. Wu et al. (2020) found that the flutter index is stable when the reduced wind speed is less than 6 and the wind attack angle is less than 3°.

As mentioned above, several effective methods have been created to analyze the total effect of the aerodynamic force in the flutter process. In this study, detailed information about the wind attack angle’s influence on the characteristics of the self-excited force and the flutter stability were revealed. First, a novel concept of “pressure flutter derivatives” was proposed to provide intuitive information about the dominant pressure fluctuations of the deck’s surrounding flow field. Combined with the previous surface flutter derivatives, a complete routine of the “distributed aerodynamic characteristics” method was established. Computational fluid dynamics (CFD) simulations were performed based on a typical bridge box deck. The figures of the pressure flutter derivatives were analyzed to understand the wind attack angles’ influence on the Scanlan flutter derivatives. Finally, the wind-induced changes of the modal damping and modal frequencies were visualized to explore the wind attack angle’s impact on the flutter stability. With these steps, a typical routine for applying this method to investigate the impact of the wind attack angle was demonstrated.

2. Theoretical derivations

Li et al. (2019) once proposed a concept of “surface flutter derivatives” to allocate the aerodynamic characteristics to the surface of an object, but how the surrounding pressure field contributed to the flutter stability was not mentioned. Therefore, a novel concept of “pressure flutter derivatives” was proposed in this study, which was used to build a connection between the pressure field’s fluctuations and the flutter characteristics.

Figure 1 shows the cross-section of a bridge deck with two DoFs: vertical and torsional. Displacements of these two DoFs are denoted as $h$ and $α$ , respectively. The self-excited wind loads are the lift force and lift torque, denoted as $L$ and $M$ , respectively.

Figure 1.

Positive directions of the forces, displacements, and wind velocity.

In quasi-steady theory, the relationship between the body’s oscillation state and the fluctuating pressure field depends only on the shape of the deck. Since the dominant fluctuation of the flow field is stimulated by the vibration of the bridge deck, the pressure’s fluctuations can be decomposed based on the deck’s motion. The pressure flutter derivatives are denoted as ${\tilde{P}}_{i, j^{+}}^{*} (i = 1 - 4)$ and are defined as follows

p_{j^{+}} - {\bar{p}}_{j^{+}} = \frac{1}{2} ρ U^{2} (k {\tilde{P}}_{1, j^{+}}^{*} \frac{\dot{h}}{U} + k {\tilde{P}}_{2, j^{+}}^{*} \frac{b \dot{α}}{U} + k^{2} {\tilde{P}}_{3, j^{+}}^{*} α + k^{2} {\tilde{P}}_{4, j^{+}}^{*} \frac{h}{b}),

(1)

where

p_{j^{+}}

is the pressure field at field point

j^{+}

{\bar{p}}_{j^{+}}

is its mean value,

ρ

is the air density,

U

is the far-field wind speed,

b

is the half width of the deck, and each dot denotes the partial difference with respect to time t. The non-dimensional reduced frequency

k

is defined as

b ω / U

, where

ω

is the circular frequency of the deck’s oscillation. It should be noticed that the “pressure flutter derivatives” are as a team, representing the projection of the pressure fluctuation of the whole flow field to the four independent or orthogonal motion components. Besides, they are functions of the non-dimensional reduced frequency. Thus, pressure flutter derivatives can be regarded as a sort of field filter. They only retain the resonance components that significantly affect the deck’s flutter performance while abandoning other confusing information related to the flow fluctuations, for example, vortex shedding with frequencies that deviate significantly from the structural oscillating frequency. These features are also true for the Scanlan’s flutter derivatives.

It is possible to calculate the surface flutter derivatives directly from the pressure flutter derivatives, as follows

F_{s e, j} = {\begin{array}{c} L_{j} \\ M_{j} \end{array}} = {\begin{array}{c} \frac{1}{2} ρ U^{2} | s_{j} | (k {\tilde{H}}_{1, j}^{*} \frac{\dot{h}}{U} + k {\tilde{H}}_{2, j}^{*} \frac{b \dot{α}}{U} + k^{2} {\tilde{H}}_{3, j}^{*} α + k^{2} {\tilde{H}}_{4, j}^{*} \frac{h}{b}) \\ \frac{1}{2} ρ U^{2} | s_{j} | | f_{j} | (k {\tilde{A}}_{1, j}^{*} \frac{\dot{h}}{U} + k {\tilde{A}}_{2, j}^{*} \frac{b \dot{α}}{U} + k^{2} {\tilde{A}}_{3, j}^{*} α + k^{2} {\tilde{A}}_{4, j}^{*} \frac{h}{b}) \end{array}} .

(2)

Vector $s_{j}$ is the outward normal vector of the surface piece $j$ , and its module $| s_{j} |$ equals the element’s length. Vector $f_{j}$ starts from the section’s torsional center and ends at the center of the surface piece $j$ , as depicted in Figure 2. As the pressure flutter derivatives are a field, the calculation only needs the values located at the deck’s surface, namely, $j$ only varies along the deck’s surface and is the subset of $j^{+}$ . The relationship can be expressed by either an inner product or a mixed product of the involved vectors or direction angles, as follows

{\tilde{H}}_{i, j}^{*} = - {\tilde{P}}_{i, j}^{*} (k) \frac{s_{j} ∙ n_{y}}{| s_{j} |} = - {\tilde{P}}_{i, j}^{*} (k) \cos (β_{s}) (i = 1 - 4) .

{\tilde{A}}_{i, j}^{*} = - {\tilde{P}}_{i, j}^{*} (k) \frac{f_{j} \times s_{j} ∙ n_{z}}{| s_{j} | | f_{j} |} = - {\tilde{P}}_{i, j}^{*} (k) \sin (β_{s} - β_{f}) (i = 1 - 4) .

(3)

Figure 2.

Demonstration of the vectors used to calculate surface flutter derivatives.

The deck section is assumed to be in the x–y plane. $n_{y}$ is the unit vector parallel to the vertical direction, and $n_{z}$ is the unit vector pointing to the outward from the plane. $β_{s}$ ( $o r β_{f}$ ) is the angle between $n_{y}$ and $s$ (or $f$ ).

The kinetic expression of the bimodal system shown in Figure 1 is equation (4). The mass, structural damping ratio, and modal circular frequency are, respectively, denoted as $m$ , $ξ$ , and $ω$ . Subscripts of $h$ and $α$ are used to distinguish the vertical and torsional degrees, respectively

{\begin{array}{c} m_{h} \ddot{h} + 2 ξ_{h} ω_{h} m_{h} \dot{h} + m_{h} ω_{h}^{2} h \\ m_{α} \ddot{α} + 2 ξ_{α} ω_{α} m_{α} \dot{α} + m_{α} ω_{α}^{2} α \end{array}} = F_{s e}

(4)

The wind-induced change of the damping ratios at the wind speed $U$ are $\sum_{j} ξ_{1, j}^{U, a e r o}$ and $\sum_{j} ξ_{2, j}^{U, a e r o}$ for the heaving mode and pitching mode, respectively. The wind-induced change of the frequencies are $\sum_{j} ω_{1, j}^{U, a e r o}$ and $\sum_{j} ω_{2, j}^{U, a e r o}$ . To allocate the aerodynamic stiffness and damping to different part of the surface, equations (5) and (6) give the calculation formulas, where $μ = ρ b^{2} / m_{h}$ , $ν = ρ b^{4} / I$ , $γ_{1} = ω_{1} / ω_{2}$ , and $γ_{2} = ω_{2} / ω_{1} .$

ω_{1, j}^{U, a e r o} = - \frac{| s_{j} |}{4 b} μ ω_{h} \int_{u = 0}^{U} ({(1 + μ H_{4}^{*, u})}^{- \frac{3}{2}} d {\tilde{H}}_{4, j}^{*, u})

ω_{2, j}^{U, a e r o} = - \frac{| s_{j} | | f_{j} |}{4 b^{2}} ν ω_{α} \int_{u = 0}^{U} ({(1 + ν A_{3}^{*, u})}^{- \frac{3}{2}} d {\tilde{A}}_{3, j}^{*, u})

(5)

ξ_{1, j}^{U, a e r o} = \int_{u = 0}^{U} (\begin{array}{c} \frac{γ_{2}^{u} μ ν H_{3}^{*, u} A_{1}^{*, u}}{{(1 - {γ_{2}^{u}}^{2})}^{2} ω_{1}^{u}} d ω_{2, j}^{u} - (\frac{{γ_{2}^{u}}^{2} μ ν H_{3}^{*, u} A_{1}^{*, u}}{{(1 - {γ_{2}^{u}}^{2})}^{2} ω_{1}^{u}} + \frac{ω_{h} ξ_{h}}{{ω_{1}^{u}}^{2}}) d ω_{1, j}^{u} \\ + \frac{μ ν A_{1}^{*, u} | s_{j} |}{4 (1 - {γ_{2}^{u}}^{2}) b} d {\tilde{H}}_{3, j}^{*, u} + \frac{μ ν H_{3}^{*, u} | s_{j} | | f_{j} |}{4 (1 - {γ_{2}^{u}}^{2}) b^{2}} d {\tilde{A}}_{1, j}^{*, u} - \frac{μ | s_{j} |}{4 b} d {\tilde{H}}_{1, j}^{*, u} \end{array})

ξ_{2, j}^{U, a e r o} = \int_{u = 0}^{U} (\begin{array}{c} \frac{γ_{1}^{u} μ ν H_{3}^{*, u} A_{1}^{*, u}}{{(1 - {γ_{1}^{u}}^{2})}^{2} ω_{2}^{u}} d ω_{1, j}^{u} - (\frac{{γ_{1}^{u}}^{2} μ ν H_{3}^{*, u} A_{1}^{*, u}}{{(1 - {γ_{1}^{u}}^{2})}^{2} ω_{2}^{u}} + \frac{ω_{α} ξ_{α}}{{ω_{2}^{u}}^{2}}) d ω_{2, j}^{u} \\ + \frac{μ ν A_{1}^{*, u} | s_{j} |}{4 (1 - {γ_{1}^{u}}^{2}) b} d {\tilde{H}}_{3, j}^{*, u} + \frac{μ ν H_{3}^{*, u} | s_{j} | | f_{j} |}{4 (1 - {γ_{1}^{u}}^{2}) b^{2}} d {\tilde{A}}_{1, j}^{*, u} - \frac{ν | s_{j} | | f_{j} |}{4 b^{2}} d {\tilde{A}}_{2, j}^{*, u} \end{array})

(6)

Finally, the overall framework of the “distributed aerodynamic characteristics” method is demonstrated in Figure 3.

Figure 3.

Demonstration of the “distributed aerodynamic characteristics” method.

3. Settings of the calculation domain

A typical box deck of the Great Belt East Bridge (GBEB) was used as the basic case in the numerical examples. From the view of methodology study, the construction status without handrails were considered in this manuscript. The computational domain (shown in Figure 4) was with a blockage ratio of 0.95%. To avoid grid distortion near the deck surface during the oscillation process, three computational regions were defined (De Miranda et al., 2014; Patruno 2015). The details of the final mesh and corresponding $y^{+}$ of the first layer are illustrated in Figure 5.

Figure 4.

Sketch of the two-dimensional (2D) computational domain.

Figure 5.

Detailed demonstration of the computational grid within the rigid zone.

For the real bridge flutter problem, the Reynolds number ( $R e$ ) is about the order of $10^{7}$ . In this paper, $10^{5}$ was selected from the view of a trade-off between the accuracy and the calculation cost (Bruno and Khris, 2003): as the typical bridge deck’s flow-separation point does not change with the Reynolds number, its aerodynamic characteristics are not very sensitive to the Reynolds number. When the Reynolds number exceeds a certain limit, the calculated results have little difference. Therefore, the unsteady Reynolds-averaged Navier–Stokes (URANS) approach was used (Brusiani et al., 2011). Although the time-averaging modeling of the turbulence cannot reproduce the flow accurately, URANS is still acceptable if only the flutter-related low-frequency fluctuations are the main concern. Furthermore, as the span-wise momentum diffusion can be appropriately treated, the overall flutter mechanism can be reasonably reproduced, even in two-dimensional simulations (Mannini et al., 2016). A blending k–ω SST turbulence model was used to reduce sensitivity to the inlet boundary condition, and the details of the involved parameters are explained by Menter (1994).

The simulations were conducted using Ansys Fluent. The semi-implicit method pressure-linked equations consistent (SIMPLEC) algorithm was used to solve the velocity–pressure coupling problem. To obtain the appropriate numerical accuracy and stability, a second-order scheme was used for the pressure equation, and the second-order upwind scheme was used for the momentum equation, turbulent kinetic energy equation, and specific dissipation rate equation. The inlet boundary was set as a velocity inlet, the incoming turbulence intensity was set as 4%, and the viscosity ratio was set as two. The outlet boundary was set as a pressure outlet. The inlet wind speed was set as $75 m / s$ and the kinematic viscosity was adjusted between cases, so that the $R e$ was maintained as $1 \times 10^{5}$ (using the reference length 𝐵) in all cases. To prevent the grid from distorting due to deck movement, diffusion and re-meshing methods were used to handle the dynamic mesh.

4. Validations

Before the formal calculations were performed, six sets of meshes were examined to identify a suitable grid density that could fully utilize the computing resources. When the grid density was adjusted from sparse to dense, the non-dimensional grid height

y^{+}

at the deck surface was changed from 16.2 to 0.9. Error studies were conducted with a static deck and a vibrating deck. The relative errors of the key parameters under different grid densities are shown in Table 1. The relative errors converged rapidly with increasing grid density. There was a balance between the calculation error and the calculation cost when

y^{+} = 1.8

Table 1.

Relative error (%) of key parameters under different grid densities.

$y^{+}$	Mean value of static force			Amplitude of dynamic force
$y^{+}$	$C_{d}$	$C_{l}$	$C_{m}$	$U r = 6$	$U r = 26$
0.9	—	—	—	—	—
1.35	0.00%	0.42%	0.29%	4.86%	0.55%
1.8	1.65%	0.63%	0.00%	5.71%	1.27%
5.4	7.64%	39.0%	2.96%	3.99%	5.13%
10.8	11.7%	82.4%	7.12%	9.78%	16.2%
16.2	26.2%	12.5%	33.5%	15.8%	36.6%

Furthermore, the static coefficients of the grid density with

y^{+} = 1.8

were compared to those obtained in other studies as shown in Table 2, showing good agreement with the previous CFD simulation results. Since handrails were ignored in all the CFD simulations listed, the

C_{d}

was relatively small compared to the results of the wind tunnel test.

Table 2.

Comparison of the numerical results and some published results.

Case	Method	$R e$	$C_{d}$	$C_{l}$	$C_{m}$
This paper	RANS (2D)	1.0 × 10⁵	0.048	0.048	−0.034
Wang (2018)	RANS (2D)	3.1 × 10⁵	0.045	0.041	−0.033
Taylor (1999)	DVM (2D)	1.0 × 10⁵	0.050	0.127	0.052
Zhu (2015)	RANS (2D)	3.1 × 10⁵	0.050	0.041	—
Larsen (1997)	Wind tunnel test (with handrails)	3.0 × 10⁵	0.077	0.067	0.028

For the dynamic case, the validation was carried out in two steps. First, the Scanlan flutter derivatives were directly compared to wind tunnel test results. However, as most papers did not present the Scanlan flutter derivatives without handrails, this comparison was not sufficient. As shown in Figure 6, $H_{i}^{*}$ and $A_{i}^{*}$ ( $i = 1 - 3$ , because the experiments did not extract $H_{4}^{*}$ and $A_{4}^{*}$ ) were compared to the experimental data (Taylor 1999). The basic tendencies matched, but the errors were significant because the deck shapes were not completely the same. Therefore, the precision needs to be examined from another perspective, with a compromise.

Figure 6.

(a, b) Extracted Scanlan flutter derivatives compared to experimental values (with handrails).

As the critical flutter speeds of the deck without handrails are known (Larsen 1993; Larsen and Walther 1997), a secondary validation of the experimental results was conducted by using the Scanlan flutter derivatives to calculate the critical flutter speed. The necessary structural characteristics are given in Table 3, which were also used in the following sections for wind attack angle analyses. A traditional frequency search method was used to seek the heaving and pitching frequencies and damping. The calculated critical flutter speed

U_{c r} = 75 m / s

was close to the experimental critical flutter speed

73 m / s

, proving that the overall effect of the simulated self-excited force was captured accurately. A grid density with

y^{+} = 1.8

behaved well in both the stationary and vibrating cases, and thus, it was chosen for the following analyses.

Table 3.

Dynamic characteristics of the Great Belt East Bridge (GBEB).

Dynamic characteristics	Denotation	Unit	Value
Equivalent mass	$m$	$t / m$	22.70
Equivalent mass moment of inertia	$I$	$t ∙ m^{2} / m$	2470
Circular frequency of heaving mode	$ω_{h}$	$r a d / s$	0.628
Damping ratio of heaving mode	$ξ_{h}$	∼	0.2%
Circular frequency of pitching mode	$ω_{α}$	$r a d / s$	1.747
Damping ratio of pitching mode	$ξ_{α}$	∼	0.2%

5. Distribution of pressure flutter derivatives

Based on the validation process described previously, a series of sinusoidal forced vibrations with different wind attack angles, reduced wind speeds (

U_{r}

), and vibrating DoFs (60 cases in total, listed in Table 4) were calculated to obtain the required data.

Table 4.

Case parameters of the forced vibrations.

Case differences	Wind attack angles	DoFs	$U_{r}$
Range	$- 3 ° t o 3 °$	Vertical and torsional	6–26
Interval	$1.5 °$	—	4

The pressure flutter derivatives (equation (1)) were introduced not only as an intermediate process to calculate the distributed aerodynamic performance but also to highlight the key fluctuations of the pressure field. The distribution of the pressure flutter derivatives depicts the relationship between the pressure field fluctuations and the vibrating deck for the vertical and torsional DoFs. Based on this critical state, Figures 7 and 8 show the corresponding pressure flutter derivatives. As shown by equation (1), ${\tilde{P}}_{4, j^{+}}^{*}$ and ${\tilde{P}}_{3, j^{+}}^{*}$ represent the components that are in phase with the displacements (vertical and torsional, respectively), and ${\tilde{P}}_{1, j^{+}}^{*}$ and ${\tilde{P}}_{2, j^{+}}^{*}$ represent the components corresponding to the velocity. This indicates that we can directly compare components ${\tilde{P}}_{1, j^{+}}^{*}$ and ${\tilde{P}}_{4^{+}}^{*}$ and components ${\tilde{P}}_{2, j^{+}}^{*}$ and ${\tilde{P}}_{3, j^{+}}^{*}$ because each pair represents the fluctuations caused by the same oscillation degree. For this reason, the color maps of the ${\tilde{P}}_{1, j^{+}}^{*} - {\tilde{P}}_{4, j^{+}}^{*}$ pairs and ${\tilde{P}}_{2, j^{+}}^{*} - {\tilde{P}}_{3, j^{+}}^{*}$ pairs were set to be the same as those in Figures 7 and 8. In addition, to show their ability to investigate the overall flutter behavior, the change percentages of the Scanlan flutter derivatives ( $H_{i}^{*}$ and $A_{i}^{*}$ , $i = 1 - 4$ ) were given based on the value at a zero wind attack angle.

Figure 7.

Wind attack angles’ impact on the fluctuations related to the vertical DoF (a) ${\tilde{P}}_{i, j}^{*}$ and the corresponding $H_{1}^{*}$ and $A_{1}^{*}$ (b) ${\tilde{P}}_{4, j}^{*}$ and the corresponding $H_{4}^{*}$ and $A_{4}^{*}$ .

Figure 8.

Wind attack angles’ impact on the fluctuations related to the vertical DoF (a) ${\tilde{P}}_{2, j}^{*}$ and the corresponding $H_{2}^{*}$ and $A_{2}^{*}$ (b) ${\tilde{P}}_{3, j}^{*}$ and the corresponding $H_{3}^{*}$ and $A_{3}^{*}$ .

As shown in Figure 7, the values of ${\tilde{P}}_{1, j^{+}}^{*}$ were much greater than those of ${\tilde{P}}_{4, j^{+}}^{*}$ , indicating that the pressure fluctuations caused by the deck’s vertical oscillations were almost in phase with the deck’s vertical velocity. Thus, the variation tendency of $H_{4}^{*}$ and $A_{4}^{*}$ seemed disordered because of their small baseline values. As ${\tilde{P}}_{1, j^{+}}^{*}$ was positive at the upper space and negative in the lower space, the wind field acted like a sticky damper, which consumed the deck’s vertical vibration energy. When reflecting to the distribution of the pressure flutter derivatives, Scanlan flutter derivatives result in large absolute values of $H_{1}^{*}$ and $A_{1}^{*}$ and small absolute values of $H_{4}^{*}$ and $A_{4}^{*}$ . This explains why the deck’s aerodynamic heaving damping $(H_{1}^{*})$ increased significantly and the aerodynamic stiffness was small. Furthermore, the pressure flutter derivatives can help explain the source of the coupling effect of the flutter. As ${\tilde{P}}_{1, j^{+}}^{*}$ was asymmetric on both sides of the deck’s central axis, the vertical motion introduced a torque that was in phase with the deck’s vertical velocity. This phenomenon corresponded to the existence of the Scanlan’s flutter derivative $A_{1}^{*}$ . In addition, ${\tilde{P}}_{1, j^{+}}^{*}$ increased significantly near the windward upper surface when the wind attack angle increased. This explained why $A_{1}^{*}$ increased with the wind attack angle and which part of the field contributed most to this phenomenon.

Unlike the contour maps shown in Figure 7, the values of ${\tilde{P}}_{2, j^{+}}^{*}$ and ${\tilde{P}}_{3, j^{+}}^{*}$ , corresponding to $α$ and $\dot{α}$ , were of the same order of magnitude, as shown in Figure 8. This means that the order of magnitude of the pressure fluctuations induced by orthogonally decomposed torsional displacement and torsional velocity was similar. As shown by the components relative to the aerodynamic damping, ${\tilde{P}}_{2, j^{+}}^{*}$ was distributed more symmetrically than ${\tilde{P}}_{3, j^{+}}^{*}$ , resulting in a smaller absolute value of $A_{2}^{*}$ . When the wind attack angle increased, a negative value zone increased rapidly in the windward upper space, deteriorating the symmetry of ${\tilde{P}}_{2, j^{+}}^{*}$ and thus resulting in a decrease in $A_{2}^{*}$ . ${\tilde{P}}_{3, j^{+}}^{*}$ was related to the aerodynamic stiffness, and it was mainly distributed in the windward surrounding field. As its values were positive at the upper field and negative at the lower field, it produced a large aerodynamic torque that was proportional to the deck’s torsional displacement, resulting in a significant decrease in $A_{3}^{*}$ . An explanation of the coupling terms of the flutter derivatives can also be provided based on the pressure flutter derivatives. Although the contour map of ${\tilde{P}}_{2, j^{+}}^{*}$ seemed more saturated, its contribution to $H_{2}^{*}$ was small, because identical signs appeared both at the upper space and the lower space, and these offset each other when accounting for the vertical force.

In summary, the pressure flutter derivatives represent the components of the pressure fluctuations, and the pressure fields are orthogonally decomposed into two components: one component’s phase identical to the deck’s displacement and the other component’s phase identical to the deck’s velocity, which constitute the motion of the deck. They can provide some insights to investigate the mechanism of the source of the traditional flutter derivatives and the reason for the change of the flutter characteristics. Although the analysis was conducted based on only one deck shape, similar routines can be used for other shapes, because the analysis method is universal.

6. Distribution of aerodynamic characteristics

To study the influence of the wind attack angle on the aerodynamic characteristics, five wind attack angles were studied, ranging from −3° to 3° with an interval of 1.5°. According to the structural dynamics characteristics listed in Table 4, the critical flutter speeds of the five cases were calculated based on traditional eigenvalue analyses, as shown in Figure 9. When the wind attack angle increased from $- 3 °$ to $3 °$ , the flutter stability decreased.

Figure 9.

Critical flutter speeds and pitching damping ratios at different wind attack angles.

The damping ratios near the critical flutter speed coincided with the variation trend of the critical flutter speed. Without loss of generality, the following discussions are based on the data obtained at a wind speed 80 m/s. Before analyzing the influence of the wind attack angle, it is beneficial to observe the distribution of the aerodynamic stiffness and aerodynamic damping at the angle $0 °$ , which was used as the reference for subsequent analyses. The distribution of the aerodynamic characteristics was obtained based on the method proposed by Li et al. (2019). The aerodynamic characteristics were represented by the wind-induced change of the modal frequencies and modal damping ratios, namely, $ω_{1, j}^{U, a e r o}$ , $ω_{2, j}^{U, a e r o}$ , $ξ_{1, j}^{U, a e r o}$ , and $ξ_{2, j}^{U, a e r o}$ . The subscript $U$ corresponded to the wind speed, which was $80 m / s$ , and the results are depicted in Figure 10. It demonstrates the distributions of the bending damping ratio, pitching damping ratio, bending stiffness, and pitching stiffness, respectively. The positive values, as indicated by the color bars in the figures, correspond to the direction pointing outward from the deck’s surface. This represents the favorable contribution to the total aerodynamic stiffness of the aerodynamic damping.

Figure 10.

Dynamic characteristics at a wind speed of 80 m/s.

For the wind-induced effect on the modal characteristics, the proportions located at the windward zones always contributed the most, while the contributions from the leeward zones were basically negligible. Specifically, for the wind-induced effect on the frequency, the whole deck had either a negative contribution to the pitching frequency or a positive contribution to the heaving frequency, except for the flow-separation zones behind the windward upper corner. This means that the contributions from the aerodynamic shape to the modal frequencies were monotonic, namely, all the contributions either tended to raise the heaving frequency or reduce the pitching frequency. As for the fluctuating pressure’s influence on the damping ratios, different patterns were evident. For the wind-induced effect on the heaving damp ratio, similar to the heaving frequency, the whole deck had a positive contribution to the heaving damping ratio. For the pitching damping ratio, the sum of the positive contributions and the negative contributions was almost zero, resulting in a slightly negative total damping ratio when the wind speed just crossed the critical value.

Since the flutter instability was dominated by the pitching mode, the changes of the torsional frequency and torsional damping ratio at different wind attack angles are given below. Making Figure 10 as the baseline, Figures 11 and 12 show the changes of the pitching aerodynamic characteristics obtained at a wind speed of 80 m/s.

Figure 11.

Effect of wind attack angle on the distribution of the pitching frequency (80 m/s).

Figure 12.

Effect of wind attack angle on the distribution of the pitching damping ratio (80 m/s).

As demonstrated in Figure 11, the wind attack angles’ influence was concentrated at the windward surfaces. When the wind attack angle changed from $0 °$ to $1.5 °$ , the overall pitching frequency decreased due to the negative contribution from the top region of the windward faring. As the angle continued to increase, the overall pitching frequency decreased further because the negative contribution was stronger. When the wind attack angle changed to the negative zones, namely, from $0 °$ to $- 1.5 °$ , the overall pitching frequency increased due to the positive contribution at the upper region of the windward fairing. However, because of the negative contributions of other regions, the overall pitching frequency did not change significantly, which showed that the critical flutter speed did not change considerably from the side.

Similar to the wind attack angles’ influence on the aerodynamic stiffness, the sensitive zone was the windward side. When the wind attack angles were positive, two negative contribution zones were evident, one located behind the windward faring vertex and the other located behind the first vertex of the upper slab. When the wind attack angle changed from $0 °$ to $1.5 °$ , the overall pitching damping ratio decreased due to the negative contribution from these two zones, making the flutter stability of the $1.5 °$ case poorer than that of the $0 °$ case. When the angle was further increased, from $0 °$ to $3 °$ , the trend remained identical. Specifically, the negative contribution from the first zone led to a stronger reduction of the flutter instability. However, when the angle changed from $0 °$ to $- 1.5 °$ , the change of the overall pitching damping was almost zero, because the negative damping ratio produced by the second zone was offset by the positive damping ratio from the first separation zone. This explains the result shown in Figure 11 that the critical flutter speeds at $0 °$ and $- 1.5 °$ were almost the same. When the angle further changed from $0 °$ to $- 3 °$ , a new dominant zone appeared at the lower surface of the windward faring. The significant negative contribution from the new separation zone caused a sudden reduction in the pitching damping.

The figures discussed above showed that the wind attack angles’ impact on the flutter aerodynamic behavior was mostly concentrated on the zones behind the windward deck vertices. Furthermore, this phenomenon was suspected to be related to the phenomenon of flow separation and reattachment, not only because their locations were identical but also because an additional dominant zone appeared at the wind attack angle of $- 3 °$ , which corresponded to a new flow-separation zone that appeared at the same location. In future analysis, the wind attack angle’s impact on the flow separation and reattachment may be studied in more detail to provide direct evidence to support this idea.

7. Conclusion

A novel concept of “pressure flutter derivatives” was proposed as an auxiliary method to provide an intuitive description for the generation of the traditional flutter derivatives. This concept can be used to understand the phase and amplitude of the flutter-related fluctuations of the pressure field. Furthermore, its relationship to the previously established “surface flutter derivatives” was proposed. A complete analysis routine was established to perform forced vibration tests to collect data, to calculate the pressure flutter derivatives to analyze the aerodynamic force source, and to calculate the surface flutter derivatives to obtain the distribution of the aerodynamic characteristics. For the problem of the wind attack angle’s impact on bridge’s flutter stability, it was found that the windward side of the deck was decisive for the flutter stability: as the wind attack angle increased, the overall damping ratio decreased mainly due to the negative contribution of the upper surface of the windward fairing. This application showed that this routine can handle not only shape changing problems (e.g., installing additional aerodynamic appendages) but also wind environment problems.

Footnotes

Acknowledgements

We thank LetPub () for its linguistic assistance during the preparation of this manuscript.

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: This work was supported by the National Science Foundation for Young Scientists of China (51808075), the Natural Science Foundation of Chongqing, China (cstc2020jcyj-msxmX0773, cstc2020jcyj-msxmX0937), the 111 Project (B18062), the National Natural Science Foundation of China (51978108), and the Fundamental Research Funds for the Central Universities (2020CDJ-LHZZ-018, 2020CDJ-LHZZ-016).

ORCID iDs

Ke Li

Shaopeng Li

Yi Hui

References

Agar

TJA

(1989) Aerodynamic flutter analysis of suspension bridges by a modal technique. Engineering Structures 11(2): 75–82.

Brusiani

Cazzoli

De Miranda

, et al. (2011) Application of the k-ω turbulence model to assess the flutter derivatives of a long span bridge. Fluid Structure Interaction 115: 231–242.

Bruno

Khris

(2003) The validity of 2D numerical simulations of vortical structures around a bridge deck. Mathematical and Computer Modelling 37(7–8): 795–828.

Chen

Kareem

(2006) Revisiting multimode coupled bridge flutter: some new insights. Journal of Engineering Mechanics 132(10): 1115–1123.

De Miranda

Patruno

Ubertini

, et al. (2014) On the identification of flutter derivatives of bridge decks via RANS turbulence models: benchmarking on rectangular prisms. Engineering Structures 76: 359–370.

Tanaka

(2000) Aerodynamic flutter analysis of cable-supported bridges by multi-mode and full-mode approaches. Journal of Wind Engineering and Industrial Aerodynamics 86(2–3): 123–153.

Iwamoto

Fujino

(1995) Identification of flutter derivatives of bridge deck from free vibration data. Journal of Wind Engineering and Industrial Aerodynamics 54–55: 55–63.

Larsen

(1993) Aerodynamic aspects of the final design of the 1624 m suspension bridge across the Great Belt. Journal of Wind Engineering and Industrial Aerodynamics 48(2): 261–285.

Larsen

Walther

(1997) Aeroelastic analysis of bridge girder sections based on discrete vortex simulations. Journal of Wind Engineering and Industrial Aerodynamics 6768(1): 253–265.

10.

, et al. (2019) An investigation into the bimodal flutter details based on flutter derivatives’ contribution along the bridge deck’s surface. Journal of Wind Engineering and Industrial Aerodynamics 192: 1–16.

11.

Wang

, et al. (2018) Flutter mechanism of flat box girder under different attack angles. Journal of Southwest Jiaotong University 4(53): 687–695.

12.

Mannini

Sbragi

Schewe

(2016) Analysis of self-excited forces for a box-girder bridge deck through unsteady RANS simulations. Journal of Fluids and Structures 63: 57–76.

13.

Menter

(1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32(8): 1598–1605.

14.

Miyata

Yamada

(1990) Coupled flutter estimate of a suspension bridge. Journal of Wind Engineering and Industrial Aerodynamics 33(1): 341–348.

15.

Ouyang

Chen

(2015) Influence of static wind additive attack angle on flutter performance of bridges. Journal of Vibration and Shock 34(2): 45–49.

16.

Patruno

(2015) Accuracy of numerically evaluated flutter derivatives of bridge deck sections using RANS: effects on the flutter onset velocity. Engineering Structures 89: 49–65.

17.

Scanlan

Tomko

(1971) Airfoil and bridge deck flutter derivatives. Journal of Asce 6(6): 1717–1737.

18.

Tang

Wang

, et al. (2017) Aerodynamic optimization for flutter performance of steel truss stiffening girder at large angles of attack. Journal of Wind Engineering and Industrial Aerodynamics 168: 260–270.

19.

Tang

Shum

(2019) Investigation of flutter performance of a twin-box bridge girder at large angles of attack. Journal of Wind Engineering and Industrial Aerodynamics 186: 192–203.

20.

Taylor

Vezza (1999) Analysis of the wind loading on bridge-deck sections using a discrete vortex method. Wind Engineering into the 21st Century, pp. 1345–52.

21.

Wang

Liu

Chen

(2018) Identification of flutter derivatives of bridge deck sections by step-by-step forced vibration with multi frequencies. Journal of Vibration and Shock 37(20): 15–23.

22.

Wang

Liao

, et al. (2020) Flutter derivatives of a flat plate section and analysis of flutter instability at various wind angles of attack. Journal of Wind Engineering and Industrial Aerodynamics 196: 104046.

23.

Yang

Xiang

(2007) Investigation on flutter mechanism of long-span bridges with 2d-3DOF method. Wind and Structures 10(5): 421–435.

24.

Zhu

(2015) Feasibility investigation on prediction of vortex shedding of flat box girders based on 2D RANS Model. China Journal of Highway and Transport 28(6): 25–32.