# Experimental Investigation and Validation on Suppressing the Unsteady Aerodynamic Force and Flow Structure of Single Box Girder by Trailing Edge Jets

^{1}

^{2}

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## Abstract

**:**

^{4}based on the incoming velocity and the height of the test model. A digital pressure measurement system was utilized to obtain and record the surface pressure that was distributed around the SBG model. The surface pressure results show that the fluctuating amplitude of the aerodynamic forces was attenuated in the controlled case at a specific range of the non-dimensional jet momentum coefficient. The Strouhal number of the controlled case also deviates from that of the original SBG model. Except for the pressure measurement experiment, a high-resolution digital particle image velocimetry system was applied to investigate the detailed flow structure behind the SBG model to uncover the unsteady vortex motion process from the SBG model with and without the trailing edge jets flow control. As the jet flow blows into the wake, the alternating vortex shedding mode is switched into a symmetrical shedding mode and the width of the wake flow is narrowed. The proper orthogonal decomposition was used to identify the energy of the different modes and obtain its corresponding flow structures. Moreover, the linear stability analysis of the flow field behind the SBG model shows that the scheme of trailing edge jets can dramatically suppress the area of unsteady flow.

## 1. Introduction

## 2. Experimental Setup and Details

^{2}in cross-section and 1050 mm in the airflow direction, possesses optically transparent walls to observe the experimental phenomenon conveniently. The honeycombs, mesh structure, and contraction section were installed upstream of the test channel to produce a high-quality uniform incoming flow entering the square test channel. The incoming flow velocity can be successively regulated at the range of 0–25 m/s. The turbulence intensity level of the free incoming flow was evaluated as less than 0.4%.

#### 2.1. Original Test Model Details

^{4}according to the incoming flow velocity of 9 m/s and the height of the SBG model. It should be declared that despite the Re of the present experiment being much less than that of the real bridge in a natural wind environment, this investigation aimed to study the control effectiveness of the control scheme of the trailing edge jets. There were two iron bars that were used to firmly fix the SBG model in the test channel to prevent displacement and vibration during the experiment. The coordinate system was set as shown in Figure 1a. The positive X and Y are defined as the airflow and upward directions, respectively. The positive Z meets the right-hand rule.

#### 2.2. Test Model with the Trailing Edge Jets

_{j}) were conducted in the experimental investigation, as shown in Table 1. Based on the flow rates and the area of the jet-hole, the average velocity jet out from the jet-hole was obtained. In the present study, the non-dimensional jet momentum coefficient (J

_{sj}) was applied to evaluate the strength of the jet flow blowing out from the jet-hole of different controlled cases [18,20,21,22], which could be defined as Equation (1).

_{sj}of the SBG model with various jet speeds was calculated and displayed in Table 1.

#### 2.3. Measurement Apparatus

^{2}and 2000 × 1600 pixels

^{2}, respectively. After obtaining the flow field photos, the cross-correlation operation was performed to extract the instantaneous velocity by adopting the interrogation window of 32 × 32 pixels and an efficient overlap rate of 50%. As a result, the spatial resolution of the instantaneous velocity in the measurement planes I and II were calculated as 1.8 × 1.8 mm

^{2}and 2.2 × 2.2 mm

^{2}, respectively. It should be noted that the measurement uncertainty for the instantaneous velocity of the PIV measurement result was evaluated to be under 5%.

## 3. Experimental Results and Discussions

#### 3.1. Surface Pressure Distribution

_{sj}= 0, the mean pressure result of the upper surface on the SBG model shows a dramatic decrease due to an inflection point that existed between the pressure measurement taps 1 and 2. The adverse pressure gradient was observed at the range of X/B from 0.11 to 0.20, as shown in Figure 4a. Therefore, the flow separation occurs in that position, causing the fluctuating pressure to have a maximal value at X/B = 0.11, as shown in Figure 4c. As the X/B was larger than 0.20, the mean pressure distribution of the upper surface had similar values, except for the position of X/B = 0.5 and near the trailing edge. The upper external surface pressure distribution fluctuation shows an enormous value when X/B gets close to 1. The reason for this is that the position nearing the trailing edge will be more influenced by the alternating vortex shedding [20,26]. The adverse pressure gradient also appears in the lower surface of the SBG model at the position of X/B = 0.31–0.41 as shown in Figure 4b, and the peak value of the fluctuating pressure coefficient was obtained at the position of X/B = 0.31, as given in Figure 4d. Therefore, the local flow separation also occurred at the corner point between the pressure measurement taps 29 and 30. The other characteristics of the lower surface pressure are similar to those of the upper surface pressure.

_{sj}= 0. The mean values of the external surface pressure of the controlled case with different non-dimensional jet momentum coefficients remain close to the uncontrolled case at the position nearing the leading edge of the SBG model. When the pressure taps were located at the trailing edge, the mean value of the external surface pressure was alleviated by the trailing edge jet flow, and the case J

_{sj}= 0.0094 exhibited the best control effect. All the controlled cases can slightly attenuate the local flow separation at X/B = 0.11 of the upper surface of the SBG model, as shown in Figure 4c, compared to case J

_{sj}= 0. However, the local flow separation was improved in the lower external surface of the controlled cases that caused a more considerable fluctuation of the pressure at X/B = 0.31, as given in Figure 4d. The fluctuation of the surface pressure close to the trailing edge was mitigated on both the upper and bottom surfaces of the SBG model with the trailing edge jets. That is because the jet flow can interrupt the alternating vortex shedding and lead to the wake flow being more steady. According to the pressure measurement results, the case J

_{sj}= 0.0094 exhibited the best control ability for alleviating the mean value and fluctuation of the surface pressure. Obtaining better control effectiveness does not need larger non-dimensional jet momentum coefficients. There is an optimal non-dimensional jet momentum coefficient to realize the control effect, which is consistent with the study by Chen et al. [20] and Gao et al. [27].

#### 3.2. Aerodynamic Force

_{sj}= 0, which was consistent with the experimental results of 0.064 that were presented by Taylor et al. [28], 0.061 by Zhang et al. [9], and the numerical simulating result of 0.060 given by Frandsen [29], was obtained as 0.062. With the increase of the J

_{sj}, both the mean value of the drag and lift coefficient shows a decrease at first and then an increase. However, the mean value of the moment coefficient of the controlled case maintained a closing value that was slightly large than the case where J

_{sj}= 0, indicating that the different J

_{sj}can not influence the moment force that is acting on the SBG model. Note that the main reason for the wind-induced vibration on the main girder is a large fluctuation of the aerodynamic force. The results of the fluctuation of the drag, lift, and moment coefficients that are acting on the SGB model with different J

_{sj}are displayed in Figure 5d–f, respectively. The fluctuation of the aerodynamic coefficients were descending at first and then ascending with the increasing of J

_{sj}. Therefore, an optimal J

_{sj}for alleviating the unsteady aerodynamic force that is being exerted in the SBG model is required. This saturation phenomenon was also found by Chen et al. [20], Gao et al. [27], and Apelt et al. [30]. When the J

_{sj}equals 0.0094, the best control ability was possessed in the present investigation. The control effect for the drag, lift, and moment coefficient fluctuation was 16.23%, 36.34%, and 38.74%, respectively. That conclusion agreed well with the results of the surface pressure analysis.

_{sj}= 0 shows a significant fluctuation during the experiment, as shown in Figure 6a. When the trailing edge jets were conducted in the SBG model with J

_{sj}= 0.0094, the fluctuation of the lift coefficient was reduced compared to the case J

_{sj}= 0. The peak value of the reduced frequency for the case J

_{sj}= 0 was obtained to be 0.285 based on the fast Fourier transform (FFT) analysis, which was similar to that of 0.28 that was presented by Taylor et al. [28] and Chen et al. [18]. The amplitude of power spectral density analysis of the case J

_{sj}= 0.0094 was lower than that of the case J

_{sj}= 0, and the peak value of reduced frequency of the lift force was also changed to 0.279, given in Figure 6b. This indicates that the trailing edge jets that were equipped in the SBG model can modify the frequency and the strength of the vortex shedding behind the test model.

#### 3.3. PIV Measurement Results

_{sj}= 0, as shown in Figure 7a. The reason is that the separation flows from the upper and bottom surfaces interact in the wake, leading to the velocity vectors colliding in this region. Therefore, the surface pressure distribution nearing the trailing edge when J

_{sj}= 0 shows a dramatic fluctuation. As the trailing edge jets is conducted in the SBG model, the distribution area and maximum value of the TKE are reduced. Moreover, the maximum value of the TKE decreases at first and then increases with the enhancement of J

_{sj,}as shown in Figure 7a–e, which can explain the changing trend of the fluctuation of the aerodynamic force that is being exerted in the SBG model. Compared to results from the target plane II, both the maximum value and the concentrated area of TKE in the target plane I of the case J

_{sj}= 0.0094 is reduced as shown in Figure 7c,f. The unsteady aerodynamic force being exerted in the cross-section of plane I is lower than that of plane II, indicating the pressure measurement result that was obtained by plane II would underestimate the control effectiveness of a fluctuating value of surface pressure that is distributed on the SBG model with trailing edge jets. Moreover, the lower TKE value illustrates that the level of turbulence mixing in the wake flow of the controlled cases is lower than the uncontrolled case [34].

_{sj}= 0. The first two POD modes represent the large-scale coherent structure, and the high-order POD modes represent the small-scale coherent structure as given by Feng et al. [37]. Therefore, the large-scale coherent structure plays an important role in the wake flow behind case J

_{sj}= 0. When the control method is applied in the SBG model, the energy of each POD mode that is exhibited immensely changes. The total energy of the first two POD modes is slightly decreased, as J

_{sj}is set as 0.0016, indicating that a little strength of the jet flow cannot significantly influence the vortex. The ratio of the energy of the first and second POD modes to the total kinetic energy for cases J

_{sj}= 0.0016, J

_{sj}= 0.0094, J

_{sj}= 0.0409, and J

_{sj}= 0.0802 are 62.0%, 33.8%, 53.3%, and 53.2%, respectively. Hence, the total energy of the first two POD modes decreases at first and then increases, as shown in Figure 9 when J

_{sj}improves. Moreover, the small-scale coherent structures in the wake flow field behind case J

_{sj}= 0.0094 occupy most of the energy for the global flow structure, which alleviates the large amplitude of the unsteady aerodynamic force that is being exerted in the SBG model compared to the case J

_{sj}= 0.

_{sj}= 0 and J

_{sj}= 0.0094 are plotted in Figure 10. For the case where J

_{sj}= 0, the first two dominant POD modes are symmetrical about the line Y/B = 0.02 and exhibit alternating vorticity that is distributed in the wake flow. Nevertheless, modes three to five vary from modes one and two: modes three and four are asymmetrical about the line Y/B = 0.02. There are two rows of counter-rotating vorticity concentrations that are displayed in the flow field behind the case J

_{sj}= 0 closed to the test model and then changed into the arrowhead structures as the flow field moves downstream, as shown in Figure 10c,d. Moreover, mode five shows some a small-scale vorticity concentration behind case J

_{sj}= 0. The antisymmetric vorticity flow fields that are calculated by POD represent the symmetric features of the vortex flow field. Conversely, the symmetrical vortex flow structures stand for an antisymmetric flow field, as Konstantinidis et al. [38] described. In consequence, modes one and two of the case J

_{sj}= 0 stand for the large-scale alternating vortex shedding forming the Karman vortex street, while modes three and four denote the small spatial scale vortices with symmetric forms, and mode five represents the irregular distribution of the vorticity structures with small geometric scale. For the SBG model with the trailing edge jets, the flow structures of POD modes show noticeable differences that are in contrast to their counterparts of the uncontrolled case. The vortex concentrations of the case J

_{sj}= 0.0094 are dramatically decreased and narrowed laterally and stretched in the flow direction, as shown in the right of Figure 10. This phenomenon of the POD modes is in connection with the elongation of the wake vortex, as shown in Figure 8b. In addition, the wake jet flow disturbs the vorticity. It divides it into two small parts at the nearing wake position as shown in modes one to four, and mode five changes to much smaller vorticity concentrating parts compared with case J

_{sj}= 0. Based on the analysis above, the wake flow structure characteristics of the case J

_{sj}= 0.0094 yield the unsteady aerodynamic force that is less than that of case J

_{sj}= 0, which is consistent with the results in Section 3.2.

_{sj}= 0 is asymmetrical in the location that is close to the SBG model and symmetrical in a further downstream position. The reason is that the geometric shape of the SBG model is asymmetrical along the line of Y/B = 0. The lowest velocity of the case J

_{sj}= 0 was less than that of the case J

_{sj}= 0.0094 at X/B = 0.03 and 0.12 because the jet flow can alleviate the velocity defect. That indicates the inner region flow field behind case J

_{sj}= 0.0094 is more stable than the case J

_{sj}= 0 [6]. Conversely, the smallest value of the velocity of the case J

_{sj}= 0 was larger than that of case where J

_{sj}= 0.0094 at X/B = 0.21 and 0.30. The velocity profile of the case J

_{sj}= 0 regained faster than that of the case J

_{sj}= 0.0094. Moreover, the width of the wake flow in case J

_{sj}= 0 was slightly wider than the case J

_{sj}= 0.0094, indicating the large drag force that was acting on the SBG model without control [20].

_{sj}= 0, There are two peak values that were observed in the wake flow of the case at X/B = 0.03–0.30, i.e., double-cusp mode. The maximum values appear when the turbulence level grows a dramatically quick ratio, mainly in the shear layers of the separation flow [33]. However, owing to the trailing edge jet flow disturbing the original flow field a the near-wake of X/B = 0.03, this profile for the controlled case exhibited a quadruple cusp mode. The fluctuation of the streamwise velocity profile of the case J

_{sj}= 0.0094 presented a double-cusp mode at X/B = 0.12–0.30. It was similar to that of the case J

_{sj}= 0, illustrating that the jet flow has a short influence range, which is identical to the investigation by Chen et al. [20]. Moreover, the peak value of the streamwise velocity fluctuation for case J

_{sj}= 0.0094 was lower than the case J

_{sj}= 0 at all the positions, leading to the unsteady drag force that was being exerted in the SBG model being mitigated.

_{sj}= 0 and J

_{sj}= 0.0094 are plotted in Figure 11c. The maximum value appears at approximately Y/B = 0.02, which is the axis of symmetry for the wake flow. The shear layers that are formed from the upper and lower surface are amalgamated and violently interact, causing a significant transverse velocity fluctuation [18,20]. The peak value of the case J

_{sj}= 0 shows an increasing trend at the locations of X/B = 0.03 to 0.21 and then a reduction when X/B is more significant than 0.21, which illustrates that the shear layer interaction in the wake flow nearing to the SBG model is weaker than that of downstream. The maximum value of the fluctuation of the transverse velocity for the case J

_{sj}= 0.0094 is obviously alleviated at all positions, indicating that the strong interaction of the shear layer decreased. That leads to the unsteady lift force that is acting on the SBG model when the trailing edge jets attenuated, as shown in Figure 5.

_{sj}= 0. The maximum value of RSS is increased at first and then decreased with the improvement of the X/B. The maximum and minimum values of the RSS are obtained at the Y/B = ~−0.01 and 0.06, respectively, because the turbulence level of the shear layer that is separated from the upper and lower surface of the SBG model is strong [33]. This causes a significant fluctuation of the lift force being exerted in the SBG model. As J

_{sj}sets as 0.0094, the RSS is mitigated at all the positions in wake flow, and the maximum and minimum values are reduced related to the uncontrolled case, which indicates that the interacting strength of the upper and lower shear layer is mitigated in the wake flow behind SBG model. Therefore, the unsteady lift force is decreased by the trailing edge jets.

#### 3.4. Linear Stability Analysis

_{sj}= 0. The $\alpha $ is the complex wave number, and the ${\alpha}_{i}$ is the imaginary part of $\alpha $. The imaginary and real parts of the complex frequency at the cusp point are 0.31 and 1.75, respectively. This indicates the velocity profiles behind case J

_{sj}= 0 at X/B = 0.08 is absolute instability, causing the TKE to exhibit a considerable value in the wake flow, as shown in Figure 7a. According to the real part of $\omega $, the Strouhal number of the wake flow can be calculated as 0.28, which is close to that which is gained by using the power spectral density analysis for the lift force that is acting on the SBG model as given in Figure 6b.

_{sj}= 0. The flow field was slightly stable at the near-wake, and the unstable region was broader for cases J

_{sj}= 0.0016, 0.0409, and 0.0802. However, for the case J

_{sj}= 0.0094, the unstable area was narrowed to X/B = 0.029, indicating the unstable range decreased about 71.84% compared to that of the case J

_{sj}= 0 as shown in the grey region of Figure 14. This phenomenon demonstrates that too strong jet flow isn’t needed for controlling the unstable flow field of the SBG model with a trailing edge jet. An appropriate jet-speed causes a steadier flow structure behind the bluff body, which is also uncovered by Gao et al. [41] in investigations of suppressing unsteady aerodynamic force that is being exerted in a circular cylinder. Therefore, a steady flow structure would excite the lower fluctuating amplitude of the aerodynamic force that is acting on the SBG model, as given in Figure 5. Moreover, the stability of the wake flow of the case J

_{sj}= 0.0094 is similar to that of the studies of leading-edge suction and trailing-edge jet (LSTJ) that are presented by Chen et al. [20]. The superiority of the present control scheme is that the energy consumption of the case J

_{sj}= 0.0094 is half of that of the control method of LSTJ because the present control method only needs the trailing edge jets, not the leading-edge suction.

## 4. Conclusions

_{sj}) serves as a monitor to evaluate the strength of the wake jet flow. The surface pressure and the PIV measurements were conducted to investigate the characteristics of aerodynamic force and flow field of the single box girder (SBG) model with various J

_{sj}. Some significant conclusions are given as follows.

_{sj}is set as 0.0094. The unsteady drag, lift, and moment forces are estimated to decrease about 16.23%, 36.34%, and 38.74%, respectively, compared to the uncontrolled case. Moreover, the vortex shedding frequency of the case J

_{sj}= 0.0094 exhibited a difference.

_{sj}= 0, leading to the more stable wake. The instantaneous wake flow shows that the control method can change the antisymmetric mode into the symmetric mode, which suppresses the alternating vortex shedding and mitigates the unsteady aerodynamic force. Moreover, the unstable wake region of case J

_{sj}= 0.0094 was narrowed by about 71.84%, compared to the uncontrolled case.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Larsen, A.; Esdahl, S.; Andersen, J.E.; Vejrum, T. Storebaelt suspension bridge-vortex shedding excitation and mitigation by guide vanes. J. Wind Eng. Ind. Aerodyn.
**2000**, 88, 283–296. [Google Scholar] [CrossRef] - Fujino, Y.; Yoshitaka, Y. Wind-induced vibration and control of Trans-Tokyo Bay Crossing Bridge. J. Struct. Eng.
**2002**, 128, 1012–1025. [Google Scholar] [CrossRef] - Yang, Y.; Zhou, R.; Ge, Y.; Zhang, L. Experimental studies on VIV performance and countermeasures for twin-box girder bridges with various slot width ratios. J. Fluids Struct.
**2016**, 66, 476–489. [Google Scholar] [CrossRef] - Zhang, H.F.; Xin, D.B.; Zhan, J.; Wang, R.; Zhou, L. Vortex-induced vibration control of a streamline box girder using the wake perturbation of horizontal axis micro-wind turbines. J. Fluids Struct.
**2022**, 108, 103444. [Google Scholar] [CrossRef] - Boberg., M.; Feltrin, G.; Martinoli, A. A novel bridge section model endowed with actively controlled flap arrays mitigating wind impact// IEEE International Conference on Robotics and Automation. In Proceedings of the 2015 IEEE International Conference on Robotics and Automation, Seattle, WA, USA, 26–30 May 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 1837–1842. [Google Scholar]
- Chen, W.L.; Chen, G.B.; Xu, F.; Huang, Y.W.; Gao, D.L.; Li, H. Suppression of vortex-induced vibration of a circular cylinder by a passive-jet flow control. J. Wind Eng. Ind. Aerodyn.
**2020**, 199, 104119. [Google Scholar] [CrossRef] - Larsen, A.; Svensson, E.; Andersen, H. Design aspects of tuned mass dampers for the Great Belt East Bridge approach spans. J. Wind Eng. Ind. Aerodyn.
**1995**, 54–55, 413–426. [Google Scholar] [CrossRef] - He, X.H.; Li, H.; Wang, H.F.; Fang, D.X.; Liu, M.T. Effects of geometrical parameters on the aerodynamic characteristics of a streamlined flat box girder. J. Wind Eng. Ind. Aerod.
**2017**, 170, 56–67. [Google Scholar] [CrossRef] - Zhang, L.Q.; Chen, G.B.; Chen, W.L.; Gao, D.L. Separation Control on a Bridge Box Girder Using a Bypass Passive Jet Flow. Appl. Sci.
**2017**, 7, 501. [Google Scholar] [CrossRef] - Zhan, J.; Xin, D.B.; Ou, J.P.; Liu, Z.W. Experimental study on suppressing vortex-induced vibration of a long-span bridge by installing the wavy railings. J. Wind Eng. Ind. Aerod.
**2020**, 202, 104205. [Google Scholar] [CrossRef] - Nagao, F.; Utsunomiya, H.; Yoshioka, E.; Ikeuchi, A.; Kobayashi, H. Effects of handrails on separated shear flow and vortex-induced oscillation. J. Wind Eng. Ind. Aerodyn
**1997**, 69–71, 819–827. [Google Scholar] [CrossRef] - Xin, D.B.; Zhan, J.; Zhang, H.F.; Ou, J.P. Control of Vortex-Induced Vibration of a Long-Span Bridge by Inclined Railings. J. Bridge. Eng.
**2021**, 26, 04021093. [Google Scholar] [CrossRef] - Battista, R.C.; Pfeil, M.S. Reduction of vortex-induced oscillations of Rio–Niterói bridge by dynamic control devices. J. Wind Eng. Ind. Aerodyn.
**2000**, 84, 273–288. [Google Scholar] [CrossRef] - Hu, C.X.; Zhao, L.; Ge, Y.J. Mechanism of suppression of vortex-induced vibrations of a streamlined closed-box girder using additional small-scale components. J. Wind Eng. Ind. Aerody.
**2019**, 189, 314–331. [Google Scholar] [CrossRef] - Li, M.; Sun, Y.G.; Jing, H.M.; Li, M.S. Vortex-induced vibration optimization of a wide streamline box girder by wind tunnel test. J. Civil. Eng.
**2018**, 22, 5143–5153. [Google Scholar] [CrossRef] - Dai, J.; Xu, Z.D.; Gai, P.P. Parameter determination of the tuned mass damper mitigating the vortex-induced vibration in bridges. Eng. Struct
**2020**, 221, 111084. [Google Scholar] [CrossRef] - El-Gammal, M.; King Hangan, H.P. Control of vortex shedding-induced effects in a sectional bridge model by span-wise perturbation method. J. Wind Eng. Ind. Aerodyn.
**2007**, 95, 663–678. [Google Scholar] [CrossRef] - Chen, G.B.; Zhang, L.Q.; Chen, W.L.; Gao, D.L.; Yang, W.H.; Li, H. Self-suction-and-jet control in flow regime and unsteady force for a single box girder. J. Bridge Eng.
**2019**, 24, 04019072. [Google Scholar] [CrossRef] - Zhang, H.F.; Xin, D.B.; Ou, J.P. Wake control of vortex shedding based on span-wise suction of a bridge section model using Delayed Detached Eddy Simulation. J. Wind Eng. Ind. Aerodyn.
**2016**, 155, 100–114. [Google Scholar] [CrossRef] - Chen, G.B.; Chen, W.L.; Gao, D.L.; Yang, Z.F. Active control of flow structure and unsteady aerodynamic force of box girder with leading-edge suction and trailing-edge jet. Exp. Therm. Fluid Sci.
**2021**, 120, 110244. [Google Scholar] [CrossRef] - Amitay, M.; Honohan, A.; Trautman, M.; Glezer, A. Modification of the Aerodynamic Characteristics of Bluff Bodies Using Fluidic Actuators. In Proceedings of the 28th Fluid Dynamics Conference, Snowmass Village, CO, USA, 29 June–2 July 1997; AIAA Paper No. 97-2004. p. 2004. [Google Scholar]
- Tensi, J.; Boué, I.; Paillé, F.; Dury, G. Modification of the wake behind a circular cylinder by using synthetic jets. J. Vis.-Jpn.
**2002**, 5, 37–44. [Google Scholar] [CrossRef] - Irwin, H.P.A.H.; Cooper, K.R.; Girard, R. Correction of distortion effects caused by tubing systems in measurements of fluctuating pressures. J. Wind Eng. Ind. Aerodyn.
**1979**, 5, 93–107. [Google Scholar] [CrossRef] - Barlow, B.; Rae, H.; Pope, A. Low-Speed Wind Tunnel Testing, 3rd ed.; Wiley: New York, NY, USA, 1999; pp. 330–375. [Google Scholar]
- Samimy, M.; Lele, S.K. Motion of particles with inertia in a compressible free shear layer. Phys. Fluids
**1991**, 3, 1915–1923. [Google Scholar] [CrossRef] [Green Version] - Chen, W.L.; Gao, D.L.; Yuan, W.Y.; Li, H.; Hu, H. Passive jet control of flow around a circular cylinder. Exp. Fluids.
**2015**, 56, 201. [Google Scholar] [CrossRef] - Gao, D.L.; Chen, G.B.; Chen, W.L.; Huang, Y.W.; Li, H. Effects of steady wake-jets on subcritical cylinder flow. Exp. Therm. Fluid Sci.
**2019**, 102, 575–588. [Google Scholar] [CrossRef] - Taylor, Z.J.; Gurka, R.; Kopp, G.A. Geometric effects on shedding frequency for bridge sections. In Proceedings of the 11th Americas conference on wind engineering 2009, San Juan, Puerto Rico, 22–26 June 2009. [Google Scholar]
- Frandsen, J.B. Comparison of numerical prediction and full-scale measurements of vortex induced oscillations. In Proceedings of the 4th International Colloquium on Bluff Body Aerodynamics and Applications, Ruhu-University of Bochum, Bochum, Germany, 11–14 September 2000. [Google Scholar]
- Apelt, C.J.; West, G.S.; Szewczyk, A.A. The effects of wake splitter plates on the flow past a circular cylinder in the range 10
^{4}≤ Re ≤ 5 × 10^{4}. J. Fluid Mech.**1993**, 61, 187–198. [Google Scholar] [CrossRef] - Chen, W.L.; Li, H.; Hu, H. An experimental study on a suction flow control method to reduce the unsteadiness of the wind loads acting on a circular cylinder. Exp. Fluids
**2014**, 55, 1707. [Google Scholar] [CrossRef] - Lim, H.C.; Lee, S.J. PIV measurements of near wake behind a U-grooved cylinder. J. Fluids Struct.
**2003**, 18, 119–130. [Google Scholar] [CrossRef] - Oruç, V. Passive control of flow structures around a circular cylinder by using screen. J. Fluids Struct.
**2012**, 33, 229–242. [Google Scholar] [CrossRef] - Benard, N.; Balcon, N.; Touchard, G.; Moreau, E. Control of diffuser jet flow: Turbulent kinetic energy and jet spreading enhancements assisted by a non-thermal plasma discharge. Exp. Fluids
**2008**, 45, 333–355. [Google Scholar] [CrossRef] - Lumley, J.L. The structure of inhomogeneous turbulent flow. In Atmospheric Turbulence and Radio Wave Propagation; Yaglom, A.M., Tatarski, V.I., Eds.; Nauka: Moscow, Russia, 1967; pp. 166–178. [Google Scholar]
- Sirovich, L. Turbulence and the dynamics of coherent structures. Part I: Coherent structures. Q. Appl. Maths.
**1987**, 45, 561–571. [Google Scholar] [CrossRef] [Green Version] - Feng, L.H.; Wang, J.J.; Pan, C. Proper orthogonal decomposition analysis of vortex dynamics of a circular cylinder under synthetic jet control. Phys. Fluids
**2011**, 23, 526. [Google Scholar] [CrossRef] - Konstantinidis, E.; Balabani, S.; Yianneskis, M. Bimodal vortex shedding in a perturbed cylinder wake. Phys. Fluids
**2007**, 19, 701. [Google Scholar] [CrossRef] - Triantafyllou, G.S.; Triantafyllou, M.S.; Chryssostomidis, C. On the formation of vortex streets behind stationary cylinders. J. Fluid Mech.
**1986**, 170, 461–477. [Google Scholar] [CrossRef] - Orszag, S.A. Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech.
**1971**, 50, 689–703. [Google Scholar] [CrossRef] [Green Version] - Gao, D.L.; Chen, W.L.; Li, H.; Hu, H. Flow around a circular cylinder with slit. Exp. Therm.
**2017**, 82, 287–301. [Google Scholar] [CrossRef]

**Figure 1.**The geometric scale of the present test model (unit: mm). (

**a**) The three-dimension view, (

**b**) cross-section in plane I, (

**c**) cross-section in plane II.

**Figure 4.**The external surface pressure distributing on the test model with and without the trailing edge jets. The mean of the upper surface (

**a**) and lower surface (

**b**) pressure distributions, RMS of the upper surface (

**c**), and lower surface (

**d**) pressure distributions.

**Figure 5.**Aerodynamic forces that are acting on the SBG model. The mean value of the drag coefficient (

**a**), lift coefficient (

**b**), and moment coefficient (

**c**). The fluctuation of the drag coefficient (

**d**), lift coefficient (

**e**), and moment coefficient (

**f**).

**Figure 6.**Time histories (

**a**) and power spectrum analysis (

**b**) of the lift force being exerted in the SBG model of cases J

_{sj}= 0 and J

_{sj}= 0.0094.

**Figure 7.**A time-averaged representation of the flow field behind SBG model. (

**a**) J

_{sj}= 0, (

**b**) J

_{sj}= 0.0016, (

**c**) J

_{sj}= 0.0094, (

**d**) J

_{sj}= 0.0409, J

_{sj}= 0.0802, and (

**f**) J

_{sj}= 0.0094. (

**a**–

**e**): plane II, (

**f**): plane I.

**Figure 10.**The POD mode of the wake flow in measurement plane I. (

**a**) mode 1, (

**b**) mode 2, (

**c**) mode 3, (

**d**) mode 4, and (

**e**) mode 5.

**Figure 11.**The velocity profile in the different locations at wake flow in measurement plane I. (

**a**) Mean of the streamwise velocity, (

**b**) fluctuation of the streamwise velocity, (

**c**) fluctuation of the transverse velocity.

**Figure 12.**Reynolds stress distributions at different positions behind test model. (

**a**) X/B = 0.03, (

**b**) X/B = 0.12, (

**c**) X/B = 0.21, and (

**d**) X/B = 0.30.

**Figure 13.**Map of lines ${\alpha}_{i}$= constant in the plane, at X/B = 0.08 for the case J

_{sj}= 0.

**Figure 14.**The imaginary part of the critical point at various positions (X/B) for different test cases.

Volumetric Flow Rate Q_{j} (m^{3}/h) | Jet Velocity U_{j}(m * s ^{−1}) | Non-Dimensional Jet Momentum Coefficients J_{sj} |
---|---|---|

0 (Original main girder model) | 0 | 0 |

1.80 | 4.42 | 0.0016 |

2.88 | 7.07 | 0.0042 |

3.60 | 8.84 | 0.0066 |

4.32 | 10.61 | 0.0094 |

5.40 | 13.26 | 0.0147 |

7.20 | 17.68 | 0.0262 |

9.00 | 22.10 | 0.0409 |

10.80 | 26.53 | 0.0590 |

12.60 | 30.95 | 0.0802 |

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## Share and Cite

**MDPI and ACS Style**

Chen, G.; Chen, W.
Experimental Investigation and Validation on Suppressing the Unsteady Aerodynamic Force and Flow Structure of Single Box Girder by Trailing Edge Jets. *Appl. Sci.* **2022**, *12*, 967.
https://doi.org/10.3390/app12030967

**AMA Style**

Chen G, Chen W.
Experimental Investigation and Validation on Suppressing the Unsteady Aerodynamic Force and Flow Structure of Single Box Girder by Trailing Edge Jets. *Applied Sciences*. 2022; 12(3):967.
https://doi.org/10.3390/app12030967

**Chicago/Turabian Style**

Chen, Guanbin, and Wenli Chen.
2022. "Experimental Investigation and Validation on Suppressing the Unsteady Aerodynamic Force and Flow Structure of Single Box Girder by Trailing Edge Jets" *Applied Sciences* 12, no. 3: 967.
https://doi.org/10.3390/app12030967