Aerodynamics of a two-dimensional bluff body with the cross-section of a train

Abstract

Experiments on the aerodynamics of a two-dimensional bluff body simplified from a China high-speed train in crosswinds were carried out in a wind tunnel. Effects of wind angle of attack α varying in [−20°, 20°] were investigated at a moderate Reynolds number Re = 9.35 × 10⁴ (based on the height of the model). Four typical behaviors of aerodynamics were identified. These behaviors are attributed to the flow structure around the upper and lower halves of the model changing from full to intermittent reattachment, and to full separation with a variation in α. An alternate transition phenomenon, characterized by an alteration between large- and small-amplitude aerodynamic fluctuations, was detected. The frequency of this alteration is about 1/10 of the predominant vortex shedding. In the intervals of the large-amplitude behavior, aerodynamic forces fluctuate periodically with a strong span-wise coherence, which are caused by the anti-symmetric vortex shedding along the stream-wise direction. On the contrary, the aerodynamic forces fluctuating at small amplitudes correspond to a weak span-wise coherence, which are ascribed to the symmetric vortex shedding from the upper and lower halves of the model. Generally, the mean amplitude of the large-amplitude mode is 3 times larger than that of the small one. Finally, the effects of Reynolds number were examined within Re = [9.35 × 10⁴, 2.49 × 10⁵]. Strong Reynolds number dependence was observed on the model with two rounded upper corners.

Keywords

alternate transition phenomenon high-speed train Reynolds number effects span-wise correlation

Introduction

Compared with other vehicles operating near the ground, high-speed trains (HSTs) are characterized by their larger slenderness ratio (length-to-height ratio) and higher commercial speed. For example, the slenderness ratio of the widely used China Railway High-speed 380BL (CRH380BL) with 16 cars is about 108, and the operating speed of the Chinese latest HST (“FuXingHao”) has been raised to more than 350 km/h. These two features make HSTs sensitive to crosswinds (Baker et al., 2009; He et al., 2014; Tian, 2019). The above trains with large slenderness ratios are akin to infinite cylinders. A detailed study can reveal some flow mechanisms underlying this unique structure. Therefore, it is of both practical and scientific interest to investigate the aerodynamic properties of a two-dimensional (2D) bluff body with the cross-section of trains in crosswinds. The effects of the lead and tail cars are acknowledged but are not in the scope of this article.

Cross-sections of trains are usually characterized by two rounded corners at their shoulders with r/b = 0.10–0.15 (r and b are the corner radius and characteristic height normal to the oncoming flow, respectively), and two sharp corners at the bottom. Similar to a square prism, rounded corner cylinder, and circular cylinder, wind angle of attack (α), rounded corner radius ratio (r/b), and Reynolds number (Re = U_∞ × b/ν, where U_∞ is the oncoming velocity and ν is the kinematic viscosity) should have a significant effect on the aerodynamics of the trains. Hence, a short review of the effects of α, r/b, and Re on prisms/cylinders is introduced as follows.

Wind angle of attack noticeably affects the aerodynamics of a square prism with both sharp and rounded corners based on the results of smoke-wire, surface-oil, and particle image velocimetry (PIV) flow visualizations. The predominant flows around the prism can be classified into three categories, that is, separated flow regime (0° < α < 15°), reattached flow regime (15° < α < 35°), and wedge flow regime (35° < α < 45°; Carassale et al., 2014; Chen et al., 2018, 2020; Huang et al., 2010; Yen and Yang, 2011). Note that the corresponding critical angles of attack to distinguish the above-mentioned flow regimes are sensitive to the blockage ratio of the wind tunnel, turbulence intensity (TI) of the oncoming flow, slenderness ratio of the prism, and so on (Carassale et al., 2014; Yen and Yang, 2011). Generally, an increase in rounded corner radius ratio r/b results in a reduction in the critical angle of attack.

Corner rounding is applied extensively in engineering practices as a passive method to reduce the aerodynamic forces, and to suppress the noticeable vortex shedding from sharp corners, for example, tall buildings, HSTs, and long-span bridges. Many previous investigations are focused on the aerodynamic characteristics of the rounded corner cylinder (Polhamus, 1958; Schewe, 1983; Tamura and Miyagi, 1999). With an increase in r/b from 0 (square prism) to 0.5 (circular cylinder), decreasing drag and lift coefficients (C_D and C_L ), as well as increasing Strouhal number (St), can be identified. In recent decades, the effects of r/b on the near wake of a rounded-corner cylinder have been revealed by PIV, laser Doppler anemometry (LDA), and numerical methods by Tamura and Miyagi (1999) and Hu et al. (2006). They found that as r/b increases, rounded corners lead to elongated but attenuated wake vortices, and the formation length of the first roll-up vortex reaching its maximum at r/b = 0.125, but the effects of r/b on the ratio of vortex wavelength to vortex lateral spacing are very limited. Similar to the circular cylinder, the flow around a cylinder with rounded corners exhibits strong Reynolds number dependence (Carassale et al., 2014; Polhamus, 1958; Schewe, 1983; Van Hinsberg et al., 2017). Within r/b = [0.08, 0.5], the critical Re, corresponding to the transition from subcritical to critical regimes and also the start of the well-known drag crisis, decreases from 1.42 × 10⁶ to 2.0 × 10⁵. Moreover, a higher TI in the oncoming flow results in a smaller critical Re, suggesting that increasing the oncoming flow turbulence has a similar effect to increasing Re. Thus, according to these previous investigations, it can be inferred that the Reynolds number may also have a strong effect on the aerodynamics of a 2D bluff body with the cross-section of the train because of its two rounded upper corners.

As the cross-section of the train is only rounded at its upper corners, the 2D bluff body studied in this article is asymmetric along the stream-wise direction. For this bluff body, the leading corner radius plays an important role in the aerodynamics of this asymmetric structure. As pointed out by Hu et al. (2006), if the corners of one lateral side are rounded, the near wake centerline will shift toward the other lateral side with sharp corners, with the wake remaining globally symmetric along the shifted centerline.

In the past few decades, numerous researchers have carried out studies on the aerodynamics of trains with different infrastructure scenarios, such as flat ground, embankment, viaduct, and long-span bridge (Baker et al., 2009; Chiu and Squire, 1992; Hemida and Krajnović, 2010; Li et al., 2019; Schetz, 2001). Most of them pointed out that infrastructure scenarios have a strong effect on the underbody flow of trains, such as the ground effect and shielding effect (Chiu and Squire, 1992). However, physical natures of the above infrastructure scenario effects are not yet understood. To find out what the infrastructure scenario effects are and to understand clearly the above effects, the aerodynamic characteristics of an isolated train body should be studied in advance.

As the cross-section of the train is more complicated relative to circular or rounded corner cylinders, the aerodynamics of this bluff body shape has not been studied in detail in previous studies. This article aims to study systematically the aerodynamic properties of a train and the influences of angle of attack, Reynolds number, and so on. Thus, the results presented here can be used as a benchmark for future studies on the aerodynamic interference between train and infrastructure scenario. Details of the experimental models and testing cases are given in section “Wind tunnel tests,” and the results and discussion are presented in section “Results and discussion.” Finally, conclusions are summarized in section “Conclusion.”

Wind tunnel tests

Experimental details

All experiments were conducted in a closed-loop low-speed wind tunnel with a square working section of 3 m in width, 3 m in height, and 15 m in length. The wind speed in the test section can be adjusted continuously from 2 to 94 m/s, with a TI that is smaller than 0.5%. The test model and definition of the coordinate system are shown in Figure 1. Two 1200 mm × 2500 mm rectangular plates were installed vertically in the test section with a separation distance of 2000 mm (Figure 1(a)). The tested model was fixed between the two plates to eliminate the effects of model supporting framework. The wind tunnel maximum blockage ratio caused by the model and support system was about 3.2%.

Figure 1.

Models: (a) tested model, (b) coordinate system, (c) train model and arrangement of pressure taps on each cross-section, and (d) locations of the five pressure-tested cross-sections (unit: mm).

As shown in Figure 1, the cross-section of the test model was simplified from a China HST. A simplified train model reduces the number of interference factors. The depth (d) and breadth (b) of the train model are 81.6 and 92.3 mm, respectively. The scale ratio of the model is 1:40, and the length is 2000 mm. The train model was made of light-activated resin using a three-dimensional printing technique.

All the test cases are listed in Table 1. The effects of wind angle of attack on the aerodynamics of the train model were investigated from −20° to 20° with an interval of 2° at U_∞ = 15 m/s. The corresponding Reynolds number is 9.35 × 10⁴ (based on b). In bridge engineering, the most interesting range of attack angles is about [−12°, 12°] (BS5400-2, 2000; He et al., 2017; JTG/T D60-01-2004, 2004). For a train–bridge system, the attack angle of the train approaching flow should at least cover the range [−12°, 12°] owing to the leading-edge separated flow of the bridge. Based on some preliminary research, it was deduced that a full understanding of the aerodynamics of the train model could be gained over α = [−20°, 20°]. Thus, a wide range of α was adopted here. At α = 0°, the effects of Reynolds number were investigated with U_∞ ranging from 15 to 40 m/s at intervals of 5 m/s, corresponding to Re = 9.35 × 10⁴ to 2.49 × 10⁵.

Table 1.

Test conditions.

Case	Model	α (°)	Wind speed (m/s)
Effect of angle of attack		−20 to 20, Δα = 2	15
Effect of Reynolds number		0	15–40, with an interval of 5

Pressure taps were installed around the train model to measure the pressure distributions. There were five pressure measurement sections in the span-wise direction (Figure 1(c) and (d)). Each tap was connected to one channel of a pressure scanner (Scanivalve, ZOC33/64PxX2) using polyvinyl chloride (PVC) tubing with an inner diameter of 0.5 mm. The length of all the PVC connecting tubes was 500 mm. Dynamic response tests showed that the tubing amplitude ratio is flat ±10% up to a maximum frequency of 51 Hz (Li et al., 2019). Because of the predominant shedding frequency of the flow around the present tested model was less than 50 Hz, the effects of the pressure tube filtering on the measurement results were neglected. Moreover, all the pressure tubes were of the same length, and thus the phase difference between taps should be the same. The sampling frequency of the pressure scanner was 625 Hz/channel. A total of 20,000 samples were collected for each tap, from which the mean and root-mean-square (RMS) values of the aerodynamics of the model were calculated as detailed in section “Data processing method.”

Data processing method

The mean and RMS pressure coefficients at each pressure tap on the train model were calculated using $C_{p} = (\bar{P} - P) / ρ U_{\infty}^{2}$ and ${C^{'}}_{p} = 2 P_{r m s} / U_{\infty}^{2}$ . $\bar{P}$ and P_RMS are the mean and RMS values of the pressure at each tap. P_∞ is the static pressure in the test section of the wind tunnel, and 0.5 $ρ U_{\infty}^{2}$ (P_d ) is the dynamic pressure of the oncoming flow. Both P _∞ and P_d were also measured by the same pressure scanner connected to a Pitot-static tube placed in the free-stream oncoming flow. The mean aerodynamic coefficients (C_D , C_L , C_M ) of each section of the train model in wind axes can be obtained from the integration of C_p by the following formulae

C_{D} \approx \frac{(\sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times cos β_{i} \times cos α + \sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times sin β_{i} \times sin α)}{b}

C_{L} \approx \frac{(- \sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times cos β_{i} \times sin α + \sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times sin β_{i} \times cos α)}{d}

C_{M} \approx \frac{(\sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times cos β_{i} \times y_{i} - \sum_{i = 1}^{N} C_{p i} \times Δ L_{i} \times sin β_{i} \times x_{i})}{d^{2}}

ΔL_i denotes the distance between consecutive half-way points of the ith tap; β_i denotes the angle between the ith tap’s surface normal and the x direction; x_i and y_i denote the coordinates of the ith tap in the x–y (train model) coordinate system, as shown in Figure 1(b) to (d). Considering that Re is relatively high, the skin friction is neglected in the calculation of C_D . Moreover, the three fluctuation coefficients ${C^{'}}_{D}, {C^{'}}_{L}, and {C^{'}}_{M}$ are the RMS values of the corresponding coefficients of drag, lift, and moment, respectively.

Results and discussion

Time-averaged aerodynamics

Figure 2 presents the span-wise aerodynamics of the five cross-sections along the train model at α = 0°. As expected, the mean and fluctuating aerodynamic coefficients along the span-wise direction are quite uniform. The results at other angles of attack are also quite uniform, which are not presented here for brevity. Thus, only the aerodynamics at the middle cross-section is discussed in sections “Time-averaged aerodynamics,” “Pressure distribution,” and “Instantaneous aerodynamics.”

Figure 2.

Distribution of aerodynamics along the span-wise direction at α = 0°: (a) mean coefficients and (b) fluctuating coefficients.

The aerodynamics of the train model tested at the middle cross-section (cross-section 3 as shown in Figure 1(d)) at Re = 9.35 × 10⁴ is shown in Figure 3. The results of three rounded square prisms with r/b = 0, r/b = 0.07, and r/b = 0.13 at Re = 2.70 × 10⁴ (Carassale et al., 2014) are also shown in Figure 3(a), (b), and (e) for comparison. Note that the cross-section of the train model is asymmetric along the x-axis direction, as shown in Figure 1(b) and (c). The upper half model could be regarded as a rounded corner cylinder with r/b ≈ 0.11, whereas the lower half is similar to a square prism.

Figure 3.

Aerodynamics of the train model. (a - c) mean drag, lift, and moment coefficients and (d - f) fluctuating drag, lift, and moment coefficients.

Generally, the present experimental results show good agreement with those of Carassale et al. (2014), which suggests a validation of the present experiments. Besides, some interesting details can be detected by taking a close look at Figure 3(a), (b), and (e). For −20° ≤ α ≤ −4°, C_D and ${C^{'}}_{L}$ decrease almost monotonically with increasing α, and reach their minimal values of 1.32 and 0.35 at α = −4°, respectively. The linear increasing relationship between C_L and α can also be identified with the maximum value of C_{L_max} = 0.90 also occurring at α = −4°. In this range of α, the rounded upper corner of the train model is rotated toward the oncoming flow, the three aerodynamic coefficients behave analogously to those of the rounded corner cylinder with r/b = 0.13 (Carassale et al., 2014), for example, a small critical |α|. On the contrary, for 10° ≤ α ≤ 20°, C_D , C_L , and ${C^{'}}_{L}$ increase gradually and get close to those of the square prism (r/b = 0) with an increase in α. Since the sharp lower corner of the train model is turned to the upstream side, the three aerodynamic coefficients share some similarities with those of a square prism, for example, a large critical |α|. For −4° < α < 10°, C_D and ${C^{'}}_{L}$ maximize at α = 4°, and then decline gradually with α departing from 4°. In this range of α, C_L reduces sharply with a negative slope, especially for α in [−4°, −2°]. This steep drop in C_L is related to sudden changes in flow around the train model, which is discussed later.

Based on the above discussion, three critical angles (α_c ) of −4°, 4° and 10° can be identified. Since the cross-section of the train model is asymmetric along the x-axis direction, the critical angles of α_c = −4° and 10° are no longer axis-symmetric at 0°. Moreover, the two critical angles are a little smaller than α_c = ± 13° found for the square prism (Carassale et al., 2014; Huang et al., 2010; Yen and Yang, 2011). These differences could be caused mainly by the curved surfaces around the train model. On the contrary, the critical angle α_c = 4° for the train model corresponds to α_c = 0° of a symmetric bluff body, that is, a square prism with or without rounded corners (Carassale et al., 2014; Huang et al., 2010; Yen and Yang, 2011) The shifting of this critical angle from 0° to 4° is primarily attributed to the asymmetric cross-section of the train model. A similar conclusion can be accessed in Hu and Zhou (2009) from another asymmetric bluff body study. Moreover, C_M , ${C^{'}}_{D}$ , and ${C^{'}}_{M}$ also behave distinctly in the four phases, as shown in Figure 3(c), (d), and (f).

Pressure distribution

Figure 4(a) to (d) and (e) to (h) presents the mean and fluctuating pressure distributions around the train model, respectively. The relative positions of the pressure taps are normalized into 0–3.50 by the height of train model (b). As illustrated in Figure 4(g), the ranges of 0–1.76 and 1.7.6–3.50 indicate the upper and lower halves of the train model, respectively. Besides, the windward and leeward faces are separately non-dimensionalized into 0–0.46, 3.15–3.50, and 1.30–2.22. With an increase in α, the pressure distributions vary significantly in the four different Phases (I–IV) as shown in Figure 4 for four ranges of angle of attack. The fundamental characteristics of the mean pressure coefficient (C_p ) of each phase are: (I) a strong suction (strong negative C_p ) on the roof with −20° ≤ α ≤ −4° (Figure 4(a)); (II) co-existing weak suctions on both the roof and bottom (Figure 4(b), −4° < α ≤ 4°); (III) a unique weak suction on the bottom (Figure 4(c), 4° < α ≤ 10°); and (IV) a strong suction on the bottom (Figure 4(d), 10° < α ≤ 20°). The fluctuating pressure coefficient $({C^{'}}_{p})$ also exhibits different behaviors from one phase to another. In Phase I, large ${C^{'}}_{p}$ is mainly located on the leeward face (1.30 < x_p < 2.22). With an increase in α to Phase II, ${C^{'}}_{p}$ on the two upstream corners rises remarkably together with it reducing significantly on the leeward face. In Phase III, ${C^{'}}_{p}$ reduces significantly over the entire model surface. In the last phase, similar behavior of ${C^{'}}_{p}$ to the Phase I behavior can be observed, but the pressure distributions are almost symmetrical in shape.

Figure 4.

Pressure distributions around the train model at various angles of attack, grouped in Phases I–IV. (a - d) mean pressure coefficient and (e - h) fluctuating pressure coefficient.

For the upper half train model (x_p ≈ 0–1.76), C_p and ${C^{'}}_{p}$ behave similarly to those of a square cylinder with rounded corners with an increasing α (Carassale et al., 2014; Chiu and Squire, 1992). From Phases I to IV, the strong suction near the upstream shoulder (x_p ≈ 0.46) decays step by step. In Phase I, the flow separation–reattachment points can be identified qualitatively based on many previous studies (Achenbach, 1968; Zdravkovich, 1997) in subcritical Reynolds regime. The separation point lies just downstream the point where C_p reaches its minimum, as highlighted by “S” in Figure 4, whereas the reattachment point lies downstream the local peak point of the ${C^{'}}_{p}$ curve and upstream the local peak point of the C_p curve, as highlighted by “A/Ar” in Figure 4. Note that the “Ar” is produced by reverse flow from the bluff body near wake, thus it is located in the trailing-edge of the train model. Generally, with an increase in|α|, points “S” and “Ar” remain almost unchanged, but point “A” moves upstream along the train roof, as highlighted by an arrow in Figure 4(a). In Phase II, the suction hump of C_p near the upstream shoulder almost disappears (Figure 4(b)), together with the occurrence of a hump of ${C^{'}}_{p}$ at the same place (Figure 4(f)). Note that it is obvious that the full reattachment of the leading-edge shedding vortices from the upstream shoulder first disappears at α ≈ −2° (Figure 4(a), (b), (e), and (f)). The steep drop in C_L identified in Figure 3(b) can be explained by this sudden change. For the other two phases (α ≥ 6°), the pressure field on the roof is flattening, indicating a full separation on the roof.

For easy and clarity representation, C_p and ${C^{'}}_{p}$ on the lower half train model (x_p ≈ 1.76–3.50) are introduced from bottom to top (Figure 4), which resembles the behavior of a square prism (Huang et al., 2010; Yen and Yang, 2011). From Phase IV to Phase I, the overall tendency of the pressure distribution is similar to this of a square prism changing from full to intermittent reattachment, and then to full separation state (Carassale et al., 2014; Huang et al., 2010; Yen and Yang, 2011). However, two distinct observations can be made when compared to the upper half train model. First, the weak separation bubble on the bottom does not vanish until α ≈ −2°, which covers both Phases III and II, suggesting the full separation flow state on the bottom of the train model is delayed relative to the square prism. This special phenomenon may be caused by the frontal short arc of the sharp corner, as shown in Figure 1. Therefore, Phase II is characterized by the concurrent suctions on both the roof and the bottom at α = −2° to 4°, which indicates a unique phenomenon not observed for a square prism with sharp or rounded corners (Carassale et al., 2014; Chiu and Squire, 1992; Huang et al., 2010; Yen and Yang, 2011). Second, the reverse flow reattachment point “Ar” cannot be identified on the bottom of the train model.

Instantaneous aerodynamics

To get an in-depth understanding of the aerodynamic characteristics of the train model in the above-mentioned four phases, typical instantaneous aerodynamic coefficients and the corresponding power spectral densities (PSDs) of C_L at α = −16°, −2°, 0°, 4°, 10°, and 16° are shown in Figures 5 and 6. As shown in the bottom of the two figures, the color-maps of the instantaneous PSDs are obtained by a wavelet transformation. An analytic complex Morlet mother wavelet is adopted here. The time axes of the two figures are normalized by U_∞ and b with a window length of 400 dimensionless units, corresponding to about 64 vortex shedding cycles.

Figure 5.

Typical instantaneous values of coefficients and lift spectra of the train model at α = −16°, −2°, 0°.

Figure 6.

Typical instantaneous values of coefficients and lift spectra of the train model at α = 4°, 10°, 16°.

For α = −16° and 16° (two representative results of Phases I and IV as highlighted by the solid lines in Figure 4), C_L of the train model is nonstationary over time as shown in Figures 5(b) and 6(j). Obviously, both large- and small-amplitude fluctuations of C_L can be identified, which occur alternately in time. Generally, the mean amplitude of the large-amplitude fluctuations is about 3 times the small ones. A similar phenomenon has also been observed by Wang and Zhou (2009), Sattari et al. (2012), Uffinger et al. (2013), and Wang et al. (2017, 2018) in the aerodynamic coefficients and near wake of a wall-mounted finite-length square cylinder in uniform flow. As suggested by the previous studies, when the train model is governed by the anti-symmetric vortex shedding, the train’s roof and bottom are controlled by vortices of different scales over time, which results in an anti-symmetric pressure distribution around the train model in the stream-wise direction. Thus, a large-amplitude fluctuation of C_L can be observed. On the contrary, when the train model is governed by the symmetric vortex shedding, the train’s roof and bottom are controlled by vortices of equivalent scales over time, which results in a symmetric pressure distribution around the train model with respect to the stream-wise direction. Thus, a small-amplitude fluctuation of C_L can be identified. As shown in Figures 5(d) and 6(l), the alternate transition between large- and small-amplitude fluctuations of C_L can be also confirmed by the wavelet maps of C_L as the vortex-shedding frequency is also non-continuous over time. Note that when C_L fluctuates with large amplitudes, the mean and fluctuating values of C_D also reach their local maxima, as well as C_M .

This alternate transition phenomenon is similar to a low-frequency sinusoidal amplitude modulation function, which may result in sudden changes in the wind load of the train from crosswinds over time. This nonstationary wind load is still not fully understood, which could be a potential threat to the operational safety of the train. However, the aerodynamics of the train are different when it is moving, compared to wind normal to the train axis as studied herein. Thus, further validations are still required to determine whether this alternate transition phenomenon occurs in a moving model test.

For α = −2° and 10° (two representative cases of Phases II and III as also highlighted by the solid lines in Figure 4), the alternate transition phenomenon observed in Phases I and IV is going to disappear, especially for C_D and C_M . Meanwhile, the vortex shedding becomes less-organized as shown in Figures 5(h) and 6(h). Moreover, for the other two typical angles of attack α = 0° and 4°, the alternate transition phenomenon almost disappears, while the vortex shedding is quite organized.

C_p of taps 10 and 30 (Figure 1) at α = −16° are presented in Figure 7, to highlight the alternate transition phenomenon of the aerodynamics of the train model. The same method used by Wu et al. (2005) is adopted here to investigate the phase difference between the two pressure signals. The results of a complex wavelet analysis corresponding to the time-averaged vortex shedding frequency are selected to calculate the phase difference. As shown in Figure 7(c), two distinct configurations of the phase difference (Δθ) can be observed, named Regimes A and B. For Regime A, Δθ varies within ±15° of 180°, which is in fair agreement with that observed in a finite wall-mounted square prism in the large-amplitude range by Sattari et al. (2012). The corresponding C_p of the taps 10 and 30 also fluctuates in the large-amplitude mode (Figure 7(a)), associated with the phase angles (θ) of the two taps being almost parallel to each other in one vortex shedding period (Figure 7(b)). On the contrary, in Regime B, the phase difference departures from its centerline of Δθ = 180°, and reaches its extreme value 0° or 360° at tU_∞/b = 85. The corresponding phase angles come across each other at the same point, as highlighted by an orange dashed line in Figure 7(b) at tU_∞/b = 85. Moreover, the RMS values of C_p of the two taps are quite small relative to those of Regime A (Figure 7(a)). Generally, the boundaries of Regimes A and B are very fuzzy, similar to that observed by Sattari et al. (2012).

Figure 7.

(a) Pressure time series of tap 10 and tap 30 at an angle of attack of −16°, (b) phase variations of the two taps, and (c) phase difference between the two taps.

As shown in Figure 8(a) and (d), the envelope of C_L , which iscalculated via the peak envelope function built in MATLAB, reveals some information of the alternate transition phenomenon. Thus, C_L and its envelope of the selected α = −16° and 4° angles, together with the amplitudes of the fast Fourier transform (FFT), are all shown in Figure 8. It is interesting to note that two predominant frequencies with broad-banded peaks appear in the FFT amplitude of C_L with α = −16°, as presented in the left column of Figure 8. The higher frequency fb/U_∞ = 0.16 is about 10 times larger than the lower one fb/U_∞ = 0.013, which are in line with that observed by Sattari et al. (2012). As for the envelope of C_L , there is just a single hump in the FFT amplitude (Figure 8(c)). The reduced frequency of this single hump is 0.013, corresponding to the lower peak in the FFT amplitude of C_L . On the contrary, the FFT amplitude of C_L with α = 4° has only a sharp and narrow peak at fb/U_∞ = 0.16. In this case, the small broad-banded hump at fb/U_∞ = 0.013, observed in the FFT amplitude of the envelope of C_L at α = −16°, disappears in Figure 8(f). Therefore, it can be concluded that the higher peak with fb/U_∞ = 0.16 corresponds to the dominant vortex shedding frequency, which is also confirmed by the wavelet time-spectra as shown in Figures 5 and 6. While the lower broad-banded peak should be the frequency of the alternate transition phenomenon between the large- and small-amplitude fluctuations of C_L .

Figure 8.

(a) and (d) Time series of C_L and its envelope, (b) and (e) FFT amplitude of C_L , (c) and (f) FFT amplitude of the envelope of C_L .

Span-wise correlation

The correlation coefficients of C_L at α = 4° and −16° are shown in Figure 9. For α = 4°, the correlation coefficients are in good agreement with that of an infinite square prism in smooth oncoming flow. The correlation coefficients at α = −16° reduces faster than those at α = 4° with an increase in the span-wise separation ΔS/b. This rapid reduction may be partially caused by the alternate transition phenomenon observed above.

Figure 9.

Span-wise correlation coefficients of lift.

The correlation coefficient between cross-sections 2 and 3, that is, ΔS/b = 1.14, is larger than 0.7, suggesting a strong span-wise coherence. With ΔS/b increasing to 1.71 (distance between cross-sections 3 and 4), the correlation coefficient at α = −16° falls below 0.7. With further increase in ΔS/b to 5.71 (distance between cross-sections 3 and 5), the correlation coefficients for both α = 4° and −16° drop considerably below 0.5.

To investigate the effects of the alternate transition phenomenon (Figures 5 and 6) on the phase difference of C_L along the span-wise direction, the above-mentioned three dimensionless span-wise separations ΔS/b = 1.14, 1.71, and 5.71 are selected for phase-difference analysis, as highlighted by the dashed lines in Figure 9.

For α = 4°, almost no alternate transition phenomenon can be observed (Figures 6 and 8). Figure 10 presents the instantaneous C_L and the corresponding phase difference Δθ for the three ΔS/bs = 1.14, 1.71, and 5.71. Obviously, Δθ for ΔS/bs = 1.14 and 1.71 changes slowly over time. Within the window length, the maximum values of Δθ at ΔS/ds = 1.14 and 1.71 are about 35° and 65°, respectively. Generally, large Δθ occurs where C_L fluctuates at relatively small amplitudes, as highlighted by a black arrow in Figure 10(c) and (d). For ΔS/b = 1.71, the maximum phase angle Δθ = 65° is almost 2 times that of Wu et al. (2005) observed in the near weak of a normal plate at the same spacing. This discrepancy may be attributed to the intermittently reattached flow on the afterbody of the train model. However, with ΔS/b increasing to 5.71, Δθ fluctuates significantly over time, suggesting a weak span-wise coherence (Szepessy, 1994), as shown in Figure 10(f). Moreover, the variation of Δθ versus ΔS/b in the time domain shows good agreement with the correlation coefficients (Figure 9), that is, a fluctuating Δθ corresponds to weak span-wise coherence.

Figure 10.

Phase difference of lift with α = 4°: (a) and (b) ΔS/b = 1.14 (distance between cross-sections 2 and 3); (c) and (d) ΔS/b = 1.71(distance between cross-sections 3 and 4); (e) and (f) ΔS/b = 5.71(distance between cross-sections 3 and 5).

For α = −16°, the alternate transition phenomenon is pronounced (Figures 6 and 8). Compared to Figure 10, Δθ in Figure 11 fluctuates more noticeably with an increase in ΔS/b, indicating weaker span-wise coherence. Two typical regimes observed Figure 7, that is, Regimes A and B, can also be detected here as highlighted by green dash lines in Figure 11. In Regime A, C_L of cross-sections 2–5 all fluctuate at large amplitudes (Figure 11(a), (c), and (e)) and Δθ varies slowly, similar to that observed in Figure 10, signifying the span-wise coherence in this regime is strong. On the contrary, C_L of all four cross-sections fluctuate at small amplitudes in Regime B. In this regime, Δθ fluctuates dramatically. Even though the phase resultant quantity in this regime is somewhat uncertain (Wu et al., 2005), a weak span-wise coherence can still be clearly detected from the fluctuating Δθ.

Figure 11.

Phase difference of lift with α = −16°: (a) and (b) ΔS/b = 1.14 (distance between cross-sections 2 and 3); (c) and (d) ΔS/b = 1.71(distance between cross-sections 3 and 4); (e) and (f) ΔS/b = 5.71(distance between cross-sections 3 and 5).

In summary, Figures 3, 9 to 11 clarify that the span-wise coherence of the pressure on the train model decays significantly with an increase in span-wise separation ΔS/b from 1.14 to 11.43 in uniform flow. The alternate transition phenomenon would be another core aggravation to this decaying behavior.

Reynolds number effect

The effects of Reynolds number on the aerodynamics of the train model at the middle cross-section are shown in Figure 12. The oncoming flow speed varies from 15 to 40 m/s, corresponding to Re = [9.35 × 10⁴, 2.49 × 10⁵].

Figure 12.

Variation of the aerodynamic properties with Re at α = 0°: (a) steady drag coefficient, (b) steady lift coefficient, (c) fluctuating lift coefficient, and (d) Strouhal number.

Figure 12 shows the variation of C_D , C_L , ${C^{'}}_{L}$ , and St as a function of Re. Some of the results in previous literature are also shown here for comparison, with r/b = 0.08–0.5, in different oncoming flow conditions. As shown in Figure 12(a), the drag crisis is a common characteristic for all the square cylinders with rounded corners. The critical Re, corresponding to the transition from subcritical to critical regimes and also the start of drag crisis, is very sensitive to the r/b. In particular, the critical Re reduces from 1.42 × 10⁶ to about 2.00 × 10⁵ with an increase in r/b from 0.08 to 0.5 (circular cylinder), as highlighted by a red arrow in Figure 12(a). Meanwhile, C_D in the subcritical Re regime drops from 1.6 to 1.2 (Polhamus, 1958; Schewe, 1983; Van Hinsberg et al., 2017). Note that the TI of the oncoming flow also plays an important role in the variation of C_D , as shown in Figure 12(a). Generally, a higher TI results in a smaller critical Re and lower value of C_D (Carassale et al., 2014; Van Hinsberg et al., 2017). For the present train model, a clear drag crisis is also observed with C_D reducing sharply from 1.70 to 1.29 in the Re range of 9.35 × 10⁴–2.49 × 10⁵, as shown in Figure 12(a). This behavior is expected due to the rounded upper corners of the train model, similar to that of the rounded square cylinder.

Figure 12(b) also presents the sudden C_L jump in the critical region, which is qualitatively similar to both circular and rounded corner cylinders. The sudden jump of C_L can be attributed to the transition of the shear layer on the upper half of the train model.

The results of ${C^{'}}_{L}$ are shown in Figure 12(c). ${C^{'}}_{L}$ goes down severely when Re increases from the subcritical region to the critical region, which is nearly identical to the results of Schewe (1983) and Van Hinsberg et al. (2017). Moreover, the TI of the oncoming flow also has a significant effect on ${C^{'}}_{L}$ of a cylinder with rounded corners.

For the circular cylinder, St jumps up in the critical regime, as shown in Figure 12(d). However, as pointed out by Zdravkovich (1997), the vortex shedding becomes less organized or completely vanishes in the critical regime. For the present tested model, St reduces slightly with an increase in Re. Moreover, the peak of the spectrum of C_L decays significantly with an increase in Re, as shown in Figure 13, which is similar to the results of Norberg and Sunden (1987). This observation suggests that the vortex shedding around the train model decays noticeably with the increase of Re from 9.35 × 10⁴ to 2.49 × 10⁵.

Figure 13.

Lift power spectra at a range of Reynolds numbers.

Figure 14 presents the distribution of pressure coefficients around the model at α = 0°. After Achenbach (1968), it may be noted that the separation point on the roof of the train model moves downstream from S1 to S3 with an increase in Re, as shown in Figure 14(a). Moreover, the backpressure on the leeward face of the train model also recovers gradually with increasing Re, which is similar to the Reynolds number effect on circular cylinders.

Figure 14.

Pressure distributions around the train model at Re = 9.35 × 10⁴–2.49 × 10⁵: (a) mean pressure coefficients and (b) fluctuating pressure coefficients.

Based on the preceding discussion, one may conclude that the aerodynamic characteristics of the train model bear significant dependence on Reynolds numbers like a square cylinder with rounded corners and a circular cylinder.

Conclusion

The aerodynamic characteristics of a 2D bluff body with the cross-section of a China HST are experimentally investigated in uniform flow. The effects of the angle of attack (α), dimensionless span-wise separation (ΔS/b), and Reynolds number (Re) are examined. The ranges of α, ΔS/b, and Re under consideration are of −20° to 20°, 1.14 to 11.43, and 9.35 × 10⁴ to 2.49 × 10⁵, respectively. Based on the results of the wind tunnel tests, the following conclusions can be drawn:

1. Four typical behaviors of the aerodynamics of the train model can be observed within −20° ≤ α ≤ 20°. For −20° ≤ α ≤ −4°, the aerodynamics of the train model shows strong similarity with those of a rounded corner cylinder. For 10° < α ≤ 20°, the model aerodynamics are in good agreement with those of a square prism. Since the train model is asymmetric along the stream-wise direction, the most organized flow behavior occurs at α = 4° instead of 0° for a symmetric bluff body. This critical angle, α = 4°, cuts the angle range of −4° < α ≤ 10° into the other two phases. These four aerodynamic phases are mainly attributed to the flow around the upper and lower halves of the train model changing from full to intermittent reattachment, and to full separation with a variation in α. Moreover, the aerodynamic characteristics of the train model can be taken as a combination of a square prism and a rounded corner cylinder.

2. The alternate transition phenomenon, characterized by an alteration between large- and small-amplitude fluctuating aerodynamics with a frequency about 1/10 of the predominant vortex shedding, can be detected. Generally, the mean amplitude of the large-amplitude mode can be 3 times that of the small one. This nonstationary aerodynamics could be a potential threat to the serviceability of the HST. The triggering mechanism of the alternate transition phenomenon could be the vortex shedding from the train model transitioning between symmetric and antisymmetric flow regimes.

3. In turbulence free oncoming flow, the aerodynamics in the longitudinal direction of the train model is quite uniform. However, the span-wise coherence of the pressures decays significantly with an increase in the span-wise separation from 1.14 to 11.43. More importantly, the alternate transition phenomenon has a strongly negative effect on the span-wise correlation of the aerodynamics of the train model.

4. The aerodynamics of the train model experiences significant Reynolds number effects resembling a cylinder with rounded corners and a circular cylinder. The critical Reynolds number, corresponding to the transition from subcritical to critical regimes, is around 1.56 × 10⁵.

Footnotes

Acknowledgements

The authors thank the reviewers for their great help on the manuscript during its review progress.

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 support of National Key R&D Program of China (2017YFB1201204). The first author would like to acknowledge the support of National Project Funded by the China Scholarship Council (201706370214).

ORCID iD

Huan Li

References

Achenbach

(1968) Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re= 5× 10⁶. Journal of Fluid Mechanics 34(4): 625–639.

Baker

Cheli

Orellano

, et al. (2009) Cross-wind effects on road and rail vehicles. Vehicle System Dynamics 47(8): 983–1022.

BS 5400-2 (2000) Steel, concrete and composite bridges —part 2: specification for loads.

Carassale

Freda

Marre-Brunenghi

(2014) Experimental investigation on the aerodynamic behavior of square cylinders with rounded corners. Journal of Fluids and Structures 44: 195–204.

Chen

Tse

Kwok

KCS

, et al. (2018) Aerodynamic damping of inclined slender prisms. Journal of Wind Engineering and Industrial Aerodynamics 177: 79–91.

Chen

Tse

Kwok

KCS

, et al. (2020) Modelling unsteady self-excited wind force on slender prisms in a turbulent flow. Engineering Structures 202: 109855.

Chiu

Squire

(1992) An experimental study of the flow over a train in a crosswind at large yaw angles up to 90. Journal of Wind Engineering and Industrial Aerodynamics 45(1): 47–74.

Wang

, et al. (2017) Effects of geometrical parameters on the aerodynamic characteristics of a streamlined flat box girder. Journal of Wind Engineering and Industrial Aerodynamics 170: 56–67.

Zou

Wang

, et al. (2014) Aerodynamic characteristics of a trailing rail vehicles on viaduct based on still wind tunnel experiments. Journal of Wind Engineering and Industrial Aerodynamics 135: 22–33.

10.

Hemida

Krajnović

(2010) LES study of the influence of the nose shape and yaw angles on flow structures around trains. Journal of Wind Engineering and Industrial Aerodynamics 98(1): 34–46.

11.

Zhou

(2009) Aerodynamic characteristics of asymmetric bluff bodies. Journal of Fluids Engineering 131(1).

12.

Zhou

Dalton

(2006) Effects of the corner radius on the near wake of a square prism. Experiments in Fluids 40(1): 106.

13.

Huang

Lin

Yen

(2010) Time-averaged topological flow patterns and their influence on vortex shedding of a square cylinder in crossflow at incidence. Journal of Fluids and Structures 26(3): 406–429.

14.

JTG/T D60-01-2004 (2004) Wind-resistant design specification for highway bridges.

15.

Wang

, et al. (2019) Aerodynamics of a scale model of a high-speed train on a streamlined deck in cross winds. Journal of Fluids and Structures 91.

16.

Norberg

Sunden

(1987) Turbulence and Reynolds number effects on the flow and fluid forces on a single cylinder in cross flow. Journal of Fluids and Structures 1(3): 337–357.

17.

Polhamus

(1958) Effect of flow incidence and Reynolds number on low-speed aerodynamic characteristics of several noncircular cylinders with applications to directional stability and spinning. Langley Field, VA: Langley Aeronautical Laboratory.

18.

Sattari

Bourgeois

Martinuzzi

(2012) On the vortex dynamics in the wake of a finite surface-mounted square cylinder. Experiments in Fluids 52(5): 1149–1167.

19.

Schetz

(2001) Aerodynamics of high-speed trains. Annual Review of Fluid Mechanics 33(1): 371–414.

20.

Schewe

(1983) On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers. Journal of Fluid Mechanics 133: 265–285.

21.

Szepessy

(1994) On the spanwise correlation of vortex shedding from a circular cylinder at high subcritical Reynolds number. Physics of Fluids 6(7): 2406–2416.

22.

Tamura

Miyagi

(1999) The effect of turbulence on aerodynamic forces on a square cylinder with various corner shapes. Journal of Wind Engineering and Industrial Aerodynamics 83(1–3): 135–145.

23.

Tian

(2019) Review of research on high-speed railway aerodynamics in China. Transportation Safety and Environment 1(1): 1–21.

24.

Uffinger

Ali

Becker

(2013) Experimental and numerical investigations of the flow around three different wall-mounted cylinder geometries of finite length. Journal of Wind Engineering and Industrial Aerodynamics 119: 13–27.

25.

Van Hinsberg

Schewe

Jacobs

(2017) Experiments on the aerodynamic behaviour of square cylinders with rounded corners at Reynolds numbers up to 12 million. Journal of Fluids and Structures 74: 214–233.

26.

Wang

Peng

, et al. (2018) Control of the aerodynamic forces of a finite-length square cylinder with steady slot suction at its free end. Journal of Wind Engineering and Industrial Aerodynamics 179: 438–448.

27.

Wang

Zhao

, et al. (2017) Effects of oncoming flow conditions on the aerodynamic forces on a cantilevered square cylinder. Journal of Fluids and Structures 75: 140–157.

28.

Wang

Zhou

(2009) The finite-length square cylinder near wake. Journal of Fluid Mechanics 638: 453–490.

29.

Miau

, et al. (2005) On low-frequency modulations and three-dimensionality in vortex shedding behind a normal plate. Journal of Fluid Mechanics 526: 117–146.

30.

Yen

Yang

(2011) Flow patterns and vortex shedding behavior behind a square cylinder. Journal of Wind Engineering and Industrial Aerodynamics 99(8): 868–878.

31.

Zdravkovich

(1997) Flow around Circular Cylinders, Vol. 1: Fundamentals. Oxford: Oxford University Press.