Flow distortion recorded by sonic anemometers on a long-span bridge: towards a better modelling of the dynamic wind load in full-scale.

The turbulent wind field around a suspension bridge deck is studied using eleven months of full-scale records from sonic anemometers mounted above the girder. Using the mean and turbulent velocity characteristics, we demonstrate that the bridge structure can significantly distort the flow. More precisely, the friction velocity, the variance of the fluctuating vertical velocity and the mean wind incidence angle are underestimated on the downstream side of the deck. The terrain is also found to influence the flow in a non-negligible way, such that turbulence characteristics differ significantly from those observed in flat and homogeneous terrains. For a hexagonal girder with a width to height ratio B/H ≈ 4.5, deck-induced flow distortion is still observed on the downwind side of the girder at a height above the road equal to 3.6H. This further supports the idea that wind measurements from a suspension bridge should rely on anemometers on both sides of the deck to mitigate flow distortion. The improved flow description combined with high-resolution acceleration records of the deck provides a simulation of the wind-induced response of the bridge with a level of accuracy that is rarely achieved in full-scale. In particular, the limits of a wind model based on flat terrain assumption as well as the limits of the strip theory are highlighted by the recorded data and the improved modelling of the bridge buffeting response.

during the post-processing of the data using the correction provided by Gill Instrument, which is a multiplication 71 scaling factor. Note that the anemometers located on H08E and H10E, are new, from an updated production lot, 72 and do not need such a correction. 73 (d) The coordinate system is rotated such that v ≈ w ≈ 0. The study in such a coordinate system can be done using 74 the planar-fit (PF) algorithm [16,17], the double rotation method or triple rotation method [18]. In the present 75 case, the high complexity of the terrain justifies the application of the double rotation technique rather than a 76 sectoral PF algorithm, which may have a limited applicability in such an environment [19]. 77 (e) Non-stationary samples are omitted in the further analysis as well as those associated with a mean wind speed 78 below 6 m s −1 . Only the second-order stationarity is considered here, i.e. the time-dependency of statistical 79 moments up to order two. For each time series, the centred moving mean and moving standard deviation are 80 studied, using a half window length of 5 min. If a local deviation from the mean or standard deviation greater 81 than 40 % is detected, the time series is assumed non-stationary and dismissed. The values of the threshold 82 parameters and window length have been chosen using simulated stationary turbulent wind velocity histories 83 that are generated using the method by Shinozuka and Deodatis [20], which showed that such a choice was 84 appropriate to differentiate between the random error, due to the finite averaging time, and the error due to 85 non-stationary fluctuations.

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(f) Following Stiperski and Rotach [15], the measurement uncertainties in the variance and covariance of the wind 87 velocity records are assessed. For the wind velocity component j = {u, v, w}, the measurement uncertainty a j for 88 the variance is defined as, where τ is the averaging time; κ j is the kurtosis of the velocity component j and u(z) is the mean wind speed at a 90 height z above the ground. Similarly, the measurement uncertainty for the covariance u w and v w between the 91 horizontal wind components and the vertical wind component are defined as: where u * is the friction velocity, calculated as advised by Weber [21]: samples with a mean wind speed above 6 m s −1 .

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(g) Turbulence characteristics are studied after removing the mean and any linear trend from the approved time 101 series.

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The atmospheric stability is estimated using the non-dimensional Obukhov length ζ, local scaling [22,23] and assuming that the virtual potential temperature is well approximated by the sonic temperature: where g = 9.81 m s −2 is the acceleration of gravity; θ v is the average virtual potential temperature; w θ v is the flux of 105 virtual potential temperature and κ ≈ 0.4 is the von Kármán constant. Only samples characterized by |ζ| < 0.1, which 106 represents near-neutral conditions, are considered in the following.

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The one-point power-spectral density (PSD) estimates of the velocity fluctuations are denoted S j , where j = In addition to the one-point velocity spectra and cross-spectra, the turbulence intensity I j , where j = {u, v, w}, is 114 studied at wind velocities large enough so that I j produce stable values. We recall that I j is defined as where σ j is the standard deviation of the velocity component j. The action of atmospheric turbulence on a wind-sensitive structure is named buffeting load and the associated 118 dynamic response is named buffeting response [25,26]. The frequency domain approach, which relies on the 119 computation of the power spectral density and the associated standard deviation of the bridge displacement and/or 120 vertical response is considered in the following.

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In the present study, only the vertical bridge acceleration response is computed, which for simplicity is evaluated forces are linearised with respect to the time-dependent angle of attack, using Taylor series up to order 1. As shown in enough to generate significant coupling effects. Finally, the strip assumption is used, i.e. the correlation of the wind forces along and across the bridge deck is taken to be identical to that of the undisturbed, incoming wind fluctuations, 132 implying that the cross-sectional aerodynamic admittance is equal to one at every frequency.

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If the flow is not perpendicular to the bridge longitudinal axis, it is said "skewed" and the angle between the normal 134 to the deck and the wind direction is the "yaw angle" β. The horizontal wind velocity component normal to the deck is The dynamic wind-induced response of the bridge deck in the vertical direction is dominated by the vertical wind 137 velocity component, so that the spectral wind load can be written as: where Xie et al. [29], the coherence of turbulence γ w is also affected by the yaw angle. In the present case, the yaw angle is

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The spectrum of the modal wind load S Q is associated with the mode shapes Φ(y) of the bridge deck, on which the 146 buffeting load is concentrated, The power spectral density of the bridge response is where H( f ) is the mechanical admittance of the system modified by the modal aerodynamic damping and stiffness.

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The mode shapes Φ(y) and eigenfrequencies of the Lysefjord bridge are computed using a continuum bridge model and analysis relying on an automated covariance-driven stochastic subspace identification algorithm [28,33].

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The standard deviation of the computed bridge displacement response is denoted σ r z (y) and is computed as The vertical turbulent wind load is modelled using the one-point spectra and coherence estimated at mid-span.

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Generally a single spectral model is desired to describe the velocity spectra on both sides of the deck. However, as 155 the spectra affected by flow distortion may have a flattened spectral peak the following spectral form is adopted: where a 1 , a 2 , b 1 and b 2 are parameters to be determined and f r is the reduced frequency defined as For a flow perpendicular to the bridge deck, the co-coherence γ w is often modelled using the Davenport model [34]: where y i and y j are two measurement positions; C is a constant named "decay coefficient" and f is the frequency.
159 However, we model here γ w as a three-parameter exponential decay function: where c 1 , c 2 and c 3 are three decay parameters permitting the co-coherence to be negative and to have a realistic 161 behaviour at large spatial separations, for which the maximum value of γ w becomes lower than unity at zero frequency.

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For a given mean wind speed and frequency, the co-coherence is assumed here to be a function of the spatial separation If the flow is not perpendicular to the bridge longitudinal axis, the 164 crosswind separation is replaced by the along-span separation: Although the approach adopted above to compute the bridge response is much simpler than in Cheynet et al. [28],  A relative difference between the mean wind speed estimated from a reference sonic anemometer and the others, 189 denoted u , is investigated in Table 1. The reference sensor is the one located on H18W or H18E for a wind from 190 south-southwest or north-northeast, respectively. Note that the discrepancies due to slightly different measurement 191 heights are not corrected for as the wind shear difference is assumed small. It should also be noted that the anemometer 192 on H10W does not measure the vertical component and, therefore, cannot provide any indication of the wind incidence 193 angle.

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For a flow from south-southwest, Table 1 shows that, on average, the absolute relative difference is lower than 4 %, 195 indicating that for this particular wind sector the deck has limited influence on the mean flow on the downwind side of 196 the bridge. The larger discrepancies observed between the mean speed measurements on H18W and H24W may be due

Sensors Wind Direction
West side East side south-southwest north-northeast The mean wind incidence angle α recorded for the two wind sectors considered are displayed in Table 1 for   whereas it is not the case for the along-wind component.

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The estimated one-point turbulence characteristics are displayed separately for a wind direction from south-220 southwest and north-northeast in Tables 2 and 3. The notation y ± x, used in the following, indicates that x is the 221 For a south-southwest flow, the ratio σ w /u * is slightly lower on the downwind side than on the upwind side, which 225 must be interpreted with caution as the value generally found in the literature in flat terrain ranges from 1.2 to 1.3 [37], 226 whereas it corresponds here to a flow likely distorted by the presence of the deck. Jensen and Hjort-Hansen [1] found a 227 ratio σ w /u * between 1.4 and 1.6 on the upwind side of the Sotra Bridge (truss girder), which is consistent with the 228 measurements conducted on the upwind side of the deck for the two wind sectors considered.

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The ratio σ w /σ u estimated on the upwind side of the bridge is slightly above 0.6, in agreement with Kristensen and σ v /σ u ≈ 1 obtained for a wind from south-southwest may, therefore, be attributed to terrain-induced flow distortion.

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For a wind from north-northeast, a more common value σ v /σ u ≈ 0.8 is found, although it is still larger than for the 235 Sotra Bridge experiment [2], which was slightly lower than 0.7. The considerable scatter obtained for σ v /u * and σ u /u * , 236 for both wind directions, may also be attributed to local terrain effects rather than the influence of the bridge structure.

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Since Table 3 shows that the presence of the deck does not clearly affect I u , the larger values of σ u /u * on the downwind 238 side of the deck are likely caused by an underestimation of u * due to flow distortion by the girder.  Both figures show that for 0.1 < f r < 1, the downwind vertical velocity spectra has a nearly constant value which is 250 considerably lower than the spectral peak of both the reference semi-empirical spectra as well as the upwind spectra.  upwind or downwind side of the girder. Note that for a flow from north-northeast, the S u spectrum on the downwind 254 side shows values that are slightly larger than on the upwind side for f r > 1, which is also observed for the flow from  For both wind sectors, the co-spectrum estimates from the downwind side of the deck are below those from the 261 upwind side at f r < 1, i.e. at the same reduced frequency as for the S w spectrum, which can be attributed to the 262 distortion of the flow by the deck. Velocity records from both sides of the deck show, however, discrepancies with the 263 theoretical slope in −7/3 at f r > 3, implying that the co-spectrum may, in addition, be influenced by the topography 264 upstream of the deck.  Figures 6 and 7 show that the ratio S w /S u obtained using the anemometers located on the downwind side of the 274 bridge is significantly lower than upwind over the entire range of reduced frequencies. If the anemometer data from the 275 upwind side are used, the ratio S w /S u converges toward a value ranging from 1.25 to 1.30 at f r > 5, indicating that for 276 a flow from south-southwest, the departure from the local isotropy may be due to a minor flow distortion on the upwind 277 side of the deck, maybe due to the non-zero mean incidence angle. Note that the left panel of fig. 6 suggests that the 278 flow recorded by the sonic anemometer mounted on H08Wb (6 m above the deck) on the upwind side may be affected, 279 to a limited extent, by the presence of the deck at 0.6 < f r < 2, contrary to the sensor on H08Wt (10 m above the deck) 280 which does not indicate any influence from the bridge deck when located upwind. 281 Figure 7 shows clearly that flow distortion is not uniform along the girder, as it is seen for flow from north-northeast, 282 that the anemometer mounted on the southern part of the bridge records the strongest distortion with a ratio S w /S u ≈ 0.8 283 at f r ≈ 10, which for z = 60 m and u = 10 m s −1 corresponds to f = 1.7 Hz. Near f r ≈ 1, which for z = 60 m and 284 u = 10 m s −1 corresponds to the first eigenfrequencies of the bridge, the ratio S w /S u is also twice as low on the 285 downstream side than on the upwind side, reinforcing the idea that the resonant buffeting response is underestimated     We recall that the west side of the deck is instrumented with more anemometers than the east side, which explains why 333 the coherence on the downstream side is displayed for a larger variety of spatial separation than upwind. Finally, the 334 right panel of fig. 11 corresponds to the estimated and fitted one-point velocity spectrum S w at mid-span on both the 335 upwind side and the downstream side of the girder. Note that the difference between the normalized PSD estimates for 336 the downwind side in fig. 10 and fig. 11 are simply due to the fact that the latter figure shows the normalized spectra 337 "as seen" by each sonic anemometer, whereas the former figure shows the spectra normalized by the variance of the 338 "undisturbed" vertical velocity on the upwind side. The co-coherence is modelled using a least-square fit of eq. (15) 339 to the full-scale co-coherence estimates. A similar procedure is done for the velocity spectrum S w using eq. (12).

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Both eqs. (12) and (15) provide an excellent fit of the co-coherence and vertical velocity spectrum, respectively. The 341 corresponding parameters for both wind sectors studied are summarized in Table 4, although only the case of a flow 342 from north-northeast is considered in the following to study the buffeting response of the bridge.  Table 4: Coefficients of the co-coherence (c 1 , c 2 , c 3 ) and one-point spectrum model (a 1 , b 1 , a 2 , b 2 ) of the vertical wind velocity component, fitted in the least-square sense on the upwind and downwind side of the deck, for the two main wind direction recorded on the Lysefjord bridge.
Wind Direction girder side Co-coherence One-point spectrum aerodynamic admittance equal to unity at every frequency may not be appropriate for the study of the wind-induced 353 bridge acceleration response. Note that the displacement response has been high-pass filtered to remove the spurious 354 low-frequency component, which comes from the limited performances of the accelerometers at low frequencies. The 355 filter applied is a 5 th order Butterworth filter with a cut-off frequency at 0.05 Hz. The strip theory assumes, in its basic form, that the wind field across the bridge is not significantly affected by the 358 presence of the deck so that the span-wise coherence of wind forces is equal to that of the undisturbed turbulence. An  In full-scale studies, the approximation of a cross-sectional admittance function set equal to 1 for all frequencies 371 (χ w ≈ 1) is commonly adopted. If the bridge displacement response is considered and if its most significant eigenfre-372 quencies are well below 1 Hz, the approximation χ w ≈ 1 is generally found acceptable. However, if the acceleration 373 response is used, the approximation χ w ≈ 1 may no longer be valid, which appears to be the case of the Lysefjord 374 bridge. This is shown in fig. 13, where the acceleration and displacement spectra of the Lysefjord bridge vertical 375 response is estimated at mid-span using a 1-hour record duration on 21-11-2017 from 23:00, with a stationary mean 376 wind speed of 10 m s −1 and a flow from north-northeast. It is clearly seen that the use of χ w ≈ 1 hampers the proper 377 decay of the acceleration response spectra with increased frequency of vibration, and thereby the overall acceleration 378 response is overestimated.

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The choice of using a simple cross-sectional aerodynamic admittance function for the Lysefjord bridge is not 380 straightforward as the deck is somewhat bluff and, therefore, the Liepmann approximation of the Sears function [47] is 381 not necessarily suitable. For a closed-box girder with a ratio H/B > 5 and 0.2 < L w /B < 2, Larose [43] proposed an 382 empirical model that is an explicit function of both K and L w /B. In fig. 13, L w /B ≈ 4 and H/B = 4.5, which is beyond 383 the scope of application for the model proposed by Larose [43]. If the latter model is used, the acceleration response 384 is underestimated. A simpler, but still suitable, cross-sectional admittance function for the Lysefjord bridge can be 385 defined as a second-order low-pass filter, independent of L w /B, with a cut-off frequency at K ≈ 0.45,  function has a much more visible effect on the acceleration response than on the displacement response. terrain. In particular, the ratio σ w /σ u might be as high as 0.7 in a narrow fjord whereas in flat and homogeneous 415 terrain σ w /σ u ≈ 0.5. For the two main wind directions considered, the flow is not affected the same way by the 416 topography, which reinforces the idea that turbulence characteristics observed on a bridge crossing a fjord or a 417 canyon should be studied separately for different sectors.

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• For a hexagonal girder with a width to height ratio B/H ≈ 4.5, mounting a sonic anemometer at a height 419 equal to 3.6H above the deck instead of 2.2H shows only minor improvements in flow distortion mitigation 420 if located downwind. To study turbulence characteristics from a long-span bridge, it is thus advised to mount 421 sonic anemometers on both sides of the girder. Such an installation is particularly important in a mountainous 422 area where the characterization of the wind conditions needs to be conducted after separation of the records in 423 different wind sectors, where some of the anemometers will inevitably be located on the downstream side of the 424 deck.

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• If the wind velocity measurements are conducted on the downstream side of the deck, the wind load is substantially 426 underestimated, compared to measurements on the upstream side, which are affected to a lower degree by the 427 flow disturbance of the bridge deck. The bridge response estimation is, however, observed to be sensitive to 428 the bridge cross-sectional admittance applied. Assuming a simple admittance function equal to unity at every 429 frequency will result in an overestimation of response when used with wind data measured upstream, but may 430 lead to a reasonable agreement between computed and recorded response when used with wind data measured 431 downstream. However, it is shown herein, that modelling the admittance function as a simple second order filter 432 will lead to a significantly improved agreement between the computed and observed full-scale bridge response,