Wind speed inference from environmental flow-structure interactions

This study aims to leverage the relationship between fluid dynamic loading and resulting structural deformation to infer the incident flow speed from measurements of time-dependent structure kinematics. Wind tunnel studies are performed on cantilevered cylinders and trees. Tip deflections of the wind-loaded structures are captured in time series data, and a physical model of the relationship between force and deflection is applied to calculate the instantaneous wind speed normalized with respect to a known reference wind speed. Wind speeds inferred from visual measurements showed consistent agreement with ground truth anemometer measurements for different cylinder and tree configurations. These results suggest an approach for non-intrusive, quantitative flow velocimetry that eliminates the need to directly visualize or instrument the flow itself.


Introduction
Fluid-structure interactions such as the bending and swaying of trees in the wind provide visual cues that contain information about the surrounding flow. If visual measurements of deflections can be used to infer quantitative estimates of local wind speeds, then common objects like trees could be used as abundant natural anemometers, requiring only non-intrusive visual access to record wind speed measurements. This would potentially be useful in applications such as data assimilation for weather forecasting, wind energy resource quantification, and understanding wildfire behavior. Recent work has examined this visual anemometry task through a data-driven approach, where a neural network based model was trained to output wind speeds based on input videos of flags and trees in naturally occurring wind (Cardona et al., 2019). However, achieving a data-driven model that would generalize to a wide variety of objects (e.g. trees of different sizes and species) could potentially require an extensive data collection campaign. Physical models for fluid-structure interactions may be advantageous in providing a framework that could be used for visual anemometry across a broader range of structures. Here, we focus on objects that can be modelled as cantilever beams under wind loading.
Flow-sensing cantilevers are found in nature. For instance, the lateral line system allows fish to sense the surrounding flow via the deflection of hair-like structures (Bleckmann and Zelick, 2009). Artificial lateral line sensors have been developed to mimic this flow sensing function as discussed by Shizhe (2014). Cantilever beam deflections have also been used to measure wind speeds specifically, such as in Tritton (1959), which used optical measurements of cantilevered quartz fiber deflections, and Kraitse and Fralick (1977), which used strain gauge measurements of millimeter-scale silicone beams. These drag-based anemometers relied on knowledge of the material properties of the beam. These physical properties could then be used in conjunction with beam bending theory to calculate the drag force on the beam and quantify the wind speed. In these cases, the beam materials were specifically chosen for sensing purposes. Natural structures such as trees have material properties that are unknown a priori, and they are more geometrically complex than single beams. However, in this work we exploit simplified models of the flow-structure interactions to avoid the need for direct consideration of these details, while still capturing the physical influence of the wind on structure deformation. By this approach, we can leverage the pervasiveness of trees and other vegetation in both rural and built environments for use in visual anemometry.
Approximate wind speed scales have been developed based on field observations of fluid-structure interactions in the past. The Fujita scale, for example, is used to infer tornado wind speed based on the damage to structures in its path (Doswell et al., 2009). This has been particularly useful since more conventional wind speed measurements are rare and difficult to obtain for tornadoes. The Beaufort scale is another well-known wind speed scale that relies on visual cues. The version of the scale adapted for use on land employs qualitative descriptions of tree behavior (e.g. branch motion or breaking of twigs) to estimate an instantaneous wind speed range, and it has also been applied to region-specific vegetation (Jemison, 1934). Visual observations of trees and other vegetation have also been used to estimate mean annual wind speeds. The Griggs-Putnam Index uses qualitative descriptions of tree deformation to categorize mean annual wind speeds into seven binned increments of 1-2 −1 (Wade and Hewson, 1979). Time series measurements of tree deformations and a physical model for the fluid-structure interactions may allow for an extension to quantitative, instantaneous flow speed measurements.
Wind-tree interactions have been widely studied (de Langre, 2008). Prior investigations have used various forms of cantilever beam models to describe tree behavior. For instance, Kemper (1968), Morgan and Cannell (1987), and Gardiner (1992) compared tree deflections to tapered cantilever beams. The relationship between wind speed and drag force on trees has also been explored, in particular with regard to the drag reduction that results from large deformations of the tree crown. Several studies have observed and quantified the drag on trees as a function of wind speed (Fraser, 1962;Mayhead, 1973;Rudnicki et al., 2004;Vollsinger et al., 2005;Kane and Smiley, 2006;Koizumi et al., 2010;de Langre et al., 2012;Manickathan et al., 2018). Despite the complexities of wind-tree interactions, this prior work suggests that the structural behavior under wind loading can be generalized in physical models.
The present work aims to infer incident wind speed measurements from observations of cantilevered cylinders and trees. We achieve this visual anemometry without a priori knowledge of the material properties of the structures, as is the case for application to natural vegetation. A model for the relationship between the drag force and mean deflection is proposed. Using this model, normalized wind speeds can be approximated based only on the measured deflections of a structure, where the normalization is based on the measured wind speed and deformation at a selected reference time. The model was tested and compared to ground truth anemometer-measured wind speeds for both cantilevered cylinders and trees in wind tunnel experiments. Given that object deflections can be observed from videos, this method can serve as a non-intrusive technique to measure normalized wind speeds, where dimensional velocities may be recovered with a single calibration measurement at a reference wind speed. This eliminates the need to directly instrument or visualize the flow for quantitative characterization.

Physical Model
A physical model was used to relate wind speed to deflections based on a force balance. The dynamic pressure, , on an object in flow is proportional to the square of the mean incident wind speed, , times the fluid density, . The mean force of the wind, , is given by multiplying by the projected frontal area, , and is therefore proportional to 2 : (1) We model the deflection of the structure following Hooke's Law:

=
(2) where is the elastic restoring force, is the elastic constant, and is the deflection. A balance of fluid dynamic and structural forces implies a relationship between the incident flow speed and the structure deformation: Furthermore, assuming that , and remain constant under the conditions of interest, the wind speed normalized by a reference condition characterized by 0 and 0 is given by: The normalized wind speed given in equation 4 is independent of the material properties of the structure, and the dimensional wind speed can be recovered given only a measurement of and the reference condition ( 0 , 0 ). Alternatively, measurements of the normalized wind speed are sufficient to describe the shape of the probability density function of the dimensional wind speed, as quantified by the Weibull shape factor, . The maximum likelihood method is commonly used to determine the Weibull distribution parameters (Seguro and Lambert, 2000), with estimated by iteratively solving the implicit equation: where 0 are the non-zero, normalized wind speed measurements.

Modification for Tree Crown Deformation
Experimental observations by Roodbaraky et al. (1994) have shown that load-deflection curves for various tree species appear to be linear, which suggests that the linear relationship proposed in equation 2 is appropriate to use for trees. However, equation 4 assumes that the frontal area of the structure is constant for all incident wind conditions. Prior observations have shown a reduced growth in drag force on trees with increasing (Fraser, 1962;Mayhead, 1973;Kane and Smiley, 2006;Koizumi et al., 2010;de Langre et al., 2012;Manickathan et al., 2018). The drag reduction is attributed to reconfiguration of the tree crown which leads to streamlining as a result of the change in area (Harder et al., 2004;Rudnicki et al., 2004;Vollsinger et al., 2005;Manickathan et al., 2018). The effect of the changing area has often been taken into account through the use of a Vogel number (Vogel, 1989), , and the drag is assumed to grow as 2+ , where has a negative value to compensate for the streamlining effect. The Vogel number has been found to vary between tree species, with typical magnitudes in the range of O ( [0.1, 1]) (de Langre et al., 2012; Manickathan et al., 2018). In general, the value of may be unknown for a tree of interest without extensive testing. Therefore, instead of accounting for reconfiguration by using a Vogel exponent, the changing instantaneous frontal area was taken directly into account. Considering that changes with wind speed, the normalized wind speed becomes: where 0 is the frontal area of the tree at reference speed 0 . Where a projection of the frontal area of the structure is not available, the change in area can be approximated by assuming ∝ ℎ 2 , where ℎ is the projected tree height. This gives the modified form of the model to determine normalized wind speeds from tree deflections:

Cantilevered Cylinder Deformation Measurements
A first set of experiments studied deflections of flexible cylinders with circular cross-section. These canonical structures contrast with more complex tree geometries studied subsequently. Experiments were carried out in a 2.06 m × 1.97 m cross-section open-circuit wind tunnel (more details can be found in Brownstein et al. (2019)). Each test cylinder was rigidly mounted to the ceiling of the tunnel, and subjected to three mean flow speeds ( = [4.5, 5.6, 6.6] ± 0.5 ms −1 ). Mean flow speeds were measured using a factory-calibrated anemometer (Dwyer Series 641RM Air Velocity Transmitter) with a digital readout. The maximum blockage ratio in the wind tunnel based on the projected area of the test cylinder and mounting apparatus was 4.0%. A schematic of the setup is shown in figure 1. The deflection for a given cylinder and wind speed was measured by tracking the free end of the cylinder in video frames collected at 240 frames per second (fps) with a resolution of 720 × 1280 pixels. The recording began with the cylinder at rest. The wind tunnel was then turned on and allowed to reach a steady-state speed. Measurements were collected for 60 seconds at each steady-state speed. The center of the cylinder surface was detected in each frame using a two-stage Hough Transform (Yuen et al., 1990) in the MATLAB Image Processing Toolbox. The tip deflection, , was determined by calculating the streamwise displacement of the center for each frame in the 60 s steady-state period with respect to its position under no wind load (figure 2). Examples of streamwise displacement versus time are shown in figure 3. The normalized wind speed, / 0 was calculated by taking the mean of equation 4. Note that while the cylinders were also free to move in the spanwise direction, spanwise displacements were an order of magnitude smaller than streamwise displacements and had a negligible mean value. Examples of cylinder free end trajectories showing both streamwise and spanwise displacements can be seen in the Supplementary Material (figure S1).
Six unique cylinders were tested. Cylinder aspect ratios ( / ) ranged from 30 to 48, and Reynolds numbers based on cylinder diameters ranged from 1.0 × 10 4 to 2.2 × 10 4 . For rigid cylinders in this range of , the drag coefficient, , should remain relatively constant, especially since the sharp drop in for cylinders is typically observed to occur at higher (O (10 5 )) (Roshko, 1961). Four of the cylinders were cut from lengths of soft, flexible PVC tubing (Masterkleer, Durometer 65A) with wall thickness = 0.6 ± 0.1 cm, length = 1.52 ± 0.01 m, and outer diameters ∈ [3.2, 5.1] ± 0.1 cm. The unprocessed PVC tubing tended to have some amount of curvature under no load. To achieve right cylindrical specimens, the PVC tubing was heated with boiling water and allowed to cool in a straightened position. To ensure that the effect of any remaining curvature leading to imperfect cylindrical shape did not systematically bias results, each PVC tube was tested three times, and rotated by 90 • about its axis in the mounting clamp in each subsequent test.
Two other cylinders of different types were tested in addition to the PVC tubes for the purpose of validating model applicability on cylinders with varying characteristics. One was a solid polyurethane  rubber rod (medium soft, Durometer 40A) with dimensions = 1.22 ± 0.01 m and = 3.2 ± 0.1 cm. The final specimen was a hollow ABS tube ( = 1.55 ± 0.01 m, = 3.2 ± 0.1 cm, = 0.3 ± 0.1 cm) with a steel compression spring at its base (spring rate of 2.8 N/mm) held by the mounting clamp at the tunnel roof. In this case, the ABS cylinder acted as a rigid body rather than a flexible cylinder like the other specimens, and the deflection was facilitated primarily by bending of the spring at its base. Table  1 summarizes the properties of the six cylinders tested.

Tree Deformation Measurements
Measurements of two natural trees were conducted in a second, larger open-circuit wind tunnel of cross-section 2.88 m × 2.88 m. Deflections were measured for the two potted trees: a conifer commonly known as a Juniper tree (Juniperus scopulorum), and a broad-leaved Bay Laurel tree (Laurus nobilis). Approximate tree dimensions are given in table 2. Trees were positioned using sand bags to weight down the pots and keep them in place. The tunnel was run in a uniform flow configuration at four distinct mean flow speeds spanning the tunnel range ( = [3.3, 6.0, 8.8, 11.4] ± 0.5 ms −1 ), which were measured using the factory-calibrated anemometer (Dwyer Series 641RM Air Velocity Transmitter).
Reynolds numbers based on a length scale of 1/2 , where is the approximate frontal area of the tree under no load, ranged from 1.7 × 10 5 to 6.3 × 10 5 . The maximum blockage ratio based on projected frontal area was 8.4%. The camera was angled perpendicular to the flow direction to capture video of the streamwise tree deflections for 110 s at each steady-state flow speed. Videos were recorded at 240 fps with a resolution of 720 × 1280 pixels. A random sample of frames was chosen to measure the tip deflections. For each 10-second clip of video data, 10 frames were randomly selected. This yielded 110 sample frames for   between visually measured speeds and ground truth values. There is strong agreement between the visual and ground truth measurements.

Normalized Wind Speeds from Tree Deflections
Visually measured normalized wind speeds versus ground truth normalized wind speeds are plotted in figure 6a for all test objects including the two trees and six cylinders. Note that cylinder datapoints are the same as they appear in figure 5, but are also included in figure 6a for ease of comparison between cylinder and tree results. The model accounting for changing frontal area (equation 7) was used to calculate the visually measured normalized wind speeds for the trees. Each of the four tunnel speeds used in the tree experiments ( = [3.3, 6.0, 8.8, 11.4] ± 0.5 ms −1 ), was taken as a reference speed 0 in combination with the other three distinct speeds for a total of 12 ground truth values. Once again, distinct markers have been used to show points calculated with each particular reference speed. The trees were subjected to a wider range of wind speeds than the cylinders based on tunnel capabilities, which led to results across a wider range of ground truth values ( / 0 ∈ [0.3, 3.5] for the trees compared to / 0 ∈ [0.7, 1.5] for the cylinders), as visible in figure 6a. The percent error of visual measurements compared to ground truth is shown in figure 6b. Dimensional wind speeds recovered by multiplying the visually measured normalized speed by the reference speed, 0 , are shown in figure 7. The accuracy of the visual measurement approach is validated in the comparison with ground truth. The visual measurements appear to agree well with ground truth measurements, but do modestly underestimate the ground truth at high wind speeds. This could be associated with the limitation in our estimate of area change, as we assumed ∝ ℎ 2 to incorporate the tree crown deformation given only a side view of the tree.

Discussion and Conclusions
Wind speed measurements inferred from structure kinematics showed strong agreement with the ground truth measurements for both the cylinders and the trees, as seen in figures 5 and 6. This suggests that flow-structure interactions can be leveraged to infer local wind conditions without a priori knowledge of the material properties of the structure, and without visualizing or instrumenting the flow itself. The agreement in normalized wind speed is particularly good at lower values of / 0 , but there is more discrepancy at higher values. As seen in figure 6b, the percent error in the visually measured normalized speed increases in magnitude as the ground truth normalized speed departs from a value of 1. One factor that may contribute the discrepancy is that the model assumptions (e.g. the linear assumption of Hooke's law) become less valid at higher wind speeds where there are larger forces acting on the structures causing larger deflections. For instance, assuming a linear relationship for a cantilever beam overestimates the tip deflection when deformations are large (Bisshopp and Drucker, 1945). This linear model would then underestimate the wind speed corresponding to a given deflection. This is reflected in the overestimates for ground truth values of / 0 < 1 and the underestimates for ground truth values of / 0 > 1 in figure 6b. A linear relationship between force and deflection was assumed here for simplicity, but it is possible that the application of a beam bending model meant for large deformations may improve agreement. Prior works have successfully described the bending behavior of sapling trunks using tapered cantilevered beams models for analyzing large deflections (Kemper, 1968;Morgan and Cannell, 1987;Gardiner, 1992). However, these more complex models require knowledge the flexural rigidity and the taper of the trunk, both of which may be unknown. As previously noted, the approximation used to quantify the changing frontal area of the trees may also be a source of error. Future studies may consider the use of a second camera aligned with the direction of flow to capture the instantaneous area.
A further limitation of the present work is the need for a calibration reference ( 0 , 0 ) to convert the normalized wind speeds to dimensional quantities. While this does not affect inference of the shape of a wind speed distribution, it does affect measurement of dimensional quantities such as the kinetic energy flux. Future work will seek to combine the present flow physics-based approach with data-driven approaches to approximate structural parameters necessary to provide the calibration reference solely from visual measurements of the structures in the flow.
This technique has potential to provide a flow-physics based method of using ubiquitous objects in characterizing flow conditions without the need for direct instrumentation of the flow. The present work focused on trees as natural, visual anemometers, but this idea could also be extended to other objects that are prevalent in important environmental flows, such as flags in the built environment (Cardona et al., 2019) and seagrass in ocean currents (Zeller et al., 2014). Figure S1 shows examples of cylinder free end trajectories. Notably, streamwise displacement values are an order of magnitude larger than spanwise displacements at all points in time, and streamwise displacements have a non-zero mean.  Figure S1. Example trajectories for the free end of a cylinder (PVC tube, = 3.8 ± 0.1 cm). Each point corresponds the position of the cylinder free end at one instant in time subject to flow speeds = 4.5 ± 0.5 ms −1 (left), = 5.6 ± 0.5 ms −1 (center), and = 6.6 ± 0.5 ms −1 (right) over the 60 s steady state period for each speed. Note that while the data was collected at 240 Hz, the data shown here have been down-sampled to 80 Hz to reduce overlapping points. All axes are identically scaled.