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R E S E A R C H A RT I C L E
Wind speed inference from environmental flow-structure
interactions
Jennifer L. Cardona
1
, Katherine L. Bouman
2
and John O. Dabiri
3*
1
Department of Mechanical Engineering, Stanford University, Stanford, California, 94305, USA
2
Computing and Mathematical Sciences & Electrical Engineering & Astronomy, California Institute of Technology, Pasadena,
California, 91125, USA
3
Graduate Aerospace Laboratories & Mechanical Engineering, California Institute of Technology, Pasadena, California, 91125,
USA
*Corresponding author. E-mail: jodabiri@caltech.edu
Keywords:
Flow imaging and velocimetry, optical based flow diagnostics, fluid-structure interactions
Abstract
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.
Impact Statement
We present a velocimetry method that infers time-dependent flow speeds using visual observations of flow-
structure interactions such as the swaying of trees. This can alleviate the need to directly instrument or visualize
the flow to quantify its speed, instead relying on preexisting objects in an environment that are deflecting due
to the incident flow. The method has the potential to turn ubiquitous objects like trees into abundant, natural,
environmental flow sensors for applications such as weather forecasting, wind energy resource quantification,
and studies of wildfire propagation.
1. 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
arXiv:2011.09609v2 [physics.flu-dyn] 18 Mar 2021
2
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 provid-
ing 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 specifi-
cally. Tritton (1959) used optical measurements of cantilevered quartz fiber deflections, and Kraitse and
Fralick (1977) 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 pur-
poses. Another example of cantilever deflection-based anemometry can be seen in Barth et al. (2005),
where the deflection of a millimeter-scale cantilever affects the position of a reflected laser, which is
calibrated to measure flow speeds.
While these prior studies have shown success in measuring flow speeds by instrumenting the flow
with cantilevers specifically designed and intended for sensing, the present work aims to extend the
concept of flow-sensing cantilevers so that it can ultimately be used with a variety of pre-existing
structures in an environment without further instrumenting the flow. Natural structures such as trees
have material properties that are unknown
a priori
, and they are more geometrically complex than
single beams. Hence, 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 prevalence 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
푚푠
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), Mor-
gan 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 rela-
tionship between the drag force and mean deflection is proposed. Using this model, normalized wind
3
speeds can be approximated based only on the measured deflections of a structure, where the normaliza-
tion 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 cylin-
ders 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 dimen-
sional 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.
2. 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 product of the fluid density,
, and the square
of the mean incident wind speed,
(Batchelor, 2000). The mean force of the wind,
, is given by
multiplying
by the projected frontal area,
, and is therefore proportional to
휌푈
2
:
=
푝퐴
/
휌푈
2
(1)
We model the deflection of the structure following Hooke’s Law:
=
휅훿
(2)
where
is the elastic restoring force,
is the elastic constant, and
is the deflection of the free end of
the cantilever. This model is applicable for small deflections under point loads or distributed loads, with
differences in the configuration of loading captured in the form of the constant of proportionality. For
a cantilever of constant cross section subject to a uniformly distributed load, the relationship between
force per unit length,
, and the tip deflection given by Euler-Bernoulli beam theory follows:
=

8
퐸퐼
4

(3)
where
is the Young’s modulus of the material,
is the area moment of inertia, and
is the length
of the beam. In this case, the elastic constant is related to the geometric and material properties of the
cantilever. The total force on the beam is equal to
푓 퐿
, and the elastic constant as defined in equation 2
is
=
8
퐸퐼
3
.
A balance of the forces in equations 1 and 2 above gives a relationship between the incident flow
speed and the structure deformation:
/
√︄
휅훿
휌퐴
(4)
Furthermore, assuming that
,
and
remain constant under the conditions of interest, the wind speed
normalized by a non-zero reference condition characterized by
0
and
0
is given by:
0
=
√︂
0
(5)
The normalized wind speed given in equation 5 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
). In the measurements described below, we examine the regime of validity of the
model in equation 5.
2.1. 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
4
2 is appropriate to use for trees. However, equation 5 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
O¹»
0
•
1
–
1
¼º
(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:
0
=
√︂
훿퐴
0
0
(6)
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
/
2
, where
is the projected tree height. This gives the modified form of the model to determine normalized wind
speeds from tree deflections:
0
=
√︄
훿ℎ
2
0
0
2
(7)
3. Experimental Methods
3.1. 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)). Mean flow speeds were measured using a factory-calibrated anemometer
(Dwyer Series 641RM Air Velocity Transmitter) with a digital readout. The anemometer was accurate
to
3%
of full scale, where full scale was set to 15 ms
1
. 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
).
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. Mean displacements were typically on the order of 250 pixels, corresponding
to physical dimensions of approximately 10 cm. The normalized wind speed,

0
was calculated by
taking the mean of equation 5. 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).
5
Top view
U
Mounting clamps
Side view
4.88 m
Flexible cylinder
Camera
Fan Array
U
Rear view
2.06 m
1.97 m
U
(a)
(a)
(b)
g
g
+
g
Figure 1.
Schematic of experimental setup to measure cylinder deflection showing (a) side and (b)
rear views. Directions of flow and gravity are indicated. Cylinder dimensions are to scale for the PVC
tube of
=
5.1

0.1 cm,
=
1.52

0.01 m (the cylinder with the largest frontal area).
δ
U
5 cm
Figure 2.
Representative frames showing the displacement of the cylinder free end (PVC tube,
=
5.1

0.1 cm). Top: cylinder surface under no wind load. Bottom: Displaced cylinder subject to incident
flow speed
=
5.6

0.5 ms
1
, with streamwise displacement,
, shown in reference to center position
under no load. Cylinder centers indicated with ‘+’.