# Scaling Laws for the Propulsive Performance of a Purely Pitching Foil in
Ground Effect

Amin Mivehchi1<EMAIL_ADDRESS>Qiang Zhang2  Melike Kurt1  Daniel B.
Quinn2  Keith W. Moored1  1Department of Mechanical Engineering and Mechanics,
Lehigh University, Bethlehem, PA 18015, USA 2Department of Mechanical
Engineering, University of Virginia, Charlottesville, VA 22094, USA

###### Abstract

Scaling laws for the thrust production and power consumption of a purely
pitching hydrofoil in ground effect are presented. For the first time, ground
effect scaling laws based on physical insights capture the propulsive
performance over a wide range of biologically-relevant Strouhal numbers,
dimensionless amplitudes, and dimensionless ground distances. This is achieved
by advancing previous scaling laws Moored & Quinn (2018) with physics-driven
modifications to the added mass and circulatory forces to account for ground
distance variations. The key physics introduced are the increase in the added
mass of a foil near the ground and the reduction in the influence of a wake
vortex system due to the influence of its image system. The scaling laws are
found to be in good agreement with new inviscid simulations and viscous
experiments, and can be used to accelerate the design of bio-inspired
hydrofoils that oscillate near a ground plane or two out-of-phase foils in a
side-by-side arrangement.

###### keywords:

##  1 Introduction

In nature, animals such as birds and flying fish use steady ground effect to
improve their cost of transport or gliding distance (Hainsworth, 1988; Rayner,
1991; Park & Choi, 2010). Similarly, some fish exploit unsteady ground effect
to improve their cost of transport or cruising speed when swimming near
substrates and sidewalls (Blake, 1983; Webb, 1993, 2002; Nowroozi et al.,
2009; Blevins & Lauder, 2013). In unsteady ground effect, fin/wing/tail/body
oscillations create time-dependent wakes and time-varying fluctuations in the
pressure field that are altered from those of an isolated swimmer or flyer.

Unsteady ground effect was first examined through the development of
analytical models for a fluttering plate in a channel Tanida (2001) and for an
oscillating wing in weak ground effect Iosilevskii (2008), but these only
apply in extreme cases, such as flying/swimming very far from or very close to
the ground. At more moderate ground proximities, experiments and computations
have shown that rigid (Quinn et al., 2014c; Mivehchi et al., 2016; Perkins et
al., 2017) and flexible (Blevins & Lauder, 2013; Quinn et al., 2014a;
Fernández-Prats et al., 2015; Dai et al., 2016; Park et al., 2017; Zhang et
al., 2017) oscillating foils and wings can have improved thrust production
with little or no penalty in efficiency when operating in unsteady ground
effect. Additionally, Kurt et al. (2019) reported the presence of a stable
equilibrium altitude for a freely-swimming pitching foil in the presence of a
ground plane that was previously observed in the lift force of constrained
flapping foils both experimentally (Mivehchi et al., 2016; Perkins et al.,
2017), and numerically Quinn et al. (2014c); Kim et al. (2017).

To understand the origins of thrust and efficiency in unsteady ground effect,
we can rely on scaling laws. The basis of many recent scaling laws lies in
classic unsteady linear theory. The theories of Theodorsen (1935), Garrick
(1936), and von Karman & Sears (1938) have become particularly useful in this
pursuit due to their clear assumptions (incompressible and inviscid flow,
small-amplitude motions, non-deforming, and planar wakes) and the
identification of the physical origins of their terms. For instance, these
theories decompose the forces acting on unsteady foils into three types: added
mass, quasi-steady, and wake-induced forces. Theodorsen’s theory was extended
by Garrick (1936) by accounting for the singularity in the vorticity
distribution at the leading edge to determine the thrust force produced and
the power required by such motions. By following Garrick (1936), Dewey et al.
(2013) and Quinn et al. (2014c, b) scaled the thrust forces of pitching and
heaving flexible panels with their added mass forces. Moored & Quinn (2018)
advanced this previous work by considering the circulatory and added mass
forces of self-propelled pitching foils as well as wake-induced nonlinearities
that are not accounted for in classical linear theory Garrick (1936). It was
shown that data generated from a potential flow solver and from experimental
measurements Ayancik et al. (2019) were in excellent agreement with the
proposed scaling laws. Similarly, Floryan et al. (2017) considered both the
circulatory and added mass forces and showed excellent collapse of
experimental data with their scaling laws for the thrust and power of a
heaving or pitching two-dimensional rigid foil. While these studies have
provided great insights into the origins of unsteady force production, they
were limited to isolated propulsors.

Here, we advanced the scaling laws for isolated purely pitching propulsors
developed by Moored & Quinn (2018) to account for the proximity to the ground.
These scaling laws provide new insight into the underlying physics of unsteady
ground effect and are verified through simulations and experiments.
Furthermore, we show that the added mass forces of the core two-dimensional
scaling relations Moored & Quinn (2018) can be modified by accounting for the
increase in the added mass of an object near a ground plane derived from
classical hydrodynamic theory Brennen (1982), and the circulatory forces can
be modified by accounting for bound and wake vortex-body interactions in
ground effect. The newly developed scaling laws offer a physical rationale for
the origins of force production, power consumption, and efficient unsteady
swimming in proximity to the ground.

Figure 1: Schematic of (a) the constrained pitching hydrofoil apparatus at
Lehigh University, and (b) the constrained hydrofoil apparatus at the
University of Virginia.

##  2 Methods

Potential flow simulations and water-channel experiments were conducted on
hydrofoils in and out of ground effect. The details of the hydrofoil geometry
and kinematics, as well as the numerical and experiment methods employed are
given below.

###  2.1 Hydrofoil Geometry and Kinematics

The hydrofoil used throughout this study has a rectangular planform shape, a
10% thick tear-drop cross-section (Quinn et al., 2014c) with a chord length of
$c=0.095$ m, and an effectively infinite aspect ratio. The hydrofoil was
actuated with a sinusoidal purely pitching motion about its leading edge of
${\theta}(t)={\theta}_{0}\sin(2\pi ft)$, where $\theta_{0}$ is the pitching
amplitude, $f$ is the frequency, and $t$ is the time. The frequency defines
the reduced frequency, $k\equiv fc/U$, and the Strouhal number, $St\equiv
fA/U$. Here, $A$ is the peak-to-peak amplitude of motion, that is,
$A=2c\sin\theta_{0}$. The amplitude of motion is reported in its dimensionless
form as $A^{*}=A/c$. One of the primary variables of the current study is the
dimensionless ground distance, $D^{*}=D/c$, where $D$ is the distance from the
leading-edge of the foil to the ground plane. The input variables used are
summarized in Table 1.

Variables/Parameters | Simulations | Exp. at UVA (EXP1) | Exp. at Lehigh U. (EXP2)
---|---|---|---
$A^{*}=\frac{A}{c}$ | $0.15\leq A^{*}\leq 0.6$ | $0.24\leq A^{*}\leq 0.52$ | $0.24\leq A^{*}\leq 0.52$
$k=\frac{fc}{U}$ | $0.1\leq k\leq 2.0$ | $0.55\leq k\leq 1.28$ | $0.77\leq k\leq 1.02$
$St=\frac{fA}{U}$ | $0.15\leq St\leq 0.60$ | $0.18\leq St\leq 0.44$ | $0.26\leq St\leq 0.63$
$D^{*}=\frac{D}{c}$ | $0.3\leq D^{*}\leq 2.0$ | $0.24\leq D^{*}\leq 1.665$ | $0.25\leq D^{*}\leq 2.6$
$\AR$ | $\infty$ |  $3$ with end-plates |  $2$ with end-plates
$\Rey=\frac{Uc}{\nu}$ | $\infty$ |  $13,$600 |  $9,$000
Table 1: Numerical and experimental variables and parameters.

For the experiments and the simulations, the time-averaged thrust and power
coefficients can be non-dimensionalized by the added mass forces and added
mass power from small amplitude theory (Garrick, 1936) or by dynamic pressure,

$C_{T}\equiv\frac{\overline{T}}{\rho S_{p}f^{2}A^{2}},\quad
C_{P}\equiv\frac{\overline{P}}{\rho S_{p}f^{2}A^{2}\overline{U}},\quad
C^{\text{dyn}}_{T}\equiv\frac{\overline{T}}{1/2\rho S_{p}U^{2}},\quad
C^{\text{dyn}}_{P}\equiv\frac{\overline{P}}{1/2\rho S_{p}U^{3}}.$ (1)

where $\rho$ is the density of the fluid medium and the two normalizations are
related through the Strouhal number by simple transformations:
$C_{T}^{\text{dyn}}=C_{T}\,(2St^{2})$ and
$C_{P}^{\text{dyn}}=C_{P}\,(2St^{2})$. The propulsive efficiency can be
defined as: $\eta\equiv C_{T}/C_{P}\equiv C_{T}^{dyn}/C_{P}^{dyn}$.

###  2.2 Numerical Method

To model the potential flow around a foil, we used a two-dimensional boundary
element method (BEM) where the flow is assumed to be irrotational,
incompressible and inviscid. By following previous studies (Katz & Plotkin,
2001; Quinn et al., 2014c; Moored, 2018), the general solution to the
potential flow problem is reduced to finding a distribution of sources and
doublets on the foil surface and in its wake. At each time step a no flux
boundary condition is enforced on the body. To solve this problem numerically,
constant strength source and doublet line elements are distributed over the
body and the wake. Each body element is assigned a collocation point, which is
located at the center of each element and shifted 1% of local thickness into
the body where a constant-potential condition is applied to enforce no flux
through the surface (i.e. Dirichlet formulation). This results in a matrix
representation of the boundary condition that can be solved for the body
doublet strengths once a wake shedding model is applied. Additionally, at each
time step, a wake boundary element is shed with a strength that is set by
applying an explicit Kutta condition, where the vorticity at the trailing edge
is set to zero. The presence of the ground is modeled using the method of
images, which automatically satisfies the no-flux boundary condition on the
ground plane.

A wake roll-up algorithm is implemented at each time step where the wake
elements are advected with the local velocity. During wake roll-up, the point
vortices, representing the ends of the wake doublet elements, must be
desingularized for numerical stability of the solution (Krasny, 1986). At a
cutoff radius of $\epsilon/c=5\times 10^{-2}$, the irrotational induced
velocities from the point vortices are replaced with a rotational Rankine core
model.

The tangential perturbation velocity component is calculated by local
differentiation of the perturbation potential. The pressure acting on the body
is found via applying the unsteady Bernoulli equation. Moreover, the mean
thrust force is calculated as the time-average of the streamwise directed
pressure forces and the time-averaged power input to the fluid is calculated
as the time average of the negative inner product of the force vector and
velocity vector of each boundary element, that is,
$P=-\int_{\mathcal{S}}\mathbf{F}_{\text{ele}}\cdot\mathbf{u}_{\text{ele}}\,d\mathcal{S}$
where $\mathcal{S}$ is the body surface.

Convergence studies found that the thrust and efficiency changes by less than
2% when the number of body panels, $N_{b}=150$, and the number of time steps
per cycle, $N_{t}=150$, were doubled independently. The current study
considered the foil’s cycle-averaged thrust and efficiency as convergence
metrics since these are the prime output variables of interest. The
computations were run over 10 flapping cycles and the time-averaged data are
obtained by averaging the last cycle. For all simulations there was less then
1% change in the thrust and efficiency after 7 flapping cycles. The two-
dimensional formulation in the current study has been validated extensively
against continuous swimming theory, numerics and experiments Quinn et al.
(2014c); Moored (2018); Akoz & Moored (2018); Kurt et al. (2019). For more
details on the numerical method see Moored (2018) and Moored & Quinn (2018).

###  2.3 Experimental Methods

New experiments (EXP1) were conducted in a closed-loop water channel (Rolling
Hills 1520) at the University of Virginia with a foil of aspect ratio $\AR=3$
and a Reynolds number of $\Rey=13,$$600$. A nominally two-dimensional flow was
achieved by installing a horizontal splitter plate and a surface plate near
the tips of the hydrofoil (Figure 1b). The gap between the hydrofoil tips and
the surface/splitter plate was less than 5 mm. Surface waves were also
minimized by the presence of the surface plate. The wall of the channel was
used as a ground plane and the dimensionless ground distance was varied within
a range of $0.35\leq D^{*}\leq 1.665$. Five different dimensionless amplitudes
were tested: $A^{*}=0.24$, $0.31$, $0.38$, $0.45$ and $0.52$ resulting in
$0.2\leq St\leq 0.45$.

A second (EXP2) previously published experimental data set Kurt et al. (2019)
is also used for further validation of the proposed scaling laws. For further
experimental details see Kurt et al. (2019). The parameters and ranges of
variables for both EXP1 and EXP2 can be found in Table 1.

##  3 Scaling laws for a pitching foil in ground effect

Figure 2: Effects of the image vortex system. (a) The down-stroke negatively-
signed vortex slows downs due to the influence of its image vortex
($l_{u}>l_{d}$). (b) The induced streamwise velocity of the shedding vortex
and its image. The induced velocity from the pair is smaller than a comparable
isolated vortex.

Moored & Quinn (2018) introduced scaling relations for the performance of two-
dimensional, self-propelled pitching hydrofoils. Here, we will briefly review
these scaling laws since they will serve as the basis with which to apply
novel modifications to the added mass and circulatory forces to account for
the proximity to the ground.

###  3.1 Scaling laws of an isolated swimmer

Moored & Quinn (2018) developed thrust and power scaling laws as a combination
of the added mass and circulatory forces from classical linear theory Garrick
(1936) with additional nonlinear terms that are not accounted for in linear
theory. For instance, the thrust coefficient defined in eq. (1), is proposed
to be proportional to the superposition of three terms,

$\displaystyle C_{T}$
$\displaystyle=c^{\prime}_{1}+c^{\prime}_{2}\phi_{2}+c^{\prime}_{3}\phi_{3}$
(2) $\displaystyle\text{with:}\quad\phi_{2}$
$\displaystyle=-\biggl{[}\frac{3F}{2}+\frac{F}{\pi^{2}k^{2}}-\frac{G}{2\pi
k}-(F^{2}+G^{2})\biggl{(}\frac{1}{\pi^{2}k^{2}}+\frac{9}{4}\biggr{)}\biggr{]},\quad\phi_{3}=A^{*},$

where $c^{\prime}_{1}$, $c^{\prime}_{2}$, and $c^{\prime}_{3}$ are constants,
and $F$ and $G$ are the real and imaginary components of Theodorsen’s lift
deficiency function, respectively Theodorsen (1935). The first and second
terms represented by $c^{\prime}_{1}$ and $c^{\prime}_{2}\phi_{2}$ are the
added mass and circulatory thrust forces, respectively, from linear theory
while the third term represented by $c^{\prime}_{3}\phi_{3}$ is not accounted
for in linear theory. The third term corresponds to the form drag that is
proportional to the time varying projected frontal area that occurs during
large-amplitude pitching oscillations. Moored & Quinn (2018) also proposed
that the power coefficient defined in equation (1) is a linear superposition
of three terms as,

$\displaystyle C_{P}$
$\displaystyle=c^{\prime}_{4}+c^{\prime}_{5}\phi_{5}+c^{\prime}_{6}\phi_{6}\quad\text{with:}\quad\phi_{5}=\frac{St^{2}}{k}\biggl{(}\frac{k^{*}}{1+k^{*}}\biggr{)},\quad\phi_{6}=St^{2}k^{*},$
(3)

where $c^{\prime}_{4}$, $c^{\prime}_{5}$, and $c^{\prime}_{6}$ are arbitrary
constants, and $k^{*}=k/(1+4St^{2})$. The first term ($c^{\prime}_{4}$) is the
added mass power from linear theory. The second term
$(c^{\prime}_{5}\phi_{5})$ is a power term that is not present in linear
theory and develops from the x-component of velocity of a pitching propulsor,
which is neglected in linear theory due to a small-amplitude assumption. For
large amplitude motions this velocity does not disappear, leading to an
additional velocity component on the bound vorticity of the propulsor and
creating an additional contribution to the generalized Kutta-Joukowski force
also known as the vortex force (Saffman 1992). The third term
$(c^{\prime}_{6}\phi_{6})$ is also a power term that is absent in linear
theory and develops during large-amplitude motions when the trailing-edge
vortices are no longer planar as assumed in the theory. As a result, the
proximity of the trailing-edge vortices induce a streamwise velocity over the
foil and an additional contribution to the vortex force. In short, the second
and third terms are described as the large-amplitude separating shear layer
and vortex proximity power terms, respectively, and both terms are circulatory
in nature. For more details on the development of the two-dimensional scaling
relations see Moored & Quinn (2018).

###  3.2 Scaling laws modifications for ground effect

In presence of the ground, it is postulated that the added mass (and thus the
added mass thrust and power terms), as well as the vortex proximity power term
will be affected and will act as the primary drivers for the observed scaling
trends with ground proximity. These two modifications for a pitching foil are
described below.

The close proximity of a solid boundary can cause a substantial increase in
the added mass of a pitching foil. This is due to the increase in fluid
acceleration between the foil and boundary Brennen (1982). Classic
hydrodynamic theory shows that for a circular cylinder with the radius $r$
moving with a distance $d$ from a solid boundary (when ${d}/{r}<1$) the added
mass increases as the cylinder moves closer to the boundary. The added mass
can be represented as the addition of the isolated added mass with an
additional added mass due to the acceleration of flow in the presence of a
solid boundary as,

$M_{a}=\pi\rho
r^{2}\left[1+\sum_{j=1}^{\infty}\frac{1}{2^{2j-1}}\left(\frac{r}{d}\right)^{2j}\right]$
(4)

Using conformal mapping, one can directly define the added mass for a specific
foil profile Korotkin (2009), however, in this study, the cylinder is mapped
to a thin airfoil with a chord length of $c=2r$ for generality. Furthermore,
since the dimensionless distance of the foil to the ground is limited by the
maximum trailing edge amplitude, we neglect the effect of higher order terms
of the series and assume the additional added mass contribution scales as the
inverse of the dimensionless distance squared and its coefficient is to be
determined. For extreme ground effect problems, the higher order terms in the
series are required. The added mass terms of equations (2) and (3) can then be
modified to

$\displaystyle c^{\prime}_{1}=c_{1}+c_{2}\zeta_{2},\quad
c^{\prime}_{4}=c_{5}+c_{6}\zeta_{6}$ (5)

where the $\zeta_{2}=\zeta_{6}=(1/D^{*})^{2}$ are the additional added mass
contributions to the thrust and power generation.

The second mechanism that significantly affects the power generation of a
pitching foil in proximity to the ground is circulatory in nature and is best
explained using the method of images, where each shedding vortex has an
oppositely-signed image vortex in the ground (see Figure 2a).The image vortex
of the near-ground negatively-signed vortex induces an additional velocity at
the trailing-edge of the foil (see figure 2b) that acts to reduce the effect
of the near-ground vortex and needs to be accounted for in the vortex
proximity term in equation (3). The power contribution due to the proximity of
the vortex is proportional to the additional lift generated from the
streamwise induced velocity as $L_{\mathrm{Prox}}\approx\rho
su_{ind}\Gamma_{w}$ based on Kutta-Joukowski theorem. The image vortex
modifies this induced streamwise velocity at the trailing edge as,

$\displaystyle
u_{\mathrm{ind}}=\Gamma_{w}\frac{f}{U}\biggl{[}\overbrace{\frac{St}{1+4St^{2}}}^{\text{Near-
Ground
Vortex}}-\overbrace{2\frac{St+4kD^{*}}{1+4(4kD^{*}+St)^{2}}}^{\text{Image
Vortex}}\biggr{]},$ (6)

where the advection speed of the vortex pair is $U$ and the $\Gamma_{w}$ is
the stragnth of the trailing edge vortex circulation. This induced streamwise
velocity decreases with decreasing ground distance due to the near
nullification of the induced velocity from the near-ground vortex by its
image.

The thrust circulatory term is also affected by this image vortex, however,
the forces acting on purely pitching foils are dominated by added mass forces,
and the circulatory modifications are not significant. The scaling laws for
the thrust and power coefficient of a purely pitching foil in proximity to the
ground are then,

$\displaystyle C_{T}$
$\displaystyle=c_{1}+c_{2}\zeta_{2}+c_{3}\zeta_{3}+c_{4}\zeta_{4},\quad
C_{P}=c_{5}+c_{6}\zeta_{6}+c_{7}\zeta_{7}+c_{8}\zeta_{8}$ (7)
$\displaystyle\text{with:}\quad\zeta_{2}$
$\displaystyle=\zeta_{6}=\frac{1}{{D^{*}}^{2}},\quad\zeta_{3}=-\biggl{[}\frac{3F}{2}+\frac{F}{\pi^{2}k^{2}}-\frac{G}{2\pi
k}-(F^{2}+G^{2})\biggl{(}\frac{1}{\pi^{2}k^{2}}+\frac{9}{4}\biggr{)}\biggr{]},$
$\displaystyle\zeta_{4}$
$\displaystyle=A^{*},\quad\zeta_{7}=\frac{St^{2}}{k}\biggl{(}\frac{k^{*}}{1+k^{*}}\biggr{)},\quad\zeta_{8}=St^{2}k^{*}-2kSt\frac{(St-4kD^{*})}{1+4(St-4kD^{*})^{2}}.$

The scaling laws can also be written in terms of the thrust and power
coefficients normalized by dynamic pressure as,

$\displaystyle C_{T}^{\mathrm{dyn}}$
$\displaystyle=2St^{2}(c_{1}+c_{2}\zeta_{2}+c_{3}\zeta_{3}+c_{4}\zeta_{4}),\quad
C_{P}^{\mathrm{dyn}}=2St^{2}(c_{5}+c_{6}\zeta_{6}+c_{7}\zeta_{7}+c_{8}\zeta_{8}).$
(8)

Figure 3: Coefficient of thrust and power as a function of reduced frequency
from the self-propelled simulations. The marker colors going from black to
white indicate the $D^{*}$ from far to close the ground, respectively, over
the range $0.3\leq D^{*}\leq 2.0$. (a) and (c) normalized based on dynamic
pressure (b) and (d) normalized based on added mass force.

##  4 Results and Discussion

The combination of computational input variables presented in Table 1 leads to
665 two-dimensional simulations with a Strouhal number range of $0.1\leq
St\leq 0.6$ and a reduced frequency range of $0.1\leq k\leq 2.0$. From these
simulations, the thrust and power coefficients as defined in equation (1) are
presented in Figure 3. As in previous work Quinn et al. (2014c), thrust and
power coefficients increase with amplitude and ground proximity (Figure 3).

Figure 4 presents the numerical data plotted as a function of the ground
effect scaling laws proposed in eqs. (7) and (8). An excellent collapse of the
data is observed showing that the scaling laws capture the physics of unsteady
ground effect in potential flows. The collapsed data can be seen to follow a
line of slope one for both the thrust and power within $\pm 2\%$ of the
predicted scaling law. The constants in the thrust law are $c_{1}=2.99$,
$c_{2}=0.06$, $c_{3}=-4.43$ and $c_{4}=-0.09$, while for the power law they
are $c_{5}=4.46$, $c_{6}=0.14$, $c_{7}=25.2$ and $c_{8}=14.13$.

To validate whether the scaling laws can also apply to viscous flows, two
experimental data sets are graphed against the scaling law predictions in
Figure 5. The first experimental data set (EXP1) shows a collapse of the data
to within $\pm 20\%$ (thrust and power) of the scaling law prediction. The
deviation of both thrust and power from the scaling law prediction is likely a
result of bending of the actuation rods at high $St$, which introduces a
heaving component to the foil motion that is not accounted for in the scaling
laws. The experimentally determined coefficients for the thrust are
$c_{1}=5.03$, $c_{2}=0.08$, $c_{3}=-5.72$, $c_{4}=-4.66$ and for the power
they are $c_{5}=4.313$, $c_{6}=0.004$, $c_{7}=0.11$ and $c_{8}=30.91$. The
second experimental data set (EXP2) shows a collapse within $\pm 22.5\%$
(thrust) and $\pm 11.5\%$ (power) of the scaling law prediction. The higher
error margin in the thrust scaling is attributed to secondary viscous effects
not accounted for in the scaling laws. The experimentally determined
coefficients for the thrust law are $c_{1}=4.29$, $c_{2}=0.09$,
$c_{3}=-14.33$, $c_{4}=-0.52$, and for the power law they are $c_{5}=8.26$,
$c_{6}=0.03$, $c_{7}=33.06$ and $c_{8}=15.38$.

Figure 4: Scaling of the (a) thrust and (b) power coefficients for all motion
amplitudes and ground proximities considered in the numerical simulations.

Our scaling relations show a good collapse of the data for a wide range of
Reynolds number from $\Rey=9,$$000$ and $\Rey=13,$$600$ in the experiments to
$Re=\infty$ in the inviscid simulations. Although, it should be noted that the
determined coefficients are different among the experimental and numerical
data sets, which highlights that the coefficients likely vary with $Re$ as
observed by Senturk & Smits (2019). In support of the proposed scaling laws
the added mass thrust and power terms are positive in both experiments and
simulations, as expected based on physical grounds. Although it is clear that
the Reynolds number can alter the coefficients, no additional terms need to be
introduced to account for data obtained at different $Re$. This supports the
previous conclusion Kurt et al. (2019) that the dominant flow physics in
ground effect are inviscid in nature. The small differences between the
scaling law agreement in the experiments and the simulations may be attributed
to secondary viscous effects.

The collapse of the data to a line of slope one for both numerical and
experimental cases confirms that the newly proposed scaling laws capture the
dominant flow physics of two-dimensional pitching propulsors in ground effect
across a wide range of $St$, $A^{*}$ and $D^{*}$.

##  5 Conclusion

New scaling laws are developed for the thrust generation and power consumption
of two-dimensional pitching propulsors in ground-effect by extending the two-
dimensional pitching scaling laws introduced by Moored & Quinn (2018) to
consider added mass and circulatory effects due to the close proximity of a
ground plane. The developed scaling laws are shown to predict inviscid
numerical data and experimental data well, within $\pm$20% of the thrust and
power data, respectively. The scaling laws reveal that both an increase in the
added mass with decreasing ground distance and a reduction in the influence of
a shed vortex by its image with decreasing ground distance are key physics to
capture in a scaling law that is valid over a wide range of ground distances,
motion amplitudes, and Strouhal numbers. These results can be extended to two
swimmers in side-by-side arrangements with an out-of-phase synchronization.
The established scaling relationships elucidate the dominant flow physics
behind the force production and energetics of pitching bio-propulsors and can
be used to accelerate the design of bio-inspired devices that swim near a
ground plane and operate in side-by-side schools.

Figure 5: Scaling of the time averaged thrust and power for (a-b) UVA data
set(EXP1) (c-d)LU data set(EXP2) for all motion amplitudes and distances from
the wall considered in experiments. The dashed lines presents 20% margins of
error.

##  6 Acknowledgements

This work was supported by the Office of Naval Research under Program Director
Dr. Robert Brizzolara on MURI grant number N00014-08-1-0642 and BAA grant
number N00014-18-1-2537, as well as by the National Science Foundation under
Program Director Dr. Ronald Joslin in Fluid Dynamics within CBET on NSF CAREER
award number 1653181 and NSF collaboration award number 1921809.

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