# Revealing the mechanism and scaling laws behind equilibrium altitudes of
near-ground pitching hydrofoils

Tianjun Han1<EMAIL_ADDRESS>Qiang Zhong2,3  Amin Mivehchi1  Daniel B.
Quinn2 and Keith W. Moored1  1 Department of Mechanical Engineering and
Mechanics, Lehigh University, Bethlehem, PA 18015, USA 2 Department of
Mechanical and Aerospace Engineering, University of Virginia, Charlottesville,
VA 22904, USA 3 Department of Mechanical Engineering, Iowa State University,
Ames, IA 50011, USA

###### Abstract

A classic lift decomposition (von Kármán & Sears, 1938) is conducted on
potential flow simulations of a near-ground pitching hydrofoil. It is
discovered that previously observed stable and unstable equilibrium altitudes
are generated by a balance between positive wake-induced lift and negative
quasi-steady lift while the added mass lift doesn’t play a role. Using both
simulations and experiments, detailed analyses of each lift component’s near-
ground behavior provide further physical insights. When applied to three-
dimensional pitching hydrofoils the lift decomposition reveals that the
disappearance of equilibrium altitudes for ${\mar A}<1.5$ occurs due to the
magnitude of the quasi-steady lift outweighing the magnitude of the wake-
induced lift at all ground distances. Scaling laws for the quasi-steady lift,
wake-induced lift and the stable equilibrium altitude are discovered. A simple
scaling law for the lift of a steady foil in ground effect is derived. This
scaling shows that both circulation enhancement and the velocity induced at a
foil’s leading edge by the bound vortex of its ground image foil are the
essential physics to understand steady ground effect. The scaling laws for
unsteady pitching foils can predict the equilibrium altitude to within $20\%$
of its value when $St<0.45$. For $St\geq 0.45$ there is a wake instability
effect, not accounted for in the scaling relations, that significantly alters
the wake-induced lift. These results not only provide key physical insights
and scaling laws for steady and unsteady ground effect, but also for two
schooling hydrofoils in a side-by-side formation with an out-of-phase
synchronization.

##  1 Introduction

Many animals improve their locomotion energetics or increase their range by
swimming or flying near the seafloor or water surface, thereby taking
advantage of ground effect (Webb, 1993; Hainsworth, 1988; Baudinette &
Schmidt-Nielsen, 1974; Blake, 1979; Park & Choi, 2010). When wings/fins move
steadily near a boundary, such as those of gliding animals or fixed-wing
aircraft, it is well-known that a steady ground effect increases their lift
and reduces their induced drag (Tremblay Dionne & Lee, 2018; He et al., 2015;
Lee & Lee, 2011; Qu et al., 2015; Yang et al., 2010, 2009; Boschetti et al.,
2010; Widnall & Barrows, 1970; Baddoo et al., 2020). However, when swimmers or
flyers oscillate their appendages near a boundary, like flatfish swimming near
the ocean floor, they experience an unsteady ground effect.

Unsteady ground effect produces a host of hydrodynamic phenomena. For example,
foils oscillating near a boundary experience a thrust or cruising speed
enhancement with no penalty to their propulsive efficiency (Wu et al., 2014;
Park et al., 2017; Dai et al., 2016; Fernández-Prats et al., 2015; Quinn et
al., 2014a), and vortex wakes are deflected near boundaries due to their image
vortices (Quinn et al., 2014b; Kurt et al., 2019; Zhong et al., 2019).
Moreover, stable equilibrium altitudes arise where an increase/decrease in a
swimmer’s altitude causes a negative/positive lift force, pulling the swimmer
back to a stable altitude (Quinn et al., 2014b; Mivehchi et al., 2016). Kurt
et al. (2019) showed that pitching foils swimming freely in the cross-stream
direction do indeed settle into a stable near-ground equilibrium altitude
under no control. By showing good agreement between the equilibrium altitudes
calculated through inviscid potential flow simulations and those measured in
viscous water channel experiments, they determined that stable equilibrium
altitudes are an inviscid phenomenon (Kurt et al., 2019). Zhong et al. (2019)
broadened our understanding of these equilibrium altitudes by studying
hydrofoils of varying aspect ratio. They discovered that near-ground
equilibrium altitudes can be stable, unstable, or absent depending upon the
aspect ratio and Strouhal number.

Now, it’s essential to understand the underlying physics that give rise to
equilibrium altitudes. In this study, we uncover the fundamental forces that
balance to generate equilibrium altitudes. Following the classic force
decomposition of von Kármán & Sears (1938) and Mccune & Tavares (1993), we
advance our understanding in two important ways: (1) we reveal the mechanism
behind equilibrium altitudes and (2) we leverage this understanding to develop
scaling laws to predict equilibrium altitudes from only the input kinematics.
Previously, Kurt et al. (2019) proposed that stable equilibrium altitudes are
generated by a balance of the added mass lift, wake-induced lift, and quasi-
steady lift. However, their force decomposition approach implicitly assumed
that the local effective flow velocity acting on a foil was unaltered due to
the presence of the ground. In this study, we no longer make this assumption
and, in fact, we determine that this effect is integral to our understanding
of the mechanism behind equilibrium altitudes. We will show that, upon deeper
investigation, equilibrium altitudes exist due to a balance between the
positive wake-induced lift and negative quasi-steady lift with added mass lift
playing precisely zero role — effectively a scallop theorem result (Purcell,
1977) for the time-averaged lift. In this way, our new understanding of the
stable equilibrium altitude mechanism supersedes that which was proposed in
Kurt et al. (2019).

##  2 Methods

###  2.1 Problem formulation

Figure 1: Illustration of a hydrofoil pitching near the ground. An inertial
frame of reference fixed to the undisturbed fluid is denoted by $(X,Z)$ while
a body-fixed frame of reference is denoted by $(x,z)$. An image system
enforces a no-flux boundary condition at the ground plane ($Z=0$).

In this study, a simple model of a pitching hydrofoil is used to represent a
swimmer propelling itself in unsteady ground effect (Figure 1). The foil is a
$7\%$ thick tear-drop shaped cross-section (Godoy-Diana et al., 2008; Quinn et
al., 2014b; Kurt et al., 2019) with a chord length of $c=0.095\,$m. The foil
undergoes sinusoidal pitching about its leading edge,

$\displaystyle\theta(t)=\theta_{0}\sin(2\pi ft),$ (1)

where $\theta(t)$ is the instantaneous pitching angle, $\theta_{0}$ is the
pitching amplitude, $f$ is the pitching frequency, and $t$ is time. The
pitching angle is considered positive for clockwise pitching rotations about
the leading edge. The trailing edge peak-to-peak amplitude $A$ is defined by

$\displaystyle A=2c\sin\theta_{0},$ (2)

and its dimensionless counterpart is $A^{*}=A/c$. The reduced frequency and
the Strouhal number are defined as

$\displaystyle k=\frac{fc}{U_{\infty}}\qquad St=\frac{fA}{U_{\infty}}$ (3)

where $U_{\infty}$ is the swimming speed of the foil. The ground proximity,
$D$, is considered as the distance from the foil’s leading edge to the ground
plane (Figure 1) and its dimensionless form is $D^{*}=D/c$. Finally, the lift
force $L$ is normalized by the dynamic pressure to give lift coefficient,

$\displaystyle C_{L}=\frac{L}{1/2\rho csU_{\infty}^{2}},$ (4)

where $\rho$ is the fluid density while the span length $s$ is set to unity
for two-dimensional simulations, and $s={\mar A}c$ for three-dimensional
simulations with ${\mar A}$ being the aspect ratio. And its time-averaged form
is given as

$\displaystyle\overline{C}_{L}=\frac{\overline{L}}{1/2\rho csU_{\infty}^{2}},$
(5)

where $\overline{L}$ is the time-averaged lift.

###  2.2 Numerical methods

The simulations model a potential flow that is assumed to be irrotational,
incompressible, and inviscid. This flow is governed by Laplace’s equation,
$\nabla^{2}\Phi^{*}=0$, where $\Phi^{*}$ is the perturbation velocity
potential in an inertial frame fixed to the undisturbed fluid. A no-flux
boundary condition, $\nabla\Phi^{*}\,\boldsymbol{\cdot\,n}=0$ where
$\boldsymbol{n}$ is the surface normal vector, must be satisfied on the body
surface and the ground plane at $Z=0$, and a far-field boundary condition that
flow perturbations decay far away from the body and its wake surfaces must be
satisfied.

Following previous studies (Katz, 1985; Quinn et al., 2014b; Moored, 2018;
Kurt et al., 2019; Zhong et al., 2019; Mivehchi et al., 2021), this potential
flow problem is solved numerically using an unsteady boundary element method
with the method of images used to enforce the ground plane boundary condition.
In this method, the foil, its wake surfaces, and their images are discretized
into a finite number of boundary elements. The general solution to the
potential flow problem is then reduced to finding a distribution of constant-
strength doublet and source elements that satisfy the body and ground plane
boundary conditions. The boundary elements also implicitly satisfy the far-
field boundary condition. The body boundary condition is imposed using a
Dirichlet condition at collocation points centered on each body element, but
pushed within the body along the element’s surface normal. Also, an explicit
Kutta condition is applied where the vorticity at the trailing edge is zero by
tuning the strength of a wake boundary element at the trailing edge. At each
time step, the trailing edge element is shed with a strength that satisfies
Kelvin’s circulation theorem and it remains constant thereafter. A wake roll-
up algorithm is implemented to advect the wake elements with the local
velocity calculated by the desingularized Biot-Savart law (Krasny, 1986; Zhu
et al., 2002; Moored, 2018; Akoz et al., 2021). Therefore, a matrix
representation of the boundary condition is formed and solved for the body
doublet strengths at each time step. The perturbation velocity over the
surface of the body is then determined by a local differentiation of the
perturbation potential.

Following the work of Moored (2018), the pressure exerted on the body is
calculated by the unsteady Bernoulli equation as

$\displaystyle P(x,z,t)=-\rho\frac{\partial\Phi^{*}}{\partial
t}+\rho(\mathbf{u_{rel}}+\mathbf{U_{\infty}})\cdot\nabla\Phi^{*}-\rho\frac{(\nabla\Phi^{*})^{2}}{2},$
(6)

where $\mathbf{u_{rel}}$ is the local surface velocity of the pitching foil
relative to the body-fixed frame of reference. The inviscid instantaneous
resultant force is then simply an integration of the instantaneous pressure
field acting over the body surface. Finally, the resultant force is decomposed
into its lift and drag directions. For more details see Moored (2018).

###  2.3 Force decomposition approach

Following the classic force decomposition of von Kármán & Sears (1938) and
Mccune & Tavares (1993), the lift force can be decomposed as the sum of the
added mass lift, quasi-steady lift, and wake-induced lift, which are denoted
by $(\cdot)_{a}$, $(\cdot)_{q}$, and $(\cdot)_{w}$, respectively. We apply
this decomposition approach directly to the pressure field defined by the
unsteady Bernoulli equation.

To calculate the added mass pressure, $P_{a}$, potential flow simulations are
employed where there are no wake elements shed and no Kutta condition
enforced. Under these conditions the first term in Equation (6) is known as
the added mass pressure. To calculate the quasi-steady pressure, $P_{q}$,
potential flow simulations are employed where there are no wake elements shed,
but there is a Kutta condition enforced (modeled in the boundary element
method with a trailing-edge panel that extends far from the foil, effectively
to infinity). Then the combined second and third terms in Equation (6) provide
the quasi-steady pressure. The combined added mass and quasi-steady pressures
are,

$\displaystyle
P_{a}(x,z,t)+P_{q}(x,z,t)=\left[\underbrace{-\rho\frac{\partial\Phi^{*}}{\partial
t}}_{\text{added
mass}}+\underbrace{\rho(\mathbf{u_{rel}}+\mathbf{U_{\infty}})\cdot\nabla\Phi^{*}-\rho\frac{(\nabla\Phi^{*})^{2}}{2}}_{\text{quasi-
steady}}\right]_{\text{no wake}}.$ (7)

The wake-induced pressure, $P_{w}$, is calculated as the difference between
the total pressure, $P$ (calculated from the full simulations described in
Section 2.2), and the combined added mass and quasi-steady pressures,

$\displaystyle P_{w}(x,z,t)=P(x,z,t)-\left[P_{a}(x,z,t)+P_{q}(x,z,t)\right].$
(8)

Finally, each force component is acquired by integrating its corresponding
pressure field over the body. Then the added mass, quasi-steady and wake-
induced lift coefficients can be defined and they sum to the total lift
coefficient,

$\displaystyle C_{L}=C_{L_{a}}+C_{L_{q}}+C_{L_{w}}.$ (9)

###  2.4 Force decomposition validation

Figure 2: Validation of the force decomposition approach against Theodorsen’s
solution across a wide reduced frequency range of $0.1\leq k\leq 10$. The
subfigure rows show: $(a)$ added mass lift, $(b)$ wake-induced lift, and $(c)$
quasi-steady lift.

Before applying the force decomposition approach to simulations of a pitching
foil in ground effect, we first validated it against Theodorsen’s solution,
which provides a test case where the added mass, quasi-steady, and wake-
induced terms are readily identified (Bisplinghoff et al., 2013). In
Theodorsen (1935), the lift force generated by a foil pitching about its
leading edge is,

$\displaystyle
L=\underbrace{\rho\pi\frac{c^{2}}{4}\left(U_{\infty}\dot{\theta}+\frac{c}{2}\ddot{\theta}\right)}_{\text{added
mass}}+\underbrace{\rho\pi
U_{\infty}c\left(U_{\infty}\theta+\frac{3c}{4}\dot{\theta}\right)}_{\text{quasi-
steady}}+\underbrace{\left[C(k)-1\right]\rho\pi
U_{\infty}c\left(U_{\infty}\theta+\frac{3c}{4}\dot{\theta}\right)}_{\text{wake-
induced}},$ (10)

where $C(k)$ is the lift deficiency function. Full boundary element
simulations and their decomposition counterparts were performed as described
in Sections 2.2 and 2.3 on a thin airfoil ($1\,\%$ thick tear-drop shape),
undergoing small pitching amplitudes of motion about its leading edge
($A^{*}=0.002$), with a frozen wake (no wake roll-up algorithm applied), and
across a wide reduced frequency range of $k=0.1,\,1,\,$ and $10$. Figure 2
shows that all three lift components precisely match the Theodorsen solution
across a wide range of $k$. Even though Theodorsen’s solution makes several
assumptions, it is worth noting that the force decomposition approach itself
is generic and has no underlying assumptions.

###  2.5 Experimental methods

Figure 3: Schematic of experiment setup equipped with removable splitter
plate.

We conducted four groups of experiments to facilitate the force decomposition
analysis, including a 2D static ground effect test, a 2D unsteady high-
frequency test (for wake deflection analysis), a 2D and a 3D unsteady ground
effect test. All hydrofoils had a tear-drop cross-section with a chord length
of $0.095$ m to ensure consistency with simulations, and were 3D-printed with
solid ABS. A stainless steel driveshaft with a diameter of $6.35$ mm was used
to pitch the hydrofoil at its leading edge, with the pitch angle $\theta$
being prescribed as $\theta_{0}\sin(2\pi ft)$. The driveshaft was driven by a
high-torque digital servo motor (Dynamixel MX64), while an absolute encoder
(US Digital A2K 4096 CPR) was utilized to verify the angle of the actual
motion. The forces and moments generated were measured using a six-axis force-
torque sensor (ATI-Mini 40: SI-40-2). The lateral forces were time-averaged
over 30 seconds or 20 oscillation cycles to determine the mean lift force for
static and unsteady tests. The incoming flow speed was set to 143 mm/s using
an ultrasonic flowmeter (Dynasonics Series TFXB), resulting a chord-based
Reynolds number of $13$,$500$. For ground effect tests, an additional splitter
plate was used instead of a tunnel sidewall to control the boundary layer
thickness (Figure 3). A boundary layer thickness of
$\delta_{\mathrm{99\%}}\approx 7.5$ mm ($\delta_{\mathrm{99\%}}/c=0.08$) was
measured using Particle Image Velocimetry at the position aligned with the
leading edge of the hydrofoil.

For all two-dimensional tests, we used a hydrofoil of aspect ratio 3 in a
nominally two-dimensional flow condition, which was achieved by installing a
horizontal splitter plate and a surface plate near the tips of the hydrofoil.
The gap between the hydrofoil tips and the surface/splitter plate was less
than 5 mm and surface waves were also minimized by the presence of the surface
plate.

To validate the 3D force decomposition, we repurposed the 3D unsteady ground
effect data that was previously published in Zhong et al. (2019). We tested
hydrofoils with four aspect ratios: 1, 1.5, 2 and 2.5, and assessed their
performance over a range of ground proximities. Each trial was repeated three
times.

##  3 Force decomposition results

|  |  |  |  | Varying $k$ |  |  |  |  |  |  | Varying $St$
---|---|---|---|---|---|---|---|---|---|---|---|---
$k$ |  |  |  |  | $0.55\,-\,1$ |  |  |  |  |  |  | $1$
$St$ |  |  |  |  | $0.3$ |  |  |  |  |  |  | $0.25\,-\,0.55$
$D^{*}$ |  |  |  |  | $0.25\,-\,1.5$ |  |  |  |  |  |  | $0.25\,-\,1.5$
Table 1:  Simulation variables investigated in the current study.

A pitching foil in ground effect and constrained to fixed ground distances was
simulated across a wide range of dimensionless ground proximities, reduced
frequencies, and Strouhal numbers, and force decomposition was applied as
described in Section 2.3. The simulation variables are detailed in Table 1.

###  3.1 Mechanism behind the generation of stable and unstable equilibria

Figure 4: Both the stable and unstable equilibrium altitudes are generated by
a balance between the positive wake-induced lift and negative quasi-steady
lift. $(a)$ Wake plots for an out-of-ground effect foil. $(b)$ Wake plots for
a near-ground foil. $(c)$ Total lift for varying $k$ with $St=0.3$. $(e)$
Total lift for varying $St$ with $k=1$. $(d)$ Zoom-in plot highlighting the
unstable equilibrium altitude. $(f)$ and $(i)$ added mass lift. $(g)$ and
$(j)$ wake-induced lift . $(h)$ and $(k)$ quasi-steady lift. Note that the
stable equilibrium altitude is marked by a solid line, the stable equilibrium
altitude for a foil unconstrained in the cross-stream direction (mass of the
foil is chosen to be $m=2.68\rho sc^{2}$) is marked by the dashed line, and
the unstable equilibrium altitude is marked by the dotted line.  Figure 5:
Force decomposition for $k=1$: (a) $St=0.25$ and (b) $St=0.55$.

The pitching foil examined in this study is effectively out of ground effect
by $D^{*}=1.5$ for cases with $St<0.45$, and by $D^{*}=2$ for cases with
higher $St$. In fact, a typical reverse von Kármán vortex street, indicative
of thrust production in an infinite domain, is observed when $D^{*}=1.5$,
$k=1$, and $St=0.3$ (Figure 4a). As the foil approaches the ground to a
distance of $D^{*}=0.4$ ($k=1$ and $St=0.3$), the wake vortices deflect away
from the ground (Figure 4b), an effect that has been attributed to the image
vortex system (Quinn et al., 2014b).

The total lift for all of the simulations is presented for varying reduced
frequency and ground distance ($St=0.3$; Figure 4c), and for varying Strouhal
number and ground distance ($k=1$; Figure 4d and 4e). A common trend emerges
in all of the in-ground effect data except when $St>0.475$: the lift is
positive close to the ground (pushing a foil away from the ground) and
negative far from the ground (pulling a foil towards the ground), crossing
zero (stable equilibrium altitude; solid lines in Figure 4) between these two
regimes. As observed previously, the stable equilibrium altitude increases
with increasing $St$ and decreasing $k$ (Kurt et al., 2019), and an unstable
equilibrium altitude emerges when $St>0.475$ (dotted line in Figure 4d), which
increases slightly with increasing $St$ in a narrow near-ground region (Zhong
et al., 2019). Altitudes below/above the unstable equilibria produce a
negative/positive lift, demonstrating that it is indeed unstable. For the
first time, this unstable equilibrium altitude is observed in simulation,
which corroborates previous experimental findings (Zhong et al., 2019).

The dashed lines in Figure 4c and 4e show the unconstrained simulation results
from Kurt et al. (2019). As discovered in Cochran-Carney (2018), the observed
difference in the equilibrium altitudes of the constrained and unconstrained
cases is ultimately caused by a difference in the unconstrained simulation’s
dimensionless mass ($m^{*}\equiv m/(\rho sc^{2})=2.68$) compared to that of
the constrained simulations ($m^{*}=\infty$). The finite mass foils of the
unconstrained simulations have a finite heaving recoil while the effectively
infinite mass foils of the constrained simulations have zero heaving recoil,
thereby altering the force production between the two cases and, consequently,
their equilibrium altitudes. Despite these differences, the constrained
equilibria show all of the same trends as the unconstrained equilibria with
only a small shift in their $D^{*}$ location. Moreover, this study will focus
on constrained simulations by design since these allow us to investigate the
forces outside of equilibrium conditions, which is essential to fully
understand the stability properties of the equilibria.

The decomposition of the lift coefficient into its added mass, wake-induced,
and quasi-steady components (Figure 4f–4k) show clear trends in the components
for all ground proximities and kinematics. The added mass lift is always
precisely zero and therefore does not play a role in generating equilibrium
altitudes (Figure 4f and 4i). For in-ground effect data, the wake-induced lift
is always positive (Figure 4g and 4j) while the quasi-steady lift is always
negative (Figure 4h and 4k), revealing that both unstable and stable
equilibria are generated by a balance between positive wake-induced lift and
negative quasi-steady lift. The magnitudes of the wake-induced and quasi-
steady lift increase with increasing $St$ and decreasing $k$.

The data contours in Figure 4 reveal the basic mechanism behind the existence
of equilibria, but they do not provide a clear picture as to why there can be
both a stable and an unstable equilibria for a given $St$ and $k$. To better
understand this feature of the data the lowest and highest $St$ cases with
$k=1$, $St=0.25$ and $St=0.55$, respectively, are presented in Figure 5. For
low $St$ (Figure 5a), the total lift only crosses zero at one $D^{*}$
producing a stable equilibrium. This equilibrium altitude is stable since for
altitudes above equilibrium the magnitude of the quasi-steady lift is greater
than the wake-induced lift leading to a net negative total lift force, and for
altitudes below equilibrium the magnitude of the wake-induced lift is greater
than the quasi-steady lift leading to net positive total lift. For high $St$
(Figure 5b), the total lift is seen to cross through two equilibrium
altitudes. Now, it becomes clear that the wake-induced lift magnitude curve
crosses through the quasi-steady lift magnitude curve at two ground distances
because the wake-induced lift has a nearly linear increase with decreasing
ground distance while the quasi-steady lift has an exponential growth. In
fact, the slopes of the lift components at the equilibrium points determines
their stability: when
$\left|d\overline{C}_{Lq}/dD^{*}\right|<\left|d\overline{C}_{Lw}/dD^{*}\right|$
the equilibrium altitude is stable and when
$\left|d\overline{C}_{Lq}/dD^{*}\right|>\left|d\overline{C}_{Lw}/dD^{*}\right|$
the equilibrium altitude is unstable.

Though this analysis has revealed the basic mechanism behind the generation of
stable and unstable equilibrium altitudes, it also raises several questions.
How does added mass lift integrate precisely to zero when we expect peak
negative added mass lift to be larger in magnitude than the peak positive
added mass lift since the foil will get closer and further from the ground
throughout an oscillation cycle? Why is the wake-induced lift positive for a
wake deflected away from a ground plane when, based on momentum flux
arguments, we would expect the opposite? Why is the quasi-steady lift negative
in nature? To address these questions, the following sections will examine
each force component more deeply.

###  3.2 Added mass lift

Figure 6: The added mass lift as a function of time for varying ground
distance with $k=1$ and $St=0.3$. Out-of-ground effect data is represented by
$D^{*}=\infty$.

As shown in the previous section, the time-averaged added mass lift is
precisely zero for all of the simulation cases. This is expected for solutions
to the linear potential flow equations with harmonic boundary conditions – a
‘scallop theorem’ result (Purcell, 1977). However, for the in-ground effect
data it is well-known that as the ground distance decreases there is an
increase in the added mass of a foil (Brennen, 1982; Mivehchi et al., 2021).
This effect should amplify both the positive and negative peaks in the added
mass lift, but should more greatly amplify the negative peak, which occurs at
the bottom of the down stroke when the foil is closest to the ground.
Considering only these peak forces, one would imagine that the time-averaged
added mass lift would be negative, yet we know that it is zero from the data
in Section 3.1, so how does the added mass lift integrate precisely to zero
and satisfy a scallop theorem?

Figure 6 presents the time-varying added mass lift at four ground distances
when $k=1$ and $St=0.3$. Indeed, as the ground distance decreases both the
positive and negative peaks grow in magnitude, and it is true that the
negative peak is amplified more than the positive peak creating a net negative
lift in the peak forces. However, it is now clear that the negative peak
becomes sharper while the positive peak becomes broader. When integrating the
lift over a full cycle, this effect counteracts the net imbalance in the
positive and negative peak forces, leading to a precisely zero time-averaged
force.

###  3.3 Wake-induced lift

Figure 7: Time-averaged lift control volume analysis. The control volume is
represented by a dashed box. (a) Upward deflected wake generated by an out-of-
ground effect foil pitching at $St=0.55$ and $A^{*}=0.3$. (b) Upward deflected
wake generated by an in-ground effect foil pitching at $St=0.3$ and
$A^{*}=0.3$. (c) Control volume analysis for the out-of-ground effect case.
(d) Control volume analysis for the in-ground effect case. The time-averaged
wake-induced lift coefficient from simulations and experiments are denoted by
the solid blue lines and blue circles, respectively. Contributions to
simulations’ wake-induced lift from the pressure difference term and the
momentum flux term in the control volume analysis are shown as green and
yellow solid lines, respectively.

In Section 3.1, the wake-induced lift was found to be positive for in-ground
effect foils regardless of their ground distance and kinematics. This is
surprising since the wake vortices deflect away from the ground and in the
time-average they also produce a momentum jet deflected away from the ground
(Quinn et al., 2014b). If one imagines a control volume around a foil and
considers only the momentum flux term, the deflection of the jet away from the
ground should have an associated negative lift force, that is, a force pushing
the swimmer toward the ground (Kurt et al., 2019), and yet the data from
Section 3.1 shows otherwise.

To understand this phenomenon more deeply we employ a control volume analysis.
To begin we will consider a simple out-of-ground effect case where a deflected
wake is also generated very much akin to the deflected wake of in-ground
effect foils. This case is useful since it will help unambiguously show the
connection among the wake-induced lift, the pressure field, and the momentum
flux, and it presents an opportunity to experimentally verify the trends in
the wake-induced lift. With this basic knowledge in mind, we will then proceed
to the in-ground effect case.

When a two-dimensional pitching foil oscillates at sufficiently high Strouhal
number, it is well-documented (Marais et al., 2012; Das et al., 2016) that
there is a wake instability that causes the wake vortices to form vortex
dipoles and, in turn, deflect the wake. Figure 7a shows an example of this
where a pitching foil is operated at $St=0.55$ and $A^{*}=0.3$. Since this
pitching foil is out of ground effect and has a symmetric motion, then the
time-averaged added mass and quasi-steady lift must both necessarily be zero.
However, due to the wake deflection the time-averaged wake-induced lift will
be non-zero and, in fact, it will be the only component thus making it
equivalent to the total lift in this specific case.

Following the work of Cleaver et al. (2012), a time-averaged control volume
analysis can be applied to the control volume (dashed box) defined in Figure
7a,

$\displaystyle\overline{L}_{w}=\underbrace{\overline{\int_{UL}\Delta
P_{UL}dA}}_{\text{Pressure difference}}\underbrace{-\overline{\int_{CS}\rho
v(\mathbf{V}\cdot\mathbf{n})dA}}_{\text{Momentum flux}},$ (11)

where the total time-averaged lift is precisely equal to the time-averaged
wake-induced lift, $\overline{L}_{w}$, the pressure difference between the
upper and lower ($UL$) control surfaces is $\Delta P_{UL}$, the velocity field
is $\mathbf{V}$, the vector normal to the control surfaces ($CS$) is
$\mathbf{n}$, and the horizontal and vertical components of the velocity field
are $u$ and $v$, respectively.

Just like the in-ground effect data from Section 3.1, both simulations and
experiments show that the wake-induced lift is in fact positive for an upward
deflected wake (Figure 7c) which, in the case of this out-of-ground effect
foil, is consistent with previous findings (Emblemsvåg et al., 2002; Liang et
al., 2011; Yu et al., 2012; Cleaver et al., 2012). Using the simulation data,
Figure 7c further shows that the wake-induced lift from the net momentum flux
term is indeed negative, as expected, while the contribution from the pressure
difference term is positive and, surprisingly, is larger in magnitude than the
momentum flux term, thereby making the wake-induced lift positive for an
upward deflected wake. In light of this analysis, momentum flux considerations
alone do not accurately discern the direction of forces generated by an
asymmetric wake; the pressure field must also be considered.

Moreover, the same control volume analysis was applied to an in-ground effect
pitching foil in Figure 7b. Since the time-averaged added mass lift is zero
(Section 3.2), the time-averaged wake-induced lift from the pressure
difference and momentum flux terms are acquired by calculating those two
contributions for the total and quasi-steady pressure and flowfields and
taking their differences as,

$\displaystyle\overline{L}_{w,\text{pressure}}=\overline{\int_{UL}\Delta
P_{UL}dA}\bigg{|}_{\text{total}}-\overline{\int_{UL}\Delta
P_{UL}dA}\bigg{|}_{\text{quasi-steady}}$ (12)
$\displaystyle\text{and}\;\;\overline{L}_{w,\text{momentum}}=-\overline{\int_{CS}\rho
v(\mathbf{V}\cdot\mathbf{n})dA}\bigg{|}_{\text{total}}+\overline{\int_{CS}\rho
v(\mathbf{V}\cdot\mathbf{n})dA}\bigg{|}_{\text{quasi-steady}}.$ (13)

Figure 7d shows that in ground effect, the momentum flux term is negative, yet
the wake-induced lift acts in the positive direction since the pressure
difference term again outweighs the momentum flux term, as in the out-of-
ground effect case. It is worth noting that compared to the out-of-ground
effect case the pressure difference term now has a more significant effect on
the wake-induced lift.

###  3.4 Quasi-steady lift

Figure 8: Asymmetry in the quasi-steady lift for a steady foil in ground
effect is caused by the velocity induced on a foil by its image body’s quasi-
steady bound circulation. $(a)$ The quasi-steady lift coefficient. Note that
the lift coefficient calculated in the simulations (using the unsteady
Bernoulli equation) is represented with solid lines, the lift coefficient
calculated from the simulations using the Kutta-Joukowski theorem is
represented by the circle markers and the experimental data is represented by
the square markers. $(b)$ The dimensionless quasi-steady circulation. $(c)$
The dimensionless effective flow velocity. $(d)$ Schematic of the components
of the effective flow velocity, which is the summation of $U_{i}$ and
$U_{\infty}$.  Figure 9: The velocity induced by the ground image body creates
asymmetry in the quasi-steady lift at finite Strouhal numbers just like the
steady foil case. $(a)$ The quasi-steady lift coefficient. $(b)$ The
dimensionless quasi-steady circulation. $(c)$ The dimensionless effective flow
velocity. $(d)$ Schematic showing the effective flow velocity is the summation
of $U_{i}$ and $U_{\infty}$.

To provide deeper insight as to why there is a net negative quasi-steady force
in the time average, we further break down the quasi-steady lift into its
constituent components by leveraging the Kutta-Joukowski theorem (von Kármán &
Sears, 1938), $L_{q}=\rho sU\Gamma_{0}$, where $L_{q}$ is the quasi-steady
lift, $s$ is the span length, $U$ is the local effective flow velocity acting
on a foil, and $\Gamma_{0}$ is the quasi-steady bound circulation of a foil.
The local effective velocity is calculated from simulations by summing the
freestream velocity, $U_{\infty}$, with the induced velocity $U_{i}$ at the
leading edge from the image body. The quasi-steady bound circulation is simply
the negative of the strength of the trailing-edge panel used to enforce the
Kutta condition in the boundary element simulations, that is,
$\Gamma_{0}=-\mu_{TE}$ Katz, Joseph; Plotkin (2005). The Kutta-Joukowski
theorem can be non-dimensionalized and written as

$\displaystyle
C_{L_{q}}=2U^{*}\Gamma_{0}^{*}\quad\;U^{*}=\frac{U}{U_{\infty}}\quad\;\Gamma_{0}^{*}=\frac{\Gamma_{0}}{cU_{\infty}}.$
(14)

When a foil is out of ground effect then $U^{*}=1$,
$\Gamma_{0}^{*}=\Gamma_{0,\infty}^{*}$, and $C_{L_{q}}=C_{L_{q,\infty}}$,
where $\Gamma_{0,\infty}^{*}$ and $C_{L_{q,\infty}}$ are the time-varying
quasi-steady bound circulation and lift coefficient for a foil in an infinite
domain. Without loss of generality, the quasi-steady bound circulation for in-
ground effect foils can be written as the multiplication of a circulation
amplification factor $\beta$ and the infinite domain quasi-steady circulation,
$\Gamma_{0}^{*}=\beta\Gamma_{0,\infty}^{*}$. We expect that both the
circulation and the effective flow velocity will be amplified or reduced for
foils in ground effect as opposed to out-of-ground effect foils.

To understand the connections among the circulation amplification factor, the
amplified/reduced effective flow velocity, and the quasi-steady lift, as well
as to verify these connections with experimental data, we begin with a simple
case of a static foil in and out of ground effect at various pitch angles
(Figure 8). For the static foil, the quasi-steady lift is precisely the total
lift, which is readily measured in experiments. Figure 8a presents the quasi-
steady lift for $D^{*}=0.3$ and $D^{*}=\infty$ from the simulations (solid
lines), which is directly determined by integrating the pressure acting on the
foil calculated from the unsteady Bernoulli equation (Section 2.2). The
simulations are also used to calculate the quasi-steady lift by using the
Kutta-Joukowski theorem (circle markers) as outlined above, and both
approaches are observed to produce exactly the same result. Also, the in-
ground effect quasi-steady lift is verified by having good agreement with
experimental data (square markers).

For the out-of-ground effect foil, the quasi-steady lift shows the classic
linear relationship with the pitch angle for $-8^{\circ}\leq\theta\leq
8^{\circ}$. However, when in ground effect the quasi-steady lift shows a
nonlinear relationship with the pitch angle, where both the negative and
positive lift regimes are amplified compared to the out-of-ground effect case,
yet the negative lift regime is amplified more than the positive lift regime.
This produces an asymmetric lift response about $\theta=0$. If one were to
quasi-statically pitch the foil through an $8^{\circ}$ amplitude sinusoidal
pitching motion this asymmetric lift response would lead to a net negative
quasi-steady lift in the time-average, just as is observed in the quasi-steady
data of finite Strouhal numbers. In fact, throughout this section we will show
that precisely the same mechanism is at play in this static case as is in the
quasi-steady cases of finite Strouhal number.

Figure 8b presents the dimensionless quasi-steady circulation. The circulation
amplification is observed to be symmetric about $\theta=0$ and appears to just
be a constant factor giving rise to a slope change in the $\Gamma_{0}^{*}$ –
$\theta$ line. This shows, at least for the static case, that $\beta$ is only
a function of $D^{*}$, not $\theta$. Moreover, the circulation amplification
is responsible for amplifying the quasi-steady lift forces in ground effect,
but is not responsible for asymmetry in the lift curve. Figure 8c shows that
the effective flow velocity increases above the freestream speed when the
quasi-steady circulation is negative (counterclockwise), while it decreases
below freestream speed when the quasi-steady circulation is positive
(clockwise). This occurs due to the velocity induced by the quasi-steady bound
vortex of the image body (Figure 8d). By consequence, increased effective flow
speed acts to amplify the negative lift of negative pitch angles (negative
bound circulation) thereby acting in concert with the circulation
amplification effect, and decreased effective flow speed acts to reduce the
positive lift of positive pitch angles (positive bound circulation), thereby
counteracting the circulation amplification effect. Therefore, it is the
change in the effective flow speed that leads to the asymmetric lift response.
This mechanism will be implicated further as the reason for the net negative
quasi-steady lift for oscillating foils from zero to finite Strouhal numbers.

Figure 9 presents the quasi-steady lift coefficient, dimensionless quasi-
steady circulation, and dimensionless effective flow velocity for a foil at a
range of ground distances with $St=0.3$. In Figure 9a, first, the quasi-steady
lift calculated by the unsteady Bernoulli equation and by the Kutta-Joukowski
theorem still show good agreement. Figure 9b shows that the negative and
positive quasi-steady circulation are symmetrically amplified with decreasing
ground distance, precisely as in the static foil case. Moreover, Figure 9c
shows that the effective flow velocity increases above and decreases below the
freestream velocity when the bound circulation is negative and positive,
respectively, also mirroring the static foil case. Consequently, the change in
effective flow velocity leads to an enhancement in the negative lift and a
reduction in the positive lift thereby generating a net negative quasi-steady
lift in the time average over one pitching cycle. Figure 9d illustrates
precisely the same effective flow altering mechanism of the image body’s
quasi-steady circulation as is in the static foil case.

###  3.5 Three-dimensional decomposition

Figure 10: The magnitude of the wake-induced lift of three-dimensional foils
degrades more with decreasing ${\mar A}$ than their quasi-steady lift leading
to the disappearance of an equilibrium altitude for ${\mar A}<1.5$. Simulation
data of pitching three-dimensional hydrofoils of varying the aspect ratio with
$St=0.25$ and $\theta_{0}=11^{\circ}$. $(a)$ Total lift as calculated from the
simulations. Note that the circle markers represent the stable equilibrium
altitudes measured by experiments. $(b)$ Decomposed lift components from the
simulations.

In Figure 10a, both $3D$ simulations and experiments observe that the stable
equilibrium altitude disappears when the aspect ratio decreases to ${\mar
A}=1$, which is reproduced from Zhong et al. (2019). Figure 10b, presents the
lift decomposition of the 3D simulations into the added mass, quasi-steady and
wake-induced components. Exactly as in the 2D simulations, a stable
equilibrium altitude arises for foils of finite ${\mar A}$ by a balance
between the positive wake-induced lift and negative quasi-steady lift. The
equilibrium altitude disappears for low aspect ratio (${\mar A}<1.5$)
hydrofoils, which instead experience a net negative time-averaged lift force.
The decomposition reveals that this occurs since the magnitude of the wake-
induced lift reduces more with decreasing ${\mar A}$ than the reduction in the
magnitude of quasi-steady lift, thereby having the negative quasi-steady lift
outweigh the positive wake-induced lift at all ground distances.

###  3.6 Scaling the equilibrium altitude

Figure 11: Scaling laws for steady and unsteady ground effect show a good
collapse across a wide range of $St$, $k$, and $D^{*}$. (a) The lift for a
steady foil in ground effect (data from Figure 8a with additional numerical
case at $D^{*}=0.25$) is compared to the steady foil scaling law. $(b)$ The
quasi-steady lift of an unsteady pitching foil (data from Figure 9) compared
to its scaling relation for $0.25\leq D^{*}\leq 1.5$, $0.25\leq St\leq 0.55$,
and $0.55\leq k\leq 1$. The marker color from dark blue to light yellow
represents increasing $St$ while the marker size from small to large
represents increasing $k$. $(c)$ Schematic to illustrate the simple flow model
for calculating the wake circulation amplification factor. $(d)$ The wake-
induced lift of an unsteady pitching foil compared to its scaling relation.
Markers outlined in red represent data with $St\geq 0.45$. $(e)$ Stable
equilibrium altitudes compared to the scaling relation. Dashed lines present
$20\%$ margins of error. These include numerical data with $0.25\leq St\leq
0.55$ and $0.55\leq k\leq 1$, and experimental data with $0.2\leq St\leq 0.4$
and $0.6\leq k\leq 0.9$, as well as experimental data from Kurt et al. (2019)
with $0.2\leq St\leq 0.6$ and $0.5\leq k\leq 1$.

The analysis of the added mass, quasi-steady, and wake-induced forces
presented above can now be leveraged to develop scaling laws that capture the
physics of ground effect and are capable of predicting the equilibrium
altitude of a pitching foil. Since the added mass forces do not play a role in
the generation of an equilibrium altitude then scaling laws for only the
quasi-steady and wake-induced lift are sought.

####  3.6.1 Quasi-steady lift

As presented in Section 3.4, the quasi-steady lift coefficient is,

$\displaystyle C_{L_{q}}\propto U^{*}\beta\Gamma_{0,\infty}^{*},$ (15)
$\displaystyle\Gamma_{0,\infty}^{*}=\pi\theta+\frac{\pi
c\dot{\theta}}{2U_{\infty}},$ (16)

where $\beta$ is the quasi-steady circulation amplification factor, $U^{*}$ is
the normalized effective flow velocity at the foil’s leading edge, and
$\Gamma^{*}_{0,\infty}$ is simply the formula for a flat plate pitching about
its leading edge from Theodorsen (1935) and Mccune & Tavares (1993). To
determine a scaling law for the quasi-steady lift, consider a simplified flow
model where the quasi-steady bound vortex is located along the chord (Figure
9d). Then a scaling relation for the circulation amplification factor and the
effective flow velocity can be determined. The circulation amplification
factor is calculated by finding the circulation amplification necessary to
enforce the Kutta condition at the foil’s trailing edge due to the influence
of the image quasi-steady bound vortex. Assuming that the distance between the
bound vortex and foil’s leading edge is $bc$ with $0\leq b\leq 1$, then the
circulation amplification factor is simply calculated as,

$\displaystyle\beta=1+\frac{(1-b)^{2}}{4{D^{*}}^{2}}\quad\text{where}\quad
b=0.55.$ (17)

To determine the placement of the bound vortex along the chord, circulation
amplification data at $D^{*}=0.5$ from an exact solution for a steady foil in
ground effect (Figure 4d in Baddoo et al. (2020)) was used to determine that
$b=0.55$. The effective flow velocity is calculated by finding the normalized
induced velocity at the leading edge, $U^{*}_{i}=U_{i}/U_{\infty}$, from the
image foil’s quasi-steady bound vortex,

$\displaystyle U^{*}=1+U^{*}_{i}\quad\text{where}\quad
U^{*}_{i}=\frac{-\Gamma_{0,\infty}^{*}\beta}{\pi}\left(\frac{D^{*}}{b^{2}+4{D^{*}}^{2}}\right).$
(18)

After organization, the time-varying quasi-steady lift becomes,

$\displaystyle
C_{L_{q}}\propto\beta\Gamma_{0,\infty}^{*}-\frac{\beta^{2}{\Gamma^{*}}^{2}_{0,\infty}D^{*}}{\pi(b^{2}+4{D^{*}}^{2})}.$
(19)

According to thin airfoil theory, the coefficient of the first term is $2$,
which gives,

$\displaystyle
C_{L_{q}}=2\beta\Gamma_{0,\infty}^{*}-c_{1}\frac{\beta^{2}{\Gamma^{*}}^{2}_{0,\infty}D^{*}}{\pi(b^{2}+4{D^{*}}^{2})},$
(20)

where $c_{1}$ needs to be determined.

As an important side note, this formula can be used to determine a lift
scaling relation for steady ground effect by neglecting the rotational lift or
virtual camber term ($\dot{\theta}$ term) in $\Gamma^{*}_{0,\infty}$. Then a
scaling law for steady ground effect can be deduced as,

$\displaystyle C^{\text{steady}}_{L}=2\pi\beta\theta-
c_{1}\frac{\pi\beta^{2}\theta^{2}D^{*}}{b^{2}+4{D^{*}}^{2}}\quad\text{where}\quad
c_{1}=2.84.$ (21)

Using the static foil numerical and experimental data from Figure 8a the
coefficient is determined to be $c_{1}=2.84$ by linear regression. Figure 11a
then compares the scaling law prediction with the data, which shows that it
can predict the steady ground effect data to within $10\%$ for the numerical
data and $15\%$ for the experimental data.

Returning to the the time-varying formula, Equation (20), we can also consider
an unsteady ground effect case where the foil undergoes sinusoidal pitching.
If this scaling relation is then time-averaged only a contribution from the
second term will remain as,

$\displaystyle\overline{C}_{L_{q}}=-c_{1}\frac{\beta^{2}\overline{{\Gamma^{*}}^{2}_{0,\infty}}D^{*}}{\pi(b^{2}+4{D^{*}}^{2})}.$
(22)

Considering that $\theta=\theta_{0}\sin(2\pi ft)$ and
$\theta_{0}=A^{*}/2=St/(2k)$ we have,

$\displaystyle\overline{C}_{L_{q}}=c_{1}\phi_{1},$ (23)
$\displaystyle\phi_{1}=-\frac{\pi^{3}}{4}\left(\frac{\beta^{2}D^{*}}{b^{2}+4{D^{*}}^{2}}\right)\left(1+\frac{1}{\pi^{2}k^{2}}\right)St^{2}.$
(24)

The coefficient $c_{1}=2.84$ determined from the static foil data is used for
scaling the quasi-steady lift of both the steady and unsteady foil cases.
Figure 11b presents an excellent collapse of the quasi-steady lift data over a
wide range of $St$, $k$ and $D^{*}$.

####  3.6.2 Wake-induced lift

The wake-induced lift can be calculated as,

$\displaystyle C_{L_{w}}\propto U^{*}\gamma\Gamma_{w,\infty}^{*},$ (25)
$\displaystyle U^{*}=1+U^{*}_{i,1}+U^{*}_{i,2}.$ (26)

where $\Gamma_{w,\infty}^{*}$ is the normalized infinite domain wake
circulation, and the effective flow velocity $U^{*}$ is composed of the
velocity $U_{i,1}^{*}$, induced by an image bound vortex (representing the
total bound vorticity, not the quasi-steady bound vorticity), and the velocity
induced by an image wake vortex, $U_{i,2}^{*}$. To determine a scaling law for
the wake-induced lift, consider a simplified flow model where a wake vortex is
placed along the chord line at a distance of $U_{\infty}/(4f)$ downstream of
the trailing edge (Figure 11c), which gives,

$\displaystyle
U^{*}_{i,1}=\frac{-\Gamma_{\infty}^{*}\gamma}{\pi}\left(\frac{D^{*}}{b^{2}+4{D^{*}}^{2}}\right)\;\;\text{and}\;\;U^{*}_{i,2}=\frac{-\Gamma_{w,\infty}^{*}\gamma}{\pi}\left[\frac{16k^{2}D^{*}}{(1+4k)^{2}+64k^{2}{D^{*}}^{2}}\right],$
(27)

where $\Gamma^{*}_{\infty}$ is the normalized infinite domain total
circulation, and $\gamma$ is the wake circulation amplification factor. By
enforcing the Kutta condition at the foil’s trailing edge, the wake
circulation amplification factor is calculated as,

$\displaystyle\gamma=\frac{1+4k(1-b)}{1+4k(1-b)-\frac{(1-b)^{2}}{(1-b)^{2}+4{D^{*}}^{2}}+\frac{4k(1-b)}{1+64k^{2}{D^{*}}^{2}}}.$
(28)

Following Moored & Quinn (2018), we have,

$\displaystyle\Gamma_{w,\infty}^{*}\propto\Gamma_{0,\infty}^{*}\frac{k^{*}}{k^{*}+1}\;\;\text{and}\;\;k^{*}=\frac{k}{1+4St^{2}},$
(29) $\displaystyle\Gamma_{\infty}^{*}\propto\Gamma_{0,\infty}^{*}.$ (30)

Similar to the quasi-steady lift, after time-averaging the wake-induced lift
only terms associated with ${\Gamma^{*}}^{2}_{0,\infty}$ remain, which gives,

$\displaystyle\overline{C}_{L_{w}}=c_{2}\phi_{2}+c_{3}\phi_{3},$ (31)
$\displaystyle\phi_{2}=-\gamma^{2}St^{2}\left(1+\frac{1}{\pi^{2}k^{2}}\right)\left(\frac{D^{*}}{b^{2}+4{D^{*}}^{2}}\right)\left(\frac{k^{*}}{k^{*}+1}\right),$
(32)
$\displaystyle\phi_{3}=-\gamma^{2}St^{2}\left(1+\frac{1}{\pi^{2}k^{2}}\right)\left[\frac{16k^{2}D^{*}}{(1+4k)^{2}+64k^{2}{D^{*}}^{2}}\right]\left(\frac{k^{*}}{k^{*}+1}\right)^{2}.$
(33)

By using the wake-induced lift data from Figure 4, the coefficients were
determined by linear regression to be $c_{2}=-136.62$ and $c_{3}=346.27$.
Figure 11d shows that a good collapse of the data can be achieved with the
exception of data with $St\geq 0.45$ (outlined in red). As shown in Figure 7a,
the wake-induced lift is significantly altered by a wake instability effect
that occurs when $St\geq 0.45$ even for out-of-ground effect foils. Since this
wake instability effect is not accounted for in the scaling laws, the data
points with $St\geq 0.45$ (Figure 11d) begin to show deviation from the
scaling law prediction.

####  3.6.3 Total lift

Now, the quasi-steady lift and wake-induced lift can be summed to acquire the
total lift, and the total lift becomes a function of $St$, $k$ and $D^{*}$,

$\displaystyle\overline{C}_{L}=c_{1}\phi_{1}+c_{2}\phi_{2}+c_{3}\phi_{3}=f\left(St,k,D^{*})\right.$
(34)

Then for a given pair of $St$ and $k$, the equilibrium altitude $D^{*}_{eq}$
can be determined by setting Equation (34) to zero. Figure 11e presents the
scaling law prediction for the equilibrium altitude. The scaling law can
predict the equilibrium altitude to within $20\%$ error except for the data
with $St\geq 0.45$. The deviation in this prediction at $St\geq 0.45$ is due
to neglecting the wake instability effect in the wake-induced lift scaling
law. For the numerical data with $St<0.45$ a $5\%$ and $10\%$ error in
predicting the quasi-steady and wake-induced lift, respectively, leads to a
$20\%$ error in predicting the equilibrium altitude. This indicates that the
equilibrium altitude relies on a delicate balance and is sensitive to small
changes in the lift components. This is also observed by Liu et al. (2023) who
found that a pitch bias angle equal to $7\%$ of the amplitude can shift the
equilibrium altitude of a pitching foil by $28\%$.

##  4 Conclusions

In this article we decomposed the lift of a near-ground pitching hydrofoil
into its added mass, wake-induced, and quasi-steady components, and determined
that their time-averaged values are always zero, positive, and negative,
respectively, across all ground proximities. This shows that both stable and
unstable equilibrium altitudes are generated by a balance between positive
wake-induced lift and negative quasi-steady lift while the added mass lift
doesn’t play a role.

Using both simulations and experiments detailed analyses are provided to
illustrate the three lift components’ near-ground behavior. In ground effect
the negative peak of the added mass lift is amplified more than the positive
peak, however, the negative peak also becomes sharper while the positive peak
becomes broader compared to a foil out of ground effect. This leads to a
scallop theorem result where the added mass lift precisely integrates to zero
over one oscillation cycle. Through a control volume analysis, the wake-
induced lift is discovered to be positive due to the positive lift from the
pressure difference on the control volume outweighing the negative lift from
the net momentum flux. This leads to the upwards deflected wake behind a near-
ground swimmer being responsible for a positive wake-induced lift. The quasi-
steady lift is further analyzed through the lens of the Kutta-Jowkouski
theorem. It’s observed that the velocity induced by the positive (negative)
quasi-steady circulation bounded to the ground image foil enhances (reduces)
the effective flow velocity at the foil’s leading edge. For a pitching foil,
this amplifies the negative quasi-steady lift during the upstroke and reduces
the positive quasi-steady lift during the downstroke leading to a net negative
quasi-steady lift. Additionally, the lift decomposition was applied to three-
dimensional pitching foil simulations of varying aspect ratio. It was
demonstrated that equilibrium altitudes of $3D$ hydrofoils are also generated
by a balance between positive wake-induced lift and negative quasi-steady
lift. Moreover, as the aspect ratio decreases, the magnitude of the wake-
induced lift drops off faster than the magnitude of the quasi-steady lift
leading to the disappearance of an equilibrium altitude when ${\mar A}<1.5$.

Using these insights, scaling laws of the quasi-steady lift, wake-induced lift
and equilibrium altitudes are developed. Additionally, a simple scaling law
for the lift of a steady foil in ground effect is discovered. The scaling laws
show good agreement to both numerical and experimental data by predicting the
equilibrium altitudes to within $20\%$ of their value as long as $St<0.45$.
For $St\geq 0.45$ significant effects from wake instability, not accounted for
in the scaling laws, arise that alter the wake-induced lift. These results not
only provide key physical insights and scaling laws for unsteady ground
effect, but also for two schooling hydrofoils in a side-by-side formation with
an out-of-phase synchronization.

##  5 Acknowledgments

This work was supported by the National Science Foundation under Program
Director Dr. R. Joslin in Fluid Dynamics, award number 1921809 as well as by
the Office of Naval Research under Program Director Dr. Robert Brizzolara on
MURI grant number N00014-22-1-2616.

##  6 Declaration of interests

The authors report no conflict of interest

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