# On the noise generation and unsteady performance of combined heaving and
pitching foils

Nathan Wagenhoffer<EMAIL_ADDRESS>Keith W Moored  Justin W Jaworski
Engineering, Pennsylvania State University Abington  Mechanical Engineering
and Mechanics, Lehigh University

###### Abstract

A transient two-dimensional acoustic boundary element solver is coupled to a
potential flow boundary element solver via Powell’s acoustic analogy to
determine the acoustic emission of isolated hydrofoils performing
biologically-inspired motions. The flow-acoustic boundary element framework is
validated against experimental and asymptotic solutions for the noise produced
by canonical vortex-body interactions. The numerical framework then
characterizes the noise production of an oscillating foil, which is a simple
representation of a fish caudal fin. A rigid NACA 0012 hydrofoil is subjected
to combined heaving and pitching motions for Strouhal numbers ($0.03<St<1$)
based on peak-to-peak amplitudes and chord-based reduced frequencies
($0.125<f^{*}<1$) that span the parameter space of many swimming fish species.
A dipolar acoustic directivity is found for all motions, frequencies, and
amplitudes considered, and the peak noise level increases with both the
reduced frequency and the Strouhal number. A combined heaving and pitching
motion produces less noise than either a purely pitching or purely heaving
foil at a fixed reduced frequency and amplitude of motion. Correlations of the
lift and power coefficients with the peak root-mean-square acoustic pressure
levels are determined, which could be utilized to develop long-range, quiet
swimmers.

##  1 Introduction

Many aquatic animals oscillate their fins to propel themselves quickly through
the ocean. Their propulsion is based on an unsteady flow paradigm that is
distinct from the steady flow paradigm that underpins the design of most man-
made underwater vehicles. Consequently, animals can excel at maneuvering and
rapid accelerations [1, 2] in addition to high-speed and high-efficiency
swimming [3, 4]. This multi-modal performance has spurred vigorous research
over the last two decades into bio-inspired autonomous underwater vehicles
(BAUVs) [5, 6, 7, 8], which would possess the additional benefit of
exceptional stealth [9, 10]. It is generally expected that fish-like swimming
motions lead to quiet acoustic signatures [9], which (even if detected) would
resemble the noise signature of real fish, making a BAUV exceptionally
difficult to detect and identify [11]. However, to date the noise signatures
of swimming animals and BAUVs have not been adequately quantified, nor have
the trade-offs between the noise signature and performance of unsteady
swimmers been examined. The competition between acoustic stealth and fluid
dynamic performance also emerges for biologically-inspired aerial vehicles
[12, 13, 14, 15] that seek novel means to suppress flow-noise generation.

Most sounds produced by fish result from aggressive actions, spawning, or
reproductive behavior [16]. The noise made during these actions can be
categorized into two broad categories: active acoustic signaling through
morphological structures and passive noise generated during swimming, feeding,
or respiration. The actively-produced noises include vocal calls, swimbladder
motions, and drumming [17, 18]. Active fish signaling is not of interest here,
as there exist quantified data for several species [17]. However, fish produce
a vortex wake by simply oscillating their fins during swimming, which is a
well-known source of noise [16]. Yet, the locomotive noise of fish during
steady-state rectilinear swimming remains relatively unquantified. Fish
locomotion produces low-amplitude hydrodynamic noise that is challenging to
record reliably and has received limited attention in the literature [16].

Instead of examining live fish, numerical tools can be used to simulate the
motions of fish and their resulting unsteady vortical wakes to estimate their
noise production. For instance, one approach to simulating the unsteady flow
field around a swimming fish is the boundary element method (BEM) [19]. This
potential flow method discretizes surfaces representing shear layers in the
flow, i.e. a solid body surface and the wake, into a collection of elements
[20, 21]. This method approximates the flow as incompressible, inviscid, and
irrotational (except on the elements). Additionally, the reduction of the
solution domain to solve only for the strengths of elements on the boundaries
enables the rapid simulation of unsteady flow phenomena. Acoustic BEMs have
been used to determine the far-field noise of an airfoil in a turbulent flow,
such as by Glegg and Devenport [22]. Their method predicts the frequency-
domain acoustic far field based on a form of the surface loading due to the
energy spectrum of the turbulent boundary layer immediately upstream of the
trailing edge. However, this method is restricted to single airfoils in steady
flows and does not account for interactions that would be present in many-
bodied systems of swimmers or fliers.

Once an unsteady flow field is simulated, an acoustic analogy is a common and
effective tool to post-process the flow data and determine the noise
production [23, 24]. Lighthill [25] first developed an acoustic analogy by
recasting the Navier-Stokes equations into a wave equation in terms of
perturbations of the fluid density. A nearly ubiquitous approach to predicting
flow noise, Lighthill’s acoustic analogy has become central to many
aeroacoustic studies [23, 24, 26, 27]. For example, Powell [28] adapted the
Lighthill analogy to consider all of the unsteady flow fluctuations as
vorticity and designated this vorticity as the forcing function for the
acoustic system. Vorticity also plays a central role in the noise theory by
Howe [29] that is adapted from Lighthill’s formulation.

Motivated by these observations, the present study makes two principal
technical advancements. First, a coupled flow-acoustic BEM is developed to
predict the noise generation and performance of hydrofoils in unsteady motion
that can also account for multi-body interactions, such as those that occur in
fish schools. Second, the novel flow-acoustic BEM will be used to examine the
performance and noise production of a hydrofoil in combined heaving and
pitching motions, which acts as a simple, two-dimensional representation of an
oscillating fish caudal fin. This paper is laid out across three main
sections. The first section describes the potential flow and acoustic boundary
element methods used in the current study. Next, the coupling between the
potential flow and acoustic solvers is presented along with validations of the
solver against available experimental data and asymptotic solutions of vortex-
body interaction noise. Finally, the performance and noise production of a
combined heaving and pitching hydrofoil are examined with respect to non-
dimensional frequency, amplitude, and heave-to-pitch ratio.

##  2 Potential Flow Boundary Element Method

This section details the two-dimensional unsteady boundary element method used
in this work. The potential flow solver is an adaptation of the panel method
described by Willis et al. [30]. The inviscid flow around hydrofoils can be
found by solving the Laplace equation with an imposed no-penetration boundary
condition on the surface,

$\nabla\phi\cdot\mathbf{\hat{n}}=0~{}~{}{\rm on}~{}~{}S_{\rm{b}},$ (1)

where $\mathbf{\hat{n}}$ is the outward unit normal of the surface. The
boundary integral equation integrates the effects of the combined distribution
of sources and doublets on the body surface $S_{\rm{b}}$ and doublets on edge
panel $S_{\rm{e}}$, with vortex particles in the wake $S_{\rm{w}}$ (cf. Fig.
1). The scalar potential may be written as

$\phi(\mathbf{t})=\int_{S_{\rm{b}}}\left[\sigma(\mathbf{s})G(\mathbf{t},\mathbf{s})-\mu(\mathbf{s})\hat{\mathbf{n}}\cdot\nabla
G(\mathbf{t},\mathbf{s})\right]\rm{d}S-\int_{S_{\rm{e}}}\mu_{\rm
e}(\mathbf{s})\hat{\mathbf{n}}\cdot\nabla G(\mathbf{t},\mathbf{s})\rm{d}S,$
(2)

where $\mathbf{s}$ is a source location, $\mathbf{t}$ is the observer
location, and
$G(\mathbf{t},\mathbf{s})=\frac{1}{2\pi}\ln|\mathbf{t}-\mathbf{s}|$ is the
two-dimensional Green’s function for the Laplace equation. The source and
doublet strengths are defined respectively as

$\displaystyle\sigma=\hat{\mathbf{n}}\cdot(\mathbf{U}+\mathbf{U}_{\rm{rel}}-\mathbf{U}_{\omega}),$
(3) $\displaystyle\mu=\phi_{I}-\phi,$ (4)

where $\mathbf{U}_{\omega}$ is velocity induced by the vortex particles in the
field, $\mathbf{U}$ is the body velocity, $\mathbf{U}_{\rm rel}$ is the
velocity of the center of each element relative to the body-frame of
reference, and $\phi_{I}$ is the interior potential of the body. At each time-
step, vorticity is defined at the trailing-edge to satisfy the Kutta
condition. The trailing edge panel is assigned the potential difference
between the upper and lower panels at the trailing edge of the foil, $\mu_{\rm
e}=\mu_{\rm{upper}}-\mu_{\rm{lower}}$. Application of the Kutta condition
ensures that the vorticity at the trailing edge of the hydrofoil is zero. An
implicit Kutta condition for the two-dimensional flow solver is similar to the
methods mentioned in [30, 31]. The iterative implicit Kutta condition employs
a Newton’s method to define the length and angle of the trailing edge panel in
order to minimize the pressure across the trailing edge. The evolution of
vorticity in the domain is governed by

$\frac{\partial\boldsymbol{\omega}}{\partial
t}+\mathbf{U}\cdot\nabla\boldsymbol{\omega}=\boldsymbol{\omega}\cdot\nabla\mathbf{U},$
(5)

where the vorticity field $\boldsymbol{\omega}$ is represented by discrete,
radially-symmetric, desingularized Gaussian vortex particles. The induced
velocity of the vortex blobs is determined by the Biot-Savart law [32],

$\mathbf{u}(\mathbf{t},t)=\sum_{i=1}^{N}\frac{\Gamma_{i}}{2\pi}\left(1-\exp\left(\frac{-|\mathbf{t-s}|}{2{r_{\rm{cut}}}^{2}}\right)\right),\\\
$ (6)

where $\Gamma_{i}$ is the circulation of the $i$th vortex particle, and
$r_{\rm{cut}}$ is the cut-off radius. Following the work of Pan et al. [33],
the cut-off radius is set to $r_{\rm{cut}}=1.3\Delta t$ for time step $\Delta
t$ to ensure that the wake particle cores overlap and a thin vortex sheet is
shed. The evolution of the vortex particle position is updated using a forward
Euler scheme [30],

$\mathbf{x}(t+1)=\mathbf{x}(t)+\mathbf{u}(\mathbf{x}(t),t)\Delta t.$ (7)

The use of discrete vortices to represent the wake instead of panels alters
the process of shedding vorticity into the wake in comparison to more
classical source-doublet methods [21]. Two edge panels are set behind a foil.
The first edge panel, set with the empirical length of
$l_{\rm{panel}}=0.4U_{\infty}\Delta t$ [21], satisfies the Kutta condition at
the trailing edge. Next, the buffer panel is attached to the edge panel and
stores information about the previous time step. Figure 1 illustrates the
distinction of the source/doublet panels across a foil, the arrangement of the
trailing-edge and buffer panels, and how the vortex particle wake behaves
behind the body.

Figure 1: Schematic of a fish with a blown up section depicting the propulsor
as the discrete geometry for the BEM. A propulsor of chord length $c$ is
discretized with source/doublet elements (blue lines with endpoints indicated
by black circles). The edge and buffer panels (green lines with endpoints
indicated by green circles) are connected to the trailing edge of the
propulsor. The vortex particle wake (blue/red circles) is shed from the
edge/buffer panels.

The vortex particles influence on the body is accounted for in the definition
of the source strength $\sigma$. The vortex particle induced velocity also
augments the pressure calculation put forth by Katz and Plotkin [21]. The
surface pressure is determined by

$\frac{P_{\infty}-P(x)}{\rho}=\left.\frac{\partial\phi_{\rm{wake}}}{\partial
t}\right|_{\rm{body}}+\left.\frac{\partial\phi}{\partial
t}\right|_{\rm{body}}-(\mathbf{U}+\mathbf{u}_{\rm{}_{rel}})\cdot(\nabla\phi+\mathbf{U}_{\omega}))+\frac{1}{2}|\nabla\phi+\mathbf{U}_{\omega})|^{2},$
(8)

where $\partial\phi_{\rm{wake}}/\partial t=\Gamma\dot{\theta}/(2\pi)$ is the
time rate of change due to a vortex particle with circulation $\Gamma$ at an
angle $\theta$ from the observation point. Equation (8) is similar to the form
put forth by Willis et al. [30], but here $\partial\phi_{\rm{wake}}/\partial
t$ is the positional change of a vortex with respect to a panel and does not
require the solution of a secondary system to find the influence of wake
vortices onto the body surface. The appendix presents validation of this
methodology against available theoretical and numerical results.

##  3 Acoustic Boundary Element Method

The Helmholtz wave equation for a homogeneous medium and its boundary
conditions are

$\displaystyle\nabla^{2}\phi+\kappa^{2}\phi=0,$ (9)
$\displaystyle\phi(\mathbf{x})=d_{n}~{}~{}{\rm on}~{}~{}S_{\rm{b}},$ (10)
$\displaystyle\frac{\partial\phi}{\partial n}(\mathbf{x})=g_{n}~{}~{}{\rm
on}~{}~{}S_{\rm{b}},$ (11)

where $\phi$ is the time-independent velocity potential, $\kappa=2\pi f/c_{0}$
is the acoustic wavenumber, $c_{0}$ is the speed of sound, $d_{n}$ is a
Dirichlet boundary condition, and $g_{n}$ is a Neumann boundary condition.

Application of Green’s second identity to Eq. (9) moves all of the information
of the system onto the boundary:

$a(\mathbf{t})\phi(\mathbf{t})=\int_{S_{\rm{b}}}\left[\frac{\partial
G(\mathbf{t},\mathbf{s})}{\partial
n}\phi(\mathbf{s})-G(\mathbf{t},\mathbf{s})\frac{\partial\phi}{\partial
n}(\mathbf{s})\right]\rm{d}S_{\rm{b}},$ (12)

where $a(\mathbf{t})=\frac{1}{2}$ on the boundary and $a(\mathbf{t})=1$ in the
exterior field. The two-dimensional acoustic Green’s functions are

$\displaystyle
G(\kappa,\mathbf{t},\mathbf{s})=\frac{\textrm{i}H_{0}^{(1)}(\kappa|\mathbf{t}-\mathbf{s}|)}{4},$
(13) $\displaystyle\frac{\partial G}{\partial
n}(\kappa,\mathbf{t},\mathbf{s})=\frac{-\textrm{i}\kappa
H_{1}^{(1)}(\kappa|\mathbf{t}-\mathbf{s}|)}{4},$ (14)

which correspond respectively to an acoustic monopole and dipole, and
$H_{n}^{(1)}$ is the Hankel function of the first kind of order $n$. In the
remainder of this work,
$G(\kappa,\mathbf{t},\mathbf{s})=G(\mathbf{t},\mathbf{s})$, as the wavenumber
is constant for each solution.

Differentiation of Eq. (12) with respect to the outward normal of the boundary
produces a quadrupole system,

$\frac{\partial\phi(\mathbf{t})}{\partial
n}=\int_{S_{\rm{b}}}\frac{\partial^{2}G(\mathbf{t},\mathbf{s})}{\partial
n_{\mathbf{t}}\partial n_{\mathbf{s}}}\phi(\mathbf{s})\rm{d}S_{\rm{b}},$ (15)

where $n_{\mathbf{s}}$ and $n_{\mathbf{t}}$ are the outward normals at the
source and observer, respectively.

A combination of Eqs. (12) and (15) for points on the boundary arrives at the
Burton-Miller formulation [27, 34]:

$\int_{S_{\rm{b}}}\left(\frac{\partial G(\mathbf{t},\mathbf{s})}{\partial
n}\phi(\mathbf{s})+\frac{1}{2}\phi(\mathbf{s})\right)\rm{d}S_{\rm{b}}+\beta\int_{S_{\rm{b}}}\frac{\partial^{2}G(\mathbf{t},\mathbf{s})}{\partial
n_{\mathbf{t}}\partial
n_{\mathbf{s}}}\phi(\mathbf{s})\rm{d}S_{\rm{b}}=\phi(x)+\beta\frac{\partial\phi(\mathbf{t})}{\partial
n},$ (16)

where $\beta=\textrm{i}/\kappa$ is a chosen coupling parameter [27]. The
frequency domain problem (16) is converted into a transient solution by the
application of the convolution quadrature method, which is detailed in the
next section.

###  3.1 Time discretization

The frequency potential operators in Eq. (16) are evaluated as convolution
integrals. Green’s functions in the frequency domain problem are Laplace
transforms of the retarded-time Green’s function, which permit their
convolution with the potential field. The potential field is evaluated by a
convolution quadrature. This methodology of time discretization can be
achieved via a convolution quadrature method put forth by Lubich [35]. A
representative example of a convolution system is

$\displaystyle\int_{0}^{t}f(t-\tau)\phi(\tau)d\tau=g(t).$

Here $f$ represents a retarded-time operator, a characteristic differential
operator of the transient wave equation, $\phi$ is some known potential
distribution, and $g(t)$ is a transient forcing function. The interested
reader may consult Hassell and Sayas [36] for a detailed explanation of the
convolution quadrature method. The retarded-time operator is a convolution
that can be discretized by splitting the time domain into $N+1$ time steps of
equal spacing, yielding $\Delta t=T/N$ and $t_{n}=n\Delta t$ for
$n=[0,1,...,N]$. The discrete convolution can be viewed as a sum of weights of
the $F$ operator at discrete times of $\phi$:

$\displaystyle F\frac{\partial\Phi(t_{n})}{\partial t}^{\Delta
t}=\sum_{j=0}^{n}w_{n-j}^{\Delta t}(F)\phi^{\Delta t}(t_{j}),$ (17)

where $F$ represents the Laplace transform of the $f$ operator, and the
superscript $\Delta t$ indicates the weight for a specific time-step size. The
series expansion can be arranged to solve for the convolution weights, $w$:

$\displaystyle F\left(\frac{\gamma(\zeta)}{\Delta
t}\right)=\sum_{n=0}^{\infty}w_{n-j}^{\Delta t}\zeta^{n},\quad|\zeta|<1,$ (18)
$\displaystyle w_{n-j}^{\Delta t}=\frac{1}{2\pi
i}\oint_{C}\frac{F(\frac{\gamma(\zeta)}{\Delta t})}{\zeta^{j+1}}d\zeta,$ (19)

where $C$ is a circle of radius $0<\lambda<1$ centered at the origin. A
second-order backwards difference function,
$\gamma(\zeta)=(1-\zeta)+\frac{1}{2}(1-\zeta)^{2}$, is used to define the
spacing of the integration. A review of other integration methods that can be
incorporated into the convolution quadrature method is presented in Hassell
and Sayas [36]. Employing a scaled inverse transform, the weights become

$\displaystyle w_{n-j}^{\Delta
t,\lambda}(F)=\frac{\lambda^{-j}}{N+1}\sum_{l=0}^{N}F(s_{l})\zeta_{N+1}^{lj},$
(20)

where $\zeta_{N+1}=\exp\left(\frac{2\pi i}{N+1}\right)$ is the discrete
Fourier transform scale, and $s_{l}=\gamma(\lambda\zeta_{N+1}^{-l})/\Delta t$
is the accompanying time-dependent complex wavenumber. The value of $s_{l}$ is
different for each time step and provides the link between the frequency-
domain solver and a transient boundary integral equation. For this
formulation, $\lambda=\Delta t^{3/N}$ is selected based on the error analysis
of Banjai and Sauter [37].

Placing (20) into the boundary value problem (16) yields a system of $N+1$
equations,

$\displaystyle\frac{\lambda^{-j}}{N+1}\sum_{l=0}^{N}F(s_{l},\mathbf{x})\hat{\phi_{l}}(\mathbf{x})\zeta_{N+1}^{lj}=\frac{\lambda^{-j}}{N+1}\sum_{l=0}^{N}\hat{g_{l}}\zeta^{lj}_{N=1},$
(21)

where $F$ is the linear combination of operators on the left hand side of Eq.
(16). Here $g_{n}$ is a discrete representation of the mixed boundary
condition. The inverse discrete Fourier transform of the convolution is

$\displaystyle\hat{\phi_{l}}=\sum_{j=0}^{N}\lambda^{j}\phi_{j}^{\lambda}\xi_{N+1}^{-lj},$
(22)

which produces the transient solution.

In summary, the convolution quadrature method [37] discretizes a transient
wave problem into a system of frequency-domain (Helmholtz) wave equations that
are uncoupled in time. This discretization allows $N+1$ independent solutions
of Eq. (16) in the frequency domain using wavenumbers $s_{l}$ that are
generated via the convolution quadrature method. The time-domain solution is
recovered by applying the inverse Fourier transform (22).

##  4 Acoustic Analogies

The Lighthill acoustic analogy [25] is an exact rearrangement of the Navier-
Stokes equation, where the resulting wave equation is forced by the so-called
Lighthill tensor, $T_{ij}$. For example, $T_{ij}$ may be used as a forcing
function in Eq. (9). The flow is assumed inviscid, without thermal losses, and
at low Mach number $(M^{2}\ll 1)$. The form of the Lighthill tensor under
these conditions is reduced to the Reynolds stress contribution,
$T_{ij}\approx\rho u_{i}u_{j}$ [38]. Taking two spatial derivatives of this
tensor yields the quadrupole source for Lighthill’s acoustic analogy. This
quadrupole is related to the vorticity field by

$\frac{\partial^{2}u_{i}u_{j}}{\partial x_{i}\partial
x_{j}}=\nabla\cdot(\boldsymbol{\omega}\times\mathbf{u})+\nabla^{2}\left(\frac{1}{2}u^{2}\right),$
(23)

where $u^{2}=\mathbf{u}\cdot\mathbf{u}$. The Powell acoustic analogy, a
derivative of the Lighthill analogy, uses this form of the velocity field as
its forcing function [28].

The Powell acoustic analogy allows the direct relation of the vorticity
defined by the flow solver to be the forcing function of the acoustic solver.
The present flow solver defines the vorticity that is bound to the body and
shed into the wake. Other vortical noise sources, such as the broadband
content in a turbulent boundary layer, may also be incorporated but are
neglected in the present work to focus on the acoustic interactions involving
incident and shed vorticity. The Powell acoustic analogy states that in free
space the forcing of the wave equation is a function of the vorticity in the
field,

$\displaystyle\partial_{t}^{2}P-\nabla^{2}P=\nabla\cdot(\boldsymbol{\omega}\times\mathbf{u}),$
(24) $\displaystyle\frac{\partial G}{\partial n}=0\quad{\rm on}\quad
S_{\rm{b}}.$ (25)

Using a Green’s function solution, and applying an integration by parts, the
pressure in the field is determined by

$P(\mathbf{t},t)=\rho\int_{S_{\rm{b}}}(\boldsymbol{\omega}\times\mathbf{v})\cdot\frac{\partial
G}{\partial n_{\mathbf{s}}}\rm{d}S_{\rm{b}},$ (26)

where $\partial G/\partial n_{\mathbf{s}}$ is the two-dimensional potential
flow Green’s function. The pressure integral in Eq. (26) is applicable
regardless of whether or not a solid body is present [39]. The use of the
potential flow Green’s function imposes instantaneously the vortical acoustic
loading onto a solid body, as in the asymptotic methods of Kambe and Kao [39,
40]. Also, the use of the flow Green’s function instead of the retarded
potential acoustic Green’s function [39], i.e.,
$G=\frac{H(t-|\mathbf{t}|/c_{0})}{2\pi\sqrt{t-|\mathbf{t}|/c_{0}}}$, removes
any singularities in time in addition to the slow decay tail created by the
Heaviside function found in the numerator. For all of the flow scenarios
observed in this work, the small products of the acoustic wavenumber with the
chord length (Helmholtz number) and with the distance between dominant wake
vortices with the foil in low-Mach-number locomotion ensures acoustically
compact vortex-body interactions [38]. Equation (26) provides the pressure at
an arbitrary point in space, for instance the collocation point of a discrete
foil. This approach is sufficient to solve a Dirichlet boundary condition, but
Eq. (25) is a no-flux Neumann boundary condition. The Neumann boundary
condition can be satisfied by arranging Euler’s equation to define the
acoustic dipole potential needed to guarantee no fluid flux through the
surface of the discrete geometry,

$\rho\frac{\partial\mathbf{u}}{\partial t}=-\nabla P.$ (27)

In Eq. (27) the velocity is taken to be the normal induced velocity from each
discrete vortex in the domain, including the discrete vortex particles that
comprise the wake and vortex values distributed along the foil found via Eq.
(2). A rearrangement of Eq. (27) that considers the outward normal pressure on
the foil results in

$\frac{\partial P}{\partial
n}(\mathbf{t})=-\rho\sum_{i=0}^{N}\frac{\partial(\mathbf{u}_{i}\cdot\hat{\mathbf{n}}(\mathbf{t}))}{\partial
t}.$ (28)

The pressure in Eq. (26) and its normal derivative in Eq. (28) are the
boundary conditions to the Burton-Miller acoustic BEM formulation in Eq. (16).

###  4.1 Validation of Acoustic Analogy for Vortex-Body Interactions

The Powell acoustic analogy is now validated as a suitable forcing function
for the one-way coupled flow-acoustic BEM. First, the canonical problem by
Crighton [41] whereby a line vortex generates noise as it passes round a half-
plane is used to verify the flow-acoustic BEM for edge scattering. Given a
point vortex that advects around a half-plane lying at $y=0$ with the edge
located at the origin, the vortex path is described analytically by

$r=\ell\sqrt{1+\left(\frac{Ut}{\ell}\right)^{2}},\quad~{}\quad\theta=2\tan^{-1}\left(-\frac{Ut}{\ell}\right),$

where $r$ and $\theta$ are the radial components of the vortex position. The
characteristic vortex velocity is $U=\Gamma/8\pi\ell$, with $\ell$ being the
distance of closest approach of the vortex to the trailing edge. Howe [38]
(see also [29]) determined the time-varying magnitude of the acoustic pressure
produced by the vortex motion near the edge using an approximate Green’s
function for a half plane with the observer in the acoustic far field:

$\displaystyle P(\mathbf{t},t)$
$\displaystyle\approx\frac{\rho\Gamma^{2}}{4\pi\ell^{2}}\left(\frac{\ell}{|\mathbf{t}|}\right)^{\frac{1}{2}}\sin\left(\frac{\theta}{2}\right)\left[\frac{\Gamma
t/8\pi\ell^{2}}{[1+(\Gamma
t/8\pi\ell^{2})^{2}]^{5/4}}\right]_{t-\frac{|\mathbf{t}|}{c_{0}}},~{}\mathbf{|t|}$
$\displaystyle\rightarrow{}\infty.$ (29)

The brackets indicate evaluation at the retarded time. Note that this model
neglects any vortex shedding by the edge.

Figure 2 illustrates the vortex path near the half plane and presents a
comparison of the analytical acoustic response from Eq. (29) against the
numerical results from the BEM developed in §§2 and 3. The solution was
created by approximating a half plane with a 10 m flat plate and passing a
vortex of strength $\Gamma$ = 1 m2 s-1 to within a distance of $\ell$ = 0.25 m
at the trailing edge in a medium with density $\rho$ = 1 kg m-3. The observer
location is $|\mathbf{t}|$ = 50 m above the trailing edge. The half plane was
approximated with a 10 m flat plate discretized with 512 boundary elements in
a cosine distribution which ensures a denser concentration of elements near
the edge. The doubling of elements on the body from 128 to 256 elements
produced less than 0.1% change in the acoustic solution as observed at
$|\mathbf{t}|$. Agreement of the acoustic responses are seen when the vortex
is near the trailing edge, i.e. $|Ut/\ell|$ ¡ 1, and the system begins to
diverge slightly outside of that range. The divergence of the solution outside
of this time is likely due to the limitation in the Howe solution that the
loading is only at the trailing edge, where as the BEM implicitly solves for
loading along the flat plate as the vortex continues to travel near the body.
The agreement between the BEM and Howe’s solution verifies the accuracy of the
acoustic portion of the BEM in modeling trailing edge scattering. The fully
coupled solver including the unsteady potential flow portion can be examined
further for vortex-body interaction cases.

Figure 2: Sound produced by vortex motion near a half plane. (a) Vortex path
round the trailing edge of a half plane. The closest position of the vortex to
the body $\ell$ occurs at $Ut/\ell=0.$ (b) Acoustic response computed by the
BEM (blue circles) and from Howe’s solution (red line).

The experimental work of Booth [42] details acoustic scattering due to vortex-
body interaction. The matched asymptotic method of Kao [40] uses this
experimental study to validate their analysis. Kao asymptotically matches the
acoustic loading from a potential flow solver to an outer far-field acoustic
solution. The selected vortex-body interaction problem sets a vortex upstream
of a NACA 0012 airfoil, with a chord of $c=0.2032$ m. The vortex has a
circulation of $\Gamma=0.52$ m2 s-1 and advects in a freestream of speed
$U_{\infty}=4.7$ m s-1 at a vertical displacement of $h=0.152c$ from the foil
centerline (cf. Fig. 3). This particular offset distance was selected because
at other distances in the experimental study the vortex impinges on the body
and breaks down. The fluid medium has a speed of sound of $c_{0}=343$ m s-1
and the density $\rho=1.225$ kg m-3. The acoustic response is then found in
front of the airfoil at $\mathbf{t}_{1}=(100c,0)$, as shown in the problem
schematic at the top of Fig. 3.

Figure 3: Acoustic emission due to a single vortex advecting past a NACA 0012
airfoil. The vortex circulation $\Gamma$, chord length $c$, observation point
$\mathbf{t}$, freestream velocity $U_{\infty}$, and offset heights $h$ are
different for each of the two validation cases. The response of $(a)$ is
observed 100 chord lengths in front of the foil $\mathbf{t}_{1}=(-100c,0)$,
while the response of $(b)$ is observed 50 chord lengths above the airfoil
$\mathbf{t}_{2}=(0,50c)$. The black lines represents the experimental results
of Booth [42], the red circle represents the matched asymptotic solution of
Kao [40], and the blue line is the result of the coupled potential flow and
transient acoustics BEM put forward in this work.

Figure 3$(a)$ compares the experimental vortex-body interaction sound results
from the work of Booth, the matched asymptotic method of Kao, and the one way
coupled potential flow and acoustic BEM presented in this study. The matched
asymptotic method and the flow-acoustic BEM have qualitatively similar
responses. The experimental acoustic response is the same order of magnitude
as the other methodologies with similar qualitative trends, albeit with more
fluctuations in its signal. The leading edge acoustic response occurs at
$t\approx 0.25$ s. It can be seen that the amplitude of this interaction is
quantitatively similar for the theoretical and BEM approaches, while
qualitatively the slope of the leading-edge response as it approaches the
minimum pressure from the flow-acoustic BEM is steeper than the matched
asymptotic method response. The trailing-edge acoustic response occurs at
$t\approx 0.30$ s, where the peak predicted response of $P\approx-0.0015$ Pa
from the matched asymptotic solution is stronger than the BEM prediction of
$P\approx-0.0005$ Pa. The increased pressure response at the trailing edge
from the matched asymptotic method can be affected by the explicit Kutta
condition applied and the manner in which the wake evolves behind the foil.
Howe [43] stated that, as a vortex passes the trailing edge of a body, the
vorticity shed into the wake tends to cancel the effect of the incoming
vorticity and mitigates the noise generation. The Kutta condition in the
potential flow BEM could be implicitly imposing the mechanism described by
Howe, which would help explain the weaker acoustic response predicted by the
flow-acoustic BEM framework. Additionally, the acoustic response in Fig.
3$(a)$ is measured in front of the foil, a location where the acoustic
pressure would be small in comparison to other measurement locations.

Further verification of the the flow-acoustic BEM is accomplished by measuring
the acoustic response where it has its maximum value. Figure 3$(b)$ presents
the measurement of the acoustic response above the foil where the peak
acoustic pressure occurs. The flow scenario has a vortex with circulation
$\Gamma=$ 0.1 m2 s-1 that is released five chord lengths upstream with
vertical offset $h=0.1c$ above the center of a NACA 0012 foil with chord $c=1$
m. The case has a freestream velocity $U_{\infty}=1$ m s-1, a sound speed of
$c_{0}=5$ m s-1, and density $\rho$ = 1 kg m-3. The acoustic response was
measured fifty chord lengths above the leading edge of the foil at
$\mathbf{t}_{2}=(0,50c)$. The matched asymptotic method predicts a leading-
edge acoustic response at $t\approx 0.25$ s, which occurs before the flow-
acoustic BEM response at $t\approx 0.35$ s. However, both approaches exhibit a
similar magnitude of response. The maximum acoustic responses of both systems
are found at $t\approx 0.45$ s with a pressure of $P=0.013$ Pa. The trailing-
edge responses at $t\approx 0.5$ s also exhibits similar magnitudes for both
approaches. The flow-acoustic BEM predicts the same order-of-magnitude
acoustic response as the experiments of Booth [42]. In addition, the flow-
acoustic BEM produces qualitatively and quantitatively similar acoustic
responses to the matched asymptotic method of Kao [40].

##  5 Acoustic Emission from Biological Swimming

Many fish swim by undulating their bodies and oscillating their caudal or tail
fins, which are in many cases responsible for the majority of their thrust
production [2, 44]. A common and simple representation of a biological swimmer
is to neglect the body and only consider the caudal fin as a combined heaving
and pitching hydrofoil [45], which is also the case in the present study.
Fish-like locomotion via a traveling wave is shown in Fig. 4$(a)$. The motion
of the rear of the foil is tracked and then treated as a discrete propulsive
foil, which is denoted in the figure as a solid black body. Figure 4$(b)$
shows how a combined heaving and pitching motion of a rigid body is used as a
proxy for the entire traveling wave system. The rigid body pitches about its
leading edge, as that is where it would be connected to the fish.

The heaving and pitching motion and the peak-to-peak amplitude of the foil are
described by

$\displaystyle h(t)=h_{0}\sin(2\pi ft),$ (30)
$\displaystyle\theta(t)=\theta_{0}\sin(2\pi ft+\phi),$ (31) $\displaystyle
A(t^{*})=2\,\text{max}\\{h(t)+c\,\sin\left[\theta(t)\right]\\},$ (32)
$\displaystyle h^{*}=2\,h(t^{*})/A(t^{*}),$ (33)
$\displaystyle\theta^{*}=2\,c\,\sin\left[\theta(t^{*})\right]/A(t^{*}),$ (34)
$\displaystyle A^{*}=A(t^{*})/c,$ (35)

where $f$ is the frequency of motion, $t$ is time, and $h_{0}$ and
$\theta_{0}$ are the maximum heaving amplitude and maximum pitching angle,
respectively. The phase delay between heave and pitch signals is $\phi=\pi/2$.
The peak-to-peak amplitude is $A(t^{*})$, and $t^{*}$ is the time at which the
foil reaches its peak amplitude. Given $A(t^{*})$ and $\theta_{0}$, $h_{0}$
can be calculated by a nonlinear equation solver. Normalizing $h(t^{*})$ and
$\theta(t^{*})$ by $A(t^{*})$ produces the identity

$h^{*}+\theta^{*}=1.$ (36)

Supposing a non-dimensional peak-to-peak amplitude $A^{*}$, the ratio of the
heaving and pitching amplitudes is solely described by the non-dimensional
heave-to-pitch ratio $h^{*}$. A purely pitching foil has a value of $h^{*}=0$,
a purely heaving foil has a value of $h^{*}=1$, and $h^{*}=0.5$ represents a
combined heaving and pitching motion where half of the total amplitude comes
from pitching and the other half comes from heaving. All other values in the
range $0<h^{*}<1$ represent combined heaving and pitching motions. The chord-
based reduced frequency $f^{*}=fc/U_{\infty}$ is the non-dimensional quantity
that describes the unsteadiness of the prescribed swimming motion.

Figure 5 shows a typical reverse von Kármán wake structure of a foil operating
at $h^{*}=0.5$, $f^{*}=0.5$, and $A^{*}=0.5$. The spacing of the wake is
dictated vertically by the amplitude of motion and horizontally by the
freestream speed and the frequency. The ratio of the vertical spacing to
horizontal spacing results in the Strouhal number, $St=fA(t^{*})/U_{\infty}$.
The example wake in Fig. 5 has a Strouhal number of $St=0.25,$ which is within
the range of typical fish swimming of $0.2<St<0.4$ [5, 46, 47]. The acoustic
pressure presented for the remainder of this work is non-dimensionalized by
dynamic pressure, $P^{*}=2P_{\rm{acoustic}}/\rho U_{\infty}^{2}.$

Figure 4: Use of a pitching/heaving foil as a proxy to undulatory locomotion.
Illustration $(a)$ shows a period of a traveling wave undulating across a NACA
0012 airfoil. The trailing edge of the foil is modeled as a separate entity
that acts as a proxy to the caudal fin of a fish. The schematic $(b)$ tracks
the motion of the ‘caudal fin’ separated from the body as a function of
pitching and heaving.  Figure 5: Typical wake of an unsteady swimmer in this
study. Wake of a foil after several cycles of motion for the values of
$h^{*}=f^{*}=A^{*}=0.5$. A foil of chord $c$ is placed in a freestream flow at
speed $U_{\infty}$. The vertical spacing of the vortices in the wake are
described as a function of the amplitude $A=c/2$ and the horizontal spacing is
a function of freestream speed and frequency, $2U_{\infty}/f$, for half of a
cycle of motion.

The flow-acoustic BEM is used to study how variations in the non-dimensional
amplitude, reduced frequency, and non-dimensional heave-to-pitch ratio alter
both the acoustic emissions and the hydrodynamic forces on the body. The
ranges of variables and parameters used in the current study are presented in
Table 1. The range of Strouhal numbers in the simulations is $0.0312\leq
St\leq 1$. The reduced frequencies and Strouhal numbers in the current study
cover the ranges associated with typical fish swimming [47, 5] and also extend
to regions where biological systems may perform fast starts or rapid turns
[2].

Input Variables: | $0.25\leq A^{*}\leq 1$ | $0\leq h^{*}\leq 1$ |
---|---|---|---
| $0.125\leq f^{*}\leq 1$ |  |
Input Parameters: |  $U_{\infty}=1$ m s-1 |  $\rho=1000$ kg m-3 |
|  $c_{0}=1000$ m s-1 |  $c=1$ m |
Table 1: Input variables and parameters used in the present study.

##  6 Results

Figure 6 presents the near-field transient acoustic pressure for a foil with
parameters of $h^{*}=0.5$, $f^{*}=0.5$, and $A^{*}=0.5$. The acoustic pressure
is determined at discrete points on circles with radii of two to five chords
away from the mid-chord of the foil at rest. A vertically-oriented pressure
dipole is generally observed. The position of the maximum acoustic pressure
shifts from the front to back as the effective angle of attack increases, as
seen in the snapshots from $t/T=0$ to $t/T=5/9$, where $T$ is the period of
motion. The sign change of the dipole strength at the middle of the period
($t/T=4/9$) corresponds to the change in effective angle of attack going from
negative to positive values. At time $t/T=4/9$, the transient acoustic
pressure field has a quadrupole shape with two of its lobes directed behind
the foil, which are an order of magnitude weaker than the response above and
below the foil.

Figure 6: Transient near-field acoustic pressure of a typical swimmer. The
non-dimensional near-field acoustic pressure $P$ is shown for a foil operating
at $h^{*}=0.5$, $f^{*}=0.5$, and $A^{*}=0.5$ at different instances of non-
dimensional time $t/T$. The acoustic pressure is found at discrete points set
around circles centered about the mid-chord when at rest. The circle radii
range from two to five chord lengths.

Figure 7a shows the acoustic response at a single observation point 50 chords
above the foil with a heave-to-pitch ratio of $h^{*}=0.375$, a reduced
frequency of $f^{*}=0.25$, and over the entire range of amplitudes used in the
current study. As expected, the pressure fluctuates harmonically in time with
the same frequency as the foil motion. For fixed heave-to-pitch ratio and
reduced frequency, the amplitude of the acoustic pressure response increases
with the amplitude of motion. A directivity plot of the root-mean-square (RMS)
pressure ($P_{\text{RMS}}$) over three motion cycles for various reduced
frequencies is shown in Fig. 7$(b)$. Regardless of the motion parameters
selected, a dipole directivity is always observed. Moreover, the peak acoustic
pressure can be observed to also increase with an increase in the reduced
frequency. Since all of the variables produce self-similar dipole acoustic
responses, the peak RMS acoustic pressure can be used as the single metric to
describe the acoustic field. The peak pressure can then be scaled by the
dynamic pressure:

$P^{*}_{\rm{Peak}}=\frac{P^{*}_{\text{RMS}}}{\frac{1}{2}\rho U^{2}}.$ (37)

Figure 7: Transient and root-mean-square (RMS) acoustic pressure levels of a
typical swimmer. $(a)$ Transient acoustic pressure response of a foil
undergoing a combined heaving/pitching motion with $h^{*}=0.375$,
$f^{*}=0.25$, and various amplitudes. The acoustic pressure is determined at
the position 50 chords above the leading-edge of the foil and shown for 3
cycles of motion. $(b)$ RMS acoustic pressure $P_{\text{RMS}}$ from a foil
undergoing a combined heaving/pitching motion with $h^{*}=0.8$, $A^{*}=0.5$,
and various reduced frequencies. The acoustic pressure response is computed on
a circle 50 chord lengths from the leading edge of the foil and is averaged
over 3 cycles of motion. The dipolar acoustic directivity is observed for all
kinematic parameters considered.

Figure 8 presents the non-dimensional peak acoustic pressure for a purely
pitching foil ($h^{*}=0$), a combined-motion foil with equal amplitude
contributions from pitching and heaving ($h^{*}=0.5$), and a purely heaving
foil ($h^{*}=1$). The global maximum in the peak acoustic pressure is found at
$h^{*}=1$, $f^{*}=1$, and $A^{*}=1$, which represents the upper bounds of all
of the parameters being explored. However, the minimum in the noise production
does not occur at the minima of the parameter set: the minimum noise occurs at
$h^{*}\approx 0.25$ for fixed values of amplitude and reduced frequency. A
purely heaving foil produces higher acoustic pressures than a purely pitching
foil, except for $f^{*}\lesssim 0.25$. The combined heaving and pitching foil
emits a weaker acoustic pressure signal than either purely heaving or pitching
foils for all combinations of reduced frequency and amplitude of motion.
Figure 8$(b)$ overlays the Strouhal number on the peak acoustic pressure,
showing that in general an increase in Strouhal number results in an increase
in the peak RMS acoustic pressure for a fixed swimming motion $h^{*}$, even
though the isolines of $St$ and $P^{*}_{\rm{peak}}$ are not precisely aligned.
Therefore, the noise level trends are not driven solely by changes in the
Strouhal number.

Figure 8: Peak acoustic pressure for $(a)$ a purely pitching foil ($h^{*}=0$),
$(b)$ a combined heaving/pitching foil ($h^{*}=0.5$), and $(c)$ a purely
heaving foil ($h^{*}=1$) as a function of reduced frequency and amplitude of
motion. Isolines of Strouhal number overlay the acoustic pressure contours in
$(b)$.

An increase in either the reduced frequency or the amplitude of motion will
result in an increase in acoustic pressure, but the motion type of the foil
can lead to lower values of acoustic pressure. In fact, a pressure minimum is
observed for a combined heaving and pitching motion for all combinations of
reduced frequency and amplitude examined in this study. Figure 9 presents a
map that shows the $h^{*}$ value leading to the lowest noise production for a
given $f^{*}$ and $A^{*}$. A purely heaving or pitching foil never produces
the lowest acoustic pressure. For $f^{*}\gtrsim 0.3$ the quietest swimming is
produced by pitch-dominated swimming motions ($h^{*}<0.5$). Only for low
reduced frequencies of $f^{*}\lesssim 0.3$ do heave-dominated motions
($h^{*}>0.5$) produce the quietest swimming, and this $h^{*}$ value is
independent of the amplitude. Lines of constant $St$ are also marked on the
figure, which denote the typical range of $St$ for swimming animals [46]. From
this map it can be seen that for swimming animals operating with low reduced
frequencies ($f^{*}<0.3$) their noise production is minimized if they utilize
heave-dominated swimming kinematics. Swimming animals with higher reduced
frequency ($f^{*}>0.3$) minimize their noise production if they utilize pitch-
dominated swimming kinematics.

Figure 9: Map showing for a given $f^{*}$ and $A^{*}$ which $h^{*}$ value
leads to the lowest noise production.

The potential flow solver also can solve for the associated performance
characteristics of these oscillating foils. The force acting on the foil is
defined by
$\mathbf{F}=\int_{S_{\rm{b}}}-(P_{\text{flow}}\hat{\mathbf{n}})\,dS$, where
$P_{\text{flow}}$ is the pressure from the flow solver as opposed to the
acoustic pressure, $\hat{\mathbf{n}}$ is the local outward normal vector and
$S_{b}$ is the foil surface. Since the potential flow method is inviscid, the
forces on the foil arise only from its external pressure distribution. The
power consumption of the oscillating motion is calculated as the negative
inner product of the force vector and velocity vector of each boundary
element, i.e.,
$P_{w}=-\int_{S_{b}}\mathbf{F}_{\rm{ele}}\cdot\mathbf{u}_{\rm{ele}}\,dS$. The
time-averaged coefficients of lift, thrust, and power may be defined as

$\displaystyle C_{L}=\frac{\overline{F_{y}}}{\frac{1}{2}\rho
U_{\infty}^{2}},\quad C_{T}=-\frac{\overline{F_{x}}}{\frac{1}{2}\rho
U_{\infty}^{2}},\quad C_{P}=\frac{\overline{P_{w}}}{\frac{1}{2}\rho
U_{\infty}^{3}},$ (38)

where $F_{x}$ and $F_{y}$ are the integrated streamwise and transverse
components of the force on the foil, respectively. The efficiency is also
defined as $\eta={C_{T}}/{C_{P}}$. In addition to the time-averaged
coefficient of lift, the maximum coefficient of lift will also prove to be a
useful metric and is defined by

$C_{L,\text{max}}=\frac{{\rm max}(\bar{F}_{y})}{\frac{1}{2}\rho
U_{\infty}^{2}}.$ (39)

Figure 10 presents a comparison of the peak RMS acoustic pressure
$P^{*}_{\rm{Peak}}$, the maximum coefficient of lift $C_{L,\text{max}}$ scaled
by a factor of $1000$ and the absolute value of the time-averaged coefficient
of lift $|C_{L}|$ scaled by a factor of $20$. These quantities are shown as
functions of the Strouhal number and non-dimensional heave-to-pitch ratio. The
lift metrics were scaled in order to make the plots the same order of
magnitude. It can be seen that the peak RMS acoustic pressure and the maximum
coefficient of lift follow the same trend for increasing $St$ and $h^{*}$. It
can also be observed that the peak RMS acoustic pressure does not follow the
same trend as the time-averaged coefficient of lift. Not surprisingly, the
maximum lift coefficient provides a better correlation with the peak RMS
acoustic pressure than the time-average coefficient of lift. In light of this
finding, the maximum coefficient of lift $C_{L,\text{max}}$ may be used as a
proxy metric for comparison of the relative acoustic emissions between
oscillating hydrofoils.

Figure 10: Comparison of lift metrics with peak acoustic pressure as a
function of $St$ and $h^{*}$. The plots show the peak RMS acoustic pressure
$P^{*}_{\rm{Peak}}$, maximum coefficient of lift scaled
$C_{L\text{max}}/1000$, and the absolute value of the time-averaged
coefficient of lift $|C_{L}|$ scaled by a factor of $20$. Subfigure $(a)$ is a
purely pitching foil, $h^{*}=0$, $(b)$ is a combined heaving and pitching
motion with $h^{*}=0.5$, and $(c)$ is a purely heaving foil, $h^{*}=1$.

Figure 11 presents a comparison among the coefficients of thrust and power,
efficiency, and the peak RMS acoustic pressure. The thrust production
increases with increasing $f^{*}$, $h^{*}$, and $A^{*}$. In figure 11, regions
of negative coefficient of thrust (and the corresponding regions of
coefficients of power, efficiency, and the peak RMS acoustic pressure) are
excluded from the contour plots shown. The power consumption also increases
with increasing $f^{*}$ and $A^{*}$; however, as $h^{*}$ increases the power
decreases to a minimum and then increases. This result indicates that combined
heaving and pitching motions use less power than purely pitching or heaving
motions for a fixed $f^{*}$ and $A^{*}$. The efficiency results show global
peaks around $h^{*}=0.85$, $f^{*}<0.2$ for all $A^{*}$. Since the
$St=f^{*}A^{*}$, the highest efficiencies occur for the lowest swimming
Strouhal numbers. Most fish swim with $0.2\leq St\leq 0.4$ [46] making it
difficult to reach the highest levels of efficiency, even with $A^{*}=1$. The
optimal $h^{*}$ to maximize efficiency will vary, depending upon the Strouhal
number of the particular swimming animal or biorobotic device.

Figure 11: Comparison of acoustic and hydrodynamic metrics. The rows
correspond to the coefficients of thrust and power, the efficiency, and the
peak acoustic pressure. The columns correspond to different $A^{*}$ values.
Each contour plot is presented as a function of $f^{*}$ and $h^{*}$.

For the first time, the noise of a biopropulsor and its performance can be
compared. The peak acoustic pressure can be observed to not follow trends of
the thrust or efficiency; however, the peak acoustic pressure does follow
similar trends with the power coefficient. This result can be explained by the
fact that the acoustic pressure is well-correlated with the maximum lift
coefficient and, consequently, with the power consumption [48]. Moreover, the
absolute value of coefficient of lift is greater than thrust for all
scenarios, further explaining the vertical acoustic dipole and the trend
between the peak RMS acoustic pressure and the maximum coefficient of lift.

These results highlight that when the power needed to move the foil is
minimized, then there is also a minimum amount of energy that can be converted
into noise, leading to the quietest acoustic signatures. In contrast, there is
a trade-off between operating at maximum propulsive efficiency and minimizing
the noise production. Furthermore, the thrust increases as the swimmer moves
from a pure pitching to a pure heaving swimming motion, which requires more
power and produces a louder acoustic signal.

##  7 Conclusion

An integrated, two-dimensional flow-acoustic boundary element solver is
developed to predict the noise generated by the vortical wake of rigid foils
in motion. The vortex-particle wake computed by the potential flow boundary
element solver furnishes the input for the transient acoustic boundary element
solver via Powell’s acoustic analogy. This one-way flow-acoustic coupling is
validated against experimental and analytical results for the acoustic
emission of a vortex gust encounter with an airfoil.

The coupled potential flow-acoustic method is used to investigate the
performance and acoustic emission of a heaving and pitching hydrofoil. The
hydrofoil is subjected to varying non-dimensional frequencies, amplitudes, and
heave-to-pitch ratios that encompass the parametric range of most swimming and
maneuvering animals. All combinations of these variables examined in this work
produce a similar dipole acoustic response, where the maximum sound pressure
levels occur directly above and below the foil. Foils in purely pitching or
purely heaving motions are found to be noisier than foils that operate with a
combined heaving and pitching motion. In fact, for fixed reduced frequency and
amplitude there exists an optimal heave-to-pitch ratio, $h^{*}$, that
minimizes the noise production. The numerical model indicates that most
swimming animals would minimize their noise production by using heave-
dominated swimming motions. As the reduced frequency increases past the regime
of swimming animals, a transition to pitch-dominated swimming motions minimize
the noise production. Moreover, the correlation between the maximum
coefficient of lift and the peak RMS acoustic pressure for all combinations of
reduced frequency, amplitude, and heave-to-pitch ratio corresponds to the
acoustic dipole response. Consequently, the trends in the coefficient of power
are well-correlated with the trends in the peak RMS acoustic pressure for
swimming motions with $h^{*}>0.25$. This result supports the conclusion that
swimming with low power consumption and a low acoustic signature can be
achieved together. In contrast, it is discovered that there is a trade-off
between swimming with high propulsive efficiency and a low acoustic signature.
These insights seek to further our understanding of swimming in nature and
could aid in the design of high-performance, quiet bio-inspired autonomous
underwater vehicles.

## Acknowledgments

The authors gratefully acknowledge financial support from the National Science
Foundation under grants 1805692 (JWJ) and 1653181 (KWM), the Office of Naval
Research under MURI grant N00014-08-1-0642 (KWM), and a Lehigh CORE grant
(JWJ, KWM).

## Appendix A

The potential flow boundary element method presented in this work is a two-
dimensional derivative of the three-dimensional method described by Willis et
al. [30]. A comparison of the BEM solution to analytic and numerical works is
performed to ensure accuracy of the method presented. Theodorsen [49] solved
analytically for the fluid-dynamic lift and moment acting on a flat-plate foil
undergoing harmonic pitching and heaving motions under the assumption of a
planar wake. Garrick [50] used these results to predict the time-averaged
thrust and efficiency of pitching and heaving motions, as well as trailing-
edge flap motions. These analytical results have recently been extended by
Jaworski [51] to also handle leading-edge flap actuation in addition to pitch,
heave, and trailing-edge flap motions, and the results from Theodorsen and
Garrick have previously been compared to computational fluid dynamic
simulations of rigid and deformable thin airfoils [52, 53].

The first validation case is against Garrick’s theory. In the numerical
simulations, a 2 %-thick tear drop foil is subjected to a purely pitching
motion of amplitude $\theta_{0}=3^{\circ}$ about the leading edge.

First, convergence studies on the number of boundary elements and time-steps
is found for a reduced frequency $f^{*}=1$. Figure 12 $(a)$ shows spatial and
$(b)$ temporal convergence for time-averaged coefficient of force. The inset
images detail the percent change (%$\Delta$) of the time-averaged value as the
number of elements or time steps per period of motion double. The spatial
convergence was conducted for a fixed temporal resolution of 150 time steps
per period. The inset of figure 12 $(a)$ shows the ${\it O}(1\%)$ difference
in the force when changing from 128 to 256 elements. The temporal convergence
study used a fix number of 256 body elements. It can be seen in figure 12
$(b)$ an ${\it O}(1\%)$ change in force when increasing from 128 to 256 time
steps per period of motion. All of the simulation results previously presented
used 150 time steps per period of motion and 256 boundary elements to define
the discrete body. Comparison to analytic solutions and convergence studies of
the acoustic BEM are presented in Appendix B. The acoustic results have been
previously conducted by Wagenhoffer et al. [11], showing the selected spatial
and temporal values of the potential flow solver will also result in converged
acoustic results.

Figure 12: Spatial and temporal convergence of potential flow BEM. (a) Plots
the time averaged coefficient of force as the number of boundary elements on
the body doubles for a fixed number of time steps.

Next, validation via comparison of Garrick’s theory to the solution of the
solver in §2 for varying reduced frequencies is detailed. Theodorsen defined
lift as

$C_{L}=\rho V^{2}c\sqrt{R^{2}+I^{2}}e^{\textrm{i}\omega t},$ (40)

and aerodynamic moment as,

$M=\frac{1}{2}\rho V^{2}c^{2}\sqrt{R^{2}+I^{2}}e^{\textrm{i}(\omega t+\phi)},$
(41)

where

$\displaystyle
R=\pi\theta_{0}\left\\{\frac{k^{2}}{2}\left(\frac{1}{8}+a^{2}\right)+\left(\frac{1}{2}+a\right)\left[F-\left(\frac{1}{2}-a\right)kG\right]\right\\},$
$\displaystyle
I=-\pi\theta_{0}\left\\{\frac{k}{2}\left(\frac{1}{2}-a\right)-\left(\frac{1}{2}+a\right)\left[G+\left(\frac{1}{2}-a\right)kF\right]\right\\},$
$\displaystyle\phi=\tan^{-1}\frac{I}{R}.$

The required power to sustain the foil motion is ${\rm Pow}=-M\dot{\theta}.$
The coefficient of thrust from Garrick [50] was corrected by Jones et al. [54]
to be

$C_{T}=\pi
k^{2}\theta_{0}^{2}\left[(F^{2}+G^{2})\left(\frac{1}{k^{2}}+\left(\frac{1}{2}-a\right)^{2}\right)-\left(\frac{1}{2}-a\right)\left(\frac{1}{2}-F\right)-\frac{F}{k^{2}}-\left(\frac{1}{2}+a\right)\frac{G}{k}\right],$
(42)

where $F$ and $G$ are the real and imaginary parts of the Theodorsen lift
deficiency function,
$C(k)=\textrm{i}H_{1}^{(1)}(k)/(H_{0}^{(1)}(k)+\textrm{i}H_{1}^{(1)}(k))$ and
$a$ is the position of the pivot point measured from the mid-chord in half-
chord intervals. Rotation about the leading edge corresponds to $a=-1$.

Figure 13 compares the potential flow BEM results against the theory of
Garrick for three different reduced frequencies that encapsulate the range of
reduced frequencies used for the study in §5. The figure shows the BEM
solution matching well to the thin-airfoil theory results over one cycle of
motion.

Figure 13: Comparison of BEM solution with solution of Garrick. Shown is a
comparison of the solution of §2 to the theory of Garrick [50] for a purely
pitching foil. From left to right are increasing values of reduced frequency,
$f^{*}=fc/U_{\infty}=[0.25,0.5,1]$. From top to bottom are comparisons of the
coefficient of lift $C_{L}$, coefficient of thrust $C_{T}$, and coefficient of
power $C_{P}$, respectively. In each plot the dashed blue line is the solution
of the potential flow method and the solid green line is the theory of Garrick
42.

The theory of Garrick is applicable to low-amplitude motion, and additional
validation against large amplitude motions is necessary to give confidence in
the potential flow solver. The work of Pan et al. [33] developed a boundary
element method to investigate leading-edge separation of heaving and pitching
foils, but that is not necessary for the work presented here as we assume that
the flow remains attached over the propulsive surface being studied. In the
work of Pan et al., thrust and efficiency are found for heaving and pitching
foils without leading-edge separation for a range of Strouhal numbers and
maximum angles of attack $\alpha_{\rm{max}}$. Figure 14 shows good agreement
between the method presented here and the work of Pan et al. for a heave-to-
chord ratio of $0.75$ and $\alpha_{\rm{max}}=35^{\circ}$ over $0.25<St<0.4$.

Figure 14: Comparison of the numerical results of the potential flow solver in
§2 with the numerical solutions by Pan et al. [33]. A combined heaving and
pitching motion with a heave-to-chord ratio of $h_{0}/c=0.75$ reaching maximum
angles of attack of $15^{\circ}$ (left) and $35^{\circ}$ (right) are performed
over Strouhal numbers ranging from 0.25 to 0.4. The results are compared with
respect to the coefficient of thrust $C_{T}$ (top) and efficiency $\eta$
(bottom). The blue squares represent the work of Pan et al. and the yellow
circles represent potential flow solver in this work.

## Appendix B

The capability of the acoustic boundary element method to model acoustic
scattering by a solid body is demonstrated and validated. A rigid circle of
radius $a$ placed at the origin that is bombarded by a harmonic field of plane
waves. The incident field of unit strength has the form

$P_{\rm{i}}(x,t)=\exp[i(\kappa r\cos\theta-\omega t)],$

where $\omega$ is the angular frequency, $\kappa$ is the wavenumber, and
$x=r\cos\theta$. The analytical result for the scattered field is [55]

$P_{\rm{s}}(x,t)=e^{i\omega
t}\sum_{n=0}^{\infty}\epsilon_{n}i^{n}\left[J_{n}(\kappa
a)-\frac{J^{\prime}_{n}(\kappa a)H_{n}(\kappa r)}{H^{\prime}_{n}(\kappa
a)}\cos n\theta\right].$ (43)

The total acoustic field is the sum of the incident and scattered fields,
$P_{\rm{t}}=P_{\rm{s}}+P_{\rm{i}}.$

Figure 15: A comparison of the analytical to BEM results of the plane wave
scatterer study with convergence studies. $(a)$ shows the fully developed
scattered field. The observation point, denoted by a black circle, is placed
at the arbitrary point $(r,\theta)=(5,\frac{\pi}{9})$. $(b)$ compares the time
history of the scattered field at the observation point for $\omega=1$ and
$\kappa=2$ with the analytic solution. $(c)$ shows the spatial convergence of
the solution, while $(d)$ shows the temporal convergence of the solution

The interaction of the harmonic incident field with the solid cylinder is as
follows. The incoming plane waves propagate in the positive $x$-direction and
make initial contact with the cylinder at $(r,\theta)=(a,\pi)$. In the area in
front of the cylinder, the plane waves are reflected back onto themselves. The
waves reflect at the front of the cylinder to create a shadow region aft of
the body. The length of the shadow region is dictated by the wavenumber, with
larger values resulting in a smaller shadow region. The $L_{2}$ error norm is
calculated over observation points placed on five circles of five points,
sampling all of the regions of the scattered field from distances of
$1a\rightarrow 10a$ from the rigid circle.

Figure 15 $(a)$ compares the transient acoustic response at a point in the
acoustic field to the analytical solution to harmonic wave forcing. Here
$\omega=1$, $\kappa=2$, and arbitrary point $(r,\theta)=(5,\frac{\pi}{9})$ are
selected for this example. Note the absence of a signal in the BEM solution
until the initial scattered wave reaches the observation point, after which
the numerical solution quickly converges to the analytical result.

Temporal and spatial discretization independence of the numerical solution are
shown in figure 15 $(c)$ and $(d)$. For the spatial convergence study, four
periods of T $=\pi$ are divided into 256 equidistant time-steps. An increasing
number of elements on the boundary were used to compare the BEM solution with
(43). The temporal convergence study (figure 15 $(d)$) had a boundary of 1024
equal length elements over a total period of T $=4\pi$. The total period is
divided into increasing numbers of equidistant time steps. Spatial convergence
occurs at approximately 512 elements, showing a relative error of less than
0.1% when using more than 256 time steps.

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