# Abstract Model of Continuous-Time Quantum Walk Based on Bernoulli
Functionals and Perfect State Transfer

Ce Wang
Yau Mathematical Sciences Center, Tsinghua University
Beijing 100084, People’s Republic of China

Abstract. In this paper, we present an abstract model of continuous-time
quantum walk (CTQW) based on Bernoulli functionals and show that the model has
perfect state transfer (PST), among others. Let $\mathfrak{h}$ be the space of
square integrable complex-valued Bernoulli functionals, which is infinitely
dimensional. First, we construct on a given subspace
$\mathfrak{h}_{L}\subset\mathfrak{h}$ a self-adjoint operator $\Delta_{L}$ via
the canonical unitary involutions on $\mathfrak{h}$, and by analyzing its
spectral structure we find out all its eigenvalues. Then, we introduce an
abstract model of CTQW with $\mathfrak{h}_{L}$ as its state space, which is
governed by the Schrödinger equation with $\Delta_{L}$ as the Hamiltonian. We
define the time-average probability distribution of the model, obtain an
explicit expression of the distribution, and, especially, we find the
distribution admits a symmetry property. We also justify the model by offering
a graph-theoretic interpretation to the operator $\Delta_{L}$ as well as to
the model itself. Finally, we prove that the model has PST at time
$t=\frac{\pi}{2}$. Some other interesting results are also proven of the
model.

Keywords. Bernoulli functionals; Continuous-time quantum walk; Perfect state
transfer; Probability distribution.

##  1 Introduction

Continuous-time quantum walks (CTQWs) are quantum analogues of classical
continuous-time random walks in probability theory, and have found wide
application in quantum computation, quantum communication as well as in
modeling physical processes [21].

In their seminal paper [10], Farhi and Gutmann initiated the study of CTQWs by
introducing their model of CTQW on a graph. Since then, many efforts have been
made to understand evolution behavior of CTQWs and related aspects (see, e.g.,
[21] and references therein). In particular, Childs [8] has described precise
correspondence between CTQWs and discrete-time quantum walks (DTQWs) on
arbitrary graphs, while Konno [14] has established a weak limit theorem for
CTQWs on the integer lattice.

Perfect state transfer (PST) is a concept with great significance in quantum
communication, which was originally introduced by Bose [3] in the context of
quantum spin chains and further studied by Christandl et al. [9]. Recently,
there has been much interest in PST in the context of CTQWs. Cheung and Godsil
[7] have addressed PST in CTQWs on cubelike graphs. Alvir et al. [1] have
investigated PST in CTQWs relative to graph Laplacians. Kempton, Lippner and
Yau [12] have considered graphs with a potential, and proven that CTQWs have
PST on such graphs. There are many other works on PST in the context of CTQWs
(see, e.g., [1, 11, 13, 15] and references therein).

Typically, a CTQW is associated with a graph $G=(V,E)$ in such a way that: the
state space of the CTQW is the space $l^{2}(V)$ of square summable complex-
valued functions on the vertex set $V$, and the evolution of the CTQW is
governed by a Schrödinger equation of the form

$\Phi_{t}=\mathrm{e}^{\mathrm{i}tH}\Phi_{0},$ (1.1)

where $H$ is the Hamiltonian that respects the topology of the graph $G$ and
$\Phi_{t}$ denotes the state of the CTQW at time $t$ with $\Phi_{0}$ being the
initial state. The CTQW is said to have PST from vertex $u\in V$ to vertex
$v\in V$ at time $t_{0}$ if $\Phi_{0}=\delta_{u}$ and
$|\langle\delta_{v},\Phi_{t_{0}}\rangle|=1$, where $\delta_{u}$ and
$\delta_{v}$ denote the basis vectors in $l^{2}(V)$ associated with $u$ and
$v$, respectively, and $\langle\cdot,\cdot\rangle$ means the inner product in
$l^{2}(V)$. Just as pointed out in [12], perfect state transfer is a rather
rare phenomenon, and constructing examples of CTQWs when this occurs can be
quite difficult.

Bernoulli functionals are measurable functions on the Bernoulli space, which
can also be viewed as functionals of a Bernoulli process. Due to the variety
in their algebraic and analytical structures, Bernoulli functionals have found
wide application in many problems in mathematical physics. For instance, the
space $\mathfrak{h}$ of square integrable complex-valued Bernoulli
functionals, together with the annihilation and creation operators on it, can
play an important role in describing the environment of an open quantum system
(see, e.g., [4, 5, 17, 18]).

In 2016, Wang and Ye [20] showed how to construct a model of DTQW (discrete-
time quantum walk) on the integer lattice $\mathbb{Z}$ with $\mathfrak{h}$ as
the coin space. In 2018, still on the integer lattice $\mathbb{Z}$, Wang et
al. [19] introduced a model of open DTQW with $\mathfrak{h}$ as the coin
space. As is well known, however, DTQWs are quite different from CTQWs
although there is some correspondence between the two types of quantum walks
[8, 21].

In this paper, motivated by the above, we would like to present a Bernoulli
functional approach to CTQWs and PST. More specifically, we would like to
construct an abstract model of CTQW in the framework of $\mathfrak{h}$ and
investigate its PST property as well as its evolution behavior. Our main work
is as follows.

First, we construct on a given subspace $\mathfrak{h}_{L}\subset\mathfrak{h}$
a self-adjoint operator $\Delta_{L}$, which we call the Laplacain on
$\mathfrak{h}_{L}$, via the canonical unitary involutions on $\mathfrak{h}$,
and by analyzing its spectral structure we find out all eigenvalues of
$\Delta_{L}$. Then, we introduce an abstract model of CTQW with
$\mathfrak{h}_{L}$ as its state space, which is governed by the Schrödinger
equation with $\Delta_{L}$ as the Hamiltonian, and define its time-average
probability distribution $\overline{P}$. We obtain an explicit expression of
$\overline{P}$, and, especially, we find $\overline{P}$ admits a symmetry
property. We justify the model by offering a graph-theoretic interpretation to
the operator $\Delta_{L}$ as well as to the model itself. Finally, we prove
that the model has PST at time $t=\frac{\pi}{2}$. Some other interesting
results are also proven.

The paper is organized as follows. In Section 2, we briefly recall the
Bernoulli space, Bernoulli functionals as well as some operators acting on
these functionals. Section 3 is our main work. Here, in Subsection 3.1 we
introduce our Laplacian $\Delta_{L}$ and analyze its spectral structure;
Subsection 3.2 describes our model of CTQW and examines properties of its
time-average probability distribution; In Subsection 3.3, we justify our model
by offering a graph-theoretic interpretation to the operator $\Delta_{L}$.
Finally in Subsection 3.4, we discuss PST in our model.

Frequently used notation. Throughout this paper, $\mathbb{N}$ denotes the set
of all nonnegative integers and $\Gamma$ its finite power set, namely

$\Gamma=\big{\\{}\sigma\mid\sigma\subset\mathbb{N},\,\\#(\sigma)<\infty\big{\\}},$
(1.2)

where $\\#(\sigma)$ means the cardinality of $\sigma$ as a set. For
$\sigma\in\Gamma$ and $k\in\mathbb{N}$, $\sigma\setminus
k\equiv\sigma\setminus\\{k\\}$, and similarly $\sigma\cup
k\equiv\sigma\cup\\{k\\}$. If $L$ is a nonnegative integer, then $\Gamma_{L}$
stands for the power set of $\mathbb{N}_{L}$, namely

$\Gamma_{L}=\\{\sigma\mid\sigma\subset\mathbb{N}_{L}\\},$ (1.3)

where $\mathbb{N}_{L}\equiv\\{0,1,\cdots,L\\}$, the truncation of $\mathbb{N}$
by $L$. Unless otherwise specified, letters like $j$, $k$ and $n$ are used to
indicate integers.

##  2 Bernoulli Functionals

In this section, we briefly recall some necessary notions, notation as well as
facts about Bernoulli functionals and operators acting on them.

Consider the set $\varOmega$ of all mappings
$\omega\colon\mathbb{N}\rightarrow\\{-1,1\\}$, which is uncountable. For each
$n\geq 0$, let $\zeta_{n}\colon\varOmega\rightarrow\\{-1,1\\}$ be the
canonical projection given by

$\zeta_{n}(\omega)=\omega(n),\quad\omega\in\varOmega.$ (2.1)

Then, one has a $\sigma$-algebra $\mathscr{F}=\sigma(\zeta_{n};n\geq 0)$ over
$\varOmega$, which is the one generated by the family of projections
$(\zeta_{n})_{n\geq 0}$. Let $(\theta_{n})_{n\geq 0}$ be a given sequence of
real numbers with the property that $0<\theta_{n}<1$ for all $n\geq 0$. By the
well-known Kolmogorov’s theorem [6], there exists a unique probability measure
$P$ on $\mathscr{F}$ such that

$P\circ(\zeta_{n_{1}},\zeta_{n_{2}},\cdots,\zeta_{n_{k}})^{-1}\big{\\{}(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{k})\big{\\}}=\prod_{j=1}^{k}\theta_{n_{j}}^{\frac{1+\epsilon_{j}}{2}}(1-\theta_{n_{j}})^{\frac{1-\epsilon_{j}}{2}},$
(2.2)

where $k$ is any positive integer, $\\{n_{j}\mid 1\leq j\leq
k\\}\subset\mathbb{N}$ with $n_{i}\neq n_{j}$ (if $i\neq j$) and
$\epsilon_{j}\in\\{-1,1\\}$, $1\leq j\leq k$. Thus, one comes to a probability
measure space $(\varOmega,\mathscr{F},P)$, which is called the Bernoulli
space. Measurable functions on $(\varOmega,\mathscr{F},P)$ are usually known
as Bernoulli functionals. In particular, square integrable functions on
$(\varOmega,\mathscr{F},P)$ are referred to as square integrable Bernoulli
functionals

Denote by $\mathfrak{h}\equiv L^{2}(\varOmega,\mathscr{F},P)$ the space of
complex-valued square integrable Bernoulli functionals with
$\langle\cdot,\cdot\rangle$ being the usual inner product given by

$\langle\xi,\eta\rangle=\int_{\varOmega}\overline{\xi(\omega)}\eta(\omega)dP(\omega),\quad\xi,\,\eta\in\mathfrak{h},$
(2.3)

where $\overline{a}$ means the complex conjugate of a complex number $a$. The
norm induced by $\langle\cdot,\cdot\rangle$ is written as $\|\cdot\|$.

Let $Z=(Z_{n})_{n\geq 0}$ be the standardized sequence of the canonical
projection sequence $(\zeta_{n})_{n\geq 0}$, namely

$Z_{n}=\frac{\zeta_{n}+1-2\theta_{n}}{2\sqrt{\theta_{n}(1-\theta_{n})}},\quad
n\geq 0.$ (2.4)

Then, $\mathfrak{h}$ has an orthonormal basis (ONB) of the form
$\\{Z_{\sigma}\mid\sigma\in\Gamma\\}$, where $Z_{\emptyset}=1$ and

$Z_{\sigma}=\prod_{j\in\sigma}Z_{j},\quad\sigma\in\Gamma,\,\sigma\neq\emptyset,$
(2.5)

where, as indicated above, $\Gamma$ is the finite power set of $\mathbb{N}$.
Usually, $\\{Z_{\sigma}\mid\sigma\in\Gamma\\}$ is known as the canonical ONB
for $\mathfrak{h}$.

######  Lemma 2.1.

Let $k\geq 0$. Then there exits a bounded operator $\Xi_{k}$ on $\mathfrak{h}$
such that

$\Xi_{k}Z_{\sigma}=\mathbf{1}_{\sigma}(k)Z_{\sigma\setminus
k}+(1-\mathbf{1}_{\sigma}(k))Z_{\sigma\cup k},\quad\sigma\in\Gamma.$ (2.6)

###### Proof.

Let $\partial_{k}$ be the annihilation operator on $\mathfrak{h}$ and
$\partial_{k}^{*}$ its adjoint (see Section 2 of [20] for details). Then, by
putting $\Xi_{k}=\partial_{k}^{*}+\partial_{k}$, we find that $\Xi_{k}$
satisfies (2.6). ∎

######  Lemma 2.2.

For all $k\geq 0$, $\Xi_{k}$ is a unitary involution, namely
$\Xi_{k}^{*}=\Xi_{k}$ and $\Xi_{k}^{2}=I$.

###### Proof.

From the proof of Lemma 2.1, we know that
$\Xi_{k}=\partial_{k}^{*}+\partial_{k}$, which implies that
$\Xi_{k}^{*}=\Xi_{k}$. A straightforward calculation gives

$\Xi_{k}^{2}=(\partial_{k}^{*}+\partial_{k})^{2}={\partial_{k}^{*}}^{2}+\partial_{k}^{*}\partial_{k}+\partial_{k}\partial_{k}^{*}+\partial_{k}^{2},$

which, together with ${\partial_{k}^{*}}^{2}=\partial_{k}^{2}=0$ as well as
the CAR in equal time (see Section 2 of [20] for details), implies
$\Xi_{k}^{2}=I$. ∎

The operator $\Xi_{k}$ is known as the canonical unitary involution on
$\mathfrak{h}$ associated with $k$. One typical property of the operator
family $\\{\Xi_{k}\mid k\geq 0\\}$ is its commutativity, namely the following
commutative relations hold true

$\Xi_{j}\Xi_{k}=\Xi_{k}\Xi_{j},\quad\forall\,j,\,k\geq 0.$ (2.7)

In view of the commutativity, one immediately finds the following products
well-defined for all $\sigma\in\Gamma$

$\Xi_{\sigma}=\prod_{k\in\sigma}\Xi_{k},$ (2.8)

where, by convention, $\Xi_{\emptyset}=I$. Using Lemma 2.2 and the
commutativity of $\\{\Xi_{k}\mid k\geq 0\\}$, one can verify that
$\Xi_{\sigma}$ remains a unitary involution for each $\sigma\in\Gamma$.

For a nonnegative integer $L\geq 0$, let
$\mathfrak{h}_{L}=\mathrm{Span}\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}$, the
subspace of $\mathfrak{h}$ spanned by
$\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}$. Then, by inheriting the inner
product $\langle\cdot,\cdot\rangle$ in $\mathfrak{h}$, $\mathfrak{h}_{L}$
itself is a complex Hilbert space of dimension $2^{L+1}$. The next lemma shows
that $\Xi_{k}$ can be viewed as an operator on $\mathfrak{h}_{L}$ whenever
$k\in\mathbb{N}_{L}$.

######  Lemma 2.3.

Let $L\geq 0$ be a nonnegative integer. Then, for each $\sigma\in\Gamma_{L}$,
$\Xi_{\sigma}$ leaves $\mathfrak{h}_{L}$ invariant. In particular, for each
$k\in\mathbb{N}_{L}$, $\Xi_{k}$ leaves $\mathfrak{h}_{L}$ invariant.

###### Proof.

It suffices to show that $\Xi_{k}Z_{\tau}\in\mathfrak{h}_{L}$ for all
$k\in\mathbb{N}_{L}$ and all $\tau\in\Gamma_{L}$. In fact, for all
$k\in\mathbb{N}_{L}$ and all $\tau\in\Gamma_{L}$, both $\tau\setminus k$ and
$\tau\cup k$ remain in $\Gamma_{L}$, which together with

$\Xi_{k}Z_{\tau}=\mathbf{1}_{\tau}(k)Z_{\tau\setminus
k}+(1-\mathbf{1}_{\tau}(k))Z_{\tau\cup k}$

implies that $\Xi_{k}Z_{\tau}\in\mathfrak{h}_{L}$. This completes the proof. ∎

######  Remark 2.1.

Let $L\geq 0$ be a nonnegative integer and $k\in\mathbb{N}_{L}$. Then, viewed
as an operator on $\mathfrak{h}_{L}$, $\Xi_{k}$ remains a unitary involution
on $\mathfrak{h}_{L}$. This is an immediate consequence of Lemma 2.2 and Lemma
2.3.

##  3 Main work

This section presents our main work. In what follows, we always assume that
$L\geq 0$ is fixed nonnegative integer. We continue to write

$\mathfrak{h}_{L}=\mathrm{Span}\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}.$ (3.1)

Note that $\mathfrak{h}_{L}$, together with the inner product
$\langle\cdot,\cdot\rangle$ in $\mathfrak{h}$, is a complex Hilbert space of
dimension $2^{L+1}$. Clearly, the collection
$\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}$ forms an ONB for $\mathfrak{h}_{L}$.
In the following, we refer to $\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}$ as the
canonical ONB for $\mathfrak{h}_{L}$.

###  3.1 Laplacian on $\mathfrak{h}_{L}$

In this subsection, we mainly introduce our Laplacian on $\mathfrak{h}_{L}$
and examine its spectral structure.

For each $k\in\mathbb{N}_{L}$, $\Xi_{k}$ can be viewed as an operator on
$\mathfrak{h}_{L}$ since it leaves $\mathfrak{h}_{L}$ invariant. We also use
$I$ as the identity operator on $\mathfrak{h}_{L}$. Consider the following
operator on $\mathfrak{h}_{L}$

$\Delta_{L}=\sum_{k=0}^{L}(I-\Xi_{k}).$ (3.2)

According to Remark 2.1, $\Delta_{L}$ is a self-adjoint (bounded) operator on
$\mathfrak{h}_{L}$. Additionally, due to their unitary property and
commutativity, $\\{\Xi_{k}\mid k\in\mathbb{N}_{L}\\}$ can naturally serve as
shift operators on $\mathfrak{h}_{L}$. In view of this, we call $\Delta_{L}$
the Laplacian on $\mathfrak{h}_{L}$.

Now let us get an insight into the spectral structure of the operator
$\Delta_{L}$. To this end, we first make some preparations.

######  Proposition 3.1.

Define
$\widehat{\Xi}^{(L)}_{\sigma}=\prod_{k=0}^{L}\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}$
for $\sigma\in\Gamma_{L}$, where
$\mathcal{E}_{\sigma}(k)=2\mathbf{1}_{\sigma}(k)-1$. Then it holds true that

$\widehat{\Xi}^{(L)}_{\sigma}\widehat{\Xi}^{(L)}_{\tau}=\left\\{\begin{array}[]{ll}2^{L+1}\widehat{\Xi}^{(L)}_{\sigma},&\hbox{$\sigma$,
$\tau\in\Gamma_{L}$,\, $\sigma=\tau$;}\\\ 0,&\hbox{$\sigma$,
$\tau\in\Gamma_{L}$,\, $\sigma\neq\tau$,}\end{array}\right.$ (3.3)

where $0$ means the null operator.

###### Proof.

By using the commutativity of the operator system $\\{\Xi_{k}\mid
k\in\mathbb{N}_{L}\\}$ on $\mathfrak{h}_{L}$ as well as the fact that each
$\Xi_{k}$ is a unitary involution on $\mathfrak{h}_{L}$, we have

$\big{(}\widehat{\Xi}^{(L)}_{\sigma}\big{)}^{2}=\prod_{k=0}^{L}\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}^{2}=\prod_{k=0}^{L}2\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}=2^{L+1}\widehat{\Xi}^{(L)}_{\sigma}.$

Next, we consider the case of $\sigma\neq\tau$, which implies that
$(\sigma\setminus\tau)\cup(\tau\setminus\sigma)\neq\emptyset$. Take
$j\in(\sigma\setminus\tau)\cup(\tau\setminus\sigma)$. Then
$\big{(}I+\mathcal{E}_{\sigma}(j)\Xi_{k}\big{)}\big{(}I+\mathcal{E}_{\tau}(j)\Xi_{k}\big{)}=0$,
which implies that

$\begin{split}\widehat{\Xi}^{(L)}_{\sigma}\widehat{\Xi}^{(L)}_{\tau}&=\prod_{k=0}^{L}\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}\big{(}I+\mathcal{E}_{\tau}(k)\Xi_{k}\big{)}\\\
&=\big{(}I+\mathcal{E}_{\sigma}(j)\Xi_{k}\big{)}\big{(}I+\mathcal{E}_{\tau}(j)\Xi_{k}\big{)}\prod_{k=0,k\neq
j}^{L}\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}\big{(}I+\mathcal{E}_{\tau}(k)\Xi_{k}\big{)}\\\
&=0.\end{split}$

Therefore (3.3) holds. ∎

######  Theorem 3.2.

As a complex Hilbert space, $\mathfrak{h}_{L}$ has an ONB of the form
$\big{\\{}\widehat{Z}^{(L)}_{\sigma}\bigm{|}\sigma\in\Gamma_{L}\big{\\}}$,
where

$\widehat{Z}^{(L)}_{\sigma}=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\gamma\setminus\sigma)}Z_{\gamma},$
(3.4)

where $\sum_{\gamma\in\Gamma_{L}}$ means to sum for all $\gamma\in\Gamma_{L}$.

###### Proof.

Since the cardinality of the system
$\big{\\{}\widehat{Z}^{(L)}_{\sigma}\bigm{|}\sigma\in\Gamma_{L}\big{\\}}$
coincides with the dimension of $\mathfrak{h}_{L}$, it suffices to show that
the system
$\big{\\{}\widehat{Z}^{(L)}_{\sigma}\bigm{|}\sigma\in\Gamma_{L}\big{\\}}$ is
orthonormal. Obviously,
$\langle\widehat{Z}^{(L)}_{\sigma},\widehat{Z}^{(L)}_{\sigma}\rangle=1$ for
all $\sigma\in\Gamma_{L}$. Now let $\sigma$, $\tau\in\Gamma_{L}$ with
$\sigma\neq\tau$. Then, from the definition of $\widehat{Z}^{(L)}_{\sigma}$,
we find

$\widehat{\Xi}^{(L)}_{\sigma}=\prod_{k=0}^{L}\big{(}I+\mathcal{E}_{\sigma}(k)\Xi_{k}\big{)}=\sum_{\gamma\in\Gamma_{L}}\Big{(}\prod_{k\in\gamma}\mathcal{E}_{\sigma}(k)\Big{)}\Xi_{\gamma}=\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\gamma\setminus\sigma)}\Xi_{\gamma},$

which together with $\Xi_{\gamma}Z_{\emptyset}=Z_{\gamma}$ yields

$\widehat{\Xi}^{(L)}_{\sigma}Z_{\emptyset}=\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\gamma\setminus\sigma)}Z_{\gamma}=\sqrt{2^{L+1}}\,\widehat{Z}^{(L)}_{\sigma}.$

Similarly, we have
$\widehat{\Xi}^{(L)}_{\tau}Z_{\emptyset}=\sqrt{2^{L+1}}\,\widehat{Z}^{(L)}_{\tau}$.
Hence, by using the self-adjoint property of $\widehat{\Xi}^{(L)}_{\sigma}$ as
well as Proposition 3.1, we get

$\big{\langle}\widehat{Z}^{(L)}_{\sigma},\widehat{Z}^{(L)}_{\tau}\big{\rangle}=\frac{1}{2^{L+1}}\big{\langle}\widehat{\Xi}^{(L)}_{\sigma}Z_{\emptyset},\widehat{\Xi}^{(L)}_{\tau}Z_{\emptyset}\big{\rangle}=\frac{1}{2^{L+1}}\big{\langle}Z_{\emptyset},\widehat{\Xi}^{(L)}_{\sigma}\widehat{\Xi}^{(L)}_{\tau}Z_{\emptyset}\big{\rangle}=0.$

This completes the proof. ∎

######  Proposition 3.3.

Let $k\in\mathbb{N}_{L}$ and $\sigma\in\Gamma_{L}$ be given. Then
$\Xi_{k}\widehat{Z}^{(L)}_{\sigma}=\mathcal{E}_{\sigma}(k)\widehat{Z}^{(L)}_{\sigma}$.

###### Proof.

By Lemma 2.2, we have
$\Xi_{k}(I+\mathcal{E}_{\sigma}(k)\Xi_{k})=\mathcal{E}_{\sigma}(k)(I+\mathcal{E}_{\sigma}(k)\Xi_{k})$,
which together with Proposition 3.1 implies that

$\begin{split}\Xi_{k}\widehat{\Xi}^{(L)}_{\sigma}&=\Xi_{k}(I+\mathcal{E}_{\sigma}(k)\Xi_{k})\prod_{j=0,j\neq
k}^{L}\big{(}I+\mathcal{E}_{\sigma}(j)\Xi_{j}\big{)}\\\
&=\mathcal{E}_{\sigma}(k)(I+\mathcal{E}_{\sigma}(k)\Xi_{k})\prod_{j=0,j\neq
k}^{L}\big{(}I+\mathcal{E}_{\sigma}(j)\Xi_{j}\big{)}\\\
&=\mathcal{E}_{\sigma}(k)\widehat{\Xi}^{(L)}_{\sigma}.\end{split}$

On the other hand, from the proof of Theorem 3.2, we find
$\widehat{Z}^{(L)}_{\sigma}=\frac{1}{\sqrt{2^{L+1}}}\widehat{\Xi}^{(L)}_{\sigma}Z_{\emptyset}$.
Thus, we finally have

$\Xi_{k}\widehat{Z}^{(L)}_{\sigma}=\frac{1}{\sqrt{2^{L+1}}}\Xi_{k}\widehat{\Xi}^{(L)}_{\sigma}Z_{\emptyset}=\frac{1}{\sqrt{2^{L+1}}}\mathcal{E}_{\sigma}(k)\widehat{\Xi}^{(L)}_{\sigma}Z_{\emptyset}=\mathcal{E}_{\sigma}(k)\widehat{Z}^{(L)}_{\sigma}.$

∎

######  Theorem 3.4.

As an operator on $\mathfrak{h}_{L}$, $\Delta_{L}$ admits a diagonal
representation of the following form

$\Delta_{L}=\sum_{\sigma\in\Gamma_{L}}2\big{(}L+1-\\#(\sigma)\big{)}\big{|}\widehat{Z}^{(L)}_{\sigma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\sigma}\big{|},$
(3.5)

where
$\big{|}\widehat{Z}^{(L)}_{\sigma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\sigma}\big{|}$
denotes the Dirac operator associated with vector
$\widehat{Z}^{(L)}_{\sigma}$.

###### Proof.

Let $\sigma\in\Gamma_{L}$ be given. Then, by using the definition of
$\Delta_{L}$ given in (3.2), and Proposition 3.3, we have

$\Delta_{L}\widehat{Z}^{(L)}_{\sigma}=\sum_{k=0}^{L}(I-\Xi_{k})\widehat{Z}^{(L)}_{\sigma}=\sum_{k=0}^{L}\big{(}1-\mathcal{E}_{\sigma}(k)\big{)}\widehat{Z}^{(L)}_{\sigma}=\Big{(}L+1-\sum_{k=0}^{L}\mathcal{E}_{\sigma}(k)\Big{)}\widehat{Z}^{(L)}_{\sigma},$

which together with $\sum_{k=0}^{L}\mathcal{E}_{\sigma}(k)=2\\#(\sigma)-(L+1)$
gives

$\Delta_{L}\widehat{Z}^{(L)}_{\sigma}=2\big{(}L+1-\\#(\sigma)\big{)}\widehat{Z}^{(L)}_{\sigma}.$
(3.6)

Thus, for any $\xi\in\mathfrak{h}_{L}$, by using Theorem 3.2 we come to

$\begin{split}\Delta_{L}\xi&=\sum_{\sigma\in\Gamma_{L}}\big{\langle}\widehat{Z}^{(L)}_{\sigma},\xi\big{\rangle}\Delta_{L}\widehat{Z}^{(L)}_{\sigma}\\\
&=\sum_{\sigma\in\Gamma_{L}}2\big{(}L+1-\\#(\sigma)\big{)}\big{\langle}\widehat{Z}^{(L)}_{\sigma},\xi\big{\rangle}\widehat{Z}^{(L)}_{\sigma}\\\
&=\sum_{\sigma\in\Gamma_{L}}2\big{(}L+1-\\#(\sigma)\big{)}\big{|}\widehat{Z}^{(L)}_{\sigma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\sigma}\big{|}\xi,\end{split}$

which together with the arbitrariness of $\xi\in\mathfrak{h}_{L}$ gives (3.5).
∎

We are now ready precisely to describe the spectral structure of the
Lalplacian $\Delta_{L}$ as follows.

######  Corollary 3.5.

As an operator on $\mathfrak{h}_{L}$, $\Delta_{L}$ has a spectrum
$\mathrm{Spec}(\Delta_{L})$ of the following form

$\mathrm{Spec}(\Delta_{L})=\big{\\{}2k\mid k\in\mathbb{N}_{L}\big{\\}}.$ (3.7)

Moreover, for each $k\in\mathbb{N}_{L}$, the eigen-space corresponding $2k$ is
exactly the one spanned by vectors
$\big{\\{}\widehat{Z}^{(L)}_{\sigma}\mid\sigma\in\Gamma_{L},\,\\#(\sigma)=L+1-k\big{\\}}$.

###  3.2 Abstract model of CTQW with $\mathfrak{h}_{L}$ being the state space

In this subsection, we establish our abstract model of CTQW and examine its
evolution properties. As usual, we use $\mathbb{R}$ to mean the set of all
real numbers.

Recall that $\Gamma_{L}$ denotes the power set of $\mathbb{N}_{L}$, namely
$\Gamma_{L}=\\{\sigma\mid\sigma\subset\mathbb{N}_{L}\\}$. For the sake of
clarity, we call elements of $\Gamma_{L}$ nodes. For two nodes $\sigma$,
$\tau\in\Gamma_{L}$, we define

$\sigma\bigtriangleup\tau=(\sigma\setminus\tau)\cup(\tau\setminus\sigma),$
(3.8)

which is usually known as the symmetric difference between $\sigma$ and
$\tau$. Note that $\Gamma_{L}$ together with $\bigtriangleup$ forms a
algebraic group. Recall also that $\\{Z_{\sigma}\mid\sigma\in\Gamma_{L}\\}$ is
called the canonical ONB for $\mathfrak{h}_{L}$.

######  Proposition 3.6.

Let $\sigma$, $\tau\in\Gamma_{L}$ be given. Then it holds true that

$\big{\langle}Z_{\sigma},\widehat{Z}^{(L)}_{\tau}\big{\rangle}=\big{\langle}\widehat{Z}^{(L)}_{\tau},Z_{\sigma}\big{\rangle}=\frac{1}{\sqrt{2^{L+1}}}(-1)^{\\#(\sigma\setminus\tau)}.$
(3.9)

###### Proof.

According to Theorem 3.2,
$\widehat{Z}^{(L)}_{\tau}=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\gamma\setminus\tau)}Z_{\gamma}$.
Thus, by using the orthogonality of the basis vectors, we come to

$\big{\langle}Z_{\sigma},\widehat{Z}^{(L)}_{\tau}\big{\rangle}=\frac{1}{\sqrt{2^{L+1}}}(-1)^{\\#(\sigma\setminus\tau)}\langle
Z_{\sigma},Z_{\sigma}\rangle=\frac{1}{\sqrt{2^{L+1}}}(-1)^{\\#(\sigma\setminus\tau)}.$

Similarly, we can show that
$\big{\langle}\widehat{Z}^{(L)}_{\tau},Z_{\sigma}\big{\rangle}$ has the same
expression as $\big{\langle}Z_{\sigma},\widehat{Z}^{(L)}_{\tau}\big{\rangle}$.
∎

######  Definition 3.1.

The continuous-time quantum walk governed by $\Delta_{L}$ (briefly, the walk
$\Delta_{L}$ or the walk) admits the following features.

  * (1)

The state space of the walk is $\mathfrak{h}_{L}$ and its states are
represented by unit vectors in $\mathfrak{h}_{L}$;

  * (2)

The evolution of the walk is governed by equation

$\xi_{t}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}\xi_{0},\quad t\in\mathbb{R},$
(3.10)

where $\xi_{t}$ denotes its state at time $t$, especially $\xi_{0}$ is its
initial state;

  * (3)

The probability $P_{t}(\sigma)$ that the walker is found at node
$\sigma\in\Gamma_{L}$ at time $t\in\mathbb{R}$ is given by

$P_{t}(\sigma)=|\langle Z_{\sigma},\xi_{t}\rangle|^{2}.$ (3.11)

In that case, the collection $\\{\,\xi_{t}\mid t\in\mathbb{R}\,\\}$ is called
the trajectory of the walk with initial state $\xi_{0}$, while, for
$t\in\mathbb{R}$, the function $\sigma\mapsto P_{t}(\sigma)$ on $\Gamma_{L}$
is referred to as the probability distribution of the walk at time $t$.

Clearly, each trajectory of the walk $\Delta_{L}$ is completely determined by
the operator $\Delta_{L}$ together with the initial state $\xi_{0}$. The next
theorem shows that the evolution of the walk $\Delta_{L}$ is even periodic in
time.

######  Theorem 3.7.

It holds true that
$\mathrm{e}^{\mathrm{i}(t+\pi)\Delta_{L}}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}$,
$\forall\,t\in\mathbb{R}$. In particular, each trajectory $\\{\,\xi_{t}\mid
t\in\mathbb{R}\,\\}$ of the walk $\Delta_{L}$ admits periodicity of the form

$\xi_{t+\pi}=\xi_{t},\quad\forall\,t\in\mathbb{R}.$ (3.12)

###### Proof.

(3.12) is a direct consequence of the equality
$\mathrm{e}^{\mathrm{i}(t+\pi)\Delta_{L}}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}$
together with (3.10). Next, we show that
$\mathrm{e}^{\mathrm{i}(t+\pi)\Delta_{L}}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}$.
However, since
$\mathrm{e}^{\mathrm{i}(t+\pi)\Delta_{L}}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}\mathrm{e}^{\mathrm{i}\pi\Delta_{L}}$,
it suffices to verify $\mathrm{e}^{\mathrm{i}\pi\Delta_{L}}=I$. In fact, by
using Theorem 3.4 and Theorem 3.2, we immediately get

$\mathrm{e}^{\mathrm{i}\pi\Delta_{L}}=\sum_{\sigma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\sigma))\pi}\big{|}\widehat{Z}^{(L)}_{\sigma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\sigma}\big{|}=\sum_{\sigma\in\Gamma_{L}}\big{|}\widehat{Z}^{(L)}_{\sigma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\sigma}\big{|}=I.$

Here the identity $\mathrm{e}^{2\mathrm{i}(L+1-\\#(\sigma))\pi}=1$ is used. ∎

For the walk $\Delta_{L}$, its probability distributions depend on its initial
state $\xi_{0}$ in general. From a physical point of view, however, the walk
should start from node $\emptyset\in\Gamma_{L}$, namely its initial state
$\xi_{0}$ should be such that $\xi_{0}=Z_{\emptyset}$, where $Z_{\emptyset}$
is the basis vector of the canonical ONB associated with node $\emptyset$,
which represents a vacuum state in physics.

######  Theorem 3.8.

Consider the walk $\Delta_{L}$. Let its initial state $\xi_{0}$ be such that
$\xi_{0}=Z_{\emptyset}$. Then, at time $t\in\mathbb{R}$, its probability
distribution admits a representation of the following form

$P_{t}(\sigma)=\frac{1}{4^{L+1}}\Big{|}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\sigma\setminus\gamma)}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\Big{|}^{2},\quad\sigma\in\Gamma_{L}.$
(3.13)

In particular, at time $t=\frac{\pi}{2}$, its probability distribution even
degenerates into a single-node distribution of the form

$P_{\frac{\pi}{2}}(\sigma)=\left\\{\begin{array}[]{ll}1,&\hbox{$\sigma=\mathbb{N}_{L}$;}\\\
0,&\hbox{$\sigma\in\Gamma_{L}$,
$\sigma\neq\mathbb{N}_{L}$.}\end{array}\right.$ (3.14)

###### Proof.

Let $t\in\mathbb{R}$ be given. Then, it follows from Theorem 3.4 that
$\mathrm{e}^{\mathrm{i}t\Delta_{L}}$ has a diagonal representation of the form

$\mathrm{e}^{\mathrm{i}t\Delta_{L}}=\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\big{|}\widehat{Z}^{(L)}_{\gamma}\big{\rangle}\\!\big{\langle}\widehat{Z}^{(L)}_{\gamma}\big{|},$
(3.15)

which, together with $\xi_{0}=Z_{\emptyset}$ as well as Proposition 3.6, gives

$\xi_{t}=\mathrm{e}^{\mathrm{i}t\Delta_{L}}\xi_{0}=\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\big{\langle}\widehat{Z}^{(L)}_{\gamma},Z_{\emptyset}\big{\rangle}\widehat{Z}^{(L)}_{\gamma}=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\widehat{Z}^{(L)}_{\gamma}.$

Thus, for $\sigma\in\Gamma_{L}$, by using Proposition 3.6 we have

$\langle
Z_{\sigma},\xi_{t}\rangle=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\big{\langle}Z_{\sigma},\widehat{Z}^{(L)}_{\gamma}\big{\rangle}=\frac{1}{2^{L+1}}\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}(-1)^{\\#(\sigma\setminus\gamma)},$

which implies that

$P_{t}(\sigma)=|\langle
Z_{\sigma},\xi_{t}\rangle|^{2}=\frac{1}{4^{L+1}}\Big{|}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#(\sigma\setminus\gamma)}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma))t}\Big{|}^{2}.$

Finally, by a straightforward calculation, we find

$\begin{split}P_{\frac{\pi}{2}}(\mathbb{N}_{L})&=\frac{1}{4^{L+1}}\Big{|}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\gamma)}\mathrm{e}^{\mathrm{i}(L+1-\\#(\gamma))\pi}\Big{|}^{2}\\\
&=\frac{1}{4^{L+1}}\Big{|}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\gamma)}(-1)^{L+1-\\#(\gamma)}\Big{|}^{2}\\\
&=\frac{1}{4^{L+1}}\Big{|}\sum_{\gamma\in\Gamma_{L}}1\Big{|}^{2}\\\
&=1,\end{split}$

which, together with the fact that the function $\sigma\mapsto
P_{\frac{\pi}{2}}(\sigma)$ is a probability distribution on $\Gamma_{L}$,
implies that $P_{\frac{\pi}{2}}(\sigma)=0$ for all $\sigma\in\Gamma_{L}$ with
$\sigma\neq\mathbb{N}_{L}$. ∎

Recall that $P_{t}(\sigma)$ is the probability that walker is found at node
$\sigma\in\Gamma_{L}$ at time $t\in\mathbb{R}$. Now consider the function
$t\mapsto P_{t}(\sigma)$ for given $\sigma\in\Gamma_{L}$. According to Theorem
3.7, this is a periodic function on $\mathbb{R}$ with $\pi$ being a period. In
view of this fact, we naturally introduce a function $\overline{P}(\cdot)$ on
$\Gamma_{L}$ in the following manner

$\overline{P}(\sigma)=\frac{1}{\pi}\int_{0}^{\pi}P_{t}(\sigma)dt,\quad\sigma\in\Gamma_{L},$
(3.16)

and call it the time-average probability distribution of the walk
$\Delta_{L}$. As can be seen, the function $t\mapsto P_{t}(\sigma)$ is
continuous for each $\sigma\in\Gamma_{L}$, which implies that
$\overline{P}(\sigma)$ is well-defined. Thus, $\overline{P}(\cdot)$ makes
sense as a probability distribution on $\Gamma_{L}$.

######  Theorem 3.9.

Let the initial state $\xi_{0}$ of the walk $\Delta_{L}$ be such that
$\xi_{0}=Z_{\emptyset}$. Then

$\overline{P}(\sigma)=\frac{1}{4^{L+1}}\sum_{(\gamma_{1},\gamma_{2})}(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})},\quad\sigma\in\Gamma_{L},$
(3.17)

where $\sum_{(\gamma_{1},\gamma_{2})}$ means to sum over the set
$\big{\\{}(\gamma_{1},\gamma_{2})\in\Gamma_{L}\times\Gamma_{L}\mid\\#(\gamma_{1})=\\#(\gamma_{2})\big{\\}}$.

###### Proof.

Let $\sigma\in\Gamma_{L}$ be given. Then, by writing

$\Sigma_{1}=\big{\\{}(\gamma_{1},\gamma_{2})\in\Gamma_{L}\times\Gamma_{L}\mid\\#(\gamma_{1})=\\#(\gamma_{2})\big{\\}}$

and $\Sigma_{2}=(\Gamma_{L}\times\Gamma_{L})\setminus\Sigma_{1}$, we can
rewrite (3.13) as

$\begin{split}P_{t}(\sigma)&=\frac{1}{4^{L+1}}\sum_{\gamma_{1}\in\Gamma_{L}}(-1)^{\\#(\sigma\setminus\gamma_{1})}\mathrm{e}^{-2\mathrm{i}(L+1-\\#(\gamma_{1}))t}\sum_{\gamma_{2}\in\Gamma_{L}}(-1)^{\\#(\sigma\setminus\gamma_{2})}\mathrm{e}^{2\mathrm{i}(L+1-\\#(\gamma_{2}))t}\\\
&=\frac{1}{4^{L+1}}\Big{[}\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}\\!\\!(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})}+\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{2}}\\!\\!(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})}\mathrm{e}^{2\mathrm{i}(\\#(\gamma_{1})-\\#(\gamma_{2}))t}\Big{]},\end{split}$

where $t\in\mathbb{R}$. Clearly,
$\int_{0}^{\pi}\mathrm{e}^{2\mathrm{i}(\\#(\gamma_{1})-\\#(\gamma_{2}))t}dt=0$
for all $(\gamma_{1},\gamma_{2})\in\Sigma_{2}$. Thus

$\overline{P}(\sigma)=\frac{1}{\pi}\int_{0}^{\pi}P_{t}(\sigma)dt=\frac{1}{4^{L+1}}\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}\\!\\!(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})},$

which is exactly the same as (3.17). ∎

Two nodes $\sigma$, $\tau\in\Gamma_{L}$ are said to be symmetric with respect
to $\mathbb{N}_{L}$ if they satisfy that

$\sigma\cap\tau=\emptyset\quad\mbox{and}\quad\sigma\cup\tau=\mathbb{N}_{L},$

which is equivalent to that $\tau=\mathbb{N}_{L}\setminus\sigma$. A function
$Q(\cdot)$ on $\Gamma_{L}$ is called symmetric if it satisfies the condition
given below

$Q(\sigma)=Q(\sigma^{c}),\quad\sigma\in\Gamma_{L},$ (3.18)

where $\sigma^{c}=\mathbb{N}_{L}\setminus\sigma$. The next theorem
characterizes symmetry of the walk $\Delta_{L}$ through its time-average
probability distribution.

######  Theorem 3.10.

Let the initial state $\xi_{0}$ of the walk $\Delta_{L}$ be such that
$\xi_{0}=Z_{\emptyset}$. Then, its time-average probability distribution
$\overline{P}(\cdot)$ is symmetric on $\Gamma_{L}$.

###### Proof.

Let $\sigma\in\Gamma_{L}$ be given. Then, by Theorem 3.9, we have

$\overline{P}(\sigma)=\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})},\quad\overline{P}(\sigma^{c})=\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{\\#(\sigma^{c}\setminus\gamma_{1})+\\#(\sigma^{c}\setminus\gamma_{2})},$

where
$\Sigma_{1}=\big{\\{}(\gamma_{1},\gamma_{2})\in\Gamma_{L}\times\Gamma_{L}\mid\\#(\gamma_{1})=\\#(\gamma_{2})\big{\\}}$.
On the other hand, for each $(\gamma_{1},\gamma_{2})\in\Sigma_{1}$, careful
calculations give

$\begin{split}(-1)^{\\#(\sigma^{c}\setminus\gamma_{1})+\\#(\sigma^{c}\setminus\gamma_{2})}&=(-1)^{\\#(\mathbb{N}_{L}\setminus(\sigma\cup\gamma_{1}))+\\#(\mathbb{N}_{L}\setminus(\sigma\cup\gamma_{2}))}\\\
&=(-1)^{2(L+1)-\\#(\sigma\cup\gamma_{1})-\\#(\sigma\cup\gamma_{2})}\\\
&=(-1)^{\\#(\sigma\cup\gamma_{1})+\\#(\sigma\cup\gamma_{2})}\\\
&=(-1)^{\\#(\gamma_{1})+\\#(\gamma_{2})+\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})}\\\
&=(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})}.\end{split}$

Thus, combining all the above, we come to

$\overline{P}(\sigma)=\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{\\#(\sigma\setminus\gamma_{1})+\\#(\sigma\setminus\gamma_{2})}=\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{\\#(\sigma^{c}\setminus\gamma_{1})+\\#(\sigma^{c}\setminus\gamma_{2})}=\overline{P}(\sigma^{c}),$

which, together with the arbitrariness of $\sigma\in\Gamma_{L}$, implies that
$\overline{P}(\cdot)$ is symmetric. ∎

######  Proposition 3.11.

Let the initial state $\xi_{0}$ of the walk $\Delta_{L}$ be such that
$\xi_{0}=Z_{\emptyset}$. Then

$\overline{P}(\emptyset)=\overline{P}(\mathbb{N}_{L})=\frac{(2L+1)!!}{(2L+2)!!},$
(3.19)

where $(2L+1)!!=(2L+1)\times(2L-1)\times\cdots\times 1$ and $(2L+2)!!$ has the
similar meaning.

###### Proof.

Write
$\Sigma_{1}=\big{\\{}(\gamma_{1},\gamma_{2})\in\Gamma_{L}\times\Gamma_{L}\mid\\#(\gamma_{1})=\\#(\gamma_{2})\big{\\}}$.
Then, a careful calculation gives

$\\#(\Sigma_{1})=\sum_{k=0}^{L+1}\binom{L+1}{k}\binom{L+1}{k}=\binom{2L+2}{L+1}.$
(3.20)

Now, by using Theorem 3.9, we have

$\overline{P}(\emptyset)=\frac{1}{4^{L+1}}\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{\\#(\emptyset\setminus\gamma_{1})+\\#(\emptyset\setminus\gamma_{2})}=\frac{1}{4^{L+1}}\sum_{(\gamma_{1},\gamma_{2})\in\Sigma_{1}}(-1)^{0}=\frac{\\#(\Sigma_{1})}{4^{L+1}},$

which together with (3.20) yields

$\overline{P}(\emptyset)=\frac{1}{4^{L+1}}\binom{2L+2}{L+1}=\frac{(2L+1)!!}{(2L+2)!!}.$

Finally, by Theorem 3.10, we know that
$\overline{P}(\mathbb{N}_{L})=\overline{P}(\emptyset)$. ∎

###  3.3 Graph-theoretic interpretation

In the present subsection, we justify our model by giving a graph-theoretic
interpretation to the operator $\Delta_{L}$ as well as to the model itself.

Consider the set $\Gamma_{L}$, which is the power set of
$\mathbb{N}_{L}\equiv\\{0,1,\cdots,L\\}$. Recall that, for $\sigma$,
$\tau\in\Gamma_{L}$, $\sigma\bigtriangleup\tau$ means the symmetric difference
between them.

######  Definition 3.2.

Let $\sigma$, $\tau\in\Gamma_{L}$. $\sigma$ is said to be adjacent to $\tau$
if $\\#(\sigma\bigtriangleup\tau)=1$. In that case, we write $\sigma\sim\tau$.

Clearly, $(\Gamma_{L},\sim)$ forms a graph, where the vertex set is
$\Gamma_{L}$ itself, while the edge set $E_{L}$ is given by

$E_{L}=\big{\\{}\\{\sigma,\tau\\}\mid\sigma,\tau\in\Gamma_{L},\,\sigma\sim\tau\big{\\}}.$
(3.21)

In the following, we simply call $(\Gamma_{L},\sim)$ the graph $\Gamma_{L}$.

######  Proposition 3.12.

Let $\sigma$, $\tau$ be vertices of the graph $\Gamma_{L}$. Then
$\sigma\sim\tau$ if and only if there exits a unique $k\in\mathbb{N}_{L}$ such
that

$\tau=\left\\{\begin{array}[]{ll}\sigma\setminus k,&\hbox{$k\in\sigma$;}\\\
\sigma\cup k,&\hbox{$k\notin\sigma$,}\end{array}\right.$ (3.22)

where $\sigma\setminus k=\sigma\setminus\\{k\\}$ and $\sigma\cup
k=\sigma\cup\\{k\\}$.

###### Proof.

The “if” part is easy to verify. Next, we verify the “only if” part. Suppose
$\sigma\sim\tau$. Then, by the definition,

$\\#(\sigma\setminus\tau)+\\#(\tau\setminus\sigma)=\\#(\sigma\bigtriangleup\tau)=1,$

which implies that $\\#(\sigma\setminus\tau)=1$ with
$\\#(\tau\setminus\sigma)=0$ or $\\#(\sigma\setminus\tau)=0$ with
$\\#(\tau\setminus\sigma)=1$, which implies that there exists
$k\in\mathbb{N}_{L}$ such that

$\sigma\setminus\tau=\\{k\\}\ \ \mbox{with}\ \
\tau\subset\sigma\quad\mbox{or}\quad\sigma\subset\tau\ \ \mbox{with}\ \
\tau\setminus\sigma=\\{k\\},$

which is equivalent to (3.22). The uniqueness of the above $k$ follows from
$\sigma\bigtriangleup\tau=\\{k\\}$. ∎

The above proposition characterizes the adjacency relation in the graph
$\Gamma_{L}$. The next proposition further shows that the graph $\Gamma_{L}$
is regular and its degree is $L+1$.

######  Proposition 3.13.

Let $\sigma\in\Gamma_{L}$ and write
$\mathcal{N}(\sigma)=\\{\tau\in\Gamma_{L}\mid\tau\sim\sigma\\}$. Then it holds
that

$\mathcal{N}(\sigma)=\big{\\{}\sigma\setminus k\mid
k\in\sigma\big{\\}}\cup\big{\\{}\sigma\cup k\mid
k\in\mathbb{N}_{L}\setminus\sigma\big{\\}}.$ (3.23)

In particular, $\\#(\mathcal{N}(\sigma))=L+1$, namely the degree of the vertex
$\sigma$ is $L+1$.

###### Proof.

Obviously, (3.23) holds for $\sigma=\emptyset$ or $\sigma=\mathbb{N}_{L}$.
Next, we suppose $\sigma\neq\emptyset$ and $\sigma\neq\mathbb{N}_{L}$. In this
case, by Proposition 3.12, we immediately have

$\mathcal{N}(\sigma)\supset\big{\\{}\sigma\setminus k\mid
k\in\sigma\big{\\}}\cup\big{\\{}\sigma\cup k\mid
k\in\mathbb{N}_{L}\setminus\sigma\big{\\}}.$

On the other hand, if $\tau\in\mathcal{N}(\sigma)$, then again by Proposition
3.12, there exits $k\in\sigma$ such that $\tau=\sigma\setminus k$ or there
exists $k\in\mathbb{N}_{L}\setminus\sigma$ such that $\tau=\sigma\cup k$,
which implies that

$\tau\in\big{\\{}\sigma\setminus k\mid
k\in\sigma\big{\\}}\cup\big{\\{}\sigma\cup k\mid
k\in\mathbb{N}_{L}\setminus\sigma\big{\\}}.$

Thus $\mathcal{N}(\sigma)\subset\big{\\{}\sigma\setminus k\mid
k\in\sigma\big{\\}}\cup\big{\\{}\sigma\cup k\mid
k\in\mathbb{N}_{L}\setminus\sigma\big{\\}}$. In summary, (3.23) remains true
for the case of $\sigma\neq\emptyset$ and $\sigma\neq\mathbb{N}_{L}$. ∎

Let $C(\Gamma_{L})$ be the set of all functions
$f\colon\Gamma_{L}\rightarrow\mathbb{C}$, where $\mathbb{C}$ denotes the set
of all complex numbers. It is known that $C(\Gamma_{L})$, together with the
usual addition and scalar multiplication as well as the usual inner product
$\langle\cdot,\cdot\rangle_{C(\Gamma_{L})}$, forms a complex Hilbert space.
Moreover, $C(\Gamma_{L})$ has an orthonormal basis of the form
$\\{e_{\sigma}\mid\sigma\in\Gamma_{L}\\}$, where $e_{\sigma}$ is the function
on $\Gamma_{L}$ given by

$e_{\sigma}(\gamma)=\left\\{\begin{array}[]{ll}1,&\hbox{$\gamma=\sigma$;}\\\
0,&\hbox{$\gamma\neq\sigma$, $\gamma\in\Gamma_{L}$.}\end{array}\right.$

Thus, as a complex Hilbert space, $C(\Gamma_{L})$ is of dimension $2^{L+1}$.
Recall that $\mathfrak{h}_{L}$ is also a complex Hilbert space of dimension
$2^{L+1}$. Therefore, there exists a unitary isomorphism $\mathsf{F}\colon
C(\Gamma_{L})\rightarrow\mathfrak{h}_{L}$ such that

$\mathsf{F}e_{\sigma}=Z_{\sigma},\quad\sigma\in\Gamma_{L},$

where $Z_{\sigma}$ is the basis vector of the canonical ONB for
$\mathfrak{h}_{L}$.

######  Definition 3.3.

The Laplacian $\widetilde{\Delta}_{L}$ of the graph $\Gamma_{L}$ is the
operator on $C(\Gamma_{L})$ defined by

$[\widetilde{\Delta}_{L}f](\sigma)=\sum_{\tau\in\mathcal{N}(\sigma)}[f(\sigma)-f(\tau)],\quad\sigma\in\Gamma_{L},$
(3.24)

where $f\in C(\Gamma_{L})$.

According to the general theory of spectral graph (see, e.g., [16]), the
Laplacian $\widetilde{\Delta}_{L}$ is a self-adjoint operator on
$C(\Gamma_{L})$. Carefully checking (3.24) and (3.23), we see that
$\widetilde{\Delta}_{L}$ can be equivalently defined as

$[\widetilde{\Delta}_{L}f](\sigma)=(L+1)f(\sigma)-\sum_{k=0}^{L}\big{[}\mathbf{1}_{\sigma}(k)f(\sigma\setminus
k)+(1-\mathbf{1}_{\sigma}(k))f(\sigma\cup k)\big{]},\quad\sigma\in\Gamma_{L},$
(3.25)

where $f\in C(\Gamma_{L})$. Next, we examine links between the Laplacian
$\widetilde{\Delta}_{L}$ of the graph $\Gamma_{L}$ and the Laplacian
$\Delta_{L}$ on $\mathfrak{h}_{L}$.

######  Theorem 3.14.

For each $k\in\mathbb{N}_{L}$, the operator $\Xi_{k}$ on $\mathfrak{h}_{L}$
has a representation on $C(\Gamma_{L})$ of the following form

$[\mathsf{F}^{-1}\Xi_{k}\mathsf{F}f](\sigma)=\mathbf{1}_{\sigma}(k)f(\sigma\setminus
k)+(1-\mathbf{1}_{\sigma}(k))f(\sigma\cup k),\quad\sigma\in\Gamma_{L},$ (3.26)

where $f\in C(\Gamma_{L})$.

###### Proof.

Since $\\{e_{\tau}\mid\tau\in\Gamma_{L}\\}$ is the orthonormal basis for
$V(\Gamma_{L})$, it suffices to show that for each $\tau\in\Gamma_{L}$ one has

$[\mathsf{F}^{-1}\Xi_{k}\mathsf{F}e_{\tau}](\sigma)=\mathbf{1}_{\sigma}(k)e_{\tau}(\sigma\setminus
k)+(1-\mathbf{1}_{\sigma}(k))e_{\tau}(\sigma\cup k),\quad\sigma\in\Gamma_{L}.$

Let $\tau\in\Gamma_{L}$ be given. Then, with $\mathsf{F}e_{\tau}=Z_{\tau}$, we
find

$\begin{split}\mathsf{F}^{-1}\Xi_{k}\mathsf{F}e_{\tau}=\mathsf{F}^{-1}\Xi_{k}Z_{\tau}&=\mathsf{F}^{-1}[\mathbf{1}_{\tau}(k)Z_{\tau\setminus
k}+(1-\mathbf{1}_{\tau}(k))Z_{\tau\cup k}]\\\
&=\mathbf{1}_{\tau}(k)e_{\tau\setminus k}+(1-\mathbf{1}_{\tau}(k))e_{\tau\cup
k},\end{split}$

which implies that

$[\mathsf{F}^{-1}\Xi_{k}\mathsf{F}e_{\tau}](\sigma)=\mathbf{1}_{\sigma}(k)e_{\tau}(\sigma\setminus
k)+(1-\mathbf{1}_{\sigma}(k))e_{\tau}(\sigma\cup k),\quad\sigma\in\Gamma_{L}.$

This completes the proof. ∎

The following theorem shows that the operator $\Delta_{L}$ on
$\mathfrak{h}_{L}$ is unitarily equivalent to the Laplacian
$\widetilde{\Delta}_{L}$ of the graph $\Gamma_{L}$.

######  Theorem 3.15.

The operator $\Delta_{L}$ on $\mathfrak{h}_{L}$ has a representation on
$C(\Gamma_{L})$ in the following manner

$\mathsf{F}^{-1}\Delta_{L}\mathsf{F}=\widetilde{\Delta}_{L},$ (3.27)

where $\widetilde{\Delta}_{L}$ is the Laplacian of the graph $\Gamma_{L}$ (see
Definition 3.3 for details).

###### Proof.

Let $f\in C(\Gamma_{L})$ be given. By the definition of the operator
$\Delta_{L}$ on $\mathfrak{h}_{L}$, we have

$\mathsf{F}^{-1}\Delta_{L}\mathsf{F}=(L+1)I-\sum_{k=0}^{L}\mathsf{F}^{-1}\Xi_{k}\mathsf{F},$

where $I$ means the identity operator on $C(\Gamma_{L})$. Thus, by suing
Theorem 3.14, we get

$\begin{split}[\mathsf{F}^{-1}\Delta_{L}\mathsf{F}f](\sigma)&=(L+1)f(\sigma)-\sum_{k=0}^{L}[\mathsf{F}^{-1}\Xi_{k}\mathsf{F}f](\sigma)\\\
&=(L+1)f(\sigma)-\sum_{k=0}^{L}[\mathbf{1}_{\sigma}(k)f(\sigma\setminus
k)+(1-\mathbf{1}_{\sigma}(k))f(\sigma\cup
k)],\quad\sigma\in\Gamma_{L},\end{split}$

which together with (3.25) yields

$[\mathsf{F}^{-1}\Delta_{L}\mathsf{F}f](\sigma)=[\widetilde{\Delta}_{L}f](\sigma),\quad\sigma\in\Gamma_{L},$

which implies that
$\mathsf{F}^{-1}\Delta_{L}\mathsf{F}f=\widetilde{\Delta}_{L}f$. It then
follows from the arbitrariness of $f\in C(\Gamma_{L})$ that
$\mathsf{F}^{-1}\Delta_{L}\mathsf{F}=\widetilde{\Delta}_{L}$. ∎

######  Remark 3.1.

According to Theorem 3.15, the operator $\Delta_{L}$ on $\mathfrak{h}_{L}$ is
unitarily equivalent to the Laplacian $\widetilde{\Delta}_{L}$ of the graph
$\Gamma_{L}$. Thus, the Schrodinger equation with $\Delta_{L}$ as the
Hamiltonian is also unitarily equivalent to that with $\widetilde{\Delta}_{L}$
as the Hamiltonian. In other words, our abstract model of CTQW described in
Definition 3.1 can be actually viewed as a model of CTQW on the graph
$\Gamma_{L}$.

###  3.4 Perfect state transfer

In the final subsection, we investigate perfect state transfer (PST) in our
abstract model of CTQW, namely the CTQW governed by $\Delta_{L}$. For brevity,
we simply call it the walk $\Delta_{L}$ below.

Let us first make clear the precise meaning of PST in the walk $\Delta_{L}$.
For nodes $\sigma$, $\tau\in\Gamma_{L}$ and $t_{0}\in\mathbb{R}$, the walk
$\Delta_{L}$ is said to have PST from $\sigma$ to $\tau$ at time $t=t_{0}$ if
it holds that

$\big{|}\langle\mathrm{e}^{\mathrm{i}t_{0}\Delta_{L}}Z_{\sigma},Z_{\tau}\rangle\big{|}=1.$
(3.28)

######  Theorem 3.16.

Let $\sigma$, $\tau\in\Gamma_{L}$ be given. Then, the following tree
conditions are equivalent mutually:

  1. (1)

$\mathrm{e}^{\mathrm{i}\frac{\pi}{2}\Delta_{L}}Z_{\sigma}=Z_{\tau}$;

  2. (2)

$\\#(\sigma\bigtriangleup\tau)=L+1$;

  3. (3)

$\tau=\mathbb{N}_{L}\setminus\sigma$.

In that case, the walk $\Delta_{L}$ has PST from $\sigma$ to $\tau$ at time
$t=\frac{\pi}{2}$.

###### Proof.

It is easy to see that under condition (1) the walk $\Delta_{L}$ has PST from
$\sigma$ to $\tau$ at time $t=\frac{\pi}{2}$. Next, we verify the equivalency
of these three conditions. Clearly, condition (2) is equivalent to condition
(3). So, we need only to show that condition (1) and condition (3) are
equivalent each other.

In fact, by using diagonal representation (3.15), we get

$\begin{split}\mathrm{e}^{\mathrm{i}\frac{\pi}{2}\Delta_{L}}Z_{\sigma}&=\sum_{\gamma\in\Gamma_{L}}\mathrm{e}^{\mathrm{i}(L+1-\\#(\gamma))\pi}\big{\langle}\widehat{Z}^{(L)}_{\gamma},Z_{\sigma}\big{\rangle}\widehat{Z}^{(L)}_{\gamma}\\\
&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\gamma)}(-1)^{\\#(\sigma\setminus\tau)}\widehat{Z}^{(L)}_{\gamma}\\\
&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\gamma)}(-1)^{-\\#(\sigma\setminus\tau)}\widehat{Z}^{(L)}_{\gamma}\\\
&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\sigma\cup\gamma)}\widehat{Z}^{(L)}_{\gamma}.\end{split}$

On the other hand, by Theorem 3.2, we have

$\begin{split}Z_{\mathbb{N}_{L}\setminus\sigma}=\sum_{\gamma\in\Gamma_{L}}\big{\langle}\widehat{Z}^{(L)}_{\gamma},Z_{\mathbb{N}_{L}\setminus\sigma}\big{\rangle}\widehat{Z}^{(L)}_{\gamma}&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#((\mathbb{N}_{L}\setminus\sigma)\setminus\gamma)}\widehat{Z}^{(L)}_{\gamma}\\\
&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{\\#((\mathbb{N}_{L}\setminus(\sigma\cup\gamma))}\widehat{Z}^{(L)}_{\gamma}\\\
&=\frac{1}{\sqrt{2^{L+1}}}\sum_{\gamma\in\Gamma_{L}}(-1)^{L+1-\\#(\sigma\cup\gamma)}\widehat{Z}^{(L)}_{\gamma}.\end{split}$

Thus, we come to a useful equality of the form
$\mathrm{e}^{\mathrm{i}\frac{\pi}{2}\Delta_{L}}Z_{\sigma}=Z_{\mathbb{N}_{L}\setminus\sigma}$.
With this equality in mind, we can immediately see the equivalence of
condition (1) and condition (3). ∎

Recall that $\mathbb{N}_{L}=\\{0,1,\cdots,L+1\\}$ and $\Gamma_{L}$ is the
power set of $\mathbb{N}_{L}$. Thus,
$\mathbb{N}_{L}\setminus\sigma\in\Gamma_{L}$ whenever $\sigma\in\Gamma_{L}$.
This together with the above proof justifies the next corollary.

######  Corollary 3.17.

For all $\sigma\in\Gamma_{L}$, it holds true that
$\mathrm{e}^{\mathrm{i}\frac{\pi}{2}\Delta_{L}}Z_{\sigma}=Z_{\mathbb{N}_{L}\setminus\sigma}$,
in particular the walk $\Delta_{L}$ has PST from node $\sigma$ to node
$\mathbb{N}_{L}\setminus\sigma$ at time $t=\frac{\pi}{2}$.

##  4 Conclusion remarks

As indicated above, we call the function $\overline{P}(\cdot)$ defined by
(3.16) the time-average probability distribution of the walk $\Delta_{L}$. In
fact, a justification for the name lies in the following observations. Just
consider the function $t\mapsto P_{t}(\sigma)$ on $\Gamma_{L}$, where
$P_{t}(\sigma)$ is the probability that the walker is found at node
$\sigma\in$ at time $t$. According to Theorem 3.7, for each
$\sigma\in\Gamma_{L}$, $t\mapsto P_{t}(\sigma)$ is a bounded periodic function
with $\pi$ being a period, which together with a limit procedure yields

$\lim_{T\to+\infty}\frac{1}{2T}\int_{-T}^{T}P_{t}(\sigma)dt=\frac{1}{\pi}\int_{0}^{\pi}P_{t}(\sigma)dt.$

Thus $\overline{P}(\cdot)$ can be redefined as

$\overline{P}(\sigma)=\lim_{T\to+\infty}\frac{1}{2T}\int_{-T}^{T}P_{t}(\sigma)dt,\quad\sigma\in\Gamma_{L},$
(4.1)

and $\overline{P}(\sigma)$ is exactly the average probability of finding the
walker at node $\sigma$ over the whole time interval
$\mathbb{R}=(-\infty,\infty)$.

## References

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Proceedings of the OMAE 202443rd International Conference on Ocean, Offshore &
Arctic Engineering OMAE2024 June 9–14, 2024 Singapore OMAE2024-125985

Zhi<EMAIL_ADDRESS>Rajeev<EMAIL_ADDRESS>
1Department of Mechanical Engineering, The University of British Columbia,
Vancouver, BC, V6T 1Z4, Canada.

# Flow-Induced Vibration of Flexible Hydrofoil in Cavitating Turbulent Flow

###### Abstract

The flow-induced vibration and cavitation dynamics of three-dimensional flow
past a cantilever flexible hydrofoil are investigated using a large eddy
simulation (LES) model, a homogeneous mixture cavitation model and the
structural modes superposition method. The present work aims to explore a
potential mechanism responsible for a propeller singing behavior, and thus
focuses on the synchronized hydroelastic coupling among the pressure pulsation
inside the flow field, the cavitation generation and the structural vibration.
To begin, we validate the tip vortex dynamics of a flexible hydrofoil against
the available experimental. Our results demonstrate that the tip vortex
shedding and the blade vibration are responsible for the intense peak in the
low-frequency tonal components of the noise source, and the trailing-edge
vortex shedding induces broadband components. Additionally, the generation of
sheet cavitation induces considerable synchronized hydrofoil vibration
(subjected to a flutter-like response), and affects the pressure fluctuations
in the flow field, which further dominate the features of the underwater noise
sources. It is suggested that the cavitation behavior and structural
vibrations co-dominate the characteristics of singing noise from a propeller
blade.

###### keywords:

Propeller singing, Hydroacoustics, Flexible blade, Flow-induced vibration,
Flutter

Documentation for asmeconf.cls: Version , .

##  1 Introduction

As a common underwater radiated noise (URN) phenomenon [1] in the marine
industry, propeller singing behavior has always been a matter of interest in
the industrial design and academic research [2, 3, 4, 5]. Its generation and
propagation pose safety and health concerns to the ship crew, passengers as
well as marine mammals [6, 7]. In addition, the potential causative factors
underpinning it, flow-induced vibration (FIV) [8, 9], could also lead to
fatigue/fracture of vessel structures. The most distinctive feature of the
propeller singing behavior is that its noise spectrum is characterized by a
narrow-band spectrum with a fairly high power level [3, 4, 5]. However, the
dynamics of two potential issues closely correlated to propeller singing,
i.e., the cavitation and the above-mentioned structural vibration of blades,
are also complex. Therefore, the exact mechanism underlying propeller singing
is still ambiguous.

As introduced above, propeller singing behavior exhibits tonal sound features
[10, 11], and specifically, the sound pressure level (SPL) of this peak is
fairly (more than 15 dB higher) than the neighboring spectral level, which
could be heard in the aft cabin of the vessel. Furthermore, the propeller
singing behavior will appear at a certain speed range, not overall range, for
the rotating propeller. However, once singing occurs, the correlated sound
will be sustained for this speed range, and the tonal frequency does not
change with the propeller rotating speed. With respect to the associated
investigation, in addition to some theoretical analysis [12, 13, 14] as well
as experimental works [15, 16, 5], high-fidelity calculations have been
applied recently to predict and analyze the URN problem and also the
accompanied vortex-shedding, cavitation, and acoustics issues. Compared to
analyzing the overall noise features for a full-scale propeller [17, 18], the
more detailed focus on the dynamics of separate blade/hydrofoil assists more
in the exploration of the mechanism underpinning URN problems, especially the
behavior of propeller singing.

The noise generation of underwater hydrofoils is closely associated with
cavitation generation [19]. Yang et al. [20] studied the unsteady cavitation
and non-cavitation flow of the three-dimensional (3D) rigid hydrofoil, with
multiphase flow solved using a large eddy simulation (LES) model coupled with
the modified Schnerr-Sauer cavitation model. Ffowcs Williams-Hawkings acoustic
method is applied for the acoustic calculation. This study stated that the
cavitation transformation and pressure fluctuations are responsible for the
low-frequency noise source generated on the body surface. The work of Wang et
al. [2] focused on the potential relationship between URN features and tip
vortex cavities. A new numerical prediction method is introduced to capture
the theoretical dispersion relations of the surface waves traveling on tip
vortex cavities. Volume fluctuation of the cavity is found to dominate the
low- and medium-frequency range of the URN spectrum. However, the detailed
intrinsic correlation between cavitation and singing behavior, i.e. the
production of tonal noise, remains not fully understood.

There are some works [21, 22] claiming that fluid-structure interactions are
responsible for the production of high-pitched sound (i.e., propeller
singing). It is also proposed that the behavior occurs due to the matching of
the main vortex shedding frequency with the blade’s structural natural
frequency. This argument is in conformance with the above-mentioned features
that singing behaviors are sustained in a certain rotor speed range with
constant tonal frequency. In this case, FIV has been suggested to be a
potential underlying factor resulting in propeller singing behavior [21, 22],
although other research work partially refuted this statement [3]. With
respect to the structural response of hydrofoil, Chae et al. [23] investigated
the FIV response of the flexible hydrofoil via numerical and experimental
works and focused on the effect of the inflow velocity, angle of attack, and
added mass effects. The induced vibration response also has an interactive
relationship with the cavitation generated on the hydrofoil surface or tip.
Sun et al. [24] investigated the cavitation process of hydrofoils with
structure response calculated using the finite element method and flow field
solved using large eddy simulation and Schnerr–Sauer model. The responses,
including structural displacements, cavitation, and pressure fluctuations, are
compared between rigid and flexible hydrofoils. Sun et al. [24] concluded that
the flexible hydrofoil exhibits smaller and more diffuse areas of cavitation
compared with the rigid hydrofoil. Dong et al. [25] further explored both the
flow dynamic and acoustic response of flexible hydrofoils under cavitation-
induced vibration and different structural materials compared. This work also
noted that material optimization is able to suppress cavitation while reducing
turbulence intensity along with the attenuation of noise generation and
propagation.

However, earlier works have not examined the issue of the potential
relationship between noise source and the three-dimensional (3D) vibration
response of a flexible blade tip in the presence of turbulent cavitating flow.
The closest research is the above-introduced work conducted by Wang et al.
[2], but the concerned tip hydrofoil is in rigid form. It is known that the
hydrofoil shape of a blade tip is tapering and is more susceptible to
deformation as well as vibration than a hydrofoil with a chord length constant
along the spanwise direction. In addition, cavitation is more likely to occur
at the tip area due to a relatively larger Reynolds number. Therefore, the FIV
study of the flexible tip hydrofoil is of great significance in analyzing the
underlying mechanism of propeller singing. Based on the above introduction,
present work will focus on the explosion of the detailed characterization of
3D FIV response of flexible tip hydrofoil considering the effect of cavitation
processing.

The paper is structured as follows: Section 2 details the numerical and
analytical methods listed above. The accuracy of the implemented models used
herein is also validated in Section 3. In Section 4, the FIV response and
potential singing behavior for the hydrofoil are analyzed. A systematic
comparison is conducted between rigid and flexible hydrofoils regarding the
flow and structural responses and the impact of cavitation is explored. In
Section 5, the key results of this study are summarized.

##  2 Numerical methodology

###  2.1 Fluid-structure interaction

With respect to the concerned configuration consisting of a flexible (elastic)
cantilevered hydrofoil submerged in a 3-D uniform water flow in this work, the
computational fluid dynamics/computational structure dynamics method is
applied to calculate the system response. The concerned fluid dynamics
consists of liquid and vapor phases and is assumed to exist as a continuous
homogeneous mixture. The phase indicator $\alpha_{l}$($\bm{x}$,$t$) represents
the phase fraction of the liquid phase in the fluid mixture. Where $\bm{x}$
and $t$ are spatial and temporal coordinates. The density $\rho$ and dynamic
viscosity $\mu$ of the fluid are considered as a weighted combination of the
liquid and vapor phases:

$\rho=\rho_{l}\alpha_{l}+\rho_{v}\left(1-\alpha_{l}\right),$ (1)
$\mu=\mu_{l}\alpha_{l}+\mu_{v}\left(1-\alpha_{l}\right),$ (2)

where $\rho_{l}$ and $\rho_{v}$ are the densities of the pure liquid and vapor
phases, respectively, and $\mu_{l}$ and $\mu_{v}$ are the dynamic viscosity of
the liquid and vapor phases.

The governing equations of the flow dynamics are the unsteady incompressible
Navier-Stokes (N-S) equations combined with the Schnerr-Sauer cavitation
model, while the boundary changes induced by the body motion are resolved
based on an arbitrary Lagrangian-Eulerian (ALE) scheme. The N-S equations in
the ALE scheme are expressed as:

$\rho\frac{\partial\bm{u}}{\partial
t}+\rho\left(\bm{u}-\bm{\hat{u}}\right)\cdot\nabla\bm{u}-\nabla\cdot\bm{\sigma}=\bm{f}_{f},\,\,\text{on}\left(\bm{x},t\right)\in\varOmega$
(3) $\frac{\partial\rho}{\partial
t}+\rho\nabla\cdot\bm{u}+\left(\bm{u}-\bm{\hat{u}}\right)\cdot\nabla\rho=0,\,\,\text{on}\left(\bm{x},t\right)\in\varOmega$
(4)
$\nabla\cdot\bm{u}=\left(\frac{1}{\rho_{l}}-\frac{1}{\rho_{l}}\right)\left(\dot{m}_{c}+\dot{m}_{v}\right)$
(5)

where $\bm{u}$ is the fluid flow velocity, $\bm{\hat{u}}$ is the velocity
component of mesh motion, $\bm{f}_{f}$ is the body force applied to the fluid,
$\bm{x}$, i.e., $(x,y,z)$, is the Cartesian coordinates in the fluid domain
$\varOmega$. $\dot{m}_{c}$ and $\dot{m}_{v}$ are the mass transfers in the
condensation and vaporization processes respectively. $\bm{\sigma}$ could be
expressed as:

$\bm{\sigma}=\bm{\sigma}^{n}+\bm{\sigma}^{sgs},$ (6)

where $\bm{\sigma}^{n}$ and $\bm{\sigma}^{sgs}$ are the Cauchy stress tensor
for a Newtonian fluid and the subgrid-scale turbulent stress tensor,
respectively.
$\bm{\sigma}^{n}=-p\bm{I}+\mu\left(\nabla\bm{u}+\left(\nabla\bm{u}\right)^{T}\right)$,
where $p$ represents the fluid pressure. $\bm{\sigma}^{sgs}$ is modeled using
a spatial filtering large eddy simulation (LES) model [26, 27].

###  2.2 Cavitation model

As the solution of the scalar transport equation, indicator $\alpha_{l}$ is
obtained as the liquid volume fraction. The scalar transport equation could
also be written in the ALE framework as:

$\frac{\partial\alpha_{l}}{\partial
t}+\left(\bm{u}-\bm{\hat{u}}\right)\cdot\nabla\alpha_{l}=\frac{\rho}{\rho_{l}\rho_{v}}\left(\dot{m}_{c}+\dot{m}_{v}\right),\,\,\text{on}\left(\bm{x},t\right)\in\varOmega$
(7)

The Schnerr-Sauer cavitation model [28] (based on the transport equations)
mathematically expresses the vaporization and condensation process and is able
to capture the cavitation phenomenon more realistically. The Schnerr-Sauer
cavitation model used in this paper is implemented in our in-house code and
expressed as:

$\dot{m}_{c}=\frac{3\rho_{l}\rho_{v}}{\rho
R_{B}}\sqrt{\frac{2}{3\rho_{l}\lvert p-p_{v}\rvert}}\cdot
C_{c}\alpha_{l}\left(1-\alpha_{l}\right)\max\left(p-p_{v},0\right)$ (8)

$\dot{m}_{v}=\frac{3\rho_{l}\rho_{v}}{\rho
R_{B}}\sqrt{\frac{2}{3\rho_{l}\lvert p-p_{v}\rvert}}\cdot
C_{v}\alpha_{l}\left(1+\alpha_{nuc}-\alpha_{l}\right)\min\left(p-p_{v},0\right)$
(9)

$R_{B}=\left(\frac{3}{4\pi
n_{0}}\frac{1+\alpha_{nuc}-\alpha_{l}}{\alpha_{l}}\right)^{1/3},$ (10)

$\alpha_{nuc}=\left(\frac{\pi n_{0}d_{nuc}^{3}}{6}\right)/\left(1+\frac{\pi
n_{0}d_{nuc}^{3}}{6}\right).$ (11)

where $C_{c}$ and $C_{v}$ are the condensation and vaporization model
coefficients, respectively (equal to 1$\times$$10^{-3}$ and
5$\times$$10^{-3}$, respectively, herein); $p_{v}$ is the liquid saturation
pressure; $R_{B}$ is the equivalent radius of the vapor volume; $d_{nuc}$ is
the local nuclei diameter (whose value is equal to 2.5$\times$$10^{-6}$
herein); and, $n_{0}$ is the number of bubbles per unit volume (whose value is
equal to 1.0$\times$$10^{13}$ herein). Detailed information on the model and
properties determination refers to the works of Kashyap & Jaiman [29] and
Sauer & Schnerr [28].

###  2.3 Structural dynamics for flexible blade

The flexible cantilever hydrofoil concerned here is regarded as a structure
with relatively small three-dimensional deformations, and the present
hydrofoil is modeled using eigenmodes obtained from solving the eigenvalue
problem derived from the linear elastic structural model. We solve the
structural displacements $\bm{\omega}^{s}(\bm{x}^{s},t)$ at co-ordinates
$\bm{x}^{s}$ using a linear elastic control equation excited by the
distributed unsteady force $f^{s}$ (acting on per unit volume). The motion of
the flexible cantilever hydrofoil is governed by the fluid forces and involves
integrating pressure and shear stress effects on the blade surface. Neglecting
the damping and shear effects, the control equations for the flexible
cantilever structure are expressed as:

$\bm{\rho}^{s}\frac{\partial^{2}\bm{\omega}^{s}}{\partial
t^{2}}=\nabla\cdot\bm{\sigma}^{s}+\rho^{s}f^{s},$ (12)

where $\rho^{s}$ is the solid density, $\bm{\sigma}^{s}$ is the Cauchy stress
tensor of the solid, which is defined as (for the linear elastic material):

$\bm{\sigma}^{s}=\lambda^{s}tr\left(\bm{\epsilon}^{s}\right)+2\mu^{s}\bm{\epsilon}^{s},$
(13)

where $\lambda^{s}$ and $\mu^{s}$ are the Lame’s parameters and
$\bm{\epsilon}$ is the infinitesimal strain tensor, which is defined as:

$\bm{\epsilon}^{s}=\frac{1}{2}\left(\nabla\bm{\omega}^{s}+\left(\nabla\bm{\omega}^{s}\right)^{T}\right).$
(14)

To obtain the eigenvalue problem, we separate the spatial and temporal
components of the displacement response as:

$\bm{\omega}^{s}\left(\bm{x}^{s},t\right)=\sum_{n=1}^{\infty}{\bm{\hat{\omega}}_{n}^{s}\left(\bm{x}^{s}\right)}e^{iw_{n}t},$
(15)

where $\bm{\hat{\omega}}_{n}^{s}$ denotes an eigenvector or mode shape of the
structure, and $w_{n}$ represents the corresponding eigenvalue or modal
frequency. Substituting Eq. 15 in Eq. 12 and neglecting body forces, we obtain
the following representation:

$-\rho^{s}w_{n}^{2}\bm{\hat{\omega}}_{n}^{s}e^{iw_{n}t}=\nabla\cdot\bm{\sigma}^{s}\left(\bm{\hat{\omega}}^{s}\right)e^{iw_{n}t},$
(16)

which owing to linear elastic modeling can be simplified to:

$\left(\mathcal{L}+\rho^{s}w_{n}^{2}\bm{I}\right)\bm{\hat{\omega}}_{n}^{s}=0,$
(17)

where $\mathcal{L}$ denotes the linear differential operator associated with
Eq. 14 and Eq. 12, completing the definition of the structural eigenvalue
problem. The boundary conditions of the cantilevered arrangement are given by:

$\bm{\omega}^{s}\left(\bm{x}^{s},t\right)\rvert_{z=0}=0,$ (18)
$\frac{\partial\bm{\omega}^{s}\left(\bm{x}^{s},t\right)}{\partial
z}\rvert_{z=0}=0,$ (19) $\bm{\sigma}^{\bm{s}}\bm{n}^{\bm{s}}\rvert_{z\neq
0}=0.$ (20)

The solution to Eq. 17 is obtained through a finite element discretization of
the structural domain. As introduced above, a mode superposition method is
considered for resolving Eq. 12 and characterizing the dynamic response of the
hydrofoil [30]. Calculix [31] software is applied herein to extract the shape
of each mode and the corresponding structural natural frequency. First-ten
dominating structural modes are considered in the present work. The structural
properties of present flexible cantilever hydrofoil are defined with Young’s
Module $E$ of 6 $\times$ $10^{8}$ $Pa$, Poisson’s ratio $\nu$ of 0.33, and
mass ratio $m^{*}$ of 7.65, resulting in the corresponding eigenvalues and
nature frequencies (for each mode) displayed in Table 1. The diagram of first-
five mode shapes is shown in Fig. 1.

Table 1: Eigenvalue $k_{i}$ and structural nature frequency $f_{n,i}$ of each mode for cantilever flexible hydrofoil concerned in present work. Mode shape |  $k_{i}$ (N/m) |  $f_{n,i}$ (Hz)
---|---|---
1 | 0.2698 $\times$ $10^{6}$ | 82.7
2 | 0.2690 $\times$ $10^{7}$ | 261.0
3 | 0.4354 $\times$ $10^{7}$ | 332.1
4 | 0.5506 $\times$ $10^{7}$ | 373.5
5 | 0.1492 $\times$ $10^{7}$ | 614.7
6 | 0.9849 $\times$ $10^{8}$ | 706.4
7 | 0.3496 $\times$ $10^{8}$ | 941.0
8 | 0.3620 $\times$ $10^{8}$ | 957.6
9 | 0.3840 $\times$ $10^{8}$ | 986.2
10 | 0.4034 $\times$ $10^{8}$ | 1010.9

A standard Lagrangian finite element technique is applied to adapt the mesh to
the new geometry of the domain. A nonlinear partitioned iterative approach
based on the nonlinear iterative force correction (NIFC) scheme is applied to
solve the fluid-structure interaction that links the fluid flow equations (3,
4) with the structural equation of motion (12). The approach for integrating
the structural equations of motion, treating the fluid-structure interface,
and solving ALE mesh motion refers to the works and introduction of
Darbhamulla & Jaiman [32], Kashyap & Jaiman [29] and Heydari et al. [30].

Figure 1: First five mode shapes of a flexible cantilever hydrofoil. The
fourth mode in panel (d) vibrates in $x-$direction.

(a)

(b)

Figure 2: Flexible hydrofoil set-up: (a) overall computational domain, and (b)
top-view of present cantilever hydrofoil. Figure 3: Overall mesh and the
expanded view of the immediate vicinity of the body surface in (a) $x-z$ and
(b) $y-z$ planes.

##  3 Computational domain and model validation

Figures 2a and 2b exhibit an overall diagram and a top view of the present
configuration of flexible NACA 662-415 elliptic hydrofoil (in cantilever
pattern with bottom-face fixed) submerged in uniform water inflow. The angle
of attack ($\alpha$), span ($S$), and maximum chord ($L$) of the concerned
hydrofoil were 9∘, 83.6 mm and 100.0 mm, respectively. With respect to the
present three-dimensional (3-D) computational domain in Figure 2a, the
position of the hydrofoil bottom surface is at the centerline of transverse
(or, cross-stream) direction (i.e., $y=$ 0), and situated at 3$S$ downstream
from the inlet boundary in $x-$direction. A Dirichlet boundary condition was
prescribed for the incident flow velocity $\bm{u}=(U_{0},0,0)$ on the inlet
face (i.e., single upstream patch) in Fig. 2a, with $U_{0}$ fixed as 10 m/s in
present work. A Neumann boundary condition is imposed on the velocity at the
outflow (outlet) boundaries, i.e., the single downstream patch of the domain,
A non-slip boundary condition is applied on the bottom patch. symmetrical
boundary conditions are applied to two-sided and up patches to avoid blockage
effects. The cavitation number is defined as
$\vartheta_{n}=(p-p_{v})/(0.5\rho_{l}(U_{0})^{2})$, where $p_{v}$ is
saturation pressure.

Table 2: Surface force coefficients for different mesh conditions with rigid hydrofoil. | Number of elements | $C_{x,rms}$ | $C_{y,rms}$ | $C_{z,rms}$
---|---|---|---|---
Mesh 1 | 2,117,540 | 0.0552 | 0.4217 | 0.0295
Mesh 2 | 4,475,608 | 0.0584 | 0.4621 | 0.0312
Mesh 3 | 6,083,799 | 0.0608 | 0.4978 | 0.0338
Mesh 4 | 7,987,184 | 0.0612 | 0.4981 | 0.0340
Figure 4: Comparison between present work and Ye et al. [16] for the
trajectory of the time-averaged tip vortex.

##  4 Results and Discussion

The mesh dependency discussion is conducted via the simulation of the present
NACA 662-415 elliptic hydrofoil in rigid/solid form. The corresponding drag
and lift coefficients ($C_{x}$ and $C_{y}$) at different mesh qualities are
calculated and the associated results are summarized in Table 2. It could be
observed that the relative differences of each parameter between mesh 1 to
mesh 2 are considerable, but all decrease to a value smaller than 0.5% as the
mesh is refined to mesh 3 (fine) and mesh 4 (very fine). As a consequence, the
strategy applied by mesh 3 is adopted in all the configurations of the present
work to achieve the best balance of calculation time and accuracy. To follow
up, we use mesh 3 to calculate the present configuration and compare the
trajectory of the time-averaged tip vortex between present results and other
accessible experimental work [16]. The results summarized in Fig. 4 indicate
high conformance between this study and other results, which indicates the
correctness of the present model. The fluid-solid interaction model for
flexible structures and the fluid model involving cavitation phenomena applied
in this study are implemented in the in-house FSI solver, whose detailed
description and model validation refer to our previous works of Kashyap &
Jaiman [29, 33] and Heydari et al. [30]. Figure 3 shows the overall and
extended view of the computational domain corresponding to mesh 3.

Figure 5: Time-histories of lift coefficients in $x-$, $y-$, and
$z-$directions for rigid hydrofoil without cavitation.

###  4.1 Rigid blade without cavitation

We first explore the configuration consisting of concerned hydrofoil in rigid
form. The time-histories of surface force coefficients in $x-$, $y-$, and $z-$
are displayed in Fig. 5. It could be observed that the mean value as well as
the fluctuating components of the pressure variation in $y-$ and
$z-$directions are maximum and minimum, respectively. Therefore, the following
contents will focus on the response of the system dynamics in the $x-$ and
$y-$directions. Torque force is not displayed here owing that rotational
displacement underlying mode 5 do not has a dominate impact on structural
response herein.

Figure 6: Spectrum of lift coefficients in $x-$, $y-$, and $z-$directions for
rigid hydrofoil without cavitation. Figure 7: Diagram of the monitor
locations. Figure 8: Spectrum of pressure fluctuation at monitors $F_{1}$,
$F_{3}$, and $F_{6}$ for rigid hydrofoil without cavitation.

The fluctuation components inside $C_{x,y,z}$ variations are extracted and the
corresponding spectrums (as a function of $St$) are shown in Figure 6. $St$
(=$fS/U_{0}$) is the Strouhal number, where $f$ is frequency. With respect to
$C_{y}$ variation, the most intense peak is located at $St$ of 0.2 (i.e., $f$
= 20 Hz). For the streamwise force $C_{x}$, besides the peak with $St$ close
to 0.2, there are other multi-components ranging from $St$ of 0.4 to about
2.0, in which the strongest peak corresponds to $St$ = 1.5. A similar
situation appears for the $C_{z}$. Those frequency components are correlated
to the structural response of the flexible hydrofoil and also pressure
fluctuations underpinning propeller singing introduced later in the present
work. In this case, we are curious about which dynamical structures each of
the above frequency components correspond to.

To answer the above question, a specific attention is placed on the pressure
variations at certain points inside fluid domain and on hydrofoil surface.
Figure 7 indicates the locations of the monitors, in which the grey-color and
yellow-color points represent the monitors on the surface and inside the flow
domain, respectively. Field monitors $F_{1,2,3,4}$ locate downstream of the
blade tip with the same vertical ($z-$) height. The spectrum for the pressure
fluctuation at yellow-marked field monitors 1, 3, and 6 (i.e., $F_{1,3,6}$)
are shown in Fig. 8. It should be noted that monitor $F_{1}$ is on the tip of
the hydrofoil, $F_{3}$ is located in the trajectory of shedding tip vortex
with $x/S$ = 0.5, and $F_{6}$ is very close to the trailing edge of the
hydrofoil with $z/S$ = 0.4.

The first as well as the second panel of Fig. 8 indicate that the dominant
frequency of the tip vortex-shedding behavior is located at $St$ = 0.2, which
is consistent with the intense frequency of the second panel in Fig. 6. This
indicates that the tip vortex-shedding mechanism underpins the lift variation
in the $y-$direction when the hydrofoil is at rigid and stationary. On the
other hand, the third panel in Fig. 8 exhibits a clear broadband feature, with
correlated $St$ varying from 0 to 4. It is expected that these broadband
components are associated with the continuous trailing-edge vortex-shedding
structure on the blade surface in the spanwise direction. Furthermore, the
chord length of the blade continues to get smaller along the spreading
direction, and this tapering geometrical feature is one of the factors for the
continuous change in the surface vortex-shedding frequency and also the
broadband features of the pressure spectrum. Those pieces of information
suggest that vortex shedding at the trailing edge and tip of the blade (at
rest status) will affect the broadband and tonal components of its
hydrodynamic noise, respectively.

Figure 9: Instantaneous isopleths of the vorticity from different views for
rigid hydrofoil without cavitation. Figure 10: Time-histories of lift
coefficients in $x-$, $y-$, and $z-$directions for flexible hydrofoil without
cavitation.

Figure 9 shows the contour of the instantaneous vorticity magnitude in the
$x-y$ and two $x-z$ orthogonal planes (from front and back views) for flow
past the rigid hydrofoil, with the distribution of pressure coefficients
depicted on it. At the front and back of the hydrofoil are the pressure side
and suction side, respectively, where the suction side will generate
cavitation later in the following cavitating calculation due to the low-
pressure distribution. Additionaly, in addition to the clear vortex shedding
trajectory behind the top of the hydrofoil tip, the suction side also showed
the behavior of fluid transitioning from laminar flow into turbulence, and
finally destabilizing into irregular vortex shedding at the trailing edge.
Those physical visualizations are consistent with the arguments on singing
features suggested above in spectra analysis.

###  4.2 Flexible blade without cavitation

Figure 11: Spectrum of lift coefficients in $x-$, $y-$, and $z-$directions for
flexible hydrofoil without cavitation.

To follow up on the above analysis, we further analyze the potential impact of
FIV on noise sources in this section. The same hydrofoil is determined as the
flexible cantilever pattern, with Young’s Module $E$ of 6 $\times$ $10^{8}$
$Pa$, Poisson’s ratio $\nu$ of 0.33, and mass ratio $m^{*}$ of 7.65. The time
histories of force coefficients $C_{x,y,z}$ and the corresponding spectrum of
correlated fluctuating components are exhibited in Figs. 10 and 11. In
addition to the broadband components, a perusal of the spectrum (cf. with Fig.
11) indicates two distinctive features: The components corresponding to $St$ =
0.2 becomes stronger for all the spectrum of $C_{x}$, $C_{y}$ and $C_{z}$; One
more intense peak of $St$ = 3.60 appears on the spectrum of $C_{x}$.

Figure 12: 3D trajectory of surface monitor $S_{2}$ for flexible hydrofoil
without cavitation. Figure 13: Trajectories of the three surface monitoring
points in the $x-y$ plane.

The motion curve of 3D displacement at surface point #2 (i.e., S2) is shown in
Fig. 12, and it is found that the amplitude in $z-$direction is relatively
small. Therefore, we continue to depict the trajectories in the $x-$ and
$y-$directions of three monitors on the surface in Fig. 13, with the blue
arrows representing the directions of the displacements. The displacement
amplitude range of the three monitors decreases as the spanwise height (i.e.,
$z-$coordinate) becomes smaller. This behavior is consistent with the
instantaneous geometric illustration of structural deformation shown in Figure
12.

The first feature is owing to the hydrofoil being subjected to forced-
vibration in the $y-$direction. Specifically, the vibration frequency is still
dominated by the energetic periodic force brought about by the tip and
trailing-edge vortex shedding. As for the structural response in the
$x-$direction, it could be observed from Fig. 5 as well as Fig. 6a that the
fluctuation energy of the lift force in $x-$direction is relatively tiny for
the rigid structure, hence it is difficult to have a substantial effect on the
structural deformation. Therefore, the structural natural frequency
corresponding to hydrofoil mode #4 (with vibration in the $x-$direction)
appears in the spectrum of $C_{x}$ (cf. with Fig. 11a). According to the noise
analysis of cylinder vibration [34], it is known that the vibration of a
structure in a certain direction will also intensify/amplify the noise level
in that direction.

Figure 14: Time-histories of structural displacements in $x-$, $y-$, and
$z-$directions at surface monitor $S_{1}$ for flexible hydrofoil without
cavitation. Figure 15: Time-histories of fluctuating components $\hat{C}_{y}$
and $\hat{y}/S$ in lift coefficients $C_{y}$ and normalized displacements
$y/S$, respectively, at surface monitor $S_{1}$ for flexible hydrofoil without
cavitation. Figure 16: Spectrum of structural displacements in $x-$ and
$y-$directions at the location of three surface monitors for flexible
hydrofoil without cavitation. Figure 17: Spectrum of pressure fluctuation at
monitors $F_{1,2,3,4,5,6}$ for flexible hydrofoil without cavitation.

Figure 14 displays the time-histories of structural displacements of monitor
$S_{2}$ in $x-$, $y-$, and $z-$directions. The fluctuating components inside
lift coefficients $C_{y}$ and normalized displacements $y/S$ (at $S_{1}$) are
both displayed inside Figure 15. It is observed that the fluctuation of the
lift coefficients on large scales is dominated by the structural vibrations,
while the structure is also in turn affected by the high-frequency components
of the lift. Furthermore, the spectrum of the structural displacements in the
$x-$ and $y-$directions for the three monitoring points $S_{1,2,3}$ are
depicted in Fig. 16. The dominating peak frequency of $x-$displacements is
located at $S_{t}$ of 3.60, consistent with that of force coefficient $C_{x}$.
Moreover, one additional intense peak with $St$ of 1.80 appears in the
displacements spectrum of both $x-$ and $y-$directions. However, from Fig. 17
it is clear that this spectrum component does not originate from the natural
frequency of any of the structural modes. In this case, it is suggested that
this frequency component with $St$ of 1.80 originates from the effect of the
periodical force owing to the vortex shedding from the hydrofoil surface. More
specifically, the deformation of the hydrofoil affects the mechanism of
trailing-edge vortex-shedding structure and induces the frequency components
with $St$ of 1.80 inside surface forces (cf. with Figure 11 a,b). This thereby
has a considerable impact on structural deformation and motion.

For the propeller singing problem, the vortex-shedding behaviors in the wake
directly affect the magnitude and frequency of the noise source. To further
explore this, we extracted the pressure variation at several points in the
flow field and displayed their spectra in Fig. 17. Four monitors,
$F_{1,2,3,4}$, are located in vicinity of the tip vortex trajectory, while the
other two points, $F_{5,6}$, are close to the trailing edge. To begin, it can
be observed that the pressure fluctuations near the tip vortex still contain
the dominant frequency of $St$ = 0.2. Moreover, pressure at all monitors also
contains the dominant frequency component, i.e., the peak corresponding to
$St$ = 1.80. Since the pressure fluctuations on the trajectory of the tip
vortex are not directly affected by the trailing-edge vortex shedding, it is
expected that they come from the effect of the blade vibration. In this case,
we infer that the overall process should perform as: deformation of the
hydrofoil leads to a change in the form of the trailing-edge vortex shedding,
and also the appearance of a new component of surface force corresponding to
$St$ = 1.80. This component has a in turn implication on the hydrofoil
vibration. The hydrofoil tip vibration then transfers this frequency component
to the pressure fluctuations in the tip vortex, which thereby affects the
noise source. Therefore, it is suggested that the vibration of the hydrofoil
structure will be closely correlated to the characteristics of the propeller
URN and also singing behaviors.

Figure 18: Instantaneous isopleths of the second invariant $Q$ from different
overviews and enlarged view on the suction side for flexible hydrofoil without
cavitation. Figure 19: Cavitation contour accompanied by the $x-y$ planes, on
which cavitation number $\vartheta_{n}$, velocity $U$, and second invariant of
the velocity gradient tensor $Q$ are depicted in panels 1, 2, and 3,
respectively.

Figure 18 exhibits the second invariant $Q$ of the velocity gradient tensor
under a Galilean transformation. It should be noted that $Q$ represents the
difference between rotation rate and strain rate. Its positive value indicates
that the vortex structure is rapidly forming at a correlated location, and
negative value means that the vortex structure is being dissipated. As can be
seen from the top view, there is a significant deformation of the hydrofoil,
and there is a clear vortex shedding behind the blade tip. In addition,
compared with the pressure side of the blade, there are a large number of
areas with positive and negative values of $Q$ on the suction side of the
blade, indicating abundant vortex structures on the suction side. Furthermore,
the structural vibration will inevitably affect the vortex generation and
shedding at the trailing-edge as well as at the tip, which in turn will have
an impact on the noise sources in the flow field. This is also demonstrated in
the above mathematical analysis.

###  4.3 Flexible blade with cavitation

To investigate the impact of cavitation generation on structural response and
noise sources, in this section, the Schnerr-Sauer cavitation model is added in
the calculation of fluid-structure interaction for the present flexible
hydrofoil compared to the setting in subsection 4.2 and other properties
maintain constant. Figure 19 exhibits the cavitation contour (with volume
fraction of 0.5) accompanied by the $x-y$ planes, on which cavitation number
$\vartheta_{n}$, velocity $U$, and second invariant of the velocity gradient
tensor $Q$ are depicted, as shown in panels 1, 2, and 3, respectively.
Firstly, a large amount of sheet cavitation occurs in present numerical
results, while no tip vortex cavitation occurs, even though all dynamical
information in the three panels indicates the presence of a tip vortex. It is
expected that the disappearance of tip vortex cavitation is because the mesh
quality in this calculation may not be sufficient to resolve the cavitation
structure, although it could accurately estimate the flow field and surface
pressure. Further work is needed in the future to achieve the capture of the
tip vortex cavitation using a higher accuracy mesh strategy. Particular
emphasis of the present work lies on the effect of sheet cavitation generation
on the structural response and noise sources.

Figure 20: Time-histories of lift coefficients in $x-$, $y-$, and
$z-$directions for flexible hydrofoil with sheet cavitation. Figure 21:
Spectrum of lift coefficients in $x-$, $y-$, and $z-$directions for flexible
hydrofoil with sheet cavitation. Figure 22: Time-histories of structural
displacements in $x-$direction at three surface monitors for flexible
hydrofoil with sheet cavitation. Figure 23: Time-histories of structural
displacements in $y-$direction at three surface monitors for flexible
hydrofoil with sheet cavitation.

The time-histories and corresponding spectrum of the lift coefficients in
three directions are displayed in Figures 20 and 21, respectively. The
formation of sheet cavitation leads to a significant change in the flow
dynamics. To begin, the shedding of sheet cavitation from the trailing edge of
the blade is regular, which leads to a dominant peak (at $St$ = 0.30), and
another harmonic component (at $St$ = 0.60) in the spectrum of the surface
force coefficients in all three directions in Fig. 20. In addition, comparing
Fig. 21 with Fig. 10, the generation of cavitation also leads to a significant
increase in the energy of the fluctuating components in surface forces, which
will result into amplified structural vibrations.

In this case, the structural displacements in $x-$ and $y-$directions for
three surface monitors are displayed in Figures. 22 and 23, respectively. A
comparison of Fig. 14 and Fig. 22 indicates that the structural vibration
amplitudes are significantly increased owing to the sheet cavitation.
Additionally, the overall structural response implies the feature of flutter
behavior, where the hydrofoil structural response resonates with variations of
lift and their harmonics herein. This is demonstrated by Fig. 24, in which the
transverse ($y-$) displacement spectrum includes identical peaks as that in
Fig. 21. The spectrum of the pressure at monitors $F_{1,2,3,4,5,6}$ in the
flow fields are plotted in Fig. 25. Compared to the non-cavitating situation
in Fig .16, the intense peaks of the spectrum at all monitoring points are
concentrated at $St$ = 0.3 and 0.6, except for the pressure spectrum at
$S_{1}$ (whose location is immediately adjacent to the blade tip), where a
peak at around $St$ = 0.2 still occurs. In conjunction with the above
discussion, such structural vibration necessarily affects the pressure
fluctuations within the flow field, which in turn affects the characterization
of the noise source. Hence, it could be concluded here the cavitation and
structural vibrations co-dominate the generation of noise sources for
propeller singing behaviors. Moreover, there is also a reciprocal effect
between cavitation structures and hydrofoil vibrations. If tip vortex
cavitation appears, it would also be influenced by structural vibration and
thereby could impact the propeller singing behavior.

Figure 24: Spectrum of structural displacements in $y-$direction at three
surface monitors for flexible hydrofoil with sheet cavitation. Figure 25:
Spectrum of pressure fluctuation at monitors $F_{1,2,3,4,5,6}$ for flexible
hydrofoil with sheet cavitation.

##  5 Conclusion

Using the recently developed cavitation FSI solver, this paper studied the
coupled dynamics of flow-induced vibration and cavitation of cantilever
flexible hydrofoils and explored the potential impact on hydro-noise sources.
The present work involved the computation of three cases: a rigid hydrofoil
without cavitation, a flexible hydrofoil without cavitation, and a flexible
hydrofoil with cavitation. The comparative analysis indicated that the
pressure fluctuations within the flow field are mainly affected by the tip
vortex shedding, the trailing-edge vortex shedding, and the structural
vibration. Among them, the tip vortex shedding and blade vibration are
correlated to the intense peak (low-frequency tonal components) of the noise
source, while the trailing-edge vortex shedding leads to broadband noise. We
further analyzed the effect of sheet cavitation’s appearance on the flow-solid
interaction response. The results demonstrated that the shedding of sheet
cavitation induces considerable periodic forces on the hydrofoil surface,
which induces flutter-like response, dominates the blade vibration, and
affects the pressure fluctuations in the flow field. The frequency of tonal
compoments inside potential noise sources are consistent with those of
cavitation shedding behaviors and structural vibration. It could be summarized
that the cavitating behaviors and structural vibrations co-dominate the
features of noise sources for propeller singing behavior. Moreover, there is
also a reciprocal effect between cavitation structures and hydrofoil
vibrations. The deformation and oscillation of hydrofoil will also in turn
modify the vortex and cavitation shedding dynamics. In the future, further
refined meshing strategy will be considered to resolve the tip vortex
cavitation and examine the underlying mechanism of noise generation and
propagation. Additionally, detailed work should be conducted to locate the
resonance range of the considered FIV system.

## Acknowledgment

The present study is supported by Mitacs, Transport Canada and Clear Seas
through a Quiet-Vessel Initiative (QVI) program. This work was made possible
by the facilities of the Shared Hierarchical Academic Research Computing
(SHARCNET) and Compute/Calcul Canada.

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