# A numerical study on the oscillatory dynamics of tip vortex cavitation

Saman Lak1  Rajeev Jaiman1<EMAIL_ADDRESS>1Department of Mechanical
Engineering, University of British Columbia, Vancouver, BC, Canada

###### Abstract

In this paper, we numerically study the mechanism of the oscillatory flow
dynamics associated with the tip vortex cavitation (TVC) over an elliptical
hydrofoil section. Using our recently developed 3D variational multiphase flow
solver, we investigate the TVC phenomenon at a Reynolds number of
$Re=8.95\times 10^{5}$ via dynamic subgrid-scale modeling and the homogeneous
mixture theory. To begin, we examine the grid resolution requirements and
introduce a length scale that considers both the tip vortex strength and the
core radius. This length scale is then employed to non-dimensionalize the
spatial resolution in the tip vortex region and establish the mesh
requirements for large eddy simulation of tip vortex cavitation. We next
perform simulations to analyze the dynamical modes of tip vortex cavity
oscillation at different cavitation numbers and compare them with the semi-
analytical solution. The breathing mode of cavity surface oscillation is
extracted from the results through the definition of an effective radius. The
temporally-averaged effective radius demonstrates that the cavity experiences
a growth region followed by decay as it progresses away from the tip. Further
examination of the characteristics of local breathing mode oscillations in the
growth and decay regions indicates the alteration of cavity’s oscillatory
behavior as it travels from the growth region to the decay region, with the
oscillations within the growth region being characterized by lower
frequencies. For representative cavitation numbers $\sigma\in[1.2,2.6]$, we
find that pressure fluctuations exhibit a shift of the spectrum towards lower
frequencies as the cavitation number decreases, similar to its influence on
breathing mode oscillations. The results indicate the existence of
correlations between the breathing mode oscillations and the pressure
fluctuations. It is found that the growth and decay regions contribute
differently to the pressure fluctuations. The low-frequency pressure
fluctuations are found to be correlated with the growth region, whereas the
breathing mode oscillations within the decay region are correlated with
higher-frequency pressure fluctuations, where the center frequency of a hump
is observed.

###### keywords:

##  1 Introduction

Noise pollution due to human activity (i.e., anthropogenic noise) poses
serious threats to the marine ecosystem (Duarte et al. (2021)). One of the
main sources of anthrophony is shipping, which generates noise through the
operation of propellers, hull vibrations, and onboard machinery (Smith &
Rigby, 2022). A significant proportion of the propeller noise is due to
cavitation phenomenon occurring in various forms such as sheet and tip vortex
cavitation. In most cases, tip vortex cavitation is the first form of
cavitation appearing on ship propellers (Zhang et al. (2015)), which arises
due to the low pressure within the vortex trailing from the tip of propeller
blades.

Tip vortex forms on lifting surfaces of finite length due to the pressure
difference between the pressure side and the suction side which disappears at
the tip. The pressure gradient on the pressure and suction sides leads to
opposite spanwise velocity components on these surfaces, which generates a
vortex trailing from the tip (Platzer & Souders (1979)). The wake sheet shed
from the blade rolls up into the tip vortex as flow moves downstream, further
strengthening the tip vortex (Batchelor (1964)). The pressure within the
vortex core may drop below the vapor pressure, leading to a cavitating tip
vortex. Tip vortex cavitation encompasses a complex dynamical interaction
among vortical motion, cavitation, and turbulence (Arndt (2002)). This complex
phenomenon can be observed in a relatively simple hydrofoil section which
produces a strong swirling flow with pressure reduction and cavitation
process. This paper is motivated by the need to provide an improved
understanding of the interplay of cavitation and vortex dynamics.

Understanding the physics of tip vortex cavitation, including the tip vortex
roll-up and formation, TVC inception, tip vortex cavity dynamics, and the
contribution of tip vortex cavitation to the underwater radiated noise (URN)
level in various operating conditions is crucial for the development of
effective TVC mitigation strategies. Numerous aspects need to be investigated
when the tip vortex flow field is to be studied, even in non-cavitating
(wetted) conditions. During the formation of the tip vortex, the tip vortex-
boundary layer interactions strongly affect the tip vortex flow field (Maines
& Arndt (1997)). Downstream of the tip, the influence of the roll-up of the
wake into the tip vortex and the complex system of streamwise vortices shed
from the tip needs to be considered (Devenport et al. (1996)). Further
downstream, viscous decay and the introduction of vortex instabilities play an
important role in the breakdown of the tip vortex (Ganesh et al. (2014)).

The occurrence of cavitation further complicates the phenomenon of tip vortex.
These complexities coupled with the high three-dimensionality and anisotropy
of the flow with extreme gradients (Asnaghi et al. (2020)), the impact of
turbulent fluctuations on the cavitation inception (Arndt & Keller (1992)),
and tip vortex wandering which gives rise to the need for special treatments
when analyzing the acquired data (Devenport et al. (1996)), makes the
investigation of tip vortex cavitating flows a complex endeavor. Despite all
of these difficulties associated with the investigation of cavitating tip
vortex flow, various researchers have studied the tip vortex cavitation
phenomenon using theoretical, experimental, and numerical methods focusing on
the contributions of TVC to underwater radiated noise.

A significant proportion of the studies aiming to investigate the
contributions of TVC to underwater radiated noise place their focus on the
oscillations of the tip vortex cavity surface. Different modes of oscillation,
namely the breathing mode, serpentine (displacement) mode, and double helical
mode, are observed on the tip vortex cavity interface. These cavity
oscillation modes are depicted in figure 1. These oscillations have been found
to affect the radiated noise (Bosschers, 2018; Wang et al., 2023). An
analytical solution for the dispersion relations of the cavity oscillation
modes was developed by Bosschers (2018). Experimental evidence for this
analytical solution was provided by Pennings et al. (2015a) by conducting
experiments on tip vortex cavitating flow over a stationary NACA66(2)-415
hydrofoil with an elliptical planform. Furthermore, different modes of tip
vortex cavitation were observed to induce different levels of instabilities
into the tip vortex altering its dynamics (Ye et al., 2023). This study
investigates the development of cavity surface oscillatory behavior as the tip
vortex cavity travels downstream to gain insight into the oscillation dynamics
of different regions of the cavity and their contribution to pressure
fluctuations in the surroundings.

Due to the limitations of experimental methods in cavitating flow
investigations, numerical studies can reveal more details about the flow and
cavity characteristics in TVC analysis. Nevertheless, obtaining sufficiently
accurate results has been a recent development due to the challenges
associated with simulating tip vortex cavitating flows. Large Eddy Simulation
(LES) was shown to outperform Reynolds-Averaged Navier-Stokes (RANS) models
for simulation of the tip vortex flow field by Asnaghi et al. (2017), and
Asnaghi et al. (2020) successfully replicated the flow field of tip vortex in
wetted conditions observed in the experiments of Pennings et al. (2015b) using
Implicit LES (ILES). In cavitating conditions, their results regarding the
azimuthal velocity distribution and the diameter of the cavitating tip vortex
also agreed with the experimental data. They also proposed mesh resolution
requirements for accurate simulation of tip vortex cavitating flows in terms
of the number of nodes across the vortex core.

To the best of the authors’ knowledge, the first numerical study focusing on
tip vortex cavity surface oscillation was carried out by Klapwijk et al.
(2022) using a partially averaged Navier-Stokes and the delayed detached eddy
simulation. Their results included some of the cavity surface oscillations,
but the breathing mode oscillations were not captured. Moreover, it is evident
from the cavity morphology they obtained in their work that the tip vortex
cavity was over-dissipated and could not capture a major proportion of the
surface oscillations. Recently, the breathing mode cavity surface oscillations
were successfully captured in the numerical study by Wang et al. (2023). The
dominance of the hydroacoustic contribution of the breathing mode was also
demonstrated in their work, with the tonal frequency and the center frequency
of the broadband hump observed in the noise spectrum being attributed to the
breathing mode at zero streamwise wavenumber and the tip vortex cavity
resonance. However, their study includes the results for only one cavitation
number. Moreover, they only considered the developed tip vortex cavity and not
the formation region, i.e., where the wake roll-up into the tip vortex is not
complete. In addition, their method for extracting the breathing mode
oscillations from the cavity surface oscillation data requires further
improvement due to the center of the cavity not being well-defined in tip
vortex cavitating flows.

In this study, using LES and the homogeneous mixture model, the tip vortex
cavitating flow over a NACA66(2)-415 stationary hydrofoil is numerically
investigated via our in-house 3D variational finite element solver. In order
to address the adequacy of mesh resolution requirements for the simulation of
tip vortex cavitating flows, a length scale is developed based on the radial
pressure gradient within the vortex core, which takes into account the
strength of the tip vortex as well as its core radius. This length scale is
then employed to non-dimensionalize the mesh resolution in the tip vortex
region to propose general mesh resolution requirements for the simulation of
tip vortex flow, rather than merely considering the number of grid nodes
across the vortex core radius. After the assessment of the numerical results
with the experimental and analytical data, a comprehensive investigation of
the breathing mode of cavity surface oscillations is carried out for three
cases with different cavitation numbers.

To extract the breathing mode oscillations, we define an effective radius
based on the cross-sectional area of the tip vortex cavity that captures the
cavity volume variations. Different regions the tip vortex cavity experiences
as it progresses away from the tip are examined using the temporally-averaged
effective radius. The spatially-averaged effective radius is analyzed to
discern the temporal variations of the vapor volume of the entire tip vortex
cavity. The spatial-temporal characteristics of local breathing mode
oscillations are further investigated to understand the characteristics of
this oscillatory mode and its development along the cavity in different
cavitation numbers. Proper Orthogonal Decomposition (POD) is employed to
further examine the streamwise variations of the breathing mode
characteristics. Next, the pressure fluctuations are probed at different
locations within the domain in all of the cases to investigate how the
contributions of tip vortex cavitation to the pressure fluctuations vary in
different cavitation numbers. The spectra of the pressure fluctuations are
then compared to the breathing mode oscillation spectra within different
regions of the tip vortex cavity to gain insight into the correlations between
the local breathing oscillations in different regions of tip vortex cavity and
the pressure fluctuations.

The remainder of the paper is structured as follows. In §2, the computational
framework employed in this work is described. The problem setup and grid
generation strategy are presented in §3. The results obtained from the
simulations are covered in §4. While the focus in §4.1 is placed on the non-
cavitating conditions, §4.2 provides an overview of the oscillation dynamics
of the tip vortex cavity. A systematic investigation of the breathing mode of
oscillation is carried out and presented in §4.3. §4.4 focuses on the pressure
fluctuations in different cavitation numbers and their correlation with
breathing mode oscillations. The key findings and concluding remarks are
summarized in §5.

Figure 1: Three main tip vortex cavity oscillation modes: breathing (left),
displacement (middle), and double helical (right))

##  2 Computational Framework

In this work, our in-house high-fidelity finite element flow solver was
employed for simulating the tip vortex cavitating flow. The mathematical
modeling and numerical formulation are briefly presented here. For further
details, the reader is referred to the original papers on the implementation
of the models used herein (Kashyap & Jaiman (2021), Jaiman et al. (2016)).

###  2.1 Multiphase flow modeling

In this work, the multiphase flow is treated as a continuous homogeneous
mixture of liquid and vapor phases. A phase indicator,
$\phi^{f}(\textbf{x},t)$, determines the phase fraction of the liquid phase at
any coordinate and time in the working fluid physical domain,
$\Omega^{f}(\textbf{x},t)$, where x and $t$ are the spatial and temporal
coordinates within the domain $\Omega^{f}(\textbf{x},t)$ with the boundary
$\Gamma^{f}$. The mixture density ($\rho^{f}$) and the mixture dynamic
viscosity ($\mu^{f}$) are considered to be linear combinations of the density
and viscosity of the liquid and vapor phases and are calculated as

$\rho^{f}=\rho_{l}\phi^{f}+\rho_{v}(1-\phi^{f})\;,$ (1)
$\mu^{f}=\mu_{l}\phi^{f}+\mu_{v}(1-\phi^{f})\;,$ (2)

where the properties with $l$ and $v$ subscripts represent the properties of
the pure liquid and pure vapor phases, respectively. In order to obtain the
value of the phase indicator $\phi^{f}$ within the domain, a scalar transport
equation is solved, which, in its conservative form, is

$\frac{\partial\phi^{f}}{\partial
t}+\phi^{f}\nabla\cdot\textbf{u}+\textbf{u}\cdot\nabla\phi^{f}=\frac{\dot{m}}{\rho_{l}}\;\indent\textrm{on}\;(\textbf{x},t)\in\Omega^{f}\;,$
(3)

where u is the fluid velocity at each coordinate $(\textbf{x},t)$, and
$\dot{m}$ is the finite mass transfer rate from the vapor phase to the liquid
phase occurring due to cavitation.

###  2.2 Cavitation modeling

In equation 3, the source term representing the mass transfer rate from the
vapor phase to the liquid phase $\dot{m}$, which includes both the effect of
condensation and evaporation, is calculated using the model proposed by
Schnerr & Sauer (2001)

$\displaystyle\dot{m}$
$\displaystyle=\frac{3\rho_{l}\rho_{v}}{\rho^{f}R_{B}}\sqrt{\frac{2}{3\rho_{l}|p^{f}-p_{v}|}}\bigg{[}C_{c}\phi^{f}(1-\phi^{f})\max(p^{f}-p_{v},0)$
(4)
$\displaystyle\quad+C_{v}\phi^{f}(1+\phi_{Nuc}-\phi^{f})\min(p^{f}-p_{v},0)\bigg{]}\;,$

where $p^{f}(\textbf{x},t)$ is the working fluid pressure, and $p_{v}$ is the
saturation pressure of the working fluid. $C_{c}$ and $C_{v}$ are the
condensation and evaporation coefficients, which were added to the model in
later numerical implementations (Ghahramani et al. (2019); Cazzoli et al.
(2016)) in order to modify the condensation and evaporation behavior with
respect to specific configurations.

In this model, the initial nuclei within the flow are assumed to be of equal
radius and homogeneously distributed within the domain. $\phi_{Nuc}$ in
equation 4 is the initial phase fraction of the bubble nuclei, which is
related to the initial volume of the nuclei per unit volume as

$\phi_{Nuc}=\frac{v_{nuc}}{1+v_{nuc}}\;,$ (5)

where $v_{nuc}$ is the initial volume of the nuclei per unit volume, which is
related to the nuclei diameter ($d_{nuc}$) and the number of nuclei per unit
volume ($n_{0}$) as $v_{nuc}=\frac{\pi n_{0}d_{nuc}^{3}}{6}$. $R_{B}$ used in
equation 4 is the equivalent radius of the vapor volume assumed to be in the
form of one bubble and is related to other parameters of the flow using the
equation

$R_{B}=\left(\frac{3}{4\pi
n_{0}}\frac{1+\phi_{Nuc}-\phi^{f}}{\phi^{f}}\right)^{1/3}.$ (6)

###  2.3 Fluid conservation of mass and momentum

The spatially filtered Navier-Stokes equations for an incompressible flow are

$\rho^{f}\frac{\partial\overline{\mathbf{u}}}{\partial
t}+\rho^{f}\overline{\mathbf{u}}\cdot\nabla\overline{\mathbf{u}}=\nabla\cdot\overline{\mathbf{\sigma}}^{f}+\nabla\cdot\mathbf{\sigma}^{sgs}+\mathbf{b}^{f}\;\;\;\textrm{on}\;(\mathbf{x},t)\in\Omega^{f}\;,$
(7)

$\nabla\cdot\overline{\mathbf{u}}=0\;\;\;\textrm{on}\;\Omega^{f}\;,$ (8)

where the overbar denotes the spatially filtered quantity, $\mathbf{b}^{f}$
denotes the body force exerted on the fluid, and the Cauchy and subgrid
stresses are represented by $\overline{\mathbf{\sigma}}^{f}$ and
$\mathbf{\sigma}^{sgs}$, respectively. The Cauchy stress for a Newtonian fluid
can be written as

$\overline{\sigma}^{f}=-\overline{p}\mathbf{I}+\mu^{f}\left(\nabla\overline{\mathbf{u}}+\left(\nabla\overline{\mathbf{u}}\right)^{T}\right)\;,$
(9)

where $\overline{p}$ represents the filtered fluid pressure. The subgrid
stress $\sigma^{sgs}$ is the additional stress term appearing in equation 7
due to the filtering applied in large eddy simulation. In order to complete
the set of equations, a model is required for the subgrid stress tensor.

###  2.4 Subgrid-scale modeling

Dynamic subgrid-scale (SGS) model is utilized in this work for modeling the
subgrid-scale stress $\mathbf{\sigma}^{sgs}$. In this model, a filter, denoted
by an overbar, is applied to the Navier-Stokes equations (7, 8), which, as
explained in §2.3, gives rise to a new stress term, the subgrid-scale (SGS)
stress
$\mathbf{\sigma}^{sgs}_{ij}=\overline{u_{i}^{f}u_{j}^{f}}-\overline{u}_{i}^{f}\overline{u}_{j}^{f}$,
to appear in the momentum equation. This stress term corresponds to the
stresses originating from the scales of motion smaller than the spatial
resolution $\Delta$. As mentioned in §2.3, since the SGS quantities are
unknown, a model for the subgrid-scale stress is required for the closure of
the problem.

The nonlinear SGS stress (Gatski & Speziale (1993); Wang & Bergstrom (2005))
is obtained using the equation

$\displaystyle\begin{split}\sigma_{ij}^{sgs}-\frac{\delta_{i}j}{3}\sigma_{kk}^{sgs}\approx&-2\mu_{t}\overline{S}_{ij}-C_{NL}6\mu_{t}^{2}/\sigma_{kk}^{sgs}\\\
&\times\left(\overline{S}_{ik}\overline{\Omega}_{kj}+\overline{S}_{jk}\overline{\Omega}_{ki}-2\overline{S}_{ik}\overline{S}_{kj}+\frac{2}{3}\overline{S}_{nk}\overline{S}_{kn}\delta_{ij}\right)\;,\end{split}$
(10)

where $\mu_{t}$ is the dynamic eddy viscosity, and
$\overline{S}_{ij}\equiv\frac{1}{2}(\frac{\partial\overline{u}_{i}^{f}}{\partial
x_{j}}+\frac{\partial\overline{u}_{j}^{f}}{\partial x_{i}})$ and
$\overline{\Omega}_{ij}\equiv\frac{1}{2}(\frac{\partial\overline{u}_{i}^{f}}{\partial
x_{j}}-\frac{\partial\overline{u}_{j}^{f}}{\partial x_{i}})$ are the resolved
strain rate and rate-of-rotation, respectively. The eddy viscosity $\mu_{t}$
in this equation is related to the mesh resolution and the resolved strain
tensor as

$\mu_{t}=\rho^{f}(C_{s}\overline{\Delta})^{2}|\overline{S}|\;,$ (11)

where $|\overline{S}|$ is the norm of the resolved strain rate tensor
calculated as $|\overline{S}|=(2\overline{S}_{ij}\overline{S}_{ij})^{1/2}$. In
order for LES to yield reasonable results close to the wall, a simple
algebraic eddy viscosity model (Balaras et al. (1996); Piomelli & Balaras
(2002)) is employed instead of equation 11 close to the wall as
$\frac{\mu_{t}}{\mu^{f}}=\kappa y^{+}_{w}(1-e^{-y_{w}^{+}}/A)$, where
$y^{+}_{w}=y_{w}u_{\tau}/\nu$ is the non-dimensional distance to the wall in
wall units with the friction velocity and the model coefficient being denoted
by $u_{\tau}$ and $\kappa$, respectively, and $A$ having a constant value of
19.

In the dynamic SGS model (Germano et al. (1991)), a test filter is defined in
addition to the grid filter. The grid filter, denoted by an overbar, has a
scale dimension of $\Delta$, and the test filter, denoted by the operation
$\widehat{(\;\;)}$, is any coarser level filter with a scale dimension of
$\widehat{\Delta}$ (Sagaut (2005)). The SGS stress at the test level
$T_{ij}=\widehat{\overline{u_{i}u_{j}}}-\widehat{\overline{u}}_{i}\widehat{\overline{u}}_{j}$
is obtained by test-filtering the equations of motion. The Leonard stress
tensor $\mathscr{L}_{ij}$, which is the stress associated with length scales
between the test filter size and the grid filter size

$\mathscr{L}_{ij}=\widehat{\overline{u}_{i}\overline{u}_{j}}-\widehat{\overline{u}}_{i}\widehat{\overline{u}}_{j}$
(12)

can be related to $T_{ij}$ and $\sigma^{sgs}_{ij}$ as

$\mathscr{L}_{ij}=T_{ij}-\sigma^{sgs}_{ij}\;.$ (13)

Using Smagorinsky eddy-viscosity model for the unknows stresses
$\sigma^{sgs}_{ij}$ and $T_{ij}$ yields

$\mathscr{L}_{ij}=-2C_{s}^{2}\bigg{(}\widehat{\Delta}^{2}\widehat{|\overline{S}|}\widehat{\overline{S}_{ij}}-\Delta^{2}|\overline{S}|\overline{S}_{ij}\bigg{)}\;,$
(14)

and the Smagorinsky constant is related to a dynamic coefficient $C_{ds}$ as
$C_{s}=\sqrt{C_{ds}}$ for $C_{ds}>0$, and $C_{s}=0$ otherwise. The dynamic
coefficient $C_{ds}$ (Lilly (1992)) is calculated as

$C_{ds}=\frac{1}{2}\frac{\langle M_{ij}\mathscr{L}_{ij}\rangle}{\langle
M_{lk}M_{lk}\rangle}\;,$ (15)

where
$M_{ij}=\widehat{\Delta}^{2}|\widehat{\overline{S}}|\widehat{\overline{S}}_{ij}-\overline{\Delta}^{2}\widehat{|\overline{S}|\overline{S}_{ij}}$
and $\langle\,.\,\rangle$ denotes some type of smoothing process such as
averaging.

##  3 Problem Setup

In this work, the cavitating flow over a stationary NACA66(2)-415 hydrofoil
with an elliptical planform is simulated. The geometry of the problem and the
working conditions are presented in §3.1, and the strategy employed for grid
generation is explained in §3.2, where the derivation of a pressure gradient-
based length scale is presented and is used for non-dimensionalization of the
mesh size within the tip vortex flow region for mesh requirements
specification.

###  3.1 Geometry and working conditions

The case numerically investigated in this study involves the flow over a
stationary NACA66(2)-415 hydrofoil, which is placed within a tunnel with a
square cross-section. The geometry and dimensions of the domain and the
hydrofoil, depicted in figure 2, are chosen identical to those of the
experiments by Pennings et al. (2015b) since their experimental data is used
herein for validation of the numerical framework. Due to manufacturing
constraints encountered in the experimental study (Pennings et al. (2015b)),
the foil employed in this investigation is truncated at the trailing edge. The
root chord length of the foil after truncation is $c_{0}=0.1256\;m$ and the
half-span of the foil is $h=0.150\;\mathrm{m}$. As shown in figure 2, the
inflow and outflow boundaries are located $5c_{0}$ and $10c_{0}$ in front of
and behind the hydrofoil, respectively. The foil’s tip is placed at the center
of the test section, having a distance of $0.150\;\mathrm{m}$ from the lateral
walls. The origin of the coordinate system is placed at the tip in the
simulations, and $x$, $y$, and $z$ directions correspond to the streamwise,
lift force, and spanwise directions, respectively, as shown in figure 2.

Figure 2: Schematic of the computational domain and boundary conditions

The inflow velocity and the Reynolds number are kept constant equal to
$U_{\infty}=6.8\;\mathrm{m/s}$ and $Re=\frac{U_{\infty}c_{0}}{\nu}=8.95\times
10^{5}$, respectively, in all of the cases, similar to the experiments. The
simulations are carried out at an angle of attack of $\alpha=9\degree$.
Simulations are conducted both for wetted and cavitating conditions. The
cavitation number ($\sigma$) is defined based on the outflow pressure as

$\sigma=\frac{p_{\infty}-p_{sat}}{\frac{1}{2}\rho U_{\infty}^{2}}\;,$ (16)

where $p_{sat}$ and $p_{\infty}$ are the saturation and the outflow pressure,
respectively. In this work, the simulations are carried out at three
cavitation numbers $\sigma=1.2$, $1.7$, and $2.6$. The primary emphasis is on
the case with a cavitation number of $\sigma=1.7$ due to the occurrence of
moderate cavitation in this scenario. The inflow velocity is set to be
constant equal to $\mathbf{u}_{in}=(U_{\infty},0,0)$. The outflow boundary
condition is weakly set as constant pressure. The bottom wall of the domain
and the hydrofoil surface are set as no-slip walls and other domain walls are
of symmetry (slip wall) boundary condition.

In order for the flow to reach statistically stationary conditions in a
shorter computation time, the initial condition was obtained from simulations
with a coarser grid and a larger time step, and the results were projected
onto the desired grid. Using this initial condition, in the first step, the
simulations were carried out for $0.2\,\mathrm{s}$ with a time step of $\Delta
t=10^{-4}\,\mathrm{s}$ in wetted conditions. Subsequently, the time step was
decreased to $\Delta t=10^{-5}\,\mathrm{s}$, which was found to be adequately
small for accurate results. Cavitation was started after $0.3\,\mathrm{s}$ of
flow time, and cavitating flow simulations were carried out for
$0.5\,\mathrm{s}$. The results obtained for the last $0.2\,\mathrm{s}$ of the
flow time were used in this work for post-processing and analysis. Due to the
computation time limitations, the cases of $\sigma=2.6$ and $\sigma=1.2$ were
simulated for $0.4\,\mathrm{s}$ and $0.45\,\mathrm{s}$, respectively,
following the onset of cavitation.

###  3.2 Grid generation strategy

It is well-known that due to the high gradients and small scales in tip vortex
cavitating flows, adequate mesh resolution in simulating such flows is of
critical significance due to the high susceptibility of such flows to
numerical diffusion (Asnaghi et al. (2020)). Therefore, an adequately fine
mesh is required within the tip vortex flow region. Due to the various
resolution requirements in different regions of cavitating tip vortex flow,
the grid needs to be meticulously designed in order to satisfy mesh
requirements while avoiding excessive computational costs. In this work, the
grid is designed and generated using Pointwise software.

(a) Streamwise distribution

(b) In-plane distribution

Figure 3: Grid distribution in the streamwise and in-plane directions

The cross-sectional views of the grid distribution employed in this work in
the streamwise and transverse directions are illustrated in figure 3.
Triangular mesh elements are generated on the hydrofoil surface satisfying the
requirements of boundary-layer-resolved LES, i.e., $\\{x^{+},z^{+}\\}<250$
(Asnaghi et al. (2020)). These triangular elements are extruded normal to the
hydrofoil surface generating prism layers as the boundary layer mesh. Due to
the importance of boundary layer flow in tip vortex cavitation simulation, the
height of prism elements is selected to ensure that the dimensionless wall-
normal distance ($y^{+}$) remains below 5. Outside the prism boundary layer,
tetrahedral elements are utilized. A refinement region is created enclosing
the hydrofoil and the region where flow dynamics are present. A refinement box
is added in the wake region to capture the effects of wake dynamics on the tip
vortex roll-up process. A cylindrical refinement region with a radius of
$r=30\;\mathrm{mm}$ is created, starting from a streamwise position slightly
upstream of the tip and extending $2c_{0}$ downstream of the tip. A smaller
refinement region along the path of the tip vortex is created enclosing the
tip vortex core to capture the extreme gradients in the tip vortex flow. This
region is created by sweeping the tip vortex trajectory by a circle with a
radius of $r=5\mathrm{mm}$, which is obtained through preliminary simulations
with a coarse cylindrical refinement.

The mesh resolution within the tip vortex region is one of the major factors
determining the accuracy of tip vortex cavitation simulation. Many researchers
have proposed various mesh requirements for simulating tip vortex cavitation
(Asnaghi et al. (2020); Ahmad et al. (2013)). All of these recommendations are
based on the number of grid points across the tip vortex diameter; however,
merely considering the number of grid points across the tip vortex diameter
does not take all the influencing parameters into account and would not
necessarily lead to sufficient resolution. Other characteristics of the tip
vortex flow, such as the circulation ($\Gamma$), need to be taken into account
as well. There might exist tip vortex flows with the same vortex core
diameters but different strengths, and the vortex flow with higher strength,
indeed, requires finer mesh resolution. Therefore, a dimensionless mesh
resolution within the tip vortex flow region ($\Delta r^{*}$) is defined
herein to account for the effect of the tip vortex strength as well as its
core radius.

In the simulation of tip vortex cavitating flows, capturing the radial
gradient of pressure $\frac{\partial p}{\partial r}$ is of utmost
significance. Therefore, the maximum radial pressure gradient together with
the fluid density $\rho$ and kinematic viscosity $\nu$ is used to define a
length scale for the tip vortex flow as

$\mathcal{L}_{\Gamma}=\biggl{(}\frac{\rho\nu^{2}}{(\frac{\partial p}{\partial
r})_{max}}\biggr{)}^{\frac{1}{3}}\;.$ (17)

Since the maximum gradient of pressure is not known prior to the simulations
and experiments, it needs to be related to the a priori known properties of
the flow, such as the Reynolds number and the lift coefficient ($C_{L}$),
which is known from the experimental data. In order to relate the maximum
radial gradient of pressure to the known parameters of the flow, the tip
vortex flow field is assumed to be axisymmetric and the radial velocity is
assumed to be zero. In this case, the radial momentum equation simplifies to
$\frac{\partial p}{\partial r}=\rho\frac{u_{\theta}^{2}}{r}$, where $r$ is the
radial distance from the vortex center, and $u_{\theta}$ is the azimuthal
velocity of the flow. The Rankine vortex model with a circulation of $\Gamma$
and a core radius of $r_{v}$ is employed to obtain the azimuthal velocity. The
Rankine vortex model assumes a solid-body-like rotation within an inner core
($r<r_{v}$), thereby having a radial velocity distribution of
$u_{\theta}(r)=\frac{\Gamma r}{2\pi r_{v}^{2}}$. The vortex flow outside the
rigid core is assumed to be an irrotational vortex
($u_{\theta}(r)=\frac{\Gamma}{2\pi r}$).

The maximum radial pressure gradient in a Rankine vortex occurs on the vortex
core boundary, i.e., $r=r_{v}$, where the azimuthal velocity in the Rankine
vortex model is $u_{\theta}=\frac{\Gamma}{2\pi r_{v}}$, the substitution of
which in the simplified radial momentum equation yields

$\left(\frac{\partial p}{\partial
r}\right)_{max}=\frac{\rho\Gamma^{2}}{4\pi^{2}r_{v}^{3}}$ (18)

for the maximum radial pressure gradient. Using the maximum radial pressure
gradient obtained from equation 18 in equation 17 and rearranging based on the
vortex Reynolds number ($Re_{\Gamma}=\Gamma/\nu$) yields

$\mathcal{L}_{\Gamma}=(2\pi)^{\frac{2}{3}}r_{v}Re_{\Gamma}^{-\frac{2}{3}}\;.$
(19)

Based on equation 19, the length scale $\mathcal{L}_{\Gamma}$ scales linearly
with the vortex core radius $r_{v}$ and scales inversely with
$Re_{\Gamma}^{\frac{2}{3}}$, indicating that as the vortex core radius
decreases or the vortex strength, represented by the vortex Reynolds number
$Re_{\Gamma}$, increases, the length scale $\mathcal{L}_{\Gamma}$ decreases,
which indicates the need for a finer mesh to be employed in the tip vortex
region.

The circulation of the tip vortex $\Gamma$ and the tip vortex core radius
$r_{v}$ are unknown before the numerical analysis or experiments. Therefore,
these quantities need to be estimated as well for the calculation of the
length scale $\mathcal{L}_{\Gamma}$. Fruman & Dugue (1992) assumed that the
local tip vortex circulation and vortex core radius are related to the wing
mid-span circulation $\Gamma_{0}$ and the boundary layer thickness $\delta$,
respectively. The mid-span circulation of the hydrofoil $\Gamma_{0}$ is
calculated using the Kutta-Jukowski theorem as
$\Gamma_{0}=\frac{1}{2}C_{L}U_{\infty}c_{0}$. Astolfi et al. (1999) stated
that the vortex core radius is directly related to the turbulent boundary
layer thickness ($\delta$) of a flat plate with a length of one blade root
chord length $c_{0}$, which can be related to the Reynolds number using the
equation $\delta=0.37c_{0}{Re}^{-0.2}$. Pennings et al. (2015a) obtained the
coefficients of proportionality $\Gamma/\Gamma_{0}$ and $r_{v}/\delta$ to be
in the range $0.44<\Gamma/\Gamma_{0}<0.49$ and $0.56<r_{v}/\delta<0.61$. In
the calculations of Astolfi et al. (1999), these values were in the ranges
$0.5<\Gamma/\Gamma_{0}<0.6$ and $0.8<r_{v}/\delta<1.0$. These values reveal
that since the goal is to find an estimation for the required mesh size within
the tip vortex core, the turbulent boundary layer thickness and the mid-span
wing circulation can be substituted for the tip vortex core radius and the tip
vortex circulation in equation 19, respectively, and all of the coefficients
can be omitted since they will merely cause scaling of the results. The
pressure gradient-based length scale $\mathcal{L}_{\Gamma}$ is described in
terms of the a priori known flow parameters as

$\mathcal{L}_{\Gamma}=\biggl{(}\frac{16\pi^{2}}{C_{L}^{2}{Re}^{2.6}}\biggr{)}^{\frac{1}{3}}c_{0}\;.$
(20)

The length scale $\mathcal{L}_{\Gamma}$ is employed to non-dimensionalize the
mesh element size within the tip vortex region $\Delta r$ as $\Delta
r^{*}=\frac{\Delta r}{\mathcal{L}_{\Gamma}}$. In this work, simulations are
carried out using different values for the non-dimensional mesh resolution
$\Delta r^{*}$ within the tip vortex region, and mesh requirements are
proposed accordingly in terms of the non-dimensional mesh resolution. The mesh
resolution values within the tip vortex flow region and their corresponding
non-dimensional values employed in the simulations carried out in this study
are presented in table 1, the results of which are presented in §4.1.

Case | Mesh size | $\Delta r^{*}$ | Number of Nodes
---|---|---|---
I | 0.250 mm | 46 | 10.3 m
II | 0.125 mm | 23 | 13.5 m
III | 0.100 mm | 18.5 | 16.9 m
Table 1: Simulation cases for mesh resolution effect study

##  4 Results and discussion

In this section, the results obtained from the simulations carried out using
our finite element solver, the details of which were discussed in §2, are
presented. In §4.1, the results of non-cavitating tip vortex flow simulations
are compared with the experimental data (Pennings et al. (2015b)), and the
effect of mesh resolution on the accuracy of the simulations is investigated.
The tip vortex cavitating flow simulation results are showcased in §4.2 and
assessed with the semi-analytical solution by Bosschers (2018). The breathing
mode of oscillation is further investigated at different cavitation numbers in
§4.3, and the pressure fluctuations within the domain are analyzed in §4.4 to
ascertain the correlations between the breathing mode oscillations of the tip
vortex cavity and the pressure fluctuations within the domain.

###  4.1 Non-cavitating flow

To study the effect of mesh resolution on the wetted tip vortex flow
simulation, and to validate the capability of the presented numerical
framework in simulating such flows, the axial ($x$ direction) and in-plane
(parallel with $yz$ plane) velocity distribution at three different downstream
locations ($x/c_{0}=0.5,\,0.75,\,1.14$) are compared qualitatively with the
experimental data, and quantitative comparison is carried out for the time-
averaged lift coefficient and the in-plane velocity distribution at the
downstream location $x/c_{0}=1.14$. Due to the effect of the lift coefficient
on the tip vortex flow field, both the lift coefficient and the velocity
profiles need to be accurately predicted in the simulations in order for the
tip vortex cavitating flow simulations to have adequate accuracy.

The data obtained from the simulations need to be processed prior to
comparison with the experimental data. First, the vortex core center is
identified at every time step assuming that the minimum pressure within the
vortex core occurs at the vortex center. In the next step, the results
obtained from 3D simulations are projected onto a 2D cross-section of the flow
orthogonal to the streamwise direction (parallel with the $yz$ plane). The
mesh projected onto the 2D plane is illustrated in figure 4(a). In the next
step, a new polar coordinate system is defined with the vortex core center as
the origin, and the data is interpolated from the 2D plane of interest onto a
polar mesh, depicted in 4(b), around the tip vortex core center using cubic
spline interpolation method. This process is carried out for different flow
times, and then the quantities obtained for various flow times on the vortex-
based polar coordinate system are averaged after aligning the tip vortex
center obtained for all flow times used in averaging. Utilizing this method of
post-processing eliminates the effect of vortex wandering on time-averaging.
For the quantitative comparison of azimuthal velocity at the downstream
location $x/c_{0}=1.14$ with the experimental measurements, the in-plane
velocity values are spatially averaged on $90\degree$ arcs between the two
perpendicular black lines illustrated in 5(b), same as the averaging method
used in Pennings et al. (2015b) since the asymmetry of the tip vortex flow
field at this location necessitates the utilization of the same averaging
method for the sake of accuracy.

(a) Triangular mesh on a 2D plane orthogonal to the free-stream flow direction
obtained from interpolation from the 3D mesh

(b) Polar grid points generated around the tip vortex center for interpolation
of the data from the 2D cross-section of the flow

Figure 4: Projected 2D triangular mesh and polar mesh used during the data
processing for comparison with the experimental data

(a) Non-dimensional Axial Velocity

(b) Non-dimensional In-plane Velocity

Figure 5: Comparison of non-dimensional velocity distribution within the tip
vortex region at downstream location of $x/c_{0}=1.14$ obtained from
simulations with different mesh sizes with the experimental measurements

As a quantitative comparison, the time- and contour-averaged azimuthal
velocity profiles predicted by the simulations employing various mesh sizes
within the tip vortex region at the cross-section $x/c_{0}=1.14$ are shown in
figure 6. This plot shows that accurately capturing the velocity distribution
of tip vortex requires extremely high mesh resolution within this region, and
utilization of inadequate resolution leads to excessive dissipation and
underprediction of the maximum azimuthal velocity and its gradient. Moreover,
employing adequately small grid elements within the tip vortex region is
necessary for the correct prediction of the tip vortex core radius. The
results presented in figure 6 indicate that the results obtained from the
simulation with $\Delta r^{*}=46$ are excessively dissipated compared to the
experimental data. Refining the grid to $\Delta r^{*}=23$ led to the overall
correct capturing of the velocity distribution, but the tip vortex flow field,
in this case, is slightly dissipated. The utilization of a grid with $\Delta
r^{*}=18.5$ led to accurate prediction of the tip vortex flow field.

Based on the qualitative and quantitative comparisons of the predicted axial
and azimuthal velocity field presented herein, it can be concluded that the
non-dimensional grid size $\Delta r^{*}$ requires to be chosen below 20 in
order for the simulations to accurately capture the dynamics of tip vortex
flow. This guideline can serve as a general mesh resolution criterion for
future works on tip vortex flow simulation. Furthermore, the agreement between
the results obtained from the simulation using a non-dimensional mesh
resolution of $\Delta r^{*}=18.5$ in the tip vortex region reveals the
capability of the developed finite element-based numerical framework in
simulating non-cavitating tip vortex flows.

Figure 6: Comparison of normalized azimuthal velocity profile at
$x/c_{0}=1.14$ obtained from simulations using various grid sizes with
experimental data

###  4.2 Overview of TVC Oscillatory Dynamics

Similar to the experimental observations (Pennings et al. (2015a)), the tip
vortex cavity observed in the simulations carried out in this work exhibits
complex surface oscillations as shown in figure 7. As explained in §1, these
surface oscillations have been found to be one of the sources of TVC noise.
Therefore, investigating the physics of these oscillations and deformations,
especially the breathing mode of oscillation, is crucial to understanding the
contributions of TVC to the URN. In this section, an overview of the
oscillations of the tip vortex cavity surface is presented for the case of
$\sigma=1.7$ using a method similar to what is commonly employed in
experimental studies (Pennings et al. (2015a)) to extract various modes of
cavity surface oscillations, which uses the diameter of the cavity as viewed
from different sides to distinguish between different modes of oscillation.
Furthermore, the results obtained using this method are compared with the
analytical solution proposed by Bosschers (2018).

Figure 7: Tip vortex cavity surface observed in simulations with a cavitation
number of $\sigma=1.7$

In order to extract various modes of the cavity surface oscillations,
illustrated in figure 1, from the complex cavity surface similar to the tip
vortex cavity illustrated in figure 7, the tip vortex cavity diameter observed
from different views is employed in experimental studies (Pennings et al.
(2015a); Bosschers (2018)). This method can be utilized in numerical studies
to attain an overview of the oscillations captured in the simulations, which
can be used as a basis for more complex post-processing of the results. The
non-dimensional tip vortex cavity radius observed from the side ($xz$ view)
and top ($xy$ view) of the domain is presented in figure 8, which clearly
indicates the presence of cavity surface oscillations in the results of the
simulations. The radius used for non-dimensionalizing the data throughout this
work, i.e., $r_{c}$, is the average of the spatially- and temporally-averaged
radius observed from the top and side views in the region of interest
($0.1<x/c_{0}<1.2$).

Figure 8 shows that the cavity surface exhibits a relatively stationary
behavior from the tip until $x/c_{0}\sim 0.5$ (formation region), and
transitions to a more dynamic behavior further downstream (developed region).
Comparing figures 8(a) and 8(b) reveals that the spatial diameter variations
of the roll-up region of the cavity ($0.1<x/c_{0}<0.5$) observed from these
two views are out-of-phase, which indicates the presence of a nearly
stationary twisting shape (double-helical mode) in this region of the cavity
similar to the observations of Ye et al. (2023). In order to extract the
cavity oscillation modes from the side-view and top-view diameter, similar to
the method employed by Bosschers (2018), the cross-power spectral density
(CPSD) is calculated from the two-dimensional Fast Fourier Transform (FFT) of
the data obtained from the two views using the equation

$S(k_{x},f)=120+10\log_{10}\left[\frac{G_{top}(k_{x},f)G^{*}_{side}(k_{x},f)}{r_{c}^{2}}\right]\;,$
(21)

where $k_{x}$ denotes the streamwise wavenumber, $G(k_{x},f)$ represents the
2D FFT of the radius data observed from the corresponding view, and
$G^{*}(k_{x},f)$ is its complex conjugate.

The wavenumber-frequency dependence of $S(k_{x},f)$ is depicted in figure
9(a). Various modes of cavity oscillation manifest themselves as curves with
higher $S(k_{x},f)$ values as shown in figure 9(a). The phase difference
between the 2D FFT of the radius data obtained from the side and top views is
also required in order to distinguish between the breathing and double helical
modes. The phase difference, illustrated in figure 9(b), is equal to
$180^{\circ}$ in double helical mode and $0^{\circ}$ in breathing mode
oscillations. The results depicted in these figures clearly indicate the
presence of the breathing and the double helical modes.

The analytical solution proposed by Bosschers (2018) is also plotted in figure
9(a) for comparison with the results obtained from the simulations in this
work. The analytical solution for the dispersion relation of the cavity
surface waves is

$\frac{2\pi
r_{c}f^{\pm}}{U_{\infty}}=\tilde{U}_{x}k_{x}r_{c}+\tilde{U}_{\theta}n\pm\sqrt{K_{\sigma}}\sqrt{\frac{-|k_{x}r_{c}|\,K^{\prime}_{n}(|k_{x}r_{c}|)}{K_{n}(|k_{x}r_{c}|)}}T_{\omega}\;,$
(22)

where $\tilde{U}_{x}$ and $\tilde{U}_{\theta}$ are the non-dimensional axial
and azimuthal velocities, respectively. $K_{n}$ and $K^{\prime}_{n}$ denote
the modified Bessel function of the second kind and its first derivative,
respectively. $K_{\sigma}$ is a non-dimensional stiffness coefficient, and
$T_{\omega}$ represents the effect of surface tension, which is neglected in
this work ($T_{\omega}=1$). The values $\tilde{U}_{x}=1.05$,
$\tilde{U}_{\theta}=0.68$ and $K_{\sigma}=0.32$ are used herein.

The results illustrated in figure 9(a) indicate that all of the four double
helical mode oscillations ($n=\pm 2^{\pm}$) are captured in the simulations,
and these modes of cavity oscillation observed in the simulations agree
reasonably well with the analytical results. The modes $n=2^{+}$ and
$n=-2^{-}$ observed in the results obtained from simulations deviate from the
analytical results, which might be resolved if the parameters in equation 22
are adjusted further. Moreover, some high-CPSD curves with $0^{\circ}$ phase
difference can be observed in figure 9(a) that are not predicted by the
analytical model.

Another dominant stripe observed in figure 9(a), which exhibits a phase
difference of $0^{\circ}$ between the two views, shows the convection of the
disturbances on the cavity surface. The convection group velocity ($c_{g}$) in
this case is found to be $c_{g}=0.9U_{\infty}$, which is similar to the value
used by Bosschers (2018), i.e., $c_{g}=0.95U_{\infty}$. In addition, figure
9(a) shows that the wavenumber-frequency lines obtained for the breathing and
double helical modes are close in this case, and the dominance of the double-
helical mode oscillations in this case hinders the detection of the breathing
mode oscillations. Only traces of the breathing mode oscillations can be seen
in the phase difference between the two views shown in figure 9(b), which
demonstrates that a more effective method is required for the extraction of
the breathing mode from the cavity oscillations. These results indicate that
the developed numerical framework in this work is capable of accurately
capturing the tip vortex cavity surface oscillations, which allows further
investigation of the cavity surface oscillations and their contribution to the
radiated noise.

(a) $xz$-plane (side view)

(b) $xy$-plane (top view)

Figure 8: Spatial-temporal variation of the non-dimensional radius observed at
$\sigma=1.7$ from: (a) side ($xz$-plane) and (b) top ($xy$-plane) views.

(a) Cross-power spectral density obtained from Equation 21

(b) Phase difference between 2D FFT of cavity diameter viewed from the side
and top views

Figure 9: Cavity oscillatory mode extraction using two orthogonal views of the
cavity

###  4.3 Breathing Mode Oscillations

A major proportion of the TVC noise, especially in low and medium frequencies,
is attributed to the mode of oscillation responsible for cavity volume
variations; i.e., the breathing mode (Wang et al. (2023)). Moreover, the
center frequency of the broadband hump observed in the URN spectrum is
hypothesized to be related to the tip vortex cavity resonance frequency
(Bosschers (2018)), and experimental evidence has been provided by Pennings et
al. (2015a) showing that this resonance frequency is the frequency of zero
group velocity of the breathing mode. These connections linking the breathing
mode to the TVC noise necessitate a comprehensive investigation of this mode
of cavity oscillation. Therefore, in this section, this mode of oscillation is
investigated in detail for three cases with different cavitation numbers.
First, in §4.3.1, a new method is developed for extracting the breathing mode
from the cavity surface obtained from the simulations by defining an effective
radius for the tip vortex cavity. The temporally- and spatially-averaged
effective radius of the cavity are analyzed in §4.3.2 and §4.3.3,
respectively. §4.3.4 focuses on the spatial and temporal development of the
effective radius delving into the characteristics of the local breathing mode
oscillations. Proper Orthogonal Decomposition (POD) is performed on the
effective radius data in §4.3.5 to further analyze the streamwise development
of the tip vortex cavity’s volume.

####  4.3.1 Breathing Mode Extraction

In order to examine the breathing mode oscillations in detail, a method is
required for extracting this mode of cavity oscillation. Klapwijk et al.
(2022) fitted an ellipse onto the transverse cross-section of the cavity
interface, and defined an effective radius as $r_{eff}=\sqrt{ab}$, where $a$
and $b$ are the semi-major and semi-minor axes of the fitted ellipse,
respectively. However, as explained by Wang et al. (2023), the method used in
Klapwijk et al. (2022) might lead to loss of deformation information when the
cross-section is non-elliptical. Such a highly non-elliptical transverse
cross-section of the cavity was also observed in our simulations as shown in
figure 11.

Figure 10: Calculation of the effective radius using the tip vortex cavity
surface data

Figure 11: Frequency-wavenumber diagram of the effective radius data
($\sigma=1.7$)

Since the breathing mode is the only cavity oscillation mode responsible for
the cavity volume variations, the idea of employing the cavity’s cross-
sectional surface area for calculating an effective radius is a sound
approach. Based on this approach, first, the transverse cross-section of the
cavity at different downstream locations is extracted, the area enclosed by
which is then calculated. Subsequently, an effective radius is calculated
based on the area enclosed by the cavity cross-section as $A_{enclosure}=\pi
r_{eff}^{2}$ as depicted in figure 11. The effective radius obtained using
this method only contains the volume variation information, and other forms of
oscillation are excluded from this quantity. Therefore, this parameter can be
employed to investigate breathing mode oscillation in the absence of the
interference of other modes.

The two-dimensional Fourier transform of the effective radius obtained using
this method for the case with a cavitation number of $\sigma=1.7$ is
illustrated in figure 11, which displays the wavenumber-frequency dependency
of the breathing mode cavity oscillations together with the analytical
solution. Based on this figure, the breathing mode oscillations obtained from
performing the extraction method on the simulation data agree well with the
analytical solution, which shows that this extraction method together with the
numerical framework employed in this work is able to accurately capture the
breathing mode oscillations.

####  4.3.2 Temporally-averaged Effective Radius

The tip vortex cavity surface can be decomposed into a mean cavity surface and
time-dependent cavity surface oscillations. The formation of the mean cavity
interface is related to the overall roll-up process of the tip vortex. In this
section, the impact of the roll-up process on the mean cavity volume is
investigated by analyzing the time-averaged effective radius. The goal here is
to gain insight into the evolution of the tip vortex cavity from its inception
to decay.

Figure 12 shows the temporally-averaged effective radius, denoted by
$<r_{eff}>$ and non-dimensionalized by $r_{c}$, with respect to the streamwise
location ($x$) non-dimensionalized by the root chord length ($c_{0}$) at three
cavitation numbers. According to the streamwise development of the temporally-
averaged effective radius in these cases, the cavity undergoes a region of
growth, where the cavity is forming and the cavity effective radius has a
growing trend overall, followed by a decay region.

A comparison of different cavitation numbers indicates that the growth region
shrinks as the cavitation number increases. In the case with $\sigma=2.6$, the
decay starts approximately at $x/c_{0}=0.12$. In the case with $\sigma=1.7$,
the growth region extends until $x/c_{0}=0.4$, and when $\sigma=1.2$, the
growth continues further until $x/c_{0}=0.53$. Furthermore, figure 12 reveals
that the decay occurs at a greater slope in higher cavitation numbers. The
most sustained cavity is observed in the case with $\sigma=1.2$, where the
slope of decay ($\frac{dr^{*}}{dx^{*}}$, where $r^{*}=\frac{r_{eff}}{r_{c}}$
and $x^{*}=\frac{x}{c_{0}}$) is approximately $0.146$. This slope of decay in
the cases of $\sigma=1.7$ and $\sigma=2.6$ increases to $0.288$ and $1.027$,
respectively.

An interesting feature of the temporally-averaged effective radius observed in
figure 12 is the spatial oscillations observed in the growth region. In the
case with $\sigma=1.7$, the cavity volume exhibits periodic spatial
oscillation with the downstream location in the growth region ($x/c_{0}<0.4$);
however, in the decay region, these spatial oscillations of the cavity volume
vanish. When $\sigma=2.6$, such oscillations are present until $x/c_{0}=0.23$,
which can be attributed to the smaller growth region in this case compared to
$\sigma=1.7$. On the other hand, as it is evident in this figure, the spatial
volumetric oscillations in the formation region vanish when the cavitation
number is lowered to $\sigma=1.2$.

The spatial variations of the temporally-averaged effective radius in the
cavity growth region are further investigated by calculating their Fourier
transform, which is illustrated in figure 13. The large-amplitude low-
wavenumber components ($\sim 0.1\,\mathrm{rad/mm}$) correspond to the overall
growth and decay of the cavity. An interesting feature observed in the case of
$\sigma=1.7$ is the hump within the wavenumber range of
$0.68\,\mathrm{rad/mm}<k<1.09\,\mathrm{rad/mm}$, which corresponds to a
wavelength range of $0.074\,c_{0}>\lambda>0.046\,c_{0}$. The center wavenumber
of this hump is $0.82\,\mathrm{rad/mm}$, corresponding to a wavelength of
$\lambda=0.061\,c_{0}$. A local maximum at this wavenumber is evident for
$\sigma=2.6$ case as well, which is less significant than the hump observed in
the case of $\sigma=1.7$ due to the spatial oscillations decaying earlier in
$\sigma=2.6$ case compared to $\sigma=1.7$. The existence of this component in
both cases further supports the idea that it stems from the characteristics of
the tip vortex flow rather than from the cavity’s characteristics.

The results presented in this section indicate that the spatial volume
variations occurring within the growth region in the cases of $\sigma=1.7$ and
$\sigma=2.6$ are caused by streamwise variations of the tip vortex flow field
in the roll-up region, which occur within the inner regions of the tip vortex
and are absent in the outer regions. Therefore, further analysis of the mean
flow field is required to understand the cause of this aspect of spatial
volume variations; however, this investigation falls outside the scope of the
present study.

Figure 12: Streamwise evolution of the temporally-averaged effective radius in
different cavitation numbers Figure 13: Fourier transform of the non-
dimensional temporally-averaged effective radius variations

####  4.3.3 Spatially-averaged Effective Radius

The volume of the tip vortex cavity is an important quantity that can reveal
any cyclic process that the cavity might undergo. In this section, the
spatially-averaged effective radius, shown as $\overline{r}_{eff}$, is
employed in order to analyze the volume variations of the entire vapor cavity
in the region of interest ($0.1<x/c_{0}<1.2$). The spatially-averaged
effective radius non-dimensionalized using $r_{c}$ is depicted in figure 14(a)
as a function of the non-dimensional time ($t^{*}=\frac{tU_{\infty}}{c_{0}}$)
for the three cavitation numbers. The spatial averaging for the case of
$\sigma=2.6$ is carried out for the streamwise location range of
$0.1<x/c_{0}<1.0$ because the cavity collapses between $x/c_{0}=1.0$ and
$x/c_{0}=1.2$ in this case. It is evident in figure 14(a) that the volume of
the cavity in the region of interest undergoes temporal variations, which
occur more intensely when $\sigma=1.7$, where cavitation occurs moderately,
compared to the other cases.

The frequency spectra of the spatially-averaged effective radius fluctuations
are illustrated in figure 14(b) obtained using FFT algorithm. The spectra of
the fluctuations demonstrate that a large proportion of the vapor volume
variations occurs in the low-frequency region below $100\,\mathrm{Hz}$ in all
of the cases. Furthermore, a distinct peak is observed in the spectrum of the
cavity volume variations in the case of $\sigma=1.7$ at a frequency of
$55\,\mathrm{Hz}$, which is nearly equal to the characteristic frequency of
the flow, i.e., $f_{ref}=\frac{U_{\infty}}{c_{0}}=54.14\,\mathrm{Hz}$. This
peak in the spectrum is evident in other cases as well, which further supports
the argument that this fluctuation component originates from the flow
characteristic frequency. Moreover, the fluctuations’ spectrum of the scenario
with $\sigma=1.7$ exhibits relatively high amplitudes at $f=25\mathrm{Hz}$ and
$f=30\mathrm{Hz}$, which reveals that the spectrum has a peak between these
frequencies. The case of $\sigma=2.6$ exhibits such a peak
($f=26.7\,\mathrm{Hz}$) as well. Therefore, this component can be attributed
to the subharmonic of the flow characteristic frequency, i.e.,
$\frac{f_{ref}}{2}=27.07\,\mathrm{Hz}$ since it is observed in two cases with
different cavitation numbers.

The spectra of the spatially-averaged effective radius fluctuations of these
cases differ in some other features. A distinct component at
$f=45\,\mathrm{Hz}$ is observed in the case of $\sigma=1.7$. Moreover, the
spectrum of the spatially-averaged effective radius fluctuations in the case
of $\sigma=2.6$ exhibits a local maximum at the second harmonic of the flow
characteristic frequency ($\sim 107\,\mathrm{Hz}$), which is not observed in
other cases. Furthermore, in this case, the spatially-averaged effective
radius possesses fluctuating components at higher frequencies
($220-240\,\mathrm{Hz}$) as well, which is not the case in lower cavitation
numbers. Since these higher frequency fluctuations are not observed in other
cases, these components might be due to the specific characteristics of the
cavity forming at this cavitation number. Overall, it is evident that the
fluctuations of the spatially-averaged effective radius in frequencies greater
than $100\,\mathrm{Hz}$ exhibit higher intensity when $\sigma=2.6$ compared to
other cavitation numbers. Overall, it is observed that increasing the
cavitation number leads to more intense vapor volume temporal variations in
higher frequencies. The results presented in this section show that the
variations of vapor volume of the tip vortex cavity are dominated by the flow
characteristics, and the characteristics of the cavity forming in different
cases have less significant impact on overall vapor volume variations.

(a) Non-dimensional spatially-averaged effective radius temporal development

(b) Frequency spectrum of the non-dimensional spatially-averaged effective
radius variations

Figure 14: Temporal dynamics of spatially-averaged effective radius

####  4.3.4 Spatial-Temporal Characteristics of Breathing Mode Oscillations

The effective radius calculated using the method explained in §4.3.1 is a
function of time and streamwise location; therefore, it can reveal the spatial
and temporal evolution of the breathing mode oscillations. The spatial and
temporal variations of the non-dimensionalized effective radius for the
cavitation number $\sigma=1.7$ are illustrated in figure 15(a), and figure
15(b) displays the non-dimensional fluctuations of the effective radius at
every streamwise location by subtracting the time-averaged effective radius
($<r_{eff}>$) at the corresponding location from the effective radius value.
The spatial oscillations observed in this case in the cavity growth region
$0.1<x/c_{0}<0.4$ can be observed in figure 15(a).

Furthermore, it is evident in figure 15(b) that the behavior of the effective
radius fluctuations alters at $x/c_{0}\sim 0.3$ in this case. It is apparent
in this figure that in the upstream region of $x/c_{0}=0.3$, some of the peaks
and troughs of effective radius fluctuations visible as red and blue stripes,
respectively, propagate upstream (shown as a dashed line in figure 15(b)), in
contrast to downstream of $x/c_{0}=0.3$ where such disturbances propagate
downstream (shown as a dot-dashed line in figure 15(b)). The slope of these
upstream and downstream propagating disturbances reveals that the propagation
speed of the disturbances propagating downstream is greater than that of the
ones propagating upstream in the region $x/c_{0}<0.3$. These differences
reveal the change of cavity oscillatory behavior as it advances from the
growth region to the decay region. Moreover, it can be observed in figure
15(a) that at a streamwise location of $x/c_{0}\sim 0.22$, the effective
radius peaks periodically approximately every
$t_{ref}=\frac{c_{0}}{U_{\infty}}$ (flow characteristic time scale), which
indicates a fluctuating component of the effective radius at a frequency of
$f_{ref}=\frac{U_{\infty}}{c_{0}}$ at this location.

(a) $\frac{r_{eff}}{r_{c}}$

(b) $\frac{r_{eff}-<r_{eff}>}{r_{c}}$

Figure 15: Spatial-temporal variations of the non-dimensional effective radius
and its fluctuations ($\sigma=1.7$)

The spectra of the effective radius fluctuations as a function of streamwise
location at different cavitation numbers are shown in figure 17. It is evident
in this figure that similar to the behavior of the spatially-averaged
effective radius fluctuations presented in §4.3.3, the major proportion of the
local fluctuations of the effective radius occurs at low frequencies (below
$300\,\mathrm{Hz}$). Furthermore, it is evident in this figure that breathing
mode oscillation characteristics alter as the cavity travels away from the
tip. Another characteristic of the breathing mode oscillations observed in the
spectra of all of the cases is the regions of high-amplitude fluctuations
within the growth region which vanish and appear periodically with streamwise
location. These regions are specified in figure 17 using magenta dashed
ellipses. Overall, according to this figure, the breathing mode oscillations
occur at lower frequencies in the growth region and higher frequency
components of breathing mode oscillation intensify as the cavity travels
downstream.

The frequency spectra of breathing mode oscillations at three streamwise
locations at different cavitation numbers are illustrated in figure 17. In the
case of $\sigma=1.7$, the breathing mode oscillations spectrum at
$x/c_{0}=0.1$ is also plotted, which reveals that the breathing mode
oscillations are small in amplitude at this location, and the intensification
of the breathing mode oscillations is apparent when the spectrum at this
location is compared to locations further downstream. This indicates that
these breathing mode oscillations are not initially present as the cavity
forms and are introduced onto the cavity surface as it flows downstream.
Moreover, a distinct peak is observed at $f=55\,\mathrm{Hz}$ at both
$x/c_{0}=0.22$ and $x/c_{0}=0.43$. The spectrum at $x/c_{0}=0.43$ exhibits two
other local maxima at $f=150\,\mathrm{Hz}$ and $300\,\mathrm{Hz}$. As the
cavity travels further downstream and develops, its spectrum displays more
peaks, an example of which is the spectrum at $x/c_{0}=0.85$, with the most
significant maximum amplitudes occurring at $f=65\,\mathrm{Hz}$,
$120\,\mathrm{Hz}$, $150\,\mathrm{Hz}$, $205\,\mathrm{Hz}$, and
$265\,\mathrm{Hz}$. This further supports that fluctuating components at
higher frequencies intensify as the cavity advances from the growth region to
the decay region.

In the case of $\sigma=1.2$, no distinct significant peak is evident in the
spectrum. Comparison of the spectrum at $x/c_{0}=1.00$ with further upstream
locations demonstrates the intensification of breathing mode oscillations as
the cavity progresses from the growth region to the decay region in this case
as well. The peaks observed in this case at $x/c_{0}=1.00$ possess frequencies
of $f=20\,\mathrm{Hz}$, $120\,\mathrm{Hz}$, $260\,\mathrm{Hz}$,
$280\,\mathrm{Hz}$, $340\,\mathrm{Hz}$.

At a cavitation number of $\sigma=2.6$, the oscillatory behavior alters from
that of lower cavitation numbers. The low-frequency oscillations are less
intense, and there is a distinct peak at $f=253\,\mathrm{Hz}$ in the spectrum
within the transition region. Overall, it can be observed that in this case,
the breathing mode spectrum extends to higher frequencies compared to the
cases with lower cavitation numbers.

Since the fine mesh region encompasses the entire cavity in the case of
$\sigma=2.6$, the end effects of the tip vortex cavity can be examined in this
case. The results indicate that the cavity undergoes a cyclic pattern of
growth and detachment at its front, which is illustrated in figure 18. The
cycle starts with elongation of the cavity, which progresses until it reaches
a critical length, followed by detachment of a part of the cavity and shedding
downstream. After the detachment, the tip vortex cavity’s end retracts until
it reaches a minimum length, and the cycle starts over with elongation. The
frequency of this cycle was found to be within the range of
$250\,\mathrm{Hz}<f<300\,\mathrm{Hz}$, which is similar to the peak
frequencies observed in the breathing mode oscillation spectrum in figure 17c
between $x/c_{0}=0.3$ and $x/c_{0}=0.6$. This highlights a potential
correlation between the breathing mode oscillation within this frequency range
and the cyclic pattern of growth and detachment observed in the tip vortex
cavity’s front.

Figure 16: Spectrum of the effective radius fluctuations with respect to
downstream location

Figure 17: Frequency spectrum of the effective radius oscillations at two
streamwise locations

Figure 18: The cyclic behavior of tip vortex cavity’s front ($\sigma=2.6$)

####  4.3.5 Decomposition of Breathing Mode Oscillations

The analysis of the effective radius reveals that the breathing mode
oscillations exhibit various characteristics in different regions of the
cavity. In this section, Proper Orthogonal Decomposition (POD) is employed to
decompose the effective radius data and obtain the main breathing mode
oscillations. The time-averaged effective radius is subtracted from the
results to extract the fluctuating components. The analysis in this section is
only carried out for the case of $\sigma=1.7$.

The energy contribution of the effective radius fluctuation modes is plotted
in figure 19(a), which shows that the fluctuations are distributed over a
large number of modes; however since the goal here is to gain insight into the
fluctuations and not reduced-order modeling, the first few modes can be
employed. Figure 19(b) depicts the first 10 modes of the effective radius
fluctuations. The change of behavior downstream of the cavity growth region,
which corresponds to $x>0.4$, is apparent in the fluctuation modes. It is also
evident from the modes that the fluctuations in the cavity growth region have
relatively small wavelengths and amplitudes compared to the decay region,
which is in line with the lower propagation speed observed within the growth
region compared to the decay region as shown in figure 17.

The amplification and the alteration in the behavior of the effective radius
fluctuations downstream of the cavity growth region indicate that some
oscillations are introduced onto the cavity interface in the roll-up process
as mentioned before. The source of these perturbations should be sought in the
temporal variations of the tip vortex flow field and the roll-up of the wake
into the tip vortex flow, which is not within the scope of this study.

(a) Singular values of the effective radius fluctuation modes

(b) First 10 modes of effective radius fluctuations obtained using POD

Figure 19: Proper orthogonal decomposition of breathing mode oscillations
($\sigma=1.7$)

###  4.4 Effect of Cavity Surface Oscillations on Pressure Fluctuations

In order to investigate the contribution of cavity surface oscillations to the
pressure fluctuations in different cases, pressure is probed at 19 points
within the domain, as shown in figure 21, on which FFT is performed
subsequently to obtain the spectra of the pressure fluctuations. The spectra
of the pressure fluctuations at these points exhibit similar behavior in all
of the cases; therefore, the pressure fluctuations spectrum at only one point,
which is located $0.5c_{0}$ away from the tip in the spanwise direction
($x=0,\,y=0,\,z=h+0.5c_{0}$), is reported herein and is shown in figure 21 for
the three cases.

Figure 20: Locations of pressure probing

Figure 21: Spectrum of the pressure fluctuations at point
($x=0,\,y=0,\,z=h+0.5c_{0}$)

It is evident in figure 21 that decreasing the cavitation number leads to more
intense pressure fluctuations. This indicates that tip vortex cavitation is,
indeed, contributing to the pressure fluctuations within the domain. Moreover,
it can be observed that as the cavitation number decreases, the spectrum
shifts towards lower frequencies overall. The pressure fluctuations at low
frequencies (below $50\mathrm{Hz}$) are relatively significant in all cases.
Besides these general features, the pressure fluctuations spectrum of each
case exhibits some unique characteristics as well.

The pressure fluctuations spectrum in the case of $\sigma=1.2$ displays a
high-amplitude peak at a frequency of $f=130\,\mathrm{Hz}$, in addition to
other local maxima at $f=40\,\mathrm{Hz}$, $60\,\mathrm{Hz}$,
$220\,\mathrm{Hz}$, and $340\,\mathrm{Hz}$. It is evident in figures 17a and
17a that a high-amplitude component is present in the breathing mode spectrum
at $f=60\,\mathrm{Hz}$ within the growth region. This peak is also observed in
the spatially-averaged effective radius spectrum in this case illustrated in
figure 14(b). The maximum observed in the pressure fluctuations spectrum at
$f=130\,\mathrm{Hz}$ is the peak of a hump in the frequency range of
$80-160\,\mathrm{Hz}$. The frequency of the center of this hump, i.e.,
$120\,\mathrm{Hz}$, is observed in figure 17a which shows relatively large
breathing mode oscillations at this frequency downstream of $x/c_{0}=0.85$.
Moreover, a peak is observed in the spatially-averaged effective radius
spectrum depicted in figure 14(b) at $f=130\,\mathrm{Hz}$. Figure 17a also
indicates the presence of breathing mode oscillations at a frequency of
$340\,\mathrm{Hz}$ within the developed region of the cavity in the case of
$\sigma=1.2$.

The pressure fluctuations spectrum in the case with a cavitation number of
$\sigma=1.7$ indicates the presence of low-frequency peaks at $f\sim
25\,\mathrm{Hz}$ and $f\sim 45\,\mathrm{Hz}$. The spectrum exhibits a
relatively broadband hump in the frequency range $160-280\,\mathrm{Hz}$, with
a peak at $f\sim 205\,\mathrm{Hz}$ and less significant peaks at
$185\,\mathrm{Hz}$, $245\,\mathrm{Hz}$, $265\,\mathrm{Hz}$, and
$285\,\mathrm{Hz}$. These peaks are present in the pressure fluctuations
spectra at all of the pressure probing locations, which indicates that these
pressure fluctuations are generated by a monopole-type source. The low-
frequency peaks at $f\sim 25\,\mathrm{Hz}$ and $f\sim 45\,\mathrm{Hz}$ were
also observed in the spatially-averaged effective radius spectrum depicted in
figure 14(b); however, the cavity volume variations at $f\sim 55\,\mathrm{Hz}$
seem to not be correlated with the pressure fluctuations. Moreover, a local
maximum at $f=205\,\mathrm{Hz}$ is also evident in the spatially-averaged
effective radius spectrum in figure 14(b).

Comparing the effective radius fluctuations spectra presented in figure 17b
with the pressure fluctuations spectrum indicates the presence of the distinct
fluctuating components at $f=205\,\mathrm{Hz}$ and $f=265\,\mathrm{Hz}$ in the
effective radius fluctuations spectrum of the location within the developed
region ($x/c_{0}=0.85$); however, this peak is not observed in the breathing
mode spectrum at $x/c_{0}=0.22$, which is in the cavity formation region. On
the contrary, the peak at $f=45\,\mathrm{Hz}$ in the pressure fluctuations
spectrum is also observed in the breathing mode spectrum at $x/c_{0}=0.22$ and
not in the spectrum at $x/c_{0}=0.85$.

In the case of $\sigma=2.6$, the pressure fluctuations spectrum exhibits a
relatively small hump within the frequency range of $220-300\,\mathrm{Hz}$
with a peak at $f=253\,\mathrm{Hz}$. The breathing mode oscillations spectrum
for this case depicted in figure 17c revealed intense breathing mode
oscillations at $f=253\,\mathrm{Hz}$ within the streamwise location range of
$0.28\,x/c_{0}\,0.55$. Moreover, the spatially-averaged effective radius
spectrum presented in figure 14(b) exhibits some local maxima at
$f=227\,\mathrm{Hz}$, $240\,\mathrm{Hz}$, $253\,\mathrm{Hz}$, and
$280\,\mathrm{Hz}$. Furthermore, the cyclic growth and detachment behavior of
the cavity’s end was found to be occurring within a frequency range of
$250-300\,\mathrm{Hz}$, which indicates that this cyclic tail effect is
potentially influencing the pressure fluctuations within the domain as well.
Figure 17c also indicates the presence of local breathing mode oscillations
within the growth region and upstream of the decay region in this case at low
frequencies from $10-40\,\mathrm{Hz}$. These oscillations might be correlated
with higher amplitudes of pressure fluctuations at frequencies lower than
$40\,\mathrm{Hz}$ in this case compared to the case of $\sigma=1.7$, wherein
such local oscillations within the growth region occur at higher frequencies.

Based on the correlations between the cavity breathing mode oscillations and
the pressure fluctuations, it can be concluded that the growth region of the
cavity is correlated with the low-frequency peaks observed in the pressure
fluctuations spectrum, and the mid-range of this spectrum, where the hump in
the pressure fluctuations spectrum occurs as well, is correlated with the
breathing mode oscillations in the decay region of the tip vortex cavity.

##  5 Concluding remarks

In this study, tip vortex cavitating flow over a stationary elliptical
NACA66(2)-415 hydrofoil was numerically investigated using a finite element-
based cavitation flow solver. To address the lack of general mesh resolution
requirements for TVC simulation, a pressure gradient-based length scale was
developed employing the Rankine vortex model, Kutta-Jukowski theorem, and the
turbulent boundary layer thickness. This new length scale was then utilized
for non-dimensionalization of the mesh resolution within the tip vortex flow
region. The simulations carried out in this study for various non-dimensional
mesh resolution values ($\Delta r^{*}$) revealed that a $\Delta r^{*}$ of 18.5
or lower should be utilized for the accurate simulation of tip vortex
cavitating flows using LES.

The oscillatory dynamics of the tip vortex cavity were investigated in the
results obtained from the simulations carried out with a mesh satisfying the
proposed resolution requirements. The simulations were conducted for three
different cases which differed only in the cavitation number. The cavity
dynamics captured in the simulations in this study were shown to agree well
with the analytical solution.

In the next step, since it is well-known that the breathing mode of
oscillation is the most influential mode of tip vortex cavity surface
oscillations concerning the generated noise compared to other modes, a new
method was developed for the extraction of this mode of oscillation from the
numerical results using a new parameter named the cavity effective radius
($r_{eff}$). Investigation of the temporally-averaged effective radius
demonstrated that the tip vortex cavity experiences different regions as it
progresses away from the tip. The results indicated that the growth region,
i.e., the region close to the tip where the cavity is forming and experiences
an overall increasing trend in volume, shrinks with increasing the cavitation
number. Furthermore, it was demonstrated that the decay occurs at an increased
slope in scenarios with higher cavitation numbers. Another observation was the
occurrence of spatial periodic cavity volume variations within the growth
region. These variations exhibit similar dominant wavenumbers between cases
with different cavitation numbers, which reveals that the dominant wavenumber
of such oscillations is related to the mean tip vortex roll-up process and not
the characteristics of the cavity forming in each specific case.

Investigation of the spatially-averaged effective radius indicated that the
overall vapor volume undergoes temporal variations, with the most intense
oscillations occurring in the moderately cavitating case, i.e., $\sigma=1.7$.
The analysis revealed that the flow characteristic frequency and its
subharmonic dominate the overall vapor volume variations, with the most
significant peak having a frequency similar to the flow characteristic
frequency in all of the cases. Moreover, the results revealed that increasing
the cavitation numbers leads to more intense vapor volume variations at higher
frequencies, with the most intense high-frequency vapor volume variations
occurring when $\sigma=2.6$.

Subsequently, the local breathing mode oscillations were investigated. The
local breathing mode oscillations exhibit different propagation
characteristics in the growth and decay regions in terms of speed and
direction. More evidence was provided for the alteration of the cavity’s
oscillatory behavior between the growth and decay regions employing Proper
Orthogonal Decomposition. Moreover, the results indicated that the breathing
mode oscillations are not inherent to the tip vortex cavity, and such
oscillations appear and intensify as the cavity travels away from the tip.
Therefore, the tip vortex cavity displays different characteristics in the
growth and decay regions. The breathing mode oscillations within the decay
region were observed to extend over a more broadband range of frequencies
compared to the growth region. Overall, the breathing mode oscillations within
the growth region were observed to occur within the low-frequency range, and
such oscillations within the decay region were found to possess higher
frequencies. The simulations revealed that at a low cavitation number, similar
to the case of $\sigma=1.2$ in this work, no significant peak was observed in
the breathing mode spectrum; however, distinct peaks were observed at higher
cavitation numbers with the most intense ones occurring in the case with
$\sigma=1.7$. Furthermore, similar to the spatially-averaged effective radius
fluctuation spectrum, the local breathing mode oscillation in the case of
$\sigma=2.6$ extended to higher frequencies compared to lower cavitation
numbers.

An interesting feature of the tip vortex cavity was observed when
$\sigma=2.6$, which was the cyclic growth and detachment behavior of the
cavity at its end. It was observed that the cavity undergoes a process of
elongation, detachment, and retraction at its end, which was found to occur
within a frequency range of $250-300\,\mathrm{Hz}$. The local breathing mode
oscillations in this working condition also displayed fluctuations within this
frequency range, which indicates a potential correlation between the cyclic
tail behavior of the tip vortex cavity and its breathing mode oscillations.
This aspect of tip vortex cavitation requires further investigation.

The pressure fluctuations within the domain were investigated in cases with
different cavitation numbers. The results indicated that decreasing the
cavitation number, which leads to a higher extent of tip vortex cavitation,
results in higher levels of pressure fluctuations overall. Another effect of
decreasing the cavitation number was found to be shifting the pressure
fluctuations spectrum towards lower frequencies, which is similar to the
behavior of breathing mode oscillations. Comparison of the pressure
fluctuations spectra at different cavitation numbers with the overall vapor
volume variations and local breathing mode oscillations at the corresponding
cavitation numbers demonstrated the existence of correlations between the
pressure fluctuations and the breathing mode of oscillation. The center
frequencies of the humps observed in the pressure fluctuations spectra were
found to be related to the breathing mode oscillations spectrum in all of the
cases.

Despite the existence of these correlations, there were some features in the
pressure fluctuations that were not related to the breathing mode of cavity
surface oscillation. Moreover, some significant fluctuating components were
observed in the breathing mode oscillations which did not affect the pressure
fluctuations despite having high amplitudes. Therefore, further investigation
is required to thoroughly understand the correlations between the breathing
mode oscillations and the pressure fluctuations.

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