# Experimental Analysis of Tip Vortex Cavitation Mitigation By Controlled
Surface Roughness

U. Svennberg  Kongsberg Hydrodynamic Research Centre, Kongsberg Maritime AB,
Kristinehamn, Sweden  A. Asnaghi  Department of Mechanics and Maritime
Sciences, Chalmers University of Technology  R. Gustafsson  Kongsberg
Hydrodynamic Research Centre, Kongsberg Maritime AB, Kristinehamn, Sweden
R.E. Bensow  Department of Mechanics and Maritime Sciences, Chalmers
University of Technology  Author to whom correspondence should be addressed:
<EMAIL_ADDRESS>
##  1 Abstract

This study presents results of experiments where roughness applications are
evaluated in delaying the tip vortex cavitation inception of an elliptical
foil. High-speed video recordings and Laser Doppler Velocimetry (LDV)
measurements are employed to provide further details on the cavitation
behaviour and tip vortex flow properties in different roughness pattern
configurations. The angular momentum measurements of the vortex core region at
one chord length downstream of the tip indicate that roughness leads to a
lower angular momentum compared to the smooth foil condition while the vortex
core radius remains similar in the smooth and roughened conditions. The
observations show that the cavitation number for tip vortex cavitation
inception is reduced by 33 % in the optimized roughness pattern compared to
the smooth foil condition where the drag force increase is observed to be
around 2 %. During the tests, no obvious differences in the cavitation
inception properties of uniform and non-uniform roughness distributions are
observed. However, the drag force is found to be higher with a non-uniform
roughness distribution.

Keywords: Roughness, Tip vortex, Mitigation, Cavitation, Suppression

##  2 Introduction

The physics of tip vortex cavitation, TVC, involves simultaneous presence of
small flow structures and phase change. Since controlling TVC is vital for
low-noise propellers [1, 2, 3], intensive efforts have been conducted both
experimentally and numerically to understand the physics of this flow. The
marine industry generally seeks only to increase the tip vortex core pressure
to alleviate cavitation, and to do so, a few approaches are proposed and
tested. These approaches generally can be classified into two groups. In the
first group, which is called active control, the tip vortex flow is changed by
injection of a solution, e.g. air, polymer, or water. In the second group so
called passive control, either the propeller geometry is modified or its
surface properties, e.g. roughness.

In the injection method, interaction between the injected solution and the
original tip vortex defines the mitigation. Consequently, the location of
injection, the type and amount of solution are decisive parameters [4, 5]. As
an example, Chahine et al. [4] observed up to 35 % delay in TVC inception
(TVCI) by injecting some combinations of polymer solutions while there was not
any significant improvement in TVCI delay by the injection of pure water or a
50 % water-glycerin viscous mixture.

Another approach to weaken a tip vortex is to include an extra geometry, e.g.
endplate or winglet, attached to the tip. The idea is to stabilize the
Reynolds shear stress and therefore, reduce the tip vortex strength [6, 7].
For tip loaded propellers (TLP), having winglet prevents formation of a single
strong tip vortex, and instead two or in some cases a few distinct weaker
vortices are generated [8]. It is expected that the evolution of the tip
vortices and their interaction defining the mitigation depend on a number of
factors, some of those being the span of the endplate and the loading
distribution on it.

Park et al. [9] proposed a semi-active control scheme tested on a model scale
propeller by attaching a flexible thread at the tip of the blade. Due to a low
pressure region of the tip vortex core, the thread is sucked into the tip
vortex. Three different thread materials were tested: steel, nylon, and
Dyneema, where only the Dyneema thread was sufficiently flexible to be sucked
into the tip vortex core. It is reported that when the thread is sucked into
the vortex, it can effectively suppress the tip vortex cavitation inception.
Amini et al. [10] investigated the effectiveness of a flexible nylon thread in
TVC mitigation of an elliptical hydrofoil. They investigated several thread
sizes (thickness and length) and conclude that certain configurations could
remarkably attenuate TVC without imposing any tangible penalties on the
hydrodynamic performances. Their conclusion highlights that the thread sizes
should be selected according to the flow conditions in a way the thread
becomes unstable and interacts with the vortex dynamically to mitigate TVC.
The study of Lee et al. [11] suggests the presence of the thread at the tip
causes a significant reduction in the streamwise velocity field around the
vortex core region and therefore mitigates TVC.

The tip geometry can be modified as an effort to restrain sheet and tip vortex
cavitation in the design of marine propellers [12, 13]. This, however, can
cause cloud cavitation which can give noise and surface erosion on tip-
modified propellers [14].

Roughening the blade, which is the main topic of this study, will also lead to
tip vortex radius increase, and therefore delay in TVCI [15]. This is believed
to be related to the turbulence generated by the roughness which destabilizes
the tip vortex and leads to its early breakdown. As discussed by Souders and
Platzer [16], a thicker turbulent boundary layer on the wing tip resulted from
the roughness increases tip vortex core radius and decreases the tip vortex
maximum tangential velocity as compared to the laminar flow case. The surface
roughness also contributes to other tip vortex properties, e.g. the roll-up
process [17]. In the study of Johnsson and Ruttgerson [18], the influence of
leading edge roughness on the tip vortex roll-up for different angles of
attack was investigated. They concluded that application of roughness on the
pressure side near the leading edge has a delaying effect on tip vortex
cavitation and leads to a substantial reduction in tip vortex strength. They
also noted an open water efficiency degradation when roughness was applied,
around 2$\%$ in their tested condition. Later, it was suggested by Philipp and
Ninnemann [19] to apply the roughness only on a small area of the blade
suction side close to the tip trailing edge region, i.e. in the area of the
tip vortex formation. This was tested by Kruger et al. [20] on a propeller
where different roughness patterns were applied on different parts of a
propeller to find the most optimum location for roughness.

Even though that experimental tests have proven the capability of using
roughness to mitigate tip vortex flows and delay cavitation, this method has
not become practically available yet. The main reason is the lack of knowledge
on how exactly roughness mitigates tip vortex. The required or optimum
roughness properties such as height, distribution, type, and pattern are the
main decisive parameters. Moreover, it should be clarified whether it is
increasing the boundary layer thickness or providing further turbulent
instabilities that leads to tip vortex mitigation. Last but not least, the
scale law on how model scale roughness studies can be extended to the full
scale propellers should be developed [21]. The current study has been
performed within a project aimed to find the answers of these questions. To
add to the knowledge, experiments for tip vortex flows and cavitation
inception around an elliptical foil have been conducted.

The tip vortex flow and cavitation inception around smooth elliptical foils
have been investigated by several researchers [22, 23, 24, 25, 26]. This type
of foils is of interest as the vortex structures around it resembles the
propeller tip vortex behaviour while making it possible to be tested in more
details both experimentally and numerically. While the previous studies were
focused on the formation and development of tip vortex over smooth surfaces,
the current measurements include the roughness application and how it changes
the tip vortex properties. In order to provide more details for analysis, the
geometry of the elliptical foil tested at Delft Technical University is used
[1, 27]. However, due to the velocity speed limitation of our cavitation
tunnel, the current foil has a scale ratio of 2.398 compared to the one tested
at Delft TU. Our recent numerical analysis [28] is used as an initial
guideline for the pattern and size of the roughness elements. The main
objective here is to evaluate these patterns in order to provide more insights
on the roughness impact in tip vortex inception and cavitating tip vortex
behaviour.

The tests are conducted within a wide range of tip vortex conditions from non-
cavitating to fully cavitating tip vortex along with varying the free-stream
velocity that is considered to evaluate the impact of Reynolds number. In
order to include the effects of the sand grains irregularities, in one test
the roughness pattern is repeated with different distributions of sand grains
on the foil. The outcome of these tests provides further knowledge on how
roughness should be applied to achieve the highest TVC mitigation with a
minimized negative impact on hydrodynamic performance. Cavitation inception
was recorded through visual inspection of high-speed recordings.

##  3 Experimental setup

The experiments are performed in the free surface cavitation tunnel at the
Kongsberg Hydrodynamic Research Center, Kristinehamn, Sweden. The cross
section of the test section is $0.8\times 0.8$ $\text{m}^{2}$ at the inlet and
$0.8\times 0.82$ $\text{m}^{2}$ at the outlet. The vertical direction was
extended gradually from inlet to outlet to compensate for the growth of the
boundary layer in order to facilitate a nearly zero-pressure gradient in the
streamwise direction. This is provided by having the top plate horizontal and
the bottom plate with a downward slope of 0.4 degrees where at the foil
section, the vertical distance is about $0.81$ m.

The tested foil is a half-model wing of elliptic planform with a NACA
$66_{2}-415$ cross section. Compared to the elliptical foil recently tested at
Delft TU [24, 27], the current foil has a scale ratio of 2.398 leading to the
root chord length $\text{C}_{0}=301.2$ mm and the span length of S=$360$ mm.
The projected surface area of the foil computed from the CAD file is
A=$8.43\times 10^{-2}$ $\text{m}^{2}$. The lift and drag coefficients are
calculated based on this area,
$\text{C}_{\text{l}}=\text{L}/(0.5\rho{\text{U}_{\text{inlet}}}^{2}A)$ and
$\text{C}_{\text{d}}=\text{D}/(0.5\rho{\text{U}_{\text{inlet}}}^{2}A)$ where L
and D are the lift and drag forces.

The sketch of the test setup is given in Fig. 1. The coordinate system, with
the foil tip at the origin, is defined with z pointing towards the inlet. The
foil is mounted on a rotatable shaft through the top plate having 1 mm gap
between the root section of the foil and the top plate to give the possibility
of varying the angle of attack. A five-component force/torque sensor
(Kongsberg dynamometer no. 54A) is mounted between the shaft and the foil to
measure the forces. The location of the camera is also highlighted in the
bottom view. The lift and drag forces are reported in the y and z directions,
respectively. The high-speed camera (FASTCAM SA-X2) is placed near the side
window while viewing the suction side of the foil.

Fig. 1: Sketch of the test tunnel and the coordinate system.

The measurements are performed with respect to the averaged free-stream
velocity ranging from 2.84 m/s to 4 m/s with typical fluctuations of $\pm
0.7\%$. The outlet cavitation number is varied from 2.6 to 8 while keeping the
inflow velocity constant. The Reynolds number is calculated based on the root
chord length and the inlet flow properties,
Re=$\rho\text{U}_{\text{inlet}}\text{C}_{0}/\mu$, and the cavitation number is
calculated based on the outlet pressure,
$\sigma=(\text{p}_{\text{outlet}}-\text{p}_{\text{sat}})/(0.5\rho\text{U}_{\text{inlet}})$.
The operating conditions were monitored during each experiment to prevent any
deviations during the measurements. The water temperature, kept around 23 ∘C
during the tests, is measured with a PT-100 sensor placed submerged in the
tunnel water downstream of the test section.

Static pressure was measured with a low pass filter at about $0.2$ Hz with a
digital absolute pressure sensor (Rosemount 3051S). The pressure was measured
in the air volume and compensated for the water depth down to the tip of the
foil, Fig. 2. The typical accuracy is $0.05\%$ of full-scale pressure
($1.1\times 10^{5}$ Pa), which is 55 Pa. The free-stream velocity is measured
using a pitot tube having the same type of the pressure transducers and
located upstream of the wing.

Fig. 2: Cavitation tunnel sketch and the location of the measurement
equipments.

The dissolved oxygen concentration (DO) was used as a measure of the amount of
dissolved gas in the water using a fluorescence-based optical sensor
(Trioxmatic 690). It has a range of 0-600 % which equals to 0-60 mg/l and the
measurements were conducted at around 40 % which corresponds to 4 mg/l. As
only dissolved oxygen is measured, this is taken as a representative indicator
for the total amount of dissolved gas.

##  4 Tested conditions

Previous LES analysis of the tip vortex flow around the foil indicates that
for considered conditions three different areas on the suction side are
important in the tip vortex formation and development [28]. These findings are
concluded based on the evaluation of the tip vortex flow streamlines and
momentum of vortical structures fed into the tip vortex core. It has been
discussed that the side of the foil where the roughness should be applied is
the side that the vortex roll-up forms, e.g. the suction side for the current
operating condition. In our later study [29], the impact of the roughness area
on the flow streamlines and separation lines are investigated. It is
highlighted that roughness provides an extra separation line close to the tip
along with the leading edge separation line on the side where the roll-up
forms. Due to the resource limitation, the roughness experimental tests are
focused only on these areas where the height is set equal to the optimum
height found in the numerical analysis, equal to 230 $\mu$m. The first
roughness pattern, called pattern (I) in this paper, consists of an area on
the leading edge (SSLE) and an area on the tip of the foil (SST). In the
roughness pattern (II), an extra area on the trailing edge (SSTE) is also
included. The description and sizes of these areas are presented in Fig. 3. In
order to include the effects of the sand grains irregularities, the tests on
the pattern (II) are repeated with different distributions of sand grains on
the foil, Fig. 4.

The cavitating tip vortex is evaluated at the cavitation numbers 1.2 and 2.6.
The tip vortex cavitation inception and development are evaluated at
cavitation numbers larger than 3.8. Most of the tests are carried out at the
free-stream velocity of 2.84 m/s. A few of the conditions is repeated at a
higher inlet velocity, i.e. 4 m/s, to evaluate the Reynolds number impact on
the tip vortex development. It also provides the opportunity of evaluating the
roughness impact on the risk for sheet cavitation at different velocities.

Fig. 3: Description of the areas where the roughness is applied, zoomed view,
suction side of the foil.

Free-stream direction (a) Uniform distribution

;

(b) Non-uniform distribution

;

(c) Sparse distribution

Fig. 4: Different distributions of sand grains for the roughness pattern (II).

##  5 Results and discussion

###  5.1 Force measurements

The first part of the results contains the smooth foil measurements at
different Reynolds numbers and angles of attack, Fig. 5. The results are
elaborated by providing comparisons with the measurements conducted by Arndt
and Keller [30] and Pennings [24]. It should be noted that these tests are
conducted on the similar foil having slightly different geometrical aspect
ratio where the root chord length of the foil used in [30] was equal to
$\text{C}_{0}=129.4$ mm, and in [24] was equal to $\text{C}_{0}=125.4$ mm.
Lift data are collected over a range of Reynolds numbers from $0.6\times
10^{6}$ to $1.13\times 10^{6}$, and the angles of attack ranging from 5∘ to
16∘. Each measurement point is averaged over at least five measured samples
having around 30 s duration. The measurements variation of each point is found
to be less that 2% which to avoid the ambiguity is not included in the figure.

In general, the measured lift and drag coefficients follow expected trends.
The differences can be related to the uncertainties of the measurements and
the devices used in different cavitation tunnels.

$4$$8$$12$$16$$0$$0.2$$0.4$$0.6$$0.8$$1$Angle of attack in
degrees$\text{C}_{\text{l}}$Re$=1.05e6$Re$=0.85e6$Re$=0.60e6$Re${}^{*}=1.13e6$Re${}^{*}=0.92e6$Re${}^{*}=0.65e6$Re${}^{{}^{\prime}}=0.85e6$

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.05$$0.1$$0.15$$\text{C}_{\text{l}}$$\text{C}_{\text{d}}$Re$=1.05e6$Re$=0.85e6$Re$=0.60e6$Re${}^{*}=1.13e6$Re${}^{*}=0.92e6$Re${}^{*}=0.65e6$

Fig. 5: Variation of the lift and drag coefficients for different angles of
attack and Reynolds numbers, smooth foil, non-cavitating condition, Re:
current tests, Re∗: extracted from Arndt and Keller [30], and
Re${}^{{}^{\prime}}$: extracted from Pennings [24].

###  5.2 Flow field

The velocity distribution at one chord length downstream of the smooth foil
measured by LDV is presented in Fig. 6. The measurements are conducted at the
inlet velocity of 4 m/s, equal to Re=$1.204\times 10^{6}$. To provide around
20 measurement points across the vortex core radius, the measurement
resolution of 60 $\mu$m is applied over the vortex core region. The
measurement resolution, then, gradually becomes coarser outside the vortex
core where the velocity gradients are smaller.

Close to the foil, the tangential velocity is highly asymmetric but gradually
becomes symmetric further downstream as the roll-up process develops. This
asymmetric behaviour can be noted from the velocity distribution in the
y-direction (noted as vertical velocity), Fig. 6(a), where the maximum
velocity is around 4 m/s and the minimum velocity is around -3 m/s suggesting
that the roll-up is not fully completed and the vortex is still developing.
The measurement section has been selected close to the tip intentionally, as
it would be cheaper to conduct numerical simulations of tip vortex flows at
closer sections.

Similar to previous studies [24, 30, 31], an accelerated axial velocity at the
vortex core, as high as 1.85 $\text{U}_{\text{inlet}}$, is observed. The axial
velocity distribution, Fig. 6(b), also suggests that the measurements have
insufficient spatial resolution to capture such a high velocity gradient
especially in the vortex core region.

(a) Vertical velocity

(b) Axial velocity

Fig. 6: Axial and vertical velocity (velocity in y-direction) distributions
for smooth foil at non-cavitating condition, Re=$1.204\times 10^{6}$,
$\alpha$= 9∘, z/$\text{C}_{0}$=1.

The variation of the normalized azimuthal velocity against the radial distance
is presented in Fig. 7. The velocity profile is plotted over the horizontal
cutline, i.e. x-direction, on the section of the LDV measurement which passes
through the vortex center. Along with the current measurements, similar
measurements on this foil are provided in this figure where the measurements
reported by Pennings [24] is conducted at $z/\text{C}_{0}$=1.14 and at
Re=$0.9\times 10^{6}$ and the measurements reported by Arndt et al. [31] is
conducted at $z/\text{C}_{0}$=1.0 and at Re=$0.52\times 10^{6}$. The
measurements include both non-cavitating condition and cavitating condition of
$\sigma=2.6$. For the rough foil tests, the roughness pattern (II) with
uniform sand grain distribution is reported. In each measurement, the
azimuthal velocity is normalized by the inlet velocity and the radial distance
is normalized by the vortex core radius. The vortex core radius is considered
as the location where the maximum azimuthal velocity occurs, and the vortex
core center is the location where the azimuthal velocity is zero. The vortex
core radius measured in this study, by Arndt et al. [31], and by Pennings [24]
are $\text{r}_{\text{vortex}}$=1.24, 1.11, 1.1 mm, respectively.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$0$$0.2$$0.4$$0.6$$0.8$$1$r/$\text{r}_{\text{vortex}}$Normalized
azimuthal velocitySmoothFoil, non-cavitating, Re=$1.204\times
10^{6}$RoughFoil, $\sigma=2.6$, Re=$1.204\times 10^{6}$SmoothFoil Pennings
[24], non-cavitating, Re=$0.9\times 10^{6}$SmoothFoil Pennings [24],
$\sigma=2.6$, Re=$0.9\times 10^{6}$SmoothFoil Arndt et al. [31], non-
cavitating, Re=$0.52\times 10^{6}$ Fig. 7: Variation of the normalized
azimuthal velocity in the vortex core region. In each condition, the radial
distance is normalized by the vortex core radius and the velocity is
normalized by the free-stream velocity. The cavitating condition is reported
for $\sigma=2.6$.

Even though the measurements are conducted at different Reynolds number
ranging from $0.52\times 10^{6}$ to $1.204\times 10^{6}$, a similar trend is
observed for the azimuthal velocity at the non-cavitating condition of the
smooth foil, especially at r/$\text{r}_{\text{vortex}}>$1\. However, inside
the vortex core, r/$\text{r}_{\text{vortex}}<$1, the difference in
measurements is more distinguishable. It is not clear whether this difference
is a result of having different measurement methods, PIV in Pennings [24], LDV
in the current tests and in Arndt et al. [31], or having different measurement
resolutions. According to the authors knowledge, in neither of the presented
data the effect of the tip vortex meandering is accounted.

As expected, the presence of cavitation inside the vortex core increases the
radius and decreases the maximum azimuthal velocity. As can be seen from
results of Pennings at $\sigma=2.6$, the radius of the vaporous tip vortex is
around 60 $\%$ larger than the non-cavitating tip vortex. It is interesting
that outside the vortex core, the trend of the velocity profiles in cavitating
and non-cavitating conditions are similar, suggesting a very little impact of
cavitation presence on the vortex roll-up. Having roughness on the foil,
however, not only changes the maximum azimuthal velocity, but also affects the
vortex roll-up and consequently the trend of the azimuthal velocity variation
in r/$\text{r}_{\text{vortex}}>$1.

By changing the distribution of pressure and boundary layers on the foil,
roughness affects the momentum distribution fed into the tip vortex. This can
be observed from Fig. 8 where the distribution of the angular momentum
obtained from the product of the normalized azimuthal velocity and the
normalized radial distance is presented. Comparison of the angular momentum
distributions between different conditions shows that the normalized angular
momentum is relatively similar in non-cavitating smooth conditions for the
range of tested Reynolds numbers. It is distinguishable that the presence of
roughness clearly decreases the angular momentum especially in the vortex core
region, i.e. r/$\text{r}_{\text{vortex}}<$5.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$0$$1$$2$$3$$4$r/$\text{r}_{vortex}$Normalized
angular momentumSmoothFoil, non-cavitating, Re=$1.204\times 10^{6}$RoughFoil,
$\sigma=2.6$, Re=$1.204\times 10^{6}$SmoothFoil Pennings [24], non-cavitating,
Re=$0.9\times 10^{6}$SmoothFoil Pennings [24], $\sigma=2.6$, Re=$0.9\times
10^{6}$SmoothFoil Arndt et al. [31], non-cavitating, Re=$0.52\times 10^{6}$
Fig. 8: Variation of the normalized angular momentum in the vortex core
region.

The comparison between streamwise and tangential velocity distributions for
the smooth and pattern (II) conditions at z/$\text{C}_{0}$=0.5 downstream of
the foil tip is presented in Fig. 9. In each condition, the velocity is
normalized by using the free-stream velocity, i.e. $\text{U}_{\text{inlet}}$.
By assuming that the vortex core center have zero rotational velocity, half of
the distance between maximum and minimum vertical velocities is considered as
the vortex radius which then is used in these figures for normalizations. The
obtained vortex core radius for the smooth foil is rv=1.24 mm, for the uniform
roughness pattern is rv=1.72 mm and for the sparse roughness pattern is
rv=1.86 mm, respectively.

The measurements indicate that in the roughened surface condition the
magnitudes of the tangential and axial velocities are decreased compared to
the smooth condition. It is also noted that the axial velocity peak happens
slightly off the center, e.g. in the smooth condition it happens at r/rv=0.5.

$-2.5$$-2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$2.5$$-1.2$$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$r/$r_{\textnormal{v}}$Normalized
azimuthal velocitySmoothUniform roughnessSparse roughness

$-3$$-2.5$$-2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$2.5$$3$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$r/$r_{\textnormal{v}}$Normalized
axial velocitySmoothUniform roughnessSparse roughness

Fig. 9: Variation of the normalized azimuthal and streamwise velocity in the
vortex core region at z/$\text{C}_{0}$=0.5 downstream of the foil tip for the
smooth and roughness pattern (II) conditions, non-cavitating, Re=$1.204\times
10^{6}$.

The sparse roughness distribution leads to slightly lower tangential velocity
compared to the uniform roughness distribution. The impact on the axial
velocity is more noticeable where lower velocity magnitude is observed in the
sparse distribution. Having a lower maximum axial velocity and larger vortex
core radius in the sparse roughness pattern indicate that the tip vortex in
this pattern is weaker, and consequently it is expected that tip vortex of
this pattern has a higher vortex core pressure.

###  5.3 Tip vortex cavitation inception

The inception point is determined through visual observations of cavitation
appearance in the tip vortex region obtained from the high-speed recordings.
The condition where the expanded bubble inside the tip vortex can be detected
by a naked eye is considered as an inception point. The inception point is
selected while lowering the cavitation number from the atmospheric pressure
and the impact of hysteresis, like the study by Amini et al. [32, 33], is not
included here.

During the tests, fresh water was added to the cavitation tunnel upstream of
the foil to provide a similar water quality. This leads to having relatively
weak water quality for all of the presented measured data. This can be noted
from Fig. 10 where the measured inception point for the smooth foil is
presented. Comparison with the measurements conducted by Arndt and Keller [30]
where different water qualities were tested indicates that the water quality
of the current tests can be considered weak, i.e. filled with enough nuclei to
initiate cavitation as soon as pressure falls below the saturation pressure.
It can be noted that the measured inception points are in good agreement with
the data provided in [30].

$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$0$$1$$2$$3$$4$$5$$6$$7$$8$$\text{C}_{\text{l}}$$\sigma_{i}$Weak
waterStrong waterCurrent measurements Fig. 10: Variation of the cavitation
inception and lift coefficient, smooth foil, Re=$8.56\times 10^{5}$, weak and
strong water data are extracted from Arndt and Keller [30].

The predicted cavitation inception points for different roughness patterns are
presented in Fig. 11. For the tested roughness patterns, application of
roughness has mitigated the TVCI by around 33$\%$. The predicted inception
point for pattern (I) is $\sigma_{i}=4.15$ and for pattern (II) with uniform
sand distribution is $\sigma_{i}=4.05$ and with the sparse distribution is
$\sigma_{i}=3.3$. No difference in the cavitation inception properties is
observed while comparing the uniform and non-uniform roughness distributions.
This, however, does not clarify whether it is the boundary layer thickness
increase or the turbulent instabilities which is the main responsible for TVC
mitigation by roughness.

It is already discussed that the tip vortex pressure of the sparse pattern is
expected to be higher than that of the uniform roughness pattern. This
corresponds to a lower cavitation inception in the sparse pattern compared to
the uniform pattern.

At the inception point of different roughness patterns, similar lift
coefficient is measured where the lift coefficient variation is less than 0.2
$\%$. The drag coefficient of the roughness patterns are higher than the
smooth foil, increased by 3 $\%$ in the pattern (I) and by 3.5 $\%$ in the
pattern (II).

Smooth(I)(II)-Uniform(II)-Sparse$0$$2$$4$$6$$8$Roughness pattern$\sigma_{i}$
Fig. 11: Variation of the cavitation inception versus different surface
roughness areas, Re=$8.55\times 10^{5}$, $\alpha$= 9∘.

###  5.4 Cavitating tip vortex

For the smooth foil condition, the cavitating tip vortex can be decomposed
into two areas. Close to the foil and within the downstream distance of 0.2
$\text{C}_{0}$ from the tip, an intermittent formation of cavitating vortex is
observed while further downstream, a relatively stationary cavitating tip
vortex is shaped. At the time instance where there is not any attached
cavitation to the foil, a small cavitating vortex forms at the tip, Fig. 12
time instance (1). This cavitating vortex grows and eventually reaches the
stationary cavitating tip vortex part, Fig. 12 time instances (1, 2 and 3). It
is interesting that the location where the stationary cavitating tip vortex
starts is relatively constant during this process, having the approximate
location of 0.18 $\text{C}_{0}$ downstream of the tip. It can be noted that
there is a clear discontinuity on the stationary cavitating tip vortex,
illustrated by the yellow arrow on the figure. When the growing attached
cavity reaches the stationary part, it separates from the foil, and is
transported downstream, time instances (7) to (14) of Fig. 12. The noise
measurements conducted by Peng et al. [34] indicates that this TVC detachment
process from the foil tip corresponds to a local peak in the tip vortex
generated noise.

(1)0.1$\text{C}_{0}$Discontinuity

(8)

(2)

(9)

(3)

(10)

(4)

(11)

(5)

(12)

(6)

(13)

(7)

(14)

Fig. 12: Cavitating tip vortex flow around the smooth foil, $\sigma=2.6$,
Re=$8.55\times 10^{5}$, $\alpha$= 9∘. The time step between the consequent
images is 1.6 ms.

When the roughness is applied on the leading edge and tip of the foil, pattern
(II), the cavitating tip vortex becomes much weaker than the one formed on the
smooth foil. The cavitating tip vortex is also observed to consist of shed
cavitating vortices formed on the tip of the foil, Fig. 13. Contrary to the
smooth foil condition, where a continuous cavitating tip vortex appears after
0.2 $\text{C}_{0}$, with pattern (II) the discontinuity between shed cavity
structures remains as they travel downstream. This can be related to the fact
that the tip vortex has become weaker in this condition and therefore cannot
create the same cavitating vortex elongation as in the smooth foil condition.
It is also noted that formation of cavitating vortex at the tip happens with
higher frequency compared to the smooth foil condition.

(1)0.1$\text{C}_{0}$

(3)

(2)

(4)

Fig. 13: Cavitating tip vortex around the foil with the roughness pattern
(II), $\sigma=2.6$, Re=$8.55\times 10^{5}$, $\alpha$= 9∘. The time step
between each image is 1.2 ms.

While in the smooth and pattern (II) conditions it is observed that cavitating
vortices are shed from the tip and fed into the downstream cavitating tip
vortex, with roughness pattern (I) no shedding behaviour close to the tip is
observed. In this case, it is found that the cavitating behaviour is more
dependent on the distributions of nuclei and their trajectories. In Fig. 14,
three different nuclei having different trajectories are highlighted. As can
be seen, depending on the location, that a nucleus enters into the low
pressure region of the tip vortex, cavitation forms at different locations.

During the tests and with naked eyes, local formation of very small bubble or
sheet cavitation around the roughness elements is observed. The analysis of
high-speed recordings shows that these tiny nucleation sites having micro- or
even nano-metric residual air pockets are formed among the roughness sands.
This, however, does not influence the characteristic of the roughness elements
in not inducing a big sheet cavity in the tested operating conditions.

(1)0.1$\text{C}_{0}$

(6)

(2)

(7)

(3)

(8)

(4)

(9)

(5)

(10)

Fig. 14: Cavitating tip vortex around the foil with the roughness pattern (I),
$\sigma=2.6$, Re=$8.55\times 10^{5}$, $\alpha$= 9∘. The time step between each
image is 1.6 ms. The colorful rectangles are used to highlight the locations
of nuclei entering into the tip vortex.

The time averaged behavior of the cavitating tip vortex derived from averaging
high-speed images over time is presented in Fig. 15. As it is discussed
before, the figure shows that the cavitating tip vortex radius of the smooth
foil is larger than of other conditions, indicative of having a stronger tip
vortex in this condition. Moreover, for the pattern (I) condition and close to
the foil, the cavitating tip vortex is very weak which relates to the
discussion provided for Fig. 14.

Smooth

Pattern (I)

Pattern (II)

Fig. 15: Time averaged cavitating tip vortex, $\sigma=2.6$, Re=$8.55\times
10^{5}$, $\alpha$= 9∘.

In Table 1, lift and drag coefficients of different conditions are presented.
As expected, the measurements indicate that having roughness will lead to a
higher drag. However, the increase depends on the roughness pattern and the
way the roughness elements are distributed. For the roughness pattern (II), it
is noted that having a non-uniform roughness elements distribution leads to a
higher drag compared to the uniform distribution. It should be noted that no
obvious difference in the cavitation behaviour or inception property is found
for the uniform and non-uniform roughness distribution for pattern (II). This
demands for more tests and analyses to clarify the impact of roughness
elements in details. The variation of lift is found to be smaller, less than
0.2 $\%$, where in the pattern (I) the lift is increased compared to the
smooth foil condition.

Table 1: Variation of the drag and lift coefficients and their related standard deviations (SD)for different surface conditions, $\sigma=2.6$, Re=$8.55\times 10^{5}$, $\alpha$= 9∘.  | Smooth | (I) | (II)-Non Uniform | (II)-Uniform | (II)-Sparse
---|---|---|---|---|---
$\text{C}_{\text{d}}$ | 0.05420 | 0.05478 | 0.05557 | 0.054540 | 0.05492
SD of $\text{C}_{\text{d}}$ | 0.00329 | 0.00577 | 0.01206 | 0.01054 | 0.01302
$\text{C}_{\text{l}}$ | 0.62875 | 0.63070 | 0.62837 | 0.62749 | 0.62768
SD of $\text{C}_{\text{l}}$ | 0.00310 | 0.00495 | 0.00871 | 0.00882 | 0.01063
Number of samples | 7 | 4 | 5 | 7 | 6

###  5.5 Roughness impact on cavitation extent

One of the main concerns with roughness application is its impact on the
cavitation extent and its regime, especially if roughness triggers the sheet
cavity into a more erosive cloudy regime. To investigate this impact,
different measurements are conducted in different operating conditions and on
different roughness patterns. The discussion here, however, is limited to the
comparison of the smooth and roughness pattern (II) having sparse sand
distribution. Representative images of cavitation extent of these two patterns
are presented in Fig. 16. At the selected cavitation number, $\sigma=1.2$, the
cavitation behaviour shows a relatively constant and strong cavitating tip
vortex while on the foil a periodic interaction of sheet and cloud cavitation
are observed. The presented figures include the minimum and maximum cavitation
extents during this periodic behaviour.

The comparison of the pictures related to the minimum cavitation extent
indicates roughness slightly increases the extent of cavitation on the leading
edge. This affects the sheet cavity extent on the foil tip and leads to a
smaller sheet cavity for the roughened foil. The most interesting effect of
roughness on the cavitation behaviour can be noted by the comparison of the
maximum cavitation extent. The presence of roughness either prevents the full
development of sheet cavity or blocks the re-entrant flow which eventually
leads to even less severe cloud cavitation. Both the extent of cloud cavity
and its harshness are observed to be reduced on the roughened foil.

(a) Smooth

(b) Pattern (II) - Sparse

Fig. 16: Minimum and maximum cavitation extent for the smooth and coarse
roughness arrangement at $\sigma=1.2$, Re=$1.204\times 10^{6}$, $\alpha$= 9∘.

###  5.6 Roughness and cavitation hysteresis

In Fig. 17 and Fig. 18, the hysteresis of cavitation are investigated on the
smooth foil and for roughness pattern (II) with sparse sand distribution. The
figures contain two columns related to the conditions where the cavitation
tunnel pressure is gradually decreased or increased while keeping the tunnel
free-stream velocity constant.

$\sigma$=1.2Free-stream direction

$\sigma$=2.6

$\sigma$=6.0

$\sigma$=9.7

$\sigma$=12.7

(a) Increasing pressure

$\sigma$=2.6

$\sigma$=4.2

$\sigma$=6.0

$\sigma$=7.0

(b) Decreasing pressure

Fig. 17: Cavitation hysteresis around the smooth foil, zoomed view of the
suction side tip.

For the smooth foil, a significant difference is found between the cavitation
behaviour of the decreasing and increasing tunnel pressure. While decreasing
the pressure, cavitation incepts at around $\sigma$=6 and further decrease
leads to a stronger TVC. At $\sigma$=1.2, the cavitation contains both a TVC
and sheet/cloud cavitation. By increasing the tunnel pressure, the sheet/cloud
cavitation disappears but the TVC remains attached to the tip until
$\sigma$=12.7. At this pressure condition, TVC suddenly separates from the tip
and after that no further TVC is observed. For the roughened foil, the
hysteresis analysis only includes $\sigma$=1.2 and $\sigma$=2.6. This,
however, was enough to indicate that the roughness does not worsen the
cavitation hysteresis in the variable pressure condition compared to the
smooth foil.

$\sigma$=1.2

$\sigma$=2.6 (a) Increasing pressure

$\sigma$=2.6 (b) Decreasing pressure

Fig. 18: Cavitation hysteresis around the roughened foil, zoomed view of the
suction side tip.

##  6 Conclusion

This paper demonstrates the possibility of using roughness in order to
mitigate cavitating tip vortex flows. Knowing that its tip vortex flow
resembles propeller tip vortex flows, an elliptical foil is selected as the
test case. The tip vortex properties of this foil at two roughness pattern
configurations are evaluated at different operating conditions. The main
findings of the study are summarized as follow,

  * •

Application of roughness is effective in delaying vortex cavitation inception.
The measurements show a decrease in the TVCI as large as 33 % in the optimized
roughness pattern compared to the smooth foil condition where the drag force
increase is observed to be around 2 %.

  * •

Application of roughness on the leading edge, tip region and trailing edge of
the suction side is found to be the most optimum configuration of roughness
pattern where the TVCI mitigation is achieved with little performance
degradation.

  * •

We did not observe any risk of increasing sheet cavity on the foil by
roughness application in the tested operating conditions. It is, however,
noted that roughness elements form tiny nucleation sites having micro- or even
nano-metric residual air pockets that generate very small nuclei continuously
and normally due to local degassing.

  * •

Roughness application not only changes the tip vortex strength and the
inception point but also the free-stream nuclei capture properties of the tip
vortex. This can be related to the fact that nuclei movements are affected by
more interactive structures generated by the roughness elements.

  * •

The TVCI is found to be similar in the uniform and non-uniform roughness
distributions for the same roughness pattern while the drag force is found to
be higher in the non-uniform roughness distribution. This, however, poses a
further question on how roughness elements should be distributed in terms of
randomness and populations to minimize the performance degradation.

  * •

The measurements highlight that in the roughened surface condition magnitudes
of the tangential and axial velocities are decreased compared to the smooth
condition. This clarifies that lower momentum has been fed into the tip vortex
in the roughened case. One possible reason for this could be increased
velocity fluctuations in the roughened cases compared to the smooth condition
which result in higher mixing rates and therefore higher viscous losses.

  * •

For a fully cavitating condition where the foil experiences a periodic
combination of cavitating tip vortex and sheet/cloud cavity, observations
indicate that roughness either prevents the full development of sheet cavity
or blocks the re-entrant flow which eventually leads to less severe cloud
cavitation and possibly less erosive cavitation regime in the roughened foil
compared to the smooth foil condition.

These conclusions are obviously to be confirmed for other ranges of the
variables that were not tested in this study.

##  7 Declarations

Funding: Financial support for this work has been provided by VINNOVA through
the RoughProp project, Grant number 2018-04085. The experimental measurements
are conducted in the free surface cavitation tunnel at the Kongsberg
Hydrodynamic Research Center, Kristinehamn, Sweden.

Availability of data and material: Raw data were generated at the cavitation
tunnel facility of Kongsberg Maritime AB, Kristinehamn, Sweden. Derived data
supporting the findings of this study are available from the corresponding
author upon reasonable request.

Conflicts of interest and Code availability: Not applicable.

##  8 Nomenclature

$\text{C}_{0}=301.2$ mm |  root chord length
---|---
$\text{S}=360$ mm |  span length
A=$8.43\times 10^{-2}$ $\text{m}^{2}$ |  projected surface area of the foil
$\rho$ |  density
$\mu$ |  dynamic viscosity
$\text{U}_{\text{inlet}}$ |  inlet velocity
p${}_{\text{outlet}}$ |  tunnel outlet pressure
p${}_{\text{sat}}$ |  saturation pressure
$\sigma=(\text{p}_{\text{outlet}}-\text{p}_{\text{sat}})/(0.5\rho\text{U}_{\text{inlet}})$ |  cavitation number
Re=$\rho\text{U}_{\text{inlet}}\text{C}_{0}/\mu$ |  Reynolds number
D |  drag force
L |  lift force
$\text{C}_{\text{d}}=\text{D}/(0.5\rho{\text{U}_{\text{inlet}}}^{2}A)$ |  drag coefficient
$\text{C}_{\text{l}}=\text{L}/(0.5\rho{\text{U}_{\text{inlet}}}^{2}A)$ |  lift Coefficient
r |  radial distance
r${}_{\text{vortex}}$ |  vortex core radius
z |  streamwise direction, Fig. 1
Do |  dissolved oxygen concentration
AOA |  angle of attack
TVC |  tip vortex cavitation
TLP |  tip loaded propeller
SSLE |  suction side leading edge, Fig. 3
SSTE |  suction side trailing edge, Fig. 3
NU |  non-uniform roughness distribution, Fig. 4

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