CAV2021 11th International Symposium on Cavitation
May 10-13, 2021, Daejon, Korea
* Corresponding Authors: [email protected]
Cavitating Flow Structure and Noise Suppression Analysis of a Hydrofoil with Wavy
Leading Edges
Mohammad-Reza Pendar 1*, José Carlos Páscoa1, Ehsan Roohi 2
1 Department of Electromechanical Engineering, University of Beira Interior, Portugal
2 School of Aerospace Engineering, Xi’an Jiaotong University, China
Abstract: The effects of the wavy leading edge (WLE) on the noise suppression mechanism due to a cavity
cloud formation around a hydrofoil, which contains condensation, detachment, collapse and shedding
phenomenon has been studied numerically. Change of the frequencies during these phenomena can be
used for cavitation detection. The oscillations global frequency modes and spectral content for two cases
of the straight leading edge (SLE) and wavy leading edge (WLE) hydrofoils are analyzed using Fourier
and continuous wavelet transformations. Produced counter -rotation vortices between the peaks of WLE
hydrofoil, by destroying the horseshoe vortex and delaying the tail vortex, changes the frequency. Here,
in addition to the noise, the hydrodynamic forces also have been discussed. To have a better
understanding in designing of the underwater vehicles with W LE hydrofoil, two important
hydrodynamic factors, noise and flow forces, had been investigated precisely. We solved the cavitating
flow in the cavitation numbers of σ=0.8at a chord-based Reynolds number of 7.2×105, using large eddy
simulation (LES) approach, as well as Kunz mass transfer model which is performed under the framework
of the OpenFOAM package.
Keywords: Cavitation noise, Wavy leading edge (WLE) hydrofoil, Cavitation, OpenFOAM, Large eddy
simulation (LES)
Introduction
Cavitation is a physical phenomenon that is multi-phase and complex that occurs when the liquid's local
pressure becomes lower than its saturated vapor pressure. In most cases, like hydraulic machinery,
cavitation damages the equipment by changing the flow structure and reducing the efficiency of the system
by making noises [1], erosion [2], unstable behaviors and vibrations like pressure fluctuation [3]. The cavity
cloud formation mechanisms including detachment, condensation, collapse, and shedding [4]. These
unsteady behaviors, particularly shedding, have a significant impact on hydrodynamic performance and
noise patterns. Earlier studies have shown that WLE hydrofoils have a relatively wide prospect of
application in underwater engineering like hydraulic machinery [5, 6], but the cavitation and noise
induction due to that must be considered in a new design.
Noise induced by wavy leading edge hydrofoils has received some attention during the last years. Up to
now, the investigation on induced noise by the hydrofoil with wavy leading-edge has focused primarily
on airfoil turbulence interaction noise [7]. But, the noise under cavitation flow is rarely examined [8]. The
cavity cloud pattern produced by wavy leading-edge hydrofoils is different from that of straight leading-
edge hydrofoils [8], so there may be some difference between their noise characteristics. Experimental noise
measurement is costly, and the results are subject to multi-factor interference [9]. In this work, cavitation
dynamics are precisely calculated, and relatively induced noises are investigated. Accurate cavitation noise
prediction is dependent on precise cavity flow simulation. Turner and Kim [ 10] calculated the aerofoil
CAV2021 11th International Symposium on Cavitation
May 10-13, 2021, Daejon, Korea
* Corresponding Authors: [email protected]
turbulence interaction noise induced by a WLE and SLE airfoil. They found that a notic eable noise
reduction occurred in the high-frequency band over the WLE foil.
Governing equations
The governing equations are the incompressible Navier -Stokes equations. The interface between the
liquid and vapor phases is captured by using the volume of fluid (VOF) method [11]. Kunz et al. [12] mass
transfer models was used for modeling of the cavitation flow . Selecting a suitable turbulence model is an
important concern in order to have an accurate cavitation flow simulation [13]. The Large Eddy Simulation
(LES) turbulence model, as described in detail in [14, 15], is appropriate to capture the flow features with
high gradient like present study. Large eddy simulation (LES) turbulence approach is based on computing
large, energy-containing eddies that are resolved on the computational grid, whereas the smaller, more
isotropic, sub-grid structures are modelled. In the current study, subgrid scale terms are modeled using
“one equation eddy viscosity” model.
Simulation setup
The dimension of the computational domain and employed boundary conditions is shown in figure 1
(a). The fully structured, orthogonal and isotropic meshes close to the WLE and SLE hydrofoil are
illustrated in figure 2. The grid size close to the hydrofoil, especially near the leading edge, where the cavity
vapor starts to form, and the vortices gradient is high, is well refined. We used around 15 million cells over
the WLE and SLE cases domain, based on the grid independency implemented in our previous work in
reference [8]. For this grid, the minimum and mean minimum values of Δy + are approximately 0.21 and 2.3.
The inbounding flow velocity is imposed as 10 U m s at the inlet boundary and a specific pressure is set
to adjust the specified cavitation number ( 2( ) / 0.5P P U ) at the outlet boundary regions. In the
simulating process, the time step and Courant number are kept less than around 10 -8 s and 0.15,
respectively. Second-order scheme accuracy is employed in discretizing all terms of equations. As a
validation propose, we examined the accuracy of the implemented numerical solution in our previous
study [8]. The lift and drag forces are compared with the experimental data of Johari et al. [5] and good
agreement was observed.
Figure 1. 3D view of the computational domain with dimension and boundary conditions.
CAV2021 11th International Symposium on Cavitation
May 10-13, 2021, Daejon, Korea
* Corresponding Authors: [email protected]
(b) (c)
Figure 2. Structured mesh distribution near the NACA 634-021 hydrofoil with WLE and SLE surface .
Results and discussion
Figure 2 shows the force coefficients distribution during the four complete cavity cycle. The most
important instants during the cycle are shown with iso-surfaces of the volume fractions ( 0.5 ) and are
labeled on the graphs. The maximum cavity length corresponds with the maximum lift coefficient (A). The maximu m
volume probably of the cavity occurs at (B). The force coefficient experiences an intense oscillation by shedding the
cavity cloud until the collapse close to the trailing edge (C). By complete collapsing of the cavity cloud, the lift values hit the minimum and higher pressure form on the hydrofoil's suction side (D).
(a)
(b)
(c)
Figure 3. The representation of the lift (b) and drag (c) coefficient distribution through four consecutive cavitation cycles for WLE
hydrofoil ( 6 , 0.8 ). The Isosurfaces of cavity cloud ( 0.5 ) is shown for critical instance in the cavity cloud evolution (a).
Due to the unsteady behavior of the cavitating flow, analysis of the Wavelet transform is necessary. The
continuous wavelet transform is computed separately for various parts of the signal in the time dimension.
Figure 4 (a) shows the continuous wavelet transform of the cavitating flow around the WLE and SLE
hydrofoil ( 6 ) with a cavitation number of 0.8. According to the figure, high range frequencies are
A
B
C
D
Condensation Detachment Shedding Collapsing
CAV2021 11th International Symposium on Cavitation
May 10-13, 2021, Daejon, Korea
* Corresponding Authors: [email protected]
damped in almost 30 (ms) for the WLE case. It can be seen that high frequencies are damped over time,
and the increase in the Strouhal number with time is apparent . The Strouhal number distribution in the
case with WLE is completely different compared to the SLE case, and a lower value is obtained. Also, figure
4 (a) illustrates the lift power spectrum density (PSD) on the surface of the WLE hydrofoil. The
corresponding scales seem to be reasonably close to Kolmogorov power law -decay (f ‒5/3) scaling.
Comparing the dominant frequency shows a higher value for the WLE case. Fourier analysis reveals that
St ~ 0.044 ( sin St f C U ) (equal to 60.3 Hzf ) and St ~ 0.033 (equal to 47.5 Hzf ) are dominant frequencies
in the cavitation flow ( 0.8 ) around the WLE and SLE hydrofoil at the studied Reynolds number.
(a)
(b)
Figure 4. (a) Wavelet transform and (b) PSD contour of cavitating flow passing over the WLE and SLE hydrofoil.
Figure 5 shows the mean pressure coefficient (a) and wall shear stress (b) distribution over the hydrofoil
surface. For the WLE case, a significant gradient of pressure produces in the hydrofoil's spanwise direction.
The mean pressure coefficient gradually decrease behind the peak and trough of the hydrofoil. The values
of the wall shear stress at the peak leading edge in the WLE case are considerably lower than the SLE case.
(a)
(b)
Figure 5. Comparison of time -averaged pressure coefficients and wall shear stress, along the hydrofoil surface, between the WLE (at the peak of
the spanwise) and the SLE hydrofoil ( 0.8 ).
f (Hz)
PS
Dof
CL
(HZ
-1)
102
103
104
105
106
100
101
102
103
104
105
WLE, = 0.8
f (Hz)
PS
Dof
CL
(HZ
-1)
102
103
104
105
106
10-1
100
101
102
103
104
SLE, = 0.8
X/CRef
wx
Me
an
0 0.2 0.4 0.6 0.8 1-20
0
20
40
60SLE, Mid-plane
WLE, Peak
WLE
SLE
f = 60.3 Hz
f = 47.5 Hz
SLE
WLE
WLE
SLE
CAV2021 11th International Symposium on Cavitation
May 10-13, 2021, Daejon, Korea
* Corresponding Authors: [email protected]
Conclusions
In the current study, the cavitation phenomenon over a hydrofoil with the WLE and the SLE was
investigated numerically, where the effect of changing the geometry was shown on the flow fluctuations
and frequency. The flow is analyzed by a wavelet transform, in addition to the Fourier transform. The
wavelet transform is appropriate for unsteady and discontinuous phenomena like cavitation, to indicate
the effects of the time dimension. Fourier analysis reveals that St~0.044 and St~0.033 are dominant
frequencies in the cavitation flow around the WLE and SLE hydrofoil at the studied Reynolds number.
Change of the frequencies during the condensation, detachment, collapse and shedding phenomena can be
used for cavitation detection. Time-averaged characteristics of flow variables are compared between the
SLE and WLE hydrofoils.
Acknowledgments
This work was supported by Project “ INTECH 4.0 –Novas Tecnologias para Fabricacao Inteligente”,
project grant no. POCI-01- 0247-FEDER-026653. The research was also partly supported by CMAST
Center for Mechanical and Aerospace Science and Technology, research unit n° 151 from Fundacao para a
Ciencia e Tecnologia (Portugal).
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