CAV2021 11th International Symposium on Cavitation

May 10-13, 2021, Daejon, Korea

* Corresponding Authors: m.reza.pendar@ubi.pt

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: m.reza.pendar@ubi.pt

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: m.reza.pendar@ubi.pt

(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: m.reza.pendar@ubi.pt

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: m.reza.pendar@ubi.pt

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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