LES study of unsteady cavitation characteristics of a 3-D hydrofoil with wavy leading edge

Highlights

• Cavitating flow around a 3-D Hydrofoil with Wavy Leading Edges (WLE) is investigated using the LES approach in OpenFOAM.

• Comparison of cavity features of wavy leading edges (WLE) hydrofoil with straight leading edge (SLE) one.

• Force coefficients, re-entrant jet and separation are assessed during the cloud cavity evolution around WLE and SLE hydrofoils.

• The mechanisms of the laminar separation bubble (LSB) and low-pressure zone behind the WLE hydrofoil are illustrated.

• Role of vorticity stretching and vorticity dilatation with the presence of cavity for the WLE and SLE hydrofoils are investigated.

Abstract

The present study seeks to conduct numerical investigations of the cavitating flow characteristics around a sinusoidal wavy leading edge (WLE) 3-D hydrofoil underlying a NACA 63₄–021 profile with an aspect ratio of 4.3. Cavitational and non-cavitational characteristics of hydrofoils are numerically examined at a chord-based Reynolds number of 7.2 × 10⁵. The sinusoidal leading edge geometries include two WLE amplitudes of 5% and 25% and two WLE wavelengths of 25% and 50% of the mean chord length. We examined the cavitating flow around the hydrofoils in different cavitation numbers, namely σ = 0.8 and σ = 1.2. The flow over the protuberances of the WLE hydrofoil is considered at varying chord lengths and a constant angle of attack α = 6°, where significant spanwise variations in all flow properties, in contrast to the straight leading edge (SLE) hydrofoil, were observed. Large eddy simulation (LES) and Kunz mass transfer models are employed to simulate the dynamic and unsteady behavior of the cavitating flow. Besides, the compressive volume of fluid (VOF) method is used to track the cavity interface. Simulation is performed under the two-phase flow solver —interPhaseChangeFoam— of the OpenFOAM package. Compared to the SLE hydrofoil, we provided an exhaustive report of the time-averaged and instantaneous fluid dynamic characteristics of the cavitating flow around the sinusoidal leading edge hydrofoil, i.e., pressure, velocity, and vorticity fields, as well as lift and drag coefficients, and turbulent kinetic energy are reported. Furthermore, detailed analyses of the instantaneous cavity leading edge and flow separation treatment, vortical structure of the flow, vorticity stretching and dilatation, details of the spanwise flow, the formation of a low-pressure zone behind the WLE hydrofoil, streamwise velocity fluctuation, and evolution of the cavity dynamics through a complete cycle are reported. Results show that early development of the laminar separation bubble (LSBs) on the suction side of WLE hydrofoil prevents significant flow separation. Furthermore, the WLE cases exhibit a significantly reduced level of unsteady fluctuations in aerodynamic forces at the frequency of periodic vortex shedding.

Introduction

Cavitation is a multi-phase incident that occurs when the local liquid pressure becomes lower than its saturated vapor pressure. Considering the performance and reliability of the modified wavy leading edge (WLE) hydrofoils in different unsteady cavitating flow regimes, i.e., the incipient, sheet, cloud, and supercavitation stages, are instrumental. Cavitation significantly influences the hydrodynamic performance due to its complex and unsteady nature; it may cause lift reduction, erosion, vibration, and noise. The mechanisms forming the cavity cloud including detachment, condensation, collapse, and shedding are not yet comprehensively understood. These unsteady behaviors of the cavity flows, especially shedding, attract the attention of many researchers, because they have a significant impact on hydrodynamic performance. Cavitation control and recognition are expected to enhance the performance and reliability of the hydrodynamic types of machinery. The aforementioned observations subsequently revealed a series of experimental and numerical investigations aimed at the simulation of the effects of wavy leading-edge on foils.

Recently, the perusal of the flippers of a particular species of whale known as the Humpback has gained popularity among researchers. Fish and Battle (1995) conducted extensive research on the Humpback whale. Wing-like geometries with a high aspect ratio and large protuberances or tubercles that are located at the leading edge of the flipper distinguish this from other kinds of flippers. Studies conducted by Fish and Battle (1995) illustrated that the humpback whale is able to travel at high speeds and make sharp U-turns. The whale flippers orientate to high angles of attack (AoAs) to perform this particular maneuver. This flipper also ensures a specific level of lift at a high AoA.

Following a brief introduction about the cavitation phenomenon and the humpback whale, we consider and report studies about some characteristics of the WLE's foils, i.e., wing-tip impact, flow behavior in the stall and pre-stall region, flow separation, vortices treatment and aerodynamic forces thematically. Then, some relevant research will be explained in full detail. Finally, the overall feature of the current study will be presented.

Various researchers compared the influence of using an infinite span or involving a wing tip. WLEs in a high angle of attack managed to limit the flow separation at the wing tip region to the outboard region of the wing. Due to this management, maximum lift coefficient increased and the stall angle could improve (Fish and Battle, 1995; Miklosovic et al., 2004; Pedro and Kobayashi, 2008; Ozen and Rockwell, 2010; Yoon et al., 2011; Weber et al., 2011; Guerreiro and Sousa, 2012).

Modifying the characteristic of the stall in hydrofoils has been one of the significant issues of various studies, especially in the last two decades (Roohi et al., 2013; Ji et al., 2013; Ji et al., 2014; Long et al., 2019). Several recent studies illustrated that although WLEs with an infinite span stall earlier than SLE, but the stall process is smoother and the foil's performance, especially lift to drag ratio in the post-stall region over a wide range of Reynolds numbers may be higher for WLEs (Miklosovic et al., 2007; Johari et al., 2007; Hansen et al., 2011; Favier et al., 2012; Zhang et al., 2013; Rostamzadeh et al., 2014; Skillen et al., 2015).

Protuberances of the leading-edge form the primary function of generating streamwise vortices intended to maintain essential lift and preventing a stall at high AoA (Esmaeili et al., 2018). The performance of WLEs is generally similar to SLE in the pre-stall region, except in a few cases that illustrate an improvement in the performance (Miklosovic et al., 2004; Guerreiro and Sousa, 2012; Zhang et al., 2013).

The investigation of pressure distributions in WLEs wings shows higher pressure at the peaks and lower pressure at the troughs. This low pressure is created due to the accelerated flow that is channeled between two contiguous peaks (Pedro and Kobayashi, 2008; Yoon et al., 2011; Zhang et al., 2013; Skillen et al., 2015; Hansen et al., 2016). Johari et al. (2007), Hansen et al. (2011), Weber et al. (2011), Favier et al. (2012), Rostamzadeh et al. (2014) and Skillen et al. (2015) claim that the separation of flow at the troughs accrued more easily compared to the peaks that stay attached in the WLE. Low pressure passed a trough collision with a huge adverse pressure gradient causes an early separation. This early flow separation in the troughs forms a laminar separation bubble (LSB).

Counter-rotating streamwise vortices that seem to originate from the trough were observed by researchers who worked on WLEs (Johari et al., 2007; Pedro and Kobayashi 2008; Weber et al., 2011; Hansen et al., 2011; Yoon et al., 2011; Favier et al., 2012; Rostamzadeh et al., 2014; Skillen et al., 2015; Hansen et al., 2016). Some of them (Favier et al., 2012) argue that the vortices are created due to the spanwise velocity gradient that caused Kelvin–Helmholtz instabilities and that produce vortices in the normal direction to the surface and are immediately tilted by the flow. Rostamzadeh et al. (2014) and Hansen et al. (2016) explained the appearance of these vortices with Prandtl's secondary flow development of the first type, where the skewness of the flow diverts the initial spanwise vortices in other directions. By reducing the unsteady fluctuations in the aerodynamic force (Favier et al., 2012; Lau et al., 2013; Skillen et al., 2015) WLE geometries seem to be appropriate for applications that operate in highly disrupting flow conditions. Miklosovic et al. (2007), Pedro and Kobayashi (2008) and Ozen and Rockwell (2010) realized that WLEs treated as spanwise fences prevent the separation of the flow at the wing-tip from spreading inboard. Others claimed that an increased momentum exchange in the boundary layer is boosted by the WLE-induced vortical structures (Fish and Battle 1995; Miklosovic et al., 2004; Pedro and Kobayashi 2008; Hansen et al., 2011).

The amplitude of the protuberances has a distinct effect on the performance of the airfoils, whereas the wavelength has little. Guerreiro and Sousa (2012) indicated that the performance sensitively to the Reynolds numbers is lower for the WLEs than the straight leading edge (SLE). Although several experimental and numerical studies have been conducted that consider non-cavitating flows over wavy leading edge (WLE) hydrofoils, there is a paucity of studies that comprehensively consider cavitating flow over the WLE hydrofoil.

Cavitation dramatically decreases the hydrodynamic performance of an underwater propeller and induces various noises, vibrations, and also erosion (Pedro and Kobayashi, 2008; Taskar et al., 2017; Chen et al., 2016; Pennings et al., 2016; Wu et al., 2015). Thus, cavitation has remained a significant focus of hydrodynamic studies during recent years (Ji et al., 2015; Wu et al., 2016; Long et al., 2018; Pendar and Roohi 2018). Miklosovic et al. (2004) tested and compared humpback whale flipper models with and without leading-edge protuberances in a wind tunnel. Johari et al. (2007) measured the effects of the sinusoidal leading edge on the loading of full-span NACA 63₄–021 foil in a water tunnel and compared it with baseline ones.

For AOAs less than the baseline stall, the lift decreased while the drag increased in the modified foils. Above this angle, the lift of the modified foils was up to 50% higher than that of the baseline foil with no drag penalty. They also observed that the amplitude of the protuberances has a considerable effect on the performance of the foils, while the wavelength has little effect on their performance. Pedro and Kobayashi (2008) numerically simulated wings inspired by the humpback whale flipper by employing detached-eddy simulations based on the Spalart–Allmaras model to scrutinize a few operating conditions of the high-aspect-ratio. The results were in high agreement with the computations and available experimental data with the benefit of granting easy access to some of the intricacies of the 3D separated flow. Weber et al. (2010) modified ship rudder geometries by leading edge protuberances and tested in a water tunnel at Reynolds numbers up to 8.8 × 10⁵. They indicated that the effects of the leading edge modifications disappear at high Reynolds numbers. They also illustrated that the tubercles can modify the location of the onset of cavitation at lower Reynolds numbers and decrease the lift and increase drag for (15 < α < 22) for (22 < α) rudders with tubercles that generate more lift than smooth rudders.

Hansen et al. (2011) investigated the influence of sinusoidal leading-edge protrusions on the performance of two 21-percent-thick NACA airfoils with different aerodynamic characteristics. They observed that reduced tubercle amplitude leads to a higher value of the lift coefficient (C_L) and a higher stall angle. In the post-stall regime, the performance with larger amplitude tubercles is more desirable. They estimated that the leading-edge protuberances behave in a similar fashion to counter-rotating vortex generators. Yoon et al. (2011) numerically investigated the effect of the wavy leading edge on the hydrodynamic characteristics of rectangular wings at Reynolds number Re = 10⁶ for ship rudder applications. Five different waviness ratios at a fixed wavelength and wavy amplitude were considered. Weber et al. (2011) conducted computational simulations on an idealized humpback whale flipper by using the same turbulence model but prioritizing the formulation of a Reynolds-averaged Navier–Stokes (RANS) that was based on the extreme computational cost of DES in comparison with RANS. These authors struggled to deal with the effects of detached vortices at post-stall conditions. Sousa and Camara (2013) performed a numerical simulation on a NASA LS (1)-0417 sinusoidal airfoil, applying wavelengths of 0.5c and amplitudes of 0.12c. They offered an explanation for the vortical structures, especially the hairpin vortices in the 3D finite wing. They also assessed the aerodynamic modification in the lift and drag coefficients.

The hydrodynamic performance improvement through drag reduction is a crucial issue that must be considered (Esmaeilifar et al., 2017). Custodio et al. (2018) investigated experimentally the cavitation around the wavy leading edges hydrofoil. Their results revealed that the cavity cloud on the WLE hydrofoils with large amplitudes was confined to the areas behind the troughs, while in the SLE case there is sheet cavity cloud along the entire span. The lift coefficient in the SLE measured greater than the WLE hydrofoil at the high angle of attack. The drag coefficient was measured as almost equal in both cases.

Inspired by the above claim, the present study numerically investigates the formation and evolution of the cavitation phenomenon over NACA 63₄–021 hydrofoils by WLE and SLE using the LES approach and the volume of fluid (VOF) technique. LES approach is quite suitable for capturing the details of the cavitating flow and depicts the flow features precisely (Passandideh-Fard and Roohi, 2008; Pendar and Roohi, 2015a, b; Pendar and Páscoa, 2019a, b; Kolahan et al., 2019; Movahedian et al., 2019; Zahiri and Roohi 2019). Cavity boundary layer interaction, surface effects, and cavity interfacial instabilities, laminar to turbulent transition, large-scale turbulence, and re-entrant jet formation are considered. The present paper is organized as follows. In Section 2, the employed computational set-up is reported in detail. The results of the numerical solutions are analyzed in Section 3 to comprehend the effects and performance of WLEs cases compared to the SLE in cavitational conditions. Finally, the concluding remarks provided summarize the study conducted and the observations made.