Experimental and numerical studies on the cavitation over flat hydrofoils with and without obstacle ＊

Ling-xin Zhang , Ming Chen Jian Deng Xue-ming Shao

1. The State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, China 2. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Hangzhou 310027, China

Abstract: To control the shedding of cavitation, an obstacle is placed on the surface of a flat hydrofoil. Both experimental and numerical studies are carried out. Images of cavitation evolution are recorded by a high-speed camera. 3-D simulations are performed to investigate the cavitating flows around the hydrofoil. The results show that the re-entrant jet plays an important role during the process of cavitation shedding. A kind of U-type shedding is identified during the evolution of the cloud cavitation. The length of the cavity is apparently reduced due to the placement of the obstacle. It is interesting to find that the cavitation shedding changes from the large-scale mode to a small-scale mode, as an obstacle is placed on the hydrofoil surface. As we can observe from both experimental and numerical results, the small-scale cavitation shedding dominates the cavitating flow dynamics, we thereby conclude that the placement of an obstacle is favorable for the inhibition of cavitation shedding.

Key words: Cloud cavitation, obstacle, re-entrant jet

Introduction

Cavitation is often observed in many hydraulic machinery systems[1-2]. It can reduce the performance or even cause the damage of the devices. To investigate the fundamental mechanism of cavitation instability, cavitation on various types of simplegeometry objects have been studied, including standard NACA-type hydrofoils, Clark-Y hydrofoils[3],cylinders[4]and spheres[5]. It is well known that the attached partial cavitation is unstable, which would most likely develop into large-scale shedding cloud cavitation due to re-entrant jets. As the cloud cavitation forms and collapses periodically, it imposes undesirable impact forces on the bodies.

There have been many hypotheses about the causes of re-entrant jet. As we know that the pressure at a cavity closure is always larger than that within the cavity, which is very favorable for the development of the re-entrant jet. In the experiments carried out by Callenaere et al.[6], the thickness of the re-entrant jet was identified to be 15% to 30% of the maximum cavity thickness. The momentum balance analysis showed that the thickness of the re-entrant jet is a linear function of the adverse pressure gradient at the closure. Besides the mechanism of adverse pressure gradient, other explanations were also proposed.Leroux et al.[7-8]suggested that the collapse of cloud cavitation leads to overpressure, which triggers the formation of re-entrant jet and thus destabilize the cavitation. Recently, Ganesh et al.[9-10]observed a shock wave propagating upstream in their wedgegeometry experiments. However, the formation of shock waves could be strongly dependent on their experimental conditions. So far, the mechanism of adverse pressure gradient is still considered to be the most accepted explanation for the cause of re-entrant jet.

To control the stability of cavitation, the most common approach is to alter the pressure distributions along the hydrofoil by optimizing the geometry of the hydrofoil section[11], or by injecting air into the area of the cavitation[12]. Another possible way is to control the re-entrant jet. For example, Kawanami et al.[13]placed an obstacle on a NACA-type hydrofoil to obstruct the re-entrant jet moving towards the leading edge. According to the results of their experiments,the drag force and cavitation noise both decrease under their experimental conditions. Che et al.[14]experimentally studied the dynamic characteristics of transitional cavity shedding on a 2-D NACA0015 hydrofoil, with symmetrical side-entrant jets created to control the cavitating flows. They declared that the jets played a dominant role in transitional cavity shedding and resulted in a two-stage shedding of transitional cavity. However, few experiments or simulations have been performed to study the cavitation on other-geometry hydrofoils.

In this paper, we aim to control the unsteady cavitating flows over a flat hydrofoil by placing an obstacle upon it. The cavitation structures without and with obstacles will be respectively recorded in the experiments. 3-D numerical simulations will be performed to analyze the flow fields in detail. The characteristics of the re-entrant jet and its interaction with the obstacle will be discussed.

1. Experimental methods

Experimental observations of cavitation are carried out in the cavitation tunnel at Zhejiang University. The tunnel is an upright close-loop structure with a rectangular test section with a cross-section of 200 mm×200 mm and 1 000 mm in length. As shown in Fig. 1, a flat hydrofoil with a 90°cone angle nose is mounted between the two side walls. The chord length of the foil isc=150 mm . A red laser is used for horizontal correction to ensure its zero attack angle. Three perspectives including top,side and bottom views can be used to observe the flow in the test section. A full-span obstacle with 0.02 m height and 0.02 m width is placed on the upper surface of the foil with a distance of 0.37cfrom the leading edge. This choice of the obstacle location accords with that given in the experiments by Kawanami et al.[13],in which an obstacle was placed at the same location with respect to the leading edge on a NACA-type hydrofoil. The images of cavitation are recorded by a Photron SA4 high-speed camera with the frame rate of 5 000 fps at a spatial resolution of 1 024×800 pixels.

The flow in the cavitation tunnel is driven by an axial flow pump, with a maximum flow velocity of 12 m/s reached in the test section. The pressure in this tunnel can be reduced to the lowest value of 0.1 atm, which function has often been used to control the cavitation number. The test section is connected to its upstream contraction section with a contraction ratio of 9, which guarantees a low turbulence uniform flow at the inlet of the test section. The turbulence intensity at the test section inlet is less than 0.2%. In the present experiments, the flow velocity at the inlet and the pressure at the outlet of the test section are set to beU=6.318 m/s andp∞=16190 Pa respectively, resulting in a cavitation number of 0.72, which is calculated as

As we understand that the scale effect is important to cavitating flows, therefore careful attention should be paid when we scale the mode test results to that of the full scale. For vortex cavitation,an empirical relation was given between cavitation number and Reynolds number[15]. However, according to the studies by Le et al.[16], for the attached cavitation, the flow structures, in particular the length of the cavitation vary little with Reynolds number,indicating that the scale effects are negligible for the attached cavitation.

Fig. 1 (Color online) Experimental setup

2. Numerical methods

Reynolds averaged Navier-Stokes (RANS)turbulence models are widely used in engineering area,which however have been found to over-predict the production of turbulence viscosity[17]. Large eddy simulation (LES) can simulate small-scale vortex shedding in the case of extremely fine grid, while its huge demand for computing resources limits its application in relatively complex flows, such as the cavitating flows. It has been widely recognized that Detached eddy simulation(DES)[18]is more capable than either unsteady RANS or LES in predicting high Reynolds number flows[18]. Here, the Spalart-Allmaras DES model[19]is employed in our numerical simulations.

The governing equations for unsteady turbulent cavitating flows are given by

The overbars in Eqs. (2)-(4) denote the averaging/filtering operation for the variables in turbulent flows,fτis the viscous stress tensor, andtτis the turbulent stress tensor, equal to the Reynolds stress near the wall and the subgrid stress far away from the wall.lρandlαare the density and the volume fraction of the liquid respectively. The density of the mixture,mρ, is a function of the liquid volume fraction

wherevρis the vapor density. The source term in Eq.(4),m˙, represents the finite phase change rate between water and vapor. The Schnerr-Sauer cavitation model[20]is used, in which the phase change rate is given by

whereαnucdenotes the initial nucleus content, and the coefficientBis

wheren0is the bubble number density per unit volume.

All the numerical simulations are performed by the open source CFD code OpenFOAM. An implicit pressure-based algorithm is developed for the computations of unsteady cavitating flows. For details of the algorithm one can refer to our previous publication[21].

3. Experimental results

In this section, the image sets of the unsteady cavitation atσ=0.72 are exhibited and discussed.Figure 2 shows top views of the experimental observation for the cavitation evolving in one cycle in the case without obstacles. As a typical cavitation shedding process, large-scale structures shedding to the wake are recorded in Figs. 2(a), 2(b), followed by the regeneration of attached partial cavities from the leading edge. Then a U-type cloud is formed at the wake flow (see Fig. 2(c)), which is easily understood by considering the side wall effects, which break the translational symmetry of the wake, leading to the faster shedding of the mid-span region than its side counterparts. Meanwhile a new cavity has reached its maximum length, as shown in Figs. 2(d), 2(e), with the re-entrant jet formed at different locations from 1/3 to 1/2 cavity length along the span, as marked in Fig. 2(d). As the re-entrant jet moves upstream towards the leading edge, it finally cuts off the attached cavity at the leading edge (see Fig. 2(f)). This process repeats every periods of cavitation shedding.

Figure 3 shows top views of the experimental observation for the cavitation evolution with an obstacle at 0.37c. Small-scale structures are observed as the rear part of the cavity is cut off and sheds into the wake, as seen in Fig. 3(a), instead of largescale shedding. This shedding process concentrates at the rear part of the cavity. In Figs. 3(b)-3(c), it is apparently shown that the maximum length of the cavity, as it evolves, is much shorter than that in the case without obstacles. Though the re-entrant jet could still be observed in front of the obstacle, even at the leading edge, as shown in Figs. 3(d, 3(e), it is not strong enough to cut off the entire attached cavity.From this series of images, we still observe that the formation and propagation of the re-entrant jet appear to be periodic. However, the small-scale shedding becomes dominant in the case with an obstacle,leading to more stabilized cavitation. To further understand the details of the flow fields, 3-D numerical simulations are performed and will be discussed in the following section.

4. Numerical results

The same geometry of the flat hydrofoil shown in Fig. 1 is used for 3-D simulations. As shown in Fig. 4,the inlet boundary is 4/3caway from the leading edge, and the outlet boundary is 3caway from the trailing edge. Mesh refinement is performed particularly around the leading edge, trailing edge and the obstacle. The whole grid number is about 10.8×106,with 120 nodes distributed along the spanwise direction.

Fig. 2 Top views of the cavitation evolution on the hydrofoil without obstacles for σ=0.72

Fig. 3 (Color online) Top views of the cavitation evolution on the hydrofoil with an obstacle for σ=0.72

With the same parameters to our experiments, we set the incoming flow to be 6.318 m/s, and fix the pressure at the outlet boundary. No-slip wall conditions are imposed on the hydrofoil surface and the two

side walls. The upper and lower boundaries of the computational domain are set to be symmetric plane.The other physical parameters are also consistent with our expe5rimental setup. The Reynolds number is 9.477×10. An adjustable time step approach based on the Courant number is used in the numerical simulations, with the upper limit of Courant number at 0.4 and the maximum time step is set to 10-6s.

Figure 5 shows the isosurfaces of the vapor volume fraction at different time instants during a cycle without obstacles, and the pressure contours on the surface have also been shown. Similar to the experimental results, in Figs. 5(a)-5(c), we observe that the attached cavitation forms and grows, as the cloud cavitation migrates downstream and then collapses. A U-type structure is clearly seen at the final stage of the cloud cavitation (see Fig.5c). At the next instant (marked witht1for further discussion),as shown in Fig. 5(d), we notice that the pressure at the closure region is relatively large, while the pressure within the cavity is approximately equal to the saturation vapor pressure. The pressure gradient drives the liquid jet moving upstream. The pressure at the closure keeps at a high level until the re-entrant jet arrives at the leading edge, marking a new cycle of cavitation shedding. As we know that the cavity length is a crucial variable to the cavitation dynamics,we therefore make a quantitative comparison between the experiments and numerical simulations. In the experiments, the cavity length is measured based on our previously developed image processing algorithm for the image sets obtained by high-speed camera[22].After averaging a sufficiently large number of instantaneous images over several cycles, the averaged vapor field is obtained, as shown in Fig. 6(a), with the averaged cavity length of 0.9098 m. We note that the vapor does not distribute evenly along the span,therefore we also make a spanwise average to get the averaged cavity length. In the numerical simulations,the time-averaged cavity length is also calculated by averaging the cavitation fields over more than 10 evolution cycles, as shown in Fig. 6(b). The numerical result of the averaged cavity length is 0.9001 m, with a difference about 1.07% from experiments.

The image processing algorithm can also be used to obtain the instantaneous vapor volumes by using the images from two views at a same time instant, as introduced in our previous publication[22]. By performing a fast Fourier transform (FFT), we get the spectrum for the fluctuation of the vapor volume. The non-dimensional frequencies or the Strouhal numbers for the experiments and the numerical simulations are 0.216 and 0.234 respectively. Here, the Strouhal number is defined asSt=fU/L, in whichfis the dominant frequency,Uin the uniform incoming v elocity andLis the chord length.

Fig. 4 The computational domain and the mesh distributions

Fig. 5 (Color online) Numerical results for the isosurfaces σof =th0e. 7v2a por volume fraction with a value of 0.7 and the pressure contours on the foil surface (without obstacles and )

To analyze the re-entrant jet in detail for the case without obstacles, we choose the instantt1, corresponding to Fig. 5(d). As shown in Fig. 7(a), at this instant, the re-entrant jet is not uniform along the span and it goes faster at the mid-span (Z4) due to the side wall effects. It is easy to understand that since the flow velocity has been affected by the wall effects, it is not possible for the re-entrant jet to synchronize along the span. However, as we look into the time-averaged fields, as shown in Fig.7(b), the pictures for different cross sections vary little, and the time-averaged cavities resemble each other in both length and thickness. This indicates that although the re-entrant jet is non-uniform along the span, the time-averaged cavitation is almost uniform due to the periodic evolution of the cavitation. We note that the averaged results are obtained by averaging the instantaneous fields over more than ten cycles.

Fig. 6 Time-averaged vapor fields

Fig. 7 Fields of vapor volume fraction on different cross sections. Note that the cross sections in (a) are non-equally spaced and marked with different spanwise (Z) coordinates, where S is the length of the half span, while that in (b) are equally spaced

Figure 8 shows the isosurfaces of the vapor volume fraction and the pressure contours on the hydrofoil surface with an obstacle at 0.37c. The cavity length is apparently reduced by the placement of the obstacle. The large-scale shedding which occurs in the case without obstacles can hardly be spotted.Instead, the small-scale shedding can be observed at the rear part of the cavity. Figures 8(a)-8(b) show the developing process of the cavitation. As the cavity approaches to the obstacle, it soon stops growth, when a re-entrant jet moves upstream. As the re-entrant jet reaches the leading edge, cavitation shedding occurs,as shown in Fig. 8(c). However, the obstacle blocks the path that the shedding cavitation travels, so that majority of the vapor mass stays in front of the obstacle. This part of the cavitation links to the attached cavity which regenerates from the leading edge. Only a small part of the vapor mass is able to move across the obstacle and collapse in the downstream. The simulated length of the cavity and the occurrence of the small-scale shedding accord well with our experimental results.

The obstacle makes great influence on the evolution of the cavitation. Figure 9 shows the instantaneous (corresponding to Fig. 8(f) and the time instant is marked with) and time-averaged fields of vapor volume fraction on several equally-spaced cross sections. Due to the nonuniformity of the re-entrant jets along the span, the growing and shedding processes vary among different cross sections. At this instant,some of the re-entrant jets are travelling upstream, while some are cutting off the cavity. As the shedding cavitation moves to the downstream, it is blocked by the obstacle. This blockage reduces the volume of the cavitation travelling further beyond the obstacle. Similar to the case without obstacles, the averaged fields on various sections are quite uniform.The averaged length of the cavity is a little longer than 0.37c, which is the location of the obstacle

Fig. 8 (Color online) Numerical results for the isosurfaces of the vapor volume fraction with a value of 0.7 and the pressure contours on the foil surface (with an obstacle at 0.37c and σ=0.72)

Fig. 9 Fields of vapor volume fraction on equally-spaced cross sections

To make a quantitative comparison with the experiments, we present the time histories of the vapor volumes for the case without obstacles and that with an obstacle in Fig. 10. It shows that the vapor volume time histories match well between experiments and numerical simulations for both with and without obstacle cases. It is seen that the vapor volumes for the cases without obstacles fluctuate periodically, with the dominant frequencies 15 Hz and 16.4 Hz identified respectively for the experiments and the numerical simulations. In comparison with its counterparts without obstacles, the vapor volumes for the cases with an obstacle are reduced remarkably in both the averaged values and the amplitudes of fluctuation.We note that the fluctuating amplitude of the vapor volume for the numerical case is lower than the experimental case without obstacles. An possible explanation for this divergence is that the spatial resolution around the hydrofoil is still not high enough,or the DES turbulence model itself can only resolve large-scale eddies.

Fig. 10 Vapor volumes in the cases without and with obstacle

Fig. 11 Time-averaged pressure distributions along the hydrofoil surface from the leading edge to the trailing edge

As we have discussed that the cavity length in the case with an obstacle is much shorter than that without obstacles, for example by comparing Fig. 9 with Fig.7. This difference can also be reflected by the pressure distributions on the hydrofoil surface. Figure 11 shows the time-averaged pressure distributions in the cases with and without obstacles. We can see that in the case without obstacles there is an adverse pressure gradient at the cavity closure. While in the case with an obstacle, it bifurcates into two branches, with one in front of the obstacle and the other behind the obstacle. The second adverse pressure gradient is exactly located at the cavity closure. According to the pressure distributions, we conjecture that the obstacle affects the cavitation in two aspects: first, the cavity length and thickness are both reduced, then instantaneously, the obstacle also weakens the re-entrant jet which should start from behind the location occupied now by the obstacle, thus suppressing the length scale of the shedding cavitation upstream of the obstacle.

5. Conclusions

In this paper, the effect of obstacle on the inhibition of cavitation shedding is studied both experimentally and numerically. High speed camera is used to capture the images of unsteady cavitation. 3-D numerical simulations are also performed to investigate the details of the cavitating flows. Comparisons are carried out between the cases with and without obstacle to show effects of obstacle.

The 3-D numerical simulations capture the main features of cavitation evolution in the cases with/without obstacle. Due to the effect of side walls, the U-type cavitation shedding is detected both in the experiment and the simulation in the case without obstacle. While in the case with an obstacle at 0.37c,only small-scale cavitation shedding can be observed behind the obstacle. Meanwhile, the cavity length and thickness both decrease apparently due to the existence of the obstacle.

The obstacle affects the flow in many ways. It changes the basic pressure distribution in the flow,especially around the obstacle. These are two adverse pressure gradients in the pressure distributions on the surface. The location of the second adverse pressure gradient limits the length of the cavity. For the transient flow, the obstacle blocks the path of the re-entrant jet as well as the shedding cavitation in the upstream, so only small-scale shedding happens behind the obstacle. It appears that the effect of obstacle on the flow is more complicated than expected. We believe that more cases at different conditions would be required to improve the understanding of effect of the obstacle.