Experimental investigation of combined vibrations for a hydrofoil-rod system at low Reynold numbers *

Ren-feng Wang , Ke Chen , Francis Nobelesse, Yun-xiang You , Wei Li

1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China

Abstract: A hydrodynamic tunnel experimental investigation and analysis of vibration characteristics for a hydrofoil in a transient regime is considered. Tests are performed for NACA 0017 models with a non-uniform section at an angle of attack AOA=5° and Reynolds numbers up to Re=7.0 ×106. This study is related to a project design of experiments in a complex facility that involves several parameters. The analysis focuses on the vibrations of a hydrofoil for different values of the bracing stiffness , the torsional stiffness , and the locations of the elastic axis a and of the center of gravity . The structural bracing response is investigated via measurements of the displacement of the free tip section of the hydrofoil using a three-axis acceleration sensor, and the torsional response of the structure is analyzed via measurements of displacements with single-axis acceleration sensors. The hydrofoil is made of reinforced plastics without flexibility, and elastic functions are performed by a spring support mechanism combined with a torsional structure. The study shows that an increase of the velocity results in different behaviors of the bracing and torsional amplitudes.Another notable result of the study is that the emergence of transition occurs simultaneously with additional peaks and changes of vibration amplitudes.

Key words: Vibrations, lifting bodies, transition, laminar separation bubbles

Introduction

As a result of its strong influence on the flow around marine structures at moderate Reynolds numbers, laminar separation bubbles (LSB) on lifting bodies have been considered in a large number of numerical and experimental investigations. A study of LSB is considered by Caster[1]for a wide range of Reynolds numbers. This study shows that low frequency motions of a bubble influence the instantaneous growth of waves. Furthermore, these slow motions distort the shape of the velocity profile, and ultimately influence the stability of the flow. A bubble,irrespective of its size, has been found to have an insignificant influence on the flow upstream from the bubble until transition appears. An experimental analysis of flow-induced vibrations for a low aspect ratio rectangular membrane wing at low Reynolds numbers from 2.4×104to 4.8×104is given by Rojratsirikul et al.[2]. This analysis shows that the combination of tip vortices and vortex shedding results in a mixture of chordwise and spanwise vibrational modes, and that the spanwise vibrational mode can always be observed at high angles of attack.

The Reynolds number has a significant influence on the overall performance of an airfoil, as well known. In particular, a laminar separation bubble becomes shorter as the Reynolds number increases.However, an increase in the Reynolds number does not necessarily increase the lift or decrease the drag[3-4]. Experimental studies are reported by Hu and Yang[5]on the transient behavior of laminar separation for an airfoil at various angles of attack and low Reynolds numbers, and suggests that a laminar separated bubble forms after the separated laminar boundary layer becomes turbulent. A large number of experiments are reported by Rist and Maucher[6]for airfoils at low Reynolds numbers in steady flows as well as in oscillating upstream. These experiments show that transitional separation bubbles are observed at low Reynolds numbers. Moreover, transition Reynolds numbers increase as the momentum thickness Reynolds numbers increase at separation,but decrease as the displacement of the separation shear layer from the wall increases. Additional experimental and numerical analysis[7-11]show that pitch-down or pitch-up motions of an airfoil promote instabilities and that the bubbles occur more upstream or more downstream than for the corresponding case of a steady angle of attack. Moreover, frequencies decrease or increase during pitch-down and pitch-up motions, and transition occurs together with the formation of uniformly distributed rollers.

Detailed experimental studies of the formation of laminar separation bubbles at low Reynolds numbers have recently been reported in published articles[12-14].These studies show that, for a semi-circular leadingedge, the growth of perturbations that follow the onset of separation ultimately leads to unsteady motions.Moreover, as the Reynolds number increases, the mean bubble length and height decrease, and Kelvin-Helmholtz vortices and unstable motions of the separation shear layer occur far downstream.Garcia et al.[15]considers the lift and drag coefficients 1C anddC of the NACA 0009 and NACA 0021 airfoils and the corresponding flow at low Reynolds numbers from 1.5×104to 5.0×104. Flow visualizations show that the flow around the NACA 0021 airfoil is completely separated and that no unsteady flow structure is found at relatively low Reynolds numbers.However, flow instabilities and Kelvin-Helmholtz vortex structures are observed at higher Reynolds numbers. A long laminar separation bubble appears in the separated shear layer for the NACA 0009 airfoil at AOA around 9°. Measurement of Cland Cdfor the NACA 0009 and NACA 0021 airfoils differs significantly. In particular, the NACA 0021 airfoil has a higher drag than the NACA 0009 airfoil. The lift of the NACA 0021 airfoil is strongly influenced by the Reynolds number, whereas the lift of the NACA 0009 airfoil is highly correlated with the angle of attack. A vortex tracking technique has been developed and validated[16-18]to measure the laminar separated bubble for a NACA 0018 airfoil at AOA=8° at low Reynolds numbers.

The foregoing brief review of the literature shows that the aerodynamic/hydrodynamic characteristics of flexible foils have been widely considered in numerous numerical and experimental studies.However, few experimental data have been reported about fluid structure interaction for complex hydrofoil-rod systems. The present study reports measurements of pitching and torsional motions of the lifting body system for several Reynolds numbers.

The influence of main parameters of the hydrofoil-rod system, specifically, the angle of attack, the bracing stiffness, the torsional stiffness, the location of the elastic axis, and the location of the center of gravity, is analyzed. The structure response to pitching and torsional motions triggered by a laminar separation bubble is also considered.

Fig. 1 (Color online) The water tunnel used in the present study

1. Experimental setup

A series of measurements have been performed in the cavitation tunnel of Shanghai Jiao Tong University. Figure 1 provides a simplified schematic of the cavitation tunnel. Its test section is 13m long with a square section 0.7 m in width and height. The highest flow velocity in the tunnel is 5 m/s with a non-uniformity that is smaller than 1%. The maximum turbulence intensity at the center of the test section is 5%. A model hydrofoil-rod system, depicted in Fig. 2,is assembled to effectively and accurately investigate the effects of the bracing stiffness, the torsional stiffness, the location of the elastic axis, the location of the center of gravity, and the inflow velocity. The values of the bracing stiffness kh, the torsional stiffness ka, the locations of the elastic axis a and of the center of gravity xaare listed in Table 1. The model system comprises a steel pedestal bolted into the ground, a steel shaft, a steel torque rod, an I-shaped frame with a rotatable plate welded in its center, and a hydrofoil. One end of the steel shaft is hooped with a universal bearing while the other end is threaded and extends horizontally into the test section.The threaded end of shaft is screwed into a tapped hole in the elastic axis position of the model hydrofoil.The torque rod is rigidly screwed onto the shaft so that they move together with the model hydrofoil. In addition, the torque rod is linked to the I-shaped frame via a set of four springs with identical spring constants.The shaft is supported at a large distance from the test section by means of a pair of small cylinders that are called universal bearing component hereafter. The lower ends of the universal bearing component are attached to the pedestal with a height adjustment screw, and the upper ends are welded onto the universal bearing as already noted. Consequently, the shaft and the hydrofoil rotate together inside the bearing, a main feature of the model system considered here.

Fig. 2 (Color online) Schematic of the apparatus

Table 1 Physical parameters of the hydrofoil-rod system

Fig. 3 (Color online) The foil models with different elastic axis positions

A NACA 0017 hydrofoil, with a span l=390 mm and a semichordon its tipside (150 mm on the root-side), is used in the experiments. The hydrofoil is mounted between the two vertical walls of the tunnel. Although the hydrofoil can be pitched to several angles of attack,most of the measurements are carried out for AOA=5° . The bracing stiffness is adjusted by selecting a proper pair of compression springs between the rotatable plate and the outside wall of the water tunnel. The upper end of each compression spring is welded onto a rotating bearing which hoops the steel shaft, and the lower end is screwed into the pedestal. As a result of the construction of the linkage,the rotations of the shaft and the hydrofoil are restrained by the rotating bearing. Moreover, their oscillatory motions are constrained by the compres-sionspringslocatedbeneaththeshaft.Horizontalmovements of the shaft are prevented by means of an upright beam welded to the pedestal. In addition, a trolley wheel on the rotating bearing can roll up and down along the beam. Because the steel shaft is supported by the compression springs and the universal bearing component, its horizontal position must be adjusted by changing the height of the universal bearing. The calibrated I-shaped frame is linked to the torque rod by means of the extension springs. Furthermore, every extension spring is attached to a corresponding slider that can move along the I-shaped frame. The torsional stiffness can be adjusted by simultaneously moving the four sliders or by selecting a different set of extension springs. Two additional model hydrofoils have been built for the purpose of considering the influence of different locations of the elastic axis. The three hydrofoils are shown in Fig. 3. These three hydrofoils have the same shape and are made from the same material. The elastic axis of the standard model hydrofoil (No. 2) is located 0.066 m= 0.48)- behind the leading edge at 1/3 of the span section away from the wing root.The elastic axes of the two additional hydrofoils are located 0.024 m (No. 1,and 0.103 m (No.3,) ydrofoil-rod system, including the weight, xaand the moment of inertia, are kept constant for the purpose of investigating the influence of the location of the elastic axis. In view of the significant influence of eccentricity on the moment of inertia, the chordwise and spanwise locations of the center of gravity are checked before assembly in order to minimize errors. The hydrofoil is made of reinforced plastics, and is hollow with a slide rail on each side, as shown in Fig. 4. As also shown in this figure, a steel tube can slide along the rails inside the hydrofoil and lead columns are equidistantly distributed along the steel tube in order to adjust the weight. The distance between the chordwise center of gravity and the elastic axis can be adjusted from 0.03 m to 0.05 m by moving the steel tube toward or away from the leading edge of the model hydrofoil. The steel tube is moved in proportion to the chord on each side to ensure that the spanwise center of gravity is always located at 1/3 span away from the root of the hydrofoil. The weight of the model hydrofoil is 2.84 kg,it can be changed by removing or adding an appropriate number of lead columns.

Fig. 4 (Color online) The foil model and the weight adjustment apparatus

2. Results

2.1 Influence of the bracing stiffness

The influence of the bracing stiffness khis analyzed by considering three different bracing springs, with spring constants 9.0×104N/m, 3.0×105N/m and 1.4×106N/m as listed in Table 1. The torsional stiffness, the locations of the elastic axis position and of the center of gravity are taken as=282 Nm/ra0.48ax for this analysis of the influence of kh.- and =0.04 m

Fig. 5 Growth of the amplitudes of the bending vibrations (a)and of the torsional vibration (b) of the hydrofoil-rod system at AOA = 5° , where = 282 N mad a k ? , a =-0.48 and = 0.04 m

The growth of the amplitude of the vibrations as V∞increases, measured from acceleration sensors placed on the wingtip (Fig. 5(a)) and on the torque rod(Fig. 5(b)), is depicted in Figs. 5(a), 5(b) for the three different bracing stiffnesses considered here. These figures, in which the amplitude is represented by the parameter A, show that the vibrations are larger, and that transition occurs for smaller speeds V∞, for larger values of kh. Specifically, transition occurs at=1.78 m/s and 2.25 m/s for9.0×104N/m,3.0×105N/m and 1.4×106N/m. Thus, a higher bracing stiffness delays the occurrence of transition.

Fig. 6 Frequency spectrums of the vibrations at the wingtip for different values of the bending stiffness and different values of the V∞ at AOA , where = a k 282 N ?m/rad , a = -0.4nd = 0.04 m

Figures 6, 7 illustrate the influence of the bracing stiffness on the bending amplitude and the torsional amplitude of the wingtip at the transition V∞, i.e., the speed at which transition occurs. Figure 6 shows the amplitudes of the vibrations that correspond to the first four vibration modes are higher for a larger bracing stiffness. The occurrence of transition that can be observed in Fig. 6 is associated with the occurrence of an additional peak near the frequency that corresponds to mode 1 in the spectrum depicted in Fig.7. Moreover, transition occurs at a frequency that is slightly higher than the frequency of mode 1 for(top of Fig. 6), whereas transition occurs at a slightly lower frequency than the mode 1 frequency for6=1.4 10 N/m× .

Fig. 7 Frequency spectrums of the vibrations on the torque rod for different values of the bending stiffness and different values of the V∞ at AOA° , where = a k 282 N ?m/rad , a = -0.4d = 0.04 m

2.2 Influence of the torsional stiffness

The influence of the torsional stiffness is studied by considering the three different sets of torsional springs, with spring constants 282 N·m/rad, 704 N·m/rad and 1 348 N·m/rad, listed in Table 1. The bracing stiffness, the locations of the elastic axis and of the location of gravity are chosen as,for this analysis of the influence of ka.and =0.04 m Figures 8(a), 8(b) depict the growth of the amplitudes of the bending (Fig. 8(a)) and torsional(Fig. 8(b)) vibrations as V∞increases for the three values of the torsional stiffness considered here. The figures show that the bending vibrations are larger,whereas the torsional vibrations are smaller, for larger values of ka. Moreover, the torsional stiffness influences the occurrence of transition.

Fig. 8 Growth of the amplitudes of the bending vibrations (a)and of the torsional vibration (b) of the hydrofoil-rod system at AOA = 5° , where =1.4 106 N/m h k × ,a = -0.48 and = 0.04 m

Fig. 9 Frequency spectrums of the vibrations at the wingtip for different values of the torsional stiffness, where = h k 1.4×106 N/m , a = -0.48 d = 0.04 m

Figure 9 depicts the spectrum of the amplitudes of vibration at the wingtip for the three values of thetorsional stiffness when V∞is around 2.0 m/s. The frequencies that correspond to the modes 1, 3 correspond to bending vibrations, whereas the frequencies of modes 2, 4 correspond to torsional vibrations. The frequencies that correspond to the modes 1, 3 and 4 are nearly the same for the three values of the torsional stiffness. However, the frequency for mode 2 is significantly influenced by the torsional stiffness. Specifically, Fig. 9 shows that the frequency of mode 2 is larger for larger values of. Fig. 9 also shows that the amplitude of the bending vibrations (modes 1, 3) are larger, whereas the torsional vibrations (modes 2, 4) are smaller, for larger values of the torsional stiffness.

2.3 Influence of the location of the elastic axis

As is illustrated in Table 1, three locations of the elastic axis are considered to analyze the influence on the vibrations of the hydrofoil-rod system. These locations are defined by(the shortest distance between the elastic axis and the leading edge of the hydrofoil),,. The bracing stiffness, the torsional stiffness and the location of gravity are taken as, =ax in this analysis of the influence ofand =0.04 m

Fig. 10 Growth of the amplitudes of the bending vibrations (a)and of the torsional vibration (b) of the hydrofoil-rod system at AOA = 5° , where =1.4 106 N/m h× ,= 282 Nrad a k ? and = 0.04m

The growth of the amplitudes of the bending (Fig.10(a)) and torsional (Fig. 10(b)) vibrations as V∞increases for the three locations of the elastic axis is illustrated in Fig.10. The amplitude of the bending vibrations of the model system foris significantly smaller than for,.The amplitude of the bending vibrations foris the largest of the three cases considered here before transition occurs.

Fig. 11 Frequency spectrums of the vibrations at the wingtip for different elastic axis positions, where =1.4× h k 106 N/m , = 282 N m/rad a k ? and = 0.04m

Figure 11 depicts the influence of the locationa of the elastic axis on the bending and torsional vibrations at the wingtip at a speed V∞around 2.25 m/s.The bending vibrations are significantly smaller than the torsional vibrations for. The spectrum of the amplitudes of the vibrations of the torque rod is depicted in Fig. 12. This figure and Fig. 10 show that the prominent additional peak, which is near the frequency corresponding to mode 2 in Fig. 12, and the transition shown in Fig. 10 occur together.

Fig. 12 Frequency spectrums of the vibrations on the torque rod for different elastic axis positions, where =1.4× h k 106 N/m 282 N m/d a k ? and = 0.04m

Fig. 13 Growth of the amplitudes of the bending vibrations (a)and of the torsional vibration (b) of the hydrofoil-rod system at AOA = 5° , where =1.4 106 N/m h k × ,= 282 N /rad a k ? and a = -0.48

2.4 Influence of the gravity center location

The influence of the location of the center of gravity xαon the vibrations of the model system is analyzed by considering three values of =ax 0.03,0.04 and 0.05 m as listed in Table 1. The bracing stiffness, the torsional stiffness and the location of the elastic axis are chosen as, =282 N ?m/rad andfor this analysis of the influence of xa.

Figure 13 depicts the growth of the amplitude of the bending vibrations (Fig. 13(a)) and of the torsional vibrations (Fig. 13(b)). The bending vibrations and the torsional vibrations are generally larger for smaller values of xa. A notable exception, however, is that the torsional vibrations grow visibly and ultimately become dominant for =0.05 mat a speed ＞2.0 m/s. This behavior agrees with the conclusion given by Herr[19]that utter is more likely to occur as a result of moving the center of gravity toward trailing edge of a foil.

Fig. 14 Frequency spectrums of the vibrations at the wingtip for different gravity center positions, where =1.4× h k 106 N/m N m a k ? and a = -0.48

Figures 14, 15 depict the spectrums of the amplitudes of the bending vibrations (Fig. 14) and of the torsional vibrations (Fig. 15) for the three values ofaconsidered here at a speed V∞around 2.0 m/s.These figures show a significant growth of the vibrations that correspond to the second torsional mode (mode 4) for =0.05 m. A corresponding rapid growth of the torsional vibrations can be observed in Fig. 13(b). This rapid growth is probably because the shedding-induced frequencies are close to the fourth natural frequency of the hydrofoil for=0.05 m[20].

Fig. 15 Frequency spectrums of the vibrations on the torque rod for different gravity center positions, where =1.4×106 N/m, =282 N ? m/rad and = -0.48

The spectrums of the amplitudes of vibrations at the wingtip (Fig. 16(a)) and at the torque rod (Fig.16(b)) for four speeds=0.70, 1.50, 2.28 and 2.47 m/s are shown in Fig. 16 for =0.05 m. Only frequencies that corresponds to the bending vibrations(modes 1, 3) can be observed in these figures for a relatively low speed =0.70 m/s. The first torsional vibrations (mode 2) become larger than the first bending vibrations (mode 1) when V∞r(nóng)eaches 1.50 m/s,and an additional peak can be observed near the frequency that corresponds to the mode 2. Figure 16(a)also shows that the torsional vibrations (modes 2, 4)grow more rapidly than the bending vibrations (modes 1, 3). This behavior is further illustrated in Fig. 17,which depicts the spectrums of the vibrations at the wingtip xα= 0.03 m for the four speeds considered here.

3. Conclusions

The influence of the bracing stiffness kh, the torsional stiffness ka, and the locationsand xaof the elastic axis and the center of gravity on the bending vibrations and the torsional vibrations of a hydrofoil-rod system in a cavitation tunnel have been experimentally investigated for Reynolds numbersRe. A three-axis acceleration sensor located on the wingtip of the hydrofoil and two single-axis acceleration sensors located on the torque rod were used for the measurements. This experimental analysis shows that the parametersNotable results of this experimental analysis are summarized below. The amplitudes of the vibrations at the frequencies that correspond to the bending vibrations (modes 1, 3) and the torsional vibrations(modes 2, 4) grow as the speed V∞increases. At relatively low speeds V∞, the mode 1 vibrations are smaller than the mode 2 vibrations. However, the reverseholdsathighspeeds.Thebendingstiffnessandax can have a significant influence upon vibrations and the flow transition, and therefore need to be considered in the design of an experimental setup to measure the flow transition. In particular, the occurrence of transition is significantly delayed for larger values of the bracing stiffness.and the torsional stiffness have similar influences on the bending vibrations of the mode l system. A smaller bracing stiffness or torsional stiffness results in larger bending vibrations of the hydrofoil for every speed V∞. However, the bracing stiffness and the torsional stiffness have opposite effects on the torsional vibrations, and a larger bracing stiffness yields smaller torsional vibrations. A shorter distance between the location of the elastic axis and the midpoint of the chord of the hydrofoil, i.e., a smaller value of a, results in higher torsional vibrations.More severe bending vibrations result if the center of gravity of the hydrofoil is closer to the trailing edge,i.e. for larger values of xa. Moreover, the bending vibrations forare significantly smaller than for a = -0.48,. The amplitudes of the bending vibrations and the torsional vibrations are smallest for. A significant growth of torsional vibrations are observed for =0.05 mas the speed V∞increases, i.e., the torsional vibrations of the hydrofoil are more sensitive to the value of V∞if the center of gravity is located closer to the trailing edge.

Further experimental studies, notably an analysis of the influence of the angle of attack AOA on the vibrations, need to be performed to fully understand fluid structure interactions for lifting bodies.