Martijn Van Rijsbergen
email: m.v.rijsbergen@marin.nl

A review

Keywords:

INTRODUCTION

Sheet cavitation is a thin layer of gas and vapour attached to the leading edge of ship propeller blades, pump rotor blades and turbine runner blades. Its inception depends on a local pressure below the saturated vapour pressure and characteristics of fluid, solid surface and flow, like nuclei content, surface roughness and state of the boundary layer [START_REF] Van Der Meulen | Cavitation inception scaling by roughness and nuclei generation[END_REF]. These characteristics can influence the form and shape of a sheet cavity, and even inhibit its appearance despite a negative pressure. Because these inception effects seem to be more pronounced on model scale than full scale, they are often designated as scale effects.

Engineering measures have been developed in model scale facilities to limit such scale effects. However, as the physical mechanisms of sheet cavitation inception are not fully understood, the limits to the methods applied in scaled experiments are not clear. In model experiments, this may result in isolated patches of cavitation, spots or bubbles instead of a continuous sheet cavity. Fortunately, valuable insights have been obtained since the last comprehensive review on the topic 25 years ago [START_REF] Rood | Review-mechanisms of cavitation inception[END_REF] by carefully designed and conducted experiments.

The objectives of this paper are to review the existing knowledge on sheet cavitation inception processes and to define areas for further research. Eventually, this should lead to model scale cavitation tests where sheet cavitation inception occurs as close as possible to the vapour pressure. Such well-controlled model tests can then be used for prediction purposes and to validate and improve computational models.

The cavitation inception process can be described in three stages, from wetted flow, via cavitation inception to limited sheet cavitation. This paper treats the first two stages. In each stage the sensitivities to fluid, surface and flow characteristics are discussed for three basic forms of attached cavitation to be introduced in the next section.

Basic forms of attached cavitation

Figure 1 shows three forms of limited, attached cavitation. These cavities have a limited extent in flow direction-clearly beyond inception-but not yet developed enough to show periodic shedding. They remain attached to the solid surface and are stationary in position as long as the ambient conditions do not change. Limited spanwise extents are shown to clearly distinguish between the forms. Also intermediate forms are often observed.

A sheet cavity is built up of a large series of closely packed spot, finger or patch cavities. A general description is given in this section. The conditions at which they occur are discussed in the following sections. has a smooth surface with characteristic divots in it and the downstream part shows a rough surface. Spot and finger cavitation have been observed on headforms by Parkin & Kermeen [START_REF] Parkin | Incipient cavitation and boundary layer interaction on a streamlined body[END_REF], Arakeri & Acosta [START_REF] Arakeri | Cavitation inception observations on axisymmetric bodies at supercritical Reynolds numbers[END_REF], on foils by Arakeri [START_REF] Arakeri | Water tunnel investigations of scale effects in cavitation detachment from smooth slender bodies and characteristics of flow past a bi-convex hydrofoil[END_REF] and Guennoun [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF], and on propeller blades by Kuiper [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF][START_REF] Kuiper | Cavitation and new blade sections[END_REF]. The third form is designated as patch cavitation. It has a rough surface all over with characteristic parallel streaks directed downstream from its leading edge. Patch cavitation has been observed on headforms by Holl & Carroll [START_REF] Holl | Observations of the various types of limited cavitation on axisymmetric bodies[END_REF], Kuhn de Chizelle et al. [START_REF] Kuhn De Chizelle | Observation and scaling of travelling buble cavitation[END_REF] and on foils by Kuiper [START_REF] Kuiper | Effects of Artificial Roughness on Sheet Cavitation[END_REF].

FLUID, SURFACE AND FLOW ASPECTS

Apart from a pressure requirement, the fluid, surface and flow have to possess other aspects to start the sheet cavitation inception process. The relevant characteristics are introduced in this section for use as building blocks in following sections.

Fluid characteristics

The most important fluid characteristics for cavitation inception are surface tension, free-stream nuclei content and concentration of dissolved gas. The latter two are discussed, as they can vary significantly in cavitation facilities.

Water contains both microscopic bubbles and particles in a size range between a few and several hundred microns. In general, bubble diameters between 10 and 400 µm are considered as relevant for cavitation inception. In cavitation tunnels, the bubble concentration in this diameter range varies between 0.1 and 10 per cm 3 , but in some tunnels concentrations of 100 to 600 per cm 3 have been measured [START_REF] Mées | Development of interferometric techniques for nuclei size measurement in cavitation tunnel[END_REF][START_REF] Billet | Cavitation nuclei measurements -a review[END_REF][START_REF] Heinke | On cavitation nuclei in water tunnels[END_REF]. Without a specific control system to regulate the bubble spectrum (size and concentration) in a tunnel, it varies with the dissolved gas content, the pressure in the test section and the running time. Since the bubble generation process is different for each tunnel design, the optimum level of the dissolved gas content ranges from 30 to 80% of the atmospheric saturated value [START_REF] Mées | Development of interferometric techniques for nuclei size measurement in cavitation tunnel[END_REF][START_REF] Billet | Cavitation nuclei measurements -a review[END_REF]. Measured solid particle concentrations in cavitation tunnels range from 5 to 400 per cm 3 [START_REF] Mées | Development of interferometric techniques for nuclei size measurement in cavitation tunnel[END_REF][START_REF] Billet | Cavitation nuclei measurements -a review[END_REF][START_REF] Heinke | On cavitation nuclei in water tunnels[END_REF][START_REF] Ebert | Interferometric particle imaging for cavitation nuclei characterization in cavitation tunnels and in the wake flow[END_REF], but often without a specification of the size range. In many cases the solid particle concentration is orders of magnitude larger than the bubble concentration.

In oceans, bubble concentrations of 1 to 3 per cm 3 are found [START_REF] Billet | Cavitation nuclei measurements -a review[END_REF]. The concentration of solid particles larger than 10 µm is 10 to 20 per cm 3 [START_REF] Ebert | Interferometric particle imaging for cavitation nuclei characterization in cavitation tunnels and in the wake flow[END_REF][START_REF] O'hern | Cavitation inception scale effects: Nuclei distributions in natural waters, Cavitation inception in a turbulent shear flow[END_REF].

Surface characteristics

Jones et al. [START_REF] Jones | Bubble nucleation from gas cavities -a review[END_REF] reviewed a number of experiments that demonstrate the existence of gas cavities on surfaces with microscopic roughness, pits or scratches. Hydrophobic surfaces can trap (sub-) microscopic bubbles upon submersion in water. These sites serve as a source for bubble nucleation at sufficiently large supersaturation of dissolved gas, negative pressures or superheats. The effects of roughness on the flow characteristics are discussed in the following section.

Flow characteristics

In this section, some flow characteristics relevant to sheet cavitation inception are discussed. Kuiper [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] mentions the possible adverse effects on cavitation inception of bubble screening and the favourable effects of a laminar separation bubble and transition to turbulence.

Bubble screening

If a bubble approaches the leading edge of a foil, it will be forced outward by the high pressure at the stagnation point and drawn towards the surface by the point of minimum pressure. Johnson & Hsieh [START_REF] Johnson | The influence of the trajectories of gas nuclei on cavitation inception[END_REF] calculated the trajectories of gas nuclei released close to the dividing streamline just upstream of a two-dimensional half body. They found that the smaller the width of the body, the smaller the maximum bubble diameter that returns to the body upstream of the point of minimum pressure. Furthermore, the lower the freestream velocity, the larger this critical diameter becomes. This effect is less strong than the effect of size. For example, a critical bubble diameter of 150 µm was found at a freestream velocity of 7.5 m/s and a body width of 5 mm. Van Rijsbergen & van Terwisga [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF] found that the experimentally determined trajectories of free-stream bubbles-approaching the leading edge of a NACA 0015 foil-deviated clearly from calculated streamlines for bubble diameters larger than 190 µm.

Laminar separation bubble

A laminar boundary layer on a foil can separate due to a strong adverse pressure gradient. As a result, the pressure gradient is strongly reduced. Instabilities grow in the separated shear layer, usually followed by transition to turbulence. Turbulent mixing entrains fluid from outside the boundary layer causing it to grow. This increases the pressure again and reattachment occurs when the pressure is approximately equal to that without separation. The recirculation zone between the points of separation and reattachment is designated as the separation bubble. Arakeri [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF] showed that the ratio of the height of the separation bubble to the momentum thickness of the boundary layer at separation decreases from 15 to 5 with increasing Reynolds number on a hemispherical headform. O'Meara [START_REF] O'meara | Experimental determination of the laminar separation bubble characteristics of an airfoil at low Reynolds numbers[END_REF] found the same trend on an airfoil. The thickness of the separation bubble varied between 2 and 4 times the displacement thickness.

Transition to turbulence

Natural transition starts when Tollmien-Schlichting waves develop in a stable laminar boundary layer with a minimum of external forcing. These linear waves break down into nonlinear, 3D instabilities. In the final phase of transition turbulent spots are formed which grow in streamwise and spanwise directions. A fully turbulent flow is formed when these spots merge in spanwise direction. At model scale, natural transition to turbulence on foils and propeller blades occurs relatively far downstream of the leading edge. Upstream of the point of minimum pressure the transition process is suppressed by the favourable pressure gradient. The transition process is stimulated downstream of the point of minimum pressure. At the same loading, only an increased Reynolds number can move transition upstream. Kuiper [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] found a shift of the transition region on smooth propeller blades from x/c = 0.6 to 0.3 by increasing the sectional Reynolds number from 5x10 5 to 10 6 . In a so-called "bypass transition" process, the exponential growth of Tollmien-Schlichting waves is bypassed. The breakdown process is forced by larger disturbances such as free-stream turbulence and surface roughness. Strips of distributed roughness-such as carborundum grains with irregular shapes and a 50% area coverage-are commonly used to trip the boundary layer in engineering applications. Isolated roughness elements with cylindrical, hemispherical or a 3D-hill geometry are used for research purposes. The flow around a roughness element can be characterized by two non-dimensional parameters, i.e., the ratio of the roughness height to the displacement thickness (k/) and the roughness Reynolds number (R k = kU k /), in which U k is the velocity at the height of the roughness.

Roughness-induced transition to turbulence occurs at the critical roughness Reynolds number, R k,crit . Based on several experiments, Kerho & Bragg [START_REF] Kerho | Airfoil boundary-layer development and transition with large leading-edge roughness[END_REF] mention a typical R k,crit value of 600, both for isolated and distributed roughness. This value holds for flat plate boundary layers without pressure gradient and k/ < 1. Values up to 1500 or higher have been found in conditions with a favourable pressure gradient and k/ > 1.

Isolated roughness element

Several aspects of the flow around an isolated roughness element can influence the cavitation inception process, such as a local pressure reduction, a laminar separation bubble and transition to turbulence.

Van Rijsbergen & Slot [START_REF] Van Rijsbergen | Bubble-induced sheet cavitation inception on an isolated roughness element[END_REF] have made steady RANS calculations of the flow around a foil with a 3D roughness element at the point of minimum pressure. The roughness element decreased the minimum pressure coefficient with 60% at a chord Reynolds number of 10 6 . This effect decreased with decreasing Reynolds numbers until at Re = 10 4 the effect was negligible. Kerho & Bragg [START_REF] Kerho | Airfoil boundary-layer development and transition with large leading-edge roughness[END_REF] give a short review of the flow regimes downstream of isolated roughness elements as observed and measured in experiments. At R k = 160 the flow is laminar without separation. At R k = 300 -350, a region of separated flow was observed downstream of the element. Matheis et al. [START_REF] Matheis | Separation and unsteady vortex shedding from leading edge surface roughness[END_REF] made RANS calculations on a 3D hill and observed a growth of the laminar separation region between R k = 190 and 330. At R k = 350 -450, hairpin vortices are generated in the shear layer on top of the element [START_REF] Kerho | Airfoil boundary-layer development and transition with large leading-edge roughness[END_REF]. At higher R k values, a turbulent wedge appears with its origin clearly downstream of the element. Turbulent spots are formed within the contours of the wedge. A relatively small increase in R k moves the origin of the turbulent wedge upstream until it reaches the element at R k,crit .

Convective diffusion

Parkin & Kermeen [START_REF] Parkin | Incipient cavitation and boundary layer interaction on a streamlined body[END_REF] observed the growth of microscopic gas bubbles on the surface of a hemispherical head model.

The bubbles were found in a narrow band of locations downstream of the point of minimum pressure. The bubbles grew to a diameter of approximately 100 µm within a few milliseconds after appearance. The growth time was found to be proportional to the calculated local pressure, which remained above the vapour pressure. Together with the milliseconds growth time this excluded vaporous cavitation. Air diffusion in still water would give growth times in the order of seconds. Parkin & Kermeen [START_REF] Parkin | The roles of convective air diffusion and liquid tensile stresses during cavitation inception[END_REF] showed that the diffusion process in flowing water-designated as convective diffusion-is much faster. An acceptable agreement between theory and experiment was found, despite of the simplifications in the model. After reaching their maximum diameter, the bubbles detached from the surface and moved downstream at approximately half the free-stream velocity. No clear explanation could be given for the fixed position during growth and detachment of the bubbles, until the work of Arakeri [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF]. He observed a similar growth of bubbles on a hemispherical headform, but only near the point of separation. Furthermore, he noticed that the bubbles became entrained into the free shear layer once the bubble size had become equal to the local separation height.

NUCLEATION

Nucleation, i.e., the formation of bubbles, does not occur spontaneously when the fluid is supersaturated with dissolved gas or the pressure decreases below the vapour pressure. The characteristics of impurities in the liquid, a surface in contact with the liquid, or a combination of these two determine the pressure at which bubbles are formed [START_REF] Caupin | Cavitation in water: a review[END_REF]. Three types of this so-called heterogeneous nucleation are described in more detail in the following sections.

Nucleation by diffusion

Groβ et al. [START_REF] Groß | Experimental and theoretical investigation of nucleation from wall-bounded nuclei in a laminar flow[END_REF] investigated the nucleation process induced by an air-supersaturated laminar flow over air-filled cylindrical pits. The diameter of the pits was between 200 and 800 µm. The liquid was silicone oil at pressures well above the vapour pressure. The nucleation rate increased with the supersaturation rate and with the wall shear stress. A model of the mass flux into the surface-bound nucleus based on shear-enhanced diffusion supported, but did not completely clarify the experimental results. Nucleation rates up to 1000 Hz were found. The size of the detached bubbles was between 10 and 1000 µm and decreased with increasing shear rate.

Bubble expansion

The rapid growth of a gas bubble is the most investigated nucleation mechanism. A free-stream bubble becomes unstable and starts to grow exponentially if the pressure falls below the Blake threshold pressure [START_REF] Brennen | Cavitation and bubble dynamics[END_REF]. This critical pressure is mainly dependent on the surface tension and the initial diameter. Bubbles at room temperature and atmospheric pressure with initial diameters of 2, 10 and 400 µm have critical pressures of respectively -41 kPa, -3.0 kPa and 2.3 kPa. The dynamic response of a bubble to a low pressure peak can be calculated with the Rayleigh-Plesset equation [START_REF] Brennen | Cavitation and bubble dynamics[END_REF]. The speed of the initial growth is mainly dependent on inertia, hence larger bubbles grow slower.

Borkent et al. [START_REF] Borkent | Nucleation threshold and deactivation mechanisms of nanoscopic cavitation nuclei[END_REF] studied the nucleation threshold of bubbles trapped in nanoscopic pits in a surface. They were able to validate the crevice model of Atchley & Prosperetti [START_REF] Atchley | The crevice model of bubble nucleation[END_REF] for cylindrical pits with diameters below 1 µm. The critical pressure of these trapped bubbles was more than 2 times lower than for free bubbles of the same size. Bremond et al. [START_REF] Bremond | Interaction of cavitation bubbles on a wall[END_REF] studied the dynamic response of bubbles in cylindrical pits with a diameter of 4 to 20 µm. On this scale, the dynamic growth of these gas pockets could be modelled as hemispheric free bubbles of the same volume.

Tribonucleation

Hayward [START_REF] Hayward | Tribonucleation of bubbles[END_REF] investigated the generation of bubbles caused by a rubbing motion between two solid surfaces immersed in water. This nucleation by rubbing was designated as tribonucleation. In a test chamber made of glass, filled with water at room temperature and a negative pressure of 0.15 bar, several solid objects were placed on an inclined wall. Once the objects started to move, a bubble was generated which expanded explosively. The occurrence of nucleation was not dependent on the dissolved gas content. Several materials were found to produce bubbles, a sliding magnet, a fast rolling steel ball and sliding soft silicone rubber. It was presumed that sliding was a critical aspect of tribonucleation. Possibly, the fast rolling steel ball had some finite slip in its contact with the wall.

Washio et al. [START_REF] Washio | Cavity generation by accelerated relative motions between solid walls contacting in liquid[END_REF] studied tribonucleation using acrylic glass objects and surfaces immersed in water. Several contact motions such as separation and collision-both normal to the contact surfaces-and sliding were tested at pressures ranging from 0 to 80 kPa. The separating motion-constant acceleration from zero speed-did not cause nucleation. A colliding motion, however, did generate a bubble, but this motion had a larger separation speed. In the sliding motion also nucleation above the vapour pressure was found. The size of the maximum developed cavity increased with decreasing pressure and increasing sliding speed.

Wildeman et al. [START_REF] Wildeman | Tribonucleation of bubbles[END_REF] investigated tribonucleation by sliding a sapphire bead over various surfaces immersed in ethanol. The tests were conducted at atmospheric pressure and 8 K below the boiling temperature of ethanol. A threshold velocity (V) for the formation of a trail of bubbles as a function of the normal force (F) was found. The threshold could be described as FV = c, where c depends on the material. For aluminium c = 9 µW with V in the order of 1 mm/s and F in the order of 10 mN. Nucleation was also found on silicon, but not on copper or glass. The occurrence of nucleation was not dependent on the dissolved gas content, just as found in [START_REF] Hayward | Tribonucleation of bubbles[END_REF]. Scanning electron microscopy recordings of the wear tracks showed fracturing on the aluminium surface and plastic deformation on the copper surface. Both aluminium and silicon have a thin, brittle oxide layer on their surfaces. Such a layer is absent on copper.

Various hypotheses for the nucleation by rubbing have been formulated and discussed in [START_REF] Wildeman | Tribonucleation of bubbles[END_REF]. The pressure drop induced by the viscous flow in the space between two separating solid surfaces could not easily explain various nucleation events in low-viscosity liquids, such as water and ethanol. An alternative hypothesis was formulated in which fracturing played a central role. The micro-crack-formed at fracturing-filled with gas and vapour before liquid could enter. The observed force-velocity threshold was interpreted as a combination of generation by fracturing, merging and dissolution of bubbles.

ATTACHED CAVITATION INCEPTION

The process of attached cavitation inception depends on the characteristics of the fluid, the surface and the flow. Both the fluid and the surface can provide nuclei. The surface and the flow affect the transport of a free-stream nucleus to the surface. Finally, the form of limited cavitation is determined by the characteristics of the surface and the flow.

Below, five inception mechanisms are described. Highspeed film and video observations on micro-scale have enabled a detailed description of the visual phenomena. Limitations to spatial and temporal resolution, however, prevented a detailed view of the instant of nucleation. Hypotheses on the key elements of the nucleation mechanisms are formulated and discussed.

Free-stream bubble & roughness element

The inception process of a streak cavity, induced by a freestream gas bubble on an isolated roughness element is shown in [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF] and [START_REF] Van Rijsbergen | Bubble-induced sheet cavitation inception on an isolated roughness element[END_REF]. Three cases with two initial diameters and two cavitation numbers are discussed here. The bubbles expanded gradually as they approached the point of minimum pressure on the foil. This growth was enhanced significantly by the low pressure at the roughness element. Calculations indicated that this pressure was below the critical pressure of the three bubbles. When the bubbles passed the roughness element they generated an attached streak cavity downstream the roughness element. At the largest cavitation number, the bubbles expanded, but did not reach their critical radii. In these cases, the largest bubble generated the largest streak. Subsequently, the bubbles travelled on while oscillating. The continued travel of these nuclei after inception was the reason in [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF] to assume that they were made of a liquid such as oil. But in [START_REF] Van Rijsbergen | Bubble-induced sheet cavitation inception on an isolated roughness element[END_REF] it was shown that their growth followed the Rayleigh-Plesset equation for a gas bubble quite well. At a slightly lower cavitation number, the bubble did grow beyond its critical diameter. A longer streak was generated and the bubble imploded without a clear rebound.

This inception process is characterized by a free-stream bubble which induces a streak cavity but largely keeps its own dynamics (by continuing travelling or imploding). It shows similarities with the travelling bubbles described in Section 3.5. The high-speed video observations of [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF] indicated that the location of inception was between the top of the roughness element and approximately 300 µm downstream. This is the region where presumably a laminar separation bubble is present [START_REF] Kerho | Airfoil boundary-layer development and transition with large leading-edge roughness[END_REF][START_REF] Matheis | Separation and unsteady vortex shedding from leading edge surface roughness[END_REF]. Instabilities in the thin, free shear layer may promote mass transfer from the passing bubble to the recirculation region as well as evaporation. This could explain that a larger passing bubble generated a larger initial streak cavity. Figure 2 shows the inception process including the hypotheses.

Figure 2.

Inception mechanism for a free-stream gas bubble and a roughness element based on [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF], flow V from left to right.

Free-stream particle & roughness element

In [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF], a class of free-stream nuclei was identified that induced a stationary cavity on a roughness element, but with clearly different characteristics than a bubble. Their form and diameter remained constant, even after the instant of inception. Their transverse deviations from calculated streamlines were opposite to the deviations of bubbles near the stagnation point and point of minimum pressure. Therefore it was concluded that these nuclei were solid particles.

In the high-speed video observations of [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF], where a free-stream particle generated an attached cavity on a roughness element, the particle had a stream-wise displacement of approximately 1 mm per frame. Some frames, however, captured the particle within the optical resolution of 10 µm from the rough surface of the foil. Therefore, it is hypothesized that nucleation occurred at the moment when the particle made contact with a roughness element. This can be designated as tribonucleation, but what is the working mechanism? If the force-velocity relation found in [START_REF] Wildeman | Tribonucleation of bubbles[END_REF] could be extrapolated to velocities in the order of 10 m/s only a normal force in the order of 1 µN would be needed. Alternatively, this relation could be interpreted as the amount of energy needed to produce a bubble. Although this relation describes the observed phenomenon, it does not give a physical explanation. Micro-scale fracturing appears to be an essential physical mechanism in the cases studied in [START_REF] Wildeman | Tribonucleation of bubbles[END_REF] and could also have been present in [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF], but it seems not likely in the experiments of [START_REF] Hayward | Tribonucleation of bubbles[END_REF] because of the smaller forces and softer materials. The collecting and merging of (sub-) micrometer surface nuclei could be an alternative working mechanism. The existence of surfacebound nuclei on the particles or roughness elementssmaller than 10 µm-cannot be excluded on the basis of the high speed images [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF].

Once an initial bubble is generated, it probably travels with the particle downstream and causes a cavity in the laminar separation bubble downstream of the roughness element as a free-stream bubble would do, see Figure 3.

Figure 3.

Inception mechanism for a free-stream particle and a roughness element based on [START_REF] Van Rijsbergen | High-speed micro-scale observations of nuclei-induced sheet cavitation[END_REF], flow V from left to right.

Bubble in surface crevice

Guennoun [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF] studied the high-frequency nucleation of bubbles from seemingly arbitrary locations along the line of minimum pressure on a 2D non-polished foil. The average surface roughness was 1 µm. An array of pressure transducers with a 10 mm spacing was mounted in streamwise direction in the foil. The pressure taps with a diameter of 1 mm were smoothly filled with a pressuresensitive compound. The pressure tap near the point of minimum pressure was found to be a reproducible nucleation site. High-speed micro-scale observations with a resolution of 30 µm could not reveal a surface-bound nucleus. But the high nucleation rate up to 8 kHz indicated that the source of the nuclei was not in the free-stream liquid, but on the surface. A micro-scale crevice in the surface was assumed to stabilize a bubble. The train of travelling bubbles could be changed into spot cavitation by increasing the angle of attack, increasing the Reynolds number or decreasing the cavitation number. The same phenomenon was observed on an elliptic foil. Now, the origin of the travelling bubbles could be traced down to a bubble entrapped by an (unintentional) 50 µm roughness element, just upstream of the point of minimum pressure. On a polished foil with an average surface roughness of 0.2 µm only finger-type cavitation patches were observed.

The experimental results in [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF] and [START_REF] Guennoun | Experimental investigation of a particular travelling bubble cavitation[END_REF] indicated two spot cavitation inception mechanisms. At a fixed rate of pressure decrease, either a trail of travelling bubbles gradually merged into a spot or, a delay in nucleation was followed by a fast growth of a single bubble into a spot. Spot cavitation inception from a trail of travelling bubbles occurred mostly at small angles of attack, see Figure 4. A decrease of the ambient pressure increased the nucleation rate and the size of the bubbles. The bubbles began to merge with each other, first at the end of the trail and then at the start. Almost simultaneously, the process of attachment to the surface began. Cavitating tails and transient patches upstream of the travelling bubbles-see Figure 7-were observed at the end of the trail. Again, this process starts at the end of the trail and moves upstream with increasing nucleation rate. The process of attachment is supported by pressure measurements. Near the nucleation site, negative surface pressures were measured when a bubble passed. This indicates that the bubbles were separated from the surface by a thin layer of liquid. Upstream of a travelling bubble, the pressure decreased, increased and decreased again. This was interpreted as a signature of a 3D boundary layer separation.

Figure 5.

Inception mechanism for a surface-bound bubble at a large angle of attack based on [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF][START_REF] Guennoun | Experimental investigation of a particular travelling bubble cavitation[END_REF], flow V from left to right.

Attached cavitation inception by the growth of a single surface-bound bubble occurs primarily at large angles of attack, see Figure 5. Two events have been reported where the downstream edge of a pressure tap served as a nucleation site. Before inception, the pressure could be decreased to -20 kPa and -48 kPa respectively. After inception the pressure increased to approximately -15 kPa. The liquid at the tap upstream of the nucleation site was clearly under tension. The pressure did not reach the vapour pressure after inception, because the tap was only marginally covered by gas and vapour. After inception-with a limited cavity on the foil-the pressure downstream of the cavity increased to a value above that in the wetted flow. This indicates the presence of a reattachment point [START_REF] Holl | Observations of the various types of limited cavitation on axisymmetric bodies[END_REF]. When the ambient pressure was decreased further, the cavity length increased and the surface pressure decreased down to the vapour pressure as soon as the sensor was covered by the cavity.

On a number of occasions, travelling bubbles were observed at one nucleation site and spot cavitation at a neighbouring site. Also large differences in nucleation rate at two neighbouring nucleation sites were observed. This indicated that local phenomena such as the initial volume of gas, the local flow and interfacial forces between liquid, solid and gas play a significant role in the nucleation process.

Guennoun et al. [START_REF] Guennoun | Experimental investigation of a particular travelling bubble cavitation[END_REF] suggested vaporization from a gas inclusion as the primary working mechanism of nucleation from a surface. The non-condensable gas only served as a catalyst for the vaporization and would remain largely in the crevice after detachment of a bubble. The nucleation models in [START_REF] Borkent | Nucleation threshold and deactivation mechanisms of nanoscopic cavitation nuclei[END_REF] and [START_REF] Bremond | Interaction of cavitation bubbles on a wall[END_REF], however, assume that expansion of noncondensable gas is the main feature. The constant vapour pressure term in their model implicates a minor contribution of the vapour production. Using the model of [START_REF] Bremond | Interaction of cavitation bubbles on a wall[END_REF], the size of the surface-bound nuclei in the two events of spot cavitation inception mentioned above can be estimated to be 2 to 3 µm. Diffusion could play a significant role in the nucleation process by providing a continuous influx of non-condensable gas into the surface-bound nucleus, as discussed in [START_REF] Groß | Experimental and theoretical investigation of nucleation from wall-bounded nuclei in a laminar flow[END_REF]. It is mentioned several times in [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF], but eventually neglected in the modelling of nucleation.

Free gas bubble & laminar separation region

Cavitation inception on a body with a laminar separation bubble occurs first as travelling bubbles in the reattachment zone, far downstream of the point of minimum pressure. Two cases of travelling bubble cavitation inception have been reported by Parkin & Kermeen in [START_REF] Parkin | Incipient cavitation and boundary layer interaction on a streamlined body[END_REF] and [START_REF] Parkin | The roles of convective air diffusion and liquid tensile stresses during cavitation inception[END_REF] and were further interpreted by Arakeri [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF]. The inception process follows on the growth and detachment of microscopic bubbles as described in Section 1.3.5.

After travelling a few millimetre in the free shear layer, the bubbles arrived in the transition region where they grew explosively to many times their original size in less than 0.2 ms and travelled on. The growth rate indicated vaporous cavitation, but the expansion occurred in a region with a higher mean pressure. Furthermore, some bubbles passed the zone without expanding. Therefore, it was suggested in [START_REF] Parkin | The roles of convective air diffusion and liquid tensile stresses during cavitation inception[END_REF] and [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF] that strong pressure fluctuations in the transition region could decrease the local pressure below the vapour pressure. Katz [START_REF] Katz | Cavitation phenomena within regions of flow separation[END_REF] measured wall pressure fluctuations with peak amplitudes up to 18% of the dynamic head in the reattachment zone. The measured characteristic periods in the order of 1 ms would be sufficient to expand bubbles with diameters up to 200 µm. As the pressure is further decreased, attached finger-type (a.k.a. band) cavities are observed with a leading edge closer to the point of separation [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF]. Katz [START_REF] Katz | Cavitation phenomena within regions of flow separation[END_REF] noticed that the transition from travelling bubble cavitation to attached band cavitation occurs very rapidly. The process starts at some point-at the same axial location-and expands laterally. He suggests that this process is induced by bubbles that are swept by the reverse flow to the stable part of the separation zone. The inception process-including hypotheses-is shown in Figure 6. Figure 6. Inception mechanism for a free gas bubble and a laminar separation region based on [START_REF] Arakeri | Viscous effects in inception and development of cavitation on axi-symmetric bodies[END_REF][START_REF] Li | Interaction of single travelling bubbles with the boundary layer and attached cavitation[END_REF], flow V from left to right.

Travelling bubble & boundary layer

Li & Ceccio [START_REF] Li | Interaction of single travelling bubbles with the boundary layer and attached cavitation[END_REF] examined the generation of attached cavities by travelling bubbles on a hydrofoil. Small bubbles were generated upstream of the leading edge of the foil which expanded while travelling over the tension region, see Figure 7.

Figure 7.

Inception process for a travelling bubble and a laminar boundary layer based on [START_REF] Kuhn De Chizelle | Observation and scaling of travelling buble cavitation[END_REF][START_REF] Katz | Cavitation phenomena within regions of flow separation[END_REF], flow V from top to bottom.

The spherical bubbles deformed to hemispherical 'caps' if the distance to the surface was approximately equal to the boundary layer thickness. With a laminar boundary layer on the foil, two cavitating 'tails' were formed at the lateral sides of a bubble. These attached patches of cavitation grew in span-wise and chord-wise direction as the bubble travelled on and connected to each other at the symmetry line. Finally, the bubble detached from the patch cavity and travelled on while the patch cavity remained.

The laminar boundary layer was considered susceptible to transition, because of the relatively high Reynolds number (1.24 x 10 6) and an adverse pressure gradient. Velocity fluctuation measurements in the boundary layer upstream of the travelling bubbles, i.e. in their wake, showed that transition to turbulence was stimulated by the bubbles. This complies with the dye visualizations of Briançon-Marjollet et al. [START_REF] Briançon-Marjollet | Transient bubbles interacting with an attached cavity and the boundary layer[END_REF]. Kuhn de Chizelle et al. [START_REF] Kuhn De Chizelle | Observation and scaling of travelling buble cavitation[END_REF] observed the development of similar patches on a range of headforms. The occurrence of patch cavities increased with the size of the headform, a larger velocity and a smaller cavitation number. In a turbulent boundary layer, a more chaotically formed patch of cavitation was generated by travelling bubbles [START_REF] Li | Interaction of single travelling bubbles with the boundary layer and attached cavitation[END_REF]. There were no clear tails observed and the patch did not persist after passage of the bubble.

In [START_REF] Li | Interaction of single travelling bubbles with the boundary layer and attached cavitation[END_REF] it was hypothesized that the travelling bubbles squeezed the boundary layer and generated streamwise vorticity which induced two turbulent spots on the spanwise edges of the bubble. These spots may lead to a local 3D boundary layer separation and fill with vapour supplied by the bubble. In the wetted flow, however, no boundary layer separation was measured. Maybe first evaporation at the surface occurs, followed by boundary layer separation [START_REF] Hoekstra | The partial cavity on a 2D foil revisited[END_REF]. Alternatively, it was suggested that the turbulent spots could stimulate previously sub-critical surface nuclei to form attached cavities. The patch cavities in [START_REF] Holl | Observations of the various types of limited cavitation on axisymmetric bodies[END_REF] and [START_REF] Kuhn De Chizelle | Observation and scaling of travelling buble cavitation[END_REF] do bear some resemblances with dye visualisations of turbulent spots by Gad-el-Hak et al. [START_REF] Gad-El-Hak | On the growth of turbulent regions in laminar boundary layers[END_REF]. The upstream part contains streamwise streaks and the downstream part is more turbulent.

SENSITIVITY STUDIES

The effects of nuclei and turbulence on sheet cavitation inception as found in sensitivity studies are discussed in this section.

Nuclei

The review of mechanisms above has shown the essential role of nuclei for attached cavitation inception. In this section several aspects of nuclei, such as the type, origin, size range and concentration are discussed.

Keller [START_REF] Keller | Investigations concerning scale effects of the inception of cavitation[END_REF] studied the effect of the free-stream nuclei spectrum on cavitation inception on various headforms. The surfaces were polished, resulting in a maximum roughness of 0.1 µm. This minimized the size of surface-bound nuclei. Degassing and filtering the water decreased the concentration of nuclei (solid and gaseous) by 35% and 95%, respectively. Degassing and filtering the water significantly lowered the cavitation inception number (σ i ). This could hold for attached or bubble cavitation since the type of incipient cavitation was not specified. Presumably, degassing also decreased the maximum bubble diameter, thus decreasing the critical pressure of the largest bubble and increasing the tensile strength of the water. The effect of filtering was interpreted by Keller as a confirmation of the nucleus model of Harvey, i.e., gas trapped in small crevices of hydrophobic particles. Alternatively, it could support the tribonucleation mechanism, see Section 2.3.

Gindroz & Billet [START_REF] Gindroz | Influence of the nuclei on the cavitation inception for different types of cavitation on ship propellers[END_REF] found a significant decrease of σ i with an increase of the tensile strength of the water for bubble and tip vortex cavitation on propeller models. Only a minor decrease in σ i was found for sheet cavitation. Apparently, the sheet cavitation inception process is less dependent on the critical pressure of free-stream bubbles than the inception processes of bubble and tip vortex cavitation. Either the expansion of free-stream bubbles is not essential in the sheet cavitation inception process or the expansion of surface-bound nuclei dominate the inception process in this case. Mées et al. [START_REF] Mées | Development of interferometric techniques for nuclei size measurement in cavitation tunnel[END_REF] found nearly 20 times more solid than gas nuclei in the same facility. This could indicate a prominent role for solid nuclei (tribonucleation) in the inception process.

Noordzij [START_REF] Noordzij | Some Experiments on Cavitation Inception with Propellers in the NSMB Depressurized Towing Tank[END_REF] found that sheet cavitation was suppressed on smooth propeller blades in MARIN's Depressurized Towing Tank (DTT). Free-stream bubbles established a fully developed sheet on all blades in one case but in another case only some additional spot cavities were observed. Kuiper [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] found a very limited improvement by seeding freestream bubbles upstream of a smooth propeller model in the DTT. With leading edge roughness on the propeller blades, however, a fully developed sheet cavity was observed. Freestream bubbles could not further improve the extent of the cavity.

Van Rees et al. [START_REF] Van Rees | An exploratory investigation of cavitation inception on the pressure side of propellers[END_REF] obtained a good agreement between the pressure side sheet cavity on a propeller with leading edge roughness in MARIN's cavitation tunnel at a standard air content and in the DTT with nuclei seeding. Without seeding no cavitation was observed. This seemed in contradiction with the findings of Kuiper. Therefore, Van Rijsbergen & van Terwisga [START_REF] Van Rijsbergen | Water quality effects on sheet cavitation inception on a ship propeller model[END_REF] investigated the same propeller model as used by Kuiper with leading edge roughness on the blades in the DTT. Only a few spots were observed without seeding at lower propeller loadings. By increasing the free-stream nuclei content, the number of spots increased until a fully developed sheet was formed. The sheet cavity was almost completely developed without nuclei seeding at higher propeller loadings. Seeding fully closed the sheet cavity.

In [START_REF] Van Der Meulen | Cavitation inception scaling by roughness and nuclei generation[END_REF][START_REF] Arakeri | Water tunnel investigations of scale effects in cavitation detachment from smooth slender bodies and characteristics of flow past a bi-convex hydrofoil[END_REF][START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] wedge-shaped spots of bubble cavitation were observed at mid-chord positions. In most cases the origin of the bubbles could be traced back to a roughness element or small surface imperfections at the leading edge of the foil or propeller blade. In [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] it was hypothesized that microscopic, turbulent low-pressure regions at the roughness elements caused dissolved gas in the water to come out of solution which formed free gas bubbles. A minimum roughness height to boundary layer thickness ratio and locally supersaturated water were assumed to be critical parameters. The same mechanism was assumed to occur for sheet cavitation inception-when the roughness elements were located in the minimum pressure region. A surface-bound nucleus as observed in [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF] at a roughness element on an elliptic foil may be the connecting element between diffusion and production of free gas bubbles. The generation of separate bubbles at small angles of attack and attached spot cavitation at large angles of attack corresponds with the findings in [START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF]. In [START_REF] Van Rijsbergen | Water quality effects on sheet cavitation inception on a ship propeller model[END_REF] the presence of small remnant free-stream nuclei is suggested. A lower minimum pressure on the blades could expand smaller nuclei at higher propeller loadings. Measurements of the free-stream nuclei spectrum and the leading edge roughness topology could provide more definitive explanations for these findings.

Surface-bound nuclei-provided by gas trapped in microscale pits-offer a controllable way of attached cavitation inception [START_REF] Bremond | Interaction of cavitation bubbles on a wall[END_REF]. The size of these pits need to be as large as possible to generate a spot cavity just below the vapour pressure. If the size is too large, however, bubbles will be generated above the vapour pressure [START_REF] Groß | Experimental and theoretical investigation of nucleation from wall-bounded nuclei in a laminar flow[END_REF]. The production of pits in the double-curved surfaces of the leading edge of a blade is challenging but there are several advantages. Contrary to distributed roughness, the pits have a minimal disturbance of the boundary layer. Furthermore, seeding of free-stream nuclei is no longer required which improves the quality of noise measurements [START_REF] Ligtelijn | MARIN Jubilee Meeting (Hydrodynamics: Computation, Model Tests and Reality)[END_REF].

Turbulence

It is commonly assumed that the boundary layer is fully turbulent at full scale which will enhance cavitation inception. Therefore, the effect of turbulence-either in the free-stream or generated by roughness-on sheet cavitation inception has been investigated on model scale.

In [START_REF] Kuiper | Cavitation inception on ship propeller models[END_REF] paint tests and cavitation tests were carried out on propeller models with two grain sizes of distributed leading edge roughness, 30 and 60 µm. The paint tests with the 30 µm roughness showed that the boundary layer was often not turbulent. Furthermore, no cavitation was found when the boundary layer was not tripped. With 60 µm roughness both a turbulent boundary layer and cavitation was observed. Therefore, it was hypothesized that cavitation inception on the roughness elements only occurs when the roughness induces a turbulent boundary layer. However, at a propeller Reynolds number of 10 6 , k = 60 µm and r/R = 0.5, R k is approximately 430. Because this value is clearly below R k,crit , this may indicate a transition further downstream. Paint tests typically lack information near the leading edge due to the high speeds. This might have masked the exact chordwise position of transition.

Ligtelijn et al. [START_REF] Ligtelijn | MARIN Jubilee Meeting (Hydrodynamics: Computation, Model Tests and Reality)[END_REF] presented a diagram-based on systematic propeller model cavitation inception tests-to determine the appropriate roughness size as a function of the sectional Reynolds number. The corresponding R k values are all well below R k,crit . Furthermore, in [START_REF] Van Rijsbergen | Water quality effects on sheet cavitation inception on a ship propeller model[END_REF] cavitation inception was found at roughness elements with R k = 210. These experiments indicate that turbulence at the roughness elements may not be a requirement for cavitation inception. However, the R k values may be sufficiently high to generate a laminar separation bubble at the roughness element. The correlation with turbulence far downstream may indicate that instabilities are already present in the free shear layer.

The effect of free-stream turbulence on cavitation inception on roughened propellers was investigated in [START_REF] Van Rees | An exploratory investigation of cavitation inception on the pressure side of propellers[END_REF]. Two propellers were tested at free-stream turbulence intensities of approximately zero (without grid) and 1% (with grid). No effect was found on the radial cavitation extent of one propeller. The other propeller showed a significant increase in cavitation extent. In the latter case, however, the grid was probably cavitating and thus generating free-stream nuclei. The effect of free-stream turbulence and leading edge roughness on cavitation inception was investigated by Korkut & Atlar [START_REF] Korkut | On the importance of the effect of turbulence in cavitation inception tests of marine propellers[END_REF]. An increase of the free-stream turbulence intensity from 3.3 to 4.5% increased σ i for sheet cavitation on smooth propeller blades. A similar increase was found for an increase of the leading edge roughness grain size. It was hypothesized that free-stream turbulence has a similar effect on the boundary layer as leading edge roughness. Alternatively it was suggested that the used wire meshes not only generated free-stream turbulence but also free-stream nuclei. In the tests with the roughened propeller blades the roughness may have provided surface-bound nuclei.

More systematic research on the effect of parameters such as k/, R k and pressure gradient on sheet cavitation inception is necessary to find the exact working mechanism.

CONCLUSIONS AND RECOMMENDATIONS

Sheet cavitation inception occurs when the local pressure is below the vapour pressure, gas and vapour dewet the surface and expand at a fixed location. If the free-stream flow provides the necessary nuclei, these need to be brought to the surface first. Surface roughness, a laminar separation bubble and turbulence enhance this process and may compensate for the screening effect. Second, the gas and vapour from the travelling bubble have to come into contact with the surface. It is hypothesized that mass transfer by shear, turbulent mixing and evaporation govern this step in the inception process. A free-stream particle has to make contact with the surface to generate a gas bubble. Roughness elements protruding the boundary layer increase the probability of contact. After advection with the flow, the generated gas bubble can induce attached cavitation just like a free-stream bubble. A surface-bound nucleus can generate an attached cavity without a laminar separation bubble. It is hypothesized that diffusion and evaporation enable this process. Finally, an attached cavity can remain at a fixed location by flow separation, either already present in the wetted flow or generated by the cavity.

High-speed micro-scale observations at increased temporal and spatial resolutions are needed to further narrow down the location of attached cavitation inception and the mechanisms involved. Experiments above and below the vapour pressure and modelling of the of mass transport, diffusion and evaporation are recommended to distinguish between their contributions to the cavitation inception process.

Gas trapped in micro-scale pits are a promising nucleation source in model scale tests. More research on the optimum size of these pits and their production at the leading edge of a blade is necessary.

Leading edge roughness may enable cavitation inception from surface-bound nuclei and from free-stream nuclei. A laminar separation bubble downstream of the roughness elements enhances both processes. Measurements of the free-stream nuclei spectrum and the leading edge roughness topology are recommended to distinguish between the two sources in model scale experiments. It is recommended to investigate the contribution of turbulence to attached cavitation inception by systematic research on the effect of the ratio of the roughness height to the displacement thickness, the roughness Reynolds number and pressure gradient on attached cavitation inception.

Figure 1 .

 1 Figure 1. Forms of limited, attached cavitation, flow direction V and relative dimensions/position to the leading edge (LE) not to scale.The first form is designated as spot cavitation. The upstream part of this wedge-shaped cavity has a smooth surface, whilst the downstream part is rough. The second form is designated as finger cavitation. The upstream part

Figure 4 .

 4 Figure 4. Inception mechanism for a surface-bound bubble at a small angle of attack based on[START_REF] Guennoun | Étude physique de l'apparition et du développement de la cavitation sur une aube isolee[END_REF][START_REF] Guennoun | Experimental investigation of a particular travelling bubble cavitation[END_REF], flow V from left to right.


Casey M Harwood
email: cmharwoo@umich.edu

Andrew J Stankovich

Yin Lu Young

Steven L Ceccio

S Y Mpo

O Tatin

Combined Experimental and Numerical Study of the Free Vibration of Surface-Piercing Struts

Keywords:

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Many high-lift devices operate in a surface-piercing configuration, including surface-piercing propellers, hydrofoils, rudders, and control surfaces. Other systems, like supercavitating propellers, dynamic positing thrusters, and submerged hydrofoils operate in close proximity to the free surface. Both cases present an increased risk of atmospheric ventilation (the entrainment of an air-filled cavity).

Additionally, recent moves toward flexible marine structures -both in research and in industry -have revealed the potential for hydro-elastic instabilities that are not adequately predicted by classical aero-elastic theory [START_REF] Chae | Dynamic response and stability of a flapping foil in a dense and viscous fluid[END_REF][START_REF] Akcabay | Influence of cavitation on the hydroelastic stability of hydrofoils[END_REF][START_REF] Akcabay | Cavity induced vibration of flexible hydrofoils[END_REF][START_REF] Akcabay | Parametric excitations and lock-in of flexible hydrofoils in two-phase flows[END_REF]. Ventilated flow around a flexible hydrofoil/strut presents an extremely complicated hydro-elastic problem. The interface between a very light fluid (air) and a very dense fluid (water) creates dramatic spatial and temporal variations in the pressure and density fields surrounding a surface-piercing body. The ramifications on the effective system added mass, damping, stiffness, and resonant properties are unknown. In this work, combined experimental and numerical studies are performed to measure the hydro-elastic response of surface-piercing struts in wetted and ventilated flows.

BACKGROUND

The issue of atmospheric ventilation on surface-piercing bodies has been investigated by [START_REF] Breslin | Exploratory study of ventilated flows about yawed surface-piercing struts[END_REF][START_REF] Rothblum | Ventilation, cavitation and other characteristics of high speed surface-piercing strut[END_REF], among others. Recently, [START_REF] Harwood | Experimental and numerical investigation of ventilation inception and washout mechanisms of a surface-piercing hydrofoil[END_REF][START_REF] Harwood | Ventilated cavities on a surface-piercing hydrofoil at moderate froude numbers: cavity formation, elimination, and stability[END_REF] performed towing-tank experiments on a surface-piercing strut constructed of aluminum. They established characteristic flow regimes, which include fully-wetted (FW) and Past research on the partial-immersion of elastic structures have been mostly limited to fully-wetted flows around bodies with simple circular or rectangular cross sections. [START_REF] Fu | Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid[END_REF] used a combined finite-element-method-boundary-element-method (FEM-BEM) model to investigate the natural and resonant modes of a flat surface-piercing plate. For partially-immersed, fully-wetted cases, a boundary-value problem was established on the fluid-domain to emulate a linear free surface and infinite fluid field, and the plate was discretized with singularity panel distribution. The in-vacuo mode shapes were determined with the FEM model and used to decouple the equations of motion. Using the pre-computed mode shapes as effective basis functions, the solution to the generalized equation of motion allowed resonant frequencies, ω, principal (modal) coordinates, and modal added mass and damping coefficients to be obtained. They found that the free surface induced a frequency-dependence into the modal added mass and damping -a consequence of the radiation potential. The frequencydependence vanished quickly with both increasing immersion depth and increasing modal frequency. Variations in the added mass, in particular, were only appreciable for values of ω S/g < 5, where S is the span of the plate and g is gravitational acceleration.

[10] presented an analysis of metallic and composite cantilevered plates, both in-water and in-vacuum, using combined numerical and analytical methods. The work showed good agreement with experimental measurements of a cantilevered surface-piercing steel plate provided in [START_REF] Lindholm | Elastic vibration characteristics of cantilever plates in water[END_REF]. The wetted frequencies were found to be much lower than the dry frequencies as a result of fluid inertial effects (added mass). The percentage reduction in resonant frequencies (compared to the dry state) varied between modes. Moreover, the changes in wet-to-dry frequency ratios were more severe for lightweight models than for heavier models. Results showed that added mass, and therefore wetted frequency, were strongly dependent on the respective mode shapes, which in turn are determined by the body shape, support-type, physical properties, and material anisotropy. Similarly, [START_REF] Motley | Free surface and solid boundary effects on the free vibration of cantilevered composite plates[END_REF] used finite element analysis to show that there is a significant reduction in resonance frequency as the immersion-depth of the plates increased, due to larger fluid added mass, and that these effects are dependent on the particular shape of the resonance mode. The results showed that as the submergence increased, the resonance frequencies decreased, asymptotically approaching the frequencies for a fully-submerged plate.

Objectives

The objective of this work is to improve the understanding of the hydro-elastic behavior of surface-piercing struts to assist in future design and control of struts and hydrofoils across a variety of operating conditions. Specifically, this study seeks to investigate the effects of partial immersion and ventilated flow on the resonant frequencies, added mass, and associated mode shapes of surface-piercing struts through combined experimental and numerical analyses. Moreover, this study aims to develop and validate a numerical approach for the simulation of wetted and ventilated surface-piercing bodies.

EXPERIMENTAL APPROACH

Geometrically-identical struts were constructed from aluminum and PVC, using the geometry of [START_REF] Harwood | Experimental and numerical investigation of ventilation inception and washout mechanisms of a surface-piercing hydrofoil[END_REF][START_REF] Harwood | Ventilated cavities on a surface-piercing hydrofoil at moderate froude numbers: cavity formation, elimination, and stability[END_REF]. The design The geometry of the struts. The aluminum and PVC struts both possess a semi-ogival section. The aluminum strut is solid. The PVC strut includes two interior channels, into which are installed shape-sensing spars (Figure 3).

possesses a semi-ogival section with a circular-arc fore-body and rectangular after-body, as shown in Figure 2. The aluminum strut was machined with a solid section. The PVC strut was constructed from two symmetric halves, which were affixed by a chemical bond and plastic-welding. Along the centerline of the PVC strut, two interior channels were machined, into which custom-built shape-sensing spars were installed.

Inferring motions of flexible structures typically requires accelerometry and/or optical methods. The former poses problems for non-harmonic motions, while the latter cannot operate in environments where fluid interfaces cause refraction (such as in the case of a ventilated cavity). A non-optical means of tracking structural deflections was developed in the form of shape-sensing spars, pictured in Figure 3.

The shape-sensing methodology invokes the beam bending equation,

z ≈ -t 2
∂ 2 Y ∂ Z , ( 1
)
where z is the surface bending strain on a beam of thickness t, Y is the coordinate in the direction of lateral beam deflection

Strain Gauge Half-Bridges Figure 3. A schematic illustration of the shape-sensing spar.

Strain gauges permit the distribution of strain along the beam to be approximated via polynomial, from which the deflection may be inferred. Two identical spars were constructed and inserted into the PVC strut at locations indicated by the blue and red regions in Figure 2. and Z is the coordinate along the initial (undeformed) length of the beam. The slim aluminum spar is instrumented with N strain gauges in a half-cosine spacing. The boundary condition, z | Z=L = 0, permits an N th -order polynomial may be fitted to the measured strains.

ˆ z (Z ) = N i=0 a i Z i . (2)
EQ 2 may be substituted into EQ 1 and twice-integrated to yield a polynomial approximation of the lateral deflection.

Ŷ (Z ) = N +2 0 A i Z i . (3)
The integration yields two additional coefficients, A 0 and A 1 , which must be determined from the beam's boundary conditions. For an ideal cantilever, A 0 = A 1 = 0, giving

A i =          0, j = 0, 1 a i ( j -2)! j! , j = 1, 2, . . . , N + 2
Two spars were constructed, denoted "A" and "B". Independent optical calibrations were carried out to correct for small differences in the strain-gauge mounting, such as orientation, adhesive thickness, and other linearizable sources of variation. The root of the spar was clamped and point loads were applied to the beam to achieve various static deflections.

A calibrated camera was used to measure the lengthwise deflections with an estimated uncertainty of 1/32 in (0.79 mm), from which the surface strains were calculated via EQ 1. A calibration matrix, unique to each spar, was fitted to minimize the squared error between the strain values measured optically and those measured by the strain gauges. The error of the preand post-calibration deformations is plotted in Figure 4a. The plotted symbols, and the linear trends fitted to them, make it clear that the calibration significantly reduced both the mean error and the variance of the error.

Under the assumption that bending along the chord is negligible (chord-wise rigidity), the deflections along the two spars -respectively ŶA (Z ) and Ŷ B (Z ) -may be used to reconstruct the deformed centerline plane of the strut. It is convenient to recast the deformations into the decoupled angular and lateral deflection of each spanwise section along the elastic axis.

θ(Z ) = arctan ŶA (Z ) -Ŷ B (Z ) X A -X B (4a) ŶE A (Z ) = ŶA (Z ) + (X E A -X A ) tan(θ(Z )), (4b)
where X A and X B respectively denote the chordwise positions of the spars inside of the strut and X E A denotes the chordwise location of the elastic axis.

In practice, the root of each spar was judged to be imperfectly clamped as a result of the compliance of the surrounding PVC, such that κ represents a dimensionless compliance factor, allowing the root of each spar to take on a non-zero slope proportional to the applied moment at Z = 0, and which must be determined in-situ. An example of the inferred bending/twisting deformation of the strut under the relaxed boundary condition (κ = 8) is shown in Figure 4b. The inferred deflections are overlaid onto calibrated photos of the strut's tip and leading edge in green dashed lines. Good agreement between the inferred and true deformations may be observed.

Y (Z ) = 0 but Y (Z ) 0 at Z = 0. The coefficient A 1 in EQ 3 was changed to A 1 = -κ ˆ z | Z=0 , where
To collect vibration data, each strut was suspended from a steel carrier in a vertical cantilevered configuration. The root of the strut was clamped and coupled to the steel structure via a six-degree-of-freedom load cell. The free tip of the strut was lowered into a 65-gallon (246 liter drum). The drum was filled with freshwater until the desired depth of immersion (h) was achieved. Immersed aspect ratios (AR h = h/c) of 0, 0.5, 1.0, and 1.5 were assessed, with AR h = 0 corresponding to the dry configuration. The motions of the strut were measured via a Polytec OFV-353 single-point laser doppler velocimeter (LDV) and -in the case of the PVC strut only -the internal shape-sensing spars. Finally, a three-axis accelerometer was attached to sub-frame to measure the vibratory response of the test structure. The strut was excited by hammer strikes, after which the decaying free-vibration response was recorded as a time-series of length M. Sampling rates of F s = 1 kHz and 5 kHz were used for the aluminum and PVC struts, respectively. The experimental setup is illustrated in Figure 5.

Discrete Fourier transforms (DFT) were used to generate frequency spectra for each recorded channel, where X is the time-series of of sampled data at times t = {t n , n = 1, 2, . . . , M }. X is a complex-valued vector corresponding to frequencies f = { f n , n = 1, 2, . . . , M }. The magnitudes of the frequency response | X | were plotted for each collected data channel, denoted by index j, and the collective frequency distributions were used to locate peaks in the frequency spectra corresponding to resonant modes (see Figure 6). The location of the peak on each channel was taken as the j th estimate of the resonant frequency, f * i, j , where i is the ordered mode number.

X = D (X ), (5)
Taking the imaginary part of Xn at a frequency yields the operating deflection shape (ODS) for frequency f n . When the sampled strain-gauge signals are considered, the ODS is an estimate of the strain distribution, z = z j , j = 1 . . . N , that oscillates sinusoidally at that frequency.

˜ z ≈ Xn = Im( Xn ), (6)
The ODS strains for each resonant condition were translated into a surface ODS ( Ỹ ) via EQs 2-4. For resonant peaks that are well-separated, the surface ODS is dominated by the underlying normal mode shape, Φ [START_REF] Schwarz | Experimental modal analysis[END_REF].

Ỹ n ≈ Φ i when f n = f * i, j (7)
The strain-gauge channels ( j = 1 . . . N) were each considered, yielding N estimates of each resonant frequency and associated ODS. Moreover, eight-to-twelve trials were conducted per value of AR h , permitting the mean and standard deviations of the resonant frequencies and ODSs to be estimated. Figure 6 shows the averaged frequency response of the strain-gauge signals at AR h = 0 in the neighborhood of the first resonant frequency. The top plot shows the magnitude of the frequency response, with a vertical line indicating the resonant peak.

The imaginary part of the frequency-response is shown in the lower plot; the ODS strains are recorded as the set of values intersecting the vertical bar.

Numerical Approach

The commercial finite element software, Abaqus, was used to analyze the first five resonant frequencies of the strut structure in the frequency domain. The Lanczos solver was used to extract the coupled structural-acoustic resonant frequencies [START_REF][END_REF].

The geometric properties of the struts (excluding the clamped root) are listed in Table 1 and material properties of the two different struts and the acoustically modeled fluid domain are listed in Table 2. The second moments of area, I xx , I yy , and J (Z-axis Rotation), are shown in the table as well as the mass m of the aluminum (AL) and PVC struts.

The strut was meshed using 3-D reduced integration, first order continuum shell elements (SC8R) [START_REF][END_REF] following the success shown in [START_REF] Kramer | Free vibration of a cantilevered composite plate in air and in water[END_REF]. The meshed strut is shown in Figure 7a with (n x , n y , n z ) indicating the number of elements in each coordinate direction. The origin of the strut is specified at the intersection of the mid-chord plane, centerline plane, and strut root (Z = 0) plane. The fluid domain was modeled as a radial projection of the strut cross section. 3-D first-order, reduced integration acoustic elements (AC3D8R) [START_REF][END_REF] were chosen for the fluid, again based on the findings of [START_REF] Kramer | Free vibration of a cantilevered composite plate in air and in water[END_REF]. To simulate ventilation, air properties were applied to a region of the acoustic fluid domain (shown in Figure 7b).

Convergence studies were performed sequentially, first refining the solid mesh and then determining the fluid domain size. The solid mesh was considered converged when further refinement on n z,y,z yielded changes in the first resonant frequency of less than 0.5%. The dimensions of the fluid domain were varied next (the fluid mesh size is defined by the converged solid mesh), and the first resonant frequency was again judged to be converged when further increases to the domain size yielded changes of less than 0.5%. The converged solid mesh parameters and fluid domain size are given in table 3.

Boundary Conditions

All degrees of freedom were fixed on the nodes of the strut's root to simulate the cantilevered test configuration. The kinematic/dynamic surface boundary condition for potential (inviscid and irrotational) flow is,

∂ 2 φ/∂t 2 + g∂φ/∂z = 0 (8)
with the velocity potential, φ, defined as,

φ(x, y, z, t) = φ o • e (i(kr-ωt)) (9)
where φ o is the complex amplitude, k is the wave-vector, ω is the angular frequency, and r is the position vector (x, y, z).

Table 3. Converged parameters of solid mesh and fluid domain, as defined in Figure 7 (strut dimensions are shown in Figure 2). When EQ 9 is evaluated at very high frequencies it can be reduced to show that

lim ω→∞ φ(x, y, z, t) = 0. ( 10
)
This approximation is only strictly valid at high frequencies, as ω → ∞. However, as shown in [START_REF] Fu | Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid[END_REF], the boundary condition was found to be approximately valid for ω S/g > 5. In the present work, dimensionless modal frequencies were found to satisfy this condition in all cases, so EQ 10 was judged to be a suitable boundary condition. Non-reflecting boundary conditions were imposed on the bottom and sides of the fluid domain to minimize reflection of energy back into the system. Finally, the contacting surfaces of the strut and fluid domain were constrained with a kinematic boundary condition.

Representing the fluid domain as an acoustic medium required the assumptions of inviscid and irrotational flow, as well as linear fluid compressibility. These assumptions lead to an equilibrium equation resembling,

∂p/∂ x + ρ f ü f = 0, ( 11
)
where ρ f is the fluid density, x is the nodal displacement vector, p is the fluid pressure and ü f is the fluid nodal acceleration vector. The constitutive behavior of acoustic fluid is,

p = -K f ∇ • u f ( 12
)
where K f is the bulk modulus of the fluid, ∇ is the gradient vector operator, and u f is the fluid nodal displacement vector. The generalized finite element equation of motion in the frequency domain for a free vibration problem without damping can be expressed as,

K s S T f s 0 K f u p -ω 2 M s 0 -S f s M f u p = 0 ( 13
)
where K s and K f are the structure and fluid stiffness, respectively, S f s represents the fluid-structure interaction term through enforcing the kinematic and dynamic conditions at the fluid-solid interface, and M s and M f are the structure and fluid added mass, respectively. u describes the displacement of the structure and the p describes the pressure throughout the acoustic medium. EQ 13 can be symmetrized and solved iteratively to yield eigenvalues (ω 2 ), and eigenvectors (Φ). This method does not yield explicit values of the added mass acting on the structure but it does provide an indirect method by which to infer the effects of fluid and cavity interactions on the frequencies and shape of the resonant modes.

In the current simulations, the air above the free surface was not modeled. Work by [START_REF] Kramer | Free vibration of a cantilevered composite plate in air and in water[END_REF] showed that the low density and high compressibility of air led to a negligible difference between simulations with and without acoustically-modeled air domains. Hence, fluid above the free surface was neglected to significantly decrease the computational cost.

RESULTS AND DISCUSSION

Numerically Predicted Mode Shapes

Mode shapes were identified using the eigen-analysis of the FEM model. The naming convention used throughout the discussion is:

[Primary axis] -[Motion type] -[# Node lines].
For example, the first mode (X-Bend-1) is primarily bending about the X-axis with a single node line. The first five ordered mode shapes for the struts in air are shown in Figure 8. Higher-order X-bending modes (X-Bend-2 and X-Bend-3) demonstrate some twisting about the Z-axis induced by the chordwise asymmetry of the section shape. Note that the mode shapes shown are indentical for both struts and in any homogenous fluid (i.e. in-air or fully-submerged in water).

Experimentally Estimated Mode Shapes

The measured operating deflection shapes (ODSs) and corresponding resonant frequencies ( f * ) of the PVC strut are shown in Figure 9. The rows correspond to values of the immersion ratio AR h , and the columns are ordered from left to right by ascending resonant frequencies within each row. Note that the Y-Bend-1 mode (also known as the "lead-lag" mode) is not captured in the experiments because the half-bridges installed on the interior spars reject axial strains. The X-Bend-1 and X-Bend-3 modes are clearly represented in the ODSs. As the immersion depth is increased, these modes appear to change only slightly (some induced twist appears in the X-Bend-3 mode), but remain recognizable. The X-Bend-2 and Z-Twist-1 modes, however, are located relatively close to one another in the frequency domain, and both underlying normal modes contribute to the ODS at nearby frequencies. As a result, the underlying modes cannot be extricated from the ODS; rather, the ODS must be regarded as weighted blend of the underlying modes. As the immersion depth is increased to AR h = 0.5, the second and third resonant modes become so closely-coupled (less than 2 Hz of separation), that the ODSs are almost indistinguishable. At AR h = 1.0, the second and third resonant ODSs are re-ordered, relative to the dry ordering, such that Z-Twist-1 actually occurs at a slightly lower frequency than does X-Bend-2. At AR h = 1.5, the original order is restored. Resonant mode shapes predicted by FEM for the cases where re-ordering occurs are shown at the bottom of the figure, corroborating the result of mode re-ordering.

Effects of Partial-Immersion on Resonance

Figure 10 depicts a comparison between the numerical and experimental resonant frequencies for both struts. The resonant frequencies of the aluminum and PVC struts are respectively shown in Figure 10a and Figure 10b. The resonance frequencies, nondimensionalized by the dry modal frequencies, f * i / f * i,dr y are shown for the aluminum and PVC struts in Figure 10c and Figure 10d, respectively. In the case of the aluminum strut, experimental data are available for the first three modes (X-Bend-1, X-Bend-2, and Z-Twist-3). For the PVC strut, experimental data are available for all of the modes pictured in the Figure 8 except Y-Bend-1.

The agreement between the numerical results and the experiments are good in most cases. A notable exception is the X-Bend-2 mode of the aluminum strut, where the FEM-predicted frequencies are consistently higher than the experimentally-observed ones. A likely explanation for the discrepancy is the root boundary condition. In the FEM model, the root of the strut was perfectly clamped, while the experimental setup may have been influenced by the compliance of the load cell, reducing the stiffness of the system. The large curvature at the root in the X-Bend-2 mode induces a large bending moment at the root, which would exacerbate the issue of an imperfect clamp. The numerical and experimental results are in close agreement for the remaining resonant modes, both for the aluminum and for the PVC struts.

In general, with increasing immersion-ratio, the resonant frequencies in Figure 10a and Figure 10b decrease, with the exception of the Y-Bend-1 mode. As Figure 10c and Figure 10d demonstrate, the reduction is more-severe for the PVC strut than the aluminum strut. Both effects are attributed to the inertial resistance of the water to the strut's motion. Added mass is the integrated inertial resistance to the local accelerations of the body, which as the name implies, may be modeled by additional mass distributed on the structure. The distribution of the added mass is purely a function of the body's motion (if free-surface waves are neglected). The resonant frequencies of the PVC strut are more-strongly reduced than the aluminum resonant frequencies because the solid-to-fluid density ratio is much smaller in the case of the PVC strut. For similar mode shapes, the fluid added mass will be nearly identical for the two struts, but will make up a greater percentage of the total system mass in the case of the PVC strut. As a result, the lighter structure is influenced to a much greater degree by the immersion into a dense fluid.

Mode shapes where deflection occurs in a direction with a large projected area (e.g. X-Bend-1) are necessarily morestrongly affected by the surrounding fluid's added mass than are mode shapes that present minimal projected area in the direction of primary motion (Y-Bend-1). This is demonstrated in Figure 10c and Figure 10d by the fact that, as the struts are immersed, the reductions to the resonant frequencies are not uniform. Namely, the first-order resonant modes (X-Bend-1 and Z-Twist-1) are affected more-severely than the higher-order modes. Moreover, note that inflections occur in the resonant frequencies of the higher-order modes. This occurs when a nodal line of the respective resonant mode is submerged. Motion along the node lines is identically zero, and so induces very little acceleration in the surrounding fluid; the result is a near-zero change in the hydrodynamic added mass.

As observed in the preceding section, the second and third modes approach one another (frequency coalescence) between AR h = 0.5 and AR h = 1.5. This occurs because the node line of the X-Bend-2 becomes submerged at approximately AR h = 1, causing the added mass component to plateau. The added mass effect is proportionally severe enough to cause a re-ordering of modes at AR h only in the case of the PVC strut. With the aluminum strut, the modes coalesce but do not re-order. Frequency coalescence is a practical concern because it signifies the sharing of energy between two modes; in extreme cases, one mode can feed energy into another in an unbounded fashion, resulting in a dynamic instability known as flutter, which can cause accelerated fatigue and/or catastrophic structural failure.

Effects of Ventilation

The dashed lines in figs 10 denote the numerical resonant frequencies when the air-filled cavity (Figure 7b) was included in the fluid domain. In all cases, the resonance frequencies increase from their fully-wetted values. In the fully-ventilated configuration, nearly half of the previously-wetted area is surrounded by air, which presents a negligible inertial opposition to motion, thus reducing the added mass. As a result, the resonant frequencies increase to a value somewhere between those in the corresponding fully-wetted condition and the dry condition.

CONCLUSIONS

Experimental and numerical investigations of the resonant behavior of aluminum and PVC struts partially immersed in water have been conducted. A simple shape-sensing methodology has been developed around an aluminum spar instrumented with strain gauges, which permits bending and twisting deformations of the PVC strut to be inferred with good accuracy. The close agreement between experimental and numerical resonant modes and frequencies validate the accuracy of the FEM model, which uses acoustic elements to simulate the air/water fluid domain.

Partial immersion in water causes the resonant frequencies of both struts to decrease, though the effect is more pronounced for the PVC strut because hydrodynamic added mass makes up a greater proportion of the effective system mass. The modal-dependence of the fluid added mass causes the second bending and first twisting frequencies to approach one another for the aluminum strut and to change order for the PVC strut. The re-ordering of modes has been shown by the numerical mode shapes and by the operating deflection shapes (ODSs) measured in the experiments. This frequency coalescence leads to concerns about coupled dynamic instability (flutter) during operation. The formation of a fully-ventilated (FV) cavity causes the resonant frequencies to increase again, as air replaces some of the water surrounding the strut, reducing the fluid added-mass.

Future Work

Future experiments will focus on the creation of single/multiinput-multi-output (SIMO/MIMO) frequency response functions matrix. An electrodynamic shaker will be used to excite the strut in in FW and FV flow conditions. It is hoped that from these data, quantitative values of modal added mass and damping, as well as improved estimates of mode shapes, will be inferred.

Figure 1 .

 1 Figure 1. (a) Fully-wetted (FW) regime, where the suction surface of the yawed strut (pictured) is completely wetted. (b) Fully-ventilated (FV) flow, where a cavity of atmospheric air is entrained around the suction surface of the yawed strut.

Figure 2 .

 2 Figure 2.The geometry of the struts. The aluminum and PVC struts both possess a semi-ogival section. The aluminum strut is solid. The PVC strut includes two interior channels, into which are installed shape-sensing spars (Figure3).

Figure 4 .

 4 Figure 4. (a) The loading condition under which errors were measured (single concentrated loads at varying spanwise locations) and the maximum percentage error for the individual spars (A and B). The errors are significantly reduced by the calibration step, and are bounded below by the propagated uncertainty in the optically-measured deflections. (b) The (post-calibration) tip-deflection and the leading edge deflection inferred from the pair of shape-sensing spars are overlaid as green dashed lines onto photos of the strut subjected to a combined bending/twisting load. A compliance factor of κ = 8 was used to relax the root boundary conditions. White solid lines indicate the initial undeformed geometry of the strut's tip and leading edge.

Figure 5 .

 5 Figure 5. Experimental setup used to measure vibratory response of struts. A large steel structure holds the surface-piercing strut in a vertically-cantilevered configuration, suspended either in air or in a 65-gallon (246 liter) drum with the free tip immersed to depth h.

Figure 6 .

 6 Figure 6. The magnitude and imaginary parts of the strain-gauge signals in the frequency domain. The first resonant mode is shown ( f * 1 = 8 Hz), indicated by the peaked response magnitude. The operating deflection shape (ODS) strains are obtained from the crossing of imaginary parts of the strain-gauge signals with f * 1 . The legend entries are the numbered strain gauges on spars A and B (N A, B = 4; N = 8).

Figure 7 .

 7 Figure 7. (a) The FEM mesh parameters of the surface-piercing strut. (b) The dimensional parameters of the fluid domain. FW and FV flows were simulated by respectively specifying the green region as water or air acoustic elements.

Mode 1 "Figure 8 .

 18 Figure 8. The first five dry (AR h = 0) mode shapes identified by FEM (identical for PVC and aluminum struts).

FEM ModesFigure 9 .

 9 Figure 9. Resonant Operating Deflection Shapes (ODSs) measured in experiments for the PVC strut. The undeformed surface is indicated by the grey patch. Vertical bars indicate ± one standard deviation in the ODS. The second and third resonant modes coalesce and "flip" order between AR h = 0.5 and AR h = 1.0. Numerically-predicted modes, shown at the bottom of the figure, also indicate a flipping of the second and third modes at AR h = 0.5 and AR h = 1.0.

Table 1 .

 1 The geometric properties of the strut, which are identical for the PVC and aluminum struts, and the masses, which vary with the strut material. Quantities exclude the root-portion of each strut.

	Property Value	I x x in 4 (m 4 ) 0.92 (3.829 × 10 -7 ) 86.99 (3.621 × 10 -5 ) 87.9 (3.659 × 10 -5 ) I y y in 4 (m 4 ) J in 4 (m 4 )	m AL slugs (kg) 1.14 (16.61)	m PV C slugs (kg) 0.56 (8.19)

Table 2 .

 2 The material properties of the PVC and aluminum struts and the fluid used in the FEM analysis.

	Material	Mass Density	Young's

Modulus Poisson Ratio Bulk Modulus lbf s 2 /in 4 (kg/m 3 ) psi (GPa) psi (GPa)

	Aluminum PVC 1.22 × 10 -4 ( 1330) 2.53 × 10 -4 (2700) Water 9.36 × 10 -5 (1000) Air 1.15 × 10 -7 (1.23)	1.00 × 10 7 (8.94) 4.00 × 10 5 (2.76) --	0.33 0.4 --	9.80 × 10 6 (67.57) 6.67 × 10 5 (3.91) 3.19 × 10 5 (2.20) 20.59 (1.42 × 10 -4 )

ACKNOWLEDGMENTS

This material is based upon work supported by the National Science Foundation Graduate Student Research Fellowship under Grant No. DGE 1256260.