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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 654 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. An Experimental Investigation of Cavitation Inception and Development of Partial Sheet Cavities on Two—Dimensional Hydrofoils. J.Astolfi, P.Dorange, J.-B.Leroux, J.-Y.Billard (Ecole Navale, France) ABSTRACT The main results of an experimental study concerning the inception and the development of partial sheet cavities on four two-dimensional hydrofoils are presented. The conditions of cavitation inception are measured together with the minimum pressure coefficient in some cases. The effect of the Reynolds number is studied. Concerning cavitation development, depending on the cavitation number or the angle of incidence various types of cavitation are observed as partial sheet cavities, bubble, fingers, patches or super cavitation patterns. For partial sheet cavities, the cavity lengths were measured on the foils for various conditions of cavitation number and angle of incidence. An attempt to correlate the cavity length data is studied at the end of the paper. INTRODUCTION The physical process associated with the inception and the development of cavitation is complex and basic experiments on two-dimensional hydrofoils remain an effective way to study the fundamentals of cavitation and to understand cavitation in more complex situations as on marine propellers for instance. Partial sheet cavity is one of the cavitation patterns that occurs on a two-dimensional hydrofoil typically near the leading edge. It corresponds to the situation for which a cavity of vapor extends over a fraction of the hydrofoil's surface. For inception of partial cavities, the two questions which arise are where and when do cavitation occur? It is generally accepted that cavitation occurs on a full-scale lifting surface at the position of the minimum pressure and when the local minimum pressure falls to or below the vapor pressure of the flowing liquid. But in many cases, particularly on scale models, the incipient cavitation number σi is found to be different (often smaller) from the opposite of the minimum pressure coefficient,—Cpmin generally obtained theoretically by computation for an inviscid flow (Arndt 1981). The main reason is that on scale models such as hydrofoils or headforms, long or short separation bubbles, occurring at the leading edge, influence the inception conditions, Arakeri (1975), Arakeri et al (1981). When at full scale, the flow separation bubbles are expected to disappear and transition will occur near the leading edge. This phenomenon is known to complicate the correlation of model and full-scale cavitation scaling, (Huang and Peterson 1976, Billet et al. 1981). Arakeri (1975), Katz (1984) and Franc and Michel (1985) indicated that attached sheet cavity development on hydrofoils requires the presence of a laminar separation. In that case, according to Katz (1984) the scenario is that “band type cavitation occurred as bubbles were entrained through the reattachment region, where they were pushed upstream by the reverse flow”. In this quiescent region, the bubbles increase progressively as the cavitation number decreases and form a vapor cavity attached near the leading edge. However, Gopalan and Katz (2000) argued that sheet cavitation can occur also on attached flow. In that case, other parameters can induce favorable conditions, for instance local pressure distribution, local surface imperfections, surface nucleus,… Concerning partial sheet cavity development, the following points need to be studied: • the mean characteristics of the vapor cavity, • the inspection of the flow near the detachment point and the surface of the cavity, • the examination of the closure region of the cavity together with the unsteadiness of the cavity, if any. Many authors have addressed these points in the past. Measurements or computations have been performed to determine the mean characteristics of sheet cavities, for example length, height or volume (Le et al 1993, Deshpande et al 1994, Kinnas et al 1994, Farhat 1994, Dang and Kuiper 1998, Dorange et al 1998). But it seems that little has been done to compare experiment and computation results intensively. Arakeri (1975) and Tassin Leger and Ceccio (1998) have studied the cavity detachment region. The latter carefully examined the separated flow over a series of bodies (including hydrofoils) near the front of the mid-chord attached cavity (occurring at a position of 37–42% of the foil chord the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 655 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. length). They observed that, in cavitating flow, the cavity separation occurs upstream of the separation of the boundary layer in the non-cavitating flow. Finally, certain experiments have focused on the closure region of sheet cavities to study the mechanisms responsible of cloud cavitation along with sheet cavity destabilization: Callenaere et al (1998), Kawanami et al (1998), Laberteaux and Ceccio (1998), Zhang et al. (1998), Kjeldsen et al. (1999), Gopalan and Katz (2000). The presence of a re-entrant jet is a reason (but probably not unique) for sheet cavity destabilization and the resulting cloud cavitation. From a numerical point of view, improvements have been progressively introduced for numerical prediction of sheet cavitation taking into account the viscous aspects of the flow, and steady or unsteady cavitation (Kubota et al 1992, Kinnas and Fine 1993, Deshpande et al 1994, Kinnas 1998 among others). Kinnas et al (1994) showed that the predicted cavity extend and volume increased with the Reynolds number (for Re=2 x 106 to Re=2 x 107) and that they are lower than the inviscid prediction Recently Brewer and Kinnas (1998) showed also that the tunnel walls (confinement) could modify cavitation development of partial sheet cavities considerably. In particular cavity lengths are found to be much larger when taking the tunnel walls into account. Although numerical simulations provide very useful information, the validation of the results obtained through numerical methods needs comparison with experimental results. All these works show that a careful experimental examination of cavitation conditions together with the inspection of the flow near the surface of foils (particularly near the leading edge for attached sheet cavitation) are still necessary to study cavitation. The aim of this paper is to present experimental results of a procedure of investigation concerning the inception and the development of partial cavities on various hydrofoils. Correlations are studied to provide empirical formulations for the cavity lengths, which may prove useful for further examination and comparison with numerical results. EXPERIMENTAL APPARATUS Test facility The experiments were conducted in the Ecole Navale Cavitation Tunnel fitted with a 1m long and h=0.192 m wide square cross test section. Velocities of up to 15 m/s and pressures between 30 mbar and 3 bar can be achieved. The experiments were conducted on the hydrofoils shown on Fig. 1. Three of them are of the NACA66 family and have determined using the data of Valentine (1974). Two of them have the same relative maximum thickness of 6%, but different chord lengths of 100 mm and 150 mm respectively. The third has a relative maximum thickness of 12% and a chord length of 100 mm. The fourth has an Eppler section (E817) with a maximum relative thickness of 11%, and was theoretically designed to improve performance with respect to cavitation inception (Eppler 1990). The theoretical cavitation-free buckets of the foils are shown on Fig. 2. The material for the foils was a polished Inox steel with a quasi zero-roughness Fig. 1 Tested foil sections. From top to bottom: NACA66–6%, NACA66–12%, Eppler E817. Excepted for the NACA66–6% foils, the foils were mounted such that the suction side of the foil was in front of the horizontal lower-wall of the test section. Note that the suction side is referred to as the most cambered side of the foil. They were clamped on one of the vertical walls of the test section. On the other wall, the foils were sustained by use of on axis (5 mm diameter) positioned at A mechanical mounting system enabled the foil to rotate at a given angle of incidence on an axis of rotation located also at Due to uncertainties in positioning the foils, it was estimated that the angle of incidence was known with an accuracy of ±0.14°. the authoritative version for attribution. Fig. 2 Theoretical curves of—Cpmin. The cavitation inception (or desinence) conditions were obtained by visual inspection of the flow field illuminated with a stroboscopic light. The angle of incidence for cavitation inception, αi(σ),

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 656 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. was determined by progressively increasing the angle of incidence for a constant cavitation number until cavitation was visible to the naked eye. Then the desinent condition, αd(σ), was determined by decreasing the angle of incidence until cavitation disappeared. For partial sheet cavities, the cavity length l, was estimated from the cavity profiles determined by image processing of visualizations obtained by a laser sheet, formed by a 1W Argon Laser source focused through a cylindrical lens. It consisted of computing the mean value of the gray level of a given pixel from about 20 acquired images and of determining the frontier of the cavity. It was then superimposed over the image of the foil without cavitation to determine the relative lengths of the cavity l*=l/c (Fig. 3) Fig. 3 Example of an image obtained from the superposition of the surface foil trace image (non cavitating flow) and the cavity image. Flow is from the right. In order to measure the near surface pressure distribution, velocity measurements were performed on both the NACA66 12% foil and the Eppler foil. The longitudinal and vertical velocities were measured on the suction side of the foil using a DANTEC two component LDA system, providing a 0.5 mm long (Z direction) and 0.04 mm wide (X and Y directions) measuring volume. It was coupled with two DANTEC enhanced Burst Spectrum Analyzers and a DANTEC Burstware software. The remote mechanical positioning system had a minimum translation step of 16 µm. The origin (X=0, Y=0) of the positioning system was at the leading edge of the foil at zero angle of incidence. Measurements were performed at Z=45 mm (spanwise direction) from the front wall of the test section. The laser beams were aligned with the spanwise direction allowing us to approach closely the surface foil. Velocities were mapped along lines normal to the foil surface. The mesh for the velocity measurements was computed previously and rotated with the foil in such a way that the measurements were performed at the same location relatively to the foil surface when changing the foil incidence. The spacing between the measurement stations was selected so that the definition of the velocity profile was accurate in the vicinity of the leading edge and of the foil surface to detect the minimum of the pressure coefficient (see Fig. 4). The normal lines were selected to be located for x* close to the leading edge where the minimum of the pressure coefficient is expected to occur. The flow outside the thin boundary layer near the surface of the foil may be considered as a potential flow. Thus the Bernoulli equation can be used to determine the coefficient of pressure from the local velocity. Assuming that the normal pressure gradient across the boundary layer is close to zero, the local pressure coefficient on the surface of the foil can be computed with Cp=1−(Ue/U∞)2, where Ue is the maximum velocity on the velocity profile along a normal to the foil surface. The maximum value Ue is assumed to be outside the boundary layer. Fig. 4 Example of the velocity field measured near the leading edge. NACA66–6% foil. Re=0.4x106. RESULTS Cavitation inception NACA 66 12% 100 mm Fig. 5 summarizes the conditions for inception and desinence of cavitation for the N66–12% foil. The abscissa denotes the cavitation number and the ordinate denotes the angle of incidence for which cavitation appears, αi(σ) Fig. 5.a, or vanishes, αd(σ) Fig. 5.b. The figure includes data obtained for the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 657 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. various Reynolds numbers together with the theoretical values of the minimum pressure coefficient, −Cpmin. As shown, a difference still exists between the inception and desinence conditions. The difference of angles between inception and desinence is of about ∆α≈0.5°. The difference between αi(σ) and αd(σ) means that at a given velocity the static pressure required to induce cavitation must be significantly lower than that required to eliminate it. However, the difference can be explain by the process of cavitation detection based on visualizations which can infer bias on the angle of cavitation inception. It can be assumed that the real cavitation inception (micro bubbles not observable) occurs before it is visually detected. Fig. 5 Experimental angle of cavitation inception Fig. 6 Angle of cavitation inception and desinence as a function of the Reynolds number for σ=1.8. (desinent) versus the cavitation number for various Reynolds numbers. (NACA66, 12%, 100 mm chord Comparison with the theoretical angle of inception for −Cpmin=1.8. length). Fig. 5 shows also that the inception (or desinent) curves for the suction side tend to be closer to the theoretical curve as the Reynolds number increases. The same tendency is observed on the pressure side (lower part of the curves), but the discrepancy between the experiment and the theory remains large even for the largest Reynolds number. The influence of the Reynolds number is depicted clearly on Fig. 6. As shown, αi (σ=1.8) and αd(σ=1.8) decrease when the Reynolds number increases, and are close to the theoretical prediction (α=4.65° for −Cpmin=1.8) when the Reynolds number is equal or larger than 0.8x106 (see also Fig. 7). Fig. 7 Relative difference between the experimental and theoretical angle of inception as a function of the Reynolds number for two hydrofoils. Fig. 8 Pressure coefficient on the suction side of the NACA66 12% foil. The pressure coefficient distribution on the suction side resulting from the velocity measurements is shown on Figs. 8 for Re=0.4 x 106 and Re=0.8 x 106. The pressure distribution exhibits a sharp peak near the leading edge. The peak magnitude (Cpmin)exp increases and its location (denoted) moves towards the leading edge when increasing the Reynolds number. The values are reported on Table 1 to be compared to the cavitation conditions. For Re=0.8 x 106, the magnitude of (−Cpmin)exp is found to be close to the desinent cavitation number and to the theoretical value of −Cpmin for an inviscid unbounded flow. For Re=0.4 x 106, (−Cpmin)exp is found to be larger than σd, both values being lower than the theoretical value of Cpmin. The difference is often attributed to the presence of a leading edge separation bubble which is not taken into account for an invcid flow. The photographs on Fig. 9 showing the cavitation pattern at the leading edge tend to the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 658 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. confirm the presence of a separation bubble. Moreover, at inception for Re=0.4x106 and Re=0.8x106, Fig. 9.a shows that a difference exists between the cavitation patterns. As shown, for Re=0.4x106, glossy longitudinal cavities characterize cavitation inception whereas, for Re=0.8x106, the inception appears as a narrow spanwise cavitation band. Moreover, the location of cavitation inception moves towards the leading edge for Re=0.8x106 in accordance with the location of the experimental minimum pressure coefficient. For the same cavitation number below cavitation inception, (see Fig. 9.b), it is shown that for Re=0.8x106, the cavitation pattern is a thin glossy cavity followed by a perturbed cavitating region downstream contrary to Re=0.4x106. This can be attributed to a laminar flow extending over a larger distance on the foil for the lowest Reynolds number. The photographs on Fig. 10, showing the development of the cavitation for the two Reynolds numbers at the same cavitation number, confirm this. It is shown (see dashed line) that the interface of the cavity with a glossy aspect (which can be representative of the laminar flow) is much larger for Re=0.4x106 than for Re=0.8x106. In the latter case, it is very small and is very close to the leading edge. This is accompanied also by a net increase of the cavitating region downstream on the foil surface. Table 1 Values and location of Cpmin (α=6°). Comparison to the desinent cavitation number and the theoretical value of Cpmin. Re (Cpmin)exp x*min σd Cpmin (inviscid) 0,4·106 −2,69 0.004 2.32 −3 0,8·106 −3,1 0.002 2.99 −3 NACA66–6% 100 mm and 150 mm. Fig. 11 shows the inception curves for the two NACA66–6% foils. As it has been observed previously, the angle of cavitation inception decreases with an increase of the Reynolds number (see also on Fig. 7 for the NACA66–6%–150mm foil). Moreover, a difference is recorded between the two foils. For a given cavitation number and for nearly the same Reynolds number, the cavitation appears first on the largest foil when increasing progressively the angle of incidence. This means that for a given angle of incidence the cavitation number at inception is larger on the largest foil. This can be due to the confinement effect, which tends to increase the peak of the minimum pressure coefficient at the leading edge. This observation is confirmed qualitatively by the theoretical minimum pressure coefficient that is computed taking into account of the tunnel walls (see Fig. 12). However, the increase of the theoretical minimum pressure coefficient is found to be smaller than the increase of the experimental inception cavitation number. Fig. 9 Photographs of cavitation. a) inception for Re=0.4x106 (left σ=2.16) and 0.8x106 (rigth σ= 2.85). b) developped cavitation σ=1.98±0.02, Re=0.4x106 (left), Re=0.8x106 (rigth) α=6°. NACA66 12% foil. Flow is from the left. Fig. 10 Photographs of cavitation development for Re=0.4 106 (left) and 0.8 106 (rigth), same cavitation number σ=1.31 ±0.01, α=6°. NACA66 12% foil. Flow is from the left. the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 659 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Fig. 11 Experimental angle of cavitation inception versus the cavitation number for the NACA66–100 mm and NACA66— 150 mm foils. Fig. 12 Theoretical values of −Cpmin with and without the tunnel walls. Eppler E817–100 mm. For the Eppler foil, Fig. 13.a shows also that the experimental cavitation-free bucket is wider than as the one that is predicted in theory. However, when it is compared to the experimental value of −Cpmin, the agreement is good, excepted as the angles of incidence is larger than 6°. In that case the experimental value of −Cpmin is found to be smaller than σi. This is clearly shown on Fig. 13.b showing the difference between the experimental cavitation number at inception and the measured minimum pressure coefficient as a function of the angle of incidence Fig. 13.a Experimental angle of cavitation inception versus the cavitation number for the E817 foil. Re=0.5 106. The discrepancy originates from the development of a separated flow region at the leading edge when the angle of incidence increases. This is shown on Fig. 14 depicting the velocity vectors for 10°. In that case, a large reverse flow region can be observed near the leading edge. Fig. 13.b Difference between the experimental inception cavitation number and the experimental minimum pressure coefficient as a function of the angle of incidence α. E817 foil, Re=0.5 106. the authoritative version for attribution. Fig. 14 Experimental velocity field on the suction side of the E817 foil showing the separated flow region at the leading edge, α=10°, Re=0.5 106.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 660 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Fig. 15 various types of cavitation on the NACA66 −12%–100 mm foil. Fig. 16.a Various types of cavitation on the NACA66—150mm foil.: 1 development of partial cavity, 2 fluctuating partial cavity, 3 pulsating sheet cavity, 4 attached shear cavitation, 5 bubbles, 6 supercavitation. Fig. 16.b type 1, σ=3, α=4°. Fig. 16.c type 2, σ=1.1, α=4°. Fig. 16.d type 4, σ=2.6, α=6° Fig. 16.e type 3, σ=0.81, α=4° the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 661 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Cavitation development Cavitation types Concerning cavitation development, several types of cavitation have been observed with various combinations of cavitation number and angle of incidence (see on Fig. 15 or Fig. 16.a and the associated photographs on Fig. 16b to 16.e). For all the foils, it was observed that partial cavities of length lower than about half the chord length have a relatively stable behavior, with a shedding of U-shaped vapor structures (Fig. 16.c). The U-shaped structures exhibit a certain degree of organization in the spanwise direction and were inclined at an angle of about 45° with the mean flow. Long sheet cavities ( larger than about 0.5) exhibit a pulsating behavior while shedding large vapor-filled structures. In this case, the position of the closure point of the cavity was strongly fluctuating and the length cavity was not determined. As the cavity closure region reached the foil trailing edge a typical behavior was observed with a periodic appearance and disappearance of the cavity at low frequency of about few Hertz (Fig. 16.e). This induced a strong fluid-structure interaction phenomenon. For relatively large angle of incidence (larger than about 4°) and large cavitation number (larger than about 2), a peculiar cavitation pattern, named attached shear cavitation, was observed (see zone 4 on Fig. 16.a). The cavitation inception appeared away from the foil surface, probably in spanwise vortices existing in the shear layer developing on the interface of the leading edge bubble separation. By decreasing the cavitation number, the cavitation pattern developed as a partial sheet cavity but with a cloudy interface. In that case, organized cavitating structures rolled up in the wake of the cavity (see Fig. 16.d). It was observed that bubble cavitation was limited to a region corresponding to low cavitation numbers and moderate angles of incidence. Some peculiar cavitation patterns appears also with bubbles as “fingers” or patches (see for instance on Fig. 11 for α of about 2° to 2.5° and σ lower than 0.5). Partial sheet cavity characteristics The data of cavity lengths are reported on Fig. 17 as a function of σ for various values of α in linear scale (Fig. 17.a) and in log-log scale (Fig. 17.b). Correlations have been investigated concerning partial cavity development. Such an attempt to correlate the data could be very useful to provide analytical tools which could be used to predict the development of partial sheet cavitation or to compare the experimental results with numerical results. Fig. 17 Relative cavity length as a function of the cavitation number for various angles of incidence and foils: a) linear scale b) log-log scale. A first correlation is shown on Fig. 18 are plotted on which the cavity lengths are plotted as a function of the ratio σ/ α (α in degrees). As shown, the data gather together very well for a given foil. In that case, the cavity length evolves as (σ/α) −m with the values of the exponent m given on Fig. 18. However, the data are separated when we compare the data for two different foils. Moreover, as shown, the exponent m varies significantly (between about 3 and 5) and an “unique” power law, describing the partial cavity development whatever the studied foil, is not determined. With σ and α as scaling parameters, the correlation does not take into account of a characteristic of the foils. A first the authoritative version for attribution. characteristic of a given foil is the angle of zero lift, αo and particularly the difference (α- αo) which can be considered also to be proportional to the lift coefficient. Fig. 19 shows the cavity length data by introducing the difference (α- αo). As shown, the data are less scattered (particularly the data of the NACA66 6% and the E817 foils are well regrouped) but there is a not significant improvement compared to Fig. 18.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 662 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Fig. 18 Relative cavity lengths as a function of σ/α. m is the exponent of a power law fitting the data for a given foil. α in degrees. Fig. 19 Relative cavity lengths as a function of σ/(α−αo) where αo is the theoretical angle of zero lift coefficient. It can be assumed that for a given foil and for a given cavitation number, the amount of the cavity development depends on the difference between the angle α at which the foil operates and αi(σ) the angle of cavitation inception for the given cavitation number. This is taken into account on Fig. 20 on which the cavity length data are plotted as a function of σ/ [(α- αi(σ)]. This representation takes into account on the inception condition (ie. l/c=0 for α=αi). In that case, it is found that the exponent m is much smaller than in the previous representation, and is close to 2. However, the value of the exponent varies significantly once again when comparing the foils. By replacing the experimental angle of inception at a given cavitation number by the theoretical value αi th (obtained from Fig. 2 for instance), it can be observed on Fig. 21 that data scattering is reduced and that only the data of the NACA66 with a chord length of 150 mm discards apart of the overall data. However, as this foil has the largest chord length, it can be assumed to be the most influenced by the tunnel walls (confinement effect). A computation of the theoretical values of −Cpmin has been carried out by taking into account of the tunnel walls (see Fig. 12). As shown on Fig. 12, for a given value of −Cpmin, the theoretical value of the inception angle is lower for the bounded flow. The difference (αith−αith conf) is found to be of about 0.4 degree in the range of the cavitation number studied. Fig. 22 shows the cavity length data as a function of σ/ [(α−αith cor (σ)] with αithcor=αith−0.4 for the NACA66–6%–150mm. It is observed that the correlation between the overall data is improved. Although data scattering still exists, it appears that all the data evolve as: (1) with an exponent which is close to −2. This correlation appears to be well representative of the cavity development on the tested foils for: 1<σ<3.5, 3.5°<α<6° and as long as the cavity lengths do not exceed half the foil chord length. Fig. 20 Relative cavity lengths as a function of σ/(α−αi(σ)) where αi(σ) is the angle of cavitation inception for a given value of σ. m is the exponent of a power law fitting the data for a given foil. the authoritative version for attribution. Fig. 21 Relative cavity lengths as a function of σ/(α−αith (σ)) where αith (σ) is the theoretical angle of cavitation inception for a given value of σ.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 663 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Fig. 22 Relative cavity lengths as a function of σ/(α−αithcor (σ)) where αithcor (σ) is the theoretical angle of cavitation inception corrected of confinement effect. CONCLUSIONS The main results of an experimental study concerning the inception and the development of partial sheet cavities on four two-dimensional hydrofoils are presented. Concerning the inception conditions, it is shown that, for a given cavitation number, the angle of incidence for cavitation inception decreases as the Reynolds number increases (between about 0.4x106 to 1.2x106). It is found to be larger than the theoretical value deduced from computation for an inviscid flow. However, the surface pressure coefficient, deduced of velocity measurements on two foils, indicate that the minimum pressure coefficient is close to the inception cavitation number excepted when a large separated region is observed at the foil leading edge. Concerning cavitation development, according to the cavitation number or the angle of incidence, various types of cavitation are observed as partial sheet cavities, bubble, fingers, patches or supercavitation patterns. For partial sheet cavities, the cavity lengths were measured on the foils for various conditions of cavitation number and angle of incidence. An attempt to correlate the cavity length data is studied at the end of the paper. This study indicates that the cavity lengths tend to evolve as (σ/(α−αi (σ))m with an exponent m close to −2 where αi(σ) is the inception angle at a given value of the cavitation number σ for a given foil. A close examination of the partial sheet cavity on one foil (NACA66–12%) shows that the aspect of the cavity can be very different when the Reynolds number increases (passing from 0.4 106 to 0.8 106). It is observed that the transition point on the cavity surface for which the interface passes from a glossy aspect to a cloudy aspect moves towards the foil leading edge when the Reynolds number increases. ACKNOWLEDGEMENTS The authors wish to express their deep appreciation for the support of the technical staff of the IRENAV and of the support of the Ecole navale, Ministry of Defense, France. A.De Longevialle who conducted a part of the experiment during his engineer project is grateful acknowledged. REFERENCES Arakeri, V. H 1975 “Viscous Effects on the Position of Cavitation Separation from Smooth Bodies”, Journal of Fluid Mechanics, Vol. 68, part 4, pp. 779–799. Arakeri, V.H., Carrol, J.A., and Holl, J.W., 1981 “A Note on the Effect of Short and Long Laminar Separation Bubbles on Desinent Cavitation”, Journal of Fluid Engineering, Vol. 103, March, pp. 28–32 Arndt, R.E.A., 1981, “Cavitation in Fluid Machinery and Hydraulic Structures”, Ann. Rev. Fluid Mech., 13:273–328. Billet, M.L., and Holl, J.W. 1981 “Scale Effects on Various Types of Limited Cavitation”, Journal of Fluid Engineering, Vol. 103, pp. 405–414. Brewer, W.H., and Kinnas, S.A. 1995 “Experimental and Computational Investigation of Sheet Cavitation on a Hydrofoil” The 2nd Joint ASME/ JSME Fluids Engineering Conference On Laser Anemometry, August 12–13, Hilton Head Island, South Carolina. Callenaere, M., Franc, J.P., Michel, J.M. 1998 “Influence of Cavity Thickness and Pressure Gradient on the Unsteady Behavior of Partial Cavities”. Third International Symposium on Cavitation, April 7–10, Grenoble, France. Dang, J., and Kuiper, G. 1998 “Re-entrant Jet Modelling of Partial Cavity Flow on Two Dimensional Hydrofoils”, Third International Symposium on Cavitation, April 7–10, Grenoble, France. Deshpande, M., Feng, J., and Merkle, C.L. 1994 “Cavity Flow Predictions Based on the Euler Equations”, Journal of Fluid Engineering, Vol. 116, pp. 36–44. Dorange, P., Astolfi, J.A., Billard, J.-Y., Fruman D.H., and Cid Tomas, I. 1998 “Cavitation Inception and Development on Two Dimensional Hydrofoils”, Third International Symposium on Cavitation, April 7–10, Grenoble, France. Eppler, R., 1990, “Airfoil Design and Data”, Springer-Verlag. Farhat, M. 1994 “Contribution à l'étude de l'érosion de cavitation: mécanismes hydrodynamiques et prédiction”, Thèse N°1273, Ecole Polytechnique Fédérale de Lausanne, Switzerland. Franc, J.-P., and Michel, J.-M. 1985 “Attached Cavitation and the Boundary Layer: the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 664 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. Experimental Investigation and Numerical Treatment”, Journal of Fluid Mechanics, Vol. 154, pp. 63–90. Gopalan S., Katz J., 2000, “Flow structure and modeling issues in the closure region of attached cavitation”, Physics of Fluids, Vol. 12., Nº4, pp. 895– 911. Huang, T.T., and Peterson, F.B. 1976 “Influence of Viscous Effects on Model/Full-Scale Cavitation Scaling”, Journal of Ship Research, Vol. 20, Nº4, pp. 215–223. Katz, J., 1984, “Cavitation phenomena within regions of flow separation”, Journal of Fluid Mechanics, Vol. 140, pp. 397–436. Kawanami, Y., Kato, H., and Yamaguchi, H. 1998 “Three-Dimensional Characteristics of the Cavities Formed on a Two-Dimensional Hydrofoil”, Third International Symposium on Cavitation, April 7–10, Grenoble, France. Kinnas, S.A. 1998 ‘The Prediction of Unsteady Sheet Cavitation', Third International Symposium on Cavitation, April 7–10, Grenoble, France. Kinnas, S.A., Mishima, S., and Brewer, W.H. 1994 “Non-linear Analysis of Viscous Flow Around Cavitating Hydrofoils”, 20th Symposium on Naval Hydrodynamics, August 21–26, University of California, Santa Barbara, C.A. Kinnas, S.A., and Fine, N.E. 1993 “A Numerical Nonlinear Analysis of the Flow around Two and Three-Dimensional Partially Cavitating Hydrofoils”. Journal of Fluid Mechanics, Vol. 254, pp. 151–181. Kjeldsen, M., Arndt, R.E.A and Effertz, M., 1999, “investigation of unsteady cavitation phenomena”, Proceedings of the 3rd ASME/JSME Joints Fluids Engineering Conference, July 18–23, 1999, San-Francisco. Kubota, A., Kato, H., and Yamaguchi, H 1992 “A New Modeling of Cavitating Flows: a Numerical Study of Unsteady Cavitation on a Hydrofoil Section”, Journal of Fluid Mechanics, Vol. 240, pp. 59–96. Le, Q., Franc, J.-P., and Michel, J.-M. 1993 “Partial Cavities: Global Behavior and Mean Pressure Distribution”, Journal of Fluid Engineering, Vol. 115, pp. 243–248. Laberteaux K R., and Ceccio, S.L. 1998 “Flow in the closure region of closed partial attached cavitation”, Third International Symposium on Cavitation, April 7–10, Grenoble, France Tassin Leger, A., and Ceccio, S.L. 1998 “Examination of the Flow near the Leading Edge of Attached Cavitation. Part 1. Detachment of Two- Dimensional and Axisymmetric Cavities”, Journal of Fluid Mechanics, Vol. 376, pp. 61–90. Valentine, D.T. 1974 “The effect of Nose Radius on the Cavitation Inception Characteristics of Two-Dimensional Hydrofoils”, Report 3813 of the Naval Ship Research and Development Center, Bethesda, Maryland 20034. Zhang, Y., Gopalan, S., and Katz, J. 1998 “On the Flow Structure and Turbulence in the Closure Region of Attached Cavitation” 22th Symposium on Naval Hydrodynamics, August, Washington D.C.. NOMENCLATURE = horizontal coordinate (streamwise) X = vertical coordinate Y = spanwise coordinate Z = coordinate along the foil chord x = foil chord length c = test section width h = mean surface roughness hr α = angle of incidence αo = theoretical zero lift angle αi = angle of incidence at cavitation inception αd = angle of incidence at cavitation desinence = free stream velocity U∞ = test section static pressure Po = vapor pressure Pv ρ = fluid density ν = kinematic viscosity = Reynolds number, U∞ c/v Re σ = cavitation number, (Po-Pv)/q σi = cavitation number at inception σd = cavitation number at desinence = mean streamwise velocity u = mean vertical velocity v modulus of local velocity, (u2+v2)0.5 = U = local potential velocity assumed to be the maximum of U on a line normal to the foil surface Ue = cavity length l = exponent m foil surface pressure coefficient, 1- (Ue/ U∞)2 = Cp = minimum of Cp Cpmin = location of Cpmin th and exp as a subscript means theoretical and experimental value. scaling by the foil chord length or the free stream velocity are denoted with as a superscript, the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 665 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. DISCUSSION R.Arndt University of Minnesota, USA Could you tell us what the design lift coefficients are for your foils. Your experiments for the 66 series foil appear to be carried out for lift coefficient greater than the design lift coefficients. For example, if you refer to Maines and Arndt, JFE 1997, you will see that for the 662–415 foil, the position of CPm moves forward to the leading edge for Cℓ>0.6 (drag bucket is 0.2–0.6). This will make quite a difference since CPm at design lift coefficient is about x/c=0.6! AUTHOR'S REPLY The main geometric characteristics of the foil are the following: • the chord length is c=100 mm or 150 mm and the span is 0.191 mm, • the maximum thickness is e=12 mm or 6 mm (τ=12% or 6%) at 45% from the leading edge, • the maximum relative camber is 2% for all the foils, the leading edge radius divided by the chord length is given by • the theoretical lift coefficient at a given angle of incidence α is given by Cl=0.1092 (1−0.83 τ) (α+2.35) for an inviscid unbounded flow, Valentine (1974). Clearly, the coordinates for the NACA 66 with 12% relative thickness are given in table 1. It is right that our experiments for the 66 series foil were carried out for lift coefficient greater than the design lift coefficient. However, this corresponded to the situation for which partial sheet cavities occurred. Table 1 Coordinates of the NACA66 foil section. x† y† suction side x† y† pressure side 0,00E+00 0,00E+00 0,00E+00 0,00E+00 −8,43E–05 8,88E–04 1,68E–04 −8,74E–04 −8,55E–05 1,79E–03 4,18E–04 −1,73E–03 −3,41E–06 2,70E–03 7,52E–04 −2,58E–03 1,62E–04 3,63E–03 1,17E–03 −3,41E–03 4,10E–04 4,58E–03 1,67E–03 −4,23E–03 7,42E–04 5,53E–03 2,25E–03 −5,03E–03 1,16E–03 6,50E–03 2,92E–03 −5,81E–03 1,65E–03 7,49E–03 3,67E–03 −6,59E–03 2,23E–03 8,48E–03 4,50E–03 −7,34E–03 2,90E–03 9,49E–03 5,41E–03 −8,09E–03 3,57E–03 1,01E–02 6,23E–03 −8,47E–03 5,87E–03 1,23E–02 8,83E–03 −1,00E–02 1,06E–02 1,58E–02 1,39E–02 −1,23E–02 2,26E–02 2,23E–02 2,64E–02 −1,61E–02 4,69E–02 3,16E–02 5,12E–02 −2,09E–02 7,13E–02 3,87E–02 7,57E–02 −2,44E–02 9,58E–02 4,46E–02 1,00E–01 −2,71E–02 1,45E–01 5,42E–02 1,49E–01 –3,12E–02 1,94E–01 6,17E–02 1,98E–01 −3,43E–02 2,43E–01 6,74E–02 2,47E–01 −3,64E–02 2,93E–01 7,18E–02 2,96E–01 −3,80E–02 3,42E–01 7,51E–02 3,44E–01 −3,90E–02 3,91E–01 7,72E–02 3,93E–01 −3,94E–02 4,41E–01 7,82E–02 4,42E–01 −3,94E–02 4,90E–01 7,80E–02 4,90E–01 −3,88E–02 5,39E–01 7,66E–02 5,39E–01 −3,75E–02 5,89E–01 7,39E–02 5,88E–01 −3,55E–02 6,38E–01 7,01E–02 6,36E–01 −3,31E–02 6,87E–01 6,49E–02 6,85E–01 −3,00E–02 7,37E–01 5,84E–02 7,34E–01 −2,65E–02 7,86E–01 5,03E–02 7,82E–01 −2,28E–02 8,35E–01 4,03E–02 8,31E–01 −1,91E–02 8,84E–01 2,90E–02 8,81E–01 −1,50E–02 9,32E–01 1,68E–02 9,30E–01 −1,01E–02 9,56E–01 1,04E–02 9,55E–01 −7,17E–03 9,81E–01 3,91E–03 9,80E–01 −3,91E–03 1,00E+00 −1,18E–03 1,00E+00 −1,18E–03 the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 666 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. DISCUSSION M.Billet and W.Straka Applied Research Laboratory The Pennsylvania State University, USA First, we would like to congratulate the authors on an outstanding experimental effort in characterizing the inception and development of partial cavities on four different hydrofoils. The authors have provided a very complete set of data for cavitation inception and desinence conditions on two-dimensional hydrofoils In the introduction, the authors mention the value of this type of basic experiment as an effective way to understand the cavitation in more complex situations such as marine propellers and we agree with this. However, an issue still remains on how these experiments can be used for three-dimensional situations. Would the authors please expand on the applicability of these data to propeller blades, especially in the presence of the spanwise loading gradients and the three- dimensionality associated with propeller designs. How will these effects influence the characteristics of the cavity such as its length and unsteadiness? The paper also illustrates the importance of “real” flow physics on cavitation by comparing the experimental results with the theoretical inviscid predictions. Please comment on the flow phenomena that result in the larger discrepancy between the experiment and predictions with negative angles of attack conditions compared to positive angles of attack. (Figures 5 and 11). AUTHOR'S REPLY Concerning the result in the larger discrepancy between the experiment and predictions with negative angles of attack conditions compared to positive angles of attack, (Figures 5 and 11), we have to say that no peculiar attention was paid on this point. However, the effect of a separation bubble that could be more important for negative angles because of the shape of the foil can be assumed. However, an other effect as the confinement effect (tunnel walls) can be also suspected. the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 667 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. DISCUSSION S.Ceccio University of Michigan, USA The authors present an interesting study of partial cavitation on two-dimensional hydrofoils. The authors compare the inception cavitation number with the minimum pressure coefficient computed for the non-cavitating flow and conclude that the −Cp, min ~ σi for flows without leading edge separation. Visual cavitation calls were made, and the authors suggest that the actual value of inception may be at higher cavitation numbers since it may be difficult to detect small micro-bubbles at inception. Could the authors provide more details on how cavitation inception was called? Also, the authors report that inception occurs at higher cavitation numbers as the attack angle is increased and leading edge separation occurs. Did the authors detect a separation bubble during the LDA measurements near the leading edge of the NACA66 hydrofoils (as shown in Figure 14 for the E817 hydrofoil)? The authors investigate the influence of Reynolds number variation on the inception and appearance of the cavity. Figure 10 suggests that the boundary layer near the region of cavity detachment has been sufficiently modified to change the appearance of the cavity as the Reynolds number is changed by a factor of two. Can the authors deduce what the state of the boundary layer is upstream of the cavity detachment? Specifically, is the boundary layer ingested by the cavity laminar, transitional, or turbulent? (This is an admittedly difficult observation to make experimentally for cavities detaching close to the leading edge). Also, are the authors confident that the freestream turbulence level will remain unchanged as the freestream velocity is increased? It is interesting to note how the extent and appearance of the cavity is influenced by the change in the Reynolds number. Could the authors comment on what physical processes they suspect are at work? AUTHOR'S REPLY Concerning a separation bubble during the LDA measurements near the leading edge of the NACA66 hydrofoils, we have to say that velocity measurement were carried out only on the NACA66 with 12% relative thickness. No observable negative velocities were detected which should be the condition for a separation bubble to exist. So the presence of a separation bubble was not clear. However, the cavitation inception formed of a thin spanwise bubble band seemed to indicate the presence of a separation bubble. Concerning the extent and appearance of the cavity which is influenced by the change in the Reynolds number. Photographs of cavitation patterns showed that such a transition is related to a transition point on the cavity surface, which moved forward to the leading edge when the Reynolds number increased. It can be assumed that this can be related to a transition from a long separation bubble to a short separation bubble when the Reynolds number increases. However, the effect of the turbulence intensity (as rightly suggested by the discuss or) which slightly increases with the velocity (Fig. 1) can be also responsible of such a phenomenon. This point will be studied closely. Figure 1 Turbulence intensity as a function of the spanwise direction measured at one chord length upstream of the leading edge of the foil for two velocities. NACA66–6%–150 mm foil. the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 668 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. DISCUSSION J.P.Franc University of Grenoble, France The authors have to be congratulated for their careful piece of experimental work on partial cavitation. The comparison of the results of various tests conducted on four different hydrofoils is particularly interesting for the analysis of the inception and development of sheet cavities on two-dimensional hydrofoils. The velocity measurements they performed on the two thicker hydrofoils (NACA 66 12% and Eppler E817) by LDV, use a very fine mesh and lead to a detailed description of the velocity field near the leading edge. In particular, the authors succeeded in determining, from LDV, the minimum of the pressure coefficient, which is known as an essential parameter for the interpretation of incipient conditions. On figure 13.b, the authors show that, for the E817 hydrofoil at a Reynolds number of .5 106, the experimental minimum pressure coefficient,—Cpmin, determined from LDV measurements, is in good agreement with the measured cavitation inception parameter, s prematurely, regarding the prediction based on LDV measurements. The authors give an interesting interpretation of this difference. From their measured velocity fields, they correlate this discrepancy to the development of a separation bubble at the leading edge, which is known to appear only at high enough angles of attack. We can conjecture that a shear layer exists between the free stream and the separation bubble, and consequently that vortical structures develop in this shear layer. These coherent structures have a given vorticity, so that the pressure in their core should be lower than the ambient pressure. As the authors have carried out detailed measurements of the velocity field near the leading edge, would it be possible to estimate a characteristic vorticity and a characteristic length of these structures? If so, it should be possible to give an estimate of the pressure drop in the core of these vortical structures and compare it to the difference between the minimum pressure coefficient and the cavitation inception parameter. Did the authors attempt such an approach? AUTHOR'S REPLY As commented by the discussor, it is right that the spatial resolution of the velocity field was good enough particularly at the leading edge. A characteristic spatial step between two consecutive velocity measurement points in the vicinity of the foil surface and near the leading edge is and Thus, an estimate of the nondimensional spanwise vorticity, given by, (1) can be computed. This was done from an interpolation of the velocity data together with a spatial numerical derivation in and directions (note that as a superscript corresponds to non-dimensional values scaled by the foil chord length and the inflow velocity). It must be pointed out that, in some cases, because one of the laser beams intersected to the foil surface, the vertical component of the velocity v was not measurable, thus, in that cases, was estimated only through the second term of Equ. 1. An example of the calculation is given in Fig. 1 showing the lines of constant spanwise vorticity for an angle of attack 10° corresponding to the case for which a flow separation was clearly detected. It can be observed a streamwise stretched vortical structure extending from the leading edge up to about along the boundary of the separated region (depicted by the dashed line on Fig. 1). Because of the vortical flow, it can be conjectured, as mentioned rightly by the discusor, that the pressure should be lower than the one deduced only from velocity measurements and the Bernoulli equation (assumption of potential flow). This can be written formally by adding a corrective term to the potential pressure coefficient taking into account of the vortical flow: (2) The first term on the right hand is the pressure coefficient deduced from the translation velocity measurements. The second one originates from the vortical flow. It can be assumed that the contribution of the vortical component can be expressed as: (3) the authoritative version for attribution.

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AN EXPERIMENTAL INVESTIGATION OF CAVITATION INCEPTION AND DEVELOPMENT OF PARTIAL SHEET CAVITIES ON 669 lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line TWO—DIMENSIONAL HYDROFOILS. which can be considered as the pressure coefficient at the core center of a vortex of radius a* and a vorticity of z*. A characteristic scale of the vorticity z* and a characteristic length scale a* can be inferred of the the vorticity profile (see Fig. 2). Here, a* corresponds to the characteristic length scale in which the vorticity is concentrated and z* is the mean value of the vorticity in this region. These values are reported on Table 1 together with an estimate of CpΩ. As shown, in that case, there is a good agreement between the inception number and the opposite of the minimum pressure coefficient taking into account of the vortical flow. Although, this result can appear as a rough estimation, it indicates that the vortical flow involved in the case of a flow separation can play a very important role in predicting cavitation inception. Figure 1 Non dimensional spanwise vorticity near the leading edge of E817 foil, =10°, Re=0.5 106. Figure 2 Vorticity profile, estimation of a characteristic length scale and a characteristic vorticity. x*= 0.04, =10°, Re=0.5 106. Table 1 Estimation of the rotational component to the minimum pressure coefficient and comparison to the inception cavitation number. x*=0.04, =10°, Re=0.5 106. a* *z (a* z*) (Cppot+CpΩ) CpΩ i 0.014 96.88 −1.84 −1.60 −3.44 3.3 5 the authoritative version for attribution.