Nia Cason

Muriel Marmursztejn

Mariapaola D'imperio

Daniele Schön
email: daniele.schon@univ-amu.fr826334las0010.

Rhythmic Abilities Correlate with L2 Prosody Imitation Abilities in Typologically Different Languages

Keywords: Music, speech, foreign language, temporal patterns, stress patterns

While many studies have demonstrated the relationship between musical rhythm and speech prosody, this has been rarely addressed in the context of second language (L2) acquisition. Here, we investigated whether musical rhythmic skills and the production of L2 speech prosody are predictive of one another. We tested both musical and linguistic rhythmic competences of 23 native French speakers of L2 English. Participants completed perception and production music and language tests. In the prosody production test, sentences containing trisyllabic words with either a prominence on the first or on the second syllable were heard and had to be reproduced. Participants were less accurate in reproducing penultimate accent placement. Moreover, the accuracy in reproducing phonologically disfavored stress patterns was best predicted by rhythm production abilities. Our results show, for the first time, that better reproduction of musical rhythmic sequences is predictive of a more successful realization of unfamiliar L2 prosody, specifically in terms of stress-accent placement.

Introduction

Rhythm in both music and language organizes events in time. Rhythm perception has been thought to be an innate predisposition of human cognition that may be similarly processed across domains (e.g., [START_REF] Besson | Comparison between language and music[END_REF][START_REF] Patel | The evolutionary biology of musical rhythm: Was Darwin wrong?[END_REF][START_REF] Patel | Exploring cognitive relations between prediction in language and music[END_REF]. This is true for both "beat" (the perception of an underlying regular pulse) and "meter" (the perception of alternating weak and strong elements; [START_REF] London | Hearing in time: Psychological aspects of musical meter[END_REF]. Notably, the patterning of weak and strong elements is a fundamental characteristic of both speech and music, which could explain why rhythm in music has been compared to stress in language [START_REF] Patel | Musical rhythm, linguistic rhythm, and human evolution[END_REF][START_REF] Schön | Short-and long-term rhythmic interventions: Perspectives for language rehabilitation[END_REF][START_REF] Magne | Speech rhythm sensitivity and musical aptitude: ERPs and individual differences[END_REF]. However, the links between rhythmic skills in music and speech have rarely been studied in the context of second language (L2) learning. The present study investigated the link between musical rhythmic competences and realization of L2 speech prosody, with the idea that better musical rhythmic skills would predict a more accurate realization of L2 prosody, specifically in terms of correct stress-accent placement (for the notion of stress-accent, see [START_REF] Beckman | Stress and non-stress accent[END_REF].

The perception of meter is crucial for language acquisition. During native language (L1) acquisition in infancy, metrical structures in speech serve as important cues in speech segmentation [START_REF] Cutler | Rhythmic cues to speech segmentation: Evidence from juncture misperception[END_REF][START_REF] Jusczyk | The beginnings of word segmentation in Englishlearning infants[END_REF][START_REF] Kuhl | Early language acquisition: Cracking the speech code[END_REF][START_REF] Morgan | Emerging integration of sequential and suprasegmental information in preverbal speech segmentation[END_REF], which has been found to be facilitated by musical skills in childhood [START_REF] François | Music training for the development of speech segmentation[END_REF]. One idea is that rhythmic entrainment (the coupling between the temporal structure of a stimulus and brain oscillatory activity) enhances the precision of auditory analyses and, as a consequence, enhances speech segmentation abilities [START_REF] Hornickel | Unstable representation of sound: A biological marker of dyslexia[END_REF][START_REF] Moreno | Musical training influences linguistic abilities in 8-year-old children: More evidence for brain plasticity[END_REF]. For instance, musical rhythmic abilities have been found to correlate with phonological abilities in preschoolers [START_REF] Moritz | Links between early rhythm skills, musical training, and phonological awareness[END_REF], with reading ability in school-age children [START_REF] Strait | Subcortical processing of speech regularities underlies reading and music aptitude in children[END_REF], and seem to be particularly relevant in decoding words requiring the use of linguistic stress [START_REF] David | Rhythm and reading development in school-age children: A longitudinal study[END_REF]. Furthermore, metrical characteristics of L1 also affect rhythm perception (Iversen, Patel, & Oghushi, 2008;[START_REF] Kusumoto | Native language determines the parsing of nonlinguistic rhythmic stimuli[END_REF] and may influence the rhythmic structure of instrumental music [START_REF] Patel | An empirical comparison of rhythm in language and music[END_REF], which suggests there to be a bidirectional influence of music and language temporal structures.

When turning to L2 learning, there is also evidence of a link between musical skills and L2 perception abilities (for reviews, see Chobert & Besson, 2013;[START_REF] Zeromskaite | The potential role of music in second language learning: A review article[END_REF][START_REF] Dittinger | Professional music training and novel word learning: From faster semantic encoding to longer-lasting word representations[END_REF], including the detection of prosodic anomalies [START_REF] Milovanov | The interplay between musical and linguistic aptitudes: A review[END_REF], prosodic pitch manipulations (Marquès, [START_REF] Marques | Musicians detect pitch violation in a foreign language better than nonmusicians: Behavioral and electrophysiological evidence[END_REF], pitch contour [START_REF] Zhao | Effect of musical experience on learning lexical tone categories[END_REF], speech imitation [START_REF] Christiner | Song and speech: Examining the link between singing talent and speech imitation ability[END_REF], and phonological perception and production (Slevc & Myiake, 2006). These findings indicate that there may be shared resources between music and L2 acquisition. However, musical aptitude has largely been determined by musical melodic/ pitch skills rather than rhythmic skills, which may be distinct [START_REF] Phillips-Silver | Born to dance but beat deaf: A new form of congenital amusia[END_REF]. Some evidence suggests that rhythmic skills may actually be a better predictor of L2 acquisition than pitch/melodic skills [START_REF] Kempe | Musical ability and non-native speech-sound processing are linked through sensitivity to pitch and spectral information[END_REF]. Indeed, recent work from Boll-Avetisyan, [START_REF] Boll-Avetisyan | Effects of musicality on the perception of rhythmic structure in speech[END_REF] found that musical rhythm aptitude, but not melody aptitude, was associated with rhythmic speech perception in native listeners.

Although links between L2 production abilities and musical rhythmic skills have rarely been demonstrated [START_REF] Christiner | Song and speech: Examining the link between singing talent and speech imitation ability[END_REF][START_REF] Christiner | Sing Chinese and tap Tagalog? Predicting individual differences in musical and phonetic aptitude using language families differing by sound-typology[END_REF], rhythm and intonation have long been regarded as one of the greatest obstacles in L2 acquisition [START_REF] Calbris | Approche rythmique intonative du français langue étrangère[END_REF]. This may be because L2 acquisition is strongly affected by the metrical characteristics of L1. Metrical "rules" that exist for L1 are used in the interpretation and segmentation of L2, even when these same rules do not apply [START_REF] Cutler | The syllable's differing role in the segmentation of French and English[END_REF], 1992;[START_REF] Otake | Mora or syllable? Speech segmentation in Japanese[END_REF]. The language-specific use of metrical cues in French, for example, has been used to explain difficulties in encoding foreign stress contrasts that exist in Spanish [START_REF] Dupoux | A destressing "deafness" in French?[END_REF][START_REF] Dupoux | Persistent stress deafness: The case of French learners of Spanish[END_REF], English [START_REF] Kolinsky | Music training facilitates lexical stress processing[END_REF][START_REF] Kolinsky | Music training facilitates lexical stress processing[END_REF]German (Schmidt-Kassow, Rothermich, Schwartze, &Kotz, 2011), which are attenuated by musical experience (Boll-Avetisyan, Bhatara, & Höhle, 2016).

In the present study, we investigated native French speakers with knowledge of English as an L2. The fundamental metrical differences between French and English can pose a challenge for native French speakers learning English. Indeed, French speakers find stress and pitch accent placement in English to be one of the most challenging points (see [START_REF] Capliez | Typologie des erreurs de production d'anglais des francophones: Segments vs. suprasegments. Recherche et pratiques pédagogiques en langues de spécialité[END_REF][START_REF] Frost | The stress site: L'accent lexical, l'anglais de spécialité et l'oral, la conception d'un outil d'apprentissage médiatisé[END_REF]. Although French is considered to be a language in which lexical stress cues are absent [START_REF] Dupoux | A destressing "deafness" in French?[END_REF][START_REF] Dupoux | Persistent stress deafness: The case of French learners of Spanish[END_REF][START_REF] Cooper | Constraints of lexical stress on lexical access in English: Evidence from native and non-native listeners[END_REF], it is characterized by the presence of stress at a phrasal level (the Accentual Phrase, or AP; see [START_REF] Jun | A phonological model of French intonation[END_REF][START_REF] Post | Tonal and phrasal structures in French intonation[END_REF]. In fact, while stress is a property of the word in English, it is a phrasal property in French. Namely, stress is located at an AP-final position in French (on the last full syllable of the AP), although an AP-initial pitch prominence is allowed in the case of an initial rise [START_REF] Delattre | L'accent final en français: Accent d'intensité, accent de hauteur, accent de durée[END_REF][START_REF] Hirst | Intonation systems: A survey of twenty languages[END_REF][START_REF] Fonagy | L'accent français, accent probabilitaire[END_REF][START_REF] Vaissière | On French prosody[END_REF]. French speakers may therefore have a less-rich representation of speech meter compared to speakers of languages in which stress position is more variable, such as English [START_REF] Di Cristo | De la métrique et du rythme de la parole ordinaire: L'exemple du français[END_REF][START_REF] Cutler | The predominance of strong initial syllables in the English vocabulary[END_REF][START_REF] Tortel | Evaluation qualitative de la prosodie d'apprenants français: Apports de paramétrisations prosodiques[END_REF]. Findings that native French speakers find it difficult to detect accentual contrasts that are not found in French support this idea (e.g., [START_REF] Dupoux | Persistent stress deafness: The case of French learners of Spanish[END_REF][START_REF] Kijak | How stressful is L2 stress? A cross linguistic study of L2 perception and production of metrical systems[END_REF][START_REF] Michelas | How deaf are French speakers to stress?[END_REF][START_REF] Bhatara | Language experience affects grouping of musical instrument sounds[END_REF][START_REF] Boll-Avetisyan | Effects of experience with L2 and music on rhythmic grouping by French listeners[END_REF].

To summarize, the link between rhythm in speech and in music is now quite well-established, but has yet to be investigated in the context of L2 prosody acquisition, especially in terms of speech production and imitation. Recent imitation studies have investigated production capabilities and individual differences in reproducing fine phonetic detail in prosody, such as tonal alignment (D'Imperio, [START_REF] D'imperio | Phonetic and phonological imitation of intonation in two varieties of Italian[END_REF] or syllable length [START_REF] Cavone | Poster presented at LabPhon 15: L1 use predicts imitation of metrical features in a typologically different L2[END_REF]. We thus investigated the relationship between rhythmic competences in music perception and production in relation to the ability for native French speakers to perceive and produce stress-accent placement in English words, specifically in word-initial and penultimate position. We hypothesized that musical rhythmic skills would be positively correlated with accuracy in the production of stress accent placement, and that this would be particularly evident for the penultimate stress, which is absent in the French system and is thus more difficult to realize. Our findings offer further support for a common processing of rhythm in language and music, and might also have practical implications for L2 learning techniques.

Materials And Methods

Participants

Twenty-three native French speakers studying at the science campus of Aix-Marseille University (16 female, age range 19-24 years, mean age 21 years) participated in the study. Participants had started learning English between the ages of 5 and 12 years (mean = 9.9 years; SD = 1.79) and had between 7 and 15 years of training in English (mean = 9.9 years; SD = 1.78). In other words, the level of English proficiency is that which French-speaking adults typically have gained from general school education. Twelve participants had received some musical training, which had started between the ages of 4 and 17 years (mean = 9.7 years, SD = 4.6; mean years of practice = 6.2, SD = 4.5). The remaining 11 participants had no musical practice (outside of the very basic school curriculum). All participants gave their written formal consent and received a bonus of one point on their English grade for the semester for participating in this study. Participants completed both musical and linguistic tests, and the order of these tests was counterbalanced across participants.

Musical Tasks

Music perception. A subset of the Musical Ear Test [START_REF] Wallentin | The musical ear test, a new reliable test for measuring musical competence[END_REF] was used to test melodic and rhythmic perception abilities. More precisely, we used the first 26 trials of each task instead of the entire 52 of the original Musical Ear Test. This was done so that all tasks could be completed in a single session. Because the Cronbach's α-consistency was very high for both melodic and rhythmic tests (0.95, reported in [START_REF] Wallentin | The musical ear test, a new reliable test for measuring musical competence[END_REF], reducing the number of items should not make a major difference. For each trial, participants were asked to decide whether two short musical phrases (melodic or rhythmic) were the same or different. When different, two musical phrases differed from one another by a single element (one note for the melodic phrases, one rhythmic change for the rhythmic phrases).

Music production. Rhythmic production. Nine rhythmic sequences with a synthesized woodblock sound were played through loudspeakers. Each sequence consisted of three to seven sounds for a total of 46 sounds/taps (see Figure A1). Participants were orally asked to reproduce each rhythm once by tapping a wooden stick on a box right after listening to the sequence. No metronome cue was provided. A microphone was positioned next to the box and the participants' performance was recorded. A change-in-slope detection algorithm was used to extract the precise tapping times. We did not want to penalize globally slower or faster reproductions, so rhythmic production accuracy was measured by calculating the ratio of the productions of the 46 taps with those of the actual stimuli. For this, all intervals between two successive taps (t n-t n1 ) were computed, and each interval was divided by the interval that preceded it (t n1-t n2 ). The ratios for the participants' productions were then compared with those for the model stimuli (m), and the mean of the absolute value of the difference for all of the ratios obtained for all nine stimuli (N) was computed for each participant using the following formula:

abs / / / n n 1 n 1 n2 n n1 n1 n2 t t t t m m m m N - - - - - ( ) ( ) ( ) ( )     ∑
This metric purposefully ignores the extent to which the tempo of the subject's drumming was similar to that of the stimulus. There was no reason to think that absolute tempo would be linked to stress placement, which is a relative phenomenon. Melodic production. Participants were presented with a subset of stimuli from Lévêque and Schön (2013) through headphones, which comprised 10 stimuli of five isochronous notes each (see Figure A2). Participants were asked to reproduce each stimulus sequence once, immediately after hearing it. The melodic structure of the stimuli was tonal, which corresponds to what may be found in a popular song repertoire. The stimuli were sung by female or male voices and were presented accordingly to the participants' gender. The pitch range of the 50 pitches to be sung was adapted to adult tessitura and stimuli were presented one octave lower for male participants. PRAAT [START_REF] Boersma | PRAAT: Doing phonetics by computer (version 5.1.34)[END_REF] was used to extract the median fundamental frequency of each note of the five-note vocal productions of the participants (discarding the transition glides). Then, the same procedure described for rhythmic production was used to compute the ratios of pitch frequency between successive intervals and to compare it to the model.

Linguistic Tasks

General English level assessment. An online test for the different levels of the Common European Framework of Reference for Languages was used (http://www.cambridgeenglish.org/test-yourenglish/adult-learners/). This consists of 25 multiple-choice questions presented in a written format that test vocabulary, grammar, and idiomatic expression knowledge to assess L2 English proficiency. Every correct item was given one point (score range 0-25).

Imitation of lexical stress in L2. This task was developed to assess the ability to correctly implement lexical stress in a foreign language. For this, we assessed the ability to implement a nuclear pitch accent associated with the second (penultimate) syllable of a trisyllabic word, which is a location that never receives primary stress in the participants' L1 (French). Participants heard sentences recorded by an American English speaker and were asked to immediately reproduce them. A script of each sentence was made available to reduce memory load. The 40 test sentences were organized in a simple Subject-Verb-Object mode and ended with an adverbial (one or two words, see Appendix B). The sentences contained either a high-frequency trisyllabic target word with a stress on the first syllable (e.g., They met the MInister yesterday) or on the penultimate syllable (e.g., They got a new aPARtment last month), which is a primary stress location that is rare in French [START_REF] Hayes | Metrical stress theory: Principles and case studies[END_REF]. The target word, selected from the CELEX database [START_REF] Burnage | CELEX: A guide for users[END_REF], was the object noun in all Subject-Verb-Object sentences and thus bore the nuclear stress of the sentence, given that they were all-focus utterances. Frequency of lexical similarity (i.e., cognate words that are similar in both languages, such as "Minister/Ministre") was controlled across the two stress conditions (stress on syllable 1, stress on syllable 2); most target words were French-English cognates (32/40). Ten sentence fillers that did not contain any trisyllabic nouns were added. The sentences were presented in a pseudo-random order. Two native English speakers listened to the participants' recordings and used a binary rating system (1 correct; 0 incorrect) to evaluate whether the participants had succeeded or not in placing the nuclear accent on the correct syllable. The procedure was blind with respect to the results in the other tasks. When the two judges did not agree on a rating, the trial received a rating of 0.5. The two judges agreed on 82% of the words, with a κ-reliability score of 0.64. To further test the reliability of the judges, we added a third judge and obtained a Cohen κ-reliability score of 0.74, which is considered as a good inter-rater agreement [START_REF] Fleiss | Statistical methods for rates and proportions[END_REF].

The decision to use judges to decide whether accent production was correct or not was dictated by the absence of solidly established values that allow us to determine whether a syllable is stressed or not. Lexical stress is defined in terms of relative perceived prominence of a syllable in a multisyllabic word (Ladd, 1996(Ladd, /2008)). Perceptually, stressed syllables are expected to be comparatively longer in duration, higher in pitch (when carrying a high or rising pitch accent), and perhaps louder than unstressed syllables [START_REF] Lehiste | Suprasegmental features of speech[END_REF]. However, there are several obstacles to objective and automatic stress identification; acoustic correlates of stress and accent are highly languagespecific (for a review, see Ortega-Llebaria & Prieto, 2011, for instance). Furthermore, stress production in L2 is subject to a variable amount of interference between L1 and L2 prosody. Hence, it is common practice to perform an auditory transcription of stress (see Michelas &D'Imperio, 2012, for French, and[START_REF] Cho | Prosodic influences on consonant production in Dutch: Effects of prosodic boundaries, phrasal accent and lexical stress[END_REF]McQueen, 2005, for Dutch), given that the perception of prominence is categorical and susceptible to linguistic-perceptual judgment.

Results

First, we explored the results for each task and, when appropriate, compared performance against chance or across conditions or tasks. A descriptive correlation analysis was then performed to see whether the different tasks were related to each other. Because several variables were found to be correlated, a factor analysis was conducted; this is a more advanced multivariate technique that allows the identification of variables that go well together and can be "summarized" in a small number of factors. Finally, a regression analysis was performed, which allowed us to test the hypothesis that performance of stress placement in ultimate and penultimate positions is best predicted by language proficiency or musical (melodic or rhythmic) skills.

Music and Language Tasks

There were no significant differences between the % of correct responses in the melody and rhythm perception tasks (Figure 1A; Wilcoxon, z = 0.5, p = 0.6). However, the population dispersion was smaller in the rhythm perception task, which indicates a more homogenous level of performance (although there was one participant with a very poor rhythmic performance). Participants were able to perform the melody and rhythm perception tasks well above chance level (one sample t-test, always p < 0.001). For music production tasks (Figure 1B), a more homogenous level of performance was also seen for rhythm compared to melody. Note that, although data are represented in terms of % distance from the model, it is impossible to directly compare melody and rhythm results due to the difference in units (Hertz vs. milliseconds, respectively). We also computed the α-Cronbach measure of reliability [START_REF] Cronbach | Coefficient alpha and the internal structure of tests[END_REF] to assess whether our measures were robust. Both melody and rhythm production showed a very high internal consistency (α = 0.93 and 0.86, respectively).

Scores for the English proficiency test (Figure 2A) showed a rather widespread distribution with a median of 13/25 (range 7-20). Thus, the population we tested did not include participants with beginner or expert English proficiency levels. For the speech imitation task (Figure 2B), participants were better able to correctly reproduce stress when it occurred on the first compared to the second syllable (Wilcoxon, z = 3.2, p = 0.001).

Correlation Analysis

A Spearman's correlation analysis (based on ranks) was performed to test the strength of relationships between variables. Correlations between all the musical and language tasks are provided in Table 1. An overview of significant values (in bold, p < 0.05, one-tailed, uncorrected) shows that accuracy in the imitation task was correlated with English proficiency level. Significant correlations were also found between the musical perception tasks (melody and rhythm) and between the musical production tasks (melody and rhythm). Rhythm production was the only musical dimension that was correlated with imitation accuracy, when the stress had to be placed on the second syllable. A post-hoc power analysis was computed using the software G Power 3.1.9.2 given α (0.05), sample size and effect size (r) and showed that significant correlations always achieved a power greater than 0.7 (range 0.7-0.9). The interdependence between these variables was further tested using a factor analysis.

Factor Analysis

The factor analysis included results from the language tasks (English proficiency level and imitation) and the music perception and production tasks (rhythm and melody). Preliminary testing showed that our model was satisfactorily adequate; The Kaiser-Meyer-Olkin index measuring the sampling adequacy gave a value greater than 0.6. Finally, following two different methods to estimate the number of factors (using the software package FACTOR, Unrestricted Factor Analysis 9.2 by Urbano Lorenzo-Seva and Pere J. Ferrando) and an eigenvalue criterion ⩾1, two factors were extracted that explained 69% of the variance (Table 2).

The first factor showed high factor loadings (i.e., correlation coefficients between variables and factors) for melody and rhythm perception, rhythm production, and imitation accuracy of the syllable with disfavored lexical stress. Thus, this first factor can be interpreted as describing musical abilities. The second factor showed high factor loadings for English proficiency scores and imitation accuracy of words with initial-syllable stress. It can thus be interpreted as a factor that describes more specific linguistic abilities.

Regression Analyses

In the regression analyses, the two outcomes of the imitation task (imitation of syllable 1 and of syllable 2) were considered as dependent variables in separate analyses. The outcome of the factor analysis was used to select the most appropriate predictors, as follows: L2 proficiency and melody perception for syllable 1 and rhythm production and rhythm and melody perception for syllable 2. When considering imitation accuracy of the first syllable, L2 proficiency significantly predicted the outcome (β = 0.61, F = 12.8, p = 0.002). When considering imitation accuracy of the penultimate syllable, only rhythm production significantly predicted the outcome (β = 0.81, F = 7.8, p = 0.01), whereas rhythm and melody perception did not reach significance (rhythm: β = 0.23, F < 1; melody: β = 0.08, F < 1). We computed f 2 Cohen values from the R 2 values of the regression and used these as effect sizes, together with α (0.05) and sample size ( 23) to compute post-hoc achieved power using G*Power 3.1.9.2. Although the post-hoc power analysis showed that significant regressors achieved a power > 0.8, results of the regression analysis should nonetheless be treated with caution due to the rather limited sample size.

Discussion

One of the greatest challenges in L2 acquisition is being able to reproduce prosodic structure, particularly rhythm and intonation [START_REF] Calbris | Approche rythmique intonative du français langue étrangère[END_REF]. We used a novel task to investigate the links between musical skills and the ability to imitate speech prosody in L2, which is an element of speech that is crucial for decoding the speech signal. French participants were more able to imitate accent placement for word-initial than for penultimate stress location in English L2. This was expected given the constraint against moving the primary stress location away from the edges of the AP in French. Most importantly, correct placement of the nuclear accent was well predicted by rhythm production scores. More precisely, participants with better rhythm reproduction ability were more able to reproduce penultimate stress by placement of a correct nuclear accent. Whereas previous studies have investigated the possible links between rhythm perception and language abilities in the context of L1 perception [START_REF] Hausen | Music and speech prosody: A common rhythm[END_REF], ours is the first study to show a link between musical rhythm production and the ability to correctly imitate stress-accent placement in an L2 that is characterized by a typologically different prosodic system. Our results suggest that rhythmic training may possibly have a facilitatory effect in learning appropriate L2 prosody (and vice versa; see [START_REF] Bhatara | Foreign language learning in French speakers is associated with rhythm perception, but not with melody perception[END_REF], for a similar conclusion on the link between foreign language experience L2 and rhythm perception).

L2 Acquisition and Musical Rhythmic Skills

Several studies have found a relationship between musicianship and L2 skills (e.g., [START_REF] Lee | Identification of Mandarin tones by English-speaking musicians and nonmusicians[END_REF]. However, studies investigating the relationship between specific musical skills and L2 have primarily focused on perception and general L2 proficiency [START_REF] Bhatara | Foreign language learning in French speakers is associated with rhythm perception, but not with melody perception[END_REF][START_REF] Boll-Avetisyan | Effects of experience with L2 and music on rhythmic grouping by French listeners[END_REF], or have looked at links between general musical experience and L2 rhythm [START_REF] Bhatara | Language experience affects grouping of musical instrument sounds[END_REF]. Other studies have focused on the musical dimension of pitch processing and have not systematically attempted to tease-apart pitch and rhythmic abilities [START_REF] Posedel | Pitch perception, working memory, and secondlanguage phonological production[END_REF][START_REF] Milovanov | Musical aptitude and second language pronunciation skills in school-aged children: Neural and behavioral evidence[END_REF][START_REF] Slevc | Individual differences in second-language proficiency: Does musical ability matter[END_REF]. Not only does this mean that there are relatively few studies concerned with temporal processing, which is crucial for language and speech comprehension [START_REF] Goswami | A temporal sampling framework for developmental dyslexia[END_REF]), but it is also questionable how pitch processing in music can be compared to pitch processing in speech (such as in tonal and intonational languages). For instance, a small inaccuracy in pitch-level production is extremely salient in music (e.g., one-third of a tone), but goes unnoticed in speech. Saying this, musical relative pitch skills may be relevant when studying factors that may impact the acquisition of tonal languages; notably, non-native musicians more accurately perceive Mandarin tones compared to nonmusicians [START_REF] Lee | Identification of Mandarin tones by English-speaking musicians and nonmusicians[END_REF][START_REF] Gottfried | Relation of pitch glide perception and Mandarin tone identification[END_REF].

Regarding the temporal aspects of speech, musical expertise has been found to have an effect on the discrimination of consonant and vowel duration in both L1 and L2 [START_REF] Sadakata | Enhanced perception of various linguistic features by musicians: A cross-linguistic study[END_REF] and the identification of segmental and tonal contrasts in L2 [START_REF] Marie | Influence of musical expertise on segmental and tonal processing in Mandarin Chinese[END_REF]. In a longitudinal study, [START_REF] François | Music training for the development of speech segmentation[END_REF] found that music training predicts more efficient speech segmentation skills in an artificial language in children. Overall, well-developed rhythmic abilities could translate into a better detection of speech regularities [START_REF] Kraus | Cognitive factors shape brain networks for auditory skills: Spotlight on auditory working memory[END_REF]. Additionally, music-L2 links are most probably bidirectional [START_REF] Bidelman | Tone language speakers and musicians share enhanced perceptual and cognitive abilities for musical pitch: Evidence for bidirectionality between the domains of language and music[END_REF]. For instance, an enhanced music rhythm perception has been found for those who have mastered an L2 [START_REF] Roncaglia-Denissen | The enhanced musical rhythmic perception in second language learners[END_REF]; this is particularly true when the L2 is rhythmically different relatively to the L1, an effect that cannot be simply explained by exposure to more complex musical rhythms (as in Turkish L1 speakers; [START_REF] Roncaglia-Denissen | The enhanced musical rhythmic perception in second language learners[END_REF]. This relationship between musical rhythm and speech perception could rely on shared cognitive functions (such as working memory), as well as on the ability to entrain to multiplexed temporal scales, from the millisecond to second levels [START_REF] Tierney | Neural entrainment to the rhythmic structure of music[END_REF][START_REF] Doelling | Cortical entrainment to music and its modulation by expertise[END_REF][START_REF] Schön | Short-and long-term rhythmic interventions: Perspectives for language rehabilitation[END_REF]. Future studies could implement a longitudinal design to demonstrate a causal effect of specific features of musical training (here, rhythm) on other features of L2 acquisition (here, accent placement).

As noted above, previous findings have been mainly concerned with L2 perception, and not L2 production. One of the only studies to have investigated the effect of music skills on L2 production showed that musical competence has an impact on phonological perception and production [START_REF] Slevc | Individual differences in second-language proficiency: Does musical ability matter[END_REF]. However, this study only tested melodic competence, and in terms of L2 production it only tested segmental phonology. The present study is the first to investigate the effects of musical expertise on lexical accent placement in L2 production, and to investigate its relationship with both melodic and rhythmic perception and production skills. Similarly to the present study, participants in [START_REF] Slevc | Individual differences in second-language proficiency: Does musical ability matter[END_REF] study (Japanese adults learning English) had an existing L2 knowledge. This means that the L2 was, nonetheless, familiar to some degree, and correct stress placement in the current study was thus also due to linguistic training. A distinction can therefore be made with studies that have looked at the effect of musical expertise on the production of a completely unknown language [START_REF] Christiner | Song and speech: Examining the link between singing talent and speech imitation ability[END_REF], 2015;[START_REF] Christiner | Sing Chinese and tap Tagalog? Predicting individual differences in musical and phonetic aptitude using language families differing by sound-typology[END_REF]. Although "sense of rhythm" was considered to play a role in the ability to imitate an unknown language (Hindi), the authors of the 2013 study found that 66% of the speech imitation variance could be explained by working memory capacity, possibly due to intrinsic high memory load when imitating a completely unknown language. In this respect, an advantage of working with a known L2, as in our case, is that it is easy to provide participants with a printed version of the heard sentences to be reproduced, thus preventing a memory load bias.

Interestingly, we did not find any relationship between melodic performance and L2 accent placement imitation success. This finding fits with the growing idea that rhythm plays a key role in speech perception and production, and supports findings that rhythm-and not melody-is most effective in language/speech therapy [START_REF] Overy | Dyslexia and music: Measuring musical timing skills[END_REF][START_REF] Goswami | A temporal sampling framework for developmental dyslexia[END_REF][START_REF] Stahl | Rhythm in disguise: Why singing may not hold the key to recovery from aphasia[END_REF][START_REF] Flaugnacco | Rhythm perception and production predict reading abilities in developmental dyslexia[END_REF]. However, an analysis of more global pronunciation skills (as in [START_REF] Christiner | A Mozart is not a Pavarotti: Singers outperform instrumentalists on foreign accent imitation[END_REF] might have revealed a relationship with melodic aptitude (e.g., see [START_REF] Milovanov | Musical aptitude and second language pronunciation skills in school-aged children: Neural and behavioral evidence[END_REF]. Future studies could more closely examine the relative contributions of different musical skills to different L2 skills, and how these contributions differ according to the L2 being learned; and also in relation to the properties of one's L1 (see [START_REF] Roncaglia-Denissen | The enhanced musical rhythmic perception in second language learners[END_REF].

The fact that L2 English proficiency (measured using the standard Cambridge test) was also predictive of a correct prosodic production may simply reflect the fact that a more thorough knowledge of the L2 implies heightened exposure, hence familiarity, with its prosody (see also [START_REF] D'imperio | Proceedings of the 18th International Congress of Phonetic Sciences (ICPhS 2015): Phonetic detail and the role of exposure in dialect imitation[END_REF]. Even more interesting is that, while we would expect English proficiency to actually be the best predictor of L2 prosody production, the best predictor of the performance on the most difficult items (i.e., penultimate stress) was rhythm production score. Future studies should investigate the relationship between musical rhythmic skills and prosody production using a known and an unknown L2 language so as to evaluate the effect of negative transfer. Indeed, previous knowledge of L1 might interfere with new learning (L2), which may result in establishing bad pronunciation habits that may not be present when testing another, unknown L2 [START_REF] Odlin | Language transfer: Cross-linguistic influence in language learning[END_REF]. It would also be interesting to compare imitation abilities with L2 that vary in their linguistic distance to L1 (such as in D'Imperio & German, 2015, in which two typologically different prosodic systems were compared). For instance, in our study, some items were lexically similar in L1 and L2, but had a different prosody (e.g., MInister/minIStre). Musical rhythmic skills might therefore play a different role when there must be a process of reconfiguration of the stress position with respect to L1 compared to when there is no lexical competition and no need for stress pattern reconfiguration.

Theoretical and Practical Implications: A Common Processing of Meter

The implications of these findings are theoretically and practically relevant. From a theoretical standpoint, our results show that metrical rhythmic patterns may be processed similarly for language and music. Namely, participants who were more accurately able to reproduce musical rhythms also showed an advantage in the realization of syllables with a disfavored stress location. This finding extends and refines previous studies showing that musical expertise is correlated with L2 skills (see Chobert &Besson, 2013, andZeromskaite, 2014, for reviews), and suggests that musical temporal aptitude may be linked to specific aspects of foreign language acquisition.

From a practical standpoint, these findings are interesting when considering the value of musicbased teaching techniques for foreign language teachers. Besides a general motivational benefit, different features of music could be used to reinforce prosodic representations of the foreign language being taught. Considering previous findings in L1 acquisition [START_REF] Cason | Rhythmic priming enhances speech production abilities: Evidence from prelingually deaf children[END_REF] and in children with reading impairments [START_REF] Bhide | A rhythmic musical intervention for poor readers: A comparison of efficacy with a letter-based intervention[END_REF][START_REF] Flaugnacco | Music training increases phonological awareness and reading skills in developmental dyslexia: A randomized control trial[END_REF], rhythmic musical training can also be predicted to impact positively on L2 phonological perception and production, too.

It has been shown that musical competence is not uniquely the result of music training, but of many other cognitive and social factors [START_REF] Swaminathan | Musical Competence is Predicted by Music Training, Cognitive Abilities, and Personality[END_REF]. However, musicians outperform nonmusicians in several musical skills, including temporal rhythmic skills [START_REF] Rammsayer | Temporal information processing in musicians and nonmusicians[END_REF][START_REF] Rammsayer | Musicians do better than nonmusicians in both auditory and visual timing tasks[END_REF][START_REF] Schaal | The rhythm span task: Comparing memory capacity for musical rhythms in musicians and non-musicians[END_REF][START_REF] Wollman | Organizational principles of multidimensional predictions in human auditory attention[END_REF]. Thus, music training or music-based L2 learning, by improving temporal processing at different levels, may in turn facilitate the learning of segmental and suprasegmental features of an L2.

Importantly, different aspects of musical training may make different contributions to L2 acquisition, depending on the language. For instance, we might expect to find similar results for Turkish L1 speakers learning English L2, because Turkish has a word-final stress [START_REF] Kabak | The phonological word and stress assignment in Turkish[END_REF]. Similarly, several other L2 skills may instead benefit from pitch training, particularly in languages in which pitch is more important (e.g., Mandarin). Finally, due to the metrical nature of French prosody, French speakers may have less-rich representations of speech meter than speakers of more prosodically variable languages, and may thus benefit more from a rhythmic music-based teaching approach than might speakers of other languages. Future studies could investigate the relationship of musical rhythmic skills and L2 prosodic skills in speakers whose L1 is more prosodically variable. To conclude, while further research is required to specify the L2-specific links with musical abilities, this study demonstrates a strong relationship between the perception and production ability of L2 meter and musical rhythmic production skills.

Figure 1 .

 1 Figure 1. A: Boxplots of the % of correct responses for the music perception tasks. B: Boxplots of the % difference from the stimuli for the music production tasks.

Figure 2 .

 2 Figure 2. A: Boxplot of the number of correct responses for the Cambridge test. B: Boxplots of the % of correct responses for the imitation task when the prominence was on the first or in the second syllable of the trisyllabic words.

Table 1 .

 1 Spearman's correlation coefficients between the dependent variables.

		L2 proficiency	Melody	Rhythm	Melody	Rhythm	Syllable 1
		level	perception	perception	production	production	Imitation
	Melody perception -0.03
	Rhythm perception	0.23	0.62
			(0.35, 1)
	Melody production -0.23	-0.45	-0.35
			(-0.8, -0.1)
	Rhythm production -0.35	-0.36	-0.5	0.14
				(-0.9, -0.15)
	Syllable 1 imitation	0.59	0.06	0.21	0.04	-0.1
		(0.34, 1)
	Syllable 2 imitation	0.47	0.33	0.16	-0.12	-0.53	0.39
		(0.1, 0.8)				(-0.9, -0.15)

Bold represents significant correlations. Values in parentheses indicate the 95% confidence interval.

Table 2 .

 2 Varimax rotated factor loadings for the language and music variables, using the option "Blank" (<I0.40I).

	Factor 1	Factor 2
	(-3.16)	(-1.71)

Min = minimum; max = maximum.

Funding

This research was supported by grants ANR-16-CONV-0002 (ILCB), ANR-11-LABX-0036 (BLRI), ANR-11-IDEX-0001-02 (A*MIDEX), and ANR 16-0012-01 (RALP) to D.S.

Utterances containing a trisyllabic word with a stress on the first syllable Utterances containing a trisyllabic word with a stress on the second syllable They spotted the criminal yesterday.

I went to the casino yesterday. Claire's going to the festival today.

They've felt some attraction lately. Lisa will change the battery tomorrow.

Justin gave some perspective yesterday. They look for happiness everyday.

They spoke to the detective yesterday. Mary will open a gallery next year.

We will create a new republic soon. James spoke to the citizen last night.

We have an appointment tomorrow. They like the atmosphere very much.

Sara hired a new assistant last month. Ken will open a factory next month.

We like potatoes a lot. Peter will write an article tomorrow.

They lost their objective in the end. They had an accident last year.

Jason's felt some improvement lately. We've found a new element lately.

They got a new apartment last month. They will take some holiday next month.

They chose the other collection yesterday. They talked about politics yesterday.

They hired a new director last month. We did the exercise last night.

They had a good impression after all. Claire will go to Africa next year.

Laura found a solution last week. Mike's changed his attitude lately.

They hired a new professor last year. They went to the hospital yesterday.

They will buy a new computer next month. They met the minister yesterday.

They started the production last year. They changed their policy last year.

We will elect a new committee next month. They met with the government last month.

They will call the inspector tomorrow.

"Fillers"

Mike will visit us tomorrow. They went to Japan last summer. Janet will bring salmon tomorrow.

She will start a new job next month. Ken never reads his mails.

They can come for dinner tonight. Jerry went to the movies last night.

I received the pictures yesterday. He used to play the drums in this band last year.

Jane traveled with her friend to India.


Kaoruko Onishi

Kunimasa Matsuda

Kazuyoshi Miyagawa

Influence of Hydrophilic and Hydrophobic Coating on Hydrofoil Performance

Keywords:

Tidal power turbines take advantage of tidal energy to generate renewable hydropower. Since the tidal turbines are fixed in the ocean, it is common to paint the blade and the structure of tidal energy generator with antifouling coating to prevent marine organisms from attaching to them. Therefore, it is important to predict the influence of the coatings on the tidal turbine's performance. In this paper, hydrophilic and hydrophobic coatings which are known to be useful in antifouling were studied from the perspective of flow field and cavitation. Cavitation was visualized with a high-speed video camera and the cavitation characteristics of blades painted with hydrophilic or hydrophobic coatings were compared. With this visualization, it was possible to observe that the hydrophilic foil and hydrophobic foil had distinctive characteristics in cavitation inception and growth. Moreover, the reliability of both coatings was evaluated in order to discuss whether these coatings were useful for long. Immersion tests were carried out to evaluate the deterioration of the coatings in pure water. In addition, magnetostriction vibratory tests were carried out to evaluate the resistance to cavitation erosion of both coatings. From these investigations, a chemical transformation of the hydrophilic coating was observed. Moreover, both coatings were easily removed when they were exposed to strong cavitation impacts.

INTRODUCTION

Nowadays, renewable energy has attracted a lot of attention due to its environmental friendliness and global warming prevention capability. [START_REF] Elghali | Marine Tidal Current Electric Power Generation Technology: State of the Art and Current Status[END_REF] Tidal power generation is one of the hydropower generation methods which relies on energy from tidal current. [START_REF] Fitirdge | The impact and control of biofouling in marine aquaculture: a review[END_REF] Since the tidal turbine is fixed in the ocean environment, marine organisms can easily attach to its blades and structure. When the marine organisms stick to the blades, the turbine performs differently from what is designed. In order to prevent marine organisms from attaching to the blades or the structure, it is known to be effective to paint the turbine with antifouling coating. [START_REF] Katsuyama | Biofouling of model turbines for tidal current power generation and the effect of anti-fouling paint[END_REF] Katsuyama et al carried out model turbine tests to evaluate the effectiveness of antifouling coatings. They compared how the marine organisms attached to the model turbines which were fixed in the ocean for more than 6 months. The turbine with antifouling paint had less marine organisms attached at the surface, compared to the turbine without antifouling paint. From this experiment, it turned out that antifouling paint was useful in reducing marine organisms which adhered to the turbine. Therefore, it is important to predict the influence of the coatings on the fluid performance if the coating is to be widely used. In this paper, hydrophilic and hydrophobic coatings which are reported to be useful in antifouling were studied from the perspective of flow field and cavitation. [START_REF] Watanabe | Drag Reduction in Flow through Square and Rectangular Ducts with highly Water-Repellent Walls[END_REF] According to Watanabe et al, highly-hydrophobic coatings contribute to reducing friction in laminar flows. [START_REF] Arndt | Effects of Surface Characteristics on Hydrofoil Cavitation[END_REF][7] Williams and Arndt conducted experiments aimed at exploring the possibility of using hydrophobic surfaces to minimize unwanted vibration due to cavitation. They compared different surface properties by using different material and surface quality. Although they found that surface characteristics have a significant effect on cavitation unsteadiness, they were unable to clarify the relation between cavitation characteristics and contact angle measurement. The purpose of this paper is to clarify the influence and mechanism of cavitation which occurs on the surface of hydrophilic and hydrophobic coatings, which show distinct contact angles. Moreover, we will discuss which of the hydrophilic or hydrophobic coating has greater influence on cavitation occurrence.

In order to accomplish this, experimental investigations of single blades were performed using a cavitation tunnel. Cavitation was visualized with a high-speed video camera and pressure fluctuation measurements which accompanied the cloud cavitation collapse was synchronized to it. The pressure was measured at the downstream of the foil. These recordings enabled the comparison of cavitation inception and growth. Furthermore, in order to discuss whether these coatings can be used practically, experiments were conducted to evaluate the reliability of both coatings. Immersion tests were carried out to measure how the coatings deteriorate in the pure water. In addition, magnetostriction vibratory test were carried out to investigate the deterioration of the coatings under cavitation condition.

Test Apparatus and Method

Characteristics of the Hydrophilic and Hydrophobic Coatings

The features of hydrophilic and hydrophobic coatings used in this research are shown in Fig. 1. The coatings are heated to be cured, which enables the coatings to adhere to the base material. The hydrophilic coating is a networked polymer with surface segregated amphoteric ions. The hydrophobic coating is a fluorine polymer which is widely used as an antifouling coating.

The wettability of the coatings was evaluated using the contact angle, it can be measured by sticking bubbles to the material surface. In this research, a contact angle goniometer was used, and air bubbles were put on the surface of the coating while sunk in pure water. The contact angles of the air bubbles were about 15deg for the hydrophilic coating, and about 100deg for the hydrophobic coating. The thickness of the coatings is about 3 ~ 4 μm.

Cavitation Tunnel

The experimental investigation was conducted using a closed loop water circuit tunnel described in Fig. 2 which contained the cavitation tunnel test section described in Fig. 3. The loop also contains two pumps and tanks upstream and downstream of the test section. The flow rate can be changed with two pumps, and the system pressure can be changed with a vacuum pump which is connected to downstream the tank. The test section has a 100mm ×100mm cross section, and is made of acrylic in order to visualize the cavitation using a high-speed video camera. A single blade which has the symmetrical NACA16-021 cross section shown in Fig. 4 is put into the test section. Its maximum chord length is 40 mm and its span length is 60mm. Its blade thickness distribution has an elliptical shape as shown in Figure 4.

The inlet pressure 𝑃 (measuring point P1 in Fig. 3) was used to calculate the cavitation number and pressure fluctuations were measured using pressure transducers on the wall. The attack angle can also be changed with a turn table at the bottom of the foil. The cavitation number used here is defined as

σ = /
where 𝑃 is the vapor pressure, ρ is the density of water and u is mean velocity.

Immersion Test Apparatus

Fig. 5 shows the immersion test facility. It consists of heater and thermostatic tank which is filled with water. The pieces were put in the bottles and immersed into the pure water. The test pieces are aluminum (A7075). The time duration of the test varied between 1 hour and 432 hours. Since the velocity of chemical reactions increases at higher temperature, the water temperature was set to 20 ℃ , 50℃,70℃ in order to accelerate the tests. Re number increases, the incipient cavitation number of each cavitation form increases. At α=10deg and α=14deg, the hydrofoil painted with hydrophobic coating (H-PHO.) has a higher incipient cavitation number. The second highest is associated with the non-coating hydrofoil (N-C.), third is the hydrophilic coating (H-PHI.). However, the incipient number of each hydrofoil was almost the same when α=20deg. In addition, when α=10deg, the difference in incipient cavitation number is smaller compared to α=14deg. Similar results have been made by Arndt et al. [START_REF] Arndt | Effects of Surface Characteristics on Hydrofoil Cavitation[END_REF]

Cavitation Growth

Table1 shows comparison of cavitation growth for each hydrofoil. The snapshots of cavitation were taken using a high-speed video camera. As the attack angle increases, the cavity get larger. Comparing cavitation growth on the hydrofoils with different coating, larger cavity occurs on the surface of the hydrophobic foil for the same cavitation number. When α=10deg, tip vortex cavitation and some small sheet cavitation were observed in the hydrophilic foil. On the other hand, tip vortex cavitation and larger sheet cavitation were observed in non-coating foil and hydrophilic foil. When α=20deg, Cloud cavitation was observed for all the foils, but the difference in the cavity size between hydrophilic and hydrophobic is smaller.

Figure 8 shows the time histories of the pressure, in which fluctuations can be observed. The pressure fluctuations of each hydrofoil differ when α=10deg, σ=0.67. The hydrophobic foil has the largest amplitude and the hydrophilic foil has the smallest. On the other hand, the amplitude of each foil is almost the same when α=20deg, σ=0.70 and σ=2.26. When α=10deg σ=2.20, the amplitude is also larger with hydrophobic foil compared to hydrophilic foil, but the difference is smaller than the result from σ=0.67.

The hydrophilic and hydrophobic foils have distinctive characteristics for cavitation inception and cavitation growth. The difference in incipient cavitation number is apparent when the attack angle and mean velocity is small. The difference in cavitation growth and pressure fluctuation is apparent when the cavitation number and the attack angle are small. On the other hand, when the attack angle is large: α=20deg, the condition at which the flow separates at the leading edge, this difference disappears.

Reliability of Hydrophilic and

Hydrophobic Coating 2.2.1. Deterioration in the Pure Water Fig. 9 shows the results of immersion test. The X-axis represents time and the Y-axis represents the contact angle. When the water condition was 50 ℃ and 70 ℃ , both hydrophilic and hydrophobic coatings were present on the surface of the material even after 430 hours. On the other hand, when the water was 20 ℃ , the contact angle of hydrophilic coating got smaller as the time passed. It can be said that hydrophilic function changed to hydrophobic coating over time at 20℃. This phenomenon is not widely known or studied, since the hydrophobic coating is not supposed to exposed to water for such a long time. We can only guess that the chemical transformation occurred at the coating surface.

Cavitation Erosion

Table 2 shows pictures of air bubbles stuck to the surface of the coatings and their contact angles. Before carrying out the test, hydrophilic piece showed 12.3 deg and hydrophobic as 108.5 deg. However, as the pieces were exposed to cavitation, the contact angle changed and after 210 seconds, the angles of both hydrophilic and hydrophobic were almost same as uncoated pieces. Both coatings lost their function due to the strong cavitation impacts force which was caused by the collapse of cavitation bubbles.

The cavitation intensity was measured using a piezoelectric sensor. The measurement was done for 20 times and each was 0.4 µsecond long. Fig. 10 shows the cavitation impacts distribution at this condition. From this result, cavitation impact Ic was calculated to 2069N2-counts using the following definition.

Ic = F * counts Fig. 11 shows the mass loss of the aluminum test piece under same vibratory test condition. From this result, the mass loss of aluminum test piece was 1.33mg after 5.5 hours of cavitation exposure. From this result, the cavitation impacts of the test condition were a lot stronger than that of cavitating turbines or pumps, in which cavitation erosion is not critical for the short time like 5 hours.

Conclusion

・Incipient cavitation number is lower for the hydrophilic coating rather than for the hydrophobic coating especially when the angle of attack is small. ・The hydrophilic coating is useful to reduce growing of cavitation especially when the angle of attack is small. ・ Both hydrophilic and hydrophobic coatings did not deteriorate over time (maximum 430 hours) at hot temperature. When immersed in water at 70℃. However, hydrophobic coating transformed to function as a hydrophilic coating at 20 degrees Celsius. ・ Both hydrophilic and hydrophobic coatings lost their function only after 210 seconds of cavitation exposure. They were not strong enough against strong cavitation impact.

1 .

 1 Characteristics of the coatings

Fig. 2 .

 2 Fig. 2. Test loop

Fig. 5 .

 5 Fig. 5. Immersion test apparatus 1.4 Magnetostriction Vibratory Apparatus

Fig. 6

 6 Fig. 6 shows external view of the apparatus a schematic diagram of it. The facility and method is based on the ASTM G 32-98 standard. The disk which is made of stainless (SUS304) was set at the tip of the vibratory horn, and test pieces were fixed facing this disk. The horn vibrates inside the open double-sided beaker which is filled with water. The water was circulated to control the temperature to 25~30℃. The amplitude of the vibratory horn was 4.8μm, and the vibratory frequency was 19.5kHz. The clearance between disk and test piece was set to 1mm. The test pieces were removed, and wettability of the coating was evaluated.

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. Pressure fluctuation (Re=2.0×10 5 )

ACKNOWLEDGEMENTS

This work was done as part of the project financially supported by the NEDO (New Energy and Industrial Technology Development Organization). We would like to thank the organizers of this work.


Kenichiro Takenaka

Tomoyuki Hayashi

Kazuyoshi Miyagawa

Application of CFD analysis for static and dynamic characteristics of hydrodynamic journal bearing

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

With recent increase in the size and speed of rotational machinery, journal bearings are becoming the bearing type of choice. The choice of journal bearings is explained by the higher loading capacity and damping than ball bearings. However, in rotors supported by journal bearings, a selfexcited vibration mechanism called oil whip occurs due to the large XY coupling effect of the dynamic characteristics during high speed rotation. This mechanism cause large vibrations that can lead to shaft system damage and accidents. Furthermore, for rotational machinery in horizontal drives, unless an appropriate bearing was designed, the rotor cannot float. This causes wear and damage of the bearings and efficiency reduction. Therefore, studies to investigate the characteristics of journal bearings have been conducted actively. [START_REF] Iwatsubo | Calculations of Dynamic Force of Parallel Annular Seal with Eccentricity[END_REF][2] [START_REF] Childs | Turbomachinery Rotordynamics, Phenomena, Modeling, and Analysis[END_REF] A typical calculation method used to investigate the bearing characteristics is the Reynolds equation, but it is unsuitable when the flow is turbulent due to high-speed rotation of the journal or when the bearing shape is complicated. In recent years, therefore, analysis of the bearing characteristics using CFD is often performed. [START_REF] Guo | Application of CFD Analysis for Rotating Machinery-Part I_Hydrodynamic, Hydrostatic Bearings and Squeeze Film Damper[END_REF] [START_REF] Panday | Numerical Unsteady Analysis of Thin Film Lubricated Journal Bearing[END_REF] However, it seems that methods such as how to create a mesh of the bearing clearance are not established yet. For example, typical size of bearing clearances are in the range of 10-5 to 10-4 [m], so the number of elements becomes enormous unless the mesh aspect ratio in the circumferential and axial direction with respect to the radial direction is increased to some extent. If it is too large, the convergence may be difficult to obtain.

In this research, the static and dynamic characteristics of water lubricated journal bearings were investigated by active control of a rotor using magnetic bearings. Oscillation is performed in the single direction of each of x and y direction to investigate the bearing dynamic characteristics. In addition, CFD analysis of the oscillation of the journal is performed using the morphing method, and it is compared with the experiment result. Two patterns of movement of the journal by morphing were carried out. One is to deform the mesh in a single direction in each of the x and y direction, and the other is to deform the mesh so as to simulate journal whirling. We compare and examine these two analyses with experimental results.

METHODS

Experiment

In order to investigate the vibration characteristics of the journal bearing, an element test apparatus simulating a journal bearing was constructed. When investigating the bearing characteristics, it is necessary to measure the fluid force acting on the bearing. Generally, in a measurement of fluid force, a method of bringing load cells into contact with the bearing housing is used. But the seal system provided in the bearing which stops fluid leakage absorbs the fluid force and prevents accurate measurements. In this test apparatus, therefore, a 2-axis force sensor that can measure the fluid force is attached to the rotor so that the fluid force acting on the rotor can be measured directly.

A cross section view of this test apparatus is shown in Figure 1 and the details of the test section is shown in Figure 2. The test apparatus has a mechanism that water pressurized by the compressor from the tank to the lower part of the test bearing.

The flow passes through the bearing clearance and is drained from the upper part of the test section at atmospheric pressure.

The rotor consists of a hollow cylinder, a 2-axial force sensor, an axial force sensor mounting flange and a magnetic bearing spindle, and the rotor is rotated by a motor in the magnetic bearing system. The axial force sensor mounting flange and the spindle are hollow, so that the cable of the axial force sensor can pass through. The cable is connected to the slip ring via the adapter at the top of the spindle.

There are two radial magnetic bearings and one thrust magnetic bearing in the magnetic bearing system, which makes it possible to float the journal bearing. Moreover, by changing the input signal of the radial magnetic bearing, it is possible to eccentrically rotate the rotor while floating it, and to oscillate at arbitrary displacement / frequency in each of the x and y directions.

The test bearing's dimensions are 𝐷 = 𝜑80, 𝐿 = 80mm, 𝐶𝑟 = 0.24mm, and its rotational speed is 𝑁 = 1500rpm for all experiments. At this time, the fluid force 𝐹 𝑥 , 𝐹 𝑦 acting on the journal is measured by the axial force sensor and the influence eccentricity and oscillation was investigated. Also, the eccentricity of the journal was measured with the eddy current displacement sensors, and pressure fluctuation at the center of the bearing was measured with a pressure transducer.

CFD

CFD analysis was performed to compare the results of static and dynamic characteristics of the journal bearings with experiments. Commercial code SCRYU / Tetra V13 was used for the calculation. The analysis area is shown in Figure 3. The computational grid consists of 2.18 million hexa elements, the total number of nodes is 2.40 million points, the number of mesh layers in the thickness(radial) direction of the bearing clearance is 8 layers, and the average mesh aspect ratio in the circumferential direction and the axial direction is 1:20. As for the boundary condition, the static pressure at the inlet and the outlet was 0Pa, and non-slip wall was applied for the bearing wall surface. When static characteristics are calculated, the mesh with eccentricity = 0 is deformed by morphing:, the number of mesh layers in the bearing clearance does not change, so it can be analyzed with high accuracy in the case of high eccentricity.

The schematic view of the moving deformation mesh is shown in Figure 4.

Next, the dynamic characteristics calculation method is described. Using an eccentric mesh, the shaft wall surface is oscillates with an amplitude of 10% of the bearing clearance. The journal movement was carried out in two patterns: unidirectional oscillations in the x and y direction respectively and a whirling. Figure 5 and Figure 6 shows the outline of these movements.

In the CFD analysis, as shown in Figure 5 and6, coordinates of the static equilibrium point are set to (e, 0) in xy coordinates, and the journal rotates at a constant angular velocity ω with a small amplitude (whirl amplitude) ∆𝑒(∆𝑒/𝐶𝑟 = 0.1) around this point. At this time, the fluid force 𝐹 𝑥 (𝑡) and 𝐹 𝑦 (𝑡) are calculated.

When the direction of eccentricity of the whirling journal coincides with the direction of the static equilibrium point, that is, when the journal center moves to (𝑒 + 𝛥𝑒 , 0) and the clearance between the journal and the bearing is minimal, the values of 𝐹 𝑥 (𝑡) and 𝐹 𝑦 (𝑡) are substituted into the following expression :

{ 𝐹 𝑥 (𝑡) 𝐹 𝑦 (𝑡) } = { 𝐹 0𝑥 𝐹 0𝑦 } -[ 𝑀 𝑥𝑥 𝑀 𝑥𝑦 𝑀 𝑦𝑥 𝑀 𝑦𝑦 ] { 𝑥̈(𝑡) 𝑦̈(𝑡) } -[ 𝐶 𝑥𝑥 𝐶 𝑥𝑦 𝐶 𝑦𝑥 𝐶 𝑦𝑦 ] { 𝑥̇(𝑡) 𝑦̇(𝑡) } -[ 𝐾 𝑥𝑥 𝐾 𝑥𝑦 𝐾 𝑦𝑥 𝐾 𝑦𝑦 ] { 𝑥(𝑡) 𝑦(𝑡) } (1)
Eqs.(1) holds only when the whirling amplitude is very small, and 𝐹 0 are the steady components of the fluid force. Also, since the locus of the center of the journal is a circle, micro displacements x and y are given as follows :

{ 𝑥 = 𝛥𝑒cos𝜔𝑡 𝑦 = 𝛥𝑒sin𝜔𝑡 (2)
If the center coordinate of the journal is (e + 𝛥𝑒, 0) at 𝑡 = 0,the clearance between the journal and the bearing is the smallest at 𝑡 = 2𝑛𝜋/𝜔, and at this time, Eqs.(1) can be transformed as follows :

{ 𝑓 𝑥 | 𝑡=2𝑛𝜋/𝜔 /
Thus, when computing the characteristic value, 12 dynamic coefficients can be derived by finding 𝐹 𝑥 (𝑡) and 𝐹 𝑦 (𝑡) when the whirling frequency 𝜔 is changed while keeping 𝛥𝑒 constant.

In the case of unidirectional oscillation, the displacement of the journal center can be described as follows :

{ 𝑥 = 𝛥𝑒cos𝜔𝑡 𝑦 = 0 or { 𝑥 = 0 𝑦 = 𝛥𝑒sin𝜔𝑡 (5)
Considering 𝑡 that cos𝜔𝑡 = 0 or sin𝜔𝑡 = 0, the following equations are obtained by substituting x and y from Eqs.(5) into Equ.(1) : Therefore, by analyzing the cases of arbitrary oscillation frequency 𝜔, it is possible to calculate the dynamic coefficients with the whirling oscillation.

{ 𝑓 𝑥 | 𝑡=2𝑛𝜋/𝜔 /𝛥𝑒 𝑓 𝑦 | 𝑡=2𝑛𝜋/𝜔 /𝛥𝑒 = 𝑀 𝑥𝑥 𝜔 2 -𝐾 𝑥𝑥 𝑀 𝑦𝑥 𝜔 2 -𝐾 𝑦𝑥 (6-a)
In order to calculate the 12 dynamic coefficients (𝑀 𝑖𝑗 , 𝐶 𝑖𝑗 , 𝐾 𝑖𝑗 ) by the whirl oscillation method, it is necessary to perform transient analyses on at least three different oscillation frequencies ω. On the other hand, in the single-directional oscillation method, it is necessary to perform analyses in at least two different excitation frequencies in the excitation directions in x and y directions, respectively, a total of 4 times. However, if the anisotropy of the bearing characteristics are small, it can be expressed as : (𝑀, 𝐶, 𝐾) 𝑥𝑥 = (𝑀, 𝐶, 𝐾) 𝑦𝑦 , (𝑀, 𝐶, 𝐾) 𝑥𝑦 = -(𝑀, 𝐶, 𝐾) 𝑦𝑥 , the unidirectional oscillation method can calculate 6 × 2 dynamic coefficients with only two analyses, which is advantageous in terms of calculation cost.

RESULTS AND DISCUSSION

Test results and analysis results of static and dynamic characteristics are shown below. Since the rotational speed of the journal in the experiment was set at 1500 [rpm], and the Reynolds number of the flow in the bearing was about 1700 in the transition area, the analysis was carried out by using the laminar flow model and the SST k-ω turbulence model.

Static Characteristics

Figure 7-a shows the relation between the Sommerfeld number and eccentricity, and Figure 7-b shows the relation between Sommerfeld number and attitude angle. It was confirmed that the difference between the analysis results using SST 𝒌 -𝝎 analysis and laminar flow analysis was slight as indicated in Figure 7-a. It is assumed that this is because the Reynolds number in the bearing was in the transition region. However, it is shown that SST k-ω calculates the bearing load somewhat smaller than laminar flow analysis. Although the two analysis results are in good agreement with the experiment, a slight deviation is confirmed at the low eccentricity, and it is found that SST k-ω is relatively consistent with the experiment.

On the other hand, looking at the relationship of the eccentricity relationship with the attitude angle, although the simulations and the experimental results generally agree, the difference between the analysis result and the experiment result is slightly larger in the high Sommerfeld region. This is consistent with the region where the analytical value and the experimental value in Figure7-a are somewhat different, and it is probably due to the low accuracy of the experiment. The lower the eccentricity ratio, the smaller the bearing load 𝑾 becomes, so the measurement accuracy decreased in all likelihood. In addition, it is confirmed that SST k-ω calculates a larger attitude angle than laminar flow analysis, but the difference is within the variation of the experimental result.

Dynamic Characteristics

The analysis results of dynamic characteristics are shown. As mentioned above, two methods, the unidirectional oscillation method and the whirl oscillation method, were analyzed, and the difference between them was investigated.

For both oscillation methods, analyses were performed for three points of oscillation frequencies 𝝎/𝜴 =0.5, 1.0, and 1.5, and the values of 𝒇 𝒙 and 𝒇 𝒚 obtained by these analyseis are expressed by equations ( 3) and ( 4), ( 6) and [START_REF] Someya | Journal-Bearing Databook[END_REF]. The dynamic coefficients 𝐌 𝐢𝐣 , 𝑪 𝒊𝒋 , 𝑲 𝒊𝒋 were obtained by fitting using the least squares method as a function of 𝝎. In addition, the eccentricity ratio of the journal was varied from 0.1 to 0.8, and the effects on the dynamic coefficient was examined. In the unidirectional oscillation method, it is possible to calculate all dynamic coefficients by analyzing only two frequencies 𝝎, but those calculated using the analysis results at 𝝎 = 0.5, 1.5 were almost the same as that calculated at 3 points, therefore those are omitted. Figures 8 to 10 show the results of laminar flow analysis, and Figures 11 to 13 show the result of turbulent flow analysis. Each dynamic coefficient is normalized by the oscillation frequency 𝝎, the oscillation amplitude ∆𝒆, and the static load 𝑾.

From the analysis results, it was confirmed that the dynamic coefficient to the eccentricity using unidirectional oscillation method is consistent to as a function of the same tendency. This tendency was confirmed in both case laminar and turbulent.

First, concerning the inertia coefficients 𝑴 𝒊𝒋 , it was confirmed that almost no difference in value caused by the oscillation method was observed. The fluid inertia force decreases as the Sommerfeld number decreases (eccentricity increases) in both of them.

Next, looking at the damping coefficients 𝑪 𝒊𝒋 , it can be seen that the value of the direct terms in the x direction(𝑪 𝒙𝒙 ) for the turbulent flow condition using the whirl method is larger than using the unidirectional method for all Sommerfeld numbers.

On the other hand, with the stiffness coefficients 𝑲 𝒊𝒋 , as the Sommerfeld number decreases, the value of the xy coupling terms are different depending on the oscillation exists. Therefore, it was confirmed that the evaluation of the bearing stability is different depending on the oscillation method. In order to investigate the difference in stability in more detail, the energy loss E of the bearing was introduced and organized.

𝑬 is expressed by the equation (8), which is the energy lost by the fluid while the journal vibrates one period. A larger 𝑬 implies higher stability.

𝐸 = 𝜋∆𝑒 2 {𝜔 2 (𝐶 𝑥𝑥 + 𝐶 𝑦𝑦 ) -(𝐾 𝑥𝑦 -𝐾 𝑦𝑥 )} (8)
The results obtained by organizing using 𝑬 are shown in Fig. 14 and Fig. 15. From these graphs, the higher the eccentricity ratio is, the larger the value of 𝑬 obtained by the whirl method compared to unidirectional, and it can be confirmed that it tends to evaluated more stably.

When 𝑪 𝒊𝒊 were calculated using the Eqs. ( 3) and ( 4) from the result obtained by the whirl method, it is influenced by 𝑴 𝒊𝒋 (𝒊 ≠ 𝒋) during the curve fitting of the analysis results. As can be seen from Figure8 and 10, the value of 𝑴 𝒚𝒙 due to the Sommerfeld number is slightly larger than the value of 𝑴 𝒙𝒚 . Therefore, it is considered that the value of 𝑪 𝒙𝒙 is influenced by the value of 𝑴 𝒚𝒙 .

The above results suggest that, it is better to use the unidirectional oscillation method when evaluating the bearing stability because the direct damping coefficients 𝑪 𝒊𝒊 are not affected by other factors.

-0.5 0 0.5

Conclusion

In order to investigate a water lublicated journal bearing using experiments and CFD, the following contents were clarified.

1. Static characteristics of the bearing in the transition zone were investigated. There were almost no differences between the analysis using the laminar flow model and the turbulent model(SST k-ω). However, the bearing load was found to be somewhat closer to the experimental results in the analyses using SST k-ω. And it was found that the static characteristics could be calculated with relatively high accuracy.

2. Dynamic characteristics of the bearing were investigated using CFD. In this analysis, journal oscillations were simulated using two mesh morphing methods, and comparisions were made between them.

2-1 Non-dimentional damping coefficient 𝐶 𝑥𝑥 𝜔⊿𝑒/𝑊 calculated by the whirl oscillation method larger in all eccentricity than the unidirectional oscillation method.

2-2 The dynamic stability of the bearing was evaluated from the viewpoint of energy loss. The result showed that the analysis results using the whirl oscillation method was tend to evaluate more stable than the unidirectional oscillation method. This tendency is particularly noticeable for high eccentricity ratio, it is considered that the value of 𝑀 𝑖𝑗 at whirl oscillation could not be ignored and could not be calculated 𝐶 𝑖𝑖 completely independently from the fluid force.

Figure 3

 3 Figure 3 Computational domain Figure 4 Moving deformation mesh

Figure 1

 1 Figure 1 Journal bearing test apparatus cross-section Figure 2 Detail of the test section

  2 -𝐾 𝑥𝑦 𝑀 𝑦𝑦 𝜔 2 -𝐾 𝑦𝑦(7-b)

Figure 5

 5 Figure 5 Whirl oscillate method Figure 6 Unidirectional oscillate method

Figure 7 -

 7 Figure 7-a Bearing load

Figure 7 -

 7 Figure 7-b Attitude angle

Figure 8 Figure 9

 89 Figure 8 Added Mass Coefficient(-:uni-axial, -: whirl)

ACKNOWLEDGMENTS

This work was done as part of the project financially supported by the NEDO (New Energy and Industrial Technology Development Organization). We would like to thank the organizers of this work.


Wakana Tsuru

Shin-Ichi Tsuda

Satoshi Watanabe

Effect of Bubble Nuclei Characteristics on a Cavitating Hydrofoil: Numerical Investigation with Homogeneous Cavitation Model

Keywords:

The aim of this study is to clarify the effects of number density and initial diameter of bubble nuclei on global behavior of cavitating flow around an isolated Clark Y hydrofoil in numerical simulation. Measurements of the number density distributions of bubble nuclei are carried out, then the number density and averaged diameter of bubble nuclei are calculated from the measurement results and are used for the parameters of cavitation models in the numerical analysis. The numerical results are compared with those obtained with defaulted values provided by a solver. As a result, it is confirmed that the number density of bubble nuclei affects time-averaged pressure distribution on the blade surface, and the fluctuation of the lift coefficient and the cavitation area. In addition, the location of the cavity leading edge in the case of using the parameters obtained by the measurements moves apparently downstream, resulting in a better agreement with actual one.

INTRODUCTION

In development stage of fluid machinery, numerical simulations for cavitating flow by CFD (Computational Fluid Dynamics) are widely performed. In the cavitation simulation, since homogeneous model has light calculation load and high practicability, it is widely applied in the field of industry. However, the models often provide false results even for simple flow around single hydrofoil [START_REF] Kato | Industry-university collaborative project on numerical predictions of cavitating flows in hydraulic machinery -part I: Benchmark test on cavitating hydrofoils[END_REF], and they cannot predict quantitatively sheet cavitation inception from free nuclei due to their nature of homogeneous flow assumption. In popular cavitation models [START_REF] Kunz | A Preconditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Prediction[END_REF][START_REF] Merkle | Computational Modeling of the Dynamics of Sheet cavitation[END_REF][START_REF] Zwart | A Two-Phase Flow Model for Predicting Cavitation Dynamics[END_REF][START_REF] Schnerr | Physical and Numerical Modeling of Unsteady Cavitation Dynamics[END_REF], model parameters such as number and radius of bubble nuclei and initial void fraction are used as tuning parameters to predict focused flow field with acceptable accuracy, whereas considerable gaps sometimes exist between the well-tuned values and actual ones. On the other hand, it has been confirmed through an experiment that behavior of sheet cavitation in growth stage as well as its inception is strongly affected by the number density distribution of bubble nuclei. Therefore, while it is hard to predict the inception of sheet cavitation, it is valuable to investigate the effects of the parameters regarding nuclei also by homogeneous approach.

The aim of this study is to evaluate the effects of the parameters, the number and the initial diameter of bubble nuclei, on global behavior of cavitating flow around an isolated Clark Y hydrofoil in numerical simulation. To obtain the number and the initial diameter of bubble nuclei, experimental measurements of the number density distributions of bubble nuclei are conducted in our cavitation tunnel. Then the numerical simulations are carried out using homogeneous cavitation model with different pairs of the number and the radius of bubble nuclei; one of which is often used as default values in a popular cavitation model, and another is obtained values by the measurements to evaluate the effects of such parameters on global behavior of cavitating flow around a hydrofoil.

METHODS

Simulation methods

Two-and three-dimensional simulations are conducted for cavitating flow around a two-dimensional Clark Y-11.7% hydrofoil which has 100mm of chord length, c, as shown in Fig. 1. In three-dimensional simulations, the half of span (full span length, s, is 81 mm) is set assuming the flow symmetry. The analytical area is shown in Fig. 2. The inlet and outlet boundaries are located at 5c far from the center of the hydrofoil. The height of the area is 2c, and the angle of attack of the hydrofoil is 8 degrees. The number of grid nodes is 230,000 for two-dimensional calculation or 3,800,000 for three-dimensional calculation. The vapor-liquid two-phase flow is treated as a homogeneous flow in which liquid-vapor mixture density is variable depending on void fraction but whose momentum is defined by an incompressible Navier-Stokes equation. The continuous equation, the momentum equation for the homogeneous flow, and the transport equation of vapor volume fraction are as follows, respectively, which are solved by ANSYS CFX 16.2,

  0 m mm u t                                 , (1)
      T mm mmm mm m u uu p u u t                 , ( 2
)
Effect of Bubble Nuclei Characteristics on a Cavitating Hydrofoil: Numerical Investigation with Homogeneous Cavitation Model -2

    vv vvm v uS t         u S   u S   u S     u S   u S         u S   , (3)
where m u , m=l(1-v)+vv, l, v, p, m and v are the velocity for mixture, the densities of mixture, liquid and vapor phases, the pressure, the mixture viscosity and the vapor phase volume fraction, respectively. For the source term of transport equation of vapor volume fraction in twodimensional simulation, Sv, two cavitation models, Schnerr-Sauer model (SS) [START_REF] Schnerr | Physical and Numerical Modeling of Unsteady Cavitation Dynamics[END_REF] and Bubble-Droplet1 Viscosity Filtering model (BD1VF) developed by Yamamoto et al. [START_REF] Yamamoto | A Simple Cavitation Model for Unsteady Simuiation and its Application to Cavitating FIow in Two-DimensionaI Convergent-Divergent Nozzle[END_REF], are employed, while only BD1VF is employed in threedimensional simulation.

In SS the continuous phase is liquid, and the source term is expressed by

  2 3 sign( ) 1 3 v vl v v vv m l pp S p pC r         . ( 4
)
This term has been derived theoretically from the bubble dynamics with as an adjusting parameter C which should be the order of unity. In this study, C=1 is set. BD1VF switches continuous phase depending on local void fraction. For the low void fraction region (v<0.5), the continuous phase is liquid, and mass source term is based on bubble dynamics as SS, while, for the high void fraction region (v>0.5), the continuous phase is vapor, and the source term is derived from Schrage's mass flux based on evaporation and condensation as Eq. ( 5),

    31 2 v v v drop g k S pp r RT      , (5)
where Rg, T and k are the gas constant, tempareture and evaporation-condensation coefficient ( = 0.5), respectively. The radii of bubble and droplet, r and rdrop, in Eqs. ( 4) and ( 5) are calculated from the expression of vapor volume fraction,     ,with the number of bubble nuclei per unit liquid volume, nnuc, as follows,

  4 1 3 v nuc v r n      , ( 6
)
1 4 3 v drop nuc v r n      . ( 7
)
To evaluate the effects of the parameters regarding bubble nuclei, two pairs of nnuc, and the initial diameter of nuclei, dnuc, are used. One is often used as default values for SS in which nnuc and dnuc are 1.6x10 13 m -3 and 2.0x10 -6 m, respectively. In addition, another pair obtained by the measurement of the number density distribution of bubble nuclei is also used, which will be described later. Moreover, to improve reproducibility of cavitation unsteadiness, BD1VF cuts the eddy viscosity of the mixture using hyperbolic tangent to have gradual change in eddy viscosity as shown in Fig. 3. For turbulence model, k- SST model is employed for the two-dimensional simulation. As shown later, it gives us better agreement of pressure distribution in noncavitating condition with experiment. However, in threedimensional simulation, the model is found to overestimate the corner separations at the roots of the blade, resulting in the deviated pressure distribution. Thus, we have decided to use k- model instead. We do not know why k- model gives better result with less corner separations, but we have taken the reproduction of flow field for our priority. The time step is set to 1x10 -4 s. The boundary conditions are set as shown in Table 1. The cavitation number, , the lift coefficient, CL, the drag coefficient, CD, and the pressure coefficient, Cp, are defined by the area-averaged pressure at 200mm upstream of the center of the blade, pref, and the velocity at 200mm upstream of the center of the blade, Uin, as follows,

2 0.5 ref v l in pp U     , ( 8
) 2 0.5 L L l in F C cs U   , (9) 2 0.5
D D l in F C cs U   , ( 10
) 2 0.5 ref p l in pp C U    , ( 11
)
where l, pv, FL and FD are the liquid density, the saturation pressure, the lift force and the drag force, respectively.

Bubble nuclei parameters

The cavitation tunnel used in this observation consists of a tank, a test section and a pump. The test section shown in Fig. Figure 5 shows the measured number density distributions of bubble nuclei, N(r), where r is radius of nuclei.

It can be seen that the number density is increased at =0.51.

Since the cavitation tunnel is a closed loop one, bubbles generated by cavitation circulate, and the number density starts to increase at around =0.5 which is close to the saturation condition of DO (DOsat=0.4). To determine the parameter of nnuc and dnuc for this simulation, the approximation formula for N(r) is from the number density distributions as shown in Fig. 5 and is expressed as follows, log ( ) 3.5 3log Nr r   .

(12) The pressure coefficient at the suction peak on the blade surface, Cpmin, is found to be around -3.5 from numerical simulation of single phase flow. The radius of nuclei, rref, which quasi-statically grows to the critical radius at Cpmin with the inlet pressure, pref, is calculated by Laplace formula. Then nnuc and dnuc are obtained as follows,

  . ( 14
)
In this simulation, the inlet pressure at inception (=1.8) is used for pref, and as a result, 3.3x10 -6 m for dnuc,and 2.4x10 8 m -3 for nnuc are obtained. The order of the obtained dnuc is similar to that of the default value, whereas the order of nnuc is much smaller than the default value by 5. These obtained and default pairs of nnuc and dnuc are applied to the simulations, and the results are compared.

RESULTS AND DISCUSSION

The pressure distributions on the blade suction surface which are time-averaged for 0.5s are shown in Fig. 6 for the twodimensional simulations with (a) SS and (b) BD1VF, and for (c) the three-dimensional simulation with BD1VF. The timeaveraged pressure distribution obtained by experiments (Matsunari et al. [START_REF] Matsunari | Experimental/Numerical Study on Cavitating Flow around Clark Y 11.7% Hydrofoil[END_REF]) is also plotted in Fig. 6. Time-averaged lift coefficient, CL, are shown in Fig. 7 for the three-dimensional simulation and the two-dimensional simulations with SS and BD1VF. In the non-cavitation condition (= 3.5), the pressure distributions of the simulations well agree with the experimental data. In CL in non-cavitation condition shown in Fig. 7, although there is a gap between the simulation results and the experimental data approximately 0.3, CL in the noncavitation condition for the three-dimensional simulation is slightly smaller than that for the two-dimensional simulation. This difference is caused by the end wall effects in threedimensional simulation. In the discrepancy of approximately 0.3 between experiment and simulations, it seems to be caused by asymmetry of the flow due to existence of tip clearance in the experiment; the tip clearance should be arranged for measurement of CL, which produces bending moment and displaces the working point of the lift force from the center of the blade in the spanwise direction. However, CL is measured under the assumption of asymmetric flow and the neglect of the bending moment.

In the cavitating condition (= 1.8) of the two-dimensional simulation with SS, the pressure in the region covered by cavity is almost equal to the vapor pressure for the both cases of large and small numbers of nuclei. However, the vapor pressure region in the case of the small number of nuclei is larger. In the two-dimensional simulation results with BD1VF, the pressure in the cavitating region is again almost equal to the vapor pressure. However, compared with SS simulation, the pressure recovery behind the cavity region is gradual, and it is more gradual with the large number of nuclei than that with the small number of nuclei. The reason for this is probably due to the unsteadiness of cavitation. In the simulation with BD1VF, the fluctuation of the cavity length is more remarkable as will be seen later. The time averaged pressure near the fluctuating cavity trailing edge takes larger value than vapor pressure, resulting in the above-mentioned gradual pressure recovery near the cavity trailing edge.

In three-dimensional simulation, the pressure inside the cavity region is almost constant at the vapor pressure as it is in the two-dimensional simulation with BD1VF. However, even with BD1VF model, the pressure recovery behind the cavity is rather steep, indicating that the unsteadiness of cavitation is weaker than that in two-dimensional simulation. The cavity length (area of vapor pressure) is almost similar in the cases with the large and small numbers of nuclei. However, looking at the leading edge area more closely, the suction pressure peak can be recognized especially in the case with the small number of nuclei. From Eqs. ( 4) and ( 5), since the vaporization speed for the small number of nuclei is slower than that for the large number of nuclei and the nuclei around the suction peak are in the growth stage, then the pressure around the suction peak can be kept lower than the vapor pressure.

The instantaneous cavity shapes of the two-dimensional simulations in the case of =1.8 are shown for SS in Fig. 8 and for BD1VF in Fig. 9. The cavity is visualized by the numerical elements colored with the local void fraction larger than 0.1. The corresponding total void volume and CL fluctuations are shown in Fig. 10. The total void volume is calculated by volume integral of v over the whole computational volume. As shown in Fig. 8, cloud cavities are not observed in SS, while the total void volume and CL fluctuate periodically. The frequencies of the void volume and CL fluctuations are close to each other, while the amplitude is apparently larger in the large number of nuclei than that in the small number of nuclei. Therefore, it is suggested that the frequencies are related to fluctuation of cavity length, and the amplitudes are to the magnitude of cavity volume fluctuation. On the other hand, in BD1VF simulation in Fig. 9, cloud cavities are observed in the both large and small numbers of nuclei. In Fig. 10(b) for the large number of nuclei, two frequency components can be recognized. One is similar to the frequency of elongating and shrinking motion observed in SS simulation, and another is high frequency component related to cloud cavity shedding. In the small number of nuclei, the void volume and CL fluctuate mainly in response to cloud cavity shedding. In the small number of nuclei, the cavity volume is smaller than that of the large number of nuclei, and therefore the component of the fluctuation of cavity length is unremarkable.

The instantaneous cavity shapes and CL fluctuations for the three-dimensional simulation are shown in Figs. 11 and12, respectively. The cavitation number is = 1.9. In the three-dimensional simulation, unsteadiness of cavitation is weaker than that in the two-dimensional simulation. The size of cavitation region looks similar between the large and small (a) SS model -6 , =3.5 N nuc =1.6*10 13 , d nuc =2.0*10 -6 , =1.8 N nuc =2.4*10 8 , d nuc =3.3*10 -6 , =1.8 numbers of nuclei, but the actual time-averaged volumes of cavity are different; 1.5x10 -6 m 3 in the large number of nuclei while 9.4x10 -7 m 3 in the small number of nuclei. In the both cases with the large and small numbers of nuclei, the threedimensionality of the cavity shape can be seen near the side wall. Since the void volume in the large number of nuclei is larger than that in the small number of nuclei, the amplitude of CL fluctuation in the large number of nuclei is larger. In the both large and small numbers of nuclei, the location of the cavity trailing edge are almost the same. However, the cavity leading edge for the small number of nuclei is located relatively downstream. As already mentioned, since the vaporization speed for the small number of nuclei is slower than that for the large number of nuclei, nuclei are still in a growth stage around the suction peak. In reality, the sheet cavity leading edge is not located at the suction peak but is often located near a separation point or a re-attachment point downstream of the suction peak where free nuclei attach to the wall and grow to sheet cavity [START_REF] Rijsbergen | A Review of Sheet Cavitation Inception Mechanisms[END_REF]. This indicates that the inception of sheet cavitation involves some heterogeneous processes. In the simulations based on the homogeneous model, nuclei exist everywhere, and they can grow at any locations with the low pressure below the vapor pressure. Even with this difference, the location of cavity leading edge in the simulation using the number of nuclei obtained by actual number density distribution of nuclei is closer to that in the actual sheet cavitation.

0 0.2 0.4 0.6 0.8 1 -1 0 1 2 3 4 x/c Pressure coefficient, -Cp Experiment, =3.0 Experiment, =1.9 3D BD1VF nuc =2.4*

CONCLUSION

The two different pairs of parameter regarding bubble nuclei are applied to the homogeneous flow simulation, and the effects of characteristics of bubble nuclei on cavitating flow are investigated. The frequencies and amplitudes of fluctuations of the lift coefficient and cavity volume are affected by the number of bubble nuclei. Although the actual inception of sheet cavitation involves some heterogeneous process, the cavity leading edge and the pressure distribution around suction peak are well simulated by the present homogeneous simulation using the number of nuclei obtained by experimental measurement. This seems to indicate the possibility of the further improvement of homogeneous cavitation model in prediction accuracy.

Nomenclature

Figure 1 .Figure 2 .

 12 Figure 1. Clark Y-11.7% hydrofoil

Figure 3 .

 3 Figure 3. Viscosity Filtering

Figure 4 .

 4 Figure 4. Test section

Figure 5 .

 5 Figure 5. Number density distribution of bubble nuclei

- 6 , 5 Figure 9 .Figure 8 .

 6598 Figure 9. Instantaneous void fraction distributions in the two-dimensional simulation with BD1VF (=1.8) (b) nnuc=2.4×10 8 m -3 , dnuc=3.3×10 -6 m (a) nnuc=1.6×10 13 m -3 , dnuc=2.0×10 -6 m

tFigure 12 .Figure 11 .

 1211 Figure12. Lift coefficients fluctuation in the three-dimensional simulation with BD1VF (=1.9)

Table 1 .

 1 Boundary conditions is 0.5mm of a tip clearance, which is not considered in the numerical simulation. The test section pressure, pref, is measured by a pressure transducer and is decreased stepwise from non-cavitation condition to super cavitation condition by a vacuum pump connected to the tank. In each stage, bubble nuclei are observed by a highspeed camera (Vision Research, Phantom V4.3) located at upstream from the center of the blade as shown in Fig.4. The camera settings are summarized in Table2. The backlighted images are recorded, and the measurements of the number density distributions are conducted by image processing. In addition to this observation, FL and FD are measured under the assumption of two-dimensional flow by strain gauges attached to cantilevered beam supporting the hydrofoil with the sampling frequency and the sampling time of 1000Hz and 40s, respectively. The dissolved air in water is evaluated by the Dissolved Oxygen (DO) which is measured before and after the experiments. In this observation and measurement,DO is approximately 1.3mg/L (20%), and Uin is set to 8.2m/s.

Boundary Condition Inflow Uin = 8.2 m/s Outflow Constant Upper and lower walls Free slip wall Blade surface No slip wall Side wall (3D) No slip wall experiment, there

Table 2 .

 2 Camera

	d	nuc	2 nuc n	ref r 	rN r dr ()

settings for nuclei observations Resolution [pixels] 256 x 512 Spatial resolution [μm/pixel] 6 Depth of field [mm] 4 Frame rate [fps] 4000 Exposure time [μs] 10 Recording time [s] 1

has a rectangular cross-section and a Clark Y-11.7% hydrofoil, which are basically identical to the configuration of the three-dimensional simulation. However, in the

Cartmill & Su (1993) Lin & Brennen, LTWT Keller & Weitendorf (1976) log N(r) = -3.5-3log r

ACKNOWLEDGMENTS

This study is partly supported by JSPS KAKENHI Grant Number JP17J04154.