PhD Student, Chaouachi Mehdi
email: chaouachimehdi@hotmail.com

ORCID Sandrine Gautier

MSc Gautier Sandrine

MSc Carnot Yoann

MD Guillemot Pierrick

PhD Pincemail Joël

Moison Yann

BSc Collin Tom

PhD Groussard Carole

PhD Vincent Sophie

Spirulina supplementation prevents exercise-induced lipid peroxidation, inflammation and skeletal muscle damage in elite rugby players Running title: Effects of spirulina in athletes

Keywords: Arthrospira, oxidative stress, high-level athletes, dietary supplement, antioxidant. Myeloperoxidase OS: Oxidative stress ox-LDL: Oxidized low-density lipoprotein SOD: Superoxide dismutase TBARS: Thiobarbituric acid reactive substances YYIRT-2: Yo-Yo intermittent recovery test level 2

 This is the first study to investigate the effects of spirulina supplementation on pro/antioxidant status, inflammation, and muscle damage markers in response to exhaustive exercise in elite team-sport athletes.

 Spirulina supplementation potentially prevents exercise-induced lipid peroxidation, inflammation, skeletal muscle damage and may accelerate the recovery of these markers.

 Spirulina supplementation could be recommended for athletes who perform a high training load in order to reduce the magnitude of oxidative stress, inflammation, and muscle damage which could help to limit performance losses and accelerate recovery after training/competitions throughout the season.

Introduction

Like most team sports, rugby is an intermittent-type activity that involves a high number of repetitive high-intensity efforts interspersed with low-intensity activities or rest [START_REF] Austin | The physical demands of Super 14 rugby union[END_REF][START_REF] Cunniffe | Time course of changes in immuneoendocrine markers following an international rugby game[END_REF][START_REF] Johnston | Applied sport science of rugby league[END_REF] .

High-intensity actions are classified as high-speed runs, sprints, acceleration/decelerations, changes of direction, and collision-based activities such as tackling, static holds, scrums, rucks, and mauls [START_REF] Austin | The physical demands of Super 14 rugby union[END_REF][START_REF] Cunniffe | Time course of changes in immuneoendocrine markers following an international rugby game[END_REF][START_REF] Johnston | Applied sport science of rugby league[END_REF][START_REF] Gabbett | Physical demands of professional rugby league training and competition using microtechnology[END_REF] .

High-impact collisions received by players, in addition to eccentric damage, elicit extensive tissue trauma [START_REF] Cunniffe | Time course of changes in immuneoendocrine markers following an international rugby game[END_REF] . The intense physical demands of rugby lead to notable skeletal muscle damage after training and competition [START_REF] Mclellan | Biochemical and endocrine responses to impact and collision during elite Rugby League match play[END_REF] . This skeletal muscle damage is also associated with an inflammatory response characterized by an elevation of proinflammatory cytokines (Interleukin-6, Interleukin-1β) and other inflammatory blood markers (C-reactive protein (CRP)) in team sports players [START_REF] Cunniffe | Time course of changes in immuneoendocrine markers following an international rugby game[END_REF][START_REF] Ispirlidis | Time-course of changes in inflammatory and performance responses following a soccer game[END_REF][START_REF] Chatzinikolaou | A microcycle of inflammation following a team handball game[END_REF][START_REF] Chatzinikolaou | The microcycle of inflammation and performance changes after a basketball match[END_REF] . This inflammation is .

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caused not only by muscle mechanical damage itself but also by oxidative stress (OS) induced by exercise [START_REF] Tiidus | Radical species in inflammation and overtraining[END_REF] . Indeed, exercise increases free radical production as a consequence of both enhanced oxygen consumption, enzyme activation, and the presence of phagocytic cells activated and recruited to the site of the initial damage [START_REF] Brancaccio | Biochemical markers of muscular damage[END_REF] . The presence of free radicals may be one of the prime factors in initiating neutrophil infiltration into muscle and the subsequent inflammatory response [START_REF] Tiidus | Lack of antioxidant adaptation to short-term aerobic training in human muscle[END_REF] . This increase in free radical production has been also implicated in the oxidative damage to macromolecules [START_REF] Mylonas | Lipid peroxidation and tissue damage[END_REF] , muscle damage [START_REF] Nikolaidis | Decreased blood oxidative stress after repeated muscle-damaging exercise[END_REF] , and the apparition of overtraining in highlytrained athletes [START_REF] Tiidus | Radical species in inflammation and overtraining[END_REF][START_REF] Mckenzie | Markers of excessive exercise[END_REF] . Furthermore, exercise-induced OS, muscle damage, and inflammation have been shown to affect neuromuscular function [START_REF] Naughton | Impact-Induced Muscle Damage and Contact Sports: Etiology, Effects on Neuromuscular Function and Recovery, and the Modulating Effects of Adaptation and Recovery Strategies[END_REF] , promote muscle fatigue [START_REF] Reid | Reactive oxygen in skeletal muscle. II. Extracellular release of free radicals[END_REF][START_REF] Endoh | Effects of muscle damage induced by eccentric exercise on muscle fatigue[END_REF] , and thus impair exercise performance [START_REF] Twist | The effects of exercise-induced muscle damage on maximal intensity intermittent exercise performance[END_REF][START_REF] Vollaard | Exercise-induced oxidative stress: myths, realities and physiological relevance[END_REF][START_REF] Ferreira | Muscle-derived ROS and thiol regulation in muscle fatigue[END_REF] . This can lead to an excessive level of accumulated fatigue over the week cycle that may lead to under-performance on match day [START_REF] Tavares | Fatigue and Recovery in Rugby: A Review[END_REF] .

To prevent and reduce these exercise-related alterations, dietary supplementation may be a promising strategy. Indeed, the consumption of protein, anti-inflammatory, and antioxidant supplements in athletes has increased enormously [START_REF] Pingitore | Exercise and oxidative stress: potential effects of antioxidant dietary strategies in sports[END_REF][START_REF] Poulios | Protein-Based Supplementation to Enhance Recovery in Team Sports: What is the Evidence[END_REF] . Protein supplements are used by athletes to promote skeletal muscle anabolism and enhance recovery of muscle function and regeneration [START_REF] Pasiakos | Effects of protein supplements on muscle damage, soreness and recovery of muscle function and physical performance: a systematic review[END_REF][START_REF] Farup | Whey protein supplementation accelerates satellite cell proliferation during recovery from eccentric exercise[END_REF] . Anti-inflammatory supplements are used to reduce inflammation related to skeletal muscle damage [START_REF] Rawson | Dietary Supplements for Health, Adaptation, and Recovery in Athletes[END_REF] while antioxidants have received interest as a useful tool in preventing or reducing OS and decreasing muscle damage [START_REF] Mcginley | Does antioxidant vitamin supplementation protect against muscle damage? Sports medicine[END_REF][START_REF] Peternelj | Antioxidant supplementation during exercise training: beneficial or detrimental? Sports medicine[END_REF][START_REF] Brisswalter | Vitamin supplementation benefits in master athletes[END_REF] .

Amongst dietary supplements, spirulina (Arthrospira) is widely used by athletes. It is a blue-green microalga known for its complex chemical makeup including high protein content (50 to 70% of its dry weight), all the essential amino and fatty-acids, and most of the vitamins and minerals [START_REF] Khan | Nutritional and therapeutic potential of Spirulina[END_REF][START_REF] Sotiroudis | Health aspects of Spirulina (Arthrospira) microalga food supplement[END_REF][START_REF] Hosseini | Nutritional and medical applications of spirulina microalgae[END_REF] , which confers to spirulina numerous interesting properties and health benefits such as antioxidant, immunomodulatory, antiinflammatory, insulin-sensitizing, anti-cancer and anti-viral activities [START_REF] Hosseini | Nutritional and medical applications of spirulina microalgae[END_REF][START_REF] Kulshreshtha | Spirulina in health care management[END_REF][START_REF] Deng | Hypolipidemic, antioxidant, and antiinflammatory activities of microalgae Spirulina[END_REF][START_REF] Wu | The antioxidant, immunomodulatory, and antiinflammatory activities of Spirulina: an overview[END_REF] . The high level of phycocyanin and β-carotene confers to spirulina, antioxidant and antiinflammatory properties which could be interesting since athletes do not typically provide enough antioxidant vitamins through their diet [START_REF] Groussard | Évaluation de l'apport en vitamines antioxydantes chez des sportifs[END_REF][START_REF] Machefer | Apports et statut en vitamines antioxydantes chez des athlètes d'endurance[END_REF][START_REF] Lako | Food habits and nutritional status of Fiji rugby players[END_REF] . Considering intense periods of training and competition, and the deficient nutritional status of elite rugby players, their performance could be impaired and antioxidant system efficiency could be decreased [START_REF] Finaud | Antioxidant status and oxidative stress in professional rugby players: evolution throughout a season[END_REF] .

Literature data is still rare and controversial about spirulina effects on exercise-induced OS and skeletal muscle damage. Indeed, Lu et al. [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF] showed positive effects of spirulina supplementation in untrained college students, since plasma concentrations of malondialdehyde (MDA) and lactate dehydrogenase (LDH) were significantly lower while the activity of blood superoxide dismutase (SOD) and glutathione peroxidase (GPx) were higher. Antioxidant effects of spirulina were also confirmed in moderately trained individuals by Kalafati et al. [START_REF] Kalafati | Ergogenic and antioxidant effects of spirulina supplementation in humans[END_REF] who demonstrated that spirulina supplementation can induce a significant increase in reduced glutathione (GSH) concentration at rest and after exercise, as well as attenuate the exercise-induced increase in lipid peroxidation (TBARS). Contrary to these conclusions, Franca et al. [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] showed that spirulina supplementation was not able to alter the magnitude of skeletal muscle damage or able to reduce OS induced by high volume and intensity of training in cycling athletes with adequate nutritional status. The literature data concerning spirulina supplementation in relation with exercise and/or training in high-level athletes are very rare. Among the three studies exposed above, only one examined elite athletes [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] by measuring changes in parameters only at rest. To our knowledge, no studies have examined the effects of spirulina supplementation in elite team-sport athletes and in response to exercise although they are known for their widespread use of ergogenic aids [START_REF] Dietz | Prediction profiles for nutritional supplement use among young German elite athletes[END_REF] .

Thus, the present study aimed to examine the effects of spirulina supplementation on pro/antioxidant status, inflammation, and muscle damage markers immediately and 24 hours after exhaustive exercise in elite rugby players. We hypothesized that due to its high concentration of proteins and antioxidants, spirulina could reduce exercise-induced OS, inflammation, and skeletal muscle damage in elite rugby players and improve recovery.

Methods

The experimental protocol was approved by a local ethics committee (NOD0172, Tours,

France -protocol number 2016-A00233-48 1. The daily dosage was 5.7g (three capsules before the three daily meals), below the 10g safety dosage recommended by Fox et al. [START_REF] Fox | Spirulina: production & potential: Edisud[END_REF] . The placebo group was given visually identical capsules with isocaloric and isoproteic content composed of 70.3 % egg proteins and 29.7 % carbohydrates.

<< Please insert Table 1 here >>

Rugby players visited the laboratory five times during the study. Anthropometric measurements were taken during the introductory laboratory visit, including height, body mass (BM), and fat mass (FM). Height and BM were collected using a wall-mounted stadiometer (Seca 206, Hamburg, Germany) and electronic scale (Terraillon, Paris, France), respectively. FM percentage was estimated from the sum of skinfolds (biceps brachii, triceps brachii, subscapular, and supra-iliac) monitored with Harpenden skinfold calipers (Baty International, West Sussex, England) using the Siri's equation [START_REF] Siri | The gross composition of the body[END_REF] .

Blood samples were obtained before (W0) and after seven weeks of supplementation (W7) both at rest (T0) and in response to exhaustive exercise (immediately after (T1), and 24 hours after (T2)). The Yoyo Intermittent Recovery Test Level 2 (YYIRT-2) was used as an exhaustive exercise to induce OS, inflammation, and muscle damage.

Supplementation compliance was assessed by counting the remaining capsules in the provided bottles at the end of the study. Gastrointestinal symptoms were monitored using a questionnaire as described by Bovenschen et al. [START_REF] Bovenschen | Evaluation of a gastrointestinal symptoms questionnaire[END_REF] . The study design is illustrated in Figure 1.

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Experimental procedures: All procedures were conducted during the competitive phase of the season (October-December 2018). All participants actively participated in the rugby-specific training sessions (muscle strength, technical and tactical training) and competitions. They were fully accustomed to the testing procedures used in this research as they routinely performed this test in the club as part of their medical/training followup.

Dietary intake: Players had to record their diet for three days including one weekend day at W0 and W7 to establish that they had similar levels of macronutrient and antioxidant intake during the period of data collection. They were asked to maintain their usual dietary habits during the recording period and to be as accurate as possible in recording the amount and type of food and fluid consumed. A color photo exhibition [START_REF] Hercberg | The SU.VI.MAX Study: a randomized, placebo-controlled trial of the health effects of antioxidant vitamins and minerals[END_REF] of commonly consumed foods and their portion sizes was used to assist in estimating amounts consumed. Diet records were analyzed using the Nutrilog software (Nutrilog®

SAS, France).

Training load: Throughout the experiment, the internal training load of all participants was monitored using the session Rate of Perceived Exertion (Session-RPE) method as described by Comyns and Flanagan [START_REF] Comyns | Applications of the Session Rating of Perceived Exertion System in Professional Rugby Union[END_REF] expressed in arbitrary units (AU). 30-minutes (min) after training sessions, players were asked to rate the global intensity of the workout session using the French version CR-10 RPE scale [START_REF] Haddad | Validity and psychometric evaluation of the French version of RPE scale in young fit males when monitoring training loads[END_REF] . The Exhaustive exercise protocol YYIRT-2: Players performed the YYIRT-2 as described by Krustrup et al. [START_REF] Krustrup | The Yo-Yo IR2 test: physiological response, reliability, and application to elite soccer[END_REF] . The YYIRT-2 examines the ability to perform repeated high-intensity exercise bout with an almost maximum aerobic energy production and a high anaerobic energy contribution [START_REF] Krustrup | The Yo-Yo IR2 test: physiological response, reliability, and application to elite soccer[END_REF][START_REF] Bangsbo | The Yo-Yo intermittent recovery test: a useful tool for evaluation of physical performance in intermittent sports[END_REF] . It consists of repeated 40 m (2 x 20 m) runs between markers set 20 m apart, at a progressively increased speed starting from 13 km/h and controlled by audio bleeps from a tape recorder. Between each running bout, the participants had a 10-s of active recovery, consisting of a 10 m (2 x 5 m) walk. When the participant failed to reach the finishing line in time twice or felt unable to continue, the total distance covered (m) was recorded and represented the test result.

Blood sampling: All blood samples were collected from an antecubital vein in two different Vacutainer tubes (EDTA and dry). Players were instructed to fast for 12 h (overnight) before T0 sampling. After blood sampling, all players were served a standardised breakfast before proceeding to the YYIRT-2 test (1 hour after breakfast).

The energy supply corresponded to 18 % of the total caloric intake. The macronutrient repartition was respectively 55%, 15%, 30% for carbohydrates, proteins, and lipids. For a player of 70-90 kg, the standardised breakfast contained: 0.125 L of orange juice, a piece of sugar, 86 g of wholemeal bread, 10 g of butter, 45 g of jam, 200 g of cottage cheese, and one kiwi. At T1, blood was collected 10 min after the end of the exhaustive exercise.

At T0, T1, and T2, the EDTA vacutainer was used to evaluate F2α-isoprostanes (F2-Isop), oxidized low-density lipoprotein (ox-LDL), myeloperoxidase (MPO) in plasma; SOD, GPX, reduced and oxidized glutathione (GSH and GSSG) in whole blood; whereas the dry Vacutainer was used to evaluate CRP, Creatine kinase (CK) and LDH in plasma.

Samples preparation: For GSSG measurement, immediately after blood collection (< 10 s), 100 µL of whole blood was added to 10 µL of a scavenger, provided in the GSSG's kit, and were immediately frozen in liquid nitrogen. The other whole blood samples were placed in separated cryotubes (50 µL for GSH measurement, and 1 mL for each SOD and GPX measurement) and were immediately frozen (< 45 s) in liquid nitrogen. For the rest of the assays, blood was immediately centrifuged at 1500g for 10 min (4° C) to separate plasma (Sigma 2-16K, Osterode, Germany). For F2-Isop, 5 µL of an ethanolic solution of butylated hydroxytoluene (0.2%) was added to 1 mL of plasma to avoid oxidation.

Aliquots were immediately frozen and stored at 80° C until analysis.

Biochemical analysis: A complete blood count was performed on the UniCel ® DxH800 automated haematology analyser (Beckman Coulter, Miami, FL, USA). GSH/GSSG ratio was determined by the GSH/GSSG-412 kit (Bioxytech, Oxis International Inc., Portland, WA, USA). Enzymatic antioxidant activities of SOD and GPX were determined by the Ransod and Ransel kits (Randox, England), respectively. The ox-LDL levels were determined spectrophotometrically with a competitive enzyme-linked immunosorbent assay (ELISA kit, Immundiagnostik, Germany). Total F2-IsoP was measured by liquid chromatography-mass spectrometry (LC/MS), as described previously [START_REF] Youssef | Exercise-induced oxidative stress in overweight adolescent girls: roles of basal insulin resistance and inflammation and oxygen overconsumption[END_REF] . The inflammation marker MPO was measured using a colorimetric kit with an enzyme-linked immunosorbent assay (Elisa K 6631, Immundiagnostik, Frankfurt am Main, Germany).

CK and LDH were measured with the Cobas® 8000 Roche analyser (Roche Diagnostics, Meylan, France) according to the manufacturer's instructions. CRP was measured with latex particle-enhanced immunoturbidimetric assay using the same modular analyzer (Cobas® 8000 C701).

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Statistical analysis: Data are presented as group mean values and standard deviations (mean ± SD). After normal distribution was examined using the Shapiro-Wilk-Test, an independent samples t-test was calculated to compare descriptive variables (at W0),

internal training load (each week), and dietary intake (at W0 and W7) between-groups.

Repeated measures were compared between the two groups using 2 (Group: SPI and PLA) × 3 (Time: T0, T1, and T2) factorial (two-way) analysis of variance at W0, and W7. If a significant Group × Time interaction was detected, pairwise comparisons were performed through simple main effect Bonferroni-corrected analysis. The significance level was set at p<0.05. All analyses were performed using SPSS for Windows, Version 20 (IBM Corp: Armonk, NY, USA).

Results

Participants' characteristics and compliance: Participants' baseline characteristics are reported in Table 2. No significant differences were found between SPI and PLA treatments for any of the descriptive parameters. SPI and PLA supplementation compliance was 98.3% and 92.3%, respectively. No adverse effects were reported during or after spirulina supplementation.

<< Please insert Table 2 here >>

Internal training load, and dietary intake: Neither internal training load nor dietary intake showed significant differences between SPI and PLA treatments. Data are shown in Table 3.

<< Please insert Table 3 here >>

The effects of spirulina supplementation: Data collected at W0 for pro/antioxidant status, inflammation and muscle damage markers are not presented in this manuscript since no significant differences were found between groups. To assess spirulina's supplementation ability to prevent the increase of OS, inflammation, and skeletal muscle damage markers after exercise, we compared T0 vs T1 while the ability to accelerate return to baseline values was examined by comparing T1 vs T2 and T2 vs T0. Values at T0 did not differ between the groups for any of the studied parameters.

Pro/antioxidant status before, and after exhaustive exercise: For Pro/antioxidant status markers (SOD, GPX, GSH/GSSG ratio, ox-LDL, and F2-Isop), no significant main effect of "Time" or "Group" and no significant "Group × Time" interaction was detected for SOD activity and ox-LDL level after spirulina supplementation. A significant main effect of "Time" was observed for only GPX activity and GSH/GSSG ratio. Indeed, GPX .

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activity significantly decreased (p<0.05) in both groups at T2 while GSH/GSSG ratio significantly increased (p<0.01) at T1 and significantly decreased (p<0.01) at T2 in both groups. However, no significant main effect of "Group" or "Group × Time" interaction was detected for these parameters. Concerning F2-Isop levels, there was a significant main effect of "Time" and a trend towards significance (p=0.077) of "Group × Time" interaction. Pairwise comparisons showed a significant F2-Isop increase (17.8%) immediately after exercise only in PLA group. No significant change was observed at T2 except a decrease with a trend towards significance (p=0.074) in PLA. Moreover, the F2-Isop level at T2 was significantly lower than at T0 in SPI group (p<0.05). The evolution of F2-Isop level at T0, T1, and T2 are illustrated in Figure 2A.

Inflammation markers before, and after exhaustive exercise: For inflammation markers (MPO and CRP), no significant main effect of "Time" or "Group" and no significant "Group × Time" interaction were detected for MPO after spirulina supplementation.

However, there was a significant main effect of "Time" and "Group × Time" interaction for CRP. As shown in Figure 2B, plasma CRP increased significantly at T1 in PLA (10.4%) with no change in SPI group. Moreover, spirulina supplementation accelerated the return to baseline values given that plasma CRP decreased significantly at T2 in SPI group (-27.5%) with a lower level compared to T0 (p<0.01) whereas, no significant change was observed in PLA compared with T1.

<< Please insert Figure 2 here >>

Skeletal muscle damage markers before, and after exhaustive exercise: For muscle damage markers (LDH and CK), no significant main effect of "Group" and no significant "Group × Time" interaction was detected for LDH after spirulina supplementation.

However, there was a significant main effect of "Time" with plasma LDH increasing significantly at T1 (p<0.01) and then significantly decreasing at T2 (p<0.01) in both groups without significant difference between groups. For CK, a significant main effect of "Time" and "Group × Time" interaction was found. As shown in Figure 2C, plasma CK increased significantly at T1 in PLA (15.5%) with no change in SPI group.

Moreover, spirulina supplementation accelerates the return to baseline values given that plasma CK decreased significantly at T2 in SPI (-34.2%) with a lower level compared to T0 (p<0.001) whereas, no significant change was observed in PLA compared with T1.

Table 4 shows the values for parameters that did not show any significant "Group × Time" interaction at T0, T1, and T2.

<< Please insert Table 4 here >>

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Accepted Article Discussion

This study is the first to examine the effects of spirulina supplementation on exerciseinduced OS, inflammation, and skeletal muscle damage markers in elite athletes. Based on the markers used in this study, our results report that spirulina supplementation potentially prevents exercise-induced lipid peroxidation, inflammation, skeletal muscle damage and may accelerate the recovery of some of these markers (F2-Isop, CRP, and CK).

Whereas very few studies evaluated spirulina supplementation in athletes, numerous publications examined the properties of this microalga in the prevention and treatment of various diseases. These studies have shown that its unusually high nutritional values confer to spirulina diverse beneficial health effects such as antioxidant, antiinflammatory, insulin-sensitizing, immunomodulatory, anti-cancer, and anti-viral activities [START_REF] Hosseini | Nutritional and medical applications of spirulina microalgae[END_REF][START_REF] Kulshreshtha | Spirulina in health care management[END_REF][START_REF] Deng | Hypolipidemic, antioxidant, and antiinflammatory activities of microalgae Spirulina[END_REF][START_REF] Wu | The antioxidant, immunomodulatory, and antiinflammatory activities of Spirulina: an overview[END_REF] . All these properties shown in patients could be interesting for team sports players considering OS, inflammation, and skeletal muscle damage that may be induced by intense training and competition [START_REF] Cunniffe | Time course of changes in immuneoendocrine markers following an international rugby game[END_REF][START_REF] Mclellan | Biochemical and endocrine responses to impact and collision during elite Rugby League match play[END_REF][START_REF] Ispirlidis | Time-course of changes in inflammatory and performance responses following a soccer game[END_REF][START_REF] Powers | Exercise-induced oxidative stress: cellular mechanisms and impact on muscle force production[END_REF] .

Intense exercise, training and competition induce OS, inflammation, and skeletal muscle damage which could impair exercise performance by affecting neuromuscular function (15) , decreasing muscle strength [START_REF] Peake | Muscle damage and inflammation during recovery from exercise[END_REF][START_REF] Byrne | The effect of exercise-induced muscle damage on isometric and dynamic knee extensor strength and vertical jump performance[END_REF] , and promoting muscle fatigue [START_REF] Reid | Reactive oxygen in skeletal muscle. II. Extracellular release of free radicals[END_REF][START_REF] Endoh | Effects of muscle damage induced by eccentric exercise on muscle fatigue[END_REF] . Since spirulina is known for its high protein content including all the essential amino acids (30- 32) , it could be of great interest for athletes especially at high level. Moreover, intense training has been shown to decrease the availability of essential amino acids, which may slow the rate of tissue repair and growth [START_REF] Kreider | Dietary Supplements and the Promotion of Muscle Growth with Resistance Exercise[END_REF] . Athletes need to ingest enough high-quality protein to maintain essential amino acid availability [START_REF] Kerksick | The effects of protein and amino acid supplementation on performance and training adaptations during ten . Accepted Article weeks of resistance training[END_REF] , to reduce skeletal muscle damage [START_REF] Nosaka | Effects of amino acid supplementation on muscle soreness and damage[END_REF][START_REF] Coombes | Effects of branched-chain amino acid supplementation on serum creatine kinase and lactate dehydrogenase after prolonged exercise[END_REF] , and to enhance recovery [START_REF] Pasiakos | Effects of protein supplements on muscle damage, soreness and recovery of muscle function and physical performance: a systematic review[END_REF] . Spirulina's composition is interesting for this population because in addition to amino acids it contains many compounds with antioxidant and anti-inflammatory virtues favorable to the process of muscle regeneration and skeletal muscle damage reduction [START_REF] Michailidis | Thiol-based antioxidant supplementation alters human skeletal muscle signaling and attenuates its inflammatory response and recovery after intense eccentric exercise[END_REF] .

To the best of our knowledge, our study is the first that used the YYIRT-2 as an exhaustive exercise to induce OS, inflammation, and skeletal muscle damage. We chose the YYIRT-2 test because it evaluates the abilities to perform intense intermittent exercise (with a large anaerobic component in combination with a significant aerobic contribution) and to recover quickly, which are essential components during the intense periods of Team-sports games [START_REF] Bangsbo | The Yo-Yo intermittent recovery test: a useful tool for evaluation of physical performance in intermittent sports[END_REF] . The YYIRT-2 is an effective testing tool able to induce OS, inflammation, and skeletal muscle damage in elite athletes because our results showed an increase in F2-Isop, CRP, and CK levels for all participants immediately after exercise at W0. Indeed, top-level athletes are well accustomed to this type of solicitation during their habitual training and competition.

Concerning the antioxidant effects of spirulina, we are the first to measure its effects on ox-LDL and F2-Isop as markers of exercise-induced lipid peroxidation in humans. F2-Isop is considered as the gold standard biomarker of lipid peroxidation [START_REF] Miu | Isoprostanes-the Gold Standard of the Oxidative Stress Assessment in Vivo[END_REF] while ox-LDL is believed to be one of the major factors involved in the pathogenesis of atherosclerosis (62) . Our results did not demonstrate any significant change for ox-LDL (Table 4) but showed a significant main effect of "Time" and a trend towards significance of "Group × Time" interaction with F2-Isop significantly increased (p<0.05) immediately after exercise only in PLA group. Moreover, the F2-Isop level at 24h after exercise was significantly lower (p<0.05) than at rest in SPI group (Figure 2A). These findings are in accordance with other studies [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF][START_REF] Kalafati | Ergogenic and antioxidant effects of spirulina supplementation in humans[END_REF] suggesting that spirulina supplementation prevents exercise-induced lipid peroxidation. In fact, Lu et al. [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF] showed that plasma concentrations of MDA were significantly decreased after spirulina supplementation.

They reported a less increase in the placebo group without significant difference between groups. The same results were reported in animal studies [START_REF] Luo | Spirulina platensis extract supplementation attenuates oxidative stress in acute exhaustive exercise: A pilot study[END_REF][START_REF] Meihua | Protective Effect of Spirulina against Cell DNA Damage and Oxidative Stress Induced by Exhaustive Exercise[END_REF] showing a less increase in MDA level after spirulina supplementation showing its ability to inhibit lipid peroxidation. In the same context, although there has been some general concern about the validity of the TBARS assay because of its lack of specificity concerning lipid peroxidation [START_REF] Janero | Malondialdehyde and thiobarbituric acid-reactivity as diagnostic indices of lipid peroxidation and peroxidative tissue injury[END_REF] , Kalafati et al. [START_REF] Kalafati | Ergogenic and antioxidant effects of spirulina supplementation in humans[END_REF] detected a significant interaction in moderately trained men, with TBARS levels increasing after exercise in the placebo group but not after spirulina supplementation. In contrast, Franca et al. [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] reported no effects of spirulina supplementation on the MDA level in elite cycling athletes subjected to high volume and intensity of training. It should be noted that the measurements in this study [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] were carried out at rest and not after exhaustive exercise. For the other markers of pro/antioxidant status (GSH/GSSG, SOD, and GPX), no significant between-group difference was found in our study (Table 4) as already described by Kalafati et al. [START_REF] Kalafati | Ergogenic and antioxidant effects of spirulina supplementation in humans[END_REF] in moderately trained, and by Franca et al. [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] in highly trained participants but only at rest. Contrary results were found by Lu et al. [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF] showing a significantly higher post-exercise GPX level after spirulina supplementation in untrained students. Similarly to Lu et al. [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF] , animal studies showed a significant increase in post-exercise GPX [START_REF] Luo | Spirulina platensis extract supplementation attenuates oxidative stress in acute exhaustive exercise: A pilot study[END_REF] , and SOD [START_REF] Luo | Spirulina platensis extract supplementation attenuates oxidative stress in acute exhaustive exercise: A pilot study[END_REF][START_REF] Meihua | Protective Effect of Spirulina against Cell DNA Damage and Oxidative Stress Induced by Exhaustive Exercise[END_REF] activity after spirulina supplementation compared to placebo recommending this microalga to enhance the antioxidant status and to inhibit OS induced by acute .

Accepted Article

exhaustive exercise. These contradictory results could be attributed to the different participants' training levels since it is well known that antioxidant enzymes are upregulated by training [START_REF] Gomez-Cabrera | Decreasing xanthine oxidase-mediated oxidative stress prevents useful cellular adaptations to exercise in rats[END_REF][START_REF] Gomez-Cabrera | Moderate exercise is an antioxidant: upregulation of antioxidant genes by training[END_REF] thus making the antioxidant effect of spirulina more difficult to detect in this population.

For inflammation markers, our results showed no significant change for MPO after spirulina supplementation (Table 4). In contrast, plasma CRP increased significantly (p<0.05) in PLA immediately after exercise without any change in SPI group. Moreover, no significant change was observed in PLA 24h after exercise whereas plasma CRP decreased significantly (p<0.01) in SPI with a lower level compared to rest value (Figure 2B). These results reflect the effect of spirulina supplementation to prevent inflammation induced by exhaustive exercise and to accelerate the return to baseline values. To our knowledge, there are no studies that examined the effects of spirulina supplementation on exercise-induced inflammation markers in humans. Anyway, our findings are consistent with the results of a recent animal study investigating spirulina supplementation on inflammation promoted by strength training [START_REF] Brito | Spirulina platensis prevents oxidative stress and inflammation promoted by strength training in rats: doseresponse relation study[END_REF] , in which the concentration of CRP decreased significantly after eight weeks of supplementation.

Concerning muscle damage markers, our findings did not show any significant difference between groups in LDH level (Table 4). However, plasma CK increased significantly (p<0.001) only in PLA immediately after exercise without any change in SPI group.

Moreover, no significant change was observed in PLA 24h after exercise whereas plasma CK decreased significantly (p<0.001) in SPI with a lower level compared to rest value (Figure 2C). These results reflect the effect of spirulina supplementation preventing exercise-induced muscle damage and accelerating damage recovery. The effects of spirulina supplementation on exercise-induced muscle damage in humans are not well documented in the scientific literature. Lu et al. [START_REF] Lu | Preventive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF] reported a decrease in CK level in the SPI group while it increased in the placebo group but without significant difference between groups. Kalafati et al. [START_REF] Kalafati | Ergogenic and antioxidant effects of spirulina supplementation in humans[END_REF] reported an increase of CK activity 24 and 48 h after exercise in both groups without significant difference while Franca et al. [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] did not show any significant change at rest for LDH or CK levels in high-level cycling athletes. Franca et al. [START_REF] Franca | Spirulina does not decrease muscle damage nor oxdidative stress in cycling athletes with adequate nutritional status[END_REF] concluded that the administration of Spirulina does not interfere in the magnitude of skeletal muscle damage in cycling athletes that practice intense training, and explained their results by the fact that athletes had an adequate nutritional status regarding macronutrients.

.

Accepted Article

Spirulina contains several active ingredients, notably phycocyanin and β-carotene that have potent antioxidant and anti-inflammatory activities. As antioxidant mechanisms of action described in the literature [START_REF] Deng | Hypolipidemic, antioxidant, and antiinflammatory activities of microalgae Spirulina[END_REF][START_REF] Wu | The antioxidant, immunomodulatory, and antiinflammatory activities of Spirulina: an overview[END_REF] , phycocyanin has the ability to scavenge free radicals, decrease nitrite production, suppress inducible nitric oxide synthase (iNOS) expression, and inhibit lipid peroxidation. In the same way, it was found that β-carotene protected against singlet oxygen-mediated lipid peroxidation, inhibited the production of nitric oxide and prostaglandin E(2), and suppressed the expression of iNOS and cyclooxygenase-2 (COX-2). Besides, phycocyanin and β-carotene inhibit proinflammatory cytokine formation, such as tumor necrosis factor α (TNFα), suppresses COX-2 expression, and decreases prostaglandin E(2) production as anti-inflammatory activities. It has been also reported to suppress the activation of nuclear factor-κB (NF-κB) through preventing degradation of cytosolic inhibitor of nuclear factor κB-α (IκB-α)

and modulate the mitogen-activated protein kinase (MAPK) activation pathways.

Conclusion

Based on the markers used in this study, our results report that spirulina supplementation potentially prevents exercise-induced lipid peroxidation, inflammation, skeletal muscle damage and may accelerate the recovery of some of these markers. Although the favourable results of our study, we recognise that there are limitations. The discrepancy in effect on markers and the small sample size would limit the validity and generalizability of our findings. Furthermore, we only explored a small number of Values are presented as mean ± standard deviation. T0, before exercise; T1, immediately after; T2, 24h after. .

Accepted Article

Figure 2: Changes in plasma F2-Isop (Fig. 2A), CRP (Fig. 2B) and CK (Fig. 2C) levels after seven weeks of supplementation determined before (T0), immediately (T1) and 24h (T2) after exercise.

Values are reported as mean ± SD. * Significant difference with T0 at p<0.05; ** p<0.01; *** p<0.001. ## Significant difference with T1 at p<0.01.

  daily training load was evaluated by multiplying the session RPE score by the training duration (min) and the weekly training load was determined by summing the daily training loads each week for each rugby player.

  biomarkers and therefore cannot know whether other relevant markers would have been affected by spirulina in a consistent way. The high density of the weekly rugby schedule results in a limited time for recovery between training sessions, which is associated with a decrease in performance. OS, muscle damage, and the consequent inflammation from rugby play a primary role in the decreased capability to perform. Based on our findings, we recommend spirulina supplementation to athletes who perform a high training load in order to reduce the magnitude of OS, inflammation, and muscle damage which could help reduce performance losses and accelerate recovery after training/competitions throughout the season. The mechanisms of action behind the above-mentioned effects of spirulina related to exercise are poorly understood. Besides, the differences in the study designs and the lack of research in the literature regarding the effects of spirulina supplementation in athletes make it difficult to make practical recommendations. More investigations are needed to elucidate this.

Figure 1 :

 1 Figure 1: Study design.

<< Please insert Figure 1 here >> .

		determine eligibility. Twenty-two participants (aged 26.0 ± 3.8 years; body weight 99.6 ±
		16.6 kg; height 183.1 ± 7.3 cm; body fat 19.5 ± 4.6 %) satisfied the inclusion criteria of
		being considered healthy in clinical examination, not having any history of surgery, and
		not taking any kind of dietary supplement for the past 3 months. Exclusion criteria
	Accepted Article	). The protocol was conducted in compliance with both Good Clinical Practices and the Declaration of Helsinki. Tests were realized at M2S lab (FINES of 31/05/17; EJ: 350051454; ET: 350051462). After being informed of the objectives, the possible risks, discomforts, and benefits, written and informed consent included tobacco use, current intake of spirulina, supplements or anti-inflammatory drugs, known chronic diseases, and allergy to spirulina or any of its components. Only 17 participants finished the study protocol, five players were injured and did not complete all steps of the protocol. Study design: A randomized, double-blind, placebo-controlled design was used. Participants were assigned to a spirulina group (SPI: n = 9), or a placebo group (PLA: n = 8). The study consisted of a daily supplementation period during 7 weeks in which the SPI group received capsules with Arthrospira platensis certified as safe and healthy, manufactured by Algae Green Value (AGV, Vannes, France). Its composition was described in Table
		was obtained from all participants before inclusion.
		Participants: Thirty elite male rugby union players belonging to the same French first
		federal division rugby club (Rennes Étudiants Club « REC ») volunteered to participate
		in this study. All players completed a clinical examination and medical interview to

Table 1 . Composition of Arthrospira platensis produced by AGV (100 g).

 1

		Unity	Amount		Unity	Amount
	Energy			Vitamins
		kJ	1530	Vitamin A	µg	<20
		kcal	363	β-carotene	mg	24,902
	Fat			Vitamin C	mg	<2,5
	Total fat	g	6,3	Vitamin B3	mg	8,5
	Saturated fat	g	3,8	Vitamin B5	mg	0,05
	Unsaturated fat	g	2,5	Vitamin B6	mg	0,09
	Monounsaturated fat	g	0,4	Vitamin B9	µg	156
	Polyunsaturated fat	g	2	Vitamin B12	µg	61,3
	Omega 3	g	0,008	Vitamin E	mg	6,84
	Omega 6	g	0,9	Minerals
	Carbohydrates			Sodium	mg	1338,2
	Total Carbohydrates	g	16,5	Phosphorus	mg	1353,8
	Dietary Fiber	g	7,4	Calcium	mg	310,4
	Proteins and amino acids		Copper	mg	1,83
	Proteins	g	63,7	Iron	mg	76,43
	Leucine	g	5,56	Magnesium	mg	434,2
	Isoleucine	g	3,47	Manganese	mg	7,06
	Valine	g	4,02	Potassium	mg	1932
				Zinc	mg	7,43

Table 2 .

 2 Participant characteristics.

		SPI (n=9)	PLA (n=8)	p-value
	Age (year)	25.2 ± 2.7	25.9 ± 3.3	0.660
	BM (kg)	97.3 ± 13.3	101.1 ± 19.0	0.635
	Height (cm)	182.3 ± 4.7	182.7 ± 6.1	0.894
	FM (%)	19.2 ± 4.9	19.9 ± 4.4	0.728
	FM (kg)	19.2 ± 6.8	20.7 ± 7.5	0.661
	LBM (kg)	78.2 ± 7.3	80.4 ± 12.4	0.649
	Distance YYIRT-2 (m)	426.7 ± 385.0	385.0 ± 95.5	0.478
	Values are presented as mean ± standard deviation. PLA, placebo; SPI, Spirulina; BM, body
	mass; FM, Fat Mass; LBM, lean body mass; YYIRT-2, Yoyo intermittent recovery test-level 2.

Table 3 . Participants' daily dietary intake and internal training load.

 3

			SPI (n=9)	PLA (n=8)	p-value (T-test)	"Group × Week" interaction
	Caloric intake	W0	2662.9 ± 522.1 2901.7 ± 651.4	0.415	0.139
	(kcal/d)	W7	2817.5 ± 559.5 2705.5 ± 544.2	0.682
	Carbohydrates	W0	2.81 ± 0.97	2.88 ± 1.09	0.891	0.084
	(g/kg/d)	W7	3.13 ± 1.01	2.67 ± 0.81	0.316
	Proteins (g/kg/d)	W0	1.40 ± 0.40	1.69 ± 0.72	0.306	0.167
		W7	1.45 ± 0.37	1.63 ± 0.72	0.541
	Fat (g/kg/d)	W0	1.18 ± 0.31	1.13 ± 0.34	0.751	0.893
		W7	1.20 ± 0.31	1.16 ± 0.44	0.851
		W1	1497.8 ± 348.5 1210.7 ± 734.0	0.310
		W2	1363.7 ± 347.5 1053.6 ± 840.0	0.325
	Internal training load (AU)	W3 W4	1224.4 ± 235.8 1339.4 ± 362.0 690.0 ± 206.2 768.8 ± 120.0	0.445 0.359	0.411
		W5	1360.4 ± 364.8 1392.1 ± 542.6	0.888
		W6	1639.6 ± 295.8 1630.5 ± 431.1	0.960
	Values are presented as mean ± standard deviation.

Table 4 . Effects of supplementation on pro/antioxidant status, inflammation and skeletal muscle damage.

 4 PLA 68.75 ± 29.97 64.88 ± 24.39 66.25 ± 24.15 0.803 SPI 57.44 ± 13.12 58.11 ± 22.73 53.33 ± 17.59 PLA 168.57 ± 41.64 197.29 ± 58.84 163.86 ± 41.77 0.871 SPI 186.29 ± 24.00 209.86 ± 17.04 175.00 ± 27.30

						"Group ×
			T0	T1	T2	Time"
						interaction
	Pro/antioxidant status
		PLA	31.65 ± 2.65	31.64 ± 3.13	31.85 ± 3.70
	SOD (UI/g Hb)					0.787
		SPI	34.06 ± 3.07	34.44 ± 2.68	34.11 ± 2.56
		PLA 509.93 ± 57.55 504.06 ± 56.71 483.00 ± 58.36
	GPX (UI/g Hb)	SPI 553.98 ± 90.62	562.41 ± 106.36	549.25 ± 106.85	0.347
	GSH/GSSG	PLA SPI	197.25 ± 190.16 205.25 ± 127.57	531.29 ± 210.85 467.38 ± 280.83	207.86 ± 89.60 174.14 ± 103.59	0.836
	Ox-LDL (U/L)
	Inflammation
		PLA	53.13±11.68	54.13±12.77	53.00±11.10
	MPO (ng/mL)					0.759
		SPI	51.63±13.81	51.75±11.47	49.00±10.43
	Skeletal muscle damage
	LDH (U/L)

Acknowledgments

.

Accepted Article

This study (SUPERF study -FANTASIO project) was supported by the Brittany Region and the competitiveness pole "Valorial, osons l'aliment plus intelligent". Spirulina was provided by the society "Algae Green Value".

We acknowledge the medical staff (Dr Pierrick Guillemot, Violaine Hamon, Dr Patrice N'Gassa), the technical staff (Yann Moison, Yoann Carnot) and all the rugby players for their involvement in the study.

Competing interests

The authors declare that they have no competing interests.

AUTHORS' CONTRIBUTION


Benjamin Lalanne
email: benjamin.lalanne@ensiacet.fr

Sébastien Tanguy

Jiří Vejrazka

Olivier Masbernat

Frédéric Risso

DROP BREAKUP MODELLING IN TURBULENT FLOWS

Keywords: Bubble and droplet dynamics, DNS, breakup modelling, turbulent flows, emulsions Mass density, [kg/m 3 ]. µ Dynamic viscosity, [Pa.s]. Ratio of densities: =, [-]. µ -], [-], [-]

This paper deals with drop and bubble break-up modelling in turbulent flows. We consider the case where the drop/bubble slip velocity is smaller than or of the order of the turbulent velocity scales, or when the drop/bubble deformation is mainly caused by the turbulent stress (atomisation is not addressed here). The deformation of a drop is caused by continuous interactions with turbulent vortices; the drop responds to these interactions by performing shape-oscillations and breaks up when its deformation reaches a critical value. Following these observations, we use a model of forced oscillator that describes the drop deformation dynamics in the flow to predict its break-up probability. Such a model requires a characterization of the shapeoscillation dynamics of the drop. As this dynamics is theoretically known only under restrictive conditions (without gravity, surfactants), CFD two-phase flow simulations, based on the Level-Set and Ghost Fluid methods, are used to determine the interface dynamics in more complex situations: deformation of a drop in the presence of gravity, bubble-vortex interactions. Results are compared with experimental data. The perspectives to apply this model to breakup in emulsification processes are also discussed.

INTRODUCTION

In industrial chemical processes, turbulent flows with dispersed deformable media (drops, bubbles) are commonly encountered and a better control of the size of the drops is desired for transport, separation issues, and maximization of interfacial area. Examples include oil industry with problems of separation of oil and water, health and food industry with the need to produce emulsions of very fine droplets in high-pressure homogenizers, nuclear energy industry with pulsed extraction columns for treatment of irradiated nuclear fuel or chemical industry for enhancement of mass transfers in gas-liquid reactors. In CFD Eulerian codes, population-balance approaches are often used to calculate the drop (or bubble) size distribution. These approaches require closure models for coalescence and breakup phenomena. The present study focuses on breakup phenomena occurring in turbulent flows. A recent review by [START_REF] Liao | A literature review of theoretical models for drop and bubble breakup in turbulent dispersions[END_REF] has shown that several models have been derived to predict breakup probability, breakup frequency, daughter-drop size distribution after breakup. Most of these existing models use a breakup criterion based on a critical Weber number : it consists in a static force balance between the hydrodynamic forces responsible for deformation (turbulent stresses at the scale of the drop) and the force of surface tension that resists to shape deformations. Performing a comparison of these models, Liao and Lucas have shown that they can predict very different breakup frequencies and daughter drop size distributions when applied to a process that is not the one for which they have been calibrated. The conclusion of their paper is that physical improvements are needed for breakup modelling. We are currently developing a new approach by modelling the deformation process of a drop in a turbulent flow using a scalar parameter that describes its deformation. In such an approach, the breakup criterion is based on a critical deformation . This new model calculates the dynamic response of drops that are excited by turbulent fluctuations of the continuous phase. As shown in [START_REF] Galinat | Dynamics of drop breakup in inhomogeneous turbulence at various volume fractions[END_REF] or in [START_REF] Maniero R | Modelling and simulation of inertial drop break-up in a turbulent pipe flow downstream of a restriction[END_REF], it gives better predictions of breakup probability than models based on a when compared to experimental breakup statistics. Section 1 of this article presents our new approach and the physics on which it is based. Then, predictions of the breakup probability with our new model are compared with experimental statistics on heptane drops in a turbulent pipe flow downstream of a restriction. This section will show that our model requires a description of the drop interface dynamics, which lies on two key-parameters: the frequency of oscillation and their damping rate . Nevertheless, these parameters are known only in a limited number of situations: in case of deformations of low amplitudes, in the absence of gravity effects and without surfactants adsorbed at the interfaces. The objective of the present paper is to show how Direct Numerical Simulations can be used to calculate and in situations of practical interest. To achieve this goal, two-phase flow simulations using the Level-Set method are performed. Section 2 describes the numerical methods. Then, after validation of the code, three examples of interface dynamics are emphasized in section 3: (a) the study of influence of gravity on and for drops, (b) first results on the effect of surface contamination on the shape-oscillations and (c) the study of the interaction between a rising bubble and a single strong vortex. Finally, the conclusion will introduce several perspectives of development of this model.

A DYNAMIC MODEL OF DROP DEFORMATION

As example of turbulent flow, let us consider a water pipe flow downstream of a restriction at a Reynolds number of 2100 (based on the orifice diameter), where turbulence is generated after the orifice, like in [START_REF] Galinat | Dynamics of drop breakup in inhomogeneous turbulence at various volume fractions[END_REF]. Fig. 1 shows experimental pictures of a colored heptane drop travelling through this flow. The velocity field has been measured by PIV. From the results, we calculate a local and instantaneous turbulent Weber number We based on the square of the fluctuating velocity between two points distant of the drop diameter (taking the maximum of the difference between vertical, transversal and diagonal components): = /!. By averaging in time, we obtain a map of the mean turbulent Weber number """"" = """"""""" /!, shown in fig. 1. It has been found that """"" is always one order of magnitude higher than other Weber numbers that characterize the inertial drop deformation due to its mean slip velocity or due to the mean flow deceleration in the flow direction after the orifice, indicating that the turbulent fluctuations are the dominant breakup cause in this flow. Nevertheless, breakup locations do not always correspond to locations of maximum values of """"" . Fig. 1 illustrates also the deformation process of a drop leading to its breakup. As mentioned by [START_REF] Risso | Oscillations and breakup of a bubble immersed in a turbulent field[END_REF], the time evolution of the surface area of the drop shows the existence of a characteristic angular frequency which is the frequency of oscillation of its shape. The time scale = 2$/ is thus characteristic of the response of an interface submitted to perturbations of its shape, occurs in turbulent flow

The problem of the low-amplitude shape deformation mathematically Scriven (1968) theoretically which are described harmonics. A to the dissipation of is associated Generally, the mode between a prolate and an oblate shape for a drop, is enough to describe the approximation. Thus, the in their papers predictions of the characteristic time scales of drop or bubble dynamics.

In order to take into account the deformation model an oscillator, forced by the turbulent fluctuations experienced by the drop when travelling the flow residence time short compar occur only if the turbulent fluctuations are intense enough, as predicted Nevertheless, if dynamic response of the drop can Indeed, in that case, the drop filters the turbulent fluctuations at its own time scale results from a coupling between turbulence and the interface dynamics We define a scalar parameter oscillation drop deformation forced oscillator:

̃= / , and that can be identified occurrence criterion denoted extension of a drop diameter, giving value can vary considered.

Note that a similar approach that describes oscillat and distorting droplets, (Taylor Analogy Breakup), is engineers to calculate breakup in low Weber number sprays (O'Rourke and [START_REF] Amsden | The TAB method for numerical calculation of spray droplet[END_REF]).

The 1D model the time evolution of the turbulent fluctuations experienced by the drop along its traj eigenfrequency damping rate of the response of an interface submitted to perturbations of its shape, turbulent flows.

problem of the oscillations of a drop submitted to amplitude shape deformation mathematically studied by Lamb (1968) and Prosperetti theoretically calculated the modes of which are described by A frequency to the dissipation of energy ensured by viscous effects is associated to the dynamics of he mode l=2, between a prolate and an oblate shape for a drop, is enough to describe the drop or bubble approximation. Thus, the in their papers for % predictions of the characteristic time scales of drop or bubble dynamics.

In order to take into account the deformation model, we propose to an oscillator, forced by the turbulent fluctuations experienced by the drop when travelling the flow residence time of the drop short compared to its response time occur only if the turbulent fluctuations are intense enough, as predicted by a static force balance approach Nevertheless, if is of same order or larger than ynamic response of the drop can Indeed, in that case, the drop filters the turbulent fluctuations at its own time scale results from a coupling between turbulence and the interface dynamics.

e define a scalar parameter , normalized drop deformation, which is oscillator:

²' ( )² * 2 + ( , ( '
)
and K is a constant that can be identified using occurrence criterion is based on .Experiments show that the maximum of a drop prior to diameter, giving an estimation of vary depending on the fluids and the process Note that a similar approach that describes oscillat and distorting droplets, (Taylor Analogy Breakup), is engineers to calculate breakup in low Weber number O'Rourke and [START_REF] Amsden | The TAB method for numerical calculation of spray droplet[END_REF]).

odel of eq. ( 1) requires the time evolution of the turbulent fluctuations experienced by the drop along its traj eigenfrequency of oscillation of the drop rate . Note that p modelling in turbulent flows of the response of an interface submitted to perturbations of its shape, in a similar way to .

oscillations of a drop submitted to amplitude shape deformation studied by Lamb (1932) Prosperetti (1980) calculated the modes of by means of series of -and a damping rate energy ensured by viscous effects the dynamics of each mode that represents the oscillation between a prolate and an oblate shape for a drop, is drop or bubble theoretical expressions and % constitute accurate predictions of the characteristic time scales of drop or In order to take into account the interface dynamics in a we propose to represent an oscillator, forced by the turbulent fluctuations experienced by the drop when travelling the flow the drop in the turbulent field to its response time occur only if the turbulent fluctuations are intense y a static force balance approach is of same order or larger than ynamic response of the drop cannot be ignor Indeed, in that case, the drop filters the turbulent fluctuations at its own time scale , and its deformation results from a coupling between turbulence and the e define a scalar parameter -amplitude of mode 2 of rmalized by d -to characterize the , which is the solution of ' ( ) * .

a constant parameter of using experimental data based on a threshold value of Experiments show that the maximum prior to breakup is about twice its an estimation of depending on the fluids and the process Note that a similar approach that describes oscillat and distorting droplets, known as the TAB model (Taylor Analogy Breakup), is commonly used by engineers to calculate breakup in low Weber number O'Rourke and [START_REF] Amsden | The TAB method for numerical calculation of spray droplet[END_REF]).

) requires the know the time evolution of the turbulent fluctuations experienced by the drop along its trajectory and (ii) the of oscillation of the drop . Note that ̃ can be obtained Miller and (1980). They have the oscillation means of series of spherical and a damping rate -( energy ensured by viscous effects each mode of order that represents the oscillation between a prolate and an oblate shape for a drop, is drop or bubble shape in first theoretical expressions given constitute accurate predictions of the characteristic time scales of drop or interface dynamics in a represent the drop an oscillator, forced by the turbulent fluctuations experienced by the drop when travelling the flow. If the in the turbulent field to its response time , breakup can occur only if the turbulent fluctuations are intense y a static force balance approach is of same order or larger than , the not be ignor Indeed, in that case, the drop filters the turbulent , and its deformation results from a coupling between turbulence and the of order l. that represents the oscillation between a prolate and an oblate shape for a drop, is in first given constitute accurate predictions of the characteristic time scales of drop or interface dynamics in a the drop as an oscillator, forced by the turbulent fluctuations f the in the turbulent field is , breakup can occur only if the turbulent fluctuations are intense y a static force balance approach. , the not be ignored. Indeed, in that case, the drop filters the turbulent , and its deformation results from a coupling between turbulence and the amplitude of mode 2 of to characterize the following It is observed that an approach based on breakup probability in the interval whereas our model based on the best criterion for this experiment). That explains why breakup locations do not necessary match to be able to describing the breakup probability in the interval on That explains why breakup locations do not necessary match to be able to describing the 14 678 interface dynamics. As already mentioned, theoretical expressions % and % for these parameters have been derived by several authors; for example, Miller and Scriven (1968) gave

% = * - = > ? > @² ABC (2) % = = > ? > @² D-2A² + E + ABC F. (3)
F and G are functions of the density and viscosity ratios given in appendix A, GH I = J I K L * G²/M I , and K L * = LRS/ TJ I + LJ U G T V.W . Expressions (2) and (3) result from an asymptotic development of the solution at large Reynolds number of oscillation C . They are valid in the linear regime of oscillation (i.e for low amplitudes of deformation), disregard the effect of gravity, and are restricted to the case of pure fluids. Thus, it is requested to model the interface dynamics in more complex configurations such as in concentrated emulsions with surfactants. One way to reach this goal is to perform Direct Numerical Simulations in order to include complexity progressively. Next section describes the numerical methods used in our CFD two-phase flow code.

NUMERICAL METHODS FOR TWO-PHASE FLOW SIMULATIONS

Configurations which are studied in the context of interface dynamics include axisymmetric and threedimensional simulations of shape-oscillations of rising drops and bubbles, interaction between a bubble and a vortex, and breakup of bubbles/drops in turbulent flows. Direct Numerical Simulations based on the Level-Set and Ghost-Fluid methods are performed, their outlines are briefly described in this section.

In the Level-Set method, the interface is numerically represented by the zero-level curve of a continuous function φ which is defined as the algebraic distance to the interface. Its displacement in a velocity field U is computed by solving an advection equation:

Xφ X + Y. Zφ = 0 (4)
In the framework of a one-fluid approach, the fluid motion is calculated by solving the incompressible Navier-Stokes equations by means of a projection method:

∇. Y = 0

(5)

Y + Y. Z Y + ∇\ φ = ∇. 2] φ ^ φ + _
where P is the pressure, ] the dynamic viscosity, the density, _ the acceleration of the gravity and D the rate of deformation tensor. I n these equations, , ] and P are discontinuous across the interface. The normal stress balance at the interface assumes that

`\a = !κ + 2 `] b c / da, (6)
where b c is the velocity normal to the interface, n is the coordinate in the direction normal to the interface, ! the surface tension and κ the interface curvature.

To handle the discontinuity of the pressure at the interface and calculate accurately its derivatives, a Ghost Fluid method has been implemented: the jump condition is extrapolated in one ghost cell on each side of the interface. The numerical formulation for the viscous term and the pressure jump at the interface follows the method detailed in [START_REF] Sussman | A sharp interface method for incompressible two-phase flows[END_REF].

An algorithm of redistanciation is used to ensure that φ remains a distance function at each time step, as described in [START_REF] Tanguy | Application of a Level-Set method for simulation of droplet collisions[END_REF].

These partial differential equations are discretized using the finite volume technique on staggered grids. Spatial derivatives are estimated with a second order central scheme while a fifth order WENO scheme is used for the convective terms, which ensures that the solution is robust. Temporal derivatives are approximated with a second-order Runge-Kutta scheme.

DNS OF INTERFACE DYNAMICS

This section illustrates how CFD is used to calculate interface dynamics in several configurations. First, the shape-oscillations of drops and bubbles are calculated and the results used to validate our numerical code.

Then, the linear shape-oscillations of rising drops are simulated in order to assess the influence of gravity on the oscillations and to extend the theoretical results in that case. Thirdly, another configuration of interaction between a rising bubble and a single vortex is studied in order to characterise the interface dynamics for nonlinear deformations as those occurring in a turbulent flow.

Validation for interface dynamics problems

The numerical methods previously presented are applied to the calculation of shape-oscillations of deformed drops and bubbles. The results obtained are validated against the linear theory of oscillation, or experiments.

Comparison with theory and mesh convergence

Axisymmetric simulations of the shape-oscillations of an initially deformed drop in the absence of gravity are performed. The imposed deformation corresponds to the mode l=2 of oscillation with a low initial amplitude 0 /C = 0.1. In order to characterise the oscillations, we define a Reynolds number of oscillation, that compares the inertial effects of deformation over the viscous damping: C efg = g !/ g C h i/ C²/µ g . Three values of C efg are investigated: 50, 100, 200. Calculations are performed on a regular mesh of 16, 32 and 64 grid points per drop radius so as to study mesh convergence of the results.

Thanks to the Level-Set function φ, the drop contour in spherical coordinates, 7 j, is extracted from the simulation at each time step, and the interface is decomposed into spherical harmonics, reading:

7 j, = C + \ cos j ,

\ being the Legendre polynomial of order l=2.

From the time evolution of , the frequency and damping rate of the oscillations can be obtained and compared against the predictions of the linear theory % and % :

= 0 cos n+ ( . ( 8
)
Table 1: Validation of the axisymmetric calculation of the shape-oscillations of non-rising drops.

Relative errors are reported in table 1 for the three C efg . Results show that the numerical code gives very accurate results on both frequency and damping rates in this range of C efg , for grids containing 32 or 64 points in a radius. Spatial convergence of the simulations is also proved by the tests. Thus, these results provide a good validation of our axisymmetric numerical code to deal with interface dynamics problems. In order to validate also the Cartesian version of the code, a 3D simulation of the shape-oscillations of a bubble is carried out at C efg = 50 on a grid containing 16 cells in a radius. Results obtained are superimposed on fig. 4 with the curves given by the axisymmetric version of the code with 8, 16, 32 and 64 grid points in a radius. Fig. 4 proves again the mesh convergence of the numerical simulations for the axisymmetric calculations, and shows that the accuracy of the oscillations calculated by the 3D Cartesian version of the code is comparable to that of the axisymmetric calculation with the same number of grid points. We can then conclude that a calculation with 16 points per bubble radius at C efg = 50, either in an axisymmetric cylindrical or in a 3D Cartesian domain, gives accurate results in accordance with the theory.

Comparison with an experiment

The code is then validated by comparison with experimental measurements of the shape-oscillations of a slow-rising bubble thanks to a high-speed camera. In the experiment, the bubble is initially attached to a capillary; its sudden translation causes the detachment of the bubble that rises in still water while performing shape-oscillations. The experiment was carried out in ultra-pure conditions in order to avoid contamination due to surfactants. The oscillations, characterized by C efg = 164, are described using the first ten modes and the initial amplitudes, obtained from the experimental shape at the instant of detachment, are larger than in the previous case ( 0 /C = 0.18 . The Bond number pq = -r ²/! is very low (pq = 0.08 to ensure that gravity has negligible effects on both the oscillations and the mean shape of the bubble (which remains spherical during the rising motion). The shape of the interface is decomposed in spherical harmonics (until order 10): Fig. 5 compares the time evolution of , h and v between the experiment and a simulation on a mesh with 32 nodes per bubble radius.

Results show an excellent agreement between the numerical simulations and the experiment for the different modes of oscillation, with a very good accuracy even for strongly damped oscillations (amplitudes less than 0.005R), validating again the use of the present code to capture bubble or drop interface dynamics.

Shape-oscillations of rising drops

We have achieved axisymmetric simulations of a drop rising in a quiescent liquid, its shape being initially elongated in the vertical direction. Thus, while rising, the drop performs shape-oscillations. The objective is here to understand the effect of the rising motion on the shape-oscillation dynamics, by comparing the frequency and damping rates values with those predicted by the linear theory of oscillation, % and % , in the absence of gravity.

The problem can be parameterized by four nondimensional numbers: ratios of density and viscosity For each drop, the rising velocity increases until the steady state is reached. The translational velocity is very slightly affected by the presence of the shape oscillations. The evolution of the drop shape with time shows that globally decreases during the acceleration stage since the drop becomes an oblate spheroid due to the rising motion. On this curve, the shape-oscillations due to the initial perturbation are also visible. In order to analyse them and conclude about the effect of the rising motion (i.e the effect of C ∞ ), we apply a high-pass filter at frequency % to the time evolution of in order to separate the evolution of the drop mean shape (which flattens with time) from that of the shapeoscillations. Then, we measure frequency and damping rate of the oscillations on the filtered signal. We observe that the frequency of oscillation is maximum for the non-rising drop and decreases slightly (-10%) when the rising velocity increases. On the contrary, we note a strong increase of the damping rate (until +300%) of the oscillations for the cases at high C ∞ . It is interesting to note that and do not keep constant values for a given set of parameters (C efg , C ∞ ) but instead they evolve monotonically with time.

This observation leads us to relate the decrease of the eigenfrequency and the increase of the damping rate with the drop instantaneous velocity, which increases with time. Thus, the main effect of the rising motion on the shapeoscillations of drops is to increase the rate of dissipation of oscillation energy, provided that the rising velocity is large enough compared to the oscillating velocity.

̃= % /2$ ̃= % /2$
For non-rising drops, the linear theory of oscillation (Miller and Scriven (1968)) shows that the dissipation takes place in boundary layers located on both sides of the interface, as illustrated in fig. 10, which represents the vorticity field of an oscillating drop. From our numerical simulations, we extract the vorticity field. For slow-rising and oscillating drops, i.e when the damping rate of the oscillations is not found to be affected by the translating motion, we observe that the vorticity of the flow is the sum of the vorticity involved in the motion of the same slow-rising drop which does not oscillate (called "pure rising flow"), and of the vorticity involved in the motion of the same oscillating drop that does not rise (called "pure oscillation flow"). Hence, pure rising and pure oscillation flows do not interact in the case of a slow-rising and oscillating drop, explaining why remains close to % . However, for rapidly rising and oscillating drops, this is not the case: we observe that vorticity contributions of rising and oscillatory motions interact, leading to an increase of the oscillation energy dissipation (cf [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF]). Consequently, the effect of the rising motion on the damping rate of oscillation can be explained by looking at the vorticity field during the oscillating motion of the droplets.

For micrometer or millimeter-sized droplets like those commonly involved in many industrial flows, the Weber number based on the rising velocity -as defined herecan hardly be larger than unity. Therefore, the present results (fig. 8 and9) show that the predictions of the linear theory of oscillation, which do not include the effect of gravity, provide good estimations of and (less than 5% of discrepancy for and less than 30% of discrepancy for

). This is a practical conclusion, useful to know the limits where the linear theory of oscillation remains valid in order to predict the time scales of the interface dynamics.

Oscillations of contaminated rising drops

Let us consider now the experimental investigation of a shape-oscillating heptane drop which is rising in non ultra-pure water, carried out by Abi [START_REF] Chebel | Shape oscillation of an oil drop rising in water: Effect of surface contamination[END_REF]. For different drop diameters (millimeter-sized droplets), the authors have measured the damping rates of the oscillations. Results are displayed in table 2 for two contrasted drop diameters. The experimental measurements show that overestimates by 200% or 300% the theoretical prediction of % , which does not consider either the rising motion or any surface contamination. Using numerical results of fig. 9, we can predict what should be the damping rate of the oscillations in the case of pure fluids and clean interfaces for these rising drops. It is found that the measured damping rates in the experiment are twice these values. Thus, another mechanism is involved: the strong increase of can be related to the presence of contaminants adsorbed at the liquid-liquid interface. Indeed, it is extremely difficult to carry out experiments with pure fluids in liquid-liquid dispersions. In that experiment, the contamination of the interfaces has been proved by calculating the experimental drag coefficient of the drops, which is found to match that of a solid sphere and not that of a drop with clean interface. We conclude from table 2 that the presence of surface-active contaminants alters significantly the shape-oscillation dynamics. This is probably related to additional tangential stresses that appear close to the interface because of contamination, which may increase strongly the dissipation within the boundary layers of oscillation. Further studies are required to characterise completely the interface dynamics of drops in the presence of surfactants, which depends on the transport properties of these adsorbed contaminants at the interfaces.

Bubble-vortex interaction

In turbulent flows, bubble (or drop) breakup can result from either an interaction with a single intense vortex or with a series of moderate vortices that make the bubble accumulate energy of deformation until breakup [START_REF] Risso | Oscillations and breakup of a bubble immersed in a turbulent field[END_REF]). Therefore, the elementary key mechanism responsible of the deformation of a bubble (or a drop) is its interaction with a vortex. An experimental study of breakup of a rising bubble in a turbulent flow, carried out by [START_REF] Ravelet | On the dynamics and breakup of a bubble immersed in a turbulent flow[END_REF], has shown that large bubble deformations caused by the turbulent fluctuations of the flow are quickly damped, and the authors have not observed shape-oscillations of the bubbles after interaction with strong vortices, contrary to what has been previously reported in microgravity conditions [START_REF] Risso | Oscillations and breakup of a bubble immersed in a turbulent field[END_REF].

In order to investigate this open question of interface dynamics in a turbulent flow in the presence of gravity, three-dimensional simulations of the interaction between a rising bubble and a single vortex have been carried out. The objective of the simulations was to study the response of the bubble after large deformations of its shape.

The non-dimensional parameters that describe the rising bubble are: the ratio of density =0.001, the ratio of viscosity µ = 0.016, the Reynolds number of oscillation C efg = 50, the Reynolds number of rising C x = 89 or 142. A Hill's vortex of same dimension than the diameter of the bubble has been chosen in order to provoke bubble large scale deformations. Indeed, it is admitted since the pioneering works of Hinze and Kolmogorov that vortices of same size as the bubble diameter are the most efficient for breakup.

The initial condition of the calculation is illustrated in fig. 11. It is the superimposition of the velocity field induced by a Hill's vortex -known analytically, involving a potential flow of characteristic velocity t , which is the initial velocity of the Hill's vortex (cf Morton ( 2004)) -and of the flow corresponding to a rising bubble at terminal velocity and characterized by C x (calculated through a preliminary simulation). To avoid singular conditions, the vortex and the bubble are off-centered from a distance t = √2/2 C. The mesh is Cartesian but non uniform far from the bubble. The bubble dynamics is captured thanks to the use of 16 grid points per bubble radius.

The intensity of the interaction between the vortex and the bubble is scaled by a Weber number based on the velocity of the vortex at the instant it encounters the bubble:

= g /!.

Fig. 12 displays pictures of the simulation in the case of a very intense interaction: = 16.5 and C x = 142. Before the interaction, the bubble is flattened with an aspect ratio of 1.84 because of its rising motion. During the interaction, we do not observe bubble breakup but large and non-axisymmetric deformations. To obtain an accurate description of the global shape of the bubble (large scales), we calculate its equivalent ellipsoid. It is defined as the ellipsoid of same inertia matrix as the bubble. In the following, we note a the length of the major semi-axis of the ellipsoid, b the length of the intermediate semi-axis, and c the length of the shorter semi-axis. Fig. 13 displays the evolution of a, b and c in the simulation that corresponds to fig. 12.

The bubble shape is initially symmetric around a vertical axis (see picture 1 of fig. 12) and its rising motion is steady. When the vortex arrives at the bubble location, it causes a large elongation of the bubble and consequently a strong increase of the length of its major axis. In the same time, its medium axis is slightly reduced, leading to a shape close to a cylinder with a large major axis and two small other axes. The maximum deformation can be seen in picture 3 of fig.

12. It is remarkable that the symmetry of the bubble has changed from an axisymmetric oblate shape before the interaction to an axisymmetric prolate shape at the maximum of deformation. Finally, the vortex leaves the bubble, which relaxes towards its equilibrium shape (pictures 4-7 of fig. 12). Its rising motion becomes unsteady and follows a zig-zag path. It can be observed in fig. 13 that the major part of the large perturbation induced by the vortex is quickly attenuated. The residual departure from the initial shape leads to slowly damped shape oscillations of low amplitudes.

Consequently, the bubble-vortex interaction can be described in two phases: the first one corresponds to a large deformation of the bubble, which is quickly damped; the second one corresponds to the relaxation of the bubble through linear oscillations. Various simulations have been carried out from different sets of parameters (different rising bubble velocities and vortex intensities). The decomposition of the interaction into two phases is relevant for every case.

During the first phase of large deformation, it is observed that the length of the major axis of the bubble is proportional to . Fig. 14 shows the time evolution of the length of the major axis for different cases, the time being normalized by the frequency of oscillation of the rising bubble given by the theory of [START_REF] Meiron | On the stability of gas bubbles rising in an inviscid fluid[END_REF]. The duration of the large deformation phase is found to be very close to one period of oscillation. These conclusions are in agreement with the experimental observations of [START_REF] Ravelet | On the dynamics and breakup of a bubble immersed in a turbulent flow[END_REF], who have also noticed that: (1) large deformations correspond to prolate bubble shapes, (2) the duration of the large deformation is equal to one period of oscillation, and (3) statistics of large bubble deformations are proportional to statistics of large turbulent fluctuations.

Hence, in case where the turbulent intensity is weak like in the experiment of Ravelet et al. (velocity fluctuations of about 20% of the mean velocity components), the bubble can be considered as a strongly damped oscillator, breakup is rare and occurs only during the interaction with a strong vortex characterized by a large .

Note that, in the simulations, breakup is observed when the bubble elongation (compare to its initial length) is about one diameter, scaling the maximum amplitude a bubble can reach before breaking up.

CONCLUSION

A new dynamic 1D approach to predict breakup probability of drops or bubbles in turbulent flows has been presented. This approach uses a scalar to describe the dynamics of deformation of a bubble/drop in a turbulent flow and predicts breakup occurrences on the basis of a critical value of deformation, contrary to most of the other existing approaches that consider a critical Weber number. The deformation dynamics is modelled as a linear oscillator forced by the turbulent fluctuations experienced by the particle along its trajectory in the flow. Two time scales are used to characterize the droposcillator: its frequency of oscillation and the damping rate . These parameters are theoretically known in limited cases: in the absence of gravity, for low-amplitude oscillations and in the absence of surfactants. The purpose of this paper was to show how CFD simulations (but also complementary experiments) can be used to determine these parameters which are characteristics of the interface dynamics. In this way, we have studied the influence of gravity on and . For liquid-liquid flows, this influence is generally low due to low density differences. In return, for gas-liquid flows, buoyancy effects have to be accounted for. Hence, simulations of the interaction between a rising bubble and a Hill's vortex reveals overdamped oscillations after a large deformation caused by the vortex. This example of simple configuration is a first step towards a better description of the complexity of breakup phenomenology in turbulent flows, where interactions between a bubble (or a drop) and vortices occur continuously and randomly.

With the aim of handling practical situations like those involved in chemical processes, other effects have to be considered, like those related to the presence of surfactants adsorbed at the interfaces, which provide them additional properties of elasticity and viscosity. Thanks to comparisons between experimental and numerical measurements of the damping rate of oscillating drops, it has been shown in this paper that the presence of contaminants can modify drastically the interface dynamics. Further studies, combining experimental and numerical tools, are required to tackle more accurately the effect of surfactants on the shapeoscillations dynamics. Moreover, to deal with breakup in concentrated emulsions (e.g. of breakup in highpressure homogenizers or in static mixers), we should also be able to determine how drop interactions alter the droplet dynamics with respect to the case of an isolated drop, following the work of [START_REF] Galinat | Dynamics of drop breakup in inhomogeneous turbulence at various volume fractions[END_REF] where break-up statistics in concentrated emulsions up to 40% have been analysed.

Currently, the proposed model can predict breakup probability of a single drop (with clean interface) travelling in a turbulent inhomogeneous flow. One objective is to extend this model to turbulent bubbly flows or emulsions with surfactants, after numerical and experimental elementary studies of the interface dynamics. Another objective is to use the calculated deformation of a drop at the instant of breakup in order to quantify its excess surface energy, and to predict the number and size of daughter drops that will be formed after breakup. We are currently focusing on the validation of such an approach to predict daughter-drop size distribution against experimental data, including cases of binary breakup and breakage into several droplets. A longer term goal is to develop new breakup kernels that could be implemented in balance population models.

Figure 1 :

 1 Figure 1: Breakup of a heptane drop downstream of a pipe restriction: deformation of the drop and map of """"" (crosses indicate breakup locations). Figure from Galinat et al. (2007).

  amplitude of mode 2 of to characterize the solution of the following ̃ parameter of the model experimental data. Breakup a threshold value of Experiments show that the maximum kup is about twice its ≅ 1/2, but th depending on the fluids and the process Note that a similar approach that describes oscillat known as the TAB model commonly used by engineers to calculate breakup in low Weber number knowledge of the time evolution of the turbulent fluctuations ectory and (ii) the of oscillation of the drop can

  of Experiments show that the maximum kup is about twice its , but this depending on the fluids and the process Note that a similar approach that describes oscillating known as the TAB model commonly used by engineers to calculate breakup in low Weber number ledge of (i) ̃ ectory and (ii) the and can

  Figure 2: model (1) the drop

Figure 2 :Figure 3 :

 23 Figure 2: Computation of the deformation of model (1) using measurements the drop (Galinat

  but from a resonant interaction that energy of deformation in model based on er is not accurate to predict this the model (1) and using drop from of the turbulence at the scale of the distance from the orifice in the pipe. Thick continuous line: experiment. Thin different Dotted lines: predictions of model (1) (2012).

Figure 4 :

 4 Figure 4: Validation of the 3D Cartesian calculation of the shape-oscillations of a non-rising bubble at C efg = 50. Figure from Lalanne (2012).

Figure 5 :

 5 Figure 5: Time evolution of , h and v : experiments (black lines) and simulations (blue lines).

  and µ , a Reynolds number of translation C ∞ = g ∞ /µ g based on the rising drop velocity ∞ , and a Reynolds number of oscillation C efg based on the oscillating velocity. For these simulations, =0.99 and µ = 1 (liquid-liquid configuration), C efg is varied between 50 and 200, and C ∞ ranges from 60 to 600. The initial deformation of the drop is set to /C = 0.1: it is low enough to ensure a linear regime of oscillation. The regular grid used for the simulation is composed of 32 nodes per drop radius. Let us examine the shape-oscillations of this rising drop. At C efg = 100, fig. 6 presents the time evolution of the Reynolds number of rising C wfg = g /µ g for four different values of C ∞ , and fig. 7 displays the corresponding time evolution of the second harmonic amplitude.

Figure 6 :

 6 Figure 6: Time evolution of Reynolds number of rising for four drops at C efg = 100, and C ∞ = 150, 200, 270, 290. Figure from Lalanne (2012).

Figure 7 :

 7 Figure 7: Time evolution of amplitude of harmonic 2 for four rising drops at C efg = 100, and C ∞ = 150, 200, 270, 290. Dotted line: calculation for a non-rising drop. Figure from Lalanne (2012).

  Fig. 8 and 9 display the results for drops at C efg = 50, 100, 200 and several C ∞ :and are presented as a function of an instantaneous Weber number = 0.5 C wfg /C efg ² that compares the magnitudes of rising and oscillatory motions. Doing so, time evolutions of both frequency and damping rate seem to collapse on single master curves. Despite a certain scattering of the results that indicates that other factors may play a role on the oscillations (drop acceleration for example), these plots tend to show that the deviation from the theory is mainly controlled by the instantaneous velocity.

Figure 8 :

 8 Figure 8: Frequency of oscillation of a rising drop normalized by the theoretical frequency for a non-rising drop, versus . Figure from Lalanne et al. (2013).

Figure 9 :

 9 Figure 9: Damping rate of oscillation of a rising drop, normalized by the theoretical damping rate for a non-rising drop, versus . Figure from Lalanne et al. (2013).

Figure 10 :

 10 Figure 10: Vorticity field (normalized by % ) of an oscillating and non-rising drop at C efg = 100. The boundary layers of oscillation visible in this figure are the main location of the dissipation of the energy of oscillation. Figure from Lalanne et al. (2013).

Figure 11 :

 11 Figure 11: Initial condition (vorticity field) for a 3Dcalculation of the interaction of a rising bubble and a Hill's vortex. Figure from Lalanne (2012).

Figure 12 :

 12 Figure12: Bubble-vortex interaction at = 16.5 and C ∞ = 142. Colors correspond to vorticity levels. Time increases with the picture numbers. Note that picture 1 is the initial condition where the bubble has an oblate shape, and picture 3 is that of maximum deformation where the bubble has a prolate shape (close to a cylinder).

Figure 13 :

 13 Figure 13: Time evolution of bubble semi-axes during its deformation by a vortex at = 16.5 and C ∞ = 142.

Figure 14 :

 14 Figure 14: Time evolution the bubble elongation defined as (aa(t=0))/R and normalized by , for several bubblevortex interactions at C ∞ = 89 or 142, and 1.5 ≤ ≤ 16.5.

Table 2 :

 2 Experimental results for oscillating and rising heptane drops in non ultra-pure water. From Abi[START_REF] Chebel | Shape oscillation of an oil drop rising in water: Effect of surface contamination[END_REF].

APPENDIX A

We give here the F and G functions, associated with eq. ( 2) and (3), corresponding to the theoretical calculation of Miller and Scriven (1968) for frequency and damping rate of the axisymmetric shape-oscillations of mode l = 2, valid for low viscous oscillations of a drop or a bubble (i.e at large Reynolds number of oscillation C ), in the case of low deformations, in zero-gravity conditions and without surfactants:


Vincent Häfliger

Eric Martin

Aaron Boone

Florence Habets

Cédric David

Pierre-André Garambois

Hélène Roux

Sophie Ricci

Anthony Thévenin

Lucie Berthon

Evaluation of regional-scale water level simulations using various river routing schemes within a hydrometeorological modelling framework for the preparation of the SWOT mission

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Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr

The ability of a regional hydrometeorological model to simulate water depth is assessed in order to prepare for the SWOT (Surface Water and Ocean Topography) mission that will observe free surface water elevations for rivers having a width larger than 50/100 m. The Garonne river (56 000 km2, in south-western France) has been selected owing to the availability of operational gauges, and the fact that different modeling platforms, the hydrometeorological model SAFRAN-ISBA-MODCOU and several fine scale hydraulic models, have been extensively evaluated over two reaches of the river. Several routing schemes, ranging from the simple Muskingum method to time-variable parameter kinematic and diffusive waves schemes with time varying parameters, are tested using predetermined hydraulic parameters. The results show that the variable flow velocity scheme is advantageous for discharge computations when compared to the original Muskingum routing method. Additionally, comparisons between water level computations and in situ observations led to root mean square errors of 50-60 cm for the improved Muskingum method and 40-50 cm for the kinematic-diffusive wave method, in the downstream Garonne river. The error is larger than the anticipated SWOT resolution, showing the potential of the mission to improve knowledge of the continental water cycle. Discharge computations are also shown to be comparable to those obtained with high-resolution hydraulic models over two reaches. However, due to the high variability of river parameters (e.g. slope and river width), a robust averaging method is needed to compare the hydraulic model outputs and the regional model. Sensitivity tests are finally performed in order to have a better understanding of the mechanisms which control the key hydrological processes. The results give valuable information about the linearity, Gaussianity and symetry of the model, in order to prepare the assimilation of river heights in the model.

  Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 13871 To link to this article : URL : https://agu.confex.com/agu/fm14/meetingapp.cgi#Paper/2900 To cite this version : Häfliger, Vincent and Martin, Eric and Boone, Aaron and Habets, Florence and David, Cédric and Garambois, Pierre-André and Roux, Hélène and Ricci, Sophie and Thévenin, Anthony and Berthon, Lucie and Biancamaria, Sylvain Evaluation of regionalscale water level simulations using various river routing schemes within a hydrometeorological modelling framework for the preparation of the SWOT mission. (2014) In: AGU Fall Meeting 2014, 15 December 2014 -19 December 2014 (San Francisco, United States).

Authors

Vincent Häfliger

Météo-France Toulouse

Eric Martin

Météo-France Toulouse

Aaron Boone

Météo-France Toulouse

Florence Habets

CNRS

Cedric David

University of California, Irvine

Pierre-André Garambois

Institut de Mécanique des Fluides de Toulouse

Hélène Roux

Institut de Mécanique des Fluides de Toulouse

Sophie Ricci

CERFACS

Anthony Thévenin

CERFACS

Lucie Berthon

CERFACS

Sylvain Biancamaria

Observatory Midi-Pyrenees Evaluation of regional-scale water level simulations using various river routing schemes within a hydrometeorological modelling framework for the preparation of the SWOT mission