L Tusell

I David

L Bodin

A Legarra

O Rafel

M López-Bejar

M Piles

Using the product threshold model for estimating separately the effect of temperature on male and female fertility 1

Keywords: fertility, heat stress, product threshold model, rabbit, temperature

Animals under environmental thermal stress conditions have reduced fertility due to impairment of some mechanisms involved in their reproductive performance that are different in males and females. As a consequence, the most sensitive periods of time and the magnitude of effect of temperature on fertility can differ between sexes. The objective of this study was to estimate separately the effect of temperature in different periods around the insemination time on male and on female fertility by using the product threshold model. This model assumes that an observed reproduction outcome is the result of the product of 2 unobserved variables corresponding to the unobserved fertilities of the 2 individuals involved in the mating. A total of 7,625 AI records from rabbits belonging to a line selected for growth rate and indoor daily temperature records were used. The average maximum daily temperature and the proportion of days in which the maximum temperature was greater than 25°C were used as temperature descriptors. These descriptors were calculated for several periods around the day of AI. In the case of males, 4 periods of time covered different stages of the spermatogenesis, the transit through the epididymus of the sperm, and the day of AI. For females, 5 periods of time covered the phases of preovulatory follicular maturation including day of AI and ovulation, fertilization and peri-implantational stage of the embryos, embryonic and early fetal periods of gestation, and finally, late gestation until birth. The effect of the different temperature descriptors was estimated in the corresponding male and female liabilities in a set of threshold product models. The temperature of the day of AI seems to be the most relevant temperature descriptor affecting male fertility because greater temperature records on the day of AI caused a decrease in male fertility (-6% in male fertility rate with respect to thermoneutrality). Departures from the thermal zone in temperature descriptors covering several periods before AI until early gestation had a negative effect on female fertility, with the pre-and peri-implantational period of the embryos being especially sensitive (from -5 to -6% in female fertility rate with respect to thermoneutrality). The latest period of gestation was unaffected by the temperature. Overall, magnitude and persistency of the temperatures reached in the conditions of this study do not seem to be great enough to have a large effect on male and female rabbit fertility.

INTRODUCTION

Animals exposed to climatic thermal stress have problems in the maintenance of their regular productive and reproductive performances. Males and females are affected in a different way by thermal stress (see review by [START_REF] Marai | Rabbits' productive, reproductive and physiological performance traits as affected by heat stress: A review[END_REF]; rabbit does exposed to heat stress reach puberty later and have reduced conception and embryonic developmental rates, whereas in the case of bucks, testosterone concentration, spermatogenesis, libido, and some seminal traits are also factors affected by heat stress. Success of AI involves different male and female physiological events that could be spe-cifically affected by temperature conditions. Therefore, the most sensitive period around insemination time and the effect of the intensity or the duration of the exposure to this environmental factor or both could also be sex dependent. The knowledge of the existence of these sexual differences concerning sensitivity to environmental temperature could lead to implement different strategies of farm and breeding management and even of selection if the goal is to obtain animals tolerant to thermal stress.

The product threshold model [START_REF] David | Product versus additive threshold models for analysis of reproduction outcomes in animal genetics[END_REF]) is an interesting model to separately establish the effect of temperature conditions on male and female fertility. This model assumes that an observed reproduction outcome is the result of the product of 2 unobserved variables corresponding to the fertility of the 2 individuals involved in the mating. This approach, compared with the classical analysis using the additive threshold model, has the advantage to provide specific estimates of the effects affecting each one of the members involved in the process. The objective of the present study was to determine the effect of temperature conditions at different periods around insemination time on male and on female rabbit fertility and to find the temperature descriptor that gives a better description of the effect of temperature on these traits.

MATERIALS AND METHODS

The research protocol was approved by the animal care and use committee of the Institut de Recerca i Tecnologia Agroalimentàries.

Fertility Data

Animals used in this study came from the Caldes line [START_REF] Gómez | The Caldes strain (Spain). Options Méditerr[END_REF], which is selected for growth rate during the fattening period. This line is bred and reared on an experimental farm belonging to the Institut de Recerca i Tecnologia Agroalimentàries in Caldes de Montbui (Barcelona, Spain). After weaning at 32 d of age, animals are housed in cages of 8 individuals with a photoperiod of 16 h light/d. They are fed ad libitum with a commercial diet (15.5% CP, 2.3% fat, 17.2% fiber) until 60 d of age. After this period, selected replacement males are located in another building of the farm, whereas replacement females remain in the same building. After 60 d of age, males and females are individually housed and fed restricted to 180 g/d of a commercial diet (16% CP, 4.3% fat, 17% fiber). Fresh water is always available.

Young males start the training to ejaculate extraction with an artificial vagina at 4.5 mo of age. One ejaculate per male is collected weekly during the next 2 wk. At 5 mo of age, males started their reproductive life and 2 ejaculates per male are collected weekly, with an interval of 30 min between collections. To prepare the AI doses, gel plugs are removed and, by visual detection, only the ejaculates with presence of urine or calcium carbonate deposits are discarded. Ejaculates are diluted 1:4 (vol/vol) in a commercial extender (Galap, IMV Technologies, Saint Ouen sur Iton, France), and the AI doses of 0.5 mL are applied within 1 h after collection.

Females follow a semi-intensive reproductive rhythm: first AI at approximately 4.5 mo of age, with subsequent 42-d reproductive cycles. At 48 h before AI, all females are treated with 15 IU of eCG (subcutaneously; Foligon, Intervet International B.V., Booxmeer, Holland). Ovulation is induced immediately after AI with 0.02 mg of Gonadorelin (intramuscularly; Fertagyl, Intervet Internacional B.V. Booxmeer, Holland). Does are placed in maternal cages and fed ad libitum 2 wk before parturition.

Diagnosis of pregnancy is made by abdominal palpation 14 d after AI, but the fertility score is assigned at birth: 1 if the female gave birth and 0 otherwise. A total of 7,625 fertility records after AI were obtained from June 2006 to October 2009.

Temperature Records

Daily temperature was automatically recorded every 30 min in a data logger (Tinytag, Gemini Data Loggers, Chichester, UK) in the male and female buildings during the same period of collection of fertility data. Both buildings have isolated roof, walls, and cooling ventilation systems to avoid animal exposure to extreme temperatures. However, because the buildings were not totally isolated, the indoor temperature records partially reflected the seasonality of the outdoor temperature. Table 1 shows summary statistics of the AI data and the daily maximum temperature records of the male and female buildings.

Three types of temperature descriptors were used: 1) the maximum temperature on the day of AI (Tmax, °C), 2) the average of maximum daily temperatures during a time period (avgTmax, °C), and 3) over the same period, the proportion of days with maximum temperature greater than 25°C (DI, percentage). This last type of descriptor encompasses the duration and intensity of the hot conditions.

To accommodate the possible nonlinear effect of these variables on AI success, the continuous values of the Tmax and avgTmax descriptors were classified into to 5 categories: ≤18°C, 19 to 22°C, 23 to 24°C, 25 to 26°C, and ≥27°C. The DI descriptors were classified into to 5 categories as well: 0 to 20%, >20 to 40%, >40 to 60%, >60 to 80%, and >80 to 100%.

The temperature descriptors were calculated for several periods covering the main physiological events that lead from gamete maturation in both sexes to gestation and birth after AI. Figure 1 shows these different events and the corresponding temperature descriptors.

Temperature Descriptors Used for Male (β ♂ )

The temperature descriptors used for the male covered the whole spermatogenesis period, which lasts ap-proximately 38 to 41 d in the adult buck [START_REF] Alvariño | Reproductive performance of male rabbits[END_REF]. Subsequently, 4 temperature descriptors from the day of AI to 40 d backward according to the main phases established in rabbits by [START_REF] Swierstra | Duration of spermatogenesis and spermatozoan transport in the rabbit based on cytological changes, DNA synthesis and labeling with tritiated thymidine[END_REF] were used (see Figure 1):

1) The avgTmax_40_20_♂ and DI_40_20_♂

were calculated in the period from d 40 to 21 before AI, which encompasses the initial stages of spermatogenesis with proliferation and maintenance of spermatogonia by mitotic divisions Correspondences between the biological processes involved in fertilization (above the chronological line) and the different temperature descriptors used in the analyses for male and female fertility (below the chronological line). Black and gray arrows refer to processes and temperature descriptors used for male and female, respectively. The term avgTmax_i_j_k is used to denote the average of maximum temperatures during a period that lasts from day i to day j for the male or female (k). The term DI_i_j_k is used to denote the temperature descriptor of proportion of days of the period from the ith to the jth day in which the maximum temperature was greater than 25°C for the male or female. The term Tmax_0_k denotes the male and female maximum temperature on the day of AI for the male or female.

and the subsequent meiosis until the formation of spermatids occurs.

2) The avgTmaxT_20_10_♂ and DI_20_10_♂ were calculated in the period ranging from d 20 to 11 before day of AI, which covers from spermatid stage until the end of spermatogenesis.

3) The avgTmax_10_0_♂ and DI_10_0_♂

were calculated in the period from d 10 until the day previous to AI and covers the maturation processes occurring during the transit and the storage of the sperm through the epididymus. 4) The Tmax_0_♂ is the maximum temperature in the day of AI, which encompasses the AI dose preparation, insemination, and changes related to the fertilization process.

Temperature Descriptors Used for Female (β ♀ )

The following temperature descriptors were used for the female (see Figure 1):

1) The avgTmax_10_0_♀ and DI_10_0_♀

were calculated in the period from d 10 before AI until the day previous to AI. That approximately covers the period of lifespan of the mature follicles [START_REF] Hill | The growth and regression of follicles in the oestrous rabbit[END_REF], which can be present in the ovarian surface until the period previous to ovulation. Under commercial conditions of AI, does are superovulated by hormonal treatment 48 h before AI [START_REF] Alvariño | Control de la reproducción en el conejo[END_REF], which improves receptivity and increases and synchronizes follicle growth as well as ovulation rate [START_REF] Maertens | Effects of PMSG induced oestrus on the performances of rabbit does: A review[END_REF].

2) The Tmax_0_♀ is the maximum temperature on the day of AI. That encompasses the ovulation event, the migration of the ova to the fertilization site, and the fertilization process [START_REF] Harper | Ovulation in the rabbit: The time of follicular rupture and expulsion of the eggs, in relation to injection of luteinizing hormone[END_REF]. The gestation length in the female rabbit lasts 30 to 32 d [START_REF] Prud'hon | La reproduction des lapins. La revue de l'élevage[END_REF].

3) The avgTmax_0_7_♀ and DI_0_7_♀

were calculated in the period from the day after AI until d 7 after AI to cover the period of periimplantation embryonic development (International Committee on Veterinary Embryological Nomenclature, 1994; [START_REF] Lee | Animal models of implantation[END_REF]. 4) The avgTmax_7_18_♀ and DI_7_18_♀

were calculated in the period from d 8 to 18 after AI, which covers the embryo and early fetal gestation (International Committee on Veterinary Embryological Nomenclature, 1994). 5) The avgTmax_18_31_♀ and DI_18_31_♀

were calculated in the period from d 19 to 31 after AI covering late gestation until birth.

Model and Statistical Analysis

The product threshold model assumes that conception after AI (the observed phenotype) occurs when both male and female are fertile. Thus, the probability of AI success is the product of the probability of 2 binary unobserved phenotypes corresponding to the male and female fertilities [START_REF] David | Product versus additive threshold models for analysis of reproduction outcomes in animal genetics[END_REF]. These hidden phenotypes can be modeled using 2 unobserved, underlying continuous variables (liabilities), each one having a fixed threshold that divides its continuous scale into 2 response categories: success and failure [START_REF] Wright | An analysis of variability in number of digits in an inbred strain of Guinea pigs[END_REF]. In matrix notation, the product threshold model used in this study can be expressed as follows:

Pr( | ( ( , , , , , ) ) , , , y p p p
X Z Z p X p = = + + × 1 1 2 1 β β β β m f ps m md f m m m m m md f Φ Φ β β β f f ps f f + + X Zp 2,
),

where y is the vector of the observed AI results (1: success to conception, 0 otherwise); Φ is the standard cumulative distribution function of a standard normal distribution; β m and β f are the vectors of temperature descriptor effects described previously for male and female, respectively; β ps is the vector of the effect of the physiological status of the female (3 levels: nulliparous, multiparous in lactation, and multiparous not in lactation at AI); p m is the vector of male effects; p md accounts for the interaction between male and day of AI; and p f is the vector of female effects. Random effects p m , p md , and p f were considered uncorrelated. Terms

X m , X 1,f , X 2,f , Z 1,m , Z 2,m
, and Z f are incidence matrices relating data with the systematic and random effects included in the model.

In our study, we have considered different models that only differed by the systematic effects β m and β f . Due to the large colinearity between temperature descriptors within sex, all these models included only 1 temperature descriptor for each hidden phenotype. Most of the possible combinations between male and female temperature descriptors were previously analyzed in several models (data not shown), and because they provided similar estimates for the effect of the male temperature descriptor, irrespective of the temperature descriptor used for the female, and vice versa, only the results of 11 different models are presented (Table 2).

The season effect was not considered in the analysis because it is greatly confounded with the environmental temperature. The age of the male was also not included because of the nonrelevant effect found in preliminary analyses. The age of the female is partially encompassed with the physiological status of the female, included in the female side of the model.

A Bayesian framework was adopted for inference. The assigned prior distributions for the parameters of the models were

p k p N i i i β ( ) ( ) ( ) ⊗ ~; | ~, p 0I σ σ 2 2 for i = m, f, md. The term k is a constant, σ i 2
is the variance for male, female, and the interaction between male and day of AI random effects, respectively. Flat bounded priors were used for variances. Thresholds and residual variances were fixed to 0 and 1, respectively. For details, see [START_REF] David | Product versus additive threshold models for analysis of reproduction outcomes in animal genetics[END_REF]. The marginal posterior distributions of the parameters of interest were derived from the joint posterior density of all the unknown parameters. The Gibbs sampler algorithm was used to estimate the marginal posterior distributions of the systematic effects and the variance components. Single chains of 200,000 iterations were run for all the analyses. The first 20,000 iterations of each chain were discarded, and samples of the parameters of interest were saved every 100 rounds. The number of discarded samples was, in all cases, much larger than the required burn-in determined by using [START_REF] Raftery | How many iterations in the Gibbs Sampler? Bayesian Statistics[END_REF] and [START_REF] Geweke | Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments[END_REF] procedures. The sampling variance of the chains was obtained by computing Monte Carlo SE [START_REF] Geyer | Practical Markov chain Monte Carlo[END_REF]. Summary statistics from the marginal posterior distributions were calculated directly from the samples saved.

Male and female unobserved phenotypic variances were obtained as the sum of all corresponding variance components of each sex. Ratios of variance were obtained dividing each variance component by the corresponding unobserved phenotypic variance.

Criteria Used to Test the Relevance of the Effect of Temperature Descriptors on Male and Female Fertilities

Rabbit comfort temperature is considered to be around 21°C [START_REF] Marai | Rabbits' productive, reproductive and physiological performance traits as affected by heat stress: A review[END_REF]. Consequently, the second level of Tmax and avgTmax descriptors would correspond to zone of thermal neutrality (ranging from 19 to 23°C) at which other levels were compared. For the DI descriptors, in absence of references in the literature concerning the effect of a persistently high tem-perature over a period on the fertility, we chose arbitrarily the second level as the reference (20 to 40% of days in the period with maximum temperature greater than 25°C). Estimates of the other levels of the temperature descriptors were obtained as contrasts to those second levels, which were set to 0. Probability of a positive/negative value of the contrast was considered. The highest posterior density intervals at 95% (HPD 95% ) of the contrasts were also computed.

RESULTS AND DISCUSSION

Table 3 shows correlations among the different temperature descriptors used in this study. As expected, the largest correlations were encountered among those temperature descriptors that were measured in the same period of time and also among those ones that were measured closer in time. Moreover, it is important to note that the temperature descriptors for the male were moderate to highly correlated with those for the female. This means that in an additive model, it would not be possible to include both factors to analyze the effects of the temperature on global fertility because there would be a colinearity effect on these estimates.

Figure 2 shows the monthly percentage of AI success and the average maximum temperature per month in the male and female buildings recorded during the experimental period of this study. A regular decrease of the AI success was observed from February to July when the temperature increased. The greatest observed AI success was reached during the coldest period of the year, and the least observed success occurred during the summertime. The detrimental effects of heat stress on conception rate have been described by several authors, especially in countries with hot climate conditions [START_REF] Marai | Tolerance of New Zealand white and Californian doe rabbits at first parity to the sub-tropical environment of Egypt[END_REF][START_REF] Yassein | Hot climate effects and their amelioration on some productive and reproductive traits in rabbit does[END_REF]. In rabbits, the number of functional sweat glands is scarce and their fur does not allow for a great amount of perspiration. The increase of respiration rate, changes in body position, and heat dissipation via ear lobes appear to be the main mechanisms to mitigate heat in this species [START_REF] Harkness | Rabbit behaviour as related to environmental stress[END_REF]. When those mechanisms are not efficient enough to dissipate the amount of heat above the zone of thermal neutrality, deterioration of growth, less resistance to diseases, and impairment in reproductive performance appear with the subsequent economic losses for breeders (reviewed by [START_REF] Marai | Rabbits' productive, reproductive and physiological performance traits as affected by heat stress: A review[END_REF].

The posterior means of the probabilities of success [START_REF] David | Product versus additive threshold models for analysis of reproduction outcomes in animal genetics[END_REF] were approximately 0.87 for male and approximately 0.83 for female fertility in all models. Equal Monte Carlo SE were obtained for both estimates in all the models (0.002). With a similar data set of the same line of rabbits and using the same product threshold model, [START_REF] David | Product versus additive model for studying AI results in several livestock populations[END_REF] found very close values for the probabilities of fertility success.

Estimates of the posterior mean of total variance of each underlying variable, ratios of variance attributable to male, and the interaction between male and day of The term avgTmax_i_j_k denotes the average of maximum temperatures during a period that lasts from day i to day j for the male or female (k). The term DI_i_j_k denotes the temperature descriptor of the proportion of days of the period from the i to the j day in which the maximum temperature was greater than 25°C for the male or female (k). The term Tmax_0_k denotes the male and female maximum temperature of the day of AI for the male or female (k).

AI in the male liability as well as the ratio of variance due to the female effect in the female liability were similar among the models. Surprisingly, in model 4 (Table 2) the posterior SD of the estimates of these figures were larger than the corresponding ones obtained in the other models.

Figures 3 and4 as well as Figures 5 and6 show estimates (posterior means) and their corresponding confidence intervals (HPD 95% ) of the effect of the temperature descriptors in male and female unobserved fertility on a 0 to 1 scale, respectively. The effect of some temperature descriptors were estimated in more than one model (obtaining the same estimates). For simplicity, only the estimates obtained in models 1 to 7 and models 1, 3 to 11 for the male and female temperature descriptors, respectively, are presented (Table 2).

Effect of Temperature Descriptors on Male Fertility

High maximum daily temperatures on the day of AI represented a loss of 6% in male fertility rate with re-Table 3. Correlations among the different temperature descriptors used to estimate the effect of thermal stress in male and female fertility Item The term avgTmax_i_j_k denotes the average of maximum temperatures during a period that lasts from day i to day j for the male or female (k). The term DI_i_j_k is used to denote the male or female temperature descriptor of proportion of days of the period from the i to the j day in which the maximum temperature was above 25°C. The term Tmax_0_k denotes the male and female maximum temperature of the day of AI. spect to thermoneutrality (Tmax_0_♂, Figure 3). This result indicates the susceptibility of the sperm to a high temperature during all the handling processes from the preparation of the AI doses until the time of insemination. Because doses of 1 male in a specific day have been exposed to the same environmental temperature, it is possible that the random effect resulting from the interaction between male and day of AI could be removing part of the effect of Tmax_0_♂. However, we preferred to include this random effect in the models because it takes into account all possible environmental factors that could have had an effect on all the AI doses prepared from one male in a specific day of AI, whereas the effect of temperature on the day of AI is common to all the doses of all the males that have been prepared on a specific day. Other results obtained in another rabbit fertility study by [START_REF] Tusell | Interaction of genotype × artificial insemination conditions for male effect on fertility and prolificacy[END_REF] indicated that the male random effect with the greatest variance was the interaction between male and day of AI. In a similar way, this effect partly encompassed the high sensitivity of the semen to the effect of environmental factors during the processes of dose preparation and application. However, a proportion of the variation obtained for this effect in the 2 models cannot be properly compared. The additive threshold model considers the existence of a unique linear variable underlying the categorical trait and the proportion of variance of a certain male or female effect is referred to phenotypic variation of this common underlying variable. However, under the product threshold model, it is assumed that 2 underlying variables exist: the liability of male fertility (nonobservable male phenotype) and the liability of female fertility (nonobservable female phenotype). In that case, the proportion of variance of a certain effect is referred to the phenotypic variation of the corresponding male or female liability. In this work, none of Posterior mean and highest posterior density interval at 95% of the effects on male fertility rate of the average maximum daily temperatures in the periods from 40 to 21 d before AI, from 20 to 11 d before AI, and from 10 to 1 d before AI (avgTmax_40_20_♂, avgT-max_20_10_♂, and avgTmax_10_0_♂, respectively) and effect on male fertility rate of the maximum temperature the day of AI (Tmax_0_♂). Estimates of each figure were obtained in models 1, 3, 5, and 7, respectively. the temperature descriptors covering spermatogenesis had a clear influence on male fertility (period from 40 to 10 d before AI, Figures 3 and4).

Low and high values of maximum temperatures covering the epididymis transit stage also did not affect male fertility (avgTmax_10_0_♂; Figure 3). However, large percentages of days with maximum temperature greater than 25°C during the epididymal transit of the sperm (DI_10_0_♂, Figure 4) produced a relevant decrease in male fertility (-7% in the contrasts with the 2 greatest levels, with a probability of a negative value of 97%). Some seminal quality traits related to male fertility [START_REF] Gadea | Sperm factors related to in vitro and in vivo porcine fertility[END_REF][START_REF] Lavara | Do parameters of seminal quality correlate with the results of on-farm inseminations in rabbits?[END_REF][START_REF] Garcia-Tomás | Reproductive performance of crossbred and purebred male rabbits[END_REF] such as sperm concentration, sperm abnormalities, and acrosome integrity were analyzed by [START_REF] Roca | Influence of constant long days on ejaculate parameters of rabbits reared under natural environment conditions of Mediterranean area[END_REF]. These authors found a negative effect of the temperature-humidity index (THI) on semen quality and suggested that spermatogenesis but not epididymal transit was affected, according to the time elapsed between the THI stress and the occurrence of impaired semen quality, under commercial conditions. However, their results should be taken with caution because season, light treatment, and THI were included as different effects in the same model, leading presumably to confounded effects. On the other hand, [START_REF] Finzi | Sperm abnormalities as possible indicators of rabbit chronic heat stress[END_REF] compared the characteristics of the spermatozoa from ejaculates obtained in consecutive weeks in males moved from thermoneutrality to heat stress conditions in a climatic chamber. They concluded that the stage of spermatid formation in the seminiferous tubules was the critical period in the formation of sperm abnormalities. The effect of this kind of seminal trait on fertility could not be observed if the sperm dosage is high [START_REF] Saacke | Relationship of seminal traits and insemination time to fertilization rate and embryo quality[END_REF], which occurs in our study. This could lead to not observing a detrimental effect on male fertility under heat stress conditions in the period from 40 to 10 d before AI. However, it has been demonstrated that, in the presence of thermal stress in both the testis and the epididymis, motile and morphologically normal sperm could be obtained, but with defective chromatin Posterior mean and greatest posterior density interval at 95% of the effects on female fertility rate of the average maximum daily temperatures in the periods from 10 to 1 d before AI, from 1 to 7 d after AI, from 8 to 18 d after AI, and from 19 to 31 d after AI (avgT-max_10_0_♀, avgTmax_0_7_♀, avgTmax_7_18_♀, and avgTmax_18_31_♀, respectively) and effect on female fertility rate of the maximum temperature the day of AI (Tmax_0_♀). Estimates of each figure were obtained in the models 5, 7, 1, 8, and 10, respectively.

(in mice; [START_REF] Banks | Impact of a mild scrotal heat stress on DNA integrity in murine spermatozoa[END_REF]. In turn, altered sperm DNA has a negative effect on fertility, especially during embryonic development [START_REF] Saacke | Relationship of seminal traits and insemination time to fertilization rate and embryo quality[END_REF][START_REF] D'occhio | Biology of sperm chromatin structure and relationship to male fertility and embryonic survival[END_REF], which could explain the negative effect of a persistently high temperature during the epididymal transit of the sperm observed in our work, if it can be demonstrated that DNA structure is more sensitive to heat stress in this stage. This kind of deficiency of the spermatozoa cannot be compensated for by increasing the sperm concentration of the AI dose.

Previous results obtained in the same line showed that cold to moderate THI indices seemed to negatively affect some quantitative and qualitative semen traits [START_REF] Garcia-Tomás | Influence of environmental temperature and relative humidity on quantitative and qualitative semen traits in rabbits[END_REF]. This result is not confirmed after analyzing fertility results of the same line in our study because no relevant detrimental effects were obtained in the contrasts with the lowest levels of the temperature descriptors. To our knowledge, there is no other information in the literature concerning the effect of cold to moderate temperatures on male reproductive performance. However, it is known that endothermic mammals can better tolerate low body temperatures than high body temperatures [START_REF] Hansen | Effects of heat stress on mammalian reproduction[END_REF].

Effect of Temperature Descriptors on Female Fertility

Although some of the temperature contrasts had a HPD 95% that did include 0, the general tendency among the female descriptors Tmax_♀ and avgTmax (except for the avgTmax_18_31_♀; Figure 5) was that departures from the thermal comfort zone had a negative effect on female fertility. Heat stress produces impairment in several physiological processes involved in female reproductive performance that lead to a decrease in conception rate [START_REF] Marai | Rabbits' productive, reproductive and physiological performance traits as affected by heat stress: A review[END_REF]. To our knowl-Figure 6. Posterior mean and greatest posterior density interval at 95% of the effects on female fertility rate of the percentage of days with maximum temperature greater than 25°C in the periods from 10 to 1 d before AI, from 1 to 7 d after AI, from 8 to 18 d after AI, and from 19 to 31 d after AI (DI_10_0_♀, DI_0_7_♀, DI_7_18_♀, and DI_18_31_♀, respectively). Estimates of each figure were obtained in the models 6, 2, 9, and 11, respectively.

edge, there is no information in the literature concerning the effect of low temperatures on female fertility that could confirm our results. [START_REF] Cheng | Development of rabbit embryos during a 96-h period of in vitro culture after superovulatory treatment under conditions of elevated ambient temperature[END_REF] observed that does exposed to high temperature before mating had a greater embryonic degeneration than those maintained in thermoneutral conditions. A reduction in the development of mature follicles and in the number of developing oocytes has been observed in rabbits during summer heat stress [START_REF] Yassein | Hot climate effects and their amelioration on some productive and reproductive traits in rabbit does[END_REF]. It has also been reported in cattle and goats that heat stress affects follicular dynamics and ovulation rate [START_REF] Doney | The effect of premating stress on the onset of oestrus and on ovulation rate in Scottish blackface ewes[END_REF][START_REF] Roth | Heat stress, the follicle, and its enclosed oocyte: Mechanisms and potential strategies to improve fertility in dairy cows[END_REF]. Although a tendency is observed in our results, we cannot confirm the detrimental effect of high temperatures during the follicular maturation period and preimplantational phase of embryos on fertility, but low temperatures during the period from 10 d to AI had a negative effect on female fertility (avgTmax_10_0_♀; Figure 5). Although estimates were imprecise, a small proportion of hot days during the 10 d before AI, as well as a large proportion of hot days during the same period, seemed to favorably affect female fertility (DI_10_0_♀; Figure 6). This surprising result needs to be confirmed and could be in part related to temperature gradients that exist in the ovarian tissues of mature animals, which are generated at least in part as a consequence of endothermic reactions within Graafian follicles [START_REF] Hunter | Pre-ovulatory Graafian follicles are cooler than neighbouring stroma in pig ovaries[END_REF].

Low and high temperatures during the day of AI negatively affect female fertility. Ovulation in does occurs approximately 10 h after the injection of LH at AI [START_REF] Harper | Ovulation in the rabbit: The time of follicular rupture and expulsion of the eggs, in relation to injection of luteinizing hormone[END_REF]. Thus, it seems that processes related to the female such as ovulation and migration of the oocytes to the ampulla of the oviduct and fertilization could be affected by the environmental temperature of the farm on the day of AI. In addition, females could be less fertile because a hyperthermic reproductive tract can also affect the fertilizing ability of the sperm and the posterior embryo survival [START_REF] Howarth | Importance of uterine environment on rabbit sperm prior to fertilization[END_REF][START_REF] Hansen | Adverse impact of heat stress on embryo production: Causes and strategies for mitigation[END_REF]. In rabbit AI, the sperm is deposited deep in the vagina and close to the cervix. Then sperm have to migrate through the cervix and the uterus, then reach the isthmus of the oviduct, and remain trapped and held in a sperm reservoir until ovulation (Harper, 1973a;[START_REF] Overstreet | Sperm transport in the reproductive tract of the female rabbit: II. The sustained phase of transport[END_REF]. Once the oocytes reach the ampulla of the oviduct, hyperactivation [START_REF] Ho | Hyperactivation of mammalian spermatozoa: Function and regulation[END_REF] and capacitation [START_REF] Dziuk | Double mating of rabbits to determine capacitation time[END_REF] of spermatozoa allow the release of sperm from the reservoir and the migration to the oocytes. Capacitated sperm is thermotactically sensitive to the gradient of temperature that exists at ovulation time between the isthmic sperm reservoir and the fertilization site. The existence of this gradient enables the migration of the sperm to the oocytes [START_REF] David | Temperature changes in the different parts of the rabbit's oviduct[END_REF][START_REF] Bahat | Sperm thermotaxis[END_REF]. This temperature-dependent process could be affected by thermal stress if excess heat is not correctly dissipated in the female, reducing the ability of the sperm to reach the oocytes.

It is important to note that most of the studies dealing with the impact of high ambient temperatures on preovulatory follicle maturation, fertilization, and embryo development were carried out in climatic chambers or in vitro under extreme conditions of high and constant temperature. However, in commercial farm conditions such as the ones used in this study, there is a wide range of daily temperature, especially in summer (Figure 2). Even in the hot season, the thermoneutrality zone is reached during the night when rabbit activity is at a maximum and animals eat [START_REF] Prud'hon | Le comportamnet alimentaire du lapin dépend beaucup de l'abreuvement[END_REF], which could enhance the heat tolerance of animals.

Departures from thermoneutrality during the preand peri-implantation embryonic development caused a decrease in female fertility (avgTmax_0_7_♀; Figure 5). A similar pattern was observed for DI_0_7_♀ (Figure 6), but the majority of the contrasts included the 0 in their HPD 95% . Several authors described that this period of gestation was very sensitive to thermal stress, leading to a decrease in the latter embryo development and survival [START_REF] Putney | Embryonic development in superovulated dairy cattle exposed to elevated temperatures between days 1 to 7 after insemination[END_REF][START_REF] Ealy | Developmental changes in embryonic resistance to adverse effects of maternal heat stress in cows[END_REF][START_REF] Hansen | Adverse impact of heat stress on embryo production: Causes and strategies for mitigation[END_REF]. However, early embryos respond differently to thermal stress depending on their age and developmental status arriving to acquire thermotolerance in more advanced stages of development [START_REF] Ealy | Developmental changes in embryonic resistance to adverse effects of maternal heat stress in cows[END_REF][START_REF] Hansen | Effects of heat stress on mammalian reproduction[END_REF]. A threshold of thermotolerance exists in rabbit preimplantation embryos exposed to high temperatures in vitro [START_REF] Makarevich | The effect of hyperthermia in vitro on vitality of rabbit preimplantation embryos[END_REF].

It seems that high and low temperatures during the embryo and early fetal gestation periods also compromise female fertility because the probability of a negative value for the contrasts between the lowest and highest levels with the reference level was 94 and 92%, respectively (avgTmax_7_18_♀; Figure 5). However, female fertility seems to be unaffected by the proportion of days with high temperature in the same period (DI_7_18_♀; Figure 6). Embryo survival in utero under heat stress conditions has been more attributable to changes in maternal physiology than to changes in the embryo itself [START_REF] Hansen | Effects of heat stress on mammalian reproduction[END_REF].

As long as the gestation advances, embryos become more resistant to the maternal exposure to heat stress [START_REF] Ealy | Developmental changes in embryonic resistance to adverse effects of maternal heat stress in cows[END_REF][START_REF] Hansen | Adverse impact of heat stress on embryo production: Causes and strategies for mitigation[END_REF]. Accordingly, a reduction in the effect of the temperature on female fertility seemed to occur as the temperature descriptor is measured later in the gestation period. Therefore, the temperature descriptors covering gestation from 18 d after AI until birth had no influence on female fertility (avgTmax_18_31_♀ and DI_18_31_♀; Figures 5 and6, respectively).

Finally, it is important to mention that because of the high correlation existing among the temperature descriptors it is difficult to establish which effects of temperature are the most important. Moreover, the season of the year was not included in the model because of the confounding effect with the temperature. Therefore, temperature descriptors encompass not only the temperature effects but also all the possible season effects associated with it, such as environmental humidity and photoperiod, that could affect the AI results. However, as noted by [START_REF] Flowers | Genetic and phenotypic variation in reproductive traits of AI boars[END_REF], it is likely that the most important effects observed were due mostly to temperature, especially in this study where photoperiod was artificially controlled.

Effect of the Physiological Status of the Female on Female Fertility

Fertility of nulliparous females is about 6% (SD 0.01) less than the multiparous females in all of the models. That agrees with the decreased values obtained for prolificacy traits in the first parities with respect to the second and subsequent parturitions in other rabbit lines [START_REF] Baselga | Genetic diversity of litter size traits between parities in rabbits[END_REF][START_REF] Piles | Genetics of litter size in three maternal lines of rabbits: Repeatability versus multiple-trait models[END_REF]. In our study, no effect of lactation was encountered. This disagrees with the negative effect observed after AI of crossbred females from 2 maternal lines [START_REF] Tusell | Different ways to model biological relationships between fertility and the pH of the semen in rabbits[END_REF], as well as results obtained after natural mating of females of the same line as the one used in this work. However, no negative effect of lactation was encountered in females of a maternal line [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF], which could mean that the effect of lactation on fertility is dependent on line and type of mating.

The product threshold model appears as an interesting model to separately estimate the effect of some environmental factors relating male and female contributions to the fertility output (such as the different temperature descriptors analyzed in this work). However, this model assumes conditional independence between male and female fertility events. This fact does not allow taking into account the estimation of the effect of interactions between male and female factors that exists in the mating process, as described by several authors in the reproduction field (e.g., ovulationassociated alterations leading to changes in the pattern of sperm motility, ova stimulation of the migration of sperm from the reservoir to the fertilization site; Harper, 1973b;[START_REF] Overstreet | Sperm transport in the reproductive tract of the female rabbit: II. The sustained phase of transport[END_REF]. Moreover, not only the male and the female but also embryos are a component contributing in any fertility outcome. Further work should be done in this direction to incorporate in the product threshold model the possibility to estimate possible interactions between the 2 fertility events.

In conclusion, the environmental temperature of the day of AI seems to be the most relevant temperature descriptor affecting male fertility because high temperature records in the day of AI caused a decrease in male fertility. This effect could be indicative of the high sperm sensitivity to temperature changes during the processes of preparation and application of the AI doses. However, some sperm physiological processes occurring around the fertilization time could also be affected by temperature. From a practical point of view, it seems to be advisable to handle the sperm to be used for AI in a rapid and careful manner to not compromise its fertilizing ability.

Departures from the thermal zone in temperature descriptors covering several periods before AI until early gestation had a negative effect on female fertility, with the pre-and peri-implantational period of the embryos being especially sensitive. The latest period of gestation seemed to be unaffected by the temperature confirming the results described previously in other studies about the gain of thermotolerance reached in the latest stages of gestation.

The product threshold model allowed determining that male and female fertility are specifically affected by temperature in different periods around the insemination time. However, the magnitude and the persistence of the temperatures reached in the commercial conditions of this study did not seem to have a large effect on male and female rabbit fertility. Nevertheless, it is possible that the effect of temperature in prolific species leads to a reduction of the number of surviving embryos that could not be detected through the analysis of the pregnancy vs. nonpregnancy result of AI. The analysis of prolificacy outcomes would be a different study, and it could not take advantage of the use of the product threshold model.

Figure 2 .

 2 Figure 2. Panel A: average maximum daily temperature recorded per month in the male and female building (Tmax_♂ and Tmax_♀, respectively) and average percentage of fertility success per month (SE in bars). Panel B: average daily temperature range recorded per month in the male and female building (Range_♂ and Range_♀, respectively).

Figure 3 .

 3 Figure 3.Posterior mean and highest posterior density interval at 95% of the effects on male fertility rate of the average maximum daily temperatures in the periods from 40 to 21 d before AI, from 20 to 11 d before AI, and from 10 to 1 d before AI (avgTmax_40_20_♂, avgT-max_20_10_♂, and avgTmax_10_0_♂, respectively) and effect on male fertility rate of the maximum temperature the day of AI (Tmax_0_♂). Estimates of each figure were obtained in models 1, 3, 5, and 7, respectively.

Figure 4 .

 4 Figure 4. Posterior mean and highest posterior density interval at 95% of the effects on male fertility rate of the percentage of days with maximum temperature greater than 25°C in the periods from 40 to 21 d before AI, from 20 to 11 d before AI, and from 10 to 1 d before AI (DI_40_20_♂, DI_20_10_♂, and DI_10_0_♂, respectively). Estimates of each figure were obtained in models 2, 4, and 6, respectively.

Figure 5 .

 5 Figure5. Posterior mean and greatest posterior density interval at 95% of the effects on female fertility rate of the average maximum daily temperatures in the periods from 10 to 1 d before AI, from 1 to 7 d after AI, from 8 to 18 d after AI, and from 19 to 31 d after AI (avgT-max_10_0_♀, avgTmax_0_7_♀, avgTmax_7_18_♀, and avgTmax_18_31_♀, respectively) and effect on female fertility rate of the maximum temperature the day of AI (Tmax_0_♀). Estimates of each figure were obtained in the models 5, 7, 1, 8, and 10, respectively.

Table 1 .

 1 Distribution of AI data and maximum temperature on the day of AI

	Fertility data

Table 2 .

 2 Different product threshold models including several temperature descriptors in the male and the female liabilities of fertility (β ♂ and β ♀ , respectively) 1

	Model	β ♂	β ♀
	1	avgTmax_40_20_♂	avgTmax _0_7_♀
	2	DI_40_20_♂	DI_0_7_♀
	3	avgTmax_20_10_♂	avgTmax _0_7_♀
	4	DI_20_10_♂	DI_0_7_♀
	5	avgTmax_10_0_♂	avgTmax_10_0_♀
	6	DI_10_0_♂	DI_10_0_♀
	7	Tmax_0_♂	Tmax_0_♀
	8	avgTmax_40_20_♂	avgTmax _7_18_♀
	9	DI_40_20_♂	DI_7_18_♀
	10	avgTmax_20_10_♂	avgTmax_18_31_♀
	11	DI_20_10_♂	DI_18_31_♀
	1

This research was supported by the Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA, Madrid, Spain) project RTA2005-00088-CO2. Part of it was carried out during a visit of the first author to INRA (Castanet-Tolosan, France), Toulouse. Llibertat Tusell Palomero received a fellowship from the INIA. The authors are grateful to the staff of Unitat de Cunicultura, Institut de Recerca i Tecnologia Agroalimentàries (Caldes de Montbui, Barcelona, Spain) for their valuable

cooperation. 2 Corresponding author: llibertat.tusell@irta.cat

Received January 31, 2011. Accepted June 29, 2011.

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François G Schmitt
email: francois.schmitt@univ-lille1.fr

Patrick Huerre

About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity

Keywords: Fluid Mechanics, Turbulence, Constitutive equation Mots-clés : Mécanique des fluides, Turbulence, Equation constitutive

Boussinesq's hypothesis is at the heart of eddy viscosity models, which are used in many different fields to model turbulent flows. In its present time formulation, this hypothesis corresponds to an alignment between Reynolds stress and mean strain tensors. We begin with historical remarks on Boussinesq's results and recall that he introduced a local averaging twenty years before Reynolds, but using an approach that prevented him from discovering Reynolds' stress tensor. We then introduce an indicator that characterizes the validity of this hypothesis. For experimental and numerical databases, when the tensors are known, this can be used to directly estimate the validity of this hypothesis. We show, using several different databases, that this hypothesis is almost never verified. We address in conclusion the analogy with kinetic theory, and the reason why this analogy cannot be applied in general for turbulent flows.

Résumé

A propos de l'hypothèse de viscosité turbulente de Boussinesq : rappels historiques et évaluation directe. L'hypothèse de Boussinesq est au coeur des modèles de viscosité, utilisés dans un grand nombre de contextes pour modéliser des écoulements turbulents. Dans sa formulation moderne, cette hypothèse correspond à un alignement entre tenseur de contrainte de Reynolds et tenseur de déformation moyen. Nous rappelons le contexte historique de l'énoncé de cette hypothèse, en soulignant que Boussinesq avait introduit une moyenne locale vingt ans avant Reynolds, mais en effectuant une erreur qui l'a privé de la mise en évidence du tenseur de Reynolds. Nous introduisons ensuite un indicateur, compris entre 0 et 1, indiquant le degré de validité de cette hypothèse. Pour des bases de données expérimentales et numériques, lorsque les différents tenseurs sont connus, ceci permet de tester directement, "a priori", cette hypothèse. Nous montrons ainsi, utilisant différentes bases de données d'écoulements turbulents, que l'hypothèse n'est presque jamais vérifiée. Nous discutons en conclusion de la théorie cinétique des gaz et de la raison pour laquelle cette analogie est discutable pour les écoulements turbulents.

Version française abrégée

Le "problème de la turbulence" touche un grand nombre de domaines, incluant l'ingénierie automobile, chimique, la combustion, l'aéronautique, la météorologie, l'océanologie, l'hydrologie, l'hydraulique fluviale, etc. Ces domaines sont à fort potentiel industriel et environnemental, et demandent souvent des réponses pratiques et quantitatives, faisant appel à des modèles. Parmi les différentes familles de modèles existant, beaucoup utilisent une moyenne de Reynolds, qui font intervenir les fluctuations instantannées à petites échelles via le tenseur de Reynolds. La modélisation intervient ici, et permet de fournir, via une hypothèse, une "fermeture" exprimant le tenseur de Reynolds en fonction de quantités moyennes. La fermeture la plus courante, fournissant le tenseur de Reynolds en fonction du champ de vitesse moyen, est celle qui est utilisée dans les modèles de viscosité turbulente, souvent appelée "hypothèse de Boussinesq". Il semble ici paradoxal de constater que la "moyenne de Reynolds" date d'une publication de 1895, tandis que l'"hypothèse de Boussinesq" permettant d'exprimer le tenseur de Reynolds est largement antérieure : la publication date de 1877, mais il s'agit d'un compte-rendu d'une séance de l'Académie des Sciences de 1872. Comment se fait-il que Boussinesq ait proposé en 1872 une fermeture pour une équation qui n'a été établie plus de 20 ans plus tard ?

La réalité historique est un peu plus complexe que ce qui transparait dans les citations dans les ouvrages actuels sur la turbulence ; nous pensons donc utile de revenir ici sur la publication originale de Boussinesq, disponible par exemple sur le site Gallica (http ://gallica.bnf.fr). Dans sa publication de 1877, Boussinesq effectue déjà une moyenne locale de l'écoulement, qu'il suppose stationnaire. Il considère les "composantes, suivant les axes, des actions exercées" à travers des plans. Il effectue alors un raisonnement qui est analogue à celui qui est utilisé en théorie cinétique, mais cette-fois ci pour les vitesses turbulentes : il obtient ainsi une équation de transport pour le champ moyen analogue à l'équation définie pour les valeurs instantanées. Boussinesq note bien que l'expression moyennée qu'il obtient obéit toujours aux équations de Navier-Stokes, mais qu'il faut remplacer le coefficient de viscosité par un nombre beaucoup plus grand, qu'il note ǫ, qui n'est plus une constante et qui dépend "encore et surtout de l'intensité de l'agitation moyenne qui s'y trouve produite". Il donne ensuite pour ce coefficient une expression qui préfigure le modèle de longueur de mélange formalisé plus tard par Prandtl. En utilisant les termes actuels, on peut constater que Boussinesq a effectué à la fois (de façon simultanée) une moyenne des équations de Navier-Stokes, et l'introduction d'une modélisation tensorielle de type viscosité turbulente, en utilisant une analogie avec la théorie cinétique. Plus tard, Reynolds a été plus précis, utilisant le même cheminement, mais fournissant de façon explicite les équations moyennées ; mais il ne mentionna pas les contributions de Boussinesq.

Dans une seconde partie, nous considérons l'hypothèse de Boussinesq formulée en utilisant les notations actuelles. Nous rappelons qu'elle correspond à un alignement entre deux tenseurs. Pour quantifier cette hypothèse, nous introduisons un indicateur qui est construit à partir d'un produit scalaire entre tenseurs. Cet indicateur est compris entre 0 et 1, et vaut 1 lorsque l'hypothèse est vérifiée. Pour pouvoir estimer cet indicateur, il faut disposer de toutes les composantes du tenseur de Reynolds et du tenseur de déformation moyen. Dans une troisième partie, nous choisissons des bases de données turbulentes permettant d'estimer cet indicateur. Nous utilisons des données de simulation numérique directe (DNS), des données de simulation à grande échelle (LES), et des données expérimentales. Les résultats obtenus à partir de ces bases de données vont tous dans le même sens : l'hypothèse de Boussinesq, qui est à la base de nombreux modèles de viscosité turbulente, est rarement vérifiée. Le problème essentiel de cette fermeture est qu'elle repose sur une analogie avec la théorie cinétique, analogie dont nous rappelons la

Introduction

The "turbulence problem" is present in many fields, including car industry, chemical engineering, combustion studies, aeronautics, meteorology, oceanography, hydrology, fluvial hydraulics, etc. These fields have a strong industrial and environmental potential, and often need practical and quantitative answers. As well known, turbulence is one of the last subjects of classical physics which is still not solved. This is why in engineering sciences, or in environmental studies, turbulence models are used [START_REF] Tennekes | A first course in turbulence[END_REF][START_REF] Wilcox | Turbulent modeling for CFD[END_REF][START_REF] Pope | Turbulent flows[END_REF][START_REF] Mathieu | An introduction to turbulent flow[END_REF][START_REF] Bernard | Turbulent flow; Analysis, Measurement, and Prediction[END_REF]. Among the different existing models, many introduce Reynolds averaging [START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF]. In the original publication, Reynolds average correspond to average Navier-Stokes equations on boxes of intermediate size. In more recent formulations, the "Reynolds averages" that are estimated correspond to an ensemble average [7]. The objectives of these approaches is to discretize the modeled equations on a grid of adequate size, so that the processing time will not be too large. This is a way to decrease the number of degrees of freedom. Thus the idea is not to use a theory to provide an instantaneous turbulent solution, but to provide a mean numerical solution on a rough grid, using a model. The important point is the non-linearity of Navier-Stokes equations: the averaged equations are not closed and are expressed using small-scale instantaneous fluctuations, through the Reynolds tensor [START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF]. The introduction of a model is then necessary to close the equations, and provide, through an appropriate hypothesis, a "closure" expressing the Reynolds stress tensor using averaged quantities.

Boussinesq and Reynolds: historical remarks

The main closure, which provides the Reynolds stress tensor using the gradient of the mean velocity field, is used in turbulent viscosity models; it is often denoted "Boussinesq's hypothesis" (see some comments in [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF], pp. 222-3). It seems here paradoxical to mention that for "Reynolds averages" a publication of 1895 is cited [START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF], whereas for "Boussinesq's hypothesis" which provides an expression for the Reynolds stress tensor, a publication of 1877 is cited, which is much earlier. This publication is in fact a report of a meeting of the French Academy of Science that was held during 1872. How is it possible that Joseph Boussinesq proposed in 1872 a closure for an equation that would be written more than 20 years later? Historical facts are somewhat more complex than what is usually mentioned in present time turbulence text books. We therefore think here that it is useful to go back to Boussinesq's original publication, which can be found in French at Gallica's web site (http://gallica.bnf.fr).

In his 1877 publication, Boussinesq performs already a local average of the flow, which is assumed stationary. He first considers temporal averages, done "during a rather short time τ " (p. 24). Soon after this, he assumes that velocity components are not correlated. He notes u 1 , v 1 , w 1 the components of the instantaneous velocity of an element of fluid, and u = u 1 its temporal average. He then introduces the acceleration u ′ 1 (Equation (2) page 28):

u ′ 1 = du 1 dt + u 1 du 1 dx + v 1 du 1 dy + w 1 du 1 dz (1)
We see here that he was not using the partial derivative notations. He then writes for the averaged acceleration u ′ = u ′ 1 (Equation (3) page 29):

u ′ = du dt + u du dx + v du dy + w du dz (2)
corresponding to the assumption

u 1 ∂ x v 1 = u 1 ∂ x v 1
, and also to

u 1 ∂ x u 1 = u 1 ∂ x u 1 .
He states further that there are some special situations (page 31): "when a group of molecules go up, we have at the same time w 1 > 0 et du1 dz > 0, whereas we have w 1 < 0 et du1 dz < 0 when the same group goes down. The product w 1 du1 dz is positive in both cases." He deduces from this that equation ( 2) is wrong in those cases, but he concludes "we see that this happens only in a relatively limited region and almost always negligible". In fact, the real situation is the opposite: here Boussinesq did not notice the apparition, through a temporal average of transport equations, of the tensor that will later be called Reynolds tensor. It is surprising in fact to discover that a few lines later, he implicitly introduces such a tensor when he considers the "components among the axes, of actions exerted" through planes of the fluid element. The average of the normal components of these actions are denoted N 1 , N 2 and N 3 , and of the tangential components T 1 , T 2 and T 3 . We recognise here the different components of the tensor -u i u j , but this identification is not mentioned by Boussinesq. Through an approach which is similar to the one done in kinetic theory, but here for turbulent velocities, he performs a development in Taylor's series and obtains the different components of the "mean actions exerted through fixed planes" (page 42). Those are expressed as (equation ( 12) page 46):

N 1 = -p + 2ǫ du dx ; N 2 = -p + 2ǫ dv dy ; N 3 = -p + 2ǫ dw dz T 1 = ǫ dv dz + dw dy ; T 2 = ǫ dw dx + du dz ; T 3 = ǫ du dy + dv dx (3)
In this equation, p is the pressure exerted on an element of fluid by the surrounding, given by (page 46):

- 1 3 (N 1 + N 2 + N 3 ) = p - 2 3 ǫ du dx + dv dy + dw dz (4)
We see here that for incompressible flows, and using usual present time notations, we have p = 2 3 K, where K is the kinetic energy, written as 2K = -(N 1 + N 2 + N 3 ). The equation (3) given by Boussinesq is in fact a tensorial closure providing stresses on the form:

- 2 3 Kδ ij + ǫ du i dx j + du j dx i (5)
Boussinesq remarks that the average expression he obtains still obeys Navier-Stokes equations, where the viscosity coefficient is replaced by a number which is much larger, denoted ǫ. This is no more a constant, and depends "mainly on the mean agitation which is produced". An expression is proposed by Boussinesq for ǫ:

ǫ = ρgΛhu 0 ( 6
)
where ρg is the weight of a volume unity, Λ is a scalar, slowly varying with h and u 0 , which are respectively a characteristic length and velocity. This relation for ǫ which is now denoted as "turbulent viscosity", or "effective viscosity", is already a mixing length formulation, as developed later by Prandtl [START_REF] Prandtl | Bericht ber Untersuchungen zur ausgebildeten Turbulenz[END_REF]. We thus see that, despite an error in the beginning concerning the correlations of velocity components, "Boussinesq's hypothesis" was originally explicitly mentioned as a tensorial relation, with a justification linked to mixing length arguments. Using present day vocabulary, we see that Boussinesq performed simultaneously an average of Navier-Stokes equations and a tensorial closure of the eddy-viscosity type, using an analogy to kinetic theory.

Boussinesq thought that he solved the turbulence problem; he did not realize his mistake in the average of Navier-Stokes equations, neither did he notice that he was assuming a strong hypothesis when he performed his analogy with kinetic theory. In the same volume, the 1877 contribution of Boussinesq is introduced by Saint Venant (a 22 pp. introduction) [START_REF] Saint-Venant | Rapport sur un mémoire de M. Boussinesq[END_REF]. The latter writes that Boussinesq solved a "true enigma". Much later, in his posthumous speech on Boussinesq's achievements, the President of the French Academy of Sciences Emile Picard mention his results as if they were exact results: "It is a remarkable result due to Boussinesq, that Navier's equations are still valid, but introducing, instead of real molecular velocities, their local averages." ( [START_REF] Picard | Discours et notices[END_REF], p. 78). On the other hand, it is surprising to notice that Boussinesq does not cite the 1895 work of Reynolds in his 1897 book. In this paper, Reynolds undertook the same averaging approach, but with a more rigourous line, introducing the stress tensor. There is a citation default in both cases, since Reynolds did not cite the 1877 results of Boussinesq on local averages.

Let us note here that neither Boussinesq nor Reynolds seem to have used the word "turbulence" in their publications: Boussinesq [START_REF] Boussinesq | Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides[END_REF][START_REF] Boussinesq | Essai sur la théorie des eaux courantes[END_REF][START_REF] Boussinesq | Théorie de l'écoulement tourbillonnant et tumultueux des liquides[END_REF] used "tumultuous movements", "eddy agitations", "movements theory", "liquid eddy theory" (in French, "mouvements tumultueux", "agitation tourbillonnaire", "théorie des remous", "théorie des tourbillons liquides", "force vive"), whereas Reynolds [START_REF] Reynolds | An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels[END_REF][START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF] used "sinuous paths", "sinuous motion", "irregular eddies", "sinuous or relative disturbance". The word turbulence seems to have been introduced in the field of fluid mechanics for the first time by William Thomson (later Lord Kelvin) in 1887 [START_REF] Lord Kelvin | On the propagation of laminar motion through a turbulently moving inviscid liquid[END_REF]. This word was not adopted by Boussinesq or Reynolds in their works posterior to 1887 (see the title of Boussinesq, 1897: Theory of tumultuous and whirling liquid flows [START_REF] Boussinesq | Théorie de l'écoulement tourbillonnant et tumultueux des liquides[END_REF]); none of them did cite Kelvin in their 1890s publications [START_REF] Reynolds | On the dynamical theory of incompressible viscous fluids and the determination of the criterion[END_REF][START_REF] Boussinesq | Théorie de l'écoulement tourbillonnant et tumultueux des liquides[END_REF]. The word turbulence was adopted in the field only later, in the beginning of the XXth century, in particular through the works of Prandtl.

Direct test of Boussinesq's hypothesis

We come back here to turbulent viscosity models. Reynolds averaging of Navier Stokes equations is written using the Reynolds stress tensor < u i u j >, where (u i ) is the fluctuating velocity, (U i ) the mean velocity, and < . > represents ensemble averaging. To achieve closure of the mean equations, it is necessary to express the Reynolds tensor from the mean velocity, and other mean quantities. Among the latter is the mean kinetic energy K = 1 2 < u i u i >. We introduce also the traceless stress tensor, also sometimes called anisotropic stress tensor: R =< u i u j > - 2 3 KI where bold notation is used for tensors, and I is the unit tensor. We also introduce the mean velocity gradient tensor A = ∂U i /∂x j and its symmetric part, giving the mean strain tensor:

S = 1 2 ∂U j ∂x i + ∂U i ∂x j (7)
whose trace vanishes for incompressible flows. Boussinesq's hypothesis corresponds then to a closure hypothesis with the following linear constitutive equation:

R = -2ν T S (8)
where ν T is a scalar coefficient called turbulent viscosity (sometimes called "effective viscosity"). This equation is a linear relation between stress and strain tensors, and is analogous to the linear constitutive equation for Newtonian flows: R ν = -2νS, where R ν is the viscous stress tensor and ν the viscosity.

In Equation ( 8), the eddy-viscosity is written for the classical K -ǫ model [START_REF] Launder | The numerical computation of turbulent flows[END_REF] using two independent turbulent quantities such as K and the dissipation ǫ: ν T = C µ K 2 /ǫ, where C µ is a non-dimensional quantity (in some recent models it is no more constant). Closure is achieved with transport equations for K and ǫ. Here we do not consider transport equations.

The main hypothesis of equation ( 8) is here the fact that the 2 tensors are proportional. This proportionality can be easily tested when the two tensors R and S are known. For this, we introduce an inner product (or "contracted product") between matrices that writes A : B = {A t B} = A ij B ij , where {.} is the trace. The associated norm is the following: A 2 = A : A. For symmetric tensors, one has A : B = {AB} and A 2 = {A 2 }. This can be used to introduce an indicator ρ RS which is defined through an inner product between tensors [START_REF] Schmitt | Experimental study of the constitutive equation for an axisymetric complex turbulent flow[END_REF]:

ρ RS = |R : S| R S (9)
which is analogous to the cosine of the angle between vectors. The ratio ρ RS is thus a number between 0 and 1, which characterizes the validity of Boussinesqs hypothesis: it is 1 when this hypothesis is valid, and the less it is valid, the more this number is close to 0 (this corresponds to "perpendicular" tensors). Pushing further the analogy with an angle between vectors, one can consider that alignment is approximatively verified for angles smaller than π/6, corresponding to a value of ρ RS larger than 0.86. This indicator is thus a direct indicator of the validity of the basic hypothesis of all linear eddy viscosity models. To estimate this number, one needs to have access to all the components of the Reynolds stress and mean strain tensors in order to estimate the invariants {RS}, {R 2 } and {S 2 }. The stress tensor is often completely or partially given in turbulent databases through turbulence intensity components and the various shear stress terms; the strain tensor is less often available, since it involves derivatives in all directions, whose estimation requires the data base to be provided on a fine enough grid. Nevertheless, except for complex 3D flows, available databases correspond to simplified geometries possessing symmetries, so that many components of the Reynolds stress and mean strain tensors vanish; this is the case for most of the databases tested here, corresponding to 2D mean flows and an axisymmetric jet flow. This indicator is applied below to numerical and experimental databases.

Application of the indicator to numerical and experimental databases

Direct numerical simulation data of simple flows

The DNS data analyzed here correspond to simple shear flows possessing only one non-zero velocity gradient Some of these data bases have been studied in details elsewhere [START_REF] Schmitt | Direct test of a nonlinear constitutive equation for simple turbulent shear flows using DNS data[END_REF]. The first database (denoted CO in the following) corresponds to a turbulent plane Couette flow at a Reynolds number Re = 1300 with a friction Reynolds number Re τ = 82 [START_REF] Bech | An investigation of turbulent plane Couette flow at low Reynolds numbers[END_REF]. The second database (denoted CF87 in the following) corresponds to a turbulent channel flow at a Reynolds number Re = 3250 corresponding to a friction Reynolds number of Re τ = 180 [START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF], and the third (denoted CF99) is the same flow at larger Reynolds number: Re = 10 4 corresponding to a friction Reynolds number of Re τ = 590 [START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Reτ = 590[END_REF]. The fourth database (denoted BL in the following) is a turbulent boundary layer on a flat plate, with zero pressure gradient, with a Reynolds number of Re = 2 10 5 or based on the momentum thickness θ, Re θ = 1410 [START_REF] Spalart | Direct simulation of a turbulent boundary layer up to Rt = 1410[END_REF]. The last DNS database corresponds to an annular pipe flow (denoted AP) with a bulk Reynolds number of Re = 2800 (based on bulk velocity and δ, the half-width of the annular gap) or friction Reynolds number Re τ = 180 [START_REF] Quadrio | Direct numerical simulation of the turbulent flow in a pipe with annular cross-section[END_REF]. The inner diameter is denoted R i and the outer is R o = 2R i . The adimensional radial distance r = (d -R i )/δ is then a number between 0 and 2. The DNS test cases chosen correspond to a panel of Reynolds numbers going from 10 3 to 10 5 , with friction Reynolds numbers from 82 to 1410. The friction Reynolds numbers above are defined using the friction velocity u τ = τ w /ρ, where τ w is the modulus of the wall shear stress. In the following, most quantities are non-dimensionalized, using u τ or u 2 τ ; the distance to the wall is expressed as usual in wall units (also a local Reynolds number) y + = yu τ /ν, (except for the AP database where natural coordinate is kept in order to visualise in the same graph the behaviour near the two walls). For these shear flows, only 1D profiles are necessary. All the profiles of the Reynolds stress tensors are usually provided. Furthermore, the only non-zero gradient is a = U ′ (y) whose knowledge provides the mean strain tensor:

S = a 2      0 1 0 0 0 0 0 0      (10)
Figure 1 represents the normalized normal stresses for the data sets, showing that the anisotropy of the stresses is pronounced. It is well-known (see e.g. [START_REF] Nisizima | Turbulent channel and Couette flows using a anisotropic k-ǫ model[END_REF][START_REF] Speziale | On nonlinear k-ℓ and k-ǫ models of turbulence[END_REF]) that since normal stresses are not identical (corresponding to anisotropy), Boussinesq's hypothesis cannot be valid. Indeed, the anisotropic stress tensor is here: [START_REF] Saint-Venant | Rapport sur un mémoire de M. Boussinesq[END_REF] and for the latter to be proportional to S, it would be necessary that diagonal terms vanish. Many authors have mentioned that Boussinesq's hypothesis is not correct for simple shear flows; our contribution in this context is here to use the representation of the indicator ρ RS (y + ) to quantify the degree of validity of Boussinesq's hypothesis (Figure 2). In this figure, we have represented the total stress and the turbulent stress alone. Very close to the wall, the viscous stress is dominant as expected, and the total stress is aligned to the strain tensor. Moreover, this figure directly shows that the viscous term has an influence on the total stress until a distance of y + = 20, but the alignment is bad for for y + > 2. Globally, for the total stress, the alignment, corresponding to the validity of Boussinesq's hypothesis, is bad for 3 < y + < 70, (corresponding roughly to the buffer layer). There is a local minimum at about the same position for all data bases, y + = 10.5 ± 0.5. The different curves superpose rather well: the data bases have approximatively the same behaviour, indicating a possible universality in the failure of this linear constitutive relation near the walls.

R =       - 2 3 K + u 2 uv 0 uv - 2 3 K + v 2 0 0 0 - 2 3 K + w 2       (
The annular pipe flow presents some interesting non-symmetry effects (between inner and outer cylin-For this ρ RS is shown in real instead of wall units in Figure 3. For this flow, this ratio has an interesting shape. Boussinesqs hypothesis is never valid (except very close to the walls for the total stress), but it is the worse at the position r 0 = 0.87 corresponding to an asymmetric center position, which is closer to the inner cylinder than the outer one. As shown in Figure 4, this position corresponds to the annulation of the mean velocity gradient tensor. The shear stress also vanishes at this position, whereas normal stresses are not isotropic, so that R is not vanishing, but is diagonal, whereas the mean strain tensor is vanishing:

R(r 0 ) =      -0.19 0 0 0 0.10 0 0 0 0.09      ; S(r 0 ) = 0 ( 12
)
The different turbulent values at this position correspond then to the non-validity of Boussinesq's hypothesis and also of any polynomial non-linear constitutive equation such as used in non-linear K-ǫ models, as discussed below.

LES data

The LES (Large Eddy Simulation) data considered here correspond to the flow over a square cylinder, with a Reynolds number of Re = 22000. This geometry was used as a test case in a workshop on LES of flows past bluff bodies held in 1995 in Germany (see a report in [START_REF] Rodi | Status of Large Eddy Simulation: Results of a Workshop[END_REF]). The data analyzed here were provided by the university of Surrey, UK. The mesh size is 257 × 241 × 64; the mean flow is 2D and averages are taken over time (56 different time sections) and the spanwise dimension (64 values). This provides 64 × 56 = 3584 samples which were used to compute mean quantities and second moments such as all components of the Reynolds stress tensor. Figure 5 shows the streamlines of the mean flow. Gradients were estimated using a second order finite difference scheme, giving access to all components of the mean strain and vorticity tensors.

This database has been used elsewhere to check the kinetic energy transport equation [START_REF] Schmitt | Direct investigation of the K-transport equation for a complex turbulent flow[END_REF]. Figure 6 represents ρ RS estimated for the whole region: light regions indicate validity of Boussinesq's hypothesis (value close to 1). An isoline represents the value = cos π/4 ≈ 0.71, corresponding to a roughly valid hypothesis. The region inside this isoline is not of wide extension. Since the tensors are very small in the inflow region, the alignment ratio is not plotted to avoid unnecessary scatter. Contrary to previous examples, this corresponds to a complex flow, with mixing and recirculation regions. A LES database has been chosen since DNS is not yet possible for such flow. The result shown here confirms previous results obtained for simple shear flows.

Experimental data

We consider also here an experimental database, corresponding to a double annular turbulent jet flow, generated by a confined double annular burner in cold conditions. Two dimensional LDV measurements have been performed on 5515 grid points close to the nozzle exit, where the flow is characterized by vortices, recirculation, high mixing rate. At each measurement position, statistics have been computed on 3000 to 16000 particles. The resulting mean flow field is axisymmetric within 2 p.cent. The nonzero components of the mean velocity gradient tensor have been computed on the fine grid, so that the above procedure can be applied to test Boussinesq's hypothesis with the indicator ρ RS computed at each grid position. The data base is available online (http://stro9.vub.ac.be/expdata); a complete description is given in [START_REF] Schmitt | LDV measurements of the flow field in the nozzle region of a confined double annular burner[END_REF]. Figure 7 below shows the streamlines of the flow analyzed. Figure 8 shows a map of ρ RS . Validity of Boussinesq's hypothesis are represented by white regions, which are of relatively limited extension. Despite important noise due to the experimental nature of the database, we can say that Boussinesq's hypothesis is not validated for such complex flow.

Discussion and conclusion

The results obtained using numerical (DNS or LES) or experimental data are very consistent in pointing the non-validity of Boussinesq's hypothesis, which is at the heart of many turbulent viscosity models. However, viscosity turbulence models such as K-ǫ model are widely used for many applications, and seem to provide satisfactory predictions. This may be seen as a contradiction. We must note in fact that these models predict rather closely only simple flows, and only as far as mean fields are concerned (mean velocities, streamflows). For example, in simple shear flows the shear stress is dominant in mean transport equations, so that the failure of the linear constitutive equation does not have important consequences for the final result. But predictions for second order moments (Reynolds stresses, kinetic energy, dissipation) are not satisfactotry, even for simple flows. And for complex flows, having high mixing rates, recirculation regions, stagnation lines, the predictions of these models can be qualitatively wrong, even for first moments (see [START_REF] Pope | Turbulent flows[END_REF] for more comments). The inaccuracies come from the transport equations and the linear constitutive equation.

The main limitation of this linear closure is the fact that it rests on an analogy with kinetic theory, analogy that can be criticised on theorical grounds, as discussed below (see also discussion in [START_REF] Tennekes | A first course in turbulence[END_REF][START_REF] Pope | Turbulent flows[END_REF][START_REF] Bernard | Turbulent flow; Analysis, Measurement, and Prediction[END_REF]).

Kinetic theory and scale separation

Eddy-viscosity turbulence models are the subject of thousands of papers, yet only a very small number of them discuss the basic theoretical weaknesses of Boussinesq and the gradient diffusion hypothesis when applied to turbulent flows (see e.g. [START_REF] Lumley | Toward a turbulent constitutive relation[END_REF][START_REF] Corrsin | Limitations of gradient transport models in random walks and turbulence[END_REF][START_REF] Egolf | Difference-quotient model: a generalization of Prandtls mixing-length theory[END_REF][START_REF] Hinze | Memory effect in a turbulent boundary-layer flow due to a relatively strong axial variation of the mean-velocity gradient[END_REF][START_REF] Bernard | Reynolds stress and the physics of turbulent momentum transport[END_REF][START_REF] Tavoularis | Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient, part I[END_REF]). As already implicitly assumed by Boussinesq, the basis of such modelling is an analogy with kinetic theory, where the viscous stress is expressed using the gradient of the mean field (see also the comments in [START_REF] Pope | Turbulent flows[END_REF] ou [START_REF] Bernard | Turbulent flow; Analysis, Measurement, and Prediction[END_REF]). The simple situation where molecules have a mean velocity of the form (U (y), 0, 0) is often taken as an illustrative example: the gradients are in fact an approximation of the finite difference quantity U (y + ℓ m ) -U (y -ℓ m ), where ℓ m is the mean-free path of molecules. Since this distance is very small compared to the scale of variation of U , the following approximation is clearly justified:

U (y + ℓ m ) -U (y -ℓ m ) = 2ℓ m ∂U ∂y (13)
For turbulent flows the same construction is implicitely assumed, taking for ℓ m the mean free path of fluid elements, i.e. the turbulent mixing length introduced by Prandtl. For this, the hypotheses below must be realised:

-each element of fluid possesses a turbulent lagrangian free path ℓ, which is a random variable whose distribution is peaked around a mean value ℓ m , the turbulent mixing length; -the turbulent mixing length is small compared to the scale of variation of mean quantities. The free path of an element of fluid in the Lagrangian framework can be defined with the Lagrangian autocorrelation function. Indeed, for each Lagrangian trajectory a characteristic time (the integral of the autocorrelation function) can be estimated, providing a characteristic length, which is the mixing length associated to the trajectory. But the second hypothesis above is less justified: since fully turbulent flows possess high non-gaussian variability, it is likely that individual free paths take a whole range of values so that their collective effect in the computation of the stress cannot resume in just one difference such as given in [START_REF] Boussinesq | Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides[END_REF]. Furthermore, their mean is interpreted as the turbulent mixing length, which is generally not small compared to the scale of variation of the mean velocity, so that the separation of scales does not hold in general for turbulent flows [START_REF] Tennekes | A first course in turbulence[END_REF].

Nonlinear developments of Boussinesq's hypothesis

Since Pope [START_REF] Pope | A more general effective-viscosity hypothesis[END_REF], it is recognized that invariant theory (and especially results obtained in the fifties by Rivlin and Spencer, see [START_REF] Spencer | Theory of invariants[END_REF]) can be used in the framework of turbulence modeling to represent the stress tensor as a development into a tensor basis composed of no more than 10 basis tensors. This provides the following general development:

R = 10 i=1 a i T i (14)
where the coefficients a i are scalar invariants of the flow, which are scalar fields having values independent of the system of reference, corresponding to traces of different tensor products [START_REF] Spencer | Theory of invariants[END_REF]. In nonlinear eddyviscosity turbulence models (see e.g. [START_REF] Yoshizawa | Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation[END_REF][START_REF] Nisizima | Turbulent channel and Couette flows using a anisotropic k-ǫ model[END_REF][START_REF] Speziale | On nonlinear k-ℓ and k-ǫ models of turbulence[END_REF]), a nonlinear constitutive relation such as ( 14) is used, with a tensorial basis T i that can be expressed on the form of products of the mean strain tensor S and of the mean vorticity tensor W. Most of these models are polynomials, of degree 2 or 3. These models generalise models of the K-ǫ family and are often called nonlinear K-ǫ models [START_REF] Pope | Turbulent flows[END_REF]. These nonlinear generalizations of Boussinesq's hypothesis help to overcome the clear limitations of the linear framework, but in a too superficial way, since they still provide closure using gradients of mean quantities; using DNS data of an annular pipe flow, we have shown above that, at a near central position, a nonlinear polynomial constitutive equation is not adequate since the strain and vorticity tensors vanish, whereas the anisotropic stress does not. This example shows that such non-linear constitutive equation cannot represent a full and consistent answer to the closure problem. The development of a new approach, more deeply connected to the multiscale and long-range correlated nature of fully developed turbulence seems desirable, providing in particular a turbulent constitutive relation of a new type. An interesting direction for further research is provided by non-local models, that take into account the history of turbulence: several models have already been proposed [START_REF] Corrsin | Limitations of gradient transport models in random walks and turbulence[END_REF][START_REF] Hinze | Memory effect in a turbulent boundary-layer flow due to a relatively strong axial variation of the mean-velocity gradient[END_REF][START_REF] Tavoularis | Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient, part I[END_REF][START_REF] Egolf | Difference-quotient model: a generalization of Prandtls mixing-length theory[END_REF][START_REF] Huang | On modelling the Reynolds stress in the context of continuum mechanics[END_REF], and an explicit expression for the kernel of the non-local model has recently been proposed and tested on simple flows [START_REF] Hamba | Nonlocal analysis of the Reynolds stress in turbulent shear flow[END_REF]. Direct analysis of DNS data, using a Lagrangian approach to follow elements of fluid, could also be a good way to better understand the mechanism inside the formation of the turbulent stress [START_REF] Bernard | Reynolds stress and the physics of turbulent momentum transport[END_REF][START_REF] Handler | On the role of accerelating fluid particles in the generation of Reynolds stress[END_REF], and improve the recent proposal of Hamba [START_REF] Hamba | Nonlocal analysis of the Reynolds stress in turbulent shear flow[END_REF]. Turbulence models have important engineering applications, and thus in this framework, having models whose predictions are only partly in agreement with reality is better than having no agreement at all. To this aim, available turbulence models possess several tuning parameters, that are determined using a posteriori validation. This type of validation is often restricted to mean profiles, and the comparison is qualitative. Depending on the type of application considered, such qualitative prediction may be enough; it is nevertheless certainly better to be able to provide good quantitative predictions for second and higher moments. Indeed, depending on the application foreseen, fluxes and even extreme and rare events can be important (e.g. for building structures able to resist strong winds); furthermore, fully developed turbulence develops intermittent fluctuations and contains long-range correlations (see reviews in [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF][START_REF] Pope | Turbulent flows[END_REF]). Such effects are not considered in classical eddy-viscosity models, and thus are not predicted. It is in this respect that direct a priori tests are useful, in order to better assess the source of the weaknesses of available models, to be able to cure these weaknesses and develop new turbulence models.

Figure 1 .

 1 Figure 1. Normal stresses for DNS databases.

Figure 2 .

 2 Figure 2. Plot of ρ RS (y + ) for DNS databases: (1) total stress (turbulent+viscous stress) and (2) turbulent stress only.

Figure 3 .

 3 Figure 3. Plot of ρ RS (r) for the DNS database AP: total stress (continuous line) and turbulent stress alone (dotted line).

Figure 4 .

 4 Figure 4. Plot of the shear stress for the AP DNS database.

Figure 5 .

 5 Figure 5. Streamlines of the flow past a square cylinder estimated from LES data.

Figure 6 .

 6 Figure 6. A map of ρ RS for the LES database. The inflow region is not plotted. The isoline corresponds to an "angle" of π/4. Only the region inside this isoline corresponds approximatively to Boussinesq's hypothesis.

Figure 7 .

 7 Figure 7. Streamlines of the double annular jet flow estimated from experimental data, showing recirculation and mixing regions.

Figure 8 .

 8 Figure 8. A map of ρ RS for the experimental double annular jet data. Only white regions correspond to validity of Boussinesq's hypothesis.

Acknowledgements thank Pr. P. Voke for providing us the LES database, Dr. M. Quadrio for sending us his DNS data, C. Hirsch, G. Mompean, B. Merci for discussions, and reviewers for interesting suggestions.


Katell Peoc'h

Etienne Levavasseur

Emilien Delmont

Alfonso De Simone

Isabelle Laffont-Proust

Nicolas Privat

Yassmine Chebaro

Céline Chapuis

Pierre Bedoucha

Jean-Philippe Brandel

Annie Laquerrière

Jean-Louis Kemeny

Jean-Jacques Hauw

Annie Laquerriere

Michel Borg

Human Rezaei

Philippe Derreumaux

Jean-Louis Laplanche

Stéphane Haïk
email: stephane.haik@upmc.fr

Substitutions at residue 211 in the prion protein drive a switch between CJD and GSS syndrome, a new mechanism governing inherited neurodegenerative disorders

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

represent 15% of patients, different phenotypes have been described depending on the variations detected at different positions within the prion protein gene. Here, we report a new mechanism governing the phenotypic variability of inherited prion diseases. First, we observed that the substitution at residue 211 with either Gln or Asp leads to distinct disorders at the clinical, neuropathological and biochemical levels (Creutzfeldt-Jakob disease or Gerstmann-Sträussler-Scheinker syndrome with abundant amyloid plaques and tau neurofibrillar pathology). Then, using molecular dynamics simulations and biophysical characterization of mutant proteins and an in vitro model of PrP conversion, we found evidence that each substitution impacts differently the stability of PrP and its propensity to produce different protease resistant fragments that may contribute to the phenotypical switch. Thus, subtle differences in the PrP primary structure and stability are sufficient to control amyloid plaques formation and tau abnormal phosphorylation and fibrillation. This mechanism is unique among neurodegenerative disorders and is consistent with the prion hypothesis that proposes a conformational change as the key pathological event in prion disorders.

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