What makes a good mate? Factors influencing male and female reproductive success in a polyphagous moth

The mating propensity of an individual is expected to depend on the costs and benefits of mating, which may vary across the sexes and across different mating opportunities. Both males and females should gain fitness either by mating with multiple mates and/or by mating with higher quality mates. Therefore, an important question in the area of sexual selection concerns what makes an optimal mate. From a female perspective, females are expected to prefer males providing direct material benefits for the present generation and/or indirect genetic benefits for their offspring in the subsequent generation. Because male contribution to these benefits can be limited, as reproduction imposes non-trivial costs on males, female benefits from mating can vary markedly as a function of the condition of their mate. In capital breeding species, in which males invest the majority of their larval resources in a single reproductive event, the females are likely to prefer to mate with virgin males in good condition (i.e. males that have developed on high quality food sources). In this study we used the European grapevine moth (Lobesia botrana) to test experimentally whether the larval nutrition and mating history of males influence their quality as mates. We provided wild L. botrana males originating from different cultivars and vineyards with unlimited access to standardized females, and examined the lifetime reproductive success of the males and the consequences for the reproductive output of females. Our results show that 'male quality' depended on both the male larval origin and mating history, and that females discriminated among males and mated more with males having high spermatophore quality (virgin males and males from certain cultivars or vineyards) to obtain substantial direct benefits.

KEY WORDS: direct benefits, larval nutrition, Lobesia botrana, male mating history, male quality, spermatophore.

The mating propensity of an individual is expected to depend on the costs and benefits of mating, which may vary across the sexes and the number of mating opportunities. When both male and female vary in their reproductive quality, the two sexes are expected to be choosy and should display higher mating preferences with partners providing higher fitness benefits.

Males and females should gain fitness either by mating with multiple mates [START_REF] Arnqvist | The evolution of polyandry: multiple mating and female fitness in insects[END_REF][START_REF] Wagner | Direct benefits and the evolution of female mating preferences: conceptual problems, potential solutions, and a field cricket[END_REF]) and/or by mating with higher quality mates. Therefore, an important question in the sexual selection area concerns what makes an optimal mate for the choosy sex. 'Mate reproductive quality' is determined by a variety of behavioral, physiological and morphological traits [START_REF] Lailvaux | Defining individual quality over lifetimes and selective contexts[END_REF][START_REF] Wilson | What is individual quality? An evolutionary perspective[END_REF]. These traits influence the propensity to mate of individuals (through precopulatory behaviors including courtship, production of sex pheromone, and mate guarding) and therefore influence their probability of being chosen as a mate and shape their realized fitness [START_REF] Simmons | Sperm competition and its evolutionary consequences in the insects[END_REF].

The benefits of mate choice depend on the quality of the chosen mate but also on the extrinsic and intrinsic conditions of the choosy individual, including its physiological state and physical and social environment. For example, some studies have shown that males mate preferentially with more fecund females [START_REF] Bonduriansky | The evolution of male mate choice in insects: a synthesis of ideas and evidence[END_REF] and tailor their ejaculate size to the level of sperm competition [START_REF] Wedell | Sperm competition, male prudence and sperm-limited females[END_REF]. In the same way, female mating behaviour is affected by a variety of intrinsic (including mating status or age) and extrinsic factors (such as predation risk, parasite infection or mate availability). Because females that fail to mate have zero fitness [START_REF] Rhainds | Female mating failures in insects[END_REF], the level of female choosiness is constrained by the risk of remaining unmated, which depends on demographic effects, low mate encounter rate, outcompetition by rivals or prereproductive death [START_REF] Kokko | Sexual selection when fertilization is not guaranteed[END_REF][START_REF] Rhainds | Female mating failures in insects[END_REF][START_REF] Rhainds | Sexual selection and mating failures: where have all the females gone?[END_REF]). Thus, female mating strategies often reflect a trade-off between maximizing the benefits of obtaining high quality mates, reducing the probability of mating failure and minimizing other mating costs [START_REF] Rhainds | Female mating failures in insects[END_REF]. By keeping in mind these trade-offs, good mates for females are those that are able to provide direct and indirect benefits [START_REF] Møller | How important are direct fitness benefits of sexual selection[END_REF]. Indirect benefits can arise from genetic traits of the chosen male (e.g. good genes), which lead to increased fitness of the resulting offspring [START_REF] Mays | Choosing mates: good genes versus genes that are a good fit[END_REF][START_REF] Tregenza | Genetic compatibility, mate choice and patterns of parentage: invited review[END_REF]. Direct benefits are related to whether the chosen male is sufficiently fertile, free of disease, or able to provide parental care, access to territories or to nutritive resources including nuptial gifts [START_REF] Choe | The evolution of social behaviour in insects and arachnids[END_REF][START_REF] Vahed | The function of nuptial feeding in insects: a review of empirical studies[END_REF]. However, the male contribution to these direct benefits can be limited, as reproduction imposes non-trivial costs on males, arising from mate location, competition, courtship, parental care, and especially ejaculate production [START_REF] Janowitz | Costing reproduction: effects of mating opportunity on mating success in male Bicyclus anynana butterflies[END_REF][START_REF] Paukku | Cost of reproduction in Callosobruchus maculatus: effects of mating on male longevity and the effect of male mating status on female longevity[END_REF][START_REF] Scharf | Reproductive trade-offs and direct costs for males in arthropods[END_REF]. Thus, female benefits from mating can be extremely variable based on the quality of their mate, because factors limiting the reproduction of males can have profound consequences for female reproductive output.

For species in which males provide females with material resources including a nutritive ejaculate (for example, spermatophores in some lepidopteran species), the influence of male mating frequency on future reproductive output can also be extremely pronounced [START_REF] Torres-Vila | Male mating history and female fecundity in the Lepidoptera: do male virgins make better partners?[END_REF][START_REF] Wedell | Sperm competition, male prudence and sperm-limited females[END_REF]. Because ejaculate production is costly [START_REF] Dewsbury | Ejaculate cost and male choice[END_REF], male performance usually declines across multiple matings, leading to diminishing reproductive returns for males (reviewed by [START_REF] Simmons | Sperm competition and its evolutionary consequences in the insects[END_REF]. Moreover, males may be limited in the amount of sperm they can transfer to a female during mating [START_REF] Marcotte | Impact of male mating history on the temporal sperm dynamics of Choristoneura rosaceana and C. fumiferana females[END_REF], and male mating history (the number of previous matings) is certainly a key factor determining female fitness, especially in species in which males can keep copulating despite being sperm depleted (Damiens & Bovin 2006[START_REF] Steiner | Mating with sperm-depleted males does not increase female mating frequency in the parasitoid Lariophagus distinguendus[END_REF]. It has been commonly assumed that males have to face trade-offs between an investment in somatic maintenance or in reproduction because they have finite resources to invest [START_REF] Barnes | Costing reproduction[END_REF][START_REF] Stearns | The evolution of life histories[END_REF]. Such trade-offs typically arise under food limitation, because male expenditure in ejaculate production is constrained in part by resource availability; consequently males have to invest in either current or future reproduction [START_REF] Simmons | Sperm competition and its evolutionary consequences in the insects[END_REF]). In capital breeders, which rely mainly on larval reserves for successful reproduction, the resources needed to produce a nutritive ejaculate can be a limiting factor. Therefore, ejaculate production could be related to the number of copulations and male larval nutrition, but few studies have reported the quantitative and qualitative relationships involved. Diet quality can have a significant influence on the rate at which males produce ejaculate, the quality of the seminal fluid proteins, and the effectiveness of the ejaculate in achieving fertilization [START_REF] Arnqvist | Postmating sexual selection: the effects of male body size and recovery period on paternity and egg production rate in a water strider[END_REF][START_REF] Gage | Sperm size or numbers? Effects of nutritional stress upon eupyrene and apyrene sperm production strategies in the moth Plodia interpunctella (Lepidoptera: Pyralidea)[END_REF][START_REF] Simmons | Ejaculate expenditure by malebush crickets decreases with sperm competition intensity[END_REF]. When males lack adequate protein sources or when they have developed on nutritionally limited host plants, critical depletion of their ejaculate generally occurs during successive matings [START_REF] Gage | Sperm size or numbers? Effects of nutritional stress upon eupyrene and apyrene sperm production strategies in the moth Plodia interpunctella (Lepidoptera: Pyralidea)[END_REF]. However, most studies have focused on the factors affecting male reproductive output following emergence, particularly during the first two mating events [START_REF] Delisle | Male larval nutrition influences the reproductive success of both sexes of the spruce budworm, Choristoneura fumiferana (Lepidoptera: Tortricidae)[END_REF][START_REF] Tigreros | Linking nutrition and sexual selection across life stages in a model butterfly system[END_REF][START_REF] Cordes | Larval food composition affects courtship song and sperm expenditure in a lekking moth[END_REF] but not on the trade-off between larval nutrition and the male entire lifetime reproductive investment.

In this context, our study goals were to assess, if (1) larval nutrition has an overall importance in male mating capacity and lifetime reproductive investment; (2) the combined effect of male larval nutrition and male mating history affect their quality as mates and (3) female prefer to mate with 'high quality mates' in order to obtain larger direct benefits. To answer these questions, we used the European grapevine moth (Lobesia botrana; Lepidoptera: Tortricidae) (Denis & Schiffermüller 1775), which is a very important pest of grapes worldwide. Several studies of this species have already shown marked effects of larval nutrition on male and female fitness [START_REF] Moreau | Grape variety affects larval performance and also female reproductive performance of the European grapevine moth Lobesia botrana (Lepidoptera: Tortricidae)[END_REF], 2007[START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF]. However, the lifetime reproductive capacity of male moths of this species remains unknown because most studies have concerned only the first mating of individuals [START_REF] Moreau | Grape variety affects larval performance and also female reproductive performance of the European grapevine moth Lobesia botrana (Lepidoptera: Tortricidae)[END_REF], 2007[START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF]see Torres-Vila et al. 1999 for an exception). In the present study, we provided wild L. botrana males that developed on different grape cultivars and in different vineyards with unlimited access to females and investigated the lifetime reproductive success of the males.

We also investigated the consequences for the reproductive output of females as a function of male larval origin and mating history. In a first step we explored variation in male reproductive investment (spermatophore size, number of sperm) during successive matings.

We predicted that (i) male reproductive investment and mating capacity would be affected by male larval nutrition on different cultivars; and (ii) male quality would depend on both their larval nutrition and mating history. In a second step, we studied the consequences of male larval nutrition and mating history on the reproductive output of females (fecundity and fertility). We predicted (iii) that female fitness would be affected by both male larval origin and mating history and (iv) that female would be more motivated to mate with males of 'high quality', thus receiving larger nutrient-rich spermatophores.

MATERIAL & METHODS

Field sampling

Lobesia botrana is a major pest of grapes. It is widely distributed in most European vineyards and is now present in the USA, where 3-4 larval generations occur each year, depending on latitude. First generation larvae of L. botrana were collected in the field during June 2013. To test for a cultivar effect within a given population, larvae were sampled from three grape (Vitis vinifera) cultivars ('Carignan', 'Mourvèdre' and 'Grenache') in the same vineyard (Perpignan, France; N 42°447.063, E 2°5256.441), ensuring the same abiotic conditions (temperature, light exposure, humidity) for larval development. The three chosen grape varieties are biochemically very different, especially in their phenolic contents [START_REF] Teissedre | Grape[END_REF]. Indeed, 'Carignan' and 'Grenache' grape extracts contains less total phenols than Mourvèdre grape extracts [START_REF] Jensen | Prediction of wine color attributes from the phenolic profiles of red grapes (Vitis vinifera)[END_REF]. To test for a geographical effect we sampled larvae from the cultivar 'Grenache' from two additional geographically distinct French vineyards; Estézargues (N 43°5649.781, E 4°3939.372) and Sénas (N 43°4354.251, E 5°145.621). Larvae were sampled at the end of the larval cycle (fifth instar), following construction of glomerulae made of flower buds aggregated in larval silk (phenology 17-25;[START_REF] Eichhorn | Phänologische entwicklungsstadien der Rebe[END_REF]. Larvae usually complete their development in a single grape bunch, and each glomerulus is only occupied by a single larva [START_REF] Torres-Vila | Physiological factors regulating polyandry in Lobesia botrana (Lepidoptera: Tortricidae)[END_REF]). To collect newly emerged adults, larvae at the end of their development were placed in large polyethylene boxes (60  40 cm, height 21 cm) in the laboratory and fed ad libitum on grape bunches from the same cultivar and site where they developed, and were incubated at 22 ± 1 °C, 60 ± 10 % RH, and under natural photoperiod conditions. The larvae were checked daily until pupation, at which time they were gently removed from their glomerulae. The pupae were weighed to the nearest 0.1 mg using a Precisa 262 SMA-FR microbalance, placed individually in glass tubes (70  9 mm diameter) stoppered with cotton wool plugs, and stored at 22 °C ± 1 °C under natural photoperiod conditions. The pupae were checked each morning, and newly emerged adults were visually sexed by examination of the ventral tip of the abdomen.

To assess the importance of male larval origin and mating history on male reproductive investment and female reproductive output, 2-day-old males of different larval origin (cultivar and site) were given daily mating access to a new 1-or 2-day-old standardized virgin female; this was continued until death of the male. The standardized females came from an inbred strain (INRA Bordeaux) maintained without diapause on a semi-artificial diet. The use of this inbred strain helps to minimize genetic variation between females and allows us to detect the male effect on female reproduction (see [START_REF] Moreau | Grape variety affects female but also male reproductive success in wild European grapevine moths[END_REF][START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF] for more details). Males used for the following mating experiments were randomly distributed into two subsamples. The first subsample was used to evaluate the effect of male origin (cultivar and site) and mating history on male lifetime reproductive investment, and was also used to monitor the male precopulatory behavior for each mating event. The second subsample was used to assess the consequences of male origin (cultivar and site) and mating history on the reproductive output of females.

Ethical Note

All experiments complied with French laws on animal experimentation. Moths were treated carefully, and the abiotic conditions (temperature, humidity and photoperiod) they experienced corresponded to the natural conditions in their native habitat. Females submitted to dissection were chilled in a freezer prior to decapitation.

Mating procedure

At dusk, one male (2 days old on the first day of the experiment) randomly selected from each test condition (cultivar or site) was placed into a mating tube (100  15 mm diameter) with a single 1-or 2-day-old standardized virgin female, and the pair were observed until mating took place, or for a maximum of 4 h in the absence of mating. The male was returned to the pupation tube after mating or at 4 h, and held under the same conditions as for moth maintenance, with water provided ad libitum. This process was repeated 24 h later in a new mating tube, and the procedure was repeated sequentially until the death of the male. A mating was considered to have been successful if a sperm-filled spermatophore was observed in the bursa copulatrix after the dissection of the female under a stereomicroscope (Nikon SMZ1500) at a magnification of 20. Matings in which males failed to transfer a spermatophore (if no spermatophore was found in the bursa copulatrix of the female, or if the female lay no eggs during her life following the observed mating) were discarded from the analysis. Various reproductive traits were measured in each subsample.

First subsample: precopulatory behaviors of males and females

For the first subsample the male and female sexual activity was videotaped (Sony HDR CX220E) until mating; only recordings of successful matings (with effective spermatophore transfer or female egg laying) were analyzed. The latency period prior to mating (the time elapsed from male/female pairing until coupling) was recorded along with the occurrence of behaviors reflecting female and male sexual motivation (as described by [START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF].

The latency to mate is a first measure that accurately reflects the reluctance or acceptance to mate in no-choice tests [START_REF] Edward | The description of mate choice[END_REF][START_REF] Muller | Direct benefits from choosing a virgin male in the European grapevine moth (Lobesia botrana)[END_REF]. Moreover, in L. botrana a female that is ready to mate signals readiness by releasing sex pheromone at dusk, which is an action that represents a fitness cost [START_REF] Harari | Fitness cost of pheromone production in signaling female moths[END_REF]). To do this the female assumes a calling position with wings raised and the pheromone gland exposed. This behavior reflects the penchant of a female to mate, and therefore we used it as a proxy of female motivation. Thus, we recorded data on the female motivation to mate (expressed as the time a female spent calling divided by the courtship duration  100). To evaluate mating ability and sexual vigor of males in courtship, we also recorded data on the percentage of male activity (the time spent in movement by the male expressed as a percent of the total courtship period).

First subsample: male reproductive performance

Immediately following mating the females were anesthetized (-25°C for 10 min), then dissected on a glass side. The bursa copulatrix containing the male spermatophore was removed and measured. The spermatophore produced by L. botrana males is very small (< 1 mg) and consequently difficult to weigh accurately. We estimated the spermatophore size by extrapolating its volume; this is a well-established method used for small moths including L. botrana [START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF][START_REF] Torres-Vila | Vine phenological stage during larval feeding affects male and female reproductive output of Lobesia botrana (Lepidoptera: Tortricidae)[END_REF]. The spermatophore length (l), width (w), and thickness (t) were measured using a stereomicroscope (Nikon SMZ1500) at a magnification of 20, and the spermatophore volume was estimated as an ellipsoid balloon [V = π/6 (l  w  t)], as described previously [START_REF] Torres-Vila | Vine phenological stage during larval feeding affects male and female reproductive output of Lobesia botrana (Lepidoptera: Tortricidae)[END_REF][START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF]. Like in all Lepidoptera, male L. botrana transfer fertile eupyrene sperm and non-fertile anucleate apyrene sperm at mating. The sperm-containing ampulla was ruptured in a drop of distillated water and the sperm mass was gently stirred to ensure dispersion. In Lepidoptera at this stage the eupyrene sperm are encysted in bundles, and each bundle contains 256 eupyrene sperm [START_REF] Cook | Effects of risks of sperm competition on the numbers of eupyrene and apyrene sperm ejaculated by the moth Plodia interpunctella (Lepidoptera: Pyralidae)[END_REF]. The number of intact bundles was counted at 40/0.65 magnification using a Nikon Eclipse E600 microscope; this number was multiplied by 256 to estimate the total number of eupyrene sperm. The solution was then washed from the slide into a 1.5 mL centrifuge tube and diluted with distilled water. Four subsamples (10 µL) were removed from the diluted sperm solution, and the number of apyrene sperm was counted by microscopy (Nikon Eclipse E600; 100 magnification). The total number of apyrene sperm was estimated by multiplying the average sperm count for the four subsamples (coefficient of variation = 12%) by the dilution factor.

We recorded: (i) male longevity; (ii) the total number of matings by males during their lifespan; (iii) the lifetime spermatophore quantity produced and the lifetime number of sperm transferred; (iv) the number of offspring sired by males during their lifetime; and (v) the spermatophore volume and the number of sperm transferred at each male mating.

Second subsample: consequences for female reproductive output

Following mating (see general mating procedure) the females were held in the mating tube and could oviposit freely on the inside surface of the glass tub. Female survival was checked daily, and following death the eggs were incubated for 7 days under the same conditions as for moth maintenance. We recorded several female traits as a function of male larval origin (cultivar or site) and mating history, including: (i) female fecundity (the number of eggs laid per female at each mating); and (v) female fertility (the proportion of hatched eggs for each mating).

Statistical analysis

All statistical tests were performed using R Software version 3.2.0 (R Core Team 2015). For each analysis we report the full model with insignificant interactions deleted, following the approach of [START_REF] Forstmeier | Cryptic multiple hypotheses testing in linear models: overestimated effect sizes and the winner's curse[END_REF]. The effect of male origin (cultivar and site) on male pupal mass was tested using a one-way ANOVA followed by Tukey's test. A cox regression was applied to assess the influence of male origin and pupal mass on male longevity. Sources of variation in the total number of matings by males during their lifetime, the lifetime quantity of spermatophore produced by males, and the number of offspring sired by males during their lifetime were identified using ANCOVAs, with male larval origin as the explanatory variable, and the number of male matings and the male and female pupal masses as covariates. Because the sperm were counted and were over-dispersed, a generalized linear model with a negative binomial distribution (NBGLM) was used to evaluate the effect of male origin on the total number of sperm transferred by males during their lifetime.

We used a general mixed model with male identity as a random effect to assess the combined effects of male mating history and larval origin on precopulatory behavior, the spermatophore size, the number of eupyrene and apyrene sperm, and the female fecundity and fertility. Male mating history was recorded as a discrete variable (1, 2, 3, 4, or 5 matings); the data for the previous 5, 6 and 7 matings were excluded because the sample size was too small for certain male origins (< 5 individuals). Pearson's χ 2 tests were used to assess the mating success of males (percentage of successful matings) as a function of larval origin (cultivar and site) and mating history. Because of non-normality, female motivation to mate (percent of time spent in the calling position) and male activity (percent of time spent in movement) were arcsine square root transformed prior to analysis. The latency period prior to mating, the male activity and the female motivation to mate were analyzed using general linear mixed models. Because data on sperm were best approximated by an over-dispersed Poisson distribution, we fitted the model with a negative binomial error structure and used the glmmADMB library to perform the analysis, which included male mating history and male larval origin as fixed effects, male and female mass as covariates, and male identity as a random factor. The proportion of eggs hatched was analyzed using the glmmPQL function with a quasi-binomial error structure.

RESULTS

Male pupal mass and longevity

Male pupal mass was affected by male larval origin (Table 1; F4,181 = 20.63, P < 0.0001).

Males from Mourvèdre in the Perpignan vineyard were larger than those from Carignan (Table 1). Among sites, the males from Grenache in Estézargues and Sénas were heavier than those from this cultivar in Perpignan. Male longevity was also influenced by the origin of the males (χ²4,181 = 23.23, P < 0.0001), and was positively correlated with male pupal mass (χ²1,181 = 26.02, p < 0.0001) (Table 1). At a given site (Perpignan), the males from Carignan died earlier than males from Grenache and Mourvèdre. However, the longevity of males from Grenache in Perpignan was similar to that of males from Grenache in Estézargues or Sénas.

As a consequence of differing longevity, males did not have the same number of mating opportunities over their lifetimes. Consequently, male mating capacity (i.e. the maximum number of matings that males undertook during their lifetimes) was positively correlated to male longevity (F1,179 = 188.01, P < 0.0001). For example, in a given vineyard the males from Carignan lived an average of 6 days, and tended to mate less often than males from Grenache or Mourvèdre, which lived for more than 7 days (Table 1 and Fig. 1). Moreover, among the males from Grenache in Sénas and Estézargues (which lived the longest), 50-75% mated 5 or more times during their lifetime, while only 30% of the males from Grenache in Perpignan mated at least 5 times (Table 1 and Fig. 1).

Male lifetime reproductive investment

The overall spermatophore quantity produced by males during their lifetime varied with the total number of matings (F1,73 = 146.36, P < 0.0001), the male larval origin (F4,73 = 9.76, P < 0.0001), and the pupal mass (F1,73 = 17.86, P < 0.0001) (Table 1). At a given site, males from Grenache transferred a greater quantity of spermatophore to females at mating (average 132.7  10 -6 mm 3 ) than males from Carignan (approximately 96.6  10 -6 mm 3 ) or Mourvèdre (approximately 125  10 -6 mm 3 ). However, males from the three geographically distinct sites transferred approximately the same quantities of spermatophore during their lifetimes. The numbers of fertile eupyrene and non-fertile apyrene sperm were positively correlated with the number of male matings (F1,73 = 57.34, P < 0.0001 and F1,73 = 16.76, P < 0.0001, respectively), but were not correlated with the male larval origin (F4,73 = 1.87, P = 0.124 and F4,73 = 0.53, P = 0.713, respectively) or the pupal mass (F1,73 = 1.44 P = 0.234 and F1,73 = 1.17 P = 0.283, respectively) (Table 1).

Consequently, the number of offspring derived from males during their lifetime was influenced by the number of male matings and the male larval origin (Fig. 2; male mating number effect: F1,95 = 151.73, P < 0.0001; male larval origin: F4,95 = 21.46, P < 0.0001; interaction term: F4,95 = 5.97, P = 0.001), but not by the male pupal mass (F1,95 = 1.25, P = 0.266). In a given vineyard, males from Mourvèdre always produced the least quantity of spermatophore and consequently produced fewer offspring than males from the two other cultivars. However, there was no geographical effect on the number of offspring sired by males, which is consistent with the observation that these males produced the same amount of spermatophore during their lifetimes. The interaction term between male mating numbers and male larval nutrition indicate that benefits of multiple copulations for a given male depended on his larval nutrition (Fig. 2). For example, males from Carignan obtained more offspring due to multiple matings during their life in comparison with males from Mourvèdre (Fig. 2).

Male reproductive investment over successive matings

For each mating opportunity, the mating success of males was relatively high (range 72.7-100%), and was not affected by male larval origin or mating history. The volume of spermatophore transferred to the female at each mating (from the first to the fifth mating) was affected by the male mating history (Fig. 3a; LR = 1548.74, P < 0.0001) and the male larval origin (Fig. 3a; LR = 43.07, P < 0.0001). It was positively correlated with male pupal mass (LR = 26.20, P < 0.0001) but not with female pupal mass (LR = 1.85, P = 0.174). The spermatophore produced by males at their first mating was 3-5 times larger than spermatophores transferred during subsequent matings, irrespective of the male larval origin (Fig. 3a). At a given site, the males from Carignan produced significantly smaller spermatophores than males from the two others cultivars, but there was no geographical effect on the spermatophore volume produced by males from Grenache among the three geographically distinct sites. The numbers of eupyrene and apyrene sperm produced by males were also affected by male mating history (Fig. 3b and3c, respectively; eupyrene sperm: LR = 76.44, P < 0.0001; apyrene sperm: LR = 105.94, P < 0.0001). However, the numbers of eupyrene or apyrene sperm were not influenced by male larval origin (eupyrene sperm: LR = 5.48, P = 0.242; apyrene sperm: LR = 2.08, P = 0.721) or by male pupal mass (eupyrene sperm: LR = 0.42, P = 0.517; apyrene sperm: LR = 0.68, P = 0.410). Males transferred more eupyrene and apyrene sperm during their first mating compared with their subsequent matings, and also transferred more sperm during their second mating compared with their fifth mating (Fig. 3b and3c).

Consequences for the reproductive output of females

The number of eggs laid by a female at each mating strongly depended on the male mating history (Fig. 4a; LR = 99.53, P < 0.0001) and larval origin (LR = 47.74.65, P < 0.0001). It was also positively correlated with the female pupal mass (Fig 4a; LR = 34.65, P < 0.0001) but not the male pupal mass (LR = 2.76, P = 0.097). At Perpignan, females mated to males from Mourvèdre laid significantly fewer eggs than females mated to males from the other two cultivars, but there was no geographical effect on the fecundity of females mated with males from Grenache at the three geographically distinct sites. Moreover, females that had copulated with virgin males (first mating) had a higher level of fecundity than females mated to nonvirgin males (subsequent matings), and females mated to males that had mated four times previously laid fewer eggs than females mated with males that had mated once or twice (Fig. 4a). Female fertility depended on the male larval origin (Fig. 4b; LR = 35.38, P < 0.0001) and was positively correlated with female pupal mass (LR = 7.77, P = 0.005) but not with male pupal mass (LR = 2.71, P = 0.099). Females mated with males from Mourvèdre had fewer hatched eggs than females mated with males from Carignan or Grenache. However, female fertility did not depend on male mating history (Fig. 4b; LR = 1.74, P = 0.783), suggesting that males provide female with sufficient sperm to fertilize the same proportion of eggs over five successive matings.

Male 'quality' and motivation to mate in both sexes

The latency period prior to mating was affected by male mating history (LR = 28.50, P < 0.0001) but not by male larval origin, or male or female pupal mass (LR = 5.06, P = 0.281, LR = 0.16, P = 0.689, and LR = 0.82, P = 0.366, respectively). Male matings occurred sooner for the first mating (13.0 min, ranged from 10.8 to 15.0 min according to the larval origin) relative to the successive matings (mean for the 2 nd , 3 rd , 4 th and 5 th mating: 15.5 min, ranged from 9.3 to 20.4 min according to the larval origin).

Male activity (expressed as the proportion of time a male spent in activity divided by the latency period prior to mating) was not influenced by male larval origin (LR = 4.81, P = 0.308). Regardless of their larval origin, males spent on average between 41.9% (males from Mourvèdre) and 55.5 % (males from Grenache) of their time in courtship. Moreover, the time spent in courtship was not dependent on male mating history (LR = 4.19, P = 0.381), ranging from 37.4% (5 th mating) to 50.3% (2 nd mating) of male courtship activity.

However, female motivation to mate (i.e. the proportion of time spent calling expressed as the time spent calling divided by the timing of onset of mating) was affected by the male larval origin (Fig. 5; LR = 39.85, P < 0.0001) and male mating history (Fig. 5; LR = 17.47, P = 0.002). Females were more motivated to mate with virgin males or males that had mated once compared with males that had mated 4 or 5 times (Fig. 5). At a given site, females in the presence of males from Mourvèdre were less motivated to mate (calling from 18 to 24% of the time according to male mating history) than females mated with males from Carignan (calling from 22 to 50% of the time depending on male mating history). Among the sites, females paired with males from Grenache in Perpignan spent less time in the calling position than females paired with males from Grenache in Estézargues or Sénas.

DISCUSSION

We found that the lifetime reproductive output of males was closely linked to their larval nutrition. Indeed, male larval nutrition on the different grape cultivars affected male longevity, male mating capacity and therefore, the number of offspring sired by males over their lifetime. As expected, male reproductive investment decreased over successive matings, and was largely affected by male larval nutrition on the different grape cultivars and among geographically distinct sites. The male spermatophore volume and the number of sperm in each ejaculate decreased from the first to subsequent matings, and these parameters were affected by the cultivar on which the male larvae were reared. These factors had major repercussions for female reproductive output. Females mated with males producing the largest spermatophore and more sperm (e.g. males from Grenache) had greater fecundity and fertility than females mated with males producing small spermatophores and less sperm (e.g. males from Mourvèdre) across different mating ranks and females were less motivated to mate with the 'lower quality' males from Mourvèdre. Moreover, females mated with virgin males (i.e. their first mating) had a greater fecundity than females mated with non-virgin males (i.e. their subsequent matings), regardless of the male larval origin. Thus, females were more motivated to mate with virgin males having high spermatophore quality than with non-virgin males, which transferred less nutritive substances and fewer sperm at mating. Our results suggest that 'male quality' depended on both male larval origin and mating history, and had major consequences for female reproductive output. Moreover, females were able to discriminate among these males, and to receive large direct benefits were more motivated to mate with males having high sperm quantity (virgin males or males from certain cultivars or geographical locations).

Larval nutrition, male mating capacity and lifetime reproductive investment

For each mating opportunity, the mating success (i.e. the probability of a male acquiring a mate) of males did not vary according to their larval origin, and remained high and constant over successive matings [min: 80%; max: 100%], suggesting that almost all males had a nonlimited mating capacity until their death (based on an inter-mating recovery period of 24 h). However, their larval origin and pupal mass strongly influenced male longevity, with small males reared on Carignan in Perpignan living for a shorter period than large males from Grenache; consequently, these males had fewer mating opportunities over their lifetime in natura. This may be partly related to the pupal mass because smaller males have less energy reserves than larger males (Muller et al., unpublished data), and cannot afford to invest in both somatic maintenance and reproductive effort [START_REF] Boggs | Understanding insect life histories and senescence through a resource allocation lens[END_REF][START_REF] Boggs | Larval food limitation in butterflies: effects on adult resource allocation and fitness[END_REF]. Indeed, males in good condition may be better competitors and have generally better mating success than males in poor condition, without incurring any survival cost [START_REF] Engqvist | Male attractiveness is negatively genetically associated with investment in copulations[END_REF][START_REF] Grandison | Amino-acid imbalance explains extension of lifespan by dietary restriction in Drosophila[END_REF]). Indeed, these males had more energy reserves to invest in somatic maintenance and/or reproduction and are expected to outcompete smaller males reared on less nutritive host plants. We found that the number of male matings was positively related to male longevity: the longer a male lived, the better its chance of reproducing several times [START_REF] Molleman | Does dietary restriction reduce life span in male fruit-feeding butterflies[END_REF]. Our results suggest that males from Grenache in Estézargues and Sénas were in the best condition, investing in both somatic maintenance (lived for > 8 days) and reproduction (had the largest number of copulations).

Male quality: a combined effect of larval nutrition and mating history

Male reproductive investment, besides to depend on the male larval nutrition, markedly decreased with increasing number of copulations, which was largely because of male inability to replenish resources in the adult stage. Between the first and subsequent matings of males, there was a > 60% decrease in spermatophore volume, confirming that male spermatophore production is very costly [START_REF] Vahed | The function of nuptial feeding in insects: a review of empirical studies[END_REF]. Thus, in L. botrana and more generally in capital breeder species, males have only a single nutrient-rich spermatophore, which is produced using energy reserves derived from larval nutrition. We previously demonstrated in a laboratory strain of L. botrana species that the first spermatophore of males plays a crucial role in female egg production [START_REF] Muller | Direct benefits from choosing a virgin male in the European grapevine moth (Lobesia botrana)[END_REF], and the present study confirms this finding in wild populations of L. botrana. This is consistent with the general assumption that male multiple mating can result in the depletion of specific ejaculate components, resulting in decreased fecundity and fertility of their mates (Pérez-Staples et al. 2008[START_REF] Wigby | Seminal fluid protein allocation and male reproductive success[END_REF]).

However, our results also indicate that the second spermatophore delivered by L. botrana males was 66-80 % smaller than the first (according to their larval origin) but the fecundity of the female mated to a once-mated male decreased by only 5-40% according to male larval origin. This suggests that spermatophore size may not be a reliable predictor of female fecundity, and that the quality of the spermatophore rather than its quantity might better explain the variation observed in female fecundity (Bissondath & Wiklund 1996[START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF]. Both the number of sperm and/or the composition of the ejaculate (e.g. accessory gland secretions) can affect female fecundity (reviewed by [START_REF] Perry | The seminal symphony: how to compose an ejaculate[END_REF]), and we recently found that protein-derived spermatophores are a key factor in female reproductive output (Muller et al., unpublished data).

The numbers of apyrene and eupyrene spermatozoids also decreased with increasing number of matings. Eupyrene and apyrene spermatogenesis is known to occur at different stages during moth development [START_REF] Friedländer | Eupyrene and apyrene sperm: dichotomous spermatogenesis in Lepidoptera[END_REF]. Eupyrene spermatogenesis typically begins during the later larval instar stages and ceases at pupation, while apyrene spermatogenesis usually starts just prior to pupation and continues throughout adulthood. All the L. botrana males sampled in this study emerged with a finite number of eupyrene sperm, and the males did not release all sperm during the first mating, but retained some for future mating opportunities, ensuring fertilization of the same proportion of female eggs over consecutive matings. However, female fertility was affected by male larval origin, with females mated with males from Mourvèdre having reduced fertility compared with females mated with males from Grenache or Carignan. Males from Mourvèdre were likely to have low quality sperm, and although they transferred the same number of eupyrene sperm as males from the other cultivars, they were not able to fertilize > 70% of female oocytes, regardless of their mating history. As with the spermatophore volume, the quantity of sperm (which is always in excess compared with the number of eggs) is probably a minor factor relative to its quality [START_REF] Snook | Sperm in competition: not playing by the numbers[END_REF][START_REF] Werner | Insect sperm motility[END_REF]. Numerous sperm traits that contribute to paternity (including sperm size, viability, and mobility) are known to influence fertilization efficiency in moths [START_REF] Morrow | The evolution of sperm length in moths[END_REF][START_REF] Perry | The seminal symphony: how to compose an ejaculate[END_REF]. In L. botrana, male larval food composition could directly influenced sperm quality, as demonstrated in other moth species [START_REF] Gage | Sperm size or numbers? Effects of nutritional stress upon eupyrene and apyrene sperm production strategies in the moth Plodia interpunctella (Lepidoptera: Pyralidea)[END_REF][START_REF] Cordes | Larval food composition affects courtship song and sperm expenditure in a lekking moth[END_REF].

Implications for the evolution of female mate choice

Because 'male quality' depends on both the male larval nutrition and male mating history, females should be able to distinguish between males of different qualities on the basis of these two factors. This study provides initial evidences that females seem to prefer to mate (1) with males originated from cultivars that enhance their reproductive performances and; (2) with virgin males rather than already mated males.

Firstly, the female motivation varied with male origin, with females being less motivated (spent less time in the calling position) to mate with males having lower spermatophore quality (those reared on Mourvèdre) than males from the other cultivars, suggesting that the females used cues (perhaps chemical fingerprints of males having different host origins) that provided information on male condition [START_REF] Costanzo | The use of chemical and visual cues in female choice in the butterfly Bicyclus anynana[END_REF][START_REF] Harris | Sperm competition and male ejaculate investment in Nauphoeta cinerea: effects of social environment during development[END_REF], and therefore spermatophore quality.

Secondly, the latency period to mating significantly increased with increasing number of matings, with non-virgin males that had already mated taking 20-25% more time to mate than virgin males. This suggests that after their first mating, males needed more time to successfully mate. This may be because of cumulative fatigue resulting from successive mating, or because the females were able to detect that these males were potentially spermdepleted, and were more reluctant to mate with them. Analysis of the precopulatory behaviors of each sex suggested that the amount of time required to mate by experienced males was not merely a result of cumulative fatigue, because males were equally active during courtship regardless of their mating history. Thus, the longer latency period prior to mating for nonvirgin males was probably the result of female reluctance to mate with previously mated males. Indeed, females were more motivated to mate with virgin males than with non-virgin males. In a recent study of L. botrana involving mate-choice experiments, we demonstrated a female preference for virgin males, which maximized the direct benefits associated with receiving large spermatophores [START_REF] Muller | Direct benefits from choosing a virgin male in the European grapevine moth (Lobesia botrana)[END_REF]).

Nevertheless, this study was a laboratory experiment and there is no yet demonstrated evidence of the existence of a female mate choice in natura in this species. Models usually predict that the level of female choosiness should depend on the importance of the cost of searching mates (ie, the proportion of lifetime devoted to searching for mates) which depends on the operational sex ratio and the encounter rate [START_REF] Bleu | Evolution of female choosiness and mating frequency: effects of mating cost, density and sex ratio[END_REF][START_REF] Etienne | How choosy should I be? The relative searching time predicts evolution of choosiness under direct sexual selection[END_REF].

Highly choosy females run the risk of remaining unmated and the level of choosiness is likely to reach a value that counterbalances the benefits of obtaining high quality males and the costs of mating and of remaining unmated [START_REF] Kokko | Sexual selection when fertilization is not guaranteed[END_REF]. Typically, females should mate fairly indiscriminately when they first mate because of the large fitness cost of not mating [START_REF] Worthington | Direct costs and benefits of multiple mating: Are high female mating rates due to ejaculate replenishment[END_REF]. However, in L. botrana, the occurrence of mating failures seems to be low [START_REF] Torres-Vila | Larval food source promotes cyclic seasonal variation in polyandry in the moth Lobesia botrana[END_REF]) and our previous mate-choice study [START_REF] Muller | Direct benefits from choosing a virgin male in the European grapevine moth (Lobesia botrana)[END_REF] indicates that virgin females have evolved the capacity to discriminate among males based on male mating experience.

Conclusion and future perspectives

Our results highlight the overall importance of larval nutrition in male mating capacity and lifetime reproductive investment, all of which could modulate the reproductive strategies of this pest. In L. botrana species, both sexes are expected to be choosy about their mating partners because both males and females greatly vary in their reproductive quality according to intrinsic and extrinsic conditions [START_REF] Moreau | Grape variety affects larval performance and also female reproductive performance of the European grapevine moth Lobesia botrana (Lepidoptera: Tortricidae)[END_REF][START_REF] Muller | Male larval nutrition affects adult reproductive success in wild European grapevine moth (Lobesia botrana)[END_REF]. Indeed, males and females have non-trivial reproductive costs [START_REF] Harari | Fitness cost of pheromone production in signaling female moths[END_REF]. Firstly, females that invest many limited resources in egg production would be expected to preferentially mate with high quality males (virgin males or males in good condition as a consequence of their larval nutrition) to obtain large direct benefits from mating (such as large and nutrient-rich spermatophores). Secondly, males are also expected to exhibit some mate choice, because spermatophore production is costly and males only produce one nutrient-rich spermatophore throughout their lifetime [START_REF] Bonduriansky | The evolution of male mate choice in insects: a synthesis of ideas and evidence[END_REF]. Moreover, in this moth species, both sexes invest in mate-finding traits; L. botrana females emit costly pheromones to attract mates [START_REF] Harari | Fitness cost of pheromone production in signaling female moths[END_REF][START_REF] Umbers | The mothematics of female pheromone signaling: strategies for aging virgins[END_REF] and males actively search mates by following these chemical signals. The occurrence of mate choice by both sexes of this species should be further investigated.

Figure 1 :

 1 Figure 1: Percentage of L. botrana males having N mating events (from 1 to > 5) with a

Figure 2 .

 2 Figure 2. Mean ± SEM of (a) the spermatophore volume, (b) the number of eupyrene sperm,

Figure 4 .

 4 Figure 4. Lifetime number of offspring sired by L. botrana males, as a function of the number

Figure 5 .

 5 Figure 5. Mean ± SEM of (a) the female fecundity (number of eggs laid) and (b) the female

Table 1 .

 1 Figure1

  Figure 5.


domain: phys spi

text:
P Grenson
email: pierre.grenson@onera.fr

S Beneddine
email: samir.beneddine@onera.fr

Analysis of shock oscillations of an external compression supersonic inlet through unsteady numerical simulations

Keywords: 1 ] x, y, z Cartesian coordinates [m] X ob Plug axial position [mm]

In this study, the ability of unsteady numerical simulations to predict the flow behavior in an external compression supersonic inlet has been investigated. The attention has been focused on the dynamical characteristic of the flow as the ingested mass-flow is progressively reduced from supercritical condition to the subcritical flow regime that features large oscillations of the shock system. This analysis is based on experimental data gathered on a rectangular inlet at M = 1.8. Numerical simulations have been performed on simplified model of a reduced size and Reynolds number in order to make it tractable for high-fidelity simulations. In the present paper, results are presented for (U)RANS and hybrid RANS-LES simulations. It is shown that URANS is unable to predict the flow oscillations which occur before the big buzz whereas hybrid RANS-LES are able to predict the flow spectra in fair agreement with the experimental data. The unsteady data computed through ZDES of one particular configuration is then employed to highlight the pressure mode associated with the little buzz phenomenon. It is shown that it features a more complicated shape than the fundamental open-closed duct pressure mode, which has generally been used to estimate buzz frequency. Eventually ZDES results are compared to a stability analysis based on a steady RANS field.

u τ

Friction velocity [ms -

I. Introduction

T role of supersonic air inlets is to decelerate the incoming supersonic air to a low subsonic flow. This operation must induce the minimum possible total pressure losses while ensuring a stable flow regime. As the amount of air required by the engine decreases below a certain threshold, the shock system ahead of the air intake may however suffer from large oscillations. This phenomenon, referred to as buzz, is highly undesirable as it can strongly alter the engine thrust (combustion extinguishment) or lead to unacceptable structural loads. Prediction of the flow stability characteristics of an air inlet at different operating conditions is therefore a critical issue and is the major concern of the present work.

Numerous studies have addressed these self-excited flow oscillations, with different explanations about their onset. According to the experiments of Ferri and Nucci [START_REF] Ferri | The Origin of Aerodynamic Instability of Supersonic Inlets at Subcritical Conditions[END_REF] on cylindrical supersonic inlet, the buzz is initiated when the vortex sheet due to the intersection between the oblique and normal shock passes below the cowl lower surface, which causes flow separation. Another explanation is due to Dailey [START_REF] Dailey | Supersonic Diffuser Instability[END_REF] who associates buzz with the flow separation on the compression surface that originates from the shock-boundary layer interaction (SBLWI). In both cases, the separation chokes the flow at the intake throat which eventually triggers the buzz.

Fisher et al. [START_REF] Fisher | On the sub-critical stability of variable ramp intakes at mach numbers around 2[END_REF] observed on a rectangular inlet that when the mass flow is decreased, small-amplitude oscillations first occur. This regime was then referred to as little buzz. With further reduction of the mass flow rate, larger amplitude oscillations take place, which corresponds to the big buzz. The frequency of the little and big buzz were found to be similar. The existence of two buzz regimes was also reported by Trapier et al. [START_REF] Trapier | Experimental Study of Supersonic Inlet Buzz[END_REF] on a rectangular mixed compression inlet. They observed the big buzz frequency to be one order of magnitude lower than the little buzz frequency. Following the idea of Newsome [START_REF] Newsome | Numerical Simulation of Near-Critical and Unsteady, Subcritical Inlet Flow[END_REF], the little buzz frequency was related to the first harmonic of the resonance modes of the inlet duct. As shown by Soltani and Spahi-Younsi [START_REF] Soltani | Buzz Cycle Description in an Axisymmetric Mixed-Compression Air Intake[END_REF], the little buzz emergence could depend on the free-stream conditions.

Several authors have attempted to study inlet buzz numerically. Different work [START_REF] Newsome | Numerical Simulation of Near-Critical and Unsteady, Subcritical Inlet Flow[END_REF][START_REF] Lu | Numerical Investigation of Inlet Buzz Flow[END_REF][START_REF] Kwak | Convergence Study of Inlet Buzz Frequency with Computational Parameters[END_REF][START_REF] Hong | Computational Study on Hysteretic Inlet Buzz Characteristics Under Varying Mass Flow Conditions[END_REF] showed that unsteady RANS (URANS) simulations succeeded in reproducing inlet buzz, with a fair prediction of the phenomenon frequency. Due to high flow Reynolds number encountered in air inlet configurations, which severely increases the required mesh size, there have been so far few attemps of numerical simulations using high-fidelity approaches such as LES or hybrid RANS-LES. Trapier et al. [START_REF] Trapier | Delayed Detached-Eddy Simulation and Analysis of Supersonic Inlet Buzz[END_REF] investigated the use of DDES (Delayed Detached Eddy Simulation) for simulating their experimental configuration. This approach not only permitted to retrieve the salient features of the big buzz but also allowed for a deeper understanding of the flow field behavior. Lyubimov and Poteknia [START_REF] Lyubimov | A study of unsteady-state operating conditions of a supersonic inlet by the RANS/ILES method[END_REF] have recently applied hybrid RANS-LES simulation on the Trapier's configuration for various flow regimes, with and without bleed system. A fair qualitative prediction of the pressure fluctuations for different throttling ratio is reported, but the limited number of points in the computational mesh restricts the scope of these results. In the aforementioned studies, no attempt in simulating the flow regime at which little buzz takes place has been reported. The most advanced study dedicated to hybrid RANS-LES simulation of little buzz is due to Candon et al. [START_REF] Candon | Acoustically Induced Shock Oscillations in a Low-Boom Inlet[END_REF]. Those author focused on a low-boom cylindrical external compression inlet for which shock oscillations have been seen to take place for a mass flow ratio close to the design configuration. Through an analysis of the time lag between unsteady pressure signals widespread across the streamwise direction of the duct, they related the shock oscillation to upstream-running compression waves from the diffuser that moved at acoustic speed. This study was however restricted to a unique mass flow ratio and the simulation duration was too short to bring conclusive results about the phenomenon frequency. It would then be interesting to assess the evolution of the shock oscillation frequency as the mass flow ratio is reduced.

Hence, the goal of the present study consists in evaluating the ability of both URANS and hybrid RANS-LES simulation strategies to reproduce flow oscillations in supersonic air inlets when the mass flow ratio is progressively decreased. The methodology is validated against an experimental data gathered on an external compression rectangular inlet. The chosen configuration exhibits the originality that the entry plane is preceded by a flat pate on which a turbulent boundary layer develops. The paper is organized as follows. The experimental setup and associated results are presented in Sec. II. Sec. III is devoted to the simulation parameters that have been selected. Comparison of the numerical data to the experiments is carried out in Sec. IV. In Sec. V, the pressure fluctuations of a little-buzzed configuration are thoroughly analyzed. Finally, a stability analysis in conducted in Sec. VI.

II. Experimental Data

A. Experimental setup

The experiment was performed in the Onera S3MA supersonic wind tunnel. It is a blow-down wind tunnel with a rectangular test section (0.86 × 0.70 m). The experimental model is sketched in Fig. 1. It consists of a rectangular duct of capture area A 1 = 70 mm × 80 mm preceded by a 10°-compression ramp and a 125 mm-long flat plate. The mass flow rate q m through the air inlet is fixed by the relative position of a conical plug X ob into the cylindrical duct section of diameter D = 84.4 mm, of equivalent area to A 1 . The transition between the 200 mm long rectangular section and the 100 mm long cylindrical one is progressively achieved over a length of 200 mm. The diffuser ramp downstream of the inlet section has a angle of 4°. The M = 1.8 developing boundary layer on the upstream plate is artificially tripped by means of a roughness band (CADCUT) located 15 mm downstream of the leading edge. The upstream stagnation pressure p i,0 is 171415 Pa and the stagnation temperature T i,0 is 310 K.

The model is equipped with static pressure taps in section 2 (green dots in Fig. 1) in order to measure the inlet performance. Additionally, unsteady pressure transducers (Kulites) are located on the compression ramp. As they lie in the neighborhood of the shock foot, they will give valuable information about the shock oscillation. Windows on the side wall of the wind tunnel allowed for both Schlieren pictures (25 im/s) and PIV measurements of the flow on the flat plate and compression ramp.

B. Results

Characteristic curve

Performance of an air inlet are generally given through the characteristic curve which expresses the pressure recovery at the engine entry plane (also called AIP, for Aerodynamic Interface Plane) as a function of the mass flow ratio. In the experiments, this plane corresponds to section 2 in Fig. 1, located ahead of the conical plug. The pressure recovery, also called efficiency η, is defined as:

η = p i,2 p i,0 (1)
The upstream stagnation pressure p i,0 is given by the static pressure in the wind tunnel air tank. Measurement of the mean stagnation pressure in the AIP would ideally require a pressure rake. As such a device was not used in the present study, the stagnation pressure p i,2 is computed from the static wall pressure p 2 and the geometrical section ratio A t /A 2 between the AIP and the sonic throat. This method relies on the hypothesis that the flow is isentropic between section 2 and the throat, which is a fair hypothesis if no shock is present in-between:

p i,2 = p s,2 1 + γ -1 2 M 2 2 γ γ-1 with M 2 = Σ -1 A t (X ob ) A 2 (2)
The mass flow through section 2 q m,2 is conventionally normalized by the theoretical maximal mass flow ratio that can be ingested by the inlet q m,max :

ε = q m,2 q m,max (3)
In the case of an external compression inlet with "shock-on-lip" oblique shocks, q m,max can be expressed as:

q m,max = 0.04042 p i,0 A 1 Σ (M 0 ) T i,0 (4)
with T i,0 the upstream stagnation temperature, which is considered constant throughout the flow. By means of Eqs. ( 1) and ( 2), the normalized mass flow ratio can be rewritten as:

ε = η A t (X ob ) A 1 Σ (M 0 ) (5)
with the throat section A t depending on the plug position X ob . The throttling ratio is defined as the ratio between the critical section A t and the fully open duct end section A exit and quantify the blocking degree of the air inlet.

The characteristic curve obtained from the experimental measurements is plotted in Fig. 2. Schlieren vizualisation and PIV mean velocity fields for various throttling ratios are also represented as they allow for the characterization of the shock pattern on the compression ramp. From the spectral results that will be presented in the next section, the dynamical feature of the shock system is also indicated (big buzz, little buzz or stable flow). Fig. 2 shows that, as the throttling ratio is incrementally reduced, three flow regimes successively take place. For high throttling ratio (TR = 0.74), only the oblique shock generated by the compression ramp is visible on the Schlieren visualization. At this condition the flow is supercritical i.e. the terminal normal shock that slows the flow from supersonic upstream condition down to subsonic condition at the AIP is located inside the duct. Because the supersonic flow is accelerated in the diffuser, the normal shock upstream Mach number is high, what induces strong stagnation pressure losses. At TR = 0.70, the normal shock lies right at the inlet section so that the flow is subsonic across the entire duct. This operational point constitutes the design point for which the pressure recovery is the highest (η = 0.83). As the mass flow ratio is reduced, the normal shock moves upstream. The shock system exhibits a classical λ-structure owing to the interaction with the boundary layer generating a separation region. In this regime, the shock system has been shown to suffer from slight oscillations that can be associated to little buzz. Below a certain throttling ratio (between TR = 0.60 and TR = 0.56), these shocks oscillations are strongly amplified, meaning that the air inlet is in the big buzz flow regime. In this regime, the flow separation occurs ahead of the compression ramp which explains the strong degradation of the pressure recovery.

Power Spectral Density

Flow unsteadiness is experimentally characterized by the power spectral density (PSD) of static pressure signals gathered by the three Kulites transducers located on the compression ramp and reported in Fig. 3. It is shown that no shock oscillation is observed for the highest throttling ratio TR = 0.74. For the design condition (TR = 0.70), the PSD of the nearest probe to the inlet section exhibits a peak at frequency f = 107 Hz. As the throttling ratio is further decreased, this peak emerges for the more upstream probes, depending on the shock location:

• at probe J for TR = 0.67;

• at probe J and F for TR = 0.65 and TR = 0.60. The fact that the frequency peak appears only on certain probes indicates that the shock oscillations remain localized near the mean shock location which is the main feature of the little buzz regime.

For the lowest throttling ratio (TR = 0.56), the three probes exhibit a peak of high energy content at the frequency f = 83 Hz.This shows that the shock oscillation has a strong amplitude and spans the entire compression ramp. This constitutes the spectral manifestation of the big buzz. Following the idea of Newsome [START_REF] Newsome | Numerical Simulation of Near-Critical and Unsteady, Subcritical Inlet Flow[END_REF], we tried to relate the observed frequencies of f = 107 Hz (little buzz) and f = 83 Hz (big buzz) to those of the acoustic modes (n = 0, 1, 2...) of an open-closed duct whose frequency is given by the following formula:

f n = (2n + 1) c 4L 1 -M 2 (6)
where M is the mean Mach number and c = γRT is the mean speed of sound in the duct. The throat side is generally considered as acoustically "closed" because the velocity at the throat is constant, provided that the throat is choked (M = 1). Formula (6) assumes a uniform distribution of both M and c in the duct, which is not really the case in the presence of a diffuser through which flow is decelerated. For TR = 0.75, we consider the mean Mach number and the static temperature to be that measured in section 2:

M = M 2 = 0.5 and T s,2 = T i,0 1 + γ-1 2 M 2 2 -1
= 295 K, which gives c = 344 m/s. The acoustic length L is the distance between the inlet section 1 and the throat section t, i.e. L = 0.5 m. With these values the fundamental frequency f n=0 of the open-closed duct is 129 Hz. Considering the hypotheses made in Eq. ( 6), this frequency is fairly close to that observed for TR between 0.60 and 0.70 and which is related to the little buzz phenomenon.

III. Simulation Parameters

A. Computational setup Performing high-fidelity simulations such as hybrid RANS-LES or LES on the experimental configuration described hereabove would require a large number of both mesh cells, due to the high Reynolds number, and time steps because of the low-frequency of the buzz phenomenon. For these reasons a reduced computational configuration has been selected for the present study. The chosen parameters are compared to the experimental ones in Tab. 1. As the buzz phenomenon is expected to be related to the SWLBI ahead of the inlet section, they have been determined so that the shock system and the relative thickness δ/H of the boundary layer on the compression ramp are similar to the experiments. The overall geometry has been scaled by a factor 1/7 in order to increase by the same factor the buzz frequency, which eventually allows to reduce the number of time step required to span a given number of buzz cycles. The freestream stagnation pressure has been raised to 300 000 Pa so that the boundary layer can still maintain turbulent. Finally, side walls have been omitted so that the z direction is considered as homogeneous (Fig. 4). Contrary to the experimental configuration, the duct section is therefore rectangular everywhere. This explains why the diffuser ramp is followed by an horizontal surface up to the sonic throat, located at the same distance from the inlet section as the plug in the experiments.

B. Numerical parameters and modeling

Computations have been performed using the Navier-Stokes structured multi-block solver FastS [START_REF] Peron | An Immersed Boundary Method for preliminary design aerodynamic studies of complex configurations[END_REF], developed at Onera. (U)RANS simulations have been conducted with both the Spalart-Allmaras (SA) [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] turbulence model which solves a unique transport equation for the eddy viscosity ν and a variant (SAc) [START_REF] Spalart | Trends in turbulence treatments[END_REF] which improves the behavior in compressible mixing layers. The Zonal Detached Eddy Simulation (ZDES) [START_REF] Deck | Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation[END_REF] has been selected as the hybrid RANS-LES approach to conduct high-fidelity simulation. This strategy has been proven accurate for prediction of different configuration of both free [START_REF] Verrière | Zonal Detached-Eddy Simulations of a Dual-Stream Jet[END_REF] and wall bounded flows [START_REF] Deck | Zonal-Detached-Eddy Simulation of the Flow Around a High-Lift Configuration[END_REF]. As for the SA model, a transport equation for ν is solved. The destruction term in this equation is modified depending on the mesh grid size ∆ and the wall distance d w so that the RANS model switches to an LES subgrid-scale one in the detached flows. A protection function f d , based the flow local properties, enforces RANS mode in attached boundary layer.

The 20.7M-cells mesh is represented in Fig. 5a. The cell dimensions ∆ in the three cartesian directions have been determined to fullfill the requirements of both:

• LES in the shear layer, expressed in vorticity thickness δ ω = (u max -u min ) ∂u/∂y max unit:

∆x = δ ω /2, ∆y = δ ω /15 and ∆z = δ ω /3 (7)
• ZDES for the boundary layers above the viscous walls, expressed in wall unit (•) + = (•) u τ /ν : ∆x + = 100, ∆y + = 1 and ∆z + = 100 (8) Fig. 5c shows the flow regions treated in RANS and LES at a particular time step. It is seen that attached boundary layers on both the lower wall (compression and diffuser ramp) and the inlet cowl correspond to RANS region while the rest of the inlet is in LES mode.

The same grid has been used for (U)RANS and ZDES simulations. Spatial discretization is realized by the AUSM+P numerical scheme [START_REF] Mary | Large Eddy Simulation of Flow Around an Airfoil Near Stall[END_REF]. The time integration is carried out by means of the second-order-accurate backward scheme of Gear. Time step ∆t = 1.0 × 10 -7 s has been chosen so that the convective CFL number, defined as: remains everywhere less than 5 (Fig. 5b).

CFL = u 2 x + u 2 y + u 2 z ∆t min(∆x, ∆y, ∆z) (9)
For the URANS simulations, the flow has been initialized with the throat section completely open i.e. the throttling ratio TR is 1.0 (see inset (a) in Fig. 7). At t = 0, the throat is suddenly closed to a given TR and the flow evolution is computed with a global time-stepping. For the ZDES simulations, the flow has been initialized with the URANS solution for the corresponding throttling ratio. Statistics are gathered after the transient has been evacuated. Depending on the throttling ratio the simulation duration spanned between 8 (for TR = 0.76) to 40 (for TR = 0.78) characteristic times τ c based on the expected shock oscillation frequency. Computations have been performed on the NEC cluster of Onera (SATOR). Calculation domain has been split in 297 blocks, balanced onto 280 Intel Xeon « Broadwell » E5-2680v4 processors. Effective CPU cost per iteration per cell was 0.7 µs.

IV. Comparison against experimental results

A. Mean flow features

The mean flow obtained for decreasing throttling ratio from TR = 0.82 to 0.72 by means of both RANS and ZDES simulations are gathered in Fig. 6.

URANS

It can been seen that, for decreasing TR, the shock system progresses upstream along the compression ramp as observed in the experiments (Fig. 2). By inspection of the shock system position, one can associate the numerical throttling ratio to the experimental one. The computational case at TR = 0.76 can for example be associated with the experimental configuration at TR = 0.65. The fact that the same shock system pattern are observed for a higher throttling ratio in the simulation originates from the absence of the side walls in the computational domain. Contrary to the experiments there is no lateral spillage.

For the URANS simulation an indication of flow unsteadiness is also reported in Fig. 6. Steady means that the URANS field converges towards a stationary solution. It should first be noticed that all the URANS computation above TR = 0.74 converges to a stationary solution despite the intense transient due to the throat closure and the global time-stepping, as illustrated in Fig. 7 for TR = 0.78. Self-excited oscillations only appear for TR below 0.74, as also shown in Fig. 7. The frequency of these oscillations is 450 Hz which corresponds approximately to the expected value of the buzz phenomenon on the reduced computational model ( f = 581 Hz). These results seem to indicate that URANS is able to reproduce the big buzz but fails to predict any fluctuations for TR which are expected to exhibit little buzz.

ZDES

As observed in Fig. 6, the shock system pattern obtained with the ZDES simulation seems to be shifted towards upstream position compared to URANS for the same throttling ratio. For instance, the flow field of TR = 0.80 with ZDES compares well to that of URANS with TR = 0.78. In Fig. 6 an indication about the unsteady behavior of the flow, that will be discussed in more detail in Sec. C, is also reported. It turns out that, as a consequence of the shift in TR between URANS and ZDES, the big buzz occurs for higher throttling ratio for ZDES.

B. Characteristic curve

Characteristic curves εη obtained with both RANS and ZDES simulations are reported in Fig. 8 the experiments. It should be mentioned that η and ε are computed by means of integrals over the plane A 2 :

η = 1 A 2 ∫ A 2 p i (x, y, z)dA p i,0 (10)
ε = ∫ A 2 ρu x (x, y, z)dA q m,max. (11)
These definitions are more accurate than their experimental counterparts (Eqs. ( 1) and ( 5)) since they do not suffer from the flow isentropy hypothesis and an isolated pressure measurement. It is seen in Fig. 8 that the mass flow ratio predicted by the numerical simulations is shifted towards ε = 1. This discrepancy with respect to the experiments is easily explained by the absence of the side walls in the computational domain that reduces the lateral spillage. The maximum efficiency at the critical point (η 0.8) is remarkably reproduced by the simulations although the computational domain is different from the experimental setup in both size and Reynolds number. The three regimes discussed in Sec. II and labeled in Fig. 2 can be fairly distinguished on the RANS simulations. It is seen that the three ZDES simulations with the highest mass flow ratio (TR = 0.82, TR = 0.80 and TR = 0.78) belong to the stable subcritical branch (II) while the lowest mass flow ratio (TR = 0.76) lies in the buzzed subcritical branch (III).

C. Unsteady flow features

A representative instantaneous field of the density gradient magnitude ∇ρ in the longitudinal plane Oxy obtained from ZDES (TR = 0.78) is represented in Fig. 4. It turns out that the resolved small-scale turbulent structures appears downstream of the diffuser ramp due to the shear layer between the high speed central flow and the separated region behind the diffuser. These structures are advected towards the sonic throat where they disappear due to the favorable pressure gradient and the mesh coarsening. There is no fine turbulent structures in the vicinity of the shock-boundary layer intersection which is, for the most part, treated in RANS (Fig. 5c).

Because we are interested in the ZDES ability to retrieve buzz features, the power spectral density of the pressure signal at probes F, I and J located on the compression ramp (Fig. 1) is plotted in Fig. 9 for the different throttling ratios. It is shown that the PSDs exhibit several peaks whose frequency is not the same depending on the throttling ratio. For TR = 0.82 down to TR = 0.78, pressure signals have both low frequency peaks, which are highlighted on the left plots in Fig. 9, and several high frequency peaks. Those high-frequency peaks are multiple of the frequency f = 12.5 kHz. The low frequency of TR = 0.80 and TR = 0.78, respectively f = 750 Hz and f = 630 Hz, are not far off the expected frequency of the little buzz on the reduced model ( f = 749 Hz) which is related to the fundamental acoustic mode (n = 0) of the duct. For the higher throttling ratio TR = 0.82, the lowest frequency is almost four times higher and does not correspond to any frequency observed in the experiments. One can suppose that the n = 4 acoustic mode is excited in that case, even though the frequency is not exactly equal to 750 Hz × 4 = 3000 Hz. As it was observed in the experiment (Fig. 3), the frequency peaks emerge on different probe depending on the throttling ratio and the related mean shock position. This is not the case for TR = 0.76 for which the three pressure probes experience the shock oscillations, corresponding to the phenomenology of the big buzz. The peak frequency is f = 550 Hz that is close to the expected big buzz frequency of f = 581 Hz. Thus the ZDES simulations have been able to reproduce the change in the frequency peak between little and big buzz.

Contrary to the experiments, the numerical simulation gives access to the mean flow properties in the duct. This allows to compute the fundamental acoustic mode of the duct from Eq. ( 6) by using the computed value of both the Mach number M and speed of sound c in the duct. In Tab. 2, the calculated fundamental frequency of the duct acoustic mode is compared to the dominant one observed for ZDES simulations and the expected frequency from the measurements. For TR = 0.78, this frequency is close to that evidenced in the ZDES simulation. The mean Mach number in the duct decreases with the throttling ratio whereas the speed of sound maintains a nearly constant value. Consequently, the acoustic frequency tends to increase with the throttling ratio, which is in opposition to what is observed both in the ZDES simulations and experimentally. This indicates that the buzz frequency cannot be simply estimated by the fundamental acoustic mode frequency of a one-dimensional open-closed duct. To get more insight on the unsteady flow features highlighted by ZDES simulations, the following section will address the spatial structure of the mode corresponding to the pressure fluctuations at the dominant frequency.

V. Detailed analysis of pressure fluctuations for TR = 0.78

In this section, we focus on the TR = 0.78 case which has been shown to exhibit little buzz at a frequency of 630 Hz. In the following, the aim is to characterize the spatial organization of the pressure fluctuations in the air inlet. For that purpose, the pressure signal from the ZDES simulation has been recorded by 20 probes widespread along the longitudinal axis of the air inlet. Fig. 10 shows their location along with the instantaneous field of the pressure fluctuations in the median longitudinal plane. The pressure fluctuations are defined as the difference between the instantaneous pressure and the local time-averaged pressure:

∆p(t) = p(t) -p (12)
It is seen in Fig. 10 that positive and negative values of the fluctuating pressure are associated with the λ-shock ahead of the inlet entrance plane, indicating its oscillating motion. Farther downstream, the fluctuating pressure field features numerous regions of alternating positive and negative values (from the longitudinal position about x = 0.03 m). Those pressure fluctuations can be related to the developing shear layer between the separated area behind the diffuser ramp and the core flow in the channel. As they are convected downstream by the main flow, the fluctuating pressure pockets grow and eventually break down into smaller scale areas.

In order to link the frequency of the pressure fluctuations to their spatial localization, a spatio-frequential map is drawn in Fig. 11. This kind of representation allows for highlighting the spectral content of the pressure signal for different locations along the longitudinal axis. For probes 0 to 4, located ahead of the compression ramp, the flow is steady and there is no pressure fluctuation. From probe 5 to the most downstream one, the pressure exhibit non-zero fluctuations. For probes 5 to 9, it is seen that pressure fluctuations exhibit a peak at a frequency of 630 Hz, which has been related to the little buzz, and several narrow-band peaks at f = 12.5 kHz, f = 25.0 and f = 37.5 kHz which are multiple of each other. Their origin is still unclear but their narrow-band feature seems to indicate an acoustic origin. The spectral content of probes 5 to 9, is similar to that observed for the probe I and J in Fig. 9c excepted that around the probe 7 a slighter peak emerges at a frequency of 2500 Hz. Such a frequency has been found to be the main shock oscillation frequency for a higher throttling ratio (Fig. 9a). It thus seems that it is still present in the flow field for lower throttling ratio. In the case of the present throttling ratio, the value of the frequency is close to the fourth multiple (630 Hz × 4 = 2520 Hz) of the little buzz frequency.

Between probe 10 and 15, there is a marked increase in the energy at high frequency (50000 Hz), with a tendency for the energy to spread from the highest frequency to the lower ones as we progress in the downstream direction (highlighted by an arrow in Fig. 11). This energy content may be associated with the shear layer which exhibits resolved turbulent structures from about the location of probe 10, as can be seen in Figs. [START_REF] Soltani | Buzz Cycle Description in an Axisymmetric Mixed-Compression Air Intake[END_REF] (numerical Schlieren field), 9c (Mach number field in the inset) and 10 (fluctuating pressure field). An empirical formula, proposed by Huerre [START_REF] Huerre | Hydrodynamic instabilities in open flows[END_REF], allows to estimate the frequency associated with a Kelvin-Helmholtz type instability in a shear layer:

f K-H = 0.135 u max -u min δ ω (13)
with u max and u min the mean flow velocity on both side of the shear layer and δ ω its vorticity thickness. From the mean velocity profile accross the channel at the location corresponding to probe 10, the estimated frequency f K-H is 45000 Hz, which is of the order of magnitude of the frequency at which turbulent energy is produced at the beginning of the shear layer, as shown in Fig. 11 for probe 10.

For probe 14 to 19, the frequency associated with the little buzz is the single one to clearly emerge from the spectrum. The little buzz can then be characterized as a global mode inasmuch as its related frequency emerges for most of the probes in the air inlet.

In order to highlight its spatial structure, a spatio-temporal map of the pressure fluctuations ∆p normalized by the local root-mean-squared pressure p RMS is drawn in Fig. 12a. Time has been nondimensionalized by the characteristic time of a little buzz period τ c = 1/ f . The pressure signals have been filtered by means of the Savitzky-Golay filter [START_REF] Savitzky | Smoothing and differentiation of data by simplified least squares procedures[END_REF] in order to remove the pressure fluctuations due to shear layer and other high-frequency phenomena. The cut-off frequency is 2000 Hz. The effect of the filter application can be assessed in Fig. 12b, which displays the time-evolution of the normalized mass flow ratio through the air inlet for both the unfiltered and filtered signal. The filtered signal clearly features a periodical variation of the mass flow ratio at the little buzz frequency. In Fig. 12a, it can been seen that the pressure varies periodically in the channel. Two distinct regions clearly exhibit such variations: the one between probe 5 and 10 and the other one between probe 12 and 19. The vertical dashed lines drawn in Fig. 12 mark the instants corresponding to peak value of the mass flow ratio. A close-up on several repetition of the periodic mode is displayed in Fig. 14a. It can be seen that the max-flow ratio instants generally correspond to a positive pressure fluctuation between probe 6 and 10 (fore part) and between probe 12 and 19 (rear part). When paying deeper attention to Figs. 12a and14a, it seems that the pressure increase in the rear part of the channel occurs before the pressure rise in the fore part of the duct. Fig. 14a also shows that the instant at which pressure fluctuations is the highest in the rear part of the duct corresponds to a (slightly) negative pressure fluctuation at probe 5, near the oscillating shock. These observations indicate that the pressure does not feature a clear overall in-phase fluctuation mode as it would be expected for the n = 0 fundamental duct acoustic mode to which is generally related the estimated frequency of buzz (equation 6). That may explain the discrepancies highlighted in Tab. 2 between the observed buzz frequencies and that estimated by eq. ( 6).

VI. Stability analysis

Unlike the good agreement between the URANS and ZDES simulations for TR associated with the big buzz phenomenon, the URANS simulations performed in the present work fail to reproduce any oscillations in the little buzz regime. Nonetheless, the following results show that the frequency of the little buzz phenomenon may still be accurately predicted from RANS results through a global stability analysis.

We focus on a RANS configuration associated with little buzz, with TR = 0.74. As explained in section IV, this flow field and the corresponding shock system pattern compares with a ZDES configuration with a higher throttling ratio: from Fig. 6, it best compares with the ZDES results with TR = 0.78, which exhibits fluctuations at a frequency f = 630 Hz. A global stability analysis then consists in computing a steady RANS field, and linearising the RANS equations about this field. For the present work, the corresponding linear operator A, the so-called Jacobian of the system, is explicitly computed by finite differences, following the numerical strategy defined in [START_REF] Beneddine | Characterization of unsteady flow behavior by linear stability analysis[END_REF]. Then, the computation of the eigenspectrum of this operator, shown in Fig. 13a, yields the set of linear global modes of the flow configuration (see for instance [START_REF] Beneddine | Characterization of unsteady flow behavior by linear stability analysis[END_REF] for further details about the stability analysis procedure).

All eigenmodes are stable (growth rate σ < 0), which is consistent with the steady nature of the results obtained with the URANS simulation for this configuration. When focusing on the least stable mode, one may observe that it displays a frequency f = 680 Hz that is close to the results from the ZDES simulation, which indicates that this mode may indeed be related to the buzz phenomenon. The spatial structure of this mode, shown in Fig. 13b, confirms this assumption. Indeed, the mode displays the expected features of the little buzz phenomenon: the fluctuations involve the expected shock oscillations and, more interestingly, pressure waves propagating in the channel of the inlet. It may be observed that the fluctuations nearby the throat are out-of-phase with those nearby the shock, which is consistent with the observations from the ZDES results. A more thorough comparison between the ZDES simulation and stability analysis is performed in Fig. 14 where a reconstructed spatio-temporal map of 5 repetitions of the pressure mode associated with frequency f = 680 Hz is compared to the a close-up view of the spatial-temporal map shown in Fig. 12a. One can first notice the varying pressure in both the fore and rear part of the air inlet with a phase shift between each other. This shift is comparable to that observed for the ZDES simulation. Secondly, both ZDES and stability analysis seem to indicate that there is low intensity pressure fluctuations at the location of probe 11. This spatial form for the pressure mode is still unexplained. One can simply notice that the location of probe 11 corresponds to the end of the subsonic diffuser ramp which induces a sudden change in the duct cross-section evolution.

VII. Conclusions and Perspectives

In the present work, unsteady RANS and ZDES simulations have been carried out on a reduced model of an external compression supersonic inlet which exhibits shock oscillations when mass flow ratio decreases below a particular threshold. The results showed that both URANS and ZDES are able to reproduce the big buzz phenomenon which is associated to large amplitude shock oscillations. Furthermore, only the ZDES strategy have permitted to retrieve small amplitude shock oscillations referred as little buzz and occurring for higher mass flow ratio. For such cases, URANS simulations quickly converged to a steady-state. ZDES has been proven to be valuable tool for a deeper understanding of both little and big buzz phenomena. In particular, the unexplained shift between little buzz and big buzz frequency is well reproduced by this strategy. A detailed analysis of the pressure fluctuations has been carried out for a given little-buzzed configuration. Comparison with big-buzzed and less throttled configurations, which have also be computed by ZDES simulations, is still ongoing work in order to explain the frequency shift.

Eventually, preliminary results from a stability analysis of a little-buzzed configuration showed that the buzz phenomenon is associated with a global mode of the linearized RANS operator, which opens the way to future studies and potential interesting insights regarding the buzz phenomenon. Future work will be dedicated to performing a large-eddy simulation (LES) of the reduced model in order to validate more quantitatively the ZDES simulations.

  Oxy plane.

Figure 1

 1 Figure 1 Experimental setup. (Green circles) Static pressure taps. (Red circles) Kulite transducers.

0. 5 0Figure 2

 52 Figure 2 Different flow regimes as the conical plug progresses into the inlet duct (decreasing throttling ratio TR). (Top) Schlieren visualization. (Middle) Mean velocity field from PIV. (Bottom) Characteristic curve ηε.

Figure 3

 3 Figure 3 Power spectral density of the static pressure signal measured at three probes on the compression surface. ( ) TR = 0.56. ( ) TR = 0.60. ( ) TR = 0.65. ( ) TR = 0.67. ( ) TR = 0.70. ( ) TR = 0.74.

Figure 4

 4 Figure 4 Computational setup: boundary conditions and sonic throat for mass flow regulation. The instantaneous field of density gradient magnitude ∇ρ from ZDES simulation is represented in Ox y plane.

  (a) Mesh distribution. (b) Convective CFL distribution. (c) RANS and LES regions (RANS =1, LES = 0).

Figure 5 Figure 6

 56 Figure 5 Numerical parameters. TR = 0.72 TR = 0.74 TR = 0.76 TR = 0.78 TR = 0.80 TR = 0.82

Figure 7

 7 Figure 7 Time-evolution of the air inlet mass flow ratio with URANS simulations. ( ) TR = 0.78. ( ) TR = 0.72. Insets correspond to instantaneous Mach number field (colormap from M = 0.0 to 1.9). (a) Initial field for URANS calculations. (b) Steady solution for TR = 0.78. Solution field corresponding to (c) the lowest and (b) highest value of ε during a buzz cycle.

Figure 8

 8 Figure 8 Comparison of characteristic curves from numerical simulations to experimental data. ( ) Experiments. ( ) ZDES. ( ) RANS SA. ( ) RANS SAc.

Figure 9 Figure 10

 910 Figure 9 Power spectral density of unsteady static pressure for three probes on the compression ramp obtained with ZDES for decreasing throttling ratio. (Left) Logarithmic plot. (Right) Linear plot in the low frequency range. ( ) Probe K. ( ) Probe I. ( ) Probe J. Probe location is depicted in Fig. 1. (Inset in lower left corner) Instantaneous Mach number distribution in z = 0.005 plane.

Figure 11

 11 Figure 11 Spatio-frequential map of the pressure fluctuations along the longitudinal axis.

Figure 12 (

 12 Figure 12 (a) Spatio-temporal map of the normalized low-pass-filtered pressure fluctuations ∆p/p RMS along the longitudinal direction. (b) Time-evolution of the normalized mass flow ratio through the air inlet. ( ) Unfiltered signal. ( ) Low-pass filtered signal. (Vertical dashed lines) Instants associated with local maximal value of the normalized mass flow ratio.

Figure 13 (

 13 Figure 13 (a) Eigenspectrum of the Jacobian operator associated to the RANS equations, linearized about the steady RANS flow field with TR = 0.74. For each eigenvalue, its real part corresponds to the growth rate σ of the corresponding mode, while the imaginary part gives its frequency f . The red dashed line highlights the frequency of the mode corresponding to the buzz phenomenon. (b) Spatial structure of the pressure fluctuations associated to the mode related to the buzz phenomenon (arbitrary amplitude). The colormap has been saturated such that the fluctuations within the channel could be seen (their amplitude is very small compared to those about the shocks).

Figure 14 (

 14 Figure 14 (a) Spatio-temporal map of the normalized low-pass-filtered pressure fluctuations ∆p/p RMS along the longitudinal direction from ZDES. (b) Reconstructed spatio-temporal map from the spatial mode associated with the least-stable frequency f = 680 Hz. . (Vertical solid lines) Instants corresponding to positive peak value of the pressure in the rear part of the duct.

Table 1 Computational reduced model parameters compared to the experimental ones.

 1

	Parameter	Experimental setup	Reduced model
	T i,0	310 K	310 K
	M 0	1.8	1.8
	p i,0	171 415 Pa	300 000 Pa
	H	70 mm	10 mm
	δ/H	4.1%	5.1%
	Re δ 2	4000	1300
	Buzz frequencies (big and little)	83 Hz and 107 Hz	581 Hz and 750 Hz
	∆t	0.2 µs	0.1 µs
	Time steps (20 buzz cycles)	1 250 000	360 000
	Mesh size (LES)	1500 M	125 M

VIII. Acknowledgments

This work has been funded by the Onera research project PR-OPOSSUM dedicated to the prevision and control of supersonic air inlets. The author would like to thank E. Garnier and J. Dandois for their insightful advice. Special thank are due to P. Duveau and J.-M. Massonnat who provided the experimental data. The author is thankful to I. Mary for his technical support with the solver FastS.