Hélène Beaunieux
email: helene.beaunieux@unicaen.fr

F Eustache

P Busson

V De La Sayette

F Viader

B Desgranges

Cognitive procedural learning in early Alzheimer's disease: impaired processes and compensatory mechanisms

Keywords: procedural memory, declarative memory, working memory, Tower of Hanoi task, learning, compensatory mechanisms

Introduction:

The aim of this study was to study cognitive procedural learning in early Alzheimer's disease (AD). Methods: Cognitive procedural learning was assessed using the Tower of Hanoi task. In order to take account of possible interactions between different systems during cognitive procedural learning, we also measured nonverbal intellectual functions, working memory and declarative memory. Results: Our results showed an apparent preservation of cognitive procedural learning in AD and a deleterious effect of the disease on verbal intelligence and declarative memory. Correlational analyses revealed a difference between AD patients and control participants in the type of processing they applied to the task. Conclusion: The noninvolvement of declarative memory would appear to be partly responsible for a slowdown in the cognitive procedural dynamics of AD patients. As the AD patients were unable to use their declarative memory, they were still in a problem-solving mode at the end of the learning protocol and had to implement higher-order cognitive processes (i.e., compensatory mechanisms) to perform the procedural task.

Introduction

In cognitive rehabilitation, it is vital to assess procedural learning in order to gauge patients' ability to learn new procedures. Given that, unlike declarative and working memory, perceptual-motor and perceptual-verbal procedural learning abilities may be preserved in Alzheimer's disease (AD) (see van Halteren-van Tilborg et al., 2007, for review), occupational rehabilitation based on cognitive procedural learning may be useful for some patients with AD [START_REF] Bourgeois | A comparison of training strategies to enhance use of external aids by persons with dementia[END_REF][START_REF] Camp | Resident-assisted Montessori programming (RAMP): Training persons with dementia to serve as group activity leaders[END_REF][START_REF] Lekeu | Training early Alzheimer patients to use a mobile phone[END_REF][START_REF] Oriani | An electronic memory aid to support prospective memory in patients in the early stages of Alzheimer's disease: a pilot study[END_REF][START_REF] Thivierge | Errorless learning and spaced retrieval techniques to relearn instrumental activities of daily living in mild Alzheimer's disease: A case report study[END_REF]. Little, however, is known about cognitive procedural learning in AD. Most studies of procedural learning in AD have focused on the learning of perceptual-motor procedures (such as the rotor test) and found preserved learning ability and performance levels for this kind of task [START_REF] Bondi | Implicit and explicit memory in Alzheimer's disease and Parkinson's disease[END_REF][START_REF] Bondi | Implicit and explicit memory following anterior communicating artery aneurysm rupture[END_REF][START_REF] Deweer | Explicit memory, procedural learning and lexical priming in Alzheimer's disease[END_REF][START_REF] Dick | Acquisition and long-term retention of a fine motor skill in Alzheimer's disease[END_REF][START_REF] Dick | Acquisition and long-term retention of a gross motor skill in Alzheimer's disease patients under constant and varied practice conditions[END_REF][START_REF] Dick | The variability of practice hypothesis in motor learning: does it apply to Alzheimer's disease?[END_REF][START_REF] Dick | Facilitating acquisition and transfer of a continuous motor task in healthy older adults and patients with Alzheimer's disease[END_REF][START_REF] Eslinger | Preserved motor learning in Alzheimer's disease: implications for anatomy and behavior[END_REF][START_REF] Heindel | Impaired learning of a motor skill in patients with Huntington's disease[END_REF][START_REF] Heindel | Neuropsychological evidence for multiple implicit memory systems: a comparison of Alzheimer's, Huntington's, and Parkinson's disease patients[END_REF]; see van Halteren-van Tilborg et al., 2007, for a review). Research on the ability to learn perceptual-verbal procedures in AD has mainly been based on data collected using a mirror-reading task. Although researchers adopting this procedure have obtained discrepant results, these can be explained to some extent by methodological differences and/or differences in disease severity [START_REF] Desgranges | Memory disorders in Alzheimer's disease and the organization of human memory[END_REF][START_REF] Deweer | Mirror reading in Alzheimer's disease: normal skill learning and acquisition of item-specific information[END_REF][START_REF] Deweer | Explicit memory, procedural learning and lexical priming in Alzheimer's disease[END_REF][START_REF] Grober | Skill learning and repetition priming in Alzheimer's disease[END_REF]Merbah et al., 2010). A handful of studies have investigated cognitive procedural learning in AD using jigsaw puzzles [START_REF] Grafman | Implicit learning in patients with Alzheimer's disease[END_REF][START_REF] Hirono | Procedural memory in patients with mild Alzheimer's disease[END_REF][START_REF] Poe | Implicit and explicit tests: evidence for dissociable motor skills in probable Alzheimer's dementia[END_REF], but they have yielded conflicting results. While Grafman et al. reported impairment of the patients' ability to improve their performances or even complete the task, [START_REF] Hirono | Procedural memory in patients with mild Alzheimer's disease[END_REF] and [START_REF] Poe | Implicit and explicit tests: evidence for dissociable motor skills in probable Alzheimer's dementia[END_REF] suggested that cognitive procedural learning is preserved. As before, these divergent findings can be explained partly by differences in the jigsaw-learning tasks, which varied in complexity and number of trials, and partly by differences in the severity of the dementia.

Above and beyond these discrepancies, which make it difficult to compare findings, none of the above-mentioned studies of procedural learning in AD considered the nonprocedural cognitive functions involved in the encoding and retrieval of information in procedural memory. And yet, research carried out in the fields of neuropsychology [START_REF] Baddeley | When implicit learning fails: amnesia and the problem of error elimination[END_REF][START_REF] Pitel | Effect of episodic and working memory impairments on semantic and cognitive procedural learning at alcohol treatment entry[END_REF][START_REF] Schmidtke | Cognitive procedural learning in amnesia[END_REF][START_REF] Schmidtke | Cognitive procedural learning in patients with fronto-striatal lesions[END_REF][START_REF] Winter | The assessment of cognitive procedural learning in amnesia: Why the Tower of Hanoi has fallen down[END_REF][START_REF] Xu | revisits the Tower of Hanoi puzzle[END_REF], neuroimaging [START_REF] Hubert | The dynamic network subserving the three phases of cognitive procedural learning[END_REF][START_REF] Hubert | Age-related changes in the cerebral substrates of cognitive procedural learning[END_REF] and experimental psychology [START_REF] Ackerman | Determinants of individual differences during skill acquisition: cognitive abilities and information processing[END_REF][START_REF] Anderson | Skill acquisition: Compilation of weak-method problem solutions[END_REF][START_REF] Beaunieux | Which processes are involved in cognitive procedural learning?[END_REF][START_REF] Beaunieux | Episodic memory deficits slow down the dynamics of cognitive procedural learning in normal aging[END_REF]Wotz, 1988) has shown that cognitive procedural learning relies on nonprocedural functions [START_REF] Eustache | MNESIS: towards the integration of current multisystem models of memory[END_REF] that can be considered through the cognitive determinants of cognitive procedural levels (i.e., the cognitive functions required to perform the procedural task at a certain point in the learning process ;[START_REF] Ackerman | Determinants of individual differences during skill acquisition: cognitive abilities and information processing[END_REF][START_REF] Beaunieux | Which processes are involved in cognitive procedural learning?[END_REF][START_REF] Beaunieux | Episodic memory deficits slow down the dynamics of cognitive procedural learning in normal aging[END_REF][START_REF] Pitel | Effect of episodic and working memory impairments on semantic and cognitive procedural learning at alcohol treatment entry[END_REF][START_REF] Schmidtke | Cognitive procedural learning in amnesia[END_REF][START_REF] Schmidtke | Cognitive procedural learning in patients with fronto-striatal lesions[END_REF][START_REF] Woltz | An investigation of the role of working memory in procedural skill acquisition[END_REF].

The contribution of cognitive functions to procedural performances has been theorized in the ACT model (Adaptive Control of Thoughts; [START_REF] Anderson | Skill acquisition: Compilation of weak-method problem solutions[END_REF]. According to this model, cognitive procedural learning occurs in three qualitatively different phases (cognitive, associative and autonomous) and involves different types of processing. Performing a new cognitive procedural task requires processes which are highly controlled in the initial, cognitive phase, and more automatic ones in the last, autonomous, phase. The associative phase is regarded as a transitional phase between the first and last ones. [START_REF] Ackerman | Determinants of individual differences during skill acquisition: cognitive abilities and information processing[END_REF] showed that during the cognitive phase, performance levels are associated with general intelligence, while in the autonomous phase individual differences in performance are mainly determined by psychomotor functions. Alongside these two functions (intelligence and psychomotor functions), [START_REF] Woltz | An investigation of the role of working memory in procedural skill acquisition[END_REF] has demonstrated that working memory is also a major cognitive determinant of individual differences in the first phase. Drawing on [START_REF] Anderson | Skill acquisition: Compilation of weak-method problem solutions[END_REF] and [START_REF] Ackerman | Determinants of individual differences during skill acquisition: cognitive abilities and information processing[END_REF] theoretical conceptions, we confirmed the existence of three qualitatively different learning phases (cognitive, associative and autonomous) in healthy young participants using the Tower of Toronto (TT) task [START_REF] Beaunieux | Which processes are involved in cognitive procedural learning?[END_REF]. We also identified specific cognitive determinants of procedural performance levels for both the cognitive phase (intellectual abilities, working and declarative memory) and the autonomous phase (psychomotor abilities). In a more recent study, we found that older participants, unlike their younger counterparts, did not spontaneously invoke declarative memory and displayed a slowdown in cognitive procedural learning, as well as delayed involvement of working memory in this learning process [START_REF] Beaunieux | Episodic memory deficits slow down the dynamics of cognitive procedural learning in normal aging[END_REF]. These results are in accordance with [START_REF] Baddeley | When implicit learning fails: amnesia and the problem of error elimination[END_REF], [START_REF] Winter | The assessment of cognitive procedural learning in amnesia: Why the Tower of Hanoi has fallen down[END_REF][START_REF] Xu | revisits the Tower of Hanoi puzzle[END_REF][START_REF] Xu | revisits the Tower of Hanoi puzzle[END_REF], who have all underlined the major contribution of declarative memory to procedural learning. These authors suggest that declarative memory of previous learning episodes can be called upon to eliminate errors on subsequent trials. Thus, declarative memory plays a role in improving procedural performance throughout the learning process. Given the well-established effect of AD on declarative memory, we can reasonably assume the dynamics of cognitive procedural learning to be disturbed in these patients and the appearance of preserved cognitive procedural learning in AD [START_REF] Dick | Acquisition and long-term retention of a fine motor skill in Alzheimer's disease[END_REF][START_REF] Dick | Acquisition and long-term retention of a gross motor skill in Alzheimer's disease patients under constant and varied practice conditions[END_REF][START_REF] Dick | The variability of practice hypothesis in motor learning: does it apply to Alzheimer's disease?[END_REF][START_REF] Dick | Facilitating acquisition and transfer of a continuous motor task in healthy older adults and patients with Alzheimer's disease[END_REF][START_REF] Hirono | Procedural memory in patients with mild Alzheimer's disease[END_REF] to be just that-mere appearance. We would expect patients with early AD to be forced by their declarative memory deficits to implement different learning strategies (i.e., compensatory mechanisms) from those implemented by controls.

The aim of this study was to investigate cognitive procedural learning in a group of patients suffering from early AD and a group of older control participants by means of a cognitive procedural test, the Tower of Hanoi (TH). We focused our study on the first two phases of cognitive procedural learning (cognitive and associative phases) because they involve working memory, declarative memory and intelligence.

Method

Participants and selection procedure

Eighteen AD patients with mild to moderate dementia and 18 controls were examined.

The patients, all right-handed, suffered from probable AD according to the criteria established by [START_REF] Mckhann | Clinical diagnosis of Alzheimer's disease: report of the NINCDS-ADRDA Work Group under the auspices of the Department of Health and Human Services Task Force on Alzheimer's Disease[END_REF]. All the patients underwent a neurological examination, standard blood tests, an electroencephalogram (EEG), and a standard neuropsychological assessment to ensure that none of them had an oral comprehension impairment. CT scans and MRI revealed no anomaly other than diffuse cerebral atrophy. The patients had no previous neurological or psychiatric history. The control participants were matched with the patients on age and education level (cf. Table 1).

Insert Table 1

All participants gave their consent to the study after detailed information was provided both to them and to their caregivers, and the study was conducted in line with the Declaration of Helsinki. All the patients were competent to give informed consent and did so.

General procedure

Our protocol comprised a cognitive procedural learning task (TH) and a set of complementary cognitive tests. The whole of the experimental protocol represented three hours of examinations for healthy controls and five hours for patients, spread over two days.

Cognitive procedural task: Tower of Hanoi (TH) task.

The Tower of Hanoi is more straightforward than its isomorphic variant the Tower of Toronto [START_REF] Beaunieux | Which processes are involved in cognitive procedural learning?[END_REF][START_REF] Beaunieux | Episodic memory deficits slow down the dynamics of cognitive procedural learning in normal aging[END_REF]. The material of the TH task was composed of a wooden block with 3 vertical pegs of equal height. Four wooden disks of decreasing diameter were stacked on the left-hand peg. The task consisted in rebuilding the pyramid of 4 disks on the right-hand peg while obeying two rules: never place a larger disk on top of a smaller one and only move one disk at a time. The rules were read out to participants and explained through examples of authorized and unauthorized moves. Before each procedural learning session, we made sure that the oral instructions were fully understood. Participants were corrected online whenever they violated a rule. Each time they completed a trial, they were congratulated on their good performance. The minimum number of moves to solve the 4-disk TH task is 15.

Learning the cognitive procedure took place over three sessions: Sessions 1 and 2 were separated by a retention period of two hours, while Sessions 2 and 3 took place 20 hours apart, including a period of sleep. Each learning session comprised five consecutive trials, meaning that participants could carry out the TH task a total of 15 times. The participants' performances on the TH task were assessed in terms of time and the number of moves needed to complete it.

Complementary tests

Intellectual abilities were measured by the Verbal IQ score of the Wechsler Adult

Intelligence Scale (WAIS-R; [START_REF] Wechsler | Echelle d'intelligence pour adultes[END_REF][START_REF] Cargnello | The WAIS-SAM: a comprehensive administrative model of modified WAIS procedures[END_REF].

Working memory was tested by means of two tasks. The phonological loop was

assessed on a task involving the serial repetition of a series of digits (digit span) taken from the WAIS-R. The visuospatial sketchpad was assessed on the reproduction of spatial sequences (visuospatial span), in a task taken from Signoret's BEM-144 [START_REF] Signoret | Batterie d'efficience mnésique[END_REF].

Declarative memory

was assessed using the California Verbal Learning Test (CVLT; [START_REF] Delis | Profiles of demented and amnesic patients on the California Verbal Learning Test: Implications for the assessment of memory disorders[END_REF][START_REF] Deweer | Explicit memory, procedural learning and lexical priming in Alzheimer's disease[END_REF]. Six scores were obtained: a learning score, two immediate recall scores (free and cued recall), two delayed recall scores (free and cued recall) and a recognition score.

Statistical analysis

Statistical analyses were conducted in three steps.

Assessment of cognitive procedural learning

Performances on the TT task were assessed by means of two variables: the number of moves and the time (in seconds) taken to complete each trial. We aggregated the data for each five-trial session, which yielded a more stable estimate of performance for each group and limited the number of correlations that subsequently had to be calculated. A multivariate analysis of variance (MANOVA) was carried out, with performance on the three sessions as the repeated measure and group as a between-participants factor. Complementary analyses on each session were conducted by mean of t-tests.

Effect of AD on complementary cognitive tasks

Unpaired Student's t-tests were carried out in order to compare the two groups' scores on the complementary cognitive tasks. For verbal intelligence and working memory, raw scores were used, while for declarative memory all the declarative raw scores were converted into standard units (z-scores) and averaged into a single composite score.

Cognitive procedural learning dynamics

In order to study the cognitive procedural learning dynamics, Pearson correlation coefficients were computed within each group to examine the relationship between procedural learning levels (total time per session) and cognitive determinants of levels of performance for the cognitive phase (verbal intellectual abilities, declarative memory and working memory; see Introduction). Finally, within the context of the correlation results, multiple regressions were performed to assess the contribution of selected variables to procedural learning scores, in order to identify the best predictor for each learning session in each group. We compared the β coefficients of the two groups to estimate whether patients invoked the same learning strategies as controls.

We chose not to consider the number of moves for the correlations and regressions, as it was not sufficiently sensitive. This variable loses its variability as soon as the participants have found the solution to the problem and thus does not reflect the automation of the cognitive procedure.

A probability level of 0.05 was adopted for all the analyses. Bonferroni corrections for multiple comparisons were applied.

Results

Assessment of cognitive procedural learning

Regarding the time taken to solve the TH task, the MANOVA showed a significant group effect, F(1, 34) = 207.7; p < .0001 and a significant session repetition effect, F(2, 68) = 41.24; p < .0001, but no significant interaction between the two, F(2, 68) = 2.5; p < .09 (Fig. 1A). T-tests conducted for each session showed that there was a significant difference between the two groups on all three sessions (Session 1: t(34) = 4.38, p <.0001; Session 2: t(34) = 3.5, p <.001 and Session 3: t(34) = 3.6, p < .0001).

Insert Figure 1

In terms of the number of moves, the MANOVA revealed no effect of group, F(1, 34) = 0.51; p = 0.47, a significant session repetition effect, F(2, 68) = 6.67; p = 0.002, and no interaction between the two, F(2, 68) = 1.37; p = 0.26 (Fig. 1B). T-tests conducted for each session showed no significant difference between the two groups on the first two sessions (Session 1: t(34) = 0.35, p = .72; Session 2: t(34) = 0.01, p =.99), but a difference approaching significance for the last session (Session 3: t(34) = 1.7, p = 0.09).

Effect of AD on complementary cognitive tasks

Unpaired Student's t-tests carried out on performances on the complementary cognitive tasks showed an overall impairment in the AD group. Patients' scores were significantly lower for the tests measuring verbal intelligence and declarative memory (Table 2). Differences in working memory scores did not reach significance.

Insert Table 2

Cognitive procedural learning dynamics

The dynamics of cognitive procedural learning was examined by analyzing the correlations in each group and for each session between the time taken to solve the TH task and verbal intelligence, working memory and declarative memory scores, each taken separately. Table 3 shows the correlations observed in each group. The pattern of significant correlations differed according to the group. In the control group, significant correlations were only found for the first session, where procedural performance levels were significantly correlated with verbal intelligence and declarative memory. Regression analyses showed that the sole predictor of procedural learning performance was declarative memory, accounting for 38% of the variance in Session 1 (Table 4). In the AD group, correlations with verbal intelligence and the phonological loop were significant for the first session (Table 3). For the two last sessions, only correlations with verbal intelligence were significant. Declarative memory was never significantly linked with procedural performance levels. Regression analyses showed that the best predictors of procedural results were the phonological loop in Session 1 and verbal intelligence in Sessions 2 and 3 (accounting for 32%, 29% and 28% of the variance, respectively). The slopes corresponding to the controls' predictor for Session 1 and the patients' predictor for Session 4 differed significantly (Table 4).

Discussion

Our results showed relative preservation of cognitive procedural learning in AD, as assessed by the TH task. Although the time taken to solve the TH task and the number of moves required suggested that the patients were slower, they seemed to learn the procedure in almost the same way as the controls, as attested to by the lack of a significant interaction between group and session for the two variables. Correlational analyses revealed a difference in the dynamics of cognitive procedural learning between AD patients and controls.

Results revealed an expected deterioration in intelligence and declarative memory [START_REF] Greenaway | Patterns of verbal memory performance in mild cognitive impairment, Alzheimer disease, and normal aging[END_REF][START_REF] Hamilton | A comparison of episodic memory deficits in neuropathologically-confirmed dementia with Lewy bodies and Alzheimer's disease[END_REF][START_REF] Lamar | Characterizing alterations in executive functioning across distinct subtypes of cortical and subcortical dementia[END_REF] in the AD patients, associated with preserved short-term memory. Comparisons of working memory scores failed to reveal any significant difference. As suggested in a recent review [START_REF] Huntley | Working memory in early Alzheimer's disease: a neuropsychological review[END_REF], these results may reflect continuing normal function of the slave systems of working memory, which only deteriorate as the disease progresses.

The deleterious effect of AD on verbal intelligence and declarative memory seems to have an impact on cognitive procedural learning dynamics. The differences between the two groups in terms of the correlations between verbal intelligence, working memory and declarative memory on the one hand, and cognitive procedural levels on the other hand, can help us to understand the effect of AD on cognitive procedural learning dynamics. In the controls, the link between procedural performance levels and intelligence and declarative memory scores in the first session confirmed the results of our previous studies and suggested that the controls were in the cognitive phase of procedural learning at that point. The absence of any correlation in the final learning session suggests that by then, they had graduated to the associative phase, which no longer requires the massive intervention of nonprocedural cognitive functions [START_REF] Beaunieux | Which processes are involved in cognitive procedural learning?[END_REF][START_REF] Beaunieux | Episodic memory deficits slow down the dynamics of cognitive procedural learning in normal aging[END_REF].

The pattern of correlations observed in AD patients was radically different. Unlike the controls, the AD patients did not seem to draw on their declarative memory at all during the first procedural learning session. Working memory and verbal intelligence alone seemed to contribute to procedural performance levels throughout the learning process.

Thus, despite the absence of major differences in procedural performances, these data reveal a clear difference in the type of processing applied by each group to perform the procedural task. AD patients seemed to draw massively on their verbal intelligence throughout the learning process and to remain in the cognitive phase right to the very the end of the task. This result indicates that, as expected, AD patients used a compensatory mechanism to attempt to attain the same learning level as the control participants. These data are in accordance with those reported by [START_REF] Schmidtke | Cognitive procedural learning in amnesia[END_REF] for amnesic patients and by [START_REF] Pitel | Effect of episodic and working memory impairments on semantic and cognitive procedural learning at alcohol treatment entry[END_REF] for alcoholic patients. They suggest that AD patients were still in a problem-solving mode at the end of the learning protocol, and that they implemented higherorder cognitive processes. Like [START_REF] Schmidtke | Cognitive procedural learning in amnesia[END_REF] and [START_REF] Pitel | Effect of episodic and working memory impairments on semantic and cognitive procedural learning at alcohol treatment entry[END_REF], we consider that the AD patients' prolonged dependence on intelligence suggests that their transition from the initial, cognitive phase to later phases of cognitive skill acquisition was delayed. This hypothesis was strengthened by the analysis of the contribution of declarative memory to procedural performance levels. Our examination of correlation patterns within the two groups showed that controls made extensive use of declarative memory in the first learning session.

AD patients, on the other hand, displayed massive impairment of declarative memory, which hampered the procedural learning process by preventing the effective correction of errors.

Because they were unable to use their declarative memory to make progress, AD patients had greater difficulty generating the cognitive procedure that had to be automated. Finally, our results emphasize the relevance of errorless learning in promoting cognitive procedural learning in AD. Errorless learning "refers to a learning condition that involves the elimination of errors during the learning process" [START_REF] Clare | Errorless learning in the rehabilitation of memory impairment: a critical review[END_REF]. Thus, the main goal of this learning technique is to compensate for the deficits of declarative memory, which is assumed to be in charge of error elimination [START_REF] Baddeley | When implicit learning fails: amnesia and the problem of error elimination[END_REF].

Most of the studies that have investigated the effect of errorless learning in AD have reported a beneficial effect on semantic learning [START_REF] Dunn | Learning face-name associations in early-stage dementia: comparing the effects of errorless learning and effortful processing[END_REF][START_REF] Haslam | How successful is errorless learning in supporting memory for high and low-level knowledge in dementia[END_REF][START_REF] Ruis | Effects of errorless and errorful face-name associative learning in moderate to severe dementia[END_REF]. To date, [START_REF] Kessels | Effects of errorless skill learning in people with mild-tomoderate or severe dementia: a randomized controlled pilot study[END_REF] are the only authors to have investigated the effect of errorless learning on procedural learning in AD in a controlled group design.

They reported a beneficial effect of errorless learning on new skill acquisition. In line with this study, our data suggest that massed learning conditions associated with an errorless learning method could be beneficial to patients in the first cognitive procedural learning phase. This learning method could allow them to reach the autonomous phase more quickly and thus benefit from the positive effects of sleep in the consolidation of information in procedural memory over the following nights [START_REF] Hauptmann | The predictive value of the leveling off of within-session performance for procedural memory consolidation[END_REF]Rauchs et al., 2004;[START_REF] Walker | A refined model of sleep and the time course of memory formation[END_REF]. Thus, we could imagine a learning method associating the positive effects of an initial massed errorless learning condition with those of a subsequent distributed learning one. Further research is now needed to put this proposal to the test from an experimental point of view. NS: not significant *: best predictor for the learning session within the group; p < .05 @: significant difference between AD patients and controls; p < .05

Figure 1 :

 1 Figure 1: Performance trends in terms of the sum of completion times (A) and the sum of the

Table 1 :

 1 Characteristics of control participants and AD patients

	Controls	Patients

a Educational level is expressed in terms of years of schooling b MDRS: Mattis Dementia Rating Scale *** p < .0001

Table 2 :

 2 Performances on the complementary testsDifferences between controls' and patients' means were evaluated by performing two-tailed t-tests. The Bonferroni correction for multiple comparisons was applied. * Significant difference between AD patients and controls.

	Cognitive function	Task	Controls	AD
			n=18	Patients
			mean (SD)	n=18	p
				mean (SD)	value
	Verbal intelligence	Verbal IQ	56.7 (12.6)	44.1 (12.1) 0.004*
	Working memory	Verbal span	5.1 (0.87)	4.5 (1.1)	0.059
	(slave systems)	Visuospatial span	4.4 (0.7)	3.9 (0.9)	0.091
		CVLT Learning (%)	62.8 (12.9)	23.6 (9.1)	<.001*
		CVLT Short-term Free recall (%) 57.3 (24.5)	5.2 (8.9)	<.001*
		CVLT Short-term Cued recall (%) 66.7 (18.7)	2.1 (27.2)	<.001*
	Declarative memory	CVLT Long term Free recall (%) 61.4 (19.7)	4.9 (11.1)	<.001*
		CVLT Long term Cued recall (%) 67.4 (19.1)	3.2 (22.5)	<.001*
		CVLT Recognition (%)	81.3 (13.3)	17.1 (34.9) <.001*
		Composite score	0.57 (0.46)	-1.1 (0.42) <.001*

Table 3 :

 3 Correlations between verbal intelligence, working memory, declarative memory and cognitive procedural learning

	Predictive independent variables		Session 1 r BP	Session 2 Session 3 r BP r BP
	Verbal intelligence		Verbal IQ	-0.48	-0.51	-0.57	-0.57
	Working memory	Phonological loop	Verbal span		-0.6
		Visuospatial sketchpad Visuospatial span
	Declarative memory		Composite score	-0.64
	Only significant correlations are reported
	AD patients' data are in bold and italics

*: p < .05; **: p < .01; ***:p < .001

Table 4 :

 4 Best predictors of procedural learning in the AD and control groups

			β coefficient		Differences in β
		Best predictors	Controls	AD patients	coefficients between groups
	Session 1	Declarative memory Phonological loop	0.64* -0.48	0.03 -0.60*	@ NS
	Session 2 Verbal intelligence	-0.26	-0.57*	NS
	Session 3 Verbal intelligence	-0.05	-0.57*	@


Benjamin Duret

Gautier Luret

Julien Reveillon

Thibaut Ménard

Alain Berlemont

François-Xavier X Demoulin
email: demoulin@coria.fr

A. Berlemont T Menard

DNS Analysis of turbulent mixing in two-phase flows

Keywords: Evaporation, Two-phase flows, turbulent mixing, passive scalar, DNS, modeling, vaporization

come   L'archive ouverte pluridisciplinaire

Introduction

The combustion of fuel issued from the evaporation of a liquid phase remains one of the major sources of energy, especially in the transport industry. Liquid injection in a combustion chamber is a critical step in fuel-air mixture preparation and its induced combustion. Hence, numerous studies have been devoted to the description and understanding of the injection process and its influence on vaporization and mixing. However, experiments are difficult, in particular in the vicinity of the dense zone of the spray. Numerical simulations could be of significant use if adapted methods are carefully developed following two main axes: the interface tracking ability and the phase change capture. These are sketched in Figure 1 along with several major liquid/gas interactions to be captured.

The last decade has seen the apparition of various numerical methods devoted to two-phase flow simulations with an accurate description of the interface position and evolution. Volume of Fluid [START_REF] Hirt | Volume of fluid (vof) method for the dynamics of free boundaries[END_REF]), Level Set [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase flow[END_REF]) and Front-tracking [START_REF] Unverdi | Computations of multi-fluid flows[END_REF]) are the most common interface tracking methods developed to carry out direct numerical simulations of two-phase flows. In such simulations, discontinuities at the interface have to be treated carefully. With this aim, numerical methods such as the delta function method [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF]) and the ghost fluid method [START_REF] Fedkiw | A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF]) have been developed.

The latter allows for applying sharp jump conditions at the interface and new works have proved it possible to accurately describe the primary atomization of liquid jets [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF], [START_REF] Desjardins | An accurate conservative level set/ghost fluid method for simulating turbulent atomization[END_REF], [START_REF] Shinjo | Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation[END_REF]).

In the framework of combustion chamber modeling, interface tracking has to be coupled with an evaporation process. Such studies are relatively recent because of the complexity of multiple interactions between the various physical phenomena that prove to cover a wide range of different space and time scales [START_REF] Tryggvason | Multiscale issues in dns of multiphase flows[END_REF][START_REF] Tryggvason | Direct numerical simulations of flows with phase change[END_REF]). Solving phase change requires the resolution of the Navier-Stokes equations, but energy and species equations have to be considered with appropriate jump conditions at the interface. The interface velocity is also affected by the phase change, as are mass and energy transfers (Figure 1). All of these constraints lead to numerous problems, in particular the difficulty to estimate the local vaporization rate, which depends on the species and temperature gradients at the interface. Recently developed numerical methods to study these flows are expensive, and only basic configurations can be considered so far. One of the first tentative attempts was dedicated to bubbly flows with a two dimensional moving mesh [START_REF] Welch | Local simulation of two-phase flows including interface tracking with mass transfer[END_REF]. The mesh, which prevented wide deformations of the interface, limited this study.

Moreover, computational costs were prohibitively expensive. Thereafter, [START_REF] Calimez | Simulation à petite échelle par une méthode v.o.f. d'écoulements diphasiques réactifs[END_REF] made a major step forward by coupling phase change with the VOF method to study droplet group combustion in two-dimensions. Strong thermodynamic hypothesis were made to reduce computational cost and complexity. The same year, [START_REF] Juric | Computations of boiling flows[END_REF] associated phase change with a front tracking method to simulate bubbly flows in two dimensions. But this method was unable, at the time, to manage significantly deformed interfaces. More recently, [START_REF] Tomar | Numerical simulation of bubble growth in film boiling using a coupled level-set and volume-of-fluid method[END_REF] studied phase change with a coupled Level Set VOF method but found it was necessary to smooth discontinuities across the interface. By using the Ghost Fluid method coupled with the Level Set method in two dimensional simulations, [START_REF] Tanguy | A level set method for vaporizing two-phase flows[END_REF] and [START_REF] Gibou | A level set based sharp interface method for the multiphase incompressible navier-stokes equations with phase change[END_REF] subsequently proposed a model to determine interface velocities aimed at solving continuity equations even including evaporation source terms. [START_REF] Strotos | Numerical investigation on the evaporation of droplets depositing on heated surfaces at low weber numbers[END_REF] analyzed droplet vaporization on a heated wall thanks to adaptive mesh refinement and the coupling between evaporation models and the VOF method. [START_REF] Schlottke | Direct numerical simulation of evaporating droplets[END_REF] also developed a model to calculate velocities at the interface for deformed droplet vaporization in three dimensions with the VOF method.

All of the above mentioned methods are very promising. However, their complexity prevents them from being used in practical geometries. In this work, we propose to simplify the interactions between the various physical phenomena while focusing on the capture of an accurate vapor field. Saturation conditions at the interface are assumed to be constant, particularly the vapor pressure.

No energy equation is resolved, and the liquid's vapor evolution is characterized by an inert scalar.

Because the flow remains inert, the vapor mass fraction is equivalent to the mixture fraction, which defines the level of mixing between vapor and air. The saturation level of vapor in the domain can normalize this mixture fraction. It leads to the variable Z, which is thus bounded between 0 (no fuel vapor) and 1 (saturated fuel vapor). Z is an inert scalar, the evolution of which is driven by a standard convection / diffusion equation in the gas phase. Z is bounded by 1 at all interface positions. A sketch to illustrate the simplified evaporation process is shown in Figure 2. This is the first step towards a simultaneous understanding of atomization, vaporization and mixing processes in dense sprays. As a preliminary work, since the mixing characteristic time is very short compared to the evaporation delay, interface regression is not considered. We focus on the properties of Z when the flow is shifting from very dense towards dispersed liquid presence, which mimics the liquid spatial evolution of an atomization process. To accurately control the various inputs of the simulation, a three-dimensional forced homogeneous turbulence of a gas / liquid flow is considered with a liquid volume fraction ranging from 1% to 99%. Both phases will be resolved in DNS with the ARCHER code [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]); the interface tracking method that has been used is a coupled Level Set/VOF method. A high-density ratio between the two phases is chosen to simulate realistic engine conditions. Both quantitative and qualitative aspects are analyzed.

In the following part of this work, the constitutive equations and numerical procedures are first described. The flow geometry is then depicted along with the various prescribed parameters.

Next, a statistical analysis of the mixture fraction is carried out by studying the evolution of the probability density function of Z. Finally, an original spectral analysis is proposed, giving rise to the discussion of the links between droplet dispersion, the evaporation process and the final topology of vapor of liquid.

Numerical considerations

Navier-Stokes and scalar equations

The joint Level Set/VOF method is coupled with a projection method to carry out the direct numerical simulation of incompressible Navier-Stokes equations:

∂V ∂t + (V.∇)V = - ∇p ρ + 1 ρ ∇.(2µD) + g + f + 1 ρ σκδ(G)n (1)
where p is the fluid pressure, V the velocity vector, g the gravity vector, µ the dynamic viscosity, and D the viscous deformation tensor. At the interface, the surface tension force can be considered based on the Dirac function δ(G): σ is the surface tension, n the normal unit vector, κ is the curvature computed from the Level Set function G (definition of κ, n and G are given in Section 2.2). The gravity term is neglected in this study. To solve the derivatives, a fifth-order scheme, WENO 5 [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF]), is used for convective terms, and a second-order central finite difference scheme is chosen for diffusive terms. (See [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF] for further details concerning the numerical procedure.) A forcing method is necessary to maintain the turbulent kinetic energy at a prescribed level. This is achieved through the source term f , which induces linear forcing [START_REF] Rosales | Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties[END_REF]). It results in f = Av' where A is the forcing coefficient, v' represents velocity fluctuations, and v is the mean flow velocity. In this paper, (•) refers to volume averaging.

A Reynolds decomposition has been applied to the velocity field V. The evolution equation of the turbulent kinetic energy k = 1 2 v' 2 may be written:

∂k ∂t + ∇.(vk) = C k + 2Ak (2)
where C k regroups the typical energetic contributions (i.e. without forcing) for the sake of clarity.

The estimation of C k involves a phase change and a liquid-gas interface. It is therefore complex to estimate A directly from equation 2, thus, the following two-stage procedure has been set up :

Stage 1 -Estimation of C n-1 k based on the A n-1 constant value :

C n-1 k = k n -k n-1 ∆t -2A n-1 k n-1 (3) Stage 2 -Computation of the forcing constant A n based on C n-1 k : 2A n k n = k c -k n τ f -C n-1 k (4)
where k c is the chosen level of kinetic energy, τ f is a characteristic relaxation time (τ f = 3∆t has been retained in this study) to avoid a sharp forcing term. Thus, the force coefficient A is computed at each time step to introduce the forcing term f in the Navier-Stokes equations.

This method maintains a statistically stationary Homogeneous Isotropic Turbulence (HIT) in the whole domain.

To correctly describe the interface, the VOF method is used jointly with the Level Set to enforce mass conservation [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]). For the sake of clarity, the mean value of the VOF method scalar is called the liquid volume fraction φ, and the local liquid volume fraction within a cell is φ. They are defined as follows: φ = V l /V t , where V l is the total volume of liquid and V t the total volume of the domain. φ = V lc /V c , where V c is the cell volume and V lc is the volume of liquid within a cell. φ respects the following transport equation:

∂φ ∂t + V.∇φ = 0 (5)
The equation is discretized following the scheme as defined by [START_REF] Sussman | A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows[END_REF]; details on the coupled second-order conservative operator split advection are not restated here.

In this work, the value of the liquid volume fraction is of primary importance. Varying the liquid volume fraction φ from 1 to 95% allowed our study to range from droplet flows to bubbly flows.

The choice of a passive scalar is representative of an evaporation process with a low temperature level at the interface, inducing a low saturation level of vapor at the interface. In such a case, assuming a local thermodynamics equilibrium at the liquid-gas interface, an evaluation of the saturation level of the vapor mass fraction Y vs can be made based on the Clausius-Clapeyron equation. The considered scalar Z in this study is the normalized form of the vapor mass fraction:

Z = Y v /Y vs .
The scalar is thus equal to 1 at the interface; it then evolves in the gas phase due to convection and diffusion. In this preliminary study, it is considered that the amount of mass of liquid transferred in the gas phase is small enough if the saturation equilibrium is reached rapidly, which is the case. Therefore, the mass of liquid in the domain remains constant while the vapor phase evolution is considered and analyzed. This assumption leads to no significant phase change, but it is of practical interest to the study of mixing and scalar fields induced by vaporization when turbulence velocity fluctuations are greater than the Stephan flow velocity. The evolution equation of the inert scalar Z may be written :

∂Z ∂t + ∇.(VZ) = ∇.(D m ∇Z) (6)
where D m is the diffusion coefficient. In this paper, D m = ν g in the whole domain, which corresponds to a Schmidt number equal to unity. A fifth-order WENO scheme is used for the convective terms. A second-order central finite difference scheme is used for diffusive terms. The time integration is performed using a third-order Runge-Kutta scheme, except at the liquid-gas interface.

A boundary condition is imposed at the interface in order to sustain the necessary value of Z = 1 6 at the liquid-gas interface, allowing the resolution of equation 6. The method is quite simple since the following relation

∂Z ∂t = 1 -Z τ w (7)
is resolved at the interface position. τ w characterizes the time to reach the thermodynamic equilibrium in the mesh cell located at the interface. This status is assumed to always be satisfied at the interface. Moreover, the mesh cells are very fine (this is necessary to capture the interface curvature) thus τ w must be very short while allowing a numerical tolerance. It has been set to τ w = 4∆t in this study. Because the numerical methods of the solver are designed to cope with strong gradients (interface capture), Z is set to 0 in the gas and 1 at the interface (and in the liquid) at the initial time of the simulation. The scalar equation is solved in the entire domain. In the liquid, Z is fixed to unity during the simulation.

Interface tracking method

In the following, the interface tracking method and the treatment of discontinuities are recalled.

More details can be found in the work of [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]. Level Set methods use a continuous function to describe the interface [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF], [START_REF] Osher | Level set methods: An overview and some recent results[END_REF]). This function is defined as the signed distance between any points of the domain and the interface. The 0 level curve of that function therefore provides the interface location. The convection equation to describe the motion of the interface in a given velocity field V reads:

∂G ∂t + V.∇G = 0 (8)
Eq. ( 8) is the hyperbolic type, and the discretization method must combine a high convergence order and robustness. Thus, for the distance function field, a 5th-order WENO scheme [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF]) is used for convective terms. The normal unit vector n and the curvature of the interface κ(G) are evaluated based on the Level Set function by :

n = ∇G |∇G| , κ(G) = ∇.n (9)
The Level Set method G no longer remains a distance function when solving Eq. ( 8) because of the velocity gradients. A redistancing algorithm [START_REF] Sussman | An improved level set method for incompressible two-phase flows[END_REF]) is thus applied to keep G as the signed distance to the interface. A numerical resolution of Eq. ( 8) combined with the redistancing algorithm can induce mass loss in under-resolved regions. This loss is the main drawback of Level Set methods. To improve mass conservation, extensions of the method have been proposed [START_REF] Sussman | A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows[END_REF], [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF], [START_REF] Olsson | A conservative level set method for two phase flow[END_REF]).

Specific studies have been developed to evaluate some of these methods [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]). It has been found that the Coupled Level Set and Volume-of-Fluid method (CLSVOF, see [START_REF] Sussman | A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows[END_REF]) is well adapted to capture atomization processes [START_REF] Ménard | Coupling level set/vof/ghost fluid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF], [START_REF] Lebas | Numerical simulation of primary break-up and atomization: Dns and modelling study[END_REF]). The main concept of this method is to benefit from the advantage of both Level

Set and VOF strategies: mass loss is limited through the VOF method, and a fine description of interface properties is kept with the Level Set. The Level Set permits the reconstruction of an accurate interface in each cell and helps in the transport of the liquid volume fraction. The location of the interface is slightly modified to retain an amount of liquid in a computational cell that corresponds to the liquid volume fraction determined by the VOF approach. This displacement is used to ensure mass conservation.

Fluid dynamics equations are solved in the context of a low Mach number approach, based on a projection method for the direct numerical simulation of incompressible Navier-Stokes equations (detailed in [START_REF] Tanguy | Application of a level set method for simulation of droplet collisions[END_REF]). The density and the viscosity depend on the sign of the Level Set function according to each phase (liquid and gas). To finalize the description of the twophase flow, jump conditions across the interface are taken into account with the Ghost Fluid (GF) method. In the GF method, ghost cells are defined on each side of the interface [START_REF] Kang | Boundary conditions capturing method for multiphase incompressible flow[END_REF], [START_REF] Liu | A boundary condition capturing method for poisson's equation on irregular domains[END_REF]). This prolongs each phase in order to allow smooth derivatives in the vicinity of the interface. As defined previously, the interface is characterized through the distance function, and jump conditions are extrapolated on a few nodes on each side of the interface. Further details on implementing the GF method to solve the Poisson equation with discontinuous coefficients can be found in the literature [START_REF] Liu | A boundary condition capturing method for poisson's equation on irregular domains[END_REF], [START_REF] Tanguy | Application of a level set method for simulation of droplet collisions[END_REF]).

Numerical configuration

The solution is carried out in a 3D cubical domain with periodic boundaries similar to the study of [START_REF] Luret | Modeling collision outcome in moderately dense spray[END_REF] and [START_REF] Luret | Spray interactions: modeling of collision/coalescence phenomena[END_REF]. This configuration extends previous numerical studies on mixing in single-phase flows (Eswaran and Pope (1988b)) to two-phase flows with a fully resolved liquid-gas formulation.

Parameters choice

Solving realistic turbulent liquid-gas flows is difficult. Several limitations exist despite the significant increase of processor capacities. To define a configuration with effective interactions between the liquid-gas interface and the turbulence, the following dimensionless parameters have been retained: gaseous Weber number We g = ρ g kL/(σ) = 1, liquid Weber number

We l = ρ l kL/(σ) = 30, liquid based Reynolds number Re l = kL/ν l = 310, liquid Ohnesorge number Oh l = √ We l /Re l = 1.77 10 -2 .
Where k is the mean kinetic energy (usually equal to k c ), σ the surface tension, ν the kinematic viscosity in the appropriate phase, ρ the density and L the box length. The grid resolution is 128 3 .

The corresponding set of fluid parameters is summarized in Table 1.

Validation

Two issues have to be addressed concerning numerical validation in turbulent liquid-gas flows:

First, numerical schemes used for single-phase flow DNS are not applicable for liquid-gas flows.

Second, determining the smallest scales in turbulent liquid-gas flows remains an unsolved problem. Consequently, validating numerical configurations for turbulent liquid-gas flows can only be achieved partially. To overcome this difficulty, a systematic procedure has been defined. The first step in this procedure involves validating the numerical method used for liquid-gas flows in a single-phase flow configuration by comparing it to classical non-dissipative DNS code. Then, the same numerical parameters are kept to carry out numerical simulations in two-phase flows.

Finally, for one liquid-gas flow configuration, the mesh resolution is doubled in all directions to study the effect of the small scales that are not captured with the lower resolution.

Single Phase flows

Numerical schemes are typically non dissipative and accurate in DNS of single phase flows. It is possible to compute converged results even for fifth-order moments by using spectral methods or high order difference schemes. The estimation of the Kolmogorov scale is wellknown in a HIT configuration, and the mesh size must be of the same order or slightly smaller than the Kolmogorov scale (1.5 η dx 1, see Eswaran and Pope (1988a)).

This kind of numerical scheme is inappropriate in the context of two-phase flows, particularly when dealing with high-density gradients. In the ARCHER code for instance, a fifth-order WENO scheme [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF]) is used as a good compromise between accuracy and robustness. A counterpart of its robustness is the possible introduction of numerical dissipative effects, which can affect small-scale turbulence. A comparison between the ARCHER code and a validated DNS code (used for single-phase flows in many configurations (ASPHODELE, [START_REF] Guichard | DNS of statistically stationary one-and two-phase turbulent combustion: a turbulent injection procedure[END_REF], [START_REF] Reveillon | Analysis of weakly turbulent dilute-spray flames and spray combustion regimes[END_REF], [START_REF] Reveillon | Effects of the preferential segregation of droplets on evaporation and turbulent mixing[END_REF] and [START_REF] Chaisemartin | Eulerian models for turbulent spray combustion with polydispersity and droplet crossing[END_REF]) has been performed to evaluate this effect. An analysis of two key quantities (turbulent kinetic energy and dissipation) reveals that both codes give similar results for the parameters and the configuration presented in this paper in the case of single-phase flows. This indicates that the single-phase flow case is sufficiently resolved with the ARCHER code for the proposed set of parameters.

Additionally, to validate the forced HIT, an estimation of the accuracy of the forcing method has been performed for the configuration described by [START_REF] Rosales | Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties[END_REF], where results have been obtained with spectral methods and high-order central difference schemes. The coefficient A of the forcing scheme (Eq. 2) is set to the same value as the one used by [START_REF] Rosales | Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties[END_REF]. The resulting turbulent kinetic energy found by [START_REF] Rosales | Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties[END_REF] is well recovered with the ARCHER code. This result confirms the ability of the ARCHER code to accurately reproduce the balance between turbulence production (forcing scheme) and dissipation.

Two-phase flows

Determining whether the resolution is high enough is more complex in two-phase flow configurations. Since a definition equivalent to the Kolmogorov scale to determine the smallest scale of the flow is not yet available, it is impossible to estimate the necessary resolution. The smallest scales remain unknown; they are not fully resolved even with the highest resolved DNS [START_REF] Shinjo | Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation[END_REF]). Hence, we have chosen to set the resolution for two-phase flows equal to the resolution determined for single-phase flow cases.

However, as a complement, a grid convergence is conducted to validate behavior of the flow studied in this article. Figure 3 shows velocity spectra of a 128 3 configuration and a 256 3 case (for φ = 5% and same box length).

Some energy has been transferred to small scales in comparison with two-phase flow cases.

This effect is dumped by the WENO 5 scheme in the 128 3 configuration. However, a large range of the spectra remains unchanged, allowing us to consider that only small scales are not well captured. The 256 3 case is shown in the appropriate figures when possible, illustrating that the main results of this paper are not significantly affected by increasing the resolution.

Initializing phase

The periodic box initialization is based on the work of [START_REF] Luret | Spray interactions: modeling of collision/coalescence phenomena[END_REF][START_REF] Luret | Modeling collision outcome in moderately dense spray[END_REF]. Eight droplets are introduced in each corner of the domain; the sum of their volume is equal to the fixed liquid volume fraction. All droplets have an initial rotational velocity in accordance with the prescribed kinetic energy. Linear forcing is inactive at the beginning of the simulation. The flow is quickly destabilized. Next, the linear forcing is activated and the turbulent kinetic energy reaches its desired level with a statistically stationary state. Then, the vaporization process is switched on.

The normalized scalar Z is used to study the mixing of vapor with the ambient air in the gas phase.

The initial scalar field is set to Z = 1 at the interface, Z = 1 in the liquid for continuity reasons, and Z = 0 in the gas.

Discussion

The scalar is analyzed from both statistical and spectral points of view. The first objective is to provide statistical information about the scalar field and the vaporization process. For combustion modeling, which is the final outcome of the atomization in a chamber, the statistical distribution of the vapor field is of primary importance. Previous works have been carried out either to characterize it with DNS of dispersed phase flows [START_REF] Reveillon | Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model[END_REF]) or to propose models [START_REF] Demoulin | Modeling of turbulent spray combustion with application to diesel like experiment[END_REF]). However, our study presents the first analysis involving a complete DNS of the fully resolved vapor field from the interface up to the far field of the spray.

Liquid/gas flow

Color levels of the instantaneous scalar field Z are presented in Figure 4 for six values of φ ranging from 1 % and 95 %. The liquid/gas interface is also shown thanks to a 3D surface representation. At this point, configurations with a small liquid volume fraction (1 % and 5 %) are considered as dispersed cases when compared with dense configurations that are greater than 90 %. This assumption has to be relativized to the common definition of dispersed flows, which considers volume fractions generally below 0.001 %. These configurations cannot be represented by interfacial flows, and gaseous DNS solvers associated with Dispersed Phase Solvers (DPS) are commonly used in this case [START_REF] Reveillon | Effects of the preferential segregation of droplets on evaporation and turbulent mixing[END_REF]). This means that the most dispersed cases studied in this paper (based on interface transport) may be considered as dense from the DNS-DPS point of view.

At low φ, (1 % and 5 %), small liquid structures with almost spherical shapes are present in the domain. These droplets are not affected by the surrounding gas because of the high-density ratio between the two phases. However, the forced turbulence causes the liquid packets to collide regularly. Small gas structures appear (bubbles) for a high liquid volume fraction (95%). They are embedded in a significant quantity of liquid. This case may be called "dispersed bubble flow".

The gas phase is strongly affected by the surrounding liquid flow, and bubbles are stretched by the shear force due to the liquid phase. The intermediate case (φ = 50%) is more complex. Both phases are present in similar proportions. The gas follows mainly the liquid motion, but collision and coalescence are less commonly observed.

Despite a relatively small Weber number value, the interface is very wrinkled in all presented cases. Since accurately resolving the interface is of crucial importance to capture breakups and vaporization rates, the usual vaporization models based on spherical droplets are not relevant. In dense cases, the gas phase volume is smaller and scalar gradients are then higher, leading to quick vaporization and mixing. The gas reaches the saturated state after a characteristic time τ vp . This time span can be estimated from the temporal evolution of the mean scalar value, presented in with any intermediary liquid volume fraction to reach the saturation. For example, when φ = 50%, then τ vp = 1.97t * . Thus, τ vp is correlated to φ but in a strongly non-linear way. The high-resolution case presents a similar evolution of the mean value of the scalar Z. A small discrepancy can be noticed showing a weak effect of small scales captured by the higher mesh resolution. However, the differences noticed for each level of liquid concentration are much greater.

Mixture fraction fluctuations

To characterize scalar mixing and the evaporation process, the scalar RMS Z ′ = (Z -Z) 2 is computed in the gas phase at each time step. The theoretical maximum value of Z ′ is given by

Z ′ m = Z(1 -Z).
This case represents a field of scalar Z with a maximum of fluctuations. Both Z ′ and Z ′ m have been plotted in Figure 6 for three different liquid volume fractions. An effect of the mesh resolution on Z ′ m can be observed, which is simply an amplification of the difference observed in Figure 5 for Z. On the contrary, the difference is attenuated as far as the RMS Z ′ is concerned, showing a small effect of mesh resolution on the result.

A comparison with the results found by Eswaran and Pope (1988b) and [START_REF] Reveillon | Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model[END_REF] can be carried out. The study of Eswaran and Pope concerned turbulent mixing in purely gaseous flows. Therefore, no source of mixture fraction fluctuations exists. Thus, an exponential decay of Z ′ is observed because of scalar dissipation, which is the sole phenomenon acting on the mixture. The study of [START_REF] Reveillon | Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model[END_REF] concerned turbulent mixing in very dispersed two-phase flows. Droplets were considered as a point source of fuel vapor. The authors showed that the scalar RMS increases initially because of the vaporization term Żw (which generates local fluctuations of vapor) and then decreases due to the scalar dissipation rate. In fact, dissipation overcomes the production of fluctuations by vaporization after a specific characteristic time. For interface tracking DNS, a similar behavior of Z ′ is observed in Figure 6. However, contrary to the study of [START_REF] Reveillon | Spray vaporization in nonpremixed turbulent combustion modeling: a single droplet model[END_REF], the mixture fraction increase is not only due to the apparition of a pocket of vapor around a group of droplets but also due to the characterization of the vapor boundary layer around each droplet or liquid structure. We have therefore made progress in the mixing description. This allows us to consider high liquid volume fractions in the study. As shown in Figure 6, the higher the liquid volume fraction, the higher the max of the RMS. This can be explained by the reduction of the gas volume available for mixing, then by a decrease of the vaporization delay, which is shorter when φ increases. Thus, the balance between dissipation and production is modified, causing higher fluctuations of the mixture fraction.

An interesting point can be made by not only observing the first two moments of the mixture fraction but also its complete statistical evolution through the evolution of its probability density function (PDF).

PDF evolution

Inert scalar PDF (denoted f (Z)) are of major importance when studying turbulent inert or reactive flows. In particular, numerous combustion models are based on the PDF of Z because it allows us to compute the filtered or averaged chemical source term through ωc = 1 0 ωc (Z * ) f (Z * )dZ * . It is generally possible to analytically reconstruct the shape of the PDF that is required to close the reaction terms based on our knowledge of the first two moments (mean and variance) of the mixture fraction [START_REF] Borghi | Turbulent combustion modeling[END_REF]). The prescribed analytical PDF shape required to describe statistical properties of Z must be able to evolve from a double delta function (high air/vapor segregation level) to a single delta function centered on the mean when perfect mixing is reached. A mathematical function that allows this kind of behavior is the beta function (see [START_REF] Libby | Turbulent reacting flows[END_REF]), which is based on an expression that uses the first two moments of Z as entry parameters: the mean Z and the variance Z ′ 2 . It gives :

f (Z) = Z a-1 (1 -Z) b-1 Γ(a)Γ(b) Γ(a + b) ( 10
)
where Γ is the gamma function. The two parameters a and b are related to the mean and variance through

a = Z c , b = 1 -Z c with c = Z 1 -Z Z ′ 2 -1 (11)
DNS of dispersed flows does not resolve the flow at the droplet length scale since they are considered as point sources of mass, momentum and energy. This implies filtering at the interface level. Thus, the considered numerical system leads to a small numerical dispersion that can affect the small scale mixing analysis. It appears that the beta function was verified to capture the turbulent mixing in many experimental and numerical works in single-phase (Eswaran and Pope (1988b)) or multiphase flows [START_REF] Reveillon | Effects of the preferential segregation of droplets on evaporation and turbulent mixing[END_REF]; [START_REF] Libby | Turbulent reacting flows[END_REF]) in the framework of dispersed droplets. However, this point should be verified when the entire scalar field is resolved.

To confirm this point, the flow is entirely resolved in both phases in this study, including the mass boundary layers located at the interface. Therefore, present results can be used to verify if beta functions are still valid when the vapor field is solved at the interface vicinity whatever the liquid volume fraction. However, we are particularly interested in results at a low liquid volume fraction, which may be used to evaluate the usual dispersed DNS mixing results. Nevertheless, there is a limitation. As stated in the introduction, the numerical cost can be exceedingly high when completely solving both phases at the interface, and the more the spray is dispersed the more expensive simulations are. Consequently, the most dispersed case presented here corresponds to a liquid volume fraction of 1%. More intensive computations are necessary to decrease this liquid volume fraction towards a smaller value, but this will be the purpose of further study. In this paper, we are focusing on the transition from dense flows ( φ = 99%) to relatively dispersed flows ( φ = 1%).

To compare exact PDF evolutions observed in interface tracking DNS with the beta PDF, the evaporation process is observed at five fixed moments, represented in Figure 7. Each moment corresponds to a specific mean value of Z: Z 1 = 0.1, Z 2 = 0.3, Z 3 = 0.5, Z 4 = 0.75 and Z 5 = 0.95, in order to study vaporization from the initialization (Z 1 ) to the saturation state (Z 5 ). The following instants: t * z 1 , t * z 2 , t * z 3 , t * z 4 , t * z 5 represent the delays necessary to reach each step. These delays differ depending on the initial liquid volume fraction φ that is considered. In the description of the results, the scalar PDF f (Z) is plotted at these five considered moments. This method to study the scalar statistically is used for all liquid volume fractions.

Comparisons of effective scalar PDF obtained from DNS (DNS-PDF) and beta-presumed PDF (β-PDF) corresponding to similar moments are shown in Figure 8 with the exception of PDF concerning high liquid volume fractions (φ = 90, 95%), in which case statistics are not determined because of the lack of samples since only few bubbles are present in the flow. DNS-PDF of the most dispersed cases (up to φ = 10%, Figure 8 -solid lines) at first present a peak corresponding to Z = 0 statistics (the initial gaseous state in the simulations). Then evaporation maintains a Z = 1 level at the interface, and mixing occurs with the joint effects of convection and diffusion. This mixing leads to a widening of the peak whose maximum value decreases at first. Because evaporation is still in progress, the mean mixture fraction level increases towards its maximum value 1, corresponding to a saturated state in the liquid. As Z increases towards 1, the maximum of the DNS-PDF begins to increase again when the mixture is at approximately a third of the saturation level. Dispersion is decreasingly effective, and the PDF tends toward a Dirac function. In the case of a liquid volume fraction equal to 50% we see a different behavior since the initial statistical peak at Z = 0 shows a very large dispersion of the statistics, and a large variety of mixture fraction levels can be observed.

If the mean and the variance of the mixture fraction are used to reconstruct the PDF based on a beta function, it is apparent (Figure 8 -dashed lines) that the β-PDF is not able to capture the mixing properties. The mixture fraction is considerably spread around the mean, and several orders of magnitude of the statistical level can be observed. A counter intuitive observation can even be made since the β-PDF closely matches with the DNS-DPS when a dense flow is considered (φ = 50%). But the more the flow is dispersed, the farther from reality the β-PDF is located. We could have supposed an inverse relationship since the dispersed case is controlled by the gaseous mixing. Moreover, different results are observed when using a DNS-DPS solver in the framework of a much more dispersed flow [START_REF] Reveillon | Effects of the preferential segregation of droplets on evaporation and turbulent mixing[END_REF]). Droplet sizes are assumed to be smaller than the Kolmogorov length scale in DNS-DPS. Consequently, they are also smaller than the mesh size. Local vaporization source terms (computed with evaporation models) are distributed on neighboring mesh nodes. The flow around and inside the droplet is not resolved, neither are boundary layers. We completely solve the flow around the droplet, including boundary layers, with interface tracking DNS. Additionally, since present two-phase flows are less dispersed, the volume occupied by scalar boundary layers over the total volume of gas available is greater than in dispersed DNS. These two facts induce the presence of specific statistical information that can be captured only with interface tracking solvers: the saturated interface leads to the permanent presence of unitary values of the mixture fraction Z. Because of a scale effect, this is almost invisible on the DNS-PDF curves plotted in Figure 8. A slight peak at Z = 1 can be observed for all curves of the case φ = 10%, for example. It seems very small compared to the main peak centered close to the mean value of Z. However, this peak is the cause of the β-PDF failure to represent turbulent mixing. The β-PDF's flattening is due to its inability to capture information at the saturated level. An example has been drawn in Figure 9, representing two PDF with exactly the same mean and variances. The solid line curve presents two singularities: first a classical Gaussian shape curve and then a Delta function centered in Z = 1. The presence of this secondary peak, which corresponds to the saturated area in the boundary layers of our DNS simulations, leads to unbalanced statistics. If the β-PDF corresponding to the solid line PDF is plotted (dashed curve), it becomes evident that it is not able to cope with the presence of the interface without a specific treatment.

PDF shapes

In Figure 10, normalized shapes of the DNS-PDF ( f (z) × Z ′ with respect to (Z -Z)/Z ′ ) are plotted for each part of the vaporization process. They reveal an important asymmetry. First, the vaporization process occurring for high values of Z affects the queue of the PDF. Even if the peak is almost invisible because of scale effects, it has been proven above to have a prevalent impact on Z statistics. As a consequence, when studying dispersed cases (φ < 10%), the peak is logically located slightly below the scalar mean value with a dispersion that is the result of the mixing process. Previous studies of purely gaseous flows (Eswaran and Pope (1988b)) have shown that mixing leads to a stationary PDF shape, close to a Gaussian shape, with a probability peak located at the mean value.

In our case, PDF tend to be self-similar over time for dispersed cases (φ = 1, 5, 10%), except during the first stage (P1 in Figure 10) of the vaporization process. This first stage is the direct result of the interface topology at the beginning of the vaporization. Then, turbulent mixing far from the boundary layer takes over, which leads to a self-similar shape of the PDF. However, if the liquid volume fraction increases, the boundary layers strongly impact the PDF, and no selfsimilarity can be observed.

Note also that the Gaussian shape observed in the framework of purely gaseous flows is not retrieved in the framework of two-phase flows. Instead, a PDF with a shape close to a lognormal function may be seen in Figure 10. This is an important result for future model development. Even if a correction is proposed, which is necessary to take the presence of an interface into account, the β-PDF would not be able to retrieve the desired shape, and a lognormal shape appears to be more appropriate.

This assumption is confirmed by the fact that the shapes tend to be self-similar over time for dispersed cases (φ = 1, 5, 10%), except during the first stage of the evaporation process. A mathematical function of this shape could replace the beta-function in this particular case. This property is lost in intermediate and dense cases. Nevertheless, similar shapes are observed for part two and three of the evaporation process: two Dirac functions at Z = 0 and Z = 1 may be observed. For part four and five a spread shape over Z values is found with a high probability (Z = 1). In these cases, the vaporization process is faster than the scalar mixing thus the PDF do not have the time to converge toward a common shape; they are more dependent on interface geometry.

Energy spectra analysis

Velocity fields

Two-phase spectra. Spectra of the velocity fluctuations of the gas/liquid mixture are presented in Figure 11. Spectra have a similar shape whatever the liquid volume fraction is. This shape is very close to those observed in single-phase flows, which is compatible with the fact that the liquid volume fraction level has no visible impact. This is an important first observation. However, this subject is not discussed in this work since a forcing procedure is used to maintain a constant turbulent kinetic energy by adding a term in the momentum equation. This term could be at the origin of the spectra similarity. To clarify this point, a different configuration with a natural production of turbulent kinetic energy, like a shear flow, has to be considered. Another stumbling block lies in the lack of development of analysis procedures of two-phase flows with a high level of liquid volume fraction and a high liquid / gas density ratio. The impact of this ratio on the turbulence spectra definition and evolution is still uncertain. Spectra of velocity fluctuations, liquid volume fractions, and vapor scalars for interfacial flows with strong deformations of the interface (including high density ratios) have never been studied prior to this work, which presents some preliminary reflections on the subject. However, this is a major topic with a wide range of applications and issues.

As a preliminary procedure, a common spectral analysis (generally applied in one-phase flows) has been carried out since there is no velocity jump at the interface position. Thus, a FFT procedure on the periodic domain is carried out to extract non-dimensional spectral density of energy visible in Figure 11. However, a phase separation procedure is necessary to discriminate both liquid and gas spectra.

Phases separation. FFT are applied to the entire domain, and the considered signal must be defined on all grid points, which complicates the analysis of phase-separated variables. Therefore, it is necessary to find a method to isolate each phase from a spectral point of view. If the velocity fluctuations are considered, the liquid volume fraction is used to separate liquid and gas in the physical space prior to FFT. Accordingly, it is possible to define

V" g = (V -V g )(1 -φ)
to study the gas phase and

V" l = (V -V l )φ
for the liquid phase. V l/g " is the velocity fluctuation's vector of the liquid (l) or gas (g) phase, respectively. V l/g is the mean velocity vector computed exclusively in the considered phase. Fluctuations of the variable in the complementary phase are nullified when a given phase is studied by using this procedure.

First, it is important to quantify the effect of nullifying velocity fluctuations by using the local liquid volume fraction φ. φ is a bimodal variable. It may be equal to either 0 or 1. Its maximum RMS may be determined thanks to φ ′ m = φ(1 -φ). Therefore, the intermediate case corresponding to a liquid volume fraction φ = 50% undergoes the strongest fluctuations. Then, a symmetry around this maximum value is respected, and a dispersed droplets case (φ = 5%) presents a similar maximum RMS of liquid volume fraction to a dispersed bubble case (φ = 95%). A spectral analysis of φ is performed in Figure 12, and this hierarchy is also observed between spectra. They have been plotted for various liquid volume fractions. The most energetic case corresponds to φ = 50%.

All spectra present an inertial range with a -5/2 slope in the log-log representation. Considering the nature of the liquid volume fraction variable, no dissipation range is observed in the high frequency region. This spectrum shape influences the phase conditioned velocity fields since they are defined as a product of the velocity with this scalar φ. This point has to be considered in the following.

Phase separated velocity spectra. At this point, it is possible to consider the phase separated velocity spectra, which have been plotted in Figure 13. All spectra have been normalized by the corresponding kinetic energy and focus is given to the various shapes that may be observed. Two major observations can be made concerning, on the one hand, the position of the maximum of the energy for each phase and, on the other hand, the inertial range.

First, let us consider the case φ = 50% where the same volume of liquid and gas are evolving in the domain. Spectra have exactly the same shape except for a slight difference in the high frequency range. If the configurations with droplets are considered (low liquid volume fraction: 1%, 5% and 10%), there is a shift of the maximum of energy, which is then separated for both phases: the gas phase presents a maximum of energy for the lowest frequency range (i.e. for the largest length scales) contrary to the liquid phase. This shift corresponds to the droplet structures that are smaller than the largest gaseous vortices. Moreover, there is a discrepancy between inertial ranges of both liquid and gas phases that do not have similar slopes. If configurations with bubbles are examined (high liquid volume fraction 90%, 95%), the maximum of energy, occurring at the low frequency range, is that of the liquid phase. The small size of the bubbles compared to the turbulent liquid vortices leads to a position of the maximum of the gas energy at a higher frequency.

If two symmetrical dispersed cases are considered, such as φ = 5% (droplets) and φ = 95% (bubbles), both phases interact very differently because of the significant difference of density between gas and liquid. Some bubbles in a liquid flow strongly undergo the liquid evolution and structures motion. Yet, droplets in a gaseous flow possess a strong inertia (in the cases considered in this work). Their motion will be weakly modified by the small structures of the flows and droplet/droplets interactions have a strong impact on their trajectory. Consequently, there is no real symmetry in the evolution of both phases when bubble and droplet dispersed cases are compared.

The characteristic length scales are different as well. To serve as an example, an estimation of the equivalent Mean Sauter Diameter (SMD) L D32 of a mono-disperse spray corresponding to the same interface density area present in the DNS case is calculated (L D32 = 6φ/Σ for droplet flows and L D32 = 6(1 -φ)/Σ for bubble flows, see [START_REF] Lebas | Numerical simulation of primary break-up and atomization: Dns and modelling study[END_REF] for details). As a reference and to demonstrate the non-symmetrical evolution of both phases the droplet SMD is equal to 29.82 µm in the case φ = 10%, whereas in the bubble dispersed case (φ = 90%) the bubble SMD is equal to 21.97 µm. This estimation is commonly used in the Euler/Lagrange formalism to characterize the transition between the Lagrangian particles (secondary atomization) and the Eulerian description (primary atomization).

The characteristic length scale of the liquid volume fraction spectrum Ł σ has been plotted in Figure 13 to compare its value to the maximum value obtained from the velocity spectrum. The more one phase is dispersed, the more its characteristic length scale becomes equivalent to the liquid volume fraction length scale. This may be a bias of the method used to distinguish between each phase velocity spectrum that uses the liquid volume fraction, but it could also be an effect of the surface tension that creates droplets (bubbles). These droplets (bubbles) enclose a pocket of liquid (gas) that evolves mainly at the same velocity, creating a scale of velocity fluctuation related to the liquid volume fraction length scale.

Scalar fluctuations

A clear definition of the scalar spectrum has to be given prior to any analysis. The mixture fraction Z is defined only in the gas phase, but we need to provide Z information in the liquid area of the flow without corrupting the resulting spectrum. Three basic approaches can be considered to separate both phases in the framework of the mixture fraction analysis: nullify scalar fluctuations by prescribing the Z values to its mean in the liquid or force Z values to either 0 or 1 in the liquid phase. The objective is to study Z fluctuations, thus the last two solutions are prohibited since artificial fluctuations would be added. These fluctuations would follow those of the phase function φ, which would corrupt the results. The first approach seems to be a good compromise. Z values along a given profile are shown in Figure 14 for a better understanding of the modified scalar field. Affecting the mean of Z in the liquid area will not affect the spectral analysis of the mixture fraction fluctuations.

It is apparent that the most important part of the vapor scalar field remains unaffected by the boundary layers. In this case, scalar mixing should remain dominated by single-phase mixing. This is possibly the reason for recovering a -5/3 slope in the spectrum.

A temporal evolution of scalar spectra is shown in Figure 15 for short times (t * ǫ [t * 0 , t * z 1 ]). Initially, the scalar spectra have a particular shape with a uniform repartition of the kinetic energy on the different scales and quickly tend to a stationary spectrum with a k -5/3 inertial range. The effect of the turbulence appears to be prevalent in comparison to the evaporation process. When looking at Figure 16, which plots the temporal evolution of scalar spectra at times t * z 1 , t * z 2 , t * z 3 , t * z 4 , t * z 5 , the same shape is retrieved.

The scalar seems to have an inertial range larger than the velocity spectra. This fact has been duplicated in single-phase flows [START_REF] Danaila | Spectrum of a passive scalar in moderate reynolds number homogeneous isotropic turbulence[END_REF]). We recall that the liquid volume fraction can also affect the shape. The energy level decreases as time increases. Because scalar values tend to the scalar mean as time increases (as we can see on the PDF shown in Section 4.1), scalar fluctuations are reduced until the saturation state is reached (corresponding to Z=1 in the entire domain).

The preceding section reveals the difficulty in studying the scalar spectrally in a two-phase flow configuration. Our first result shows that it is possible to recover single-phase flow behavior in the dispersed limit. For a higher liquid volume fraction, the influence of the complementary phase spectrum hides the vapor scalar behavior.

Conclusion

This paper presents the first attempt to study mixing under the constraint of a turbulent liquidgas flow and evaporation with full resolution of both phases. It is evident that further works are needed to fully explore such complex phenomena. Among them, the effect of forced turbulence, mesh resolution, and the methods used to analyze each phase separately are of primary interest.

However, by comparison to previous works that used a discrete approach for the minority phase, the present work gives valuable indications for modeling purposes.

DNS have been performed in homogeneous isotropic turbulence with a full resolution of both phases. Original results have been found when studying the scalar from both statistical and spectral points of view. Exact PDF shapes obtained with DNS are much closer to a lognormal shape than the commonly presumed beta PDF. Moreover, the interface mass boundary layer introduces a statistical flaw that is not captured by any presumed PDF. This is an important result since presumed beta PDF are often used in combustion modeling to describe the equivalence ratio field and to close the chemical source term.

Spectral analysis has been conducted for the first time in such turbulent liquid-gas and interfacial flows. A method has been proposed to isolate each phase in the spectral domain based on the local liquid volume fraction. The majority phase spectrum corresponds to the total velocity, except at a high frequency range, which has not been considered. Its behavior remains similar to that of a single-phase flow since the density is not considered in the computation of the spectra. Vapor scalar spectra in the gas phase have been analyzed. A specific procedure has been developed to extract the gas phase information. This procedure has first been validated qualitatively with a dispersed droplet case (φ = 5%) where the vapor spectrum shows a similar behavior to scalar spectra in single-phase flows. The evaporation process does not seem to affect the spectrum shape but it has a direct influence on the energy level.

Gt+1 Gt

Vr ωvp Liq Gas

Figure 5 .

 5 Figure 5. t * is the time normalized by a characteristic turbulent time τ turb = L/u ′ based on the velocity root mean square u ′ and the box length L. Dense liquid flows (φ = 95%) reach the saturation level (τ vp = 0.38t * ) very quickly while a comparatively long delay (τ vp = 10.2t * ) is 12

Figure 1 :

 1 Figure 1: Phase change phenomena at the interface. V r is the regressing velocity, ωvp is the local vaporization rate, G is the Level Set function.

Figure 2 :Figure 3 :

 23 Figure 2: Phase change modeling at the interface in the present work.Table 1: Parameters of the simulation (S.I. units)

Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :

 4567810111213 Figure 4: Instantaneous scalar field with interface visualization (in white) at Z = 0.5, for various liquid volume fractions φ. The first column represents the scalar field for low φ (1,5,10% from top to bottom), and the second column at high φ (50,90,95% from top to bottom).

Figure 14 :

 14 Figure 14: Left: Z and φ values over X axis, square : φ, delta : Z. Right: Scalar iso-contour in thin lines, liquid phase representation in black, Z values extraction line in bold black.

Figure 16 :

 16 Figure16: Non-dimensional temporal evolution of Z spectra for each phase of the vaporization process, φ = 5%. Notations are those used in Figure6.

  Figure 15: Non-dimensional temporal evolution of Z spectra, for φ = 5%. t * 0 is the time representing the initial scalar field, t * ǫ represents short times: t* ǫ [t * 0 , t * z 1 ]. t * z 1is the time corresponding to the first phase of the evaporation process (Z 1 ).

						t* z 1
		1e-06			t* z 2
						t* z 3
						t* z 4
	E z (k) / Z' 2	1e-07			t* z 5 t* 0
		1e-08
						t* z 1
			1e-06	0,1	k dx / 2 π	t* ε t* 0 k -5/3	1
	2
	E z (k) / Z'	1e-07
			1e-08
			1e-09	0,1	k dx / 2 π

Acknowledgement

This work was granted access to the HPC resources of IDRIS under the allocation 2010-x2010026153 made by GENCI (Grand Equipement National de Calcul Intensif).