Jean-Michel Servet
email: jean-michel.servet@graduateinstitute.ch

Bruno Tinel
email: btinel@univ-paris1.fr

The behavioural and neoliberal foundations of randomisations 1

Keywords: A13, B41, C93, G41. EFM Classification codes:

Randomized controlled trials by behavioural economists pretend to be pragmatic and only interested in what really works to solve practical problems but in reality they have notorious normative and ideological aspects.

Key points:

Behavioural RCTs ignore contexts and composition effects and reflect the biases of those who perform assessments.

Behavioural randomizers presume without demonstrating that market exchanges are the most effective form of regulation for societies in all situations of social life.

The positive or negative incentives ("nudges") offered by behavioural economics aim to normalize the behaviour of consumers, users, employees or small/independent producers. They are part of a set of power devices by which individual behaviours are shaped and forced, without their knowledge, to conform to dominant class interests.

Introduction

A symposium on the assessment of public policies was held at the French National Assembly in June 2018 as a prefiguration of the national assessment office that will help elected representatives make their decisions. 2 During the preliminary word, Esther Duflo has been presented by a MP as "the world reference regarding evaluation". 3 In her introductory lecture, Duflo said her randomized evaluations (or randomized controlled trials, RCT) conducted at J-Pal are "inspired by behavioural economics". 4 As a support to this statement, she contended that poor people (who are prime targets of these assessments) display behavioural biases that she considers as a-temporal and universal amongst such kind of population. In her Ely lecture given at the Annual Meeting of the American Economic Association in January 2017, she said a good economist should act and think like "a good plumber". This text actually teaches us little about the art of good plumbing, but this rhetorical process, which is less a metaphor than a way of establishing a benign complicity with the reader, or the listener, while overplaying a kind of empiricist modesty, serves among other things to anchor the analysis into behaviourism: "To summarize, economists have the disciplinary training to make good plumbers: economics trains us in behavioral science, incentives issues, and firm behavior" (Duflo, 2017:20). The aim here is therefore to remedy the supposed behavioural biases displayed by the poor by elaborating the "good" incentives ("a smart 'nudging' policy", ibid., p. 6) not on the basis of major theoretical or abstract principles but in a putative pragmatic way based on an empirical and concrete knowledge provided by behavioural economics.

2 An op-ed in Liberation signed by about fifty researchers (Batifoulier et al., 2018) highlights the lack of diversity of the methods presented at this conference. A few weeks earlier, another newspaper column in Le Monde by nine researchers including three "Nobel" in economics [START_REF] Bédécarrats | L'aide au développement passe par des réformes[END_REF] had highlighted the same methodological flaws. Another one has also been published in The Guardian in July [START_REF] Alkire | Buzzwords and tortuous impact studies won't fix a broken aid system[END_REF]. 3 https://youtu.be/4wiO7Iv37Ys?t=130 4 https://youtu.be/4wiO7Iv37Ys?t=2754

In an article published in the Journal of African Economics, Harrison had criticized the current randomized experiments as lacking the rigour of laboratory behavioural economic tests [START_REF] Harrison | Randomisation and its discontents[END_REF]. But he did not question the notion of behavioural bias shared with J-Pal randomisers among others. Thus, Harrison does not dispute the fact that randomisers, albeit not exemplary, can be part of behavioural economics.

RCT economists are not all behavioural economists and not all behaviourists practice randomized evaluation, especially Richard Thaler. This article focuses on the intersection between both groups. While they pretend to be largely pragmatic and only interested in what really works to solve practical problems it is crucial to point out their normative content and show that their approach is in fact based on clear-cut neoliberal ideological and political grounds [START_REF] Servet | L'économie comportementale en question[END_REF]. Most of the critical arguments highlighted in this article are not new.

They have already been emphasised many times about mainstream economics and methodological individualism. The contribution of this article is to show the concrete form of some of these flaws in the work of behavioural randomisers.

The first section emphasizes that randomized behavioural experiments give a primacy to the individual. Despite intended pragmatism, these studies do not escape the long-time acknowledged limits of methodological individualism. The second section shows that these studies crucially centre, both positively and normatively, on economic and financial interdependencies by the Market to the detriment of public collective management or civil society controlling commons. In the third section, this paper proposes a discussion on the role of these behavioural randomized studies in the reproduction of the state in the neoliberal era.

To do this, the discussion goes beyond the common opposition between the state and the market. The expertise provided by behavioural randomisers helps to subject people to incentives (positive or negative) that lead them to best fulfil the creed of a "good economy" inspired by neoliberalism. In doing so, the state becomes an agent of the Market.

Some concrete shortcomings of methodological individualism

The design of the experiments and the processing of the data are carried out from an individualistic conception of social interactions which ignores contexts and composition effects and reflects the prejudices of those who are realizing these randomized evaluations.

The data recorded by the technical assistants of the randomisers during the experiments are collected from individuals or households. These form the basis of each survey. The context is only mentioned in an exceptional way so the information communicated is then very rudimentary. This explains why randomisers do not seek to do interdisciplinary cooperation.

Their science is essentially that of enumeration [START_REF] Supiot | La gouvernance par les nombres[END_REF]. But it is limited to the collection of data from the statistically selected populations and their treatment. The aggregated data that would be captured in the community (meso level) and in the society (macro level) where these populations live are ignored [START_REF] Maxwell | Poor Economics: a review[END_REF].

For example, the randomized evaluations presented in the American Economic Journal: Applied Economics in January 2015, measure the impact of microcredit on those who benefit from a loan as regard to control groups, a supposed similar population that does not benefit the treatment. Changes in income, investment or particular types of goods and services are examined. But the effect of possible competition distortions between micro-entrepreneurs induced by the benefit of that particular loan is not taken into account. As a result of such a positive discrimination, the beneficiary of the credit can for instance acquire a larger stock of raw materials at a lower cost or accept that her/his customers pay later on. If the beneficiary experiences an increase in turnover, and consecutively in income, consumption, savings and investments, it is reasonable to ask whether this is not to the detriment of micro-entrepreneurs neighbouring the same activity but not receiving a loan, as there is nothing in the experiment to prejudge an increase in aggregate demand. If the game is zero-sum at the sector level or community level, no positive economic impact can appear at the meso and global levels. Some authors (Banerjee et al., 2015: 24) mention that there can be general equilibrium issues but they do not incorporate them into the treatment and analysis.

Let us illustrate this primacy granted to the individual in behavioural RCTs by that led by Somville and Vandewalle in India on the incentive to savings [START_REF] Somville | Saving by Default: Evidence from a Field Experiment in Rural India[END_REF]. It examines the effects on savings behaviour of a series of payments randomly paid in cash for some and as a deposit for others on a three months period. 5 The focus is essentially on whether the money is retained or not by the respondent. If the money is not kept by the individual, the authors assume that the expenditure took place under the pressure of some short term need. 6 In their perspective, a person is presumed as always subject to "bad behaviour" such as invariably rejecting choices that would be beneficial to her/him or to choose the satisfaction of immediate pleasures rather than more "useful" expenses later.

However, it is possible to imagine that these sums of money that individuals no longer possess were not necessarily spent idly but lent to their entourage for example. For the authors, such a practice amounts to an expense or transfer deemed negative. They pay no attention to what this pocketed money during the trial can represent or mean for the respondents. The investigators ignore the fact that the "individuals" they study are inserted in a local community where they live on a daily basis of credit and debt relationships that entangle the economic and the social spheres. In this context, the loans made to the entourage are not to be necessarily interpreted by the researcher as a negative element induced by "social pressure". Lending to a close relation who is in dire need immediately opens the possibility that she/he or another close relation, responding to the same social and economic logic, will lend in her/his turn. Since the constitution of the discipline in the eighteenth century, the tribe of economists was formed against all these forms of solidarity and community logic. The so-called pragmatic modernity of behaviourists in general and of randomisers in particular shows here the weight of economic ideology.

In addition, the gap between the needs revealed by the populations and those supposed and ascribed by the designers of the evaluation can be very significant. Indeed, the central position of the individual in the trials does not automatically imply a perfect match between the goods or services disseminated for the tests and the needs of the populations tested [START_REF] Jatteau | Faire preuve par le chiffre ? Le cas des expérimentations aléatoires en économie[END_REF]. Frequently, alternative distribution modalities of a good or a service are tested as for instance the supply of electricity by solar panels, generators or an electric network (Anant et alii, 2014). Randomisers test a service, for example drinking water supply, wondering whether it is better to charge through meters at home or through paid public terminals where users would get supplies. But they do not wonder if people would prefer a public rather than private water supply for example. The designers of the experiment choose what is on the menu and these choices are not trivial, but reflect their own preferences without giving the tested population the opportunity to issue another option. The satisfaction of a need is not compared to the satisfaction of other potentially alternative needs. Of course, another survey could offer other options, but such a bias in the questionnaire has a strong influence on the final result; it is a basic methodological question in all the social sciences. To what extent such experiments are really different from a mere market study on the price level of solvent demand? In addition, the utility of the good or service under test is not called into question by the authors of the tests because the goods tested are presumed to express a progress or a development whose content is not itself put into question. However, witnesses and randomized test subjects say that, after being tested, mosquito nets, condoms, chlorine tablets purifying water, and so on are frequently left abandoned (Jatteau, 2013:20). After distribution, insecticide-treated mosquito nets were not placed over the beds of humans but were used to protect goats from mosquito bites, for example. This illustrates the mechanisms of diversion, appropriation of "development aids" and the many sham behaviours they elicit.

These drawbacks are certainly no less important than what has been denounced in the past about official development assistance.

To understand the origin of these failures, one must analyse the process from the initiative of the test to the final decision to test. In the best case, the instigation comes from a development organization in contact with the populations, which wonders for example why this supply of services or goods meets only mitigated success. In these cases, RCTs turn merely to be a way to improve the organisation's marketing campaigns. In the worst case (given actual local needs), the initiative comes from researchers who have received support from a donor and contact local organizations to apply the test in exchange for certain benefits such as compensation for their field agents, or vehicles left after the experiment, etc. Most of the time, this information on funding and budget allocation is not indicated in articles reporting on experiments.

One of the authors of this article (Servet) has reviewed, for a public funder, a research project in western Cameroon to randomly test the transformation of local rotating savings and loan associations into risk and solidarity hedging instruments. In this area, however, tontines with the largest financial volume are auctioned, which is incompatible with a logic of mutual behaviour. This illustrates the strong misunderstanding that these researchers had of the milieu in which they planned to work. The referee learned later on that the other evaluator had also expressed a very negative opinion on this project because of the great ignorance of the field.

Another example reported by Arthur Jatteau [START_REF] Chartrain | [END_REF] concerns an attempt to install sanitary equipment in rural villages in Kenya. Once the equipment was put in place, researchers who supervised the study were asked to return to the villages one year later to observe whether the villagers still used the equipment. However, the researchers refused on the pretext that utilization rates could be too low, or even zero, and that they needed tangible results for their future publications. The decision was then made to return only a few weeks after the installation of the equipment to take advantage of the novelty effect they would elicit. This temporal element reveals a deep discrepancy between the method and the object.

While development takes decades, surveys are carried out in a few months without questioning the stability and the duration of the behaviours observed. The need to publish at all costs and as quickly as possible pushes these "researchers" to neglect the deep facts that are part of the slowness and duration of social life.

Randomized trials, as well as computer-based tests, being focused on the individual do not capture the effects of effects, i.e. systemic relations. There exist loop side effects with positive or negative feedbacks. Sometimes these effects are even overlooked at the individual level. For example, in the mosquito net trial, the evaluators (Cohen and Dupas, 2010) did not care about the type of insecticide coating the mosquito nets. Mosquito nets are usually impregnated with pyrethroids but it appeared that this product could cause adverse effects on the cognitive development of the child for pregnant women. 7 Similarly, in the tests on the effects of a vermifuge on school absenteeism, the randomisers did not wonder whether or not a greater presence at school would really increase school performance, mistakenly prejudging a positive impact.8

A pro-market twist?

Behavioural randomisers presume that market exchanges are the most effective form of regulation for societies reduced to their economic base. In the RCT they conduct in India, Somville and Vandevalle consider that a reduction in the use of cash for payments on accounts would reduce durably the consumptions considered "not essential". As a result, this would cause an increase in household savings. This financial "nudge" is supposed to induce poverty reduction, a small impulse with a great result. Unfortunately, this enchanting sequence is probably illusory. All suggest that the savings of these poor households are more or less rapidly absorbed by the shocks they suffer in the absence of protection against risks, especially in the event of illness or death of a member who used to provide a large share of the family income. 9 This need for protection implies the establishment of public support for the development of both health insurance institutions and health services. 10 It is difficult to imagine that care structures and prevention systems can be effectively established by the mere use of "market laws" and the free expression of individual preferences. In fact, this has never been observed in history: all the existing systems of social protection and prevention have been set up by public authorities and by community or mutual organisations. Before the 20th century and the nationalization of insurance schemes, these were associations which took charge of the voluntary contributions of their members to certain risks (Laville, 2010 for the 19th). Neither the public nor community associations made it a profit logic.

Let's take another example, among hundreds of other possible, to account for the mercantile bias of randomisers: a study on the use of a toilet emptying system in Dakar. 11 As indicated in the title, this study is about testing private and commercial solutions to imagine how to encourage them. What is at stake is a new pumping system that is more efficient than manual emptying. The question is how to get more households willing to pay more for something less polluting and faster? Less than a third of households use mechanical pumping, which is costly compared to shovel evacuation, the equivalent of five to three days' wages. The authors explain their study by highlighting the health consequences of the poor disposal of 9 In the same vein, [START_REF] Bryan | Risking together[END_REF] show how households are becoming the shock absorbers of last resort including in advanced economies. 10 For an analysis of the case of the United States cf. Zelizer (1979). 11 This study in progress is led by Jean-François Houde, Molly Lipscomb and Laura Schechter, cf.

Market Structuring of Sludge Management for the Benefit of Vulnerable Households in Dakar, https://www.poverty-action.org/study/market-structuring-sludge-management-benefit-vulnerablehouseholds-dakar.

wastewater and excrement, resulting in particular in increased infant mortality. Here stops their concern for the collective. All major European cities have built in their development phase and have since maintained a public sewage system and treatment plants to evacuate and wash the different categories of wastewater. Has not such a system proved its efficiency in meeting the needs of the greatest number? The question of a more modern evacuation of excrements will arise anyway. In Dakar, 59.3percent of houses have a septic tank, 6.4percent have a sealed pit, 3.5percent use latrines and 0.1percent have no sewage system, 30.6percent

of homes are already connected to a sewer (SRSD Dakar, 2013, chap. 5:51). 12 It is true that the local press regularly emphasizes the lack of maintenance of this system but the development of such a public solution is not even mentioned by randomisers. They do not even consider dry toilet although this is the most environmentally friendly solution. 13 The only solution under test is a private and mercantile modality. The test involves 4000 households who will receive a reduction of 10 or 50percent on the cost of mechanical pumping. The effect on neighbours who do not benefit from the subsidised service is also tested. The emphasis on the market and the individual dimension of the test is reinforced by the type of financial arrangements envisaged, in particular the timing of the payment and its linking with savings accounts to cover this type of expense. Other tests, like those conducted in Morocco (Devoto et al., 2012) to encourage the installation of drinking water meters at home have the same commercial bias, which tends to legitimise the involvement of water suppliers like Veolia to expand their clientele. The same question arises about the tests relating to the domestic production of electricity by means of solar panels where private 12 The summary of the study indicates that it takes place in Dakar. It should be noted that the sewer is much less used outside the department of Dakar. In its suburbs, the most common mode of sewage disposal is the septic tank; it concerns 99percent of concessions in Guédiawaye, 62.18percent in Pikine and 91percent in Rufisque. 13 One might think that this solution is only valid for rural areas. But the equipment in dry toilets of the premises of the Charles Léopold Mayer Foundation for the Progress of Humankind in the XIth arrondissement of Paris shows the possible use in city.

companies seem always preferred to a supply of electricity by networks articulating private and collective logic and supposing public investments [START_REF] Sudarshan | Welfare Benefits of Decentralized Solar Energy for the Rural Poor in India[END_REF].

The pro-Market bias for its presumed competitive mechanisms is pervasive in a great deal of tests conducted, for example, in India and Indonesia to eliminate or restrict corruption in the distribution of public aid [START_REF] Benjamin | Monitoring corruption: evidence form a field experiment in Indonesia[END_REF][START_REF] Banerjee | E-governance, Accountability, and Leakage in Public Programs: Experimental Evidence from a Financial Management Reform in India[END_REF]. In these approaches, the market is not understood as a social construct but as a "mechanism" of supplies and demands whose price process is never clearly analysed. 14 In many cases, the authors of these tests assume that government intervention at both the national and local levels inevitably leads to rent capture, which has a negative effect on economic activity. The appropriation of the state by private interests is presumed somehow inherent to public intervention and new distribution procedures are supposed to eliminate or restrict it. These studies imagine state operating rules considered as normal without calling into question the nature of the State in these societies.

Nor do they question the possible clannish functioning of the State, the economy of gift and counter-gift in social relations, the level of remuneration of civil servants in relation to other social categories, etc. It is also important to consider the direction and distribution of financial flows induced by what is usually called "corruption". If the extracted resources are then invested abroad, their effect and significance is probably not the same as if they are used to fuel expenditures in the local economy and provide jobs. In this case, buying citizens' votes can be a redistributive system of wealth. Inspired by a sort of fair price ideal (including regarding public expenditures), randomisers are unable to grasp such systemic effects.

However, these politico-cultural and economic elements should be taken seriously in order to hope to eliminate corruption in a global way. 15 Some tests, for example on the effectiveness of accompanying unemployed people to find a job, note that public institutions are as effective as private institutions while their costs are lower. 16 One could then conclude that randomisers are not systematically victims of a promarket bias but a thorough reading of the papers reveals anyway a strong prejudice that the market is the most effective regulation modality of societies. 17

Public intervention at the service of the Market

Putting public intervention into the service of the Market, understood as the competition of private interests and property (Hillenkamp and Servet, 2015), is a central element of the neoliberal configuration. Commentators often oppose behavioural economists to neoliberals in that the former rely heavily on public structures (national as well as multilateral). Just as State / Market opposition is very much a rhetorical and political game rather than a historical and organizational reality (Andreau et ali., 2006[START_REF] Théret | Régimes économiques de l'ordre politique : esquisse d'une théorie régulationniste des limites de l'Etat[END_REF][START_REF] Elias | The Civilizing Process[END_REF], the mistake here is to think that neoliberals would systematically oppose the State. They oppose a State socialising part of the activity by producing directly goods and services, but not a State that contributes to the institution of what they consider to be the proper functioning of competition and the defence of private property rights. They want a State as sustainer of the Market order (Mitchell and Fazi, 2017, chap.5). Rather than only economic activities, the 16 Cf. Crépon et ali, (2011) and[START_REF] Marc | Évaluation par expérimentation aléatoire de dispositifs de reclassement innovants pour les demandeurs d'emploi[END_REF][START_REF] Marc | Évaluation par expérimentation aléatoire de dispositifs de reclassement innovants pour les demandeurs d'emploi[END_REF]. Crepon et ali. (2011:3) develop a promarket justification: "this finding remains hard to interpret, because the counselling market has only recently been opened to the private sector: it is possible that private providers price over their marginal cost in a market in infantry where competition is limited". Yet, in the initial French version of the 2009 Report, this explanation is absent: "the weaker performance of PPOs could be explained by the need to train newly recruited staff in recent structures: however, in our sample of 12 cohorts, we do not observe a strengthening of the overall PPOs impact through time" (our translation). 17 In their opinion, all the dimensions composing the social are reduced to the economic dimension which economy is itself analysed through a sort of folkish methodological individualism. The preceding remarks are also valid for the extension of the tests to the political field and to the choices exercised by voters who are then considered as consumers of services and public goods. These tests were conducted in India [START_REF] Banerjee | Do informed voters make better choices? Experimental evidence from urban India[END_REF]), Indonesia (Olken 2007, 2009), Brazil [START_REF] Ferraz | Exposing corrupt politicians: the effects of Brazils' publicity released audits on electoral outcomes[END_REF][START_REF] Ferraz | Exposing corrupt politicians: the effects of Brazils' publicity released audits on electoral outcomes[END_REF]Mexico (O. Chong et al., 2011). The fact that randomizers do not systematically reject public intervention creates an illusion for those who ignore the relationship between the State and the market in neoliberal thinking.

neoliberal doctrine considers that all human activities have to be conceived in terms of market and competition of private interests. Hence, the State has to be both the builder and the protector of a genuine market-based society.

In the classic vision that goes from Smith to Samuelson through Walras and many others, the state is responding to the market and private sector failures (resulting from externalities, increasing returns etc.) and thus acting directly in place of a private monopoly. In the postwar period, economists like Musgrave and others have developed public economics around the idea that when neoclassical price theory doesn't apply because real conditions are too far from required hypotheses, then the government can operate directly instead of private initiatives and (under certain conditions) achieve a second best optimum. Since its beginning, the neoliberal doctrine is based on a rejection of such kind of mixed economy on the grounds that the direct intervention of the state is the ferment of socialism [START_REF] Audier | Le colloque Lippmann. Aux origines du néolibéralisme[END_REF][START_REF] Jones | Masters of the Universe: Hayek, Friedman, and the birth of neoliberal politics[END_REF]. This doctrine has two sides. On the one hand, the notion of state failures [START_REF] Kahn | State Failure in Weak States: a Critique of New Institutionalist Explanations[END_REF] leads to the rejection of any form of nationalization of activity (i.e. production and trade) or social provisioning. On the other hand, the notion of the State-as-a-regulator (or sustainer of the Market order) ensures the institution of the market and capitalist relations, which are not spontaneous but need to be protected against the reactions of the society itself as noticed long time ago by [START_REF] Polanyi | The Great Transformation[END_REF]. 18 Most of contemporary behavioural economists and randomisers are contributing to the toolkit of the neoliberal State-as-a-regulator. It is from this perspective that the positive or negative incentives ("nudges") offered by behavioural economics make sense. They aim to normalize the behaviour of consumers, users, employees or small/independent producers. From a Foucaldian point of view, these "nudges" are part of a set of power devices by which 18 A pionneer contribution in economics is [START_REF] Coase | The problem of social cost[END_REF]. Against this political vision, Karl Polanyi has developed the opposite idea that society must develop specific protections to survive the generalization of market exchanges (Hillenkamp, Laville 2013; Laville 2010, Laville et alii 2017, [START_REF] Maucourant | The Institution, the Economy and the Market: Karl Polanyi's Institutional Thought for Economists[END_REF][START_REF] Mendell | Karl Polanyi et le processus institué de démocratisation politique[END_REF].

individual behaviours are modelled and forced, without their knowledge, to conform to the class interests that dominate them. 19The founders of neoliberalism have affirmed the importance of a politically strong state to enable the most meticulous regulation of production and trade. Yesterday, this intervention was conceivable for these thinkers by a constraint that could go up to a dictatorship. Today, the constraint can be soft thanks to the proposed and tested incentives. Experts who claim to be behavioural economists are there to think about these incentives and measure their effects.

They claim that their contribution is only depoliticized technical support. They would be the ones that enable policy makers to make informed choices. And they turn out to be neoliberal policy experts. The difference between behavioural randomisers, on the one hand, and patented neoliberals, on the other, is that the former claim they are pragmatic, while the latter think of the need for structuring theoretical thinking (see table 1 below). Ironically, the former can be put under the criticism of Friedrich von Hayek himself: "Applying to each task the 'social techniques' most appropriate to its solution, unfettered by any dogmatic belief, seems to some the only manner of proceeding worthy of a rational and scientific age… Those self-styled modern 'realists' have only contempt for the old-fashioned reminder that if one starts unsystematically to interfere with the spontaneous order there is no practicable halting point and that it is therefore necessary to choose between alternative systems. They are pleased to think that by proceeding experimentally and therefore 'scientifically' they will succeed in fitting together in piecemeal fashion a desirable order by choosing for each particular desired result what science shows them to be the most appropriate means of achieving it" (Hayek 1973: 57-8).

[

insert table 1]

Conclusion

Behavioural randomisers are now enjoying a success that is not dissociated from the general There is still a need to go beyond the narrow nature of an individual-centred approach to include systemic effects, trends and counter-trends. It is necessary to uncover an alleged pragmatism and to warn about the dangers of this increasingly dominant method in social sciences and correlatively in the evaluation of social and development policies. It is not about claiming that there is no worse methodology than RCTs but their practice by behavioural economists is questionable and leads to biased results. Better random tests could be carried out taking into account the context, historicity, feedback, social and macroeconomic determinants, links and interactions with community and society; by not predetermining in advance the result on the researchers' preferred choices, by not presuming that the researcher knows better than the interviewee what is good for him / her; by taking the time to study the feedback effects (positive or negative) of the survey and its recommendations, by being more aware that the respondents "play" with the interviewers; by taking into account social habits over time, and making interdisciplinary cooperation; and by taking into account composition effects (what is true at the micro level is not necessarily so for society as a whole). Last but not least, this method cannot replace other methods in the social sciences, be they qualitative or quantitative. It is therefore not rational for the RCTs to capture a gigantic share of public resources intended to finance development projects.

  neoliberal context. After a first development in the 1960s, the method of behavioural tests carried out in the form of experiments of the "great society" in the United States has been pushed into the background(Jatteau, 2017) before being reborn in the 2000s and reaching the heights today with the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel prize for Richard Thaler in 2015 and for Abhijit Banerjee, Esther Duflo and Michael Kremer in 2019. Before the reign of neoliberalism in the 1980s, these experiments failed to gain a foothold in prestigious academic institutions, as is the case today. A deep reason for their faint seduction in the past is undoubtedly that, at that ante-neoliberal time, it was doubtful for many to forget society and to pretend to solve "social" problems by treating only individual behaviour. The conviction of the collective determinants was then strong.

For a detailed critical review, seeServet (2018: 73, ff.).

Barnerjee and Duflo (2011) use the same prejudice as an assumption on individual behaviour.

About the effects of pyrethroids, seeViel et ali. (2015).

[START_REF] Miguel | Worms: identifying impacts on education and health in the presence of treatment externalities[END_REF].

For an introduction to this old but overlooked question by mainstream economics, see for instance[START_REF] Berta | On perfect competition: definitions, usages and foundations[END_REF],[START_REF] Bénicourt | Is Anything Worth Keeping in Microeconomics?[END_REF] and[START_REF] Kirman | Market Mechanisms in Real Markets: Why did markets disappear from Economic Analysis? School and Workshop on Market Mechanism Design[END_REF].

This point is less addressed to RCTs than to neoliberalism in general, and it is not a question here of saying which research method could eliminate all corruption in a global manner.

In the late 1970s, the word "nudge" was not yet used as it is today, but[START_REF] Foucault | The birth of biopolitics: lectures at the Collège de France[END_REF] expresses this idea when he writes "Homo oeconomicus is the interface of government and the individual" (p. 252-3).


A Al-Kebsi

R Mose

Y Hoarau

2019-ICE-0136

Ice accumulation in aircraft is modelled using the Level-Set (LS) method. Current two-dimensional and threedimensional icing models are limited regarding complex re-meshing due to ice accretion. The Level-Set method allows fully multi-step simulation of ice accretion. The solid boundary is treated implicitly or explicitly. The solid body can be defined via a characteristic level set function in the former case, and as a layer or list of points in the later. Consequently, the boundary conditions for the airflow are applied through a penalization term, or by direct forcing. Correspondingly, the droplet transport is computed using an Eulerian approach in an Immersed Boundary Method (IBM) coupled with Level-Set (IBM-LS) framework, in which the droplet fields impinge on an explicit layer of cells defined by the LS function using a discrete IBM formulation of the interface.

INTRODUCTION

Aircraft icing can reduce flight safety in extreme weather conditions. According to the US National Transport Safety Board, this is one of the main causes of flight accidents ( [START_REF] Reehorst | National transportation safety board aircraft accident investigation supported[END_REF]). Icing occurs by the impingement of suspended water droplets on aircraft structures. As a result, the designed aerodynamic surfaces are deformed by the introduction of new solid attached to the surface. The design of adequate de-icing mechanisms requires a thorough knowledge of the icing phenomenon itself. An experimental icing study cannot exceed the scope of a handful of simple cases due to its complexity and cost. On the other hand, the use of numerical modelling makes it possible to simulate all possible configurations while studying a spectrum of different parameters.

Simple ice accretion codes treat the icing process as a single layer in a quasi-steady state. A single icing step is assumed to be divided in four intermediate consecutive steps: 1. Dry airflow topology is evaluated. 2. Suspended water droplets are transported impacting against the surface. 3. A liquid film is generated on the surface; the thermodynamic balance of the film is calculated in order to evaluate ice accretion rate. [START_REF] Beaugendre | Development of a second generation in-flight ic-Figure 8: Deformed Geometry (advected φ) obtained on the second fine mesh using 1,2 and 5 ice shots. ing simulation code[END_REF]. The geometry is finally deformed by the accumulation of ice.

All the parameters evaluated through steps one to three are assumed not to change during the ice accumulation, only the flow parameters and variables of the initial clean geometry are used until the end of the ice accumulation.

The dry airflow simulation is usually achieved using the two-dimensional panel method. Recently, several codes used by the industry are coupled with a Navier-Stokes solver, replacing the traditional panel method as in FENSAP-ICE (McGill University) [START_REF] Beaugendre | Development of a second generation in-flight ic-Figure 8: Deformed Geometry (advected φ) obtained on the second fine mesh using 1,2 and 5 ice shots. ing simulation code[END_REF] or LEWICE3D [START_REF] Bidwell | Users manual for the NASA Lewis three-dimensional ice accretion code (LEWICE 3D)[END_REF].

The droplet transport can be modelled by either a Lagrangian or an Eulerian approach. In the former, we formulate the droplet dynamics equation derived from Newton's second law [START_REF] Sang | Numerical simulation of icing effect and ice accretion on three-dimensional configurations[END_REF] while in the later we advect a droplet concentration and momentum fields. Following that, the impact rate is calculated geometrically in the Lagrangian formulation or from direct determination in the Eulerian approach [START_REF] Scott | Navier-stokes solution to the flowfield over ice accretion shapes[END_REF]. The Eulerian approach is more advantageous than the traditional Lagrangian approach, mainly due to its simplicity with multi-body and three-dimensional geometries and its direct integration to existing PDE based solvers.

The third step is to evaluate the mass and energy balance of the liquid film accumulated on the surface by the impinging droplets. The simplest method, called the Messinger model, is based on the mass and energy balance [START_REF] Bernard | Equilibrium temperature of an unheated icing surface as a function of air speed[END_REF]. In the Messinger model, the liquid film is computed from the stagnation point and then looping to further downstream cells towards the trailing edge. An improvement of this Method is the Messinger's iterative model [START_REF] Zhu | 3D ice accretion simulation for complex configuration basing on improved messinger model[END_REF], in which it is possible to treat multiple stagnation points and thus three-dimensional geometries. Other recognizable improvements are based on the resolution of ordinary or partial differential equations where the ODE-PDE model the mass and energy conservation. Such method was initially proposed by [START_REF] Myers | Extension to the Messinger model for aircraft icing[END_REF] based on an ODE, and then extended for multi-body geometries by [START_REF] Ozgen | Ice accretion simulation on multi-element aifoils using extended messingermodel[END_REF]. This extension to the Messinger model added the ability to solve a conduction heat transfer equation on the ice and water substrates. Ultimately, [START_REF] Bourgault | Development of a shallow-water icing model in fensapice[END_REF] proposed a PDE method, based on the Messinger model and the shallow water equation, abbreviated as SWIM. Such PDE model integrates perfectly within a Navier-Stokes air solver coupled with an Eulerian droplet model. The main result obtained from this step is the ice accumulation rate, represented as an ice thickness or an icing velocity.

Lastly, having evaluated the ice thickness, the deformed geometry is obtained. Many techniques exist in the literature to obtain the deformed geometry [START_REF] Montreuil | ECLIPPS: 1. threedimensional CFD prediction of the ice accretion[END_REF], [START_REF] Cao | Numerical simulation of three-dimensional ice accretion on an aircraft wing[END_REF], [START_REF] Verdin | Multistep results in icecremo2[END_REF], [START_REF] Hasanzadeh | Wing aerodynamic performance analysis and stall prediction using canice2d-ns icing code[END_REF] such as grid movement techniques, or re-meshing.

An improvement on the single step ice assumption is to model the icing phenomenon as a multi-layer problem, wherein, the ice forms step-by-step, layer-by-layer as a piecewise function of time. At the end of each layer, the flow field properties (steps 1-3) are recalculated. Consequently, the mesh requires to be regenerated at each step. Numerous studies showed that an improvement was achieved when using multi-step icing, notably for glaze ice configurations. However, application are mainly restricted to two dimensional cases, since the automatic grid regeneration is nearly impossible without resulting in low quality grids. Only few three-dimensional cases are mentioned in the literature, and they are not well explained in terms of convergence and mesh quality, and are usually accompanied with poor heat transfer evaluation and poor ice formation calculation.

In this study, we develop an approach to avoid remeshing. Such approach would overcome most of the limitations of re-meshing and most importantly would inspire Quasi-non-Steady ice accretion modelling. The core of such an approach is embedded grid techniques, namely Level-Set (LS) [START_REF] Osher | Fronts Propagating with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations[END_REF], [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF] and Immersed Boundary Method (IBM) [START_REF] Peskin | Flow patterns around heart valves: A numerical[END_REF]. We focused on the use of Level-Set method.

In a Level-Set framework, the solid body is defined implicitly by a characteristic scalar function. The Level-Set was first used in icing to evaluate the ice shedding trajectories by [START_REF] Beaugendre | Computation of ice shedding trajectories using cartesian grids, penalization, and level sets[END_REF]. In this work, the level zero represents the evolving ice/air interface. Also, the inside zone could be used to represent the accreted ice and one can evaluate the heat transfer in this substrate. Likewise, the out-side zone represents the external flow zone. By refining close to the wall, a high quality flow can be obtained. The mesh in question can start from a body-fitted mesh or from any mesh since the grid is embedded and distance normal to the wall is not conditioned. Such embedded-grid method integrates perfectly with the Eulerian formulation of PDEs, overcoming meshing issues. Embedded-grids can be performed using structured grids [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF][START_REF] Mittal | Immersed boundary methods[END_REF][START_REF] Hu | A conservative interface method for compressible flows[END_REF]. The no slip boundary condition at the wall can be achieved through a penalty term, in which the solid is treated as an impermeable medium in the so called Brinckman-Navier-Stokes equations.

In previous papers [START_REF] Pena | Development of a three-dimensional icing simulation code in the NSMB flow solver[END_REF][START_REF] Laurendeau | A single step ice accretion model using level-set method[END_REF], we developed and validated an icing simulation code in the NSMB flow solver. The Ice solver consisted of four modules: compressible air solver, Eulerian droplet solver, SWIM solver, ALE grid regeneration module. The compressible Navier-Stokes air solver is now implemented with a penalization term to reproduce the no slip boundary condition at the wall. The Eulerian droplet module is integrated with an IBM-LS formulation where the impingement boundary condition is applied to an explicit representation of the solid wall. Currently, we are limited to rime ice configurations, since the third module is still under development. These different modules are integrated within the Level-Set framework, where the zero level represents the advancement of the iced surface. The Level-Set equations are discretised using a fifth order spatial WENO scheme and a third order time Runge-Kutta scheme to ensure consistency. The LS is redistanced after each ice layer by solving a reinitialization equation. The icing velocity evaluated at the interface is propagated throughout the whole domain by solving an additional PDE. The implementation of turbulence wall laws to model the turbulent boundary layer is still under progress. All the modules were developed for multi-block grids parallelised with the MPI environment. Chimera superimposed grids are supported as well since they are used to ensure fine grids close to the wall.

NUMERICAL METHOD AND GOVERNING EQUATIONS

The ice accretion modules are developed in the NSMB Solver (Navier-Stokes Multi-block solver) [START_REF] Jb Vos | Recent Advances in Aerodynamics inside the NSMB (navier-stokes multi-block) Consortium[END_REF][START_REF] Hoarau | Recent Developments of the Navier Stokes Multi Block (NSMB) cfd solver[END_REF]. The NSMB solver uses structured, multi-block, chimera grids, supports grid motion, contains parallel compressible and incompressible NS solvers and is able to treat embedded grid techniques such as IBM and LS.

PENALIZED COMPRESSIBLE FLOW

In the airflow solver module the solid body is defined implicitly through the LS function. In this implicit representation or LS disposition the solid is given by a characteristic function χ s , which is set to a smoothed Heaviside function H. The smoothed Heaviside function in turn uses the signed distance LS function in our model H(-φ). Setting the LS function as a signed distance enhances the numerical accuracy [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF][START_REF] Mulder | Computing interface motion in compressible gas dynamics[END_REF]. A penalty term is added as a source term to the NS equations to respect the no-slip boundary condition at the wall [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF][START_REF] Abgrall | An immersed boundary method using unstructured anisotropic mesh adaptation combined with levelsets and penalization techniques[END_REF]. This term resembles the continuous Boundary Condition forcing used in IBM. The only difference is that the Heaviside function is constructed from the LS signed distance function. The penalized compressible NS equations take the form seen in Eq. 1

∂ρa ∂t + ∇ ⋅ (ρ a u a ) = 0 ∂ρaua ∂t + ∇ ⋅ (ρ a u a ⊗ u a ) = ∇ ⋅ π + f u ∂ρaea ∂t + ∇ ⋅ ((ρ a e a + p) u a ) = ∇ ⋅ (πu a + q) + f e f u = 1 η χ s (ρ a u a -ρ a u s ) f e = 1 η χ s θ s ρ a ( (T ) -(T s )) + (ρ a u a -ρ a u s ) ⋅ u a . (1)
The variable ρ represents the air density, u the dry air flow velocity, e the specific total energy, p the pressure, q the heat flux, and π the stress tensor. The new additional terms f u and f e represents the penalty terms, where 1 η is a penalization parameter, and χ s is the characteristic function of the solid. As mentioned earlier, the characteristic function is a smoothed Heaviside function computed from the signed distance LS function, both of which are given in Eq. 2,3

H(φ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0 φ < - 0.5 1 + φ + sin(πφ ) π φ ≤ 1 φ > (2)
χ s = H(-φ) (3)
EULERIAN DROPLET TRANSPORT In the droplet transport module developed in NSMB, we used an Eulerian representation [START_REF] Laurendeau | A single step ice accretion model using level-set method[END_REF]. Wherein, a system of conservation equations of the droplet velocity and volume fraction are solved. The droplets are assumed to have a spherical shape, which flattens under high Reynolds numbers. Furthermore, the size is assumed to have a (LANGMUIR "D") distribution. Readers could refer to [START_REF] Laurendeau | A single step ice accretion model using level-set method[END_REF] where all the other assumptions are justified, or to [START_REF] Scott | Navier-stokes solution to the flowfield over ice accretion shapes[END_REF] who proposed the method. The governing conservation equations are given in Eq. 4

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂αw ∂t + ∇.(α w u w ) = 0 ∂αwuw ∂t + ∇. (α w u w ⊗ u w ) = F D + F BG F D = α w C D Re d 24K (u a -u w ) F BG = α w 1 -ρa ρw 1 F r 2 g (4)
The variable α w represents the non-dimensionalized water volume fraction, u w non-dimensionalized water droplet velocity field, u a non-dimensionalized air velocity field, ρ w water density, and g gravity vector. The term K = ρdU ∞ 18Lµ is an inertia parameter, L represents the reference length, and d is the droplet median diameter. Re d , the droplets Reynolds number for a spherical particle relative to the air flow phase is defined based on the slip velocity as in Eq. 5.

Re d = ρd u a -u w µ a (5)
The variable C D is the drag coefficient of the droplets and is given empirically as a function of the droplets shape.

For spherical droplets we use the form suggested by [START_REF] Schiller | A drag coefficient correlation[END_REF].

For super large droplets (SLD), the droplets are assumed to deform into discs. This results in a modification of the drag coefficient. A more elaborate drag coefficient takes the form in equation 6

C d = (C d,sphere + ee (C d,disk -C d,sphere )) × Re d (6)
The variable ee is a weight factor and is given by another empirical equation 7 and is a measure of the deformation of the droplet from a sphere to a disk, C d,sphere the drag coefficient of a sphere, and C d,disk that of a disk.

ee = 1 -(1 + 0.007W e 0.5 ) -6 (7)
W e is the Weber number which measures the relative importance of the droplet's inertia to its surface tension. It is given by equation 8.

W e = ρ d u a -u w 2 D σ (8)
The variable σ is the droplets surface tension approximated at 0.0756 N/m for water.

Because of the assumption that all droplets are captured by the surface and the fact that the model treats the icing in quasi-steady state, special treatment is required for the droplets boundary condition. Droplets are assumed to disappear on the solid body in zones where they impact against the solid body. In other words, the solid acts as a Neumann outlet. However, in dry zones where droplets do not impact against the solid body, the solid should act as a Dirichlet BC. The droplet velocity vector is checked at the interface and the appropriate BC is applied locally accordingly. This method applied in [START_REF] Jung | Numerical modeling for eulerian droplet impingement in supercooled large droplet conditions[END_REF] is given by equation 9

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ if u ⋅ n < 0 W et → N euman ∂u ∂n = 0, ∂α ∂n = 0 if u ⋅ n > 0 Dry → Dirichlet u = 0, α = 0 (9)
In a Level-Set disposition this adaptive boundary condition cannot be applied implicitly, no continuous penalty term can achieve such behavior, at least to our knowledge. An explicit forcing of the boundary condition using a IBM representation of the LS implicit function can easily achieve this adaptive BC. Consequently, we use the first layer of cells adjacent to the interface to directly enforce the required boundary condition.

IBM-LS Technique: Droplet BC Consider Fig. 1. The adaptive BC is imposed on the cells marked with x and are henceforth called immersed boundary IB points. The procedure is summarized in the following points:

(1) First we detect this layer of points outside the interface by detecting the change of sign(φ) with neighbors. An array containing their indexes is saved IBp ind .

(2) Each point is mirrored against the solid surface resulting in an image point (im) further from the interface at a distance ∆ from the IB point along the normal direction. The normal direction is available from the signed-distance LS function n = ∇φ. The distance ∆ is calculated from the cell size to ensure that the image point (im) is close to a fluid cell and outside the IB point.

(3) The coordinates of these image points are calculated via Eq. 10 and are saved in an array → X im .

(4) The distance to the wall at these image points which is the same as the LS function is calculated via Eq. 11 and saved in a third array φ im .

(5) An algorithm searches for the fluid cell closest to the image point. The indexes of the closest point are saved in an array CP ind .

(6) This closest point is used to interpolate the state vector Q to the image point. The gradient of the droplet state vector is calculated at our list of closest points ∇Q cp and is then used to interpolate the state vector Q cp to the image point as given in Eq. 12.

(7) The required BC is imposed by checking the droplet velocity component normal to the wall. Consequently, we impose Q IB = Q ip for the wet zones where u w ⋅n < 0 and Q IB = Q ip * φ ip φ im for the dry zones where u w ⋅ n > 0.

(8) If and only if the Level-Set is moved by advection, resulting in a new interface, the arrays IBp ind , → X im , φ im , and CP ind are re-evaluated.

→ X im = → X W p + ∇φ W p * ∆ ( 10
)
φ im = φ W p + ∆ (11)
Q ip = Q cp -∇Q cp ⋅ (∆ ⋅ ∇φ) (12)
This method was also developed using two image points to better apply the Neumann BC on the wall. Another variation of the method was also developed, where we reconstruct the flux at the wall instead of at the cell center as proposed by [START_REF] Capizzano | A eulerian method for water droplet impingement by means of an immersed boundary technique[END_REF]. However, we do not follow the method depicted in [START_REF] Capizzano | A eulerian method for water droplet impingement by means of an immersed boundary technique[END_REF], where he extrapolates from the image point to the wall without enforcing a zero gradient. We found that the first variation presented here was more stable than the two others. Collection Efficiency The collection efficiency is directly calculated using the formula in Eq. 13.

β = αu.n (13)
In the IBM-LS configuration this variable can be calculated at either the first layer of cells outside which are accessible through the array W P ind , or at the cells whose Dirac Delta δ(φ) is higher than zero and whose LS function φ is positive.

THERMODYNAMIC MODELLING OF THE LIQUID FILM

The thermodynamic model developed in NSMB is based on the work of [START_REF] Bourgault | Development of a shallow-water icing model in fensapice[END_REF], [START_REF] Beaugendre | FENSAP-ICE's three-dimensional in-flight ice accretion module: ICE3D[END_REF] called the Shallow-Water for Ice Modelling SWIM. The SWIM module is based on conservation equations of mass and energy. The interested reader can refer to [START_REF] Laurendeau | A single step ice accretion model using level-set method[END_REF] where the implementation of the model is fully explained. In an embedded grid framework, we are currently still developing the appropriate variation of the model. Consequently, only dry rime ice configurations are being studied so far, where the impinging mass is assumed to solidify on impact. Thus, resultant mass rate of ice accretion ṁice is evaluated through Eq. 14.

ṁice = ṁimp = LW C ⋅ V ∞ ⋅ β ( 14
)
The variable LW C represents the liquid water content and V ∞ the free stream or far fields velocity.

LEVEL-SET FUNCTION The Level-Set approach was developed in [START_REF] Laurendeau | A single step ice accretion model using level-set method[END_REF] to track the ice air interface evolution. The ice/air interface is represented implicitly by the zero iso-contour of the Level-Set function φ. The LS function φ is set as a signed distance function [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF]. A signed distance function is given by: φ( → x ) = d( → x ). It is initialized in the computational domain as follows:

• φ = d, in the outside zone (air)

• φ = -d, in the inside zone (ice,solid)

• Γ = x φ(x, t) = 0, the interface
The variable d is the distance to the interface Γ.

The LS approach replaces the need for an initial solid with body fitted mesh. However, one can still start from an initial solid with a body fitted mesh; in such case, the negative part of the LS function, the inside zone, can be used to define the ice substrate where it is possible to solve a heat transfer equation. The accumulation of ice resulting in deformation of the solid/air interface is achieved by advecting the LS function. Consequently, The Level-Set advection equation given in Eq. 15.

∂φ ∂t + → v ice ∇.φ = 0 (15)
The variable → v ice represents the LS velocity field. This velocity is equal to the ice accretion velocity on the interface. The velocity at the interface is calculated in Eq. 16.

→ v wall = ṁice ρ ice .∇φ (16)
This velocity calculated at the interface is then propagated in the normal direction through the whole domain. An important assumption made here is that ice forms normal to the solid. To propagate the icing velocity in the normal direction an additional PDE system is solved as given in Eq. 17

∂w ∂t + sign(φ)( → ∇φ) ⋅ ∇w = 0 (17)
The variable w represents the propagation of v wall normal to the wall. The steady state solution of Eq. 17 gives the required Level-Set velocity w steady = → v ice . The resulting velocity field is fed to Eq. 15 to advance the geometry by advecting the LS function φ.

When the LS function is advected (after each ice layer), the signed distance feature of the LS function φ is no more guaranteed. While advecting the LS function (during a single ice layer but at iteration steps of Eq. 15), the signed distance feature can get deteriorated based on the grid and the discretization methods used. Reinitializing φ while being advected can restore the values around the interface to signed distance. Advecting a signed distance function is numerically stable and ameliorates conservation. Reinitializing φ at the end of each ice layer is essentially required to calculate the new solid characteristic function χ s from the smoothed Heaviside function H(φ). To recover Φ( → x ) = d( → x ) we solve the reinitialization equation Eq. 18 proposed by [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF]. This equation can be rewritten in the form in Eq. 19 to allow a conservative finite volume discretization.

∂φ ∂t + ∇φ = 1 (18)
∂φ ∂t + sign(φ) ∇φ ∇φ ⋅ ∇φ = sign(φ) (19)
Where the sign(φ) is approximated numerically by the smooth function in Eq.20.

sign(φ) = φ φ 2 + ∇φ 2 ∆x 2 (20)
It was shown that Eq. 19 does not respect local mass conservation at the interface. In other words, while redistancing φ in the domain the cells in which the zero interface passes do not conserve the sides of the interface. The distance being calculated perfectly all around the interface, it does not guarantee that the interface would not move locally in the cells through which it passes. In ice accretion simulation, the ice mass or icing rate is the most critical variable. An alteration of the local amount of ice recurring each time φ is re-distanced would make the whole approach lose its appeal. Many methods exist in the literature to preserve the interface. We chose to use the method advised by [START_REF] Solomenko | Mass conservation and reduction of parasitic interfacial waves in level-set methods for the numerical simulation of two-phase flows: A comparative study[END_REF], initially proposed by [START_REF] Sussman | An efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow[END_REF], and given in Eq. 21

∂φ ∂t + sign(φ 0 ) ∇φ ∇φ ⋅ ∇φ = sign(φ) + λδ(φ 0 ) ∇φ 0 (21)
The variable λ represents the Correction Factor that enforces local mass conservation and is given in Eq. 22, and δ(φ 0 ) is the smoothed Dirac Delta function given in Eq. 23

λ = - ∫ Ω i,j,k δ(φ 0 )(φ n+1 -φ 0 ) ∆t ∫ Ω i,j,k δ 2 (φ 0 ) ∇φ 0 (22)
δ(φ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 φ < - 0.5 (1 + sin(πφ )) φ ≤ 1 φ > (23)
Both integrals in Eq.22 are an integration over the cell Ω i,j,k and are evaluated from a 9-point stencil in 2d Eq. 24 or from a 27-point stencil in 3d as given in Eq. 25

Ωi,j f = ∆x 2 24 ⎛ ⎝ 16f i,j + 1 m,n=-1;(m,n)≠(0,0) f i+m,j+n ⎞ ⎠ (24)
Ω i,j,k f = ∆x 3 78 ⎛ ⎝ 52f i,j,k + 1 m,n,q=-1;(m,n,q)≠(0,0,0) f i+m,j+n,k+q ⎞ ⎠ (25)
Eq. 19 is first solved as a prediction step then Eq. 21 corrects φ. The additional source term impacts the φ only at the interface where δ(φ) > 0. So one can avoid the computational cost of the correction step by calculating the correction term only at points where δ(φ) > 0.

The LS equations [START_REF] Bernard | Equilibrium temperature of an unheated icing surface as a function of air speed[END_REF][START_REF] Montreuil | ECLIPPS: 1. threedimensional CFD prediction of the ice accretion[END_REF][START_REF] Mulder | Computing interface motion in compressible gas dynamics[END_REF] are discretized using a WENO5 spatial scheme and a RK3 temporal scheme. To limit communication time in the MPI environment, only two ghost cells are communicated between neighbouring blocks, and only two ghost cells are interpolated and communicated for chimera blocks. Thus, the WENO5 scheme falls to a TVD scheme with a flux limiter on the block connectivity. This is because the WENO5 stencil needs three neighbouring cells, whereas the TVD scheme used uses only two neighbouring cells.

The final algorithm is shown in Fig. 2.

CURRENT RESULTS

In this section, icing simulations are performed on twodimensional NACA0012 airfoil. The flow field is solved explicitly. The flow is restricted to laminar, since wall laws for the LS embedded grids are under development. We only present rime ice configurations. Three Cartesian meshes have been generated. The coarse mesh contains 318×288 cells in a single block and the size of cells close to the wall is 0.008, the second mesh 1300 × 1000 with cell size 0.002 and in 64 blocks, and the third mesh a chimera mesh 16800 × 16800 with cell size 0.0006 and in 128 blocks.

LEVEL-SET EQUATIONS The signed distance function is first initialized to -1 inside and +1 outside. The distance is then manually initialized in a small band around the interface. Without this manual initialization, the interface is highly deformed. The manual initialization compares the distance between a cell and all the points used to generate the NACA0012 profile. Using this well initialized φ we solve the reinitialization equation 21. If we use the equation 19 without the correction term the interface gets deteriorated as seen in figure 3. The optimized result obtained with equation 21 is seen in figure 4 Reinitialization of the LS function φ is carried out after each icing layer. The time step is a function of the cell sizes since the temporal scheme is a RK3 explicit scheme. Since the advecting velocity is the gradient of φ which has a magnitude of

Figure 1 :

 1 Figure 1: Schematics of IBM-LS framework. ∎: flow domain, x: IB cells layer, ○: solid domain.

Figure 2 :

 2 Figure 2: IBM-LS multi-layer ice accretion algorithm diagram.

Figure 3 :

 3 Figure 3: Re-distancing the well initialized φ without local conservation.

Figure 4 :

 4 Figure 4: Re-distancing the well initialized φ with conservation

Figure 5 :

 5 Figure 5: The x component of v ice propagated normal to the wall for the coarsest grid.

Figure 6 :

 6 Figure 6: The z component of v ice propagated normal to the wall for the second multi-block grid and the resulting φ.

Figure 7 :

 7 Figure 7: Deformed Geometry (advected φ) obtained on the coarse mesh using 1,2,5 and 10 ice shots.