Jean-Matthieu Etancelin

Marlène Murris-Espin

Philippe Poncet

PARTICLE METHODS FOR THE DYNAMICS OF POROUS BIOFILMS WITH HETEROGENEOUS RHEOLOGY AND ITS INTERACTION WITH LUNG EPITHELIUM

CONTEXT: UPSCALING FLOW AROUND LUNG CELLS

Multiscale description of human lung from trachea (∼ 2cm) to its epithelium at the 6-7th generations (∼ 10µm high), one cilia being 100nm wide (sketch inspired from [START_REF] Chatelin | A Hybrid Grid-Particle Method for Moving Bodies in 3D Stokes Flow with Variable Viscosity[END_REF]).

FROM DNS OF THE PORE SCALE REACTIVE FLOWS

The superficial velocity u follows the Darcy-Brinkman-Stokes equation in a porosity field ε, which defines a pure liquid phase where ε = 1 and a solid matrix where ε is small :

ε -1 ∂(ρu) ∂t + ε -1 div ε -1 ρu ⊗ u -ε -1 div (2µD(u)) + µ * K -1 u = f -∇p (1)
along with div u = 0, using the usual notations. The permeability K of the lower scale is driven by the Kozeny-Carman relation

K(ε) = K 0 ε 3 (1 -ε) -2 .
Given the scales, it reduces to

-div (2µD(u)) + µ * K -1 0 (1 -ε) 2 ε 2 u = ε(f -∇p) (2)
An acid of concentration C dissolves a solid of molar volume m at rate r following

∂C ∂t + div (ε -1 u C) -div (d m ε 1+β ∇ε -1 C) = -r(C, ε) and ∂ε ∂t = -m r(C, ε) (3)
where d m is molecular diffusion of the acid and β β β is the index of tortuosity within the solid.

Using a particle method for (3) means that we describe the fluid by a set of volumes v p , concentrations c p , and locations ξ p . Then, one gets the particle formulation of (3) :

dC p dt = div (d m ε 1+β ∇ε -1 C) -Cu • ∇ε -1 ξ p , dξ p dt = ε -1 u(ξ p ), dv p dt = div (ε -1 u) ξ p v p (4)
with back and forth high order interpolations from the grid used to solve (2).

t=10.5min t=12.3h t=42.6h

Dissolution of a 5% porous calcite body (1.43mm wide) at pH=2, following model (2, 3). See [START_REF] Etancelin | Improvement of remeshed Lagrangian methods for the simulation of dissolution processes at pore-scale[END_REF] for more details and the validation of the numerical method (4) in [START_REF] Molins | Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set[END_REF]. The resulting porosity/permeability is displayed to the right.

TO THE DNS OF LUNG CELLS WITH PCL/MUCUS BIOFILM

The lung epithelium (the ciliated cells B(t)) is surrounded by a Newtonian pericelial liquid (PCL) of viscosity µ ∞ and a miscible shear-thinning mucus composed of mucin proteins at concentration C.

Model (2, 3) is still valid but in this case, the fluid is not reacting (r = 0) and K(ε) = 1 B(t) /ε. Here (2) and (3) are coupled by means of the Carreau relationship defining the viscosity of the heterogeneous non-Newtonian fluid (see [START_REF] Chatelin | Numerical and experimental investigation of mucociliary clearance breakdown in cystic fibrosis[END_REF] for instance) :

µ(α, D) = µ ∞ + µ ∞ B µ ∞ α d 1-N 0 -1 1 + 2 D : D d 2 0 α(N -1)/2 ( 5
)
where D is the shear-rate and α = C/C 0 the renormalized concentration of mucin (d 0 is the cut-off frequency and C 0 the maximum admissible concentration of mucins).

In this model, only two "patient specific" parameters remain: the bulk viscosity B and the shear-thinning index N (see the clinical context thereafter).

Cilia B(t) and fluid domain Ω(t) in motion (picture to the left, from [START_REF] Chatelin | Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport[END_REF]), driving the mucus (isosurfaces of mucin concentration on the picture to the right, see [START_REF] Sanchez | Analysis of 3D non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance[END_REF] for more details).

FROM THE UPSCALING OF THE PORE SCALE DIFFUSION

Diffusion in this sandstone, which is 21% porous, leads to a tortuosity index of β = 0.68 β = 0.68 β = 0.68. This means that the upscaled diffusion coefficient in an homogeneous medium is ε β times the molecular diffusion (taken from [START_REF] Hume | A velocity-vorticity method for highly viscous 3D flows with application to digital rock physics[END_REF]).

TO THE UPSCALING OF THE DIFFUSION WITHIN LUNG CELL

Diffusion within ciliated cell with time-dependent geometry is upscaled and computed using the same technique as above. It is required in order to estimate the mucin slip-through dynamics. The resulting tortuosity index β(t) is shown to be relatively time independent.

Cilia carpet with time-dependent geometry (left picture, from [START_REF] Chatelin | Hybrid grid-particle methods and Penalization: A Sherman-Morrison-Woodbury approach to compute 3D viscous flows using FFT[END_REF]) and resulting time-dependant effective diffusion and porosity (right picture). This leads to a tortuosity index at β ≃ 0.4 β ≃ 0.4 β ≃ 0.4 when not in flat shape configuration.

CLINICAL CONTEXT: MONITORING OF THERAPIES FOR CYSTIC-FIBROSIS

Rheology coefficients B and N have been measured from expectoration samples of cystic fibrosis patients. Experiments were carried out on a Mars III rheometer equipped with a cone and plate geometry (angle 1 o , diameter 35mm) which required a small volume of fluid (0.2ml).

It has been shown in [START_REF] Chatelin | Numerical and experimental investigation of mucociliary clearance breakdown in cystic fibrosis[END_REF] that numerical simulation of mucus dynamics are coherent with clinical data: slow mucus motion coincides with CF patients, while the highest mucus motions correspond to healthy people.

  (B, N ) phase diagram with clinical data from [5] (figure to the left). Examples of robust estimation of coefficients B and N by rheometry, using aliquoting of mucus samples (figure to the right) .


Keywords: ARF = acute respiratory failure, ICU = Intensive care unit, SARS-CoV-2 = severe acute respiratory syndrome coronavirus 2, VV-ECMO = veno-venous extracorporeal membrane oxygenation air evacuation, COVID-19, Mayotte, Reunion Island, SARS-CoV-2, transportation

Conclusion: Emergency air evacuation of patients with ARF due to SARS-CoV-2 was associated with severe hypoxemia but remained feasible. In cases of ARF due to SARS-CoV-2 requiring emergency air evacuation, sedated patients receiving invasive mechanical ventilation and curare should be prioritized over nonintubated patients. It is noteworthy that patients with SARS-CoV-2 pneumonia related to the 501Y.V2 variant were very severe despite their young age.

Introduction

Medical evacuation is the extraction by an air (helicopter, plane), land (ambulance) or naval unit of a person suffering from a health problem.

Medical aviation represents a stage in the history of medicine and more particularly in aeronautical medicine.

During wars, the speed with which the patients are evacuated in order to apply early treatment to them has always been considered a determining factor in their survival. This is how medical aviation was born and developed during the wars of the 20th century, benefiting both from improved aircraft performance and from progress in hospital medicine.

The knowledge acquired on the battlefields was then put at the service of the entire civilian and military population. It has been of great help in managing epidemic peaks due to the spread of coronavirus-2 internationally, as well as that of the Indian Ocean where our study was made.

Mayotte (270,000 inhabitants) and Reunion Island (845,000 inhabitants) are 2 French overseas departments located in the Indian Ocean at a distance of 10,000km from Paris. Until January 2021, Mayotte was relatively spared by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic. On January 6, 2021, the incidence of deaths caused by SARS-CoV-2 was 20.7 per 100,000 inhabitants on the island (compared to 116.4 per 100,000 inhabitants in metropolitan France). [START_REF]COVID-19: point épidémiologique à Mayotte[END_REF][START_REF]Coronavirus (COVID-19) -Santé publique France[END_REF] However, the incidence of infection with SARS-CoV-2 rose rapidly from 268 per 100,000 inhabitants in week 3 to >800 per 100,000 inhabitants in weeks 5 and 6, an increase that has been attributed to the SARS-CoV-2 501Y.V2 variant. [START_REF] Tegally | Detection of a SARS-CoV-2 variant of concern in South Africa[END_REF] In February 2021, the only hospital of the island, Mayotte Hospital, [START_REF]COVID-19: point épidémiologique à Mayotte[END_REF] was overwhelmed by the explosion in the number of cases, as it is equipped with only 16

intensive care unit (ICU) beds for a population of 270,000 inhabitants. All evaluated patients had positive confirmatory respiratory sample with SARS-CoV-2 by real time reverse transcription polymerase chain reaction using kit targeting IP2 and IP4 regions and N gene. Samples from nasopharyngeal swabs or patient's respiratory specimens were extracted using NucliSens easyMAG system (BioMérieux). The genomes were secondly obtained by sequencing using the Oxford Nanopore technology and the Artic Network's overlapping amplicon protocol. [START_REF]Protocol: Real-Time RT-PCR Assays for the Detection of SARS-CoV-2[END_REF][START_REF] Corman | Detection of 2019 novel coronavirus (2019-nCoV) by real-time RT-PCR[END_REF]

Therapeutic management

In accordance with our protocol, all patients with ARF due to SARS-CoV-2 were treated with: a 3 rd generation cephalosporin or piperacillin-tazobactam after realization of respiratory sample; dexamethasone at a dosage of 6mg/day for 10days; systematic deworming with ivermectine or albendazole; en-hanced systematic anticoagulation according to the guidelines of the French Society of Hemostasis and the French Society of Anesthesia and Intensive Care. [START_REF] Susen | Traitement anticoagulant pour la prévention du risque thrombotique chez un patient hospitalisé avec COVID-19 et surveillance de l'hémostase. Proposition du groupe GIHP et du GFHT 3 avril[END_REF] High flow nasal cannula oxygenation was initiated in patients who need standard oxygen ≥9L/min to maintain peripheral arterial oxygenation saturation ≥92%. The decision of the timing of intubation and mechanical ventilation was not protocolized but determined by the ICU team.

Details of air evacuation

The protocol of air evacuation had been established by the prehospital emergency medical system of La Reunion and Mayotte and there was no specific training for these transports. In order to be transported, patients had to meet the following criteria: FiO 2 <0.6, PEEP <15, and norepinephrine <1 mg/h. The use of curares in patients receiving invasive mechanical ventilation was not protocolized and was left to the discretion of the medical team. As there was no patient isolation unit and additional decontamination against SARS-CoV-2 was required between air transport.

Data collection

Clinical and biological data along with information on comorbidities were collected on admission to ICU.

Complications occurring during air transport and in ICU were recorded.

Survival at hospital discharge was also recorded

Outcomes

The primary outcome was the occurrence of severe hypoxemia during air transport (ie, oxygen saturation (spO 2 ) <88% during >15 minutes).

The secondary outcomes were the need for veno-venous extracorporeal membrane oxygenation (VV-ECMO), duration of invasive mechanical ventilation, in-ICU length of stay and hospital mortality.

Other complications during air transport were evaluated: oxygen desaturation requiring increase of oxygen requirements or introduction of curares, hypotension (mean arterial pressure <65 mmHg) requiring positive inotropic therapy or fluid challenge >500 mL of crystalloid, endotracheal tube or vascular catheter dislodgement, arrhythmia, cardiac arrest, agitation requiring modification of sedation.

Statistical analysis

Categorical variables were expressed as total number (percent-age) and continuous variables as median (25 th -75 th percentiles). Categorical variables were compared using Fisher exact test. Continuous variables were compared using the nonparametric Wilcoxon test or Mann-Whitney U test, as appropriate. A P value <.05 was considered significant. Statistical analyses were conducted using SPSS 15.0 (SPSS Inc, Chicago, IL). For all patients the indication for emergency air evacuation was to free up ICU beds in the hospital of Mayotte.

Results

Over

The characteristics of patients are shown in Tables 1 and2. The median age of patients was 55 (46-65) years. Of the 38 evacuated patients, 44.7% had arterial hypertension, 36.8% were diabetic, and 34.2% had a body mass index >30kg/m 2 .

Fifteen patients (39.5%) were screened for the SARS-CoV-2 501Y.V2 variant, all of whom tested positive.

The median time from onset of symptoms to air evacuation was 12 (9-14) days. For patients receiving invasive mechanical ventilation, the median time from orotracheal intubation to air evacuation was 4 (2-7) days. Eight patients (21.1%) were treated with noradrenaline during air transport.

A total of 13 patients (34.2%) developed an episode of severe hypoxemia during air transport.

Delay between air transfer or delay between orotracheal intubation in mechanically ventilated patients and onset of symptoms were not associated with severe hypoxemia during air transport (P>.9) (Table 1). In univariate analysis, the only factors predicting the occurrence of severe hypoxemia during air transport was lack of treatment with curare (P=.012), lack of invasive mechanical ventilation (P=.003)

(Table 1).

The median paO2/FiO2 ratio was significantly lower on admission to our ICU (140 [102-192] mmHg) than on departure from Mayotte Hospital (165 [150-200] mmHg, P = .022) (Table 2).

Three patients developed hypotension requiring positive inotropic therapy (mean arterial pressure <65 mmHg) and fluid challenge >500mL of crystalloid during air transport.

Agitation requiring modification of sedation occurred in 2 patients (5.3%) and oxygen desaturation requiring increase of fraction of inspired oxygen, sedation enhancement and introduc-tion of neuromuscular blocking agents occurred in 8 patients (21%).

No cardiac arrest, arrhythmia, endotracheal tube or vascular catheter dislodgement occurred during air transport.

The median plasma lactate concentration on admission to ICU was 1.3 (1.1-1.6) mmol/L (Table 2).

Evolution in ICU

One patient had a fatal cardiac arrest due to pulmonary embolism in our ICU within 24 hours of departure from Mayotte Hospital.

One patient who presented with acute coronary syndrome in Mayotte underwent coronary angiography in our ICU. The intervention revealed severe 3-vessel disease requiring revascularization.

Of the 35 patients who were not on VV-ECMO support on departure from Mayotte, 6 (17.1%) required VV-ECMO support for refractory acute respiratory distress syndrome in our ICU. Overall, 9 (23.7%) of 38 patients received VV-ECMO support for refractory acute respiratory distress syndrome (Table 3).

Pulmonary embolism and hospital-acquired pneumonia occurred in 8 patients (21.1%) and 19 patients (50%), respectively. The median length of stay in ICU was 21 (13-46) days, with no significant difference between patients with and without an episode of severe hypoxemia during air transport (P=.584) (Table 3).

After a median follow-up of 56 (49-66) days, 7 deaths (18.4%) occurred, with no significant difference between patients with and without an episode of severe hypoxemia during air transport (P=.385) (Table 3).

Discussion

In our study, 34.2% of patients with ARF due to SARS-CoV-2 developed an episode of severe hypoxemia during air transport.

The respiratory status of patients deteriorated considerably between departure and arrival, as attested by the significant decrease in the median paO2/FiO2 ratio.

The only factors significantly associated with severe hypoxemia during air transport were lack of treatment with curare and lack of invasive mechanical ventilation.

At the beginning of the SARS-CoV-2 pandemic, international guidelines recommended the early intubation of infected patients with respiratory distress [START_REF] Poston | Management of critically ill adults with COVID-19[END_REF] because noninvasive ventilation was believed to cause secondary contamination of caregivers via aerosolization. [START_REF] Tran | Aerosol generating procedures and risk of transmission of acute respiratory infections to healthcare workers: a systematic review[END_REF] This risk has since been shown to be limited, and noninvasive ventilation is now recommended over invasive ventilation. [START_REF] Li | High-flow nasal cannula for COVID-19 patients: low risk of bio-aerosol dispersion[END_REF][START_REF] Wang | The experience of high-flow nasal cannula in hospitalized patients with 2019 novel coronavirus-infected pneumonia in two hospitals of Chongqing, China[END_REF] However, in our study, almost all patients were intubated in Mayotte before being evacuated, as noninvasive mechanical ventilation proved ineffective due the severity of their condition. Note that this procedure was in accordance with international guidelines for air evacuation, which recommend stabilizing patients at risk of respiratory deterioration through preventive orotracheal intubation, whether or not they are infected with SARS-CoV-2. [START_REF] Teichman | International aeromedical evacuation[END_REF] Studies have shown that the transfer of critically ill patients receiving mechanical ventilation can increase ventilation duration and patient mortality. [START_REF] Barratt | Effect of non-clinical interhospital critical care unit to unit transfer of critically ill patients: a propensitymatched cohort analysis[END_REF][START_REF] Parmentier-Decrucq | Adverse events during intrahospital transport of critically ill patients: incidence and risk factors[END_REF] However, the long-distance transfer of patients with ARF appears to be safe when performed by specialized and dedicated medical teams. [START_REF] Uusaro | Safe long-distance interhospital ground transfer of critically ill patients with acute severe unstable respiratory and circulatory failure[END_REF] The Parenmark and Walther study showed a large proportion of ICU-to-ICU transfers and an increased odds of dying for those transferred due to other reasons than repatriation. [START_REF] Parenmark | [END_REF] Another example is the Larosa study . [START_REF][END_REF] who attest the Chilean Air Force experience and show that air transport of mechanically ventilated patients with COVID-19 infection has been shown to be a safe way of transport, with no in-flight deaths and an in-hospital mortality of 12%, which compares favorably with the inhospital mortality of similar patients who did not undergo air transport.

In the study by Pavain et al, the inter-hospital transfer of patients with SARS-CoV-2 pneumonia receiving mechanical ventilation was not associated with an increase in mortality. [START_REF] Painvin | Inter-hospital transport of critically ill patients to manage the intensive care unit surge during the COVID-19 pandemic in France[END_REF] In our study, no excess mortality was found in patients who developed an episode of severe hypoxemia during air transport, even though a significant deterioration of patients' respiratory status was observed on arrival. In our practice, the use of curares was not protocolized and was left to the discretion of the medical team. Nevertheless, lack of treatment with curare was associated with severe hypoxemia during air transport. From a medical perspective, several patients in our study received a personal benefit from air evacuation. Thus, 9 patients with refractory acute respiratory distress syndrome (23.7%) were able to receive VV-ECMO support in Reunion Island, where 16 machines are presently available compared to 2 machines in Mayotte. Moreover, 1 patient who presented with acute coronary syndrome in Mayotte was able to undergo coronary angiography in Reunion Island; the intervention revealed three-vessel disease, prompting treatment with coronary revascularization. From an ethical perspective; however, the benefits of emergency air evacuation existed only at the collective level (eg, reduction in the need for triage). [START_REF] Painvin | Inter-hospital transport of critically ill patients to manage the intensive care unit surge during the COVID-19 pandemic in France[END_REF][START_REF] Turc | Collective aeromedical transport of COVID-19 critically ill patients in Europe: a retrospective study. Collective aeromedical transport of COVID-19 critically ill patients in Europe[END_REF] The lack of ethical benefit at the personal level explains why many patients and/or their families refused the evacuation.

[24] Lastly, it should be stressed that patients in our case series were critically ill, with almost a quarter of them requiring VV-ECMO support despite their young age. This could due to the increased virulence of variant of concern, which has been highlighted in several studies. [START_REF] Challen | Risk of mortality in patients infected with SARS-CoV-2 variant of concern 202012/1: matched cohort study[END_REF][START_REF] Pearson | Estimates of Severity and Transmissibility of Novel SARS-CoV-2 Variant 501Y.V2 in South Africa[END_REF] The other issues to be highlighted are the numerous transfers to manage to reach the ICU at the hospital of Reunion Island (ICU to ambulance, ambulance to airplane .

. . ) and the discomfort of the protective equipment in a tropical zone where temperature and hygrometry are very high at that season which corresponded to austral summer.

Our study has many limitations. Biases may have been introduced due to the retrospective nature of the study. In addition, the number of evaluated patients was relatively small. Yet, in their study of patients with SARS-CoV-2 pneumonia evacuated by air, Pavain et al [START_REF] Painvin | Inter-hospital transport of critically ill patients to manage the intensive care unit surge during the COVID-19 pandemic in France[END_REF] examined only 13 patients, none of whom were on VV-ECMO support; moreover, travel time was shorter than in our study. Similarly, the study by Turc et al [START_REF] Turc | Collective aeromedical transport of COVID-19 critically ill patients in Europe: a retrospective study. Collective aeromedical transport of COVID-19 critically ill patients in Europe[END_REF] evaluated only 36 patients (with mechanical ventilation but without VV-ECMO support) who traveled for a much shorter period of time (70minutes) than in our study.

In conclusion, emergency air evacuation of patients with ARF due to SARS-CoV-2 501Y.V2 variant was associated with respiratory complications but remained feasible.

In

Total flight time was 2

 2 hours and total travel time between the 2 hospitals was 6hours. Patients were evacuated on board an Embraer 135 aircraft specially equipped for medical evacuation missions (12 seats and 3 stretchers). The vast majority of flights carried 3 patients, 2 of whom received invasive mechanical ventilation and 1 received conventional oxygen therapy. The medical crew, was composed of 3 doctors and 3 nurses, experienced in inter-hospital transport. Both the cockpit and a delimited area at the bottom of the aircraft were considered "clean"; the rest of the aircraft was deemed "contaminated." The medical team was fully dressed in protective clothing for the duration of the flight including examination gloves, filtering facepiece class 2 mask (FFP2) and goggles for eye protection. Each patient was assigned 1 doctor, 1 nurse, and 1 emergency kit composed of a vacuum mattress, an aspirator, a defibrillator, a transport ventilator, and suitcases with ventilation material and emergency medication (Fig.1).

  the study period, 43 patients with SARS-CoV-2 pneumonia were evacuated by air from Mayotte Hospital in Mamoudzou to Félix Guyon University Hospital in Reunion Island. Of these, 38 patients (84.4%) presenting with ARF were transferred directly to our ICU and included in the study cohort. During air transport, 4 patients (10.5%) received conventional oxygen therapy (between 6 L/min and 9 L/min) before departure and 34 patients (89.5%) received invasive mechanical ventilation; 3 (8.8%) of the ventilated patients also received VV-ECMO support.

  cases of ARF due to SARS-CoV-2 requiring emergency air evacuation, sedated patients receiving invasive mechanical ventilation and curare should be prioritized over non-intubated patients. It is notewor-thy that patients with SARS-CoV-2 pneumonia related to the 501Y.V2 variant were very severe despite their young age. Évacuations sanitaires d'urgence de patients atteints d'une défaillance respiratoire due au SARS-CoV-2 depuis Mayotte vers l'île de la Réunion Résumé Introduction : En février 2021, une explosion des cas de syndrome de détresse respiratoire aiguë due au coronavirus 2 (SARS-CoV-2) a submergé le seul hôpital à Mayotte. Nous avons rapporté une série de cas de patients atteints d'insuffisance respiratoire aiguë (IRA) due au SRAS-CoV-2 qui ont été évacués par voie aérienne de Mayotte vers l'île de La Réunion. Méthode : Cette étude observationnelle rétrospective a évalué tous les patients consécutifs atteints d'IRA due au SARS-CoV-2 qui ont été évacués par voie aérienne du CH de Mayotte vers l'unité de réanimation du CHU Félix Guyon à La Réunion entre le 2 février et le 5 mars 2021. Résultats : Au total, 43 patients atteints de pneumonie à SRAS-CoV-2 ont été évacués par voie aérienne, pour un temps de vol total de 2 heures et un temps de trajet total de 6 heures. Parmi ceuxci, 38 patients (88,4 %) avec un âge médian de 55 (46-65) ans ont présenté une IRA et ont été hospitalisés dans notre unité de soins intensifs. Quinze patients ont été dépistés pour le variant SARS-CoV-2 501Y.V2, tous testés positifs. Treize patients (34,2 %) ont développé un épisode d'hypoxémie sévère pendant le transport aérien, et le rapport médian paO2/FiO2 était plus faible à l'admission en réanimation (140 [102-192] mmHg) qu'au départ (165 [150-200], P= .022). Les facteurs associés à une hypoxémie sévère pendant le transport aérien étaient l'absence de traitement par curare (P = 0,012) et l'absence de ventilation mécanique invasive (P = 0,003). Neuf patients (23,7 %) ont reçu une assistance à l'oxygénation par membrane extracorporelle veino-veineuse dans notre unité de soins intensifs. Sept décès (18,4 %) sont survenus à l'hôpital. Conclusion : L'évacuation aérienne d'urgence des patients atteints d'IRA due au SRAS-CoV-2 était associée à une hypoxémie sévère mais restait faisable. En cas d'IRA due au SRAS-CoV-2 nécessitant une évacuation aérienne d'urgence, les patients sous sédation sous ventilation mécanique invasive et curare doivent être prioritaires sur les patients non intubés. Il est à noter que les patients atteints de pneumonie SARS-CoV-2 liée au variant 501Y.V2 étaient très sévères malgré leur jeune âge. Discipline : Médecine d'urgence, réanimation médicale Mots-Clés : évacuation aérienne, COVID-19, Mayotte, La Réunion, SARS-CoV-2, transport

TABLE DES MATIERES :

 DES

	SOMMAIRE DES TABLEAUX ET DES FIGURES :
	1. Introduction Figure 1: Air evacuation of patients	p.4 p.8
	2. Methods	p.5
	2.1 Study Design	p.5
	2.2 Therapeutic management	p.6
	2.3 Details of air evacuation	p.6
	2.4 Data collection	p.7
	2.5 Outcomes	p.8
	2.6 Statistical analysis	p.9
	3. Results	p.9
	3.1 Evolution in ICU	p.13
	4. Discussion	p.14

Table 1 :

 1 Characteristics at hospital departure of the 38 patients with acute respiratory failure due to SARS-CoV-2 p.11

Table 2 :

 2 Characteristics at hospital admission (Reunion Island) of the 38 patients with acute respiratory failure due to SARS-CoV-2 p.12

Table 3 :

 3 Evolution during the stay in intensive care unit p.[START_REF] Von Elm | STROBE InitiativeStrengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement: guidelines for reporting observational studies[END_REF]

2. Methods 2.1. Study design

	located in the Indian Ocean. As a result, patients needing heart, lung or liver
	transplantation must be transferred to Paris, over a distance of 10,000 km. It is to be
	noted that medical teams in both Reunion Island and Mayotte have experience with
	emergency air evacuations, including for patients receiving extracorporeal membrane
	oxygenation support. [5,6] It was necessary to urgently transfer patients with acute
	respiratory failure (ARF) due to SARS-CoV-2 from Mayotte to the only ICU in Indian
	Ocean equipped with many VV-ECMO supports. To date, few studies have examined
	the outcome of patients with SARS-CoV-2 pneumonia evacuated by air, [7-9] and
	only a handful of them have focused on intensive care patients. [10,11] Here, we
	report a case series of patients with ARF due to SARS-CoV-2 who were evacuated
	by air from Mayotte to Reunion Island.
	This retrospective observational study evaluated all consecutive patients with ARF
	due to SARS-CoV-2 who were evacuated by air from Mayotte Hospital in
	Mamoudzou to the ICU of Félix Guyon University Hospital in Reunion Island between
	February 2 and March 5, 2021. All patients or their legally authorized representative
	were verbally informed about the study and could refuse to participate; moreover,
	they received a written information notice about the process of data collection. The
	study was approved by the Ethics Committee of the French Society of Infectious
	Disease and Tropical Medicine (CER-MIT), and was declared to the Commission
	nationale de l'informatique et des libertés (French Data Protection Agency MR004, #
	2206739). This study complies with the Strengthening the Reporting of Observational
	studies in Epidemiology recommendations statement. [12]

The medical infrastructure in Reunion Island meets European standards (P3 laboratories, 16 extracorporeal membrane oxygenation machines, coronary angiography, surgeons trained to perform all types of surgeries, among others). [4] When Mayotte Hospital became overwhelmed with cases of SARS-CoV-2 pneumonia, several patients were transferred to Félix Guyon University Hospital in Reunion Island, in a context of solidarity among islands of the Indian Ocean region. All patients had to be evacuated by air due to the emergency situation. There is no transplantation center (except kidney) on the French overseas territories of Reunion Island and Mayotte which are

Emergency air evacuation of patients with acute respiratory failure due to SARS-CoV-2 from

Mayotte to Reunion Island Abstract Background: In February 2021, an explosion of cases of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pneumonia overwhelmed the only hospital in Mayotte. To report a case series of patients with acute respiratory failure (ARF) due to SARS-CoV-2 who were evacuated by air from Mayotte to Reunion Island.

Method


What do we know when we know that a theory is consistent ?

Gilles Dowek

École polytechnique and INRIA LIX, École polytechnique, 91128 Palaiseau Cedex, France. Gilles.Dowek@polytechnique.fr, http://www.lix.polytechnique.fr/~dowek Abstract. Given a first-order theory and a proof that it is consistent, can we design a proof-search method for this theory that fails in finite time when it attempts to prove the formula ⊥?

1 Searching for proofs in a theory 1.1 Who knows that higher-order logic is consistent? It is well known that higher-order logic can be presented as a first-order theory, i.e. that there exists a first-order theory H and a function Φ translating closed formulas of higher-order logic to closed formulas of the language of H such that the sequent ⊢ A is provable in higher-order logic if and only if the sequent H ⊢ ΦA is provable in first-order logic (see, for instance, [?]). Thus, instead of using a proof-search method specially designed for higher-order logic, such as higher-order resolution [?,?], it is possible to use a first-order method, such as resolution, to search for proofs in higher-order logic.

However, this reduction is inefficient. Indeed, if we attempt to prove the formula ⊥ with higher-order resolution, the clausal form of the formula ⊥ is the empty set of clauses, from these clauses, we can apply neither the higher-order resolution rule that requires two clauses to be applied, nor any other rule of higher-order resolution, that all require at least one clause. Thus, this attempt to prove the formula ⊥ fails immediately. In contrast, the axioms of H give an infinite number of opportunities to apply the resolution rule and thus when searching for a proof of H ⊢ ⊥, the search space in infinite.

Thus, we can say that higher-order resolution "knows" that higher-order logic is consistent, because an attempt to prove the formula ⊥ fails in finite time, while first-order resolution does not.

A proof-search method for the theory H

There is an link between higher-order logic and higher-order resolution and another link between higher-order logic and the first-order theory H. But can we establish a direct link between higher-order resolution and the theory H, without referring to higher-order logic?

The answer is positive because the translation Φ can be inverted: there exists a function Ψ translating closed formulas of the language of H to closed formulas of higher-order logic such that the sequent H ⊢ B is provable in first-order logic if and only if the sequent ⊢ Ψ B is provable in higher-order logic. Thus, a way to search for a proof of a sequent H ⊢ B is to apply higher-order resolution to the sequent ⊢ Ψ B. Thus, independently of higher-order logic, higher-order resolution can be seen as special proof-search method for a the first-order theory H.

As Ψ ⊥ = ⊥ this first-order proof-search method immediately fails when attempting to prove the sequent H ⊢ ⊥. Thus, this method is much more efficient than applying first-order resolution to the sequent H ⊢ ⊥ as it "knows" that the theory H is consistent.

1.3 A proof-search method for a theory T Can we generalize this to other theories than H? Given an arbitrary first-order theory T and a proof that T is consistent, can we always build in the theory T, i.e. exploit the consistency of T to design a proof-search method that fails in finite time when required to prove the formula ⊥?

Of course, as we are not interested in the trivial solution that first tests if the formula to be proven is ⊥ and then applies any method when it is not, we have to restrict to proof-search methods that do not mention the formula ⊥.

It is clear that the consistency of the theory T is a necessary condition for such a method to exist: if T is inconsistent, a complete proof-search method should succeed, and not fail, when attempting to prove the formula ⊥. The main problem is to know if this hypothesis is sufficient.

Resolution modulo

Resolution modulo

Resolution modulo is a proof-search method for first-order logic that generalizes higher-order resolution to other theories than the theory H. Some axioms of the theory H are equational axioms. How to build in equational axioms is well-known: we drop equational axioms and we replace unification by equational unification modulo these axioms (see, for instance, [?,?]). Equational unification modulo the equational axioms of H is called higher-order unification.

From a proof-theoretical point of view, this amounts to define a congruence on formulas generated by the equational axioms and to identify congruent formulas in proofs. For instance, if we identify the terms 2 + 2 and 4, we do not need the axiom 2+2 = 4 that is congruent to 4 = 4, but when we substitute the term 2 for the variable x in the term x + 2, we obtain the term 4. We have called deduction modulo the system obtained by identifying congruent formulas in proofs.

But not all axioms can be expressed as equational axioms. For instance, if the axiom of arithmetic S(x) = S(y) ⇒ x = y can be replaced by the equivalent equational axiom P red(S(x)) = x, the axiom ¬0 = S(x), that has no one-point model, cannot be replaced by an equational axiom.

Thus, we have extended deduction modulo by identifying some atomic formulas with not atomic ones. For instance, identifying formulas with the congruence generated by the rewrite rules N ull(0) -→ ⊤ and N ull(S(x)) -→ ⊥ is equivalent to having the axiom ¬0 = S(x).

When we have such rewrite rules operating directly on formulas, equational resolution has to be extended. Besides the resolution rule, we need to add another rule called Extended narrowing. For instance, if we identify the formula P (1) with ¬P (0), we can refute the set of clauses {¬P (x)}, but to do so, we have to be able to substitute the term 1 for the variable x in the clause ¬P (x), deduce the clause P (0) and conclude with the resolution rule. More generally, the Extended narrowing rule allows to narrow any atom in a clause with a propositional rewrite rule. The proposition obtained this way must then be put back in clausal form. Equational resolution extended with this rule is called ENAR -Extended Narrowing and Resolution -or resolution modulo for short.

When we orient the axioms of H as rewrite rules and use resolution modulo, we obtain exactly higher-order resolution.

Proving completeness

Proving the completeness of higher-order resolution, and more generally of resolution modulo, is not very easy. Indeed higher-order resolution knows that higher-order logic is consistent, i.e. it fails in finite time when attempting to prove the formula ⊥. Thus, a finitary argument shows that the completeness of higher-order resolution implies the consistency of higher-order logic, and by Gödel's second incompleteness theorem, the completeness of higher-order resolution cannot be proved in higher-order logic itself. This explains that some strong proof-theoretical results are needed to prove the completeness of higher-order resolution, at least the consistency of higher-order logic. The completeness proof given by Andrews and Huet [?,?,?] uses a result stronger than consistency: the cut elimination theorem for higher-order logic.

In the same way, the completeness of resolution modulo rests upon the fact that deduction modulo the considered congruence has the cut elimination property. Indeed, when the congruence is defined by rules rewriting atomic formulas to non-atomic ones, deduction modulo this congruence may have the cut elimination property or not. For instance, deduction modulo the rule P -→ Q∧R has the cut elimination property, but not deduction modulo the rule P -→ Q ∧ ¬P [?] and resolution modulo this second rule is incomplete.

Is it possible to weaken this cut elimination hypothesis and require, for instance only consistency? The answer is negative: the rule P -→ Q ∧ ¬P is consistent, but resolution modulo this rule is incomplete. More generally, Hermant [?] has proved that the completeness of resolution modulo a congruence implies cut elimination for deduction modulo this congruence.

A resolution strategy

At least in the propositional case, resolution modulo can be seen as a strategy of resolution [?].

For instance, consider the rule P -→ Q ∧ R. The Extended narrowing rule allows to replace an atom P by Q ∧ R and to put the formula obtained this way back in clausal form. With this rule, from a clause of the form C ∨ P we can derive the clauses C ∨ Q and C ∨ R and from a clause of the form C ∨ ¬P we can derive the clause C ∨ ¬Q ∨ ¬R.

We can mimic this rule by adding three clauses ¬P ∨Q, ¬P ∨R, P ∨¬Q∨¬R and restricting the application of the resolution rules as follows: (1) we cannot apply the resolution rule using two clauses of the this set (2) when we apply the resolution rule using one clause of this set the eliminated atom must be the underlined atom. Notice that this set of clauses is exactly the clausal form of the formula P ⇔ (Q ∧ R). This strategy is in the same spirit as hyper-resolution, but the details are different.

If we apply the same method with the formula P ⇔ (Q ∧ ¬P ), we obtain the three clauses ¬P ∨ Q, ¬P ∨ ¬P , P ∨ ¬Q ∨ P with the same restriction and, like resolution modulo, this strategy is incomplete: it does not refute the formula Q.

The fact that this strategy is complete for one system but not for the other is a consequence of the fact that deduction modulo the rule P -→ Q ∧ R has the cut elimination property, but not deduction modulo the rule P -→ Q ∧ ¬P .

Understanding resolution modulo as a resolution strategy seems to be more difficult when we have quantifiers. Indeed, after narrowing an atom with a rewrite rule, we have to put the formula back in clausal form and this involves skolemization.

From consistency to cut elimination

We have seen in section ?? that the theory T = {P ⇔ (Q ∧ ¬P )} is consistent, but that resolution modulo the rule P -→ (Q ∧ ¬P ) is incomplete.

Thus, it seems that the consistency hypothesis is not sufficient to design a complete proof-search method that knows that the theory is consistent. However the rule P -→ (Q ∧ ¬P ) is only one among the many rewrite systems that allow to express the theory T in deduction modulo. Indeed, the formula P ⇔ (Q ∧ ¬P ) is equivalent to ¬P ∧ ¬Q and another solution is to take the rules P -→ ⊥ and Q -→ ⊥. Deduction modulo this rewrite system has the cut elimination property and hence resolution modulo this rewrite system is complete. In other words, the resolution strategy above with the clauses ¬P , ¬Q is complete and knows that the theory is consistent.

Thus, the goal should not be to prove that if deduction modulo a congruence is consistent then it has the cut elimination property, because this is obviously false, but to prove that a consistent set of axioms can be transformed into a congruence in such a way that deduction modulo this congruence has the cut elimination property. To stress the link with the project of Knuth and Bendix [?], we call this transformation an orientation of the axioms.

A first step in this direction has been made in [?] following an idea of [?]. Any consistent theory in propositional logic can be transformed into a polarized rewrite system such that deduction modulo this rewrite system has the cut elimination property.

To do so, we first put the theory T in clausal form and consider a model ν of this theory (i.e. a line of a truth table ).

We pick a clause. In this clause there is either a literal of the form P such that ν(P ) = 1 or a literal of the form ¬Q such that ν(Q) = 0.

In the first case, we pick all the clauses where P occurs positively P ∨ A 1 , ..., P ∨ A n and replace these clauses by the formula (¬A 1 ∨ ... ∨ ¬A n ) ⇒ P . In the second, we pick all the clauses where Q occurs negatively ¬Q∨B 1 , ..., ¬Q∨B n and replace these clauses by the formula Q ⇒ (B 1 ∧ ... ∧ B n ). We repeat this process until there are no clauses left. We obtain this way a set of axioms of the form A i ⇒ P i and Q j ⇒ B j such that the atomic formulas P i 's and Q j 's are disjoint.

The next step is to transform these formulas into rewrite rules and this is difficult because they are implications and not equivalences. But, this is possible if we extend deduction modulo allowing some rules to apply only to positive atoms and others to apply only to negative atoms. This extension of deduction modulo is called polarized deduction modulo. We get the rules P i -→ + A i and Q j -→ -B j . Using the fact that the P i 's and the Q j 's are disjoint, it is not difficult to prove cut elimination for deduction modulo these rules [?].

So, this result is only a partial success because resolution modulo is defined for non-polarized rewrite systems and orientation yields a polarized rewrite system. There may be two ways to bridge the gap: the first is to extend resolution modulo to polarized rewrite systems. There is no reason why this should not be possible, but this requires some work. A more ambitious goal is to produce a non-polarized rewrite system when orienting the axioms. Indeed, the axiom P ⇒ A can be oriented either as the polarized rewrite rule P -→ -A or as the non-polarized rule P -→ (P ∧ A), and similarly the axiom A ⇒ P can be oriented as the rule P -→ (P ∨ A). But the difficulty here is to prove that deduction modulo the rewrite system obtained this way has the cut elimination property.

Bridging this gap would solve our initial problem for the propositional case. Starting from a consistent theory, we would build a model of this theory, orient it using this model, i.e. define a congruence and resolution modulo this congruence would be a complete proof search method for this theory that knows that the theory is consistent.

But, this would solve only the propositional case and for full first-order logic, everything remains to be done.

We have started this note with a problem in automated deduction: given a theory T and a proof that it is consistent, can we design a complete proof-search method for T that knows that T is consistent? We have seen that this problem boils down to a problem in proof theory: given a theory T and a proof that it is consistent, can we orient the theory into a congruence such that deduction modulo this congruence has the cut elimination property?

We seem to be quite far from a full solution to this problem, although the solution for the propositional case seems to be quite close.

Some arguments however lead to conjecture a positive answer to this problem: first the fact that the problem seems almost solved for propositional logic, then the fact that several theories such as arithmetic, higher-order logic, and some version of set theory have been oriented. Finally, we do not have examples of theories that can be proved to be non orientable (although some counter examples exist for intuitionistic logic). However, some theories still resist to being oriented, for instance higher-order logic with extensionality or set theory with the replacement scheme.

A positive answer to this problem could have some impact on automated theorem proving, as in automated theorem proving, like in logic in general, axioms are often a burden.


domain: phys spi

text:
B Franc ¸ois

F Gand

HYBRID RANS/LES SIMULATIONS OF SECONDARY FLOWS IN A LINEAR CASCADE OF TURBINE BLADES WITH EXTERNAL TURBULENCE

Keywords: TURBINE, SECONDARY FLOW, HYBRID RANS/LES, TURBULENCE

Turbine blades are highly aerodynamically loaded, which leads to secondary flows in the vicinity of the hub/shroud walls. These secondary flows are the cause of significant aerodynamic losses in the turbine row, that need to be accurately predicted by the design engineer. Also, turbine rows operate in a flow with a high intensity of free-stream turbulence. Predicting such flows is quite challenging for RANS models that may lack accuracy. In the present work, a hybrid RANS/LES approach, named Zonal Detached Eddy Simulation, is assessed on a linear cascade of turbine blades submitted to a significant external turbulence. The selected configuration is the MAGPI configuration (Re=560 000) for which measurements are available downstream of the blades and on the blade walls. In this work, both the ZDES approach and the methodology for injecting turbulence are presented. The physical consistency of ZDES solutions is analysed and discussed. Finally, ZDES results are compared to the measurements and with a RANS solution: pressure losses maps downstream of the blade simulated with ZDES show a fair agreement with experiments both in shape and magnitude, while the RANS solution overestimates the pressure losses. The comparison of the pressure distribution on the blade wall near the lower wall (where secondary flows develop) exhibits also a better matching of the ZDES results with the measurements.

INTRODUCTION

Turbine blades are submitted to strong aerodynamic loads. This leads to strong pressure gradients at the vicinity of shroud walls generating secondary flows responsible for the main losses in a turbine row. These flows are three-dimensional, they generate a strong mixing of momentum with the main flow [START_REF] Cui | Numerical Investigation of Contrasting Flow Physics in Different Zones of a High-Lift Low-Pressure Turbine Blade[END_REF]). The prediction of the aerodynamic losses is a concern for the engineer who needs accurate methods to design high-performance and efficient machines. RANS methods are widely used in design cycle in industry, because they offer the best compromise between computational cost and accuracy. For such complex flows, the choice of the turbulence model remains delicate because of its strong influence on the results. Also, some turbulence models face difficulties for modeling large external turbulence rates. If not modelled, the turbulence can be partly resolved for a higher level of accuracy (Large Eddy Simulations approaches) but at the expense of a much larger computational cost. In turbine flows, Wall-Resolved LES (WRLES) is often prefered because the turbulence in the boundary layer is essentially resolved [START_REF] Fiore | Description of the flow in a linear cascade with an upstream cavity part 1: Influence of turbulence[END_REF]) and not modelled as in Wall-Modeled LES approaches. However, the numerical cost of WRLES is high because of the mesh required in the boundary layer. An alternative way is to resort to hybrid RANS/LES approaches, which solve the attached boundary layer with RANS while the rest is solved with LES. The mesh effort is reduced compared to WRLES, so does the numerical effort [START_REF] Tucker | Hybrid les approach for practical turbomachinery flows part i: Hierarchy and example simulations[END_REF]). This strategy is followed here using the Zonal Detached Eddy Simulation of [START_REF] Deck | Recent Improvements in the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF]. Yet, this approach has never been assessed on turbine configurations in which secondary flows and external turbulence are present. The configuration studied in this work is a linear cascade of a turbine from the EU project MAGPI. Measurements downstream of the cascade and on the blade walls are available and will be compared to the numerical results.

The present paper is organized as follows. First the configuration is presented as well as the secondary flow topology. Second, the numerical methodes are described: the Zonal Detached Eddy Simulation and the approach for injecting turbulent fluctuations to simulate the external turbulence in the computational domain. Third, the flow solutions are analysed to verify their consistency and the distribution of RANS/LES solving areas. Finally, the numerical results are compared to the measurements and discussed. The influence of the mesh on the numerical results is also discussed.

CONFIGURATION

Geometry and main flow features MAGPI (Main Annulus Gas Path Interactions) project is a European Commission funded project (2006)(2007)(2008)(2009)(2010)(2011). The project addressed the interactions between main gas path and secondary flow systems in commercial gas turbines. Experiments were performed on turbine disk rim and compressor manifold cavity heat transfer, hot gas ingestion, and spoiling effects of cooling air flow and their impact on turbine and compressor performance, as well as a reduction of secondary air losses. The experimental data was used for better understanding of the complex flow phenomena and improvements of platform and cavity design.

The MAGPI configuration investigated in the present work is a linear cascade composed of five nozzle guide vanes (NGV) installed at Karlsruhe University, Germany [START_REF] Schuler | Investigation of the Influence of Different Rim Seal Geometries in a Low-Pressure Turbine[END_REF]). The rig is set in an open circuit, which includes an upstream honeycomb settling chamber, a centrifugal blower, and a Venturi pipe to target desired inflow conditions. Upstream of the blade leading edge, the rim seal is included in a cavity module linked to the test-section allowing to set different rim seal designs. Purge flows of various mass flow rate are supplied to the cavity with a tangential component with respect to the span direction that mimic the driving effect of the rotor disk on the sealing flow in the cavity. The blade design represents a typical low-pressure turbine. In the present work, only the configuration without cavity is studied. Figure 1 gives an illustration of the test rig with the linear cascade and a summary of the main geometrical and flow features.

Flow topology

In order to illustrate and better understand the general flow topology observed in the MAGPI configuration prior to analyzing in details the simulations, Figure 2 presents the main secondary flow structures in the inter-blade channel, obtained from a RANS simulation1 . For the sake of clarity, only half on the channel height is displayed. Firstly, the incoming lower wall boundary layer impinges the blade leading-edge (LE) creating a horseshoe vortex. In the suction side, this vortex is convected along the blade wall, labeled as 1b in Fig. 2-a. On the pressure side, this vortex becomes a passage vortex crossing the whole inter-blade channel, labeled as 1a in Fig. 2-a. This passage vortex is driven by the strong azimuthal pressure gradient due to the large turning of the flow in the channel and the low momentum of the flow at the lower wall. This passage vortex impacts the adjacent blade suction side and is then convected along the suction side, labelled as 2 in Fig. 2-b. Inspection of the friction map (not shown here for conciseness) shows that the interaction of the passage vortex with the suction side of the blade do not cause flow separation (independently of the turbulence model). Near the suction side of the blade, the passage vortex causes then, in its vicinity, the creation of another vortex rotating in the opposite direction (colored in red), visible in Fig. 2-b and Fig. 2-d. On the first half of the pressure side, a flow separation occurs near the lower wall, downstream of the start of the passage vortex. This flow separation is labelled as 3 in Fig. 2-c. Downstream of this flow separation, a second passage vortex is created and is convected again across the whole channel because of the azimuthal pressure gradient. Nevertheless, it does not interact with the adjacent blade as observed in Fig. 2-d and labelled as 3 but with its wake. Figure 2-d illustrates the two main vortices responsible for the main aerodynamic losses of the linear cascade: the vortex colored in blue is the passage vortex (coming from the horseshoe vortex and labelled as 1a in Fig. 2-a) while the vortex colored in red is induced by the passage vortex (1a) (see Fig. 2-b), usually called "wall vortex".

HYBRID RANS/LES APPROACH

The hybrid RANS/LES method used in this work is the Zonal Detached Eddy Simulation (ZDES) developed at ONERA. The reader is referred to [START_REF] Deck | Recent Improvements in the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF] for the complete formulaiton of ZDES, which main ideas are recalled herefater. This approach has been used with success to simulate a wide range of applications of industrial interest (aircraft, space launchers, jet flows and turbomachinery) . ZDES has three operating modes. It is important to bear in mind that modes 1 and 2 of ZDES belong to the so-called "natural Detached Eddy Simulation" approaches in which the whole attached boundary layers are treated in RANS and the freeshear flows are treated in LES. Conversely, mode 3 of ZDES is a Wall Modelled LES approach in which the outer part of attached boundary layers is resolved in LES while the inner part is treated in RANS.

In the present paper, the formulation ZDES mode 2 is applied to the whole computational domain. ZDES mode 2 provides an "automatic" operating option (referred to as mode 2 in the following) for which the switch between RANS and LES regions is dynamically set by the model. Indeed, the ZDES mode 2 was developed to deal with separations over smooth surfaces and works as a global RANS/LES approach where the model itself defines the RANS and DES areas, which makes the treatment of complex geometries straightforward for the user. To ensure that attached boundary layers are treated in RANS independently of the grid density, a shielding function f d is used. This function was defined in the framework of Delayed-DES (DDES) [START_REF] Spalart | A New Version of Detached-Eddy Simulation, Resistant to Ambiguous Grid Densities[END_REF]) to be equal to 0 in the boundary layer and 1 elsewhere:

f d = 1 -tanh (8r d ) 3 , r d = ν + ν U i,j U j,i κ 2 d 2 w (1)
where ν is the kinematic viscosity, U i,j is the velocity gradient and κ is the Karman constant.

The hybrid length scale for ZDES mode 2 then reads:

dII ZDES = d w -f d × max 0, d w -C DES ∆ II ZDES (2)
which ensures that dII ZDES = d w in attached boundary layers where f d = 0, i.e. the underlying RANS model (the SA model is used in this work) is forced in these areas. Conversely to DDES, the mode 2 of ZDES provides a specific definition of the subgrid length scale according to the flow resolution and a threshold value f d0 = 0.8 determined in [START_REF] Deck | Recent Improvements in the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF]. When f d < f d0 , dII ZDES = ∆ max (∆ max being the characteristic mesh length necessary to ensure the correct behaviour of f d ). When f d > f d0 , the subgrid length scale revolves to dII

ZDES = ∆ vol = 3 ∆ x ∆ y ∆ z .

MESH AND NUMERICAL PARAMETERS Mesh

Figure 3: Mesh used in the simulations. One point over two is shown in the illustration.

Figure 3 presents the mesh used in the simulations and the main mesh features. The computational domain contains a single channel blade. It extends to one axial chord upstream of the blade and three axial chords downstream of the blade. These choices allow the convection of the secondary flows up to at least two axial chords. Near the outlet boundary condition, a buffer area with large mesh cells was set. The largest mesh size is 11 mm. At the inlet boundary condition, flow quantities measured experimentally are imposed (pressure stagnation, temperature stagnation, velocity direction). All walls are treated with a no-slip adiabatic boundary condition. The total number of mesh cells is 13 millions. The RANS and ZDES simulations were performed on the same mesh.

Flow Solver and Numerical Parameters

The flow solver is the elsA software [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF]), owned by ONERA and Safran. This compressible flow solver is based on a cell centered finite volume technique and structured multiblock meshes. For the steady RANS computations, the time scheme used is a standard 1st order implicit Backward Euler and for the unsteady ZDES simulations, the time scheme is achieved by an implicit second order scheme based on a Newton algorithm. The timestep has been set to 3 × 10 -7 s. The choice was mainly driven by acoustic CFL number considerations and computational cost (the selected timestep is small enough to capture the turbulent structures at stake). It corresponds to a convective CFL number around 36 in the smallest cells in the boundary layer and of 0.1 for cells with the length of 1 mm. This timestep samples the convection time of a particle from the leading-edge to the trailing-edge (TE) of the blade by 7000. For each timestep, five sub-iterations are used in the inner-loop of the time scheme and the residuals decrease by one order of magnitude. The convective flux discretization is treated with the AUSM (Advection Upstream Splitting Method) scheme with a third order limiter. The turbulence model used in the RANS part of ZDES is the turbulence model from [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF].

METHODS TO ACCOUNT FOR THE EXTERNAL TURBULENCE IN ZDES

Setting the correct inlet turbulence rate To account for the freestream turbulence, perturbations modeling this turbulence need to be injected at the inlet of the computational domain. Measurements indicate that the turbulence rate is roughly equal to 6% at the leading edge position. Upstream of the blade at a distance of at seven chords, the turbulence is generated by a grid composed of bars of 8 mm with an aperture of 25 mm. The turbulence impinging the blade is assumed to be homogeneous and isotropic.

According to [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF] and used by [START_REF] Fiore | Description of the flow in a linear cascade with an upstream cavity part 1: Influence of turbulence[END_REF], the time evolution of the turbulent kinetic energy can be written as:

k(t) = k(t 0 ) 1 + (C ϵ -1) t τ 0 -1 Cϵ-1 (3)
with τ 0 = k(t 0 )/ϵ(t 0 ) and C ϵ = 1.92. The Taylor assumption is used to relate time and spatial evolution. It comes then using the relation x = u 0 t where u 0 is the mean velocity:

k(x) = k(x 0 ) 1 + (C ϵ -1) x u 0 τ 0 -1 Cϵ-1 (4)
with τ 0 = k(x 0 )/ϵ(x 0 ). The dissipation rate can be related to a turbulent length scale L t with the relation L t = C Lt k 3/2 /ϵ where C Lt = 0.4 [START_REF] Pope | Turbulent Flows[END_REF]).

For the present configuration, the velocity is equal to u 0 = 41.4 m/s., the flow angle α = 38 • . The distance between the inlet and the leading-edge of the blade (following the flowpath) is roughly 95 mm. Given that the uncertainties regarding the turbulent length scale for the upstream turbulence, two length scales were tested in the simulations: 8 mm and 4 mm. To meet the measured turbulence rate at the leading-edge position, the estimated turbulence rate imposed at the inlet based on Eq. 4 for turbulent length scales L = 8 mm and L = 4 mm are respectively 7% and 8.5%.

Methodology for injecting external turbulence

The methodology to account for external turbulence is based on the approach of [START_REF] Jarrin | Reconstruction of Turbulent Fluctuations for Hybrid RANS/LES Simulations using a Synthetic-Eddy Method[END_REF], named Synthetic Eddy Method (SEM). In order to generate an unsteady and turbulent velocity field at the inlet of the computational domain, the principle of SEM is to superimpose a large number of turbulent structures defined analytically by their velocity field over a mean velocity field provided by the user. Thus, the velocity field in each point of the inlet boundary condition u i is defined as follows:

u i (x, y, z, t) = U i + 1 √ N N k=1 ϵ k j A ij V B σ -3 T x -x k (t) σ T y -y k (t) σ T z -z k (t) σ i = 1, 2, 3
where:

• Provided data from the user are the mean velocity field U i , the Choleski decomposition of the targeted Reynolds tensor A ij and the length scales of the coherent structures σ.

• N , V B are respectively the number of structures and a control volume (computed automatically)

• ϵ k j : a random value equal to 1 or -1 • (x k (t), y k (t), z k (t)) are the coordinates of the turbulent structure k following an assumption of frozen turbulence

• T is a shape function for the coherent structures, defined as

T (x) = 3 2 (1 -|x|) if x < 1, it equals 0 otherwise.
For the present study, the Reynolds stress is assumed homogeneous and isotropic such as R ij = (T u × U 0 ) 2 × δ ij where T u is the turbulence rate, U 0 a reference velocity, δ ij the Kronecker symbol. After 15 time passages from LE to TE (i.e. 10 5 iterations), the mean mass flow rate and the mean forces are converged. Figure 4 presents the evolution of the magnitude of the instantaneous velocity over time on one numerical probe located at mid-span, at the center of the inter-blade channel and at the same axial position as the leading-edge of the blades. For conciseness, only the signal for ZDES with L = 8mm is shown. The red curve depicts the instantaneous velocity modulus signal whereas the green curve is the time-averaged quantities over a sliding window. The sliding window width is equal to ten time passages of a particle convected from the leading-edge to the trailing-edge of the blade. Both signals are normalized by the final time-averaged value: it gives direct access to the variation in percent of the signals.

RESULTS AND DISCUSSION OF THE ZDES SIMULATION Convergence

The time in abscissa is normalized by this time passage. Figure 4 shows the mean values are converged and varies less than a few percents over the periods. Figure 5 illustrates the injection in the inlet plane and the convection of the turbulent structures for two ZDES simulations with prescribed external turbulence. It enables to verify that the shape and size of turbulent structures at the inlet and their evolution through the computational domain are qualitatively correct. The purpose of injecting external turbulence is to reproduce the experimental conditions in which a turbulence rate of 6% was measured on the leading-edge plane of the cascade. Calculations of this turbulence rate from the simulations show that ZDES L = 8 mm and ZDES L = 4 mm actually reached a turbulence rate at the LE of 5.5% and 4.5% respectively. With the decaying turbulence law, the turbulence rate is slighty underestimated in the simulations compared to the measurements. Nevertheless, it remains quite significant and it is believed to have a noticeable influence on the results.

Convection of the injected turbulent structures

Behavior of the ZDES shielding function and analysis of the flow Analysis without external turbulence

The present ZDES formulation was never applied to a turbine configuration with secondary flows. As a validation step of the approach, the shielding function maps (indicating RANS or LES solving) are inspected. Figure 6-a presents a flow visualisation with secondary flows colored by the helicity (blue denotes counterclockwise rotation while red denotes clockwise rotation). Close-up views are shown in subfigures within Fig. 6-a. The same passage vortex labelled 1a and described in Fig. 2-a and a flow separation (also present in RANS) on the pressure side on the blade can be observed. On all subfigures in Fig. 6-a, a second color map is superimposed on a plane located at x = 0.2C x in which the shielding function field f d is displayed. Dark magenta color indicates that the area is solved with a RANS modeling, otherwise it is solved with LES. The dark magenta area nearby the wall indicate the presence of the boundary layer which is, as expected, treated in RANS; the rest of the domain is treated in LES. Locally, Fig. 6-b shows that the presence of the passage vortex nearby the lower wall and interacting with its boundary layer : this implies the boundary layer is no longer a canonical attached boundary layer in its upper part and for this reason, the method switches to a LES solving. This is a consistent behavior of the approach which automatically detects this kind of situation. In Eq. 1, the additional vorticity brought by the main vortex decreases the variable r d and modifies the value of the shielding function to 1 (LES solving). Same behavior was observed by Franc ¸ois et al. [2020] with tip vortices nearby a shroud wall on a fan. Fig. 6-a,b (bottom-right) shows that the flow separation is well solved in LES mode near the wall. This is also an expected feature of the approach. For conciseness, the colormap of the shielding function is shown only on one plane but additional analysis have demonstrated a relevant repartition of the RANS and LES solving area in the whole computational domain.

Effect of the external turbulence

The validation work is continued with the analysis of the effect of external turbulence on the RANS/LES areas. Using the same pattern as Fig. 6-b for shielding function visualisation, Fig. 6c,d presents ZDES with external turbulence for two turbulent length scales. Observations of the boundary layer on the lower wall show that the RANS area reduces due to the presence of the external turbulence. The presence of eddies give rise to strong velocity gradient at the border of the boundary layer. Compared to simulations without external turbulence, the variable r d becomes smaller and the model switches to LES in these part. This explains why the RANS area is smaller with external turbulence in ZDES simulations. This expected behavior is observed in the whole computational domain.

COMPARISON TO THE MEASUREMENTS AND RANS RESULTS

Pressure losses downstream of the blades Figure 7 presents stagnation pressure maps at a quarter-chord downstream of the cascade for the experimental data, a RANS simulation using the Spalart-Allmaras turbulence model and two ZDES simulations accounting for external turbulence with two turbulent length scales. In the same figure, another prediction using ZDES with a mesh of higher density (27 millions points) than the one presented in Fig. 3 is added. The convection of the passage vortex, described in Fig. 2, generates stagnation pressure losses at positions z/H ≈ 0.2 and z/H ≈ 0.8 on the channel height. The pressure losses due to the wake are also visible on the whole channel height at y/s ≈ 0. In the simulations, the RANS results with the SA turbulence model predicts a correct shape of the maps of losses but not their magnitude. Pressure losses are overestimated in both the wakes and the passage vortex where the ZDES results reproduces quite accurately the pressure losses maps, both in shape and magnitude. In a companion paper, Uncu et al. [2023] show similar overestimations of pressure losses with other turbulence models (relying on the Boussinesq approximation and a Reynolds Stress Model), although these discrepancies with measurements are somehow compensated when comparing the radial profiles of pressure losses due to the azimuthal average. In Fig. 7, a slight overestimation of the pressure losses can be observed compared to the measurements near the lower wall. Comparison of Fig. 7-c and Fig. 7-e show the predicted maps of pressure losses are almost identical with two different meshes. First, the RANS and ZDES results on mesh with 13 millions points (named M13) are compared. At mid-span, the results are identical between the three ZDES and match quite well with the measurements: the same under-estimation of the blade loading as RANS results is observed. This is an expected behavior of the method because at this channel height, the boundary layer remains attached and solved in RANS with ZDES. Also, the presence of external turbulence has no effect on the C p distribution for this blade height. At blade height of 4% and 6%, both ZDES results with external turbulence are very similar. Both show a significantly better agreement of the results with the measurements compared to RANS results and to ZDES without external turbulence. The results of the latter are quite similar to the RANS results. At blade height of 6%, the bump observed at a position of 60% chord in the measurements on the suction side is reproduced with the same intensity with ZDES but at a slightly more upstream position than in the measurements whereas the tested RANS model over-estimates this bump (such as the ZDES without external turbulence). Among the two ZDES accounting for external turbulence, the simulation using the turbulent length scale of 8mm (and having the closest turbulence rate to the measurements when meeting the blade leading-edge) is in closest agreement with experiments. At blade height of 4%, a bump is observed on the C p distribution with the RANS model while it is absent with the ZDES results with mesh M13. Figure 9 shows that the main reason of this discrepancy is the position of the low pressure area which is more distant (and of higher intensity) to the hub for the RANS model than for the ZDES results.

Wall pressure on blade

The results of a ZDES simulation on a refined mesh M27 are also plotted in Fig. 8 and Fig. 9, which provides some insights on the mesh sensitivity of the results. At h/H = 6% and h/H = 50%, the discrepancies between the two meshes are quite low as observed in Fig. 8. At h/H = 4%, the discrepancies between the two ZDES with different densities is as large as the ones with RANS results. Inspection of the pressure distribution on the whole suction side surfaces in Fig. 9 shows that the reason for this is again that the trajectory of the passage vortex along the suction side (identified by the low pressure area) is slightly different. The low pressure area is located closer to the hub surface for the simulation using the mesh of 27 millions points and intersect the isosurface at h/H = 4% whereas for the ZDES using the mesh of 13 millions points, it does not intersect this iso-surface.

CONCLUSIONS

The present paper proposes an assessment of a hybrid RANS/LES approach, the Zonal Detached Eddy Simulation (ZDES), on the aerodynamic simulation of turbine flow with external turbulence on a linear cascade. The flow topology around turbine blades is quite complex because secondary flows are present and generate additional losses. Their prediction is important in an industrial design process. The ZDES results have shown that first, the approach works correctly on such flows with a consistent distribution between RANS and LES solving areas. Accounting for external turbulence with this approach for a turbomachinery flow was also new in this work and the flow analysis shows that the method behaves as expected. These observations increase the maturity of the approach for turbomachinery flows. Furthermore, the ZDES results presents a very good agreement with the experimental results on both the pressure losses map and on the pressure distribution on the blades. Regarding the last point, the comparison with the C p measurements must be done with care because the numerical predictions are strongly dependent of the trajectories of the secondary flows. Three major conclusions can be drawn from this work. First, accounting the external turbulence for this configuration significantly improves the results, especially for secondary flows. Second, the ZDES results provide better prediction results than RANS simulations using Spalart-Allmaras model. Third, the mesh influence study has revealed a low mesh sensitivity of the flow solution with two meshes. Future work will be dedicated to the simulation of this configuration with a cavity flow interacting with the main flow.

Figure 1 :

 1 Figure 1: Test rig of the MAGPI configuration from Schuler et al. [2011] (left) and main geometrical and flow features (right)

Figure 2 :

 2 Figure 2: Flow topology of the secondary flows from a RANS simulation illustrating the main flow physics of the MAGPI configuration. Blue and red colors indicate respectively counterclockwise and clockwise vortices with respect to the X-axis. Legend from (a) applies to figures b,c,d.

Figure 4 :

 4 Figure 4: Instantaneous and time-averaged (10 periods) signal of the magnitude of the velocity over time iteration for the ZDES with turbulent length scale L = 8 mm.

Figure 5 :

 5 Figure 5: Illustration of the turbulent structures (Q criterion colored by the axial momentum) for the ZDES for length scales (left) L = 8mm and (right) L = 4mm

  Fig. 6-b is identical to Fig. 6-a except the secondary flows isocontours have been removed for the better visualisation of the shielding function colormap.

Figure 6 :

 6 Figure 6: Plots of secondary flows with shielding function contour maps at plane x = 20% without external turbulence (a & b), with external turbulence (T u = 6%) and two turbulence length scales : (c) L = 8 mm , (d) L = 4 mm. Same legend applies for all the figures.

Figure 7 :

 7 Figure 7: Comparison of stagnation pressure map (P i -P i,amb ) at a quarter axial chord downstream of the cascade. P i,amb refers to the stagnation pressure at one axial chord upstream of the leading-edge of the cascade. MXX indicates the number of million cells in the mesh used.

Figure 8 :

 8 Figure 8: Pressure coefficients at h/H = 4% (left), h/H = 6% (center) and h/H = 50% (right) from ZDES

Figure 9 :

 9 Figure 9: Non-dimensionned time-averaged pressure on the suction side of the blade from RANS and ZDES simulations (Tu=6%,L = 8mm) using two mesh densities.

F

  . Uncu, B. Franc ¸ois, R. Barrier, Buffaz N., and Le Guyader S. RANS Prediction of the Aerodynamic Losses in a Linear Turbine Cascade with an Upstream Cavity and a Purge Flow. In 15th European Turbomachinery Conference, 2023.

the secondary flow topology does not vary much according to the turbulence model

ACKNOWLEDGEMENTS

The authors wish to thank Fatih Uncu for the fruitful discussions on the experimental database of the MAGPI configuration and on the RANS modeling and Safran Helicopter Engines for providing this experimental database in the framework of the PhD thesis of Fatih Uncu. The authors are grateful to the EU commission who funded the experiments in the framework of the MAGPI project (Grant number 30874).