Erol Licsandru

Istvan Kocsis

Yue-Xiao Shen

Samuel Murail

Yves-Marie Legrand

Arie Van Der Lee #

Daniel Tsai

Marc Baaden

Manish Kumar

Mihail Barboiu

Salt-excluding artificial water channels exhibiting enhanced dipolar water and proton translocation

Keywords: Artificial water channels, self-assembly, aquaporins, proton transport

Aquaporins (AQPs) are biological water channels known for fast water transport (~10 8 -10 9 molecules/s/channel) with ion exclusion. Few synthetic channels have been designed to mimic this high water permeability, and none reject ions at a significant level. Selective water translocation has previously been shown to depend on water-wires spanning the AQP pore that reverse their orientation, combined with correlated channel motions. No quantitative correlation between the dipolar orientation of the water-wires and their effects on water and proton translocation has been reported. Here, we use complementary X-ray structural data, bilayer transport experiments and molecular dynamics (MD) simulations to gain key insights and quantify transport. We report artificial imidazole-quartet water channels with 2.6-Å pores, similar to AQP channels, that encapsulate oriented dipolar water-wires in a confined chiral conduit. These channels are able to transport ~10 6 water molecules per second, which is within two orders of magnitude of AQPs' rates, and reject all ions except protons. The proton conductance is high (~5 H+/s/channel) and approximately half that of the M2 proton channel at neutral pH. Chirality is a key feature influencing channel efficiency.

Introduction

Water is essential for life. [1][2][3] Under physiological conditions, its molecular-scale hydrodynamics are of crucial relevance to biological functions. 2,3 Changes in water's properties between conventional bulk water and biologically confined water are not limited to size because the water's behavior also becomes quite different. 2 The strong ordering of water molecules at the interface of phospholipid bilayers renders its oriented dipolar structure different from that of bulk water. 4 Moreover, oriented poreconfined water dipoles and correlation motions strongly determine the water and ionic transport rates and the selectivity of biological pores, such as aquaporins (AQPs) [5][6][7] and gramicidin A. 8 In this context, we previously reported that natural AQPs can be mimicked using simpler artificial water channels. 9,10 We noted that artificial water channels formed from the supramolecular organization of alkylureido-ethylimidazole compounds (Fig. 1) present significant water/proton permeabilities. 11 These compounds form imidazole-quartet (I-quartet) water channels, i.e., stacks of four imidazoles and two water molecules, that can mutually stabilize oriented dipolar water-wires within 2.6-Å pores (Fig. 1b). This channel diameter is very close to the narrowest constriction (2.8 Å) observed in very efficient AQP biological water-transport channels. 5 Here, we show that these novel I-quartet channels exhibit enhanced water and proton transport rates with complete ion rejection. This feature has not been observed for other artificial water channels, which typically have larger pore diameters (4-5 Å), that approach AQPs' permeability but do not achieve any salt rejection. [12][13][14][15] We focused on two hypothesized requirements for the realization of more effective artificial water channels using the self-assembling I-quartet motif: (i) More stable synergistically interacting I-quartets and water-wires within the bilayer membrane environment will lead to stabilized channels with higher permeability. (ii) Enhancing the dipolar organization of the water-wires will play an important role in improving water transport and proton translocation.

Results and discussion

Considering the requirements mentioned above, a close examination of the I-quartet structures led us to propose the following novel strategies to control I-quartet water channel formation and stabilization within bilayers: a) HC4, HC6, and HC8 compounds, which have variable alkyl chain lengths, may induce variable morphological hydrophobic stabilization by promoting external wrapping around I-quartets. b) Chiral S-HC8 and R-HC8 compounds may enhance the net chiral selfassembly of I-quartets and thereby strengthen the water dipolar orientation within the channels. Six compounds were prepared for the studies described here (Figs. 1c-h). We treated the corresponding isocyanates with histamine (CH3CN/N,N-dimethylacetamide, 120 °C, 5 h) to crystallize HC4, HC6, HC8, R-HC8 and S-HC8 as white powders. The nuclear magnetic resonance (NMR) and electrospray ionization mass spectrometry (ESI-MS) spectra agree with the proposed formulas (Supplementary materials). We obtained colorless single crystals of HC4, HC6, HC8, R-HC8, and S-HC8 after recrystallization from water at room temperature. The X-ray structures reveal the expected I-quartets, all (except HC4) containing oriented water-wires (Fig. 1c-h) (a detailed discussion of the X-ray structural results will be published elsewhere). In all structures, the H-atoms of the water molecules present 100% occupancy, and the water molecules of one channel adopt the same dipolar orientation. The overall structures of achiral compounds HC6 and HC8 are centrosymmetric, and thus, waterwires of opposite water dipolar orientations are present in successive channels. In contrast to achiral compounds, the noncentrosymmetric structures of chiral compounds R-HC8 and S-HC8 show unique dipolar orientations for both channels (Fig. S1).

Water-transport properties. HC4, HC6, HC8, R-HC8 and S-HC8 assemble into functional water-permeable channels in liposomes. Their water transport rates are strongly dependent on the hydrophobic behaviors of the alkyl tails grafted onto the I-quartets.

We conducted two assays, adding the compounds into the liposomes either before or after lipid-mixing. In the first, pre-mix assay, the imidazole I-quartets were mixed with lipids before the liposomes formed. We carefully optimized the lipid systems' phosphatidylcholine, phosphatidylserine and cholesterol (PC/PS/Chl) composition (molar ratio of 4/1/5) and used a low temperature (10 °C) to maintain low lipid background permeability (~4 μm/s). This low background permeability allowed us to resolve the water transport through these channels in stoppedflow light-scattering experiments. 16 Under hypertonic conditions driven by outwardly directed osmotic gradients, the shrinkage of the liposomes increased the light-scattering signal. We fitted a sum of two exponential functions to the latter data. The kinetics of vesicle shrinkage indicated that some channels had low permeability. Therefore, we implemented a method used in recent studies of low-water-permeable systems, 15 including aquaporin AQP0 17 . This analysis yielded two shrinkage rate constants (a larger constant, k1, and a smaller constant, k2) and, thus, two permeability values. For HC4 and HC6, k1 was independent of the compound concentrations, while k2 increased with the addition of the self-assembled compounds (Fig. 2a); this phenomenon was reversed when we added S-HC8 channels to the vesicles. This result indicates that the superstructures assembled by HC4 and HC6 are probably not stable contributing less to the overall bilayer permeability than the lipid background (Fig. S2), whereas HC8 and its chiral isomers R-HC8 and S-HC8 increased the overall permeability relative to the background lipid permeability. Therefore, we used the smaller k2 to calculate the permeabilities for HC4 and HC6 assemblies. The larger k1 was used to characterize the permeabilities of HC8 and its chiral isomers R-HC8 and S-HC8. The net permeabilities of HC4 and HC6 assemblies at a channel-to-lipid weight ratio of 1 (CLR=1) were 0.012±0.003 μm/s and 0.027±0.015 μm/s, respectively, whereas for HC8 and its chiral isomers R-HC8 and S-HC8, the permeabilities were 1.0±0.3 μm/s, 2.8±0.7 μm/s and 4.1±0.2 μm/s, respectively (Fig. 2b). The permeabilities of the I-quartet channels were confirmed in the post-mix assay, in which channels were injected into pre-formed liposomes with the aid of dimethylsulfoxide (DMSO). The temperature in the post-mix experiments was ~20 °C, leading to a relatively high background permeability (~25 μm/s). However, it was still possible to differentiate the channels' contributions to water transport (Fig. S3A). We obtained post-mix results (Fig. S3B) similar to the pre-mix results (Fig. 2b). HC4 and HC6 assemblies maintained low permeability, but HC8 and its isomers channels were more permeable toward water.

When dissolved in non-ionic surfactants, these I-quartet channels show a unique UV-absorbance signal at 230 nm that we used to calculate the bilayer insertion efficiency (Fig. S4). For HC8 and its chiral isomers R-HC8 and S-HC8, the insertion efficiency was 12.7±2.6%, 68.8±2.9% and 53.1±6.8%, respectively (Table S1). We calculated the single-channel permeability of each channel type based on the insertion, actual lipid concentration and channel configuration in the lipids, as previously described 15 (Table S2, see Supporting information for detailed calculations). In the shrinking mode, HC8, R-HC8 and S-HC8 had permeabilities of 1.4(±0.4)´10 6 , 7.9(±2.1)´10 5 and 1.5(±0.1)´10 6 water molecules/s/channel, respectively, which are only two orders of magnitude lower than that of AQPs (~10 8 -10 9 water molecules/s/channel). [4][5][6] Proton-transport properties. Proton-transport experiments were performed in two independent assays (Fig. 3), both of which showed that the transport activity of the I-quartet channels increased with the alkyl chain length.

In the first fluorescence spectroscopy assay, [START_REF] Matile | In Analytical Methods in Supramolecular Chemistry[END_REF] we encapsulated HPTS, a pH-sensitive fluorescent dye, inside vesicles, added channels to the vesicle solution and monitored the fluorescence change over time upon exposure to low pH. The fluorescence intensity was monitored for 500 s, and the fluorescence change rate followed a typical dose-response profile (Fig. 3b and Fig. S5).

The calculated EC50 values showed the effectiveness of each channel-lipid system for proton transport, with lower values indicating higher transport efficiencies. [START_REF] Matile | In Analytical Methods in Supramolecular Chemistry[END_REF] The most active S-HC8 channels had the lowest EC50 (11.2 µM) (Fig. 3c and Table S3). Based on the EC50, we observed no activity for HC4, weak activity for HC6 and fairly strong activity for HC8, whereas the activities of the S-HC8 and R-HC8 optical isomers were one order of magnitude higher than those of the achiral HC8 channels (Table S3 and Fig. S5). Tests using an equimolar mixture of the S-HC8 and R-HC8 isomers displayed reduced transport activity. All of the determined Hill numbers were below 1, suggesting that the presented channels belong to the type II class (Table S3). [START_REF] Matile | In Analytical Methods in Supramolecular Chemistry[END_REF] To complement these results, we conducted kinetic measurements of proton-transport activity over 50 s using stopped-flow techniques. We subjected the liposomes with I-quartet channels to a pH gradient across the bilayer and assessed the proton flux by monitoring the fluorescence intensity of encapsulated fluorescent dextran (Fig. 3d). As shown in Fig. S6, the addition of HC4 channels did not contribute to the fluorescence response, whereas HC6 channels slightly increased the proton-exchange rates across the lipid membranes. In contrast, the kinetics were found to increase by orders of magnitude for HC8, and both the R-HC8 and S-HC8 channels displayed slightly higher transport kinetics than HC8. For R-HC8, we observed that the second phase of the fluorescence increase saturated in less than 5 s (Fig. S6). This behaviour was significantly different from that of S-HC8, which showed continuously increasing fluorescence without saturation as the concentration increased (especially for the 380-μM sample). This observation implies that the R-HC8 chiral channels had an immediately higher kinetic response to the proton transport, whereas the S-HC8 chiral channels required time to exhibit dynamic behaviours within the lipid bilayer, leading to slower initial transport rates and, subsequently, higher proton permeability, as observed for the proton activity over long periods of time.

We further estimated that the single-channel proton transport rates (Fig. 3e, see the Supporting materials for detailed calculations) of the HC4, HC6, HC8, R-HC8 and S-HC8 channels were 0.1±0.2, 0.9±0.4, 2.4±0.5, 5.6±0.5 and 5.1±1.3 protons/s/channel, respectively. The proton fluxes (especially for the R-HC8 and S-HC8 channels) were similar to that of the M2 proton channel determined under the same experimental conditions at pH=7. [START_REF] Leiding | [END_REF] Our initial assumption, which postulated that water-wires are structural descriptors that could be used to un-derstand the M2 channel's proton-transport function 11 , is confirmed by the proton fluxes observed in these systems. A recent computational study 20 based on a high-resolution structure of the M2 proton channel quantified how water-wires factor into the process of proton translocation.

The bilayer membrane transport experiments allowed us to draw the following conclusions:

(i) Water/proton transport activities increase substantially as the grafted alkyl chain length increases (HC4< HC6<< HC8), implying a more stable hydrophobic shell wrapping around the I-quartets at the interface with the bilayers. The increased efficiency of the I-quartets could be synergistically attributed to the enhanced hydrophobic stabilization of the channels within the bilayer membrane environment based on the optimal molecular packing of the hydrophobic tails. Although HC4 and HC6 show channel structures in crystals forms, but they did not behave like highly water-permeable or proton conductive channels in bilayer membranes. This is a structure-activity argument and is relatted to their stability in the bilayer membrane which will be further described in the simulation part bellow. Indeed, the tail length seemed to be optimal for octyl chains (longer chains led to precipitation when mixed with lipids). Molecular dynamics (MD) simulations described subsequently support this assertion.

(ii) The I-quartets of the chiral R-HC8 and S-HC8 compounds show a unique dipolar orientation of water-wires within all the channels and exhibit high water/proton conductivities, with a preference for the S-HC8 isomer. The explanation may relate to the channel chirality: The chiral S-HC8 channels may be more compatible with the naturally occurring chiral L-lipids than the isostructural HC8 channels. 21 Table S1 shows insertion efficiency measurements supporting this assertion. Moreover, we noted increased water/proton activity of the chiral R-HC8 I-quartets under kinetic conditions, whereas the overall conductivity of the S-HC8 I-quartets was higher under thermodynamic equilibrium conditions.

Ion (Na + and Cl -) transport across the bilayer membranes incorporating the HC8 and S-HC8 I-quartet channels was assessed using standard fluorescence procedures, and no response versus the control experiment and a lack of variation of the transport activity with the concentration of I-quartets (Fig. S7) were observed. In addition, the apparent permeability showed no significant difference between the two hypertonic solutions (+200-mM NaCl). We calculated the reflection coefficient for NaCl as the ratio of the measured water permeability in the presence of salt to that in the presence of a completely retained sucrose osmolyte. The coefficient (113±15%, n=4) indicated that this channel (HC8) achieved approximately 100% salt rejection in the current lipid system. 22 Molecular simulations. We performed MD simulations to understand the structure/activity relationships of the artificial water channels formed by HC6, HC8, R-HC8 and S-HC8 Iquartets. We embedded small (ca. 3 nm wide) I-quartet channel patches based on the X-ray structures in a lipid bilayer environment with the same composition as that used in the experiments to begin each simulation (Table S4). The MD results showed that I-quartets preferentially located within the bilayer membrane region and stabilized water channels in all simulations, although the degrees of water-occupancy, transport and structuring varied. We observed that the structural dynamics of Iquartet channels was dependent on the lateral pressure applied to the membrane, with higher pressure favoring the structuring of the water-wires and stabilizing the structure of the I-quartets (Fig. S8). For our comparative investigations, we chose 10 atm of lateral pressure as a good compromise to maintain aggregate structuring and allow us to achieve reasonable sampling and water activity statistics (Figs. S9 andS10) within the simulation's microsecond timescale.

We observed that the chiral R-HC8 and S-HC8 I-quartets form the most stable and structured systems, followed by the significantly less structured HC8 I-quartet (Fig. 4a), in accordance with the experimentally observed bilayer insertion and water/proton-conduction behaviours. The latter instability echoes the influence of the alkyl chain length, with improved packing and a more stable hydrophobic shell formation occurring with longer chains. After the initial relaxation, the simulations of the chiral S-HC8 and R-HC8 I-quartet systems show slightly twisted superstructures. The twist angles between the upper and lower leaflets were 18.4° ± 2.1 and 7.6 ° ± 4.0, respectively, over the second half of the simulation, whereas the twist was negligible for HC6 and HC8 (Fig. S11). Both the chiral S-HC8 and R-HC8 I-quartets with bent structures contained more structured water-wires than the achiral linear HC8 and HC6 Iquartets and exhibited higher stability over time (Figs. 4,S12, and S13 and supplementary movies). We quantified the waterwire preservation by analyzing the water dipole orientations within the membrane, as shown in Fig. 4B. The R-HC8 and S-HC8 systems show a strong preference for a dipole orientation of ~25°, whereas the HC8 system shows two orientations at the beginning of the simulation and tends towards a near-random distribution at the end. The dipole orientation is slightly more pronounced for the S-HC8 system compared to the R-HC8 system (Fig. S14). The MD simulations provided further insights into the dynamic behaviors of the I-quartets and water molecules under confined conditions in a lipid bilayer environment. Although the HC6, HC8, R-HC8, and S-HC8 molecules deviate from their initial positions, the chiral channels preserve their overall integrity, but the achiral ones lose it. Water transport is another important property of these aggregates. Because the model system size is small, we focused on the central channels to make inferences regarding water-transport trends.

These buried paths are less affected by boundary effects at the lipid/I-quartet interface than the peripheral ones. Despite the limited time scale and the absence of an osmotic driving force, we observed significant spontaneous permeation through R-HC8 compared to the reference HC8 system (Fig. S15), whereas the incipient water transport in the S-HC8 channels was near zero. This observation is consistent with the experimental results, shedding light on the fast kinetic response of the R-HC8 channels presenting a certain motional behavior within the bilayer. In contrast, the slower thermodynamic response is related to the incipient stability of the S-HC8 channels within the L-lipids and needs time to reach dynamic behaviors and to achieve functional water permeability within the bilayer. This is highlighting the importance of the dynamics of the channels superstructures which may remain stable and dynamic enough to allow water motion inside the water channels.

Conclusions

Overall, the self-assembled I-quartet channels, obtained using simple chemistry, can be assembled into robust artificial water channels and remain stable in lipid membranes. The complementary MD and experimental studies presented here provide the first molecular-level insights into the functional dynamics of artificial water channels in the lipid membrane environment. Their permeability (~ 10 6 water molecules/s/channel) is only two orders of magnitude lower than that of AQPs. The I-quartets are dimensionally compatible with encapsulated water molecules, both measuring 2.6-2.8 Å in diameter, which is very similar to the narrowest pore dimension of natural AQPs. 4- 6 Accordingly, we postulate that a pore with a diameter of ~ 3 Å is a critical prerequisite for salt-ion-selective water translocation. The observed total ionic-exclusion behaviour of these channels suggests that they hold significant promise for the incipient development of the first innovative materials based on artificial synthetic scaffolds mimicking the functions of natural water channels.

These channels' lipid affinities influence their stability in the membrane and consequently constitute a means to tune their activity. In all cases, the activity of the chiral I-quartets is superior to that of the isostructural non-chiral I-quartets. This observation suggests that the lipids may exert a conformational preference, stabilizing the chiral superstructures. Additionally, the chiral I-quartets preserve the total dipolar orientation and ordering of the water-wires. The mutual interactions between the inner chiral surfaces of the I-quartets, which result in the total dipolar orientation of the water-wires, 23,24 probably impose an internal dielectric polarization within the pore, acting as a constitutive driving force and activating water-and proton-translocation in a way that differs significantly from their non-oriented or bulk counterparts.

These findings may inspire the incorporation of I-quartets into future membrane materials for desalination and water-purification applications. 25 Artificial I-quartet water channels have a 2.6 Å pore encapsulating oriented dipolar water-wires in a confined chiral channel. They transport ~10 6 water molecules per second/channel approaching those of AQPs within two orders of magnitude and reject all ions except protons. They also exhibit high proton conductance (~5 H + /s/channel) close to that of the highly conductive M2 proton channel (~10 H + /s/channel at neutral pH).

Figure 1 .

 1 Figure 1. Artificial I-quartet water channels and their chiral isomers pack differently. a. Side and b. top views of I-quartet water/proton channels that self-assemble into tetrameric-tubular architectures confining dipolar oriented water-wires. Top views of water confining Iquartet channels in stick representations (N, blue; C, grey; O, red; H, white), taken from crystal structures of C. Butylureido-ethylimidazole HC4, D. Hexylureido-ethylimidazole HC6, E. Octylureido-ethylimidazole HC8, F. Bis(ethylimidazole ureido)hexyl HC6H, 11 G. S-Octylureido-ethylimidazole S-HC8 and R-octylureido-ethylimidazole R-HC8. These structures show that packing and self-assembly are dramatically different between chiral and achiral isomers, but all channels have pores with a diameter of 2.6 Å.

Figure 2 .

 2 Figure 2. Water-transport activity of I-quartet channels in liposomes. a. Representative stopped-flow light-scattering experimental traces of liposome pre-mix assays at a CLR (HC6 and S-HC8) of 1 after abrupt exposure to a hypertonic solution of 200-mM NaCl. b. The water permeabilities of the liposomes containing different I-quartet channels at CLR=1 were measured under hypertonic conditions from the premix assays. c. The net water permeability of the HC8 channels in liposomes at a CLR of 1. The liposomes (1 mg/mL, with a molar ratio of PC/PS/Chl=4/1/5) were abruptly exposed to a hypertonic solution of 200-mM NaCl or 400-mM sucrose from the pre-mix assays. The data shown in b and c are the averages of at least triplicates, and the error bars represent the standard deviation.

Figure 3 .

 3 Figure 3. Proton-transport activity of I-quartet channels in liposomes. a. Liposomes encapsulating pH-sensitive dyes (8-hydroxypyrene-1,3,6-trisulfonic acid [HPTS] and fluorescent dextran) showed fluorescence quenching because of proton export via imidazole channel and K + import via Valinomicin channel (see Supplementary materials for details). b. Ratiometric fluorescence (I/I0) curves of the assembled R-HC8 channels with different channel concentrations in the HPTS assay. The signal was normalized at t=500 s. c. Calculated values of EC50 (in µM) determined in the HPTS assay, representing the concentration at which the given compound (including racemic RacHC8) achieved half of its total transport capabilities. d. The pH inside the liposomes (pH=6.4) with the incorporated HC8 channels increased over time upon exposure to a higher pH environment (pH=7.4) in kinetic experiments. The experiments were performed on a stopped-flow instrument using a fluorescent D-3305 dextran dye as a pH sensor. Higher HC8 channel concentrations correlated with faster transport kinetics. e. Protontransport rates of different I-quartet channels calculated from the fluorescent dextran assay. The data shown are the averages of at least triplicates, and the error bars represent the standard deviation.

Figure 4 .

 4 Figure 4. MD simulations of the I-quartet channels illustrate how the hydrophobic chain length and chirality influence the structuring of the hydrophobic shells around the water-wires in these artificial channels. Data for HC8, R-HC8 and S-HC8 channels at 10 atm are shown. a. IMIDN17 atom radial distribution function for various time spans of each simulation. b. Cumulative water dipole orientation distributions within the membrane region at various time spans indicate the highest stabilization for the S-HC8 channels. c. Isosurface water density within the membrane region cumulated at the beginning (0-50 ns) and the end (400-450 ns) of the simulations of the HC8, R-HC8 and S-HC8 channels at 10 atm.

ACKNOWLEDGMENT

This work was supported by funds from ITN DYNANO, PITN-GA-2011-289033 (http://www.dynano.eu)

ASSOCIATED CONTENT

Supporting Information. Materials and Methods, Figs. S1 to S15, Tables S1 to S4, Supplimentary References. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author

* mihail-dumitru.barboiu@univ-montp2.fr

Author Contributions

The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript. / ‡ These authors contributed equally.


domain: spi phys

text:
Antoine Dufau
email: antoine.dufau@safrangroup.com

Julien Marty

Daniel Man

Estelle Piot

HIGH-FIDELITY SIMULATIONS OF THE FLOW AROUND T106C CASCADE AT LOW REYNOLDS NUMBER: THE EFFECTS OF FREESTREAM TURBULENCE AND STAGGER ANGLE

Keywords:

Flows in low-pressure turbines are highly dependent on the laminar-turbulent transition of the boundary layer and the separation of the laminar boundary layer. This study focuses on the flow around the T106C low-pressure turbine cascade blade for a low Reynolds number 100, 000 and a moderately low turbulence intensity 0.8%. For this paper, three direct numerical simulations (DNS) are carried out to understand the discrepancies observed between the simulations and the experimental data. To that end, the effect of freestream turbulence intensity and blade stagger angle are investigated. A first simulation is performed without freestream turbulence, then a second one is performed with a modified stagger angle, and finally a third one is performed with an injection of synthetic turbulence at the inlet plane based on the synthetic eddy method (SEM). The results are compared to experimental data and RANS simulations. These comparisons show that there are still discrepancies between the flow predicted by the DNS and the experimental measurements. A comparison between DNS and RANS simulations highlights some leads to better understand these discrepancies.

INTRODUCTION

Flow conditions in the low-pressure turbines (LPT) are defined by low Reynolds numbers. Consequently, the boundary layer on a LPT blade is largely laminar and so it tends to separate in the presence of high adverse pressure gradient. Moreover, the current design trend tends towards a reduction of the aircraft engines mass. As the low-pressure turbine represents about 30% of the engine mass, a solution is to reduce at much as possible the number of blades in LPT stages or the number of stages of the LPT. As a result of this trend, the aerodynamic load on each blade is increased and so the adverse pressure gradient on the suction side of the blades. Therefore, the performance of a LPT is strongly dependent on the transition of the boundary layer on the suction side of the blade.

LPT design requires an accurate tool to predict the blade loading and the losses due to the presence of laminar separation bubble. This bubble can be open, long or short, according to its shape, but also to the effect on pressure distribution over the walls. As LPT design is based on numerous daily simulations, the RANS simulations are the most commonly used. However, they require a turbulence model and also a laminar-turbulent transition model because of the low Reynolds number. One can cite the transition models of Langtry et al. [START_REF] Langtry | A Correlation-Based Transition Model Using Local Variables-Part II: Test Cases and Industrial Applications[END_REF], of Kubacki and Dick [START_REF] Kubacki | An algebraic intermittency model for bypass, separation-induced and wakeinduced transition[END_REF] or of Arnal et al. [START_REF] Arnal | Overview of laminar-turbulent transition investigations at ONERA Toulouse[END_REF]. Experiments over a flat plate or in linear cascade configurations [START_REF] Michálek | Aerodynamic Performance of a Very High Lift Low Pressure Turbine Airfoil (T106C) at Low Reynolds and High Mach Number With Effect of Free Stream Turbulence Intensity[END_REF][START_REF] Lengani | A POD-Based Procedure for the Split of Unsteady Losses of an LPT Cascade[END_REF] are a very useful tool to better understand the transition mechanism involved in LPT and to improve the modeling of the transition process [START_REF] Kubacki | An Extended Version of an Algebraic Intermittency Model for Prediction of Separation-Induced Transition at Elevated Free-Stream Turbulence Level[END_REF]. Nevertheless, the data coming from experiments can be limited. In that case, direct numerical simulations (DNS) are a powerful way to generate accurate data, even if the computational cost can be high. For example, Bernardos et al. [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles Using k-ω Turbulence Models[END_REF] used a DNS to develop and calibrate an algebraic transition model based on a non-local value.

First DNS carried out on turbine airfoil rely on a incompressible configuration, with a very low Mach number, such as the T106A. Wu and Durbin [START_REF] Wu | Evidence of longitudinal vortices evolved from distorted wakes in a turbine passage[END_REF] investigated the evolution of passing wakes in a turbine passage and their effect on separation bubble. The wakes are generated through a boundary condition. Michelassi et al. [START_REF] Michelassi | Analysis of DNS and LES of Flow in a Low Pressure Turbine Cascade with Incoming Wakes and Comparison with Experiments[END_REF] highlighted also the presence of elongated structures and showed that the wake impingement inhibits laminar separation, removing the shear layer generation and appearance of Kelvin-Helmholtz instability. This phenomena is also found in the work of Wissink [START_REF] Wissink | DNS of separating, low Reynolds number flow in a turbine cascade with incoming wakes[END_REF]. More recently, Sandberg et al. [START_REF] Sandberg | Compressible Direct Numerical Simulation of Low-Pressure Turbines-Part I: Methodology[END_REF][START_REF] Michelassi | Compressible Direct Numerical Simulation of Low-Pressure Turbines-Part II: Effect of Inflow Disturbances[END_REF] have performed compressible DNS computations for several flow conditions on T106A with and without passing wakes generated by moving bars and with different Reynolds numbers and turbulence intensities. Other studies [START_REF] Garai | DNS of Flow in a Low-Pressure Turbine Cascade Using a Discontinuous-Galerkin Spectral-Element Method[END_REF][START_REF] Alhawwary | DNS and LES of the flow over the T106C turbine using the high-order FR/CPR method[END_REF] have performed compressible DNS computations of the flow around low-pressure blades getting good accordance with the experiemental data. All these works deal with DNS carried out on a low Mach number configuration. For a higher Mach number configuration, we can cite the T106C airfoil experimentally tested at von Karman Institute [START_REF] Michálek | Aerodynamic Performance of a Very High Lift Low Pressure Turbine Airfoil (T106C) at Low Reynolds and High Mach Number With Effect of Free Stream Turbulence Intensity[END_REF]. Hillewaert et al. [START_REF] Hillewaert | Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low-Reynolds number in the T106C highlift low pressure turbine cascade[END_REF] performed simulations on this T106C blade with a discontinuous Galerkin method. Although the simulations are DNS, they showed that some discrepancies remain when the results are compared with the experimental data. More recently, one can also cite the work of Alhawwary and Wang [START_REF] Alhawwary | DNS and LES of the flow over the T106C turbine using the high-order FR/CPR method[END_REF] who performed a similar study.

This literature survey shows that a limited number of compressible DNS have been performed. Therefore, the objective of the present paper is to perform compressible DNS on the T106C blade. Three direct numerical simulations (DNS) of the flow around T106C cascade have been performed. The goal of this study is to understand the differences between the first simulation and the experimental data. These investigations highlights the effect of freestream turbulence and stagger angle on the flow around a low-pressure turbine blade. Then, the last DNS computation has been compared to a RANS simulation which is in good accordance with experimental data. This comparison leads to some new hypothesis concerning the differences between the DNS and the experiments. The test case investigated here is the T106C cascade known as a very high-lift mid-loaded low-pressure turbine blade and experimentally tested in the framework of the European Project TATMo at the von Karman Institute [START_REF] Michálek | Aerodynamic Performance of a Very High Lift Low Pressure Turbine Airfoil (T106C) at Low Reynolds and High Mach Number With Effect of Free Stream Turbulence Intensity[END_REF][START_REF]Aerodynamic Performance of a Very High Lift Low Pressure Turbine Airfoil (T106C) at Low Reynolds and High Mach Number With Effect of Free Stream Turbulence Intensity[END_REF]. The velocity peak over suction side is located at mid-curvilinear abscissa followed by a strong flow deceleration toward the trailing edge. It should be noticed that the adverse pressure gradient is stronger than the one in modern LPT. Table 1 summarizes the parameters of the T106C profile. This configuration has been tested for a several Reynolds numbers and turbulence intensities. For this study, we consider only one case with no turbulence grid and the second lowest exit isentropic Reynolds number Re 2,is = 100, 000. The natural freestream turbulence intensity (FTI) of the facility is 0.8% and the turbulent integral length scale measured is 4 mm. This value of turbulent integral length scale is stated by Michalek et al. [START_REF]Aerodynamic Performance of a Very High Lift Low Pressure Turbine Airfoil (T106C) at Low Reynolds and High Mach Number With Effect of Free Stream Turbulence Intensity[END_REF] and Hillewaert et al. [START_REF] Hillewaert | Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low-Reynolds number in the T106C highlift low pressure turbine cascade[END_REF]. The exit isentropic Reynolds number Re 2,is is based on the exit isentropic velocity V 2,is and the chord c. The exit isentropic Mach number M 2,is is based on the ratio between inlet stagnation pressure P 0,1 and exit static pressure P s,2 . The measurements have been made on the center blade of the cascade at mid-span. As the aspect ratio is high, a hypothesis of a 2D flow is made concerning the experimental measurements. We also make this hypothesis in our study while comparing simulations to the measurements.

DESCRIPTION OF T106C CASCADE

NUMERICAL METHOD Direct numerical simulations

For this study, three DNS simulations have been performed using the ONERA in-house solver FastS. The solver FastS has already been used successfully for others studies [START_REF] Dandois | Large-eddy simulation of laminar transonic buffet[END_REF][START_REF] Lugrin | Transition scenario in hypersonic axisymmetrical compression ramp flow[END_REF][START_REF] Gleize | Numerical Simulation Of NACA4412 Airfoil In Pre-stall Conditions[END_REF] and it has been created from the solver FLU3M [START_REF] Pechier | Magnus effect over finned projectiles[END_REF][START_REF] Raverdy | High-Resolution Large-Eddy Simulation of Flow Around Low-Pressure Turbine Blade[END_REF] by developping a Python interface and an optimized HPC performance. This solver is based on an hybrid MPI/OpenMP framework. The time scheme used is the second-order accurate backward scheme of Gear. An optimization of the computing time is done with a Newton algorithm with a spatially adaptive sub-iteration number [START_REF] Mary | Large eddy simulation of flow around an airfoil near stall[END_REF] in which the numerical dissipation is nullified whenever a numerical wiggle is detected. The global time step used for the three simulations is 4.3 • 10 -7 s which equals about 10 -3 convective time units τ and leads to a maximal CFL number about 25 in the smallest cells of the grid.

Two meshes were created for this study: one for each stagger angle. The mesh used for DNS n o 2 is shown in figure 1. The meshes are a combination of an O-type grid around the blade and four H-type grids. Two others H-type grid are used downstream and upstream to extrude the mesh up to the inlet and outlet planes. Moreover, two buffer zones are placed upstream (only for case without injection of turbulence) and downstream (for all cases) in order to suppress acoustic wave reflection on the inlet and outlet boundary conditions. These buffer zones are H-type grids based on an extrusion of the inlet and outlet planes with a growing cell size in x direction (expansion ratio set to 1.15). The buffer zone length equals to 1•c ax . For the design of the mesh, the non-dimensional wall cell lengths were fixed in order to respect x + < 10, y + < 2 and z + < 5. Figure 2 Moreover, the number of cells was fixed so that the expansion ratio does not exceed 1.02. The 2D mesh is extruded along z-direction with 151 point for DNS n o 1 and 301 points for DNS n o 2 and n o 3 which represents about respectively 15% and 30% of the axial chord c ax . The fig-ure 3 shows the evolution of the Pearson correlation coefficient based on velocity magnitude as a function of the span position (z) difference in the turbulent boundary layer on the suction side near the trailing edge (green point in figure 1). To that end, this coefficient is computed from the sets of velocity magnitude extracted at 30 positions that are distributed regularly along span direction. This coefficient r based on two sets of velocity magnitude U z=0 and U z ) is defined by the ratio between their covariance cov and the product of their standard deviations σ :

r U (z) = cov(U z=0 ,U z ) σ (U z=0 )σ (U z )
This observation traduces that the flow is not correlated in spanwise direction and so the span length is large enough in the present simulations. The injection of synthetic turbulence in DNS n o 3 enforces a wider span because of the high integral length scale of the turbulence. The total mesh size is about 140 millions of points for DNS n o 1 and 280 millions of points for DNS n o 2 and n o 3.

The inlet plane of the mesh is located at 1.75•c ax of the leading edge and the outlet plane is located at 2. • c ax of the trailing edge. The inlet boundary condition is a subsonic inflow condition for which five variables are specified: stagnation pressure, stagnation enthalpy and the three components of velocity direction unit vector. The outlet boundary condition is a subsonic outflow condition for which static pressure is specified. As we make the hypothesis of a 2D cascade flow, periodic conditions are used in y and z directions. An adiabatic no-slip condition is used at wall.

Periodicity with Synthetic Eddy Method

To be closer to the experimental conditions, we have chosen to inject turbulence at the inlet condition of DNS n o 3. For that purpose, the method introduced by Jarrin et al. [START_REF] Jarrin | A synthetic-eddy-method for generating inflow conditions for large-eddy simulations[END_REF] has been developed in a ONERA in-house tool [START_REF] Gand | Zonal detached eddy simulation (zdes) using turbulent inflow and high order schemes: Application to jet flows[END_REF][START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (zdes) and synthetic turbulent inflow[END_REF]. The main idea of this synthetic eddy method (SEM) is the generation of an inlet map of velocity fluctuations with the contribution of several synthetic eddies located in a box that surrounds the inlet condition. Theses contributions are then summed to a prescribed mean field. Each eddy is defined by its position and three others others variables ε k that defines the sign of the contribution on each axis-component of the velocity field. In this study, Gauss shape functions are used for the contributions of the eddies and all the eddies have the same length scale σ equals to the experiment integral length scale (L Tu = 4mm). At each time step, the position of the eddies is updated by using the Taylor's frozen hypothesis and the mean velocity field.

In the original formulation, the size of the box is based on the size of the inlet plane to which the length scale of the eddies σ is added in each direction in order to obtain a uniform Non-dimensional cell length (in wall unit)

x + y + z + FIGURE 2: Non-dimensional wall units computed from DNS n o 1 results map of velocity fluctuations. Moreover, when an eddy crosses one of the boundaries of the box, it is removed and a new eddy is created at a random position on the opposite side of the box and with random contribution signs ε k . To take account of periodicity and generate a periodic inlet map, we have modified this formulation. Our method is similar to the one proposed by Muller-Schindewolffs and Herbst [START_REF]The Effect of Turbulent Scales on Low-Pressure Turbine Aerodynamics: Part A -An Optimized Turbulent Boundary-Condition[END_REF]. First, the size of the box of eddies was reduced to the exact size of the inlet plane in the two periodic directions. Then, at each generation of inlet map, the box of eddies is duplicated and translated 8 times in order to surround the inlet condition as presented in figure 4. Conse- quently, this technique is directly using the periodic hypothesis to fill the gap at the borders of the box created by the reduction of the size of the box. To fully respect the periodic hypothesis, an other modification has been made on the convection of the eddies. When a eddy crosses a periodic box boundary, it is translated at the same position at the opposite box boundary and it keeps the same contribution signs ε k . To sum up, these two modifications are still following the initial concept while respecting the periodic hypotheses of our configuration. Figure 5 is showing contours of fluctuating x-component velocity of an inlet map generated with our modified SEM to take into account the periodic conditions. The map has been duplicated 8 times to assess its periodicity. x /U 1 of the periodic inlet map generated with Synthetic Eddy Method -duplicated 8 times

RANS simulations

The RANS simulation presented in this study have been carried out with ONERA solver elsA [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF][START_REF] Plot | The high level of maturity of the elsa cfd software for turbomachinery aerodynamics applications[END_REF] co-owned by Airbus, Safran and ONERA.The second order in space upwind scheme of Roe, an implicit time scheme and a scalar LUSSOR implicit method are used. A local time step is used by specifying a CFL number increasing from 1 to 10 in the 1000 first iterations of the simulations. The simulation have been performed during 40 000 iterations leading to a reduction of residuals by more than 5 orders of magnitude. As in [START_REF] Dufau | Simulation of Passing Wakes Inducing Unsteady Boundary Layer Transition Around Low-Pressure Turbine Blade[END_REF], the turbulence is modeled by Wilcox's kω turbulence model [START_REF] Wilcox | Reassessment of the scaledetermining equation for advanced turbulence models[END_REF] and the transiton is modeled with the Menter-Langtry model [START_REF] Langtry | A Correlation-Based Transition Model Using Local Variables-Part II: Test Cases and Industrial Applications[END_REF][START_REF] Menter | A Correlation-Based Transition Model Using Local Variables-Part I: Model Formulation[END_REF] which is composed of two transport equations: one for the numerical intermittency γ and one for the transition Reynolds number Re θ . For the turbulence model, the production term of kinetic energy is based on vorticity to avoid leading edge anomaly. In the present study with Re 2,is = 100, 000, the inlet turbulence intensity Tu and the inlet eddy viscosity ratio µ t /µ were fixed to, respectively, 0.95% and 10 [32] in order to both obtain a turbulence intensity Tu LE of around 0.8% at leading edge and follow the turbulence intensity decrease that was measured in the experiments.

RESULTS

Direct numerical simulations

As this study aims at identifying the numerical parameters necessary to get a good agreement with experimental data, the first DNS computation is carried out without any stagger angle correction, without any synthetic turbulence and with a span of 15% of the axial chord as indicated in table 2. Based on previous RANS computations [START_REF] Minot | Advanced numerical setup for separation-induced transition on high-lift low-pressure turbine flows using the γ-r θ t˜model[END_REF], the inflow angle is set to 35°instead of the experimental inlet angle of 32.7°. In comparison, after a preliminary analysis, Hillewaert et al. [START_REF] Hillewaert | Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low-Reynolds number in the T106C highlift low pressure turbine cascade[END_REF] set the inlet angle to 34.7°. The statistical convergence can be assessed with the figure 6 in which are plotted the cumulative mean value of the axial velocity and the cumulative RMS value of the fluctuations of axial velocity computed from a signal recorded by a gauge located in the wake, near the blade trailing edge. The quantities are non-dimensionalized by their value computed from the all signal.Considering the time step, the axial chord and the velocity scale, the transient ends after 100 convective time units τ and the time-average is then performed over 350 convective time units τ for all three DNS that are presented here. The convective time unit τ is assessed from the blade chord c and the average outflow velocity magnitude U To have an idea of the quality of the inlet SEM boundary condition, one can observe in figure 7 the power spectral density of y-component of velocity upstream of the leading edge (red position in figure 1). The slope is a bit lower than the classical spectrum slope -5/3 based on Kolmogorov's theory for isotropic turbulence. Moreover, the decrease of turbulence intensity from the inlet injection plane to the leading edge is plotted in figure 8. The turbulence intensity at the leading edge is slightly higher than the experimental one equal to 0.8%. Also, the power spectral density of y-component upstream in the blade wake (green position in figure 1) can be observed in figure 9. At this point, the spectrum slope is close to the classical slope -5/3 for isotropic turbulence.

Figure 10 shows the time-averaged isentropic Mach number over the blade, for both DNS and experimental data. The comparison between the first simulation and experiments highlights a mismatch in terms of pressure gradient in the laminar part of the boundary layer over suction side of the blade. While the isentropic Mach number distributions are similar close to the leading edge, the slope differs from 5% of axial chord up to 20%. Thus, the agreement in terms of isentropic Mach number distribution is not as good as expected. Nevertheless, the peak of isentropic Mach number is close to the experimental one, suggesting that the discrepancies between the simulation and the experiments could came from a wrong incidence angle and not from a wrong prediction of the flow physics.

Considering the observations about DNS n o 1 and the idea of Hillewaert et al. [START_REF] Hillewaert | Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low-Reynolds number in the T106C highlift low pressure turbine cascade[END_REF], we have chosen to increase the stagger angle of the T106C blade by 1 degree (configuration DNS n o 2). Thanks to this slight modification, the evolution of isentropic Mach number over the first half of the suction side is in good agreement with experiments. Except in the first ten percents of the axial chord, the distribution of the isentropic Mach number over the pressure side is not influenced by the stagger angle nor by the freestream turbulence. Nonetheless, separation and reattachment points over suction side are not in accordance with the experiments yet. Indeed, the DNS n o 2 gives similar results to DNS n o 1. Consequently, the mismatch on the isentropic Mach number in the vicinity of the laminar separation bubble may not be a consequence of the mismatch on favorable pressure gradient in the laminar part. However, another parameter could explain discrepancies with experimental data : the freestream turbulence. In experiments, the freestream turbulence intensity without turbulence grid is close to 0.8% while the current simulations are carried-out without any synthetic turbulence. Therefore, a third DNS is conducted by introducing synthetic turbulence following the methodology previously described. The discrepancies appear only downstream the peak of isentropic Mach number so the freestream turbulence does not have a significant influence on the laminar boundary layer. As expected, the separation and reattachment points are, respectively, moved downstream and upstream in comparison with the DNS n o 2 performed without freestream turbulence. Thus, the length of the laminar separation bubble is reduced. However, the pressure recovery at the end of the separation bubble, i.e., the decrease of isentropic Mach number, is still very prompt compared to the experimental data. Moreover, the isentropic Mach number at trailing edge suggests that the laminar separation bubble is open in the experiments but it is closed in all three DNS. These aspects will be further discussed. Given these observations, the hypothesis of three-dimensional effects in the experimental data could explain the difficulty to reproduce the measurements with the simulations.

The validation of DNS is also carried out through the comparison of the kinetic energy loss profile between the experiments and the DNS. These profiles are plotted in figure 11. As in the experiments, the pitch is nondimensionalized by the distance between the two blades 0.08836m and its y-origin is located at 0.247 axial chord from the y-position of the leading edge. Despite the mismatch in terms of isentropic Mach number over the first part of the suction side, the profile predicted by the DNS is close to the experimental one. However, the loss is slightly underestimated and the profile is too thin. The increase of the stagger angle by one degree, thickens the profile, but the loss peak is still slightly underestimated, which is coherent with the prediction of a close bubble. To take into account both the modified trailing edge position and the modified outlet angle, the profile must be shifted by 0.055 in order to keep the relative position to the trailing edge in the experiments. These corrected profiles are also plotted on figure 11. Thus, the wake profile is rather in good agreement with experiments. The adding of freestream turbulence is responsible for a shorter bubble and thus, for a thinner profile characterized by a lower loss peak.

Besides the validation of the DNS by comparison to timeaveraged experimental data, the numerical simulations must be verified in terms of flow physics and behavior of boundary conditions. First, figure 12 depicts the nondimensionalized spanwise component (Z-axis) of the instantaneous velocity around the T106C blade. As the time-averaged value of this component must be equal to zero, the instantaneous value is representative of the fluctuations and can be used to investigate the behavior of the synthetic eddy method. In the present case, the size of coherent structures injected at the inlet is close to the integral thickness used in SEM (L Tu = 4mm) and the evolution of these structures in the blade passage is coherent with the literature: bowing, reorientation, elongation and stretching due to the difference of velocity field between suction and pressure sides [START_REF] Wu | Evidence of longitudinal vortices evolved from distorted wakes in a turbine passage[END_REF][START_REF] Stieger | The Unsteady Development of a Turbulent Wake Through a Downstream Low-Pressure Turbine Blade Passage[END_REF]. The destabilization of the shear layer resulting from the laminar boundary layer separation is also well visible. The Kelvin-Helmholtz instability causes the structure breakdown into turbulence allowing a possible boundary layer reattachment and pressure recovery. More downstream, once the mesh stretching starts, the coherent structures are progressively dissipated. Figure 13 shows the contours of density gradient magnitude. Not only this confirms the prediction of the role of Kelvin-Helmholtz instability in the laminar-turbulent transition onset, but this also show the dissipation of coherent structures in the stretching area. This confirms the good behavior of boundary conditions used in this study.

Comparison to RANS computations

In order to better understand the discrepancies between DNS and experimental data, RANS computations are carried out with the same operating conditions, without (RANS β = 0 o ) and with (RANS β = 1 o ) the stagger angle modification.

The simulation RANS β = 0 o is the one that will be compared to the experiments and the DNS simulations in the following part because of its good match with the experimental isentropic mach distribution. However, simulation RANS β = 1 o is useful to observe a noticeable discrepancy between flows pre- Figure 14 shows the distribution of isentropic Mach number over the blade wall. The laminar part of the boundary layer predicted by RANS β = 0 o computation is in good accordance with the experiment. Nonetheless, as the stagger angle is not modified, this result is quite surprising and will be discussed later. However, concerning the second part of the suction side, the agreement with experimental data is slightly better for the β = 0 o computation than the DNS computations. Indeed, the analysis of the isentropic Mach number distributions suggests that the RANS β = 0 o computation predicts an open bubble with a slow pressure recovery. On the contrary, the DNS predicts a closed and long laminar separation bubble with a harsh pressure recovery; this is consistent with the results obtained by Hillewaert et al. [START_REF] Hillewaert | Assessment of a high-order discontinuous Galerkin method for the direct numerical simulation of transition at low-Reynolds number in the T106C highlift low pressure turbine cascade[END_REF]. In the experiments, the pressure recovery is stronger than in the RANS β = 0 o results but smoother than in the DNS results. Moreover, the value of the experimental isentropic Mach number close to the trailing edge suggests an open bubble. As Reynolds number equal to 100 000 is the experimental limit between open and closed bubble, this could explain the discrepancies observed between the numerical and experimental data.

Figure 15 shows the kinetic energy loss profile along the pitch axis, for the experiments, the RANS β = 0 o computation and the DNS n o 3. The DNS underestimates the loss while the RANS β = 0 o overestimates it. This is coherent with the fact that the bubble is open for RANS β = 0 o and closed for DNS. However, the location of the loss peak predicted by DNS is in good agreement with the experimental one, around 0.45 pitchwise while the RANS simulation predicts a peak around 0.55 pitchwise.

Figure 16 depicts the iso-contours of turbulent kinetic energy k over the suction, in the boundary layer frame, and shows the zero isoline of tangential velocity u // . Above this isoline, the flow goes from leading edge to trailing edge while below, the and a smaller area with strong turbulence production for DNS. This is consistent with the results of Bouchard et al. [START_REF] Bouchard | Validation of correlations-based transition modeling strategies applied to the spalart-allmaras turbulence model for the computation of separation-induced transition[END_REF], who showed that separation bubbles predicted by RANS simulations are both too thick and too long. This is coherent with observations made on isentropic Mach number distributions. It can explain that RANS over-predicts the opening of the laminar separation bubble. Regarding the DNS results, the location and the topology of the high turbulent kinetic energy zone are also consistent with the work of Bouchard et al.. It should be noticed that, despite the synthetic turbulence at the inlet, the level is sufficiently small so that it is not visible in the bottom figure. The modification of the stagger angle influences slightly the level of turbulent kinetic energy: the peak is located downstream and the level is a little bit higher. The time-averaged flow of the DNS and the steady RANS simulations have been post-processed in order to compute the integral thicknesses, the shape factor and the skin friction. The evolution of these quantities accordingly to the axial location normalized by the axial chord are plotted in figure 17. It should be noticed that the modification of axial chord due to the increase of stagger angle is taken into account. After a strong rise around the leading edge, the boundary layer thickness evolves smoothly and there is no significant discrepancy between RANS and DNS, and between the three DNS (see figure 17a). This is also visible on the momentum thickness evolution (see figure 17a). Nevertheless, two families of curves are visible on the shape factor distribution. In the first part of the boundary layer development, the shape factor of RANS computation is similar to the DNS n o 1 without the stagger angle change and without freestream turbulence. As the freestream turbulence intensity is quite small, this can be explained by the insensitivity of the RANS model to

CONCLUSIONS

In this study, three DNS simulations of the T106C blade have been performed in an iterative way. A first one with the same geometry as the experiments and no freestream turbulence, a second one with a modified stagger angle and a third one with an injection of turbulence. For this third simulation, a modification of the synthetic eddy method has been introduced in order to take into account periodic conditions while generating synthetic turbulence. Modifying the blade stagger angle to match the laminar part of the experimental isentropic Mach number was successful. However, the prediction of the separation bubble by the third DNS with a modified stagger angle and an injection of freestream turbulence is still not in accordance with the open bubble observed in the experiments. Nevertheless, in the experiments, the exit isentropic Reynolds number (100,000) has been highlighted as the limit between the long bubble case and the open bubble case. A way to investigate the remain discrepancies between the experiments and the DNS computation could be to perform a fourth DNS computation at a slightly smaller exit isentropic Reynolds number such as 95,000. One could expect that such a DNS simulation predicts an open bubble similar to the one observed in the experiments.

In the last part of this study, we have made a comparison between the third DNS simulation and a RANS simulation which is in good accordance with the experimental data concerning the isentropic Mach number distribution. The observations made from this comparison are in agreement with the hypothesis of a too close bubble predicted by the DNS simulation compared to the experimental measurements and the RANS results. Moreover, this comparison highlights the over-prediction of the opening of the bubble by the RANS simulation. This could be explained by a too weak production of turbulent kinetic energy in the upper and read part of the bubble.

As future works, DNS with a higher freestream turbulence intensity and a higher Reynolds number used in experiments will be performed in order to investigate the effect of these parameters on the laminar separation bubble. Two other aspects can also be combined: the effect of passing wakes generated by bars and compressibility effects as future low-pressure turbines will rotate at higher speed.

ACKNOWLEDGMENT

All simulations have been carried out with elsA software co-owned by AIRBUS, SAFRAN, and ONERA. The authors wish to thank the Von Karman Institute who provided experimental results reported in this study in the framework of the European research programs TATMo (www.tatmo.eu) and UTAT (Unsteady Transition in Axial Turbines) in which ONERA was also involved. This work was partially granted access to the HPC resources of TGCC Joliot-Curie/Irene and GENCI (allocation A0112A13039).

FIGURE 1 :

 1 FIGURE 1: Mesh for DNS n o 2 (without SEM) with the buffer zones in red -1 point over 15 in each direction

FIGURE 3 :

 3 FIGURE 3: Pearson correlation coefficient on Velocity Magnitude r U between 30 points along span direction in the wake blade -DNS n°3

FIGURE 4 :

 4 FIGURE 4: Virtual duplication of the box of eddies for periodic SEM

FIGURE 5 :

 5 FIGURE 5: Fluctuations of x-component velocity u ′ x /U 1 of the periodic inlet map generated with Synthetic Eddy Method -duplicated 8 times

2 .

 2 DNS β ( o ) Tu (%) ∆Z (%c ax )

FIGURE 6 :

 6 FIGURE 6: Evolution of u x at one probe in the wake near the traling edge. Cumulative average value u x and RMS value u ′ x

FIGURE 7 :

 7 FIGURE 7: Power spectral density of y-component of velocity upstream of the leading edge -DNS n°3

FIGURE 8 :

 8 FIGURE 8: Evolution of turbulence intensity Tu from the inlet plane to the leading edge -DNS n°3

FIGURE 9 :

 9 FIGURE 9: Power spectral density of y-component of velocity in the blade wake -DNS n°3

FIGURE 10 :

 10 FIGURE 10: Isentropic Mach number distribution around T106C blade from experiment and DNS simulations. Experimental uncertainties are plotted for suction side

  β = 0° Tu LE = 0 DNS 2β = 1° Tu LE = 0 DNS 2β = 1° Tu LE = 0 -corrected pitch DNS 3β = 1° Tu LE = 0.8 DNS 3β = 1° Tu LE = 0.8 -corrected pitch

FIGURE 11 :

 11 FIGURE 11: Kinetic energy loss in the wake, for experiments and DNS simulations

FIGURE 12 :

 12 FIGURE 12: Instantaneous iso-contours of nondimensionalized z component of velocity u ′ z /U 2 for simulation DNS n o 3

FIGURE 13 :

 13 FIGURE 13: Instantaneous iso-contours of || ⃗ ∇ρ|| • c/ρ for simulation DNS n o 3

FIGURE 14 :FIGURE 15 :

 1415 FIGURE 14: Comparison of isentropic Mach number distribution around T106C blade from between RANS and DNS n o 3 simulations. Experimental uncertainties are plotted for suction side

FIGURE 16 :

 16 FIGURE 16: Iso-contours of turbulent kinetic energy k on the suction side. Isoline of tangential velocity u //

FIGURE 17 :

 17 FIGURE 17: Integral thicknesses, shape factor and skin friction over the upstream part of the suction side for RANS and DNS computations

TABLE 1 :

 1 T106C characteristics and flow conditions

TABLE 2 :

 2 List of Direct Numerical Simulations