T Jaumier

S Vincent

L Vincent

R Desmorat

Creep and damage anisotropies of 9%Cr and 14%Cr ODS steel cladding

Two Oxide Dispersion Strengthened (ODS) steel tubes, obtained by means of hot extrusion and cold rolling, are studied as potential materials for cladding applications in sodium-cooled fast reactors. A 9%Cr martensitic tube (Fe-9Cr-1W-0.3Ti-0.25Y2O3) with isotropic microstructure is compared to a 14%Cr ferritic tube (Fe-14Cr-1W-0.3Ti-0.25Y2O3) with elongated grains and a strong texture. Tile and notched ring specimens are used to perform uniaxial creep tests at 650°C in the longitudinal and transverse directions of the tubes. The 14%Cr ODS steel tube exhibits strong mechanical anisotropies, while the 9%Cr presents only a slight anisotropy in creep resistance, presumably due to the influence of specimen geometry on mechanical fields. Both intergranular and ductile zones have been observed on the fracture surfaces of the uniaxial creep specimens, suggesting the competition of two damage mechanisms. Damage for both tubes appeared anisotropic, with orientated cracks in the rolling directions on the ring specimens and the 14%Cr tile specimens. Internal pressure creep tests also performed at 650°C show complementary results for both tubes in agreement with ring specimens' ones, validating their use for hoop properties assessment. Overall, while the 14%Cr ODS steel tube is strong in the longitudinal direction, the 9%Cr tube proves as resistant in the transverse direction.

Introduction

The development of advanced nuclear systems such as Gen. IV sodium cooled fast reactors relies upon the availability of reliable structural materials. Oxide Dispersion Strengthened (ODS) steels are suitable candidates for fuel cladding applications due to low irradiation induced swelling [START_REF] Alamo | Assessment of ODS-14%Cr ferritic alloy for high temperature applications[END_REF][START_REF] Kim | Defect and void evolution in oxide dispersion strengthened ferritic steels under 3.2 MeV Fe+ ion irradiation with simultaneous helium injection[END_REF], good creep strength [START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF][START_REF] Ukai | Alloying design of oxide dispersion strengthened ferritic steel for long life FBRs core materials[END_REF][START_REF] Ukai | Development of Oxide Dispersion Strengthened Ferritic Steels for FBR Core Application, (I)[END_REF][START_REF] Ratti | Développement de nouvelles nuances d'aciers ferritiques-martensitiques pour le gainage d'éléments combustibles des réacteurs à neutrons rapides au sodium[END_REF] and corrosion resistance [START_REF] Yvon | Structural materials challenges for advanced reactor systems[END_REF].

ODS steel tubes are mainly obtained by mechanical alloying steel with yttria powder and consolidating the result by hot extrusion, followed by cold-pilgering punctuated by intermediate heat treatments to allow a reduction of the material hardness [START_REF] Toualbi | Assessment of a new fabrication route for Fe-9Cr-1W {ODS} cladding tubes[END_REF][START_REF] Dubuisson | {ODS} Ferritic/martensitic alloys for Sodium Fast Reactor fuel pin cladding[END_REF][START_REF] Ukai | Perspective of ODS alloys application in nuclear environments[END_REF]. These different steps and the chemical composition of the material, in particular the chromium content, determine the resulting microstructure. For alloys with chromium content below 12% Cr, there is a phase transformation above 850°C and isotropic martensitic or ferritic microstructure can be obtained after the phase transformation. For alloys with chromium content above 13% Cr, there is no phase transformation; only ferritic microstructures can be obtained [START_REF] Ukai | 10 -Oxide dispersionstrengthened/ferrite-martensite steels as core materials for Generation {IV} nuclear reactors[END_REF]. Furthermore, abnormal grain growth leading to reduced mechanical properties at recrystallization temperatures (1400°C) for high chromium ferritic ODS steels makes such treatments difficult to control [START_REF] Alamo | Assessment of ODS-14%Cr ferritic alloy for high temperature applications[END_REF][START_REF] Sallez | Recrystallization, abnormal grain growth and ultrafine microstructure of ODS ferritic steels[END_REF]. The resulting anisotropic microstructure with elongated grains in the rolling direction and a strong texture lead to anisotropic mechanical behaviour for high chromium ODS steels [START_REF] Ukai | Development of Oxide Dispersion Strengthened Ferritic Steels for FBR Core Application, (I)[END_REF][START_REF] Li | Effects of fabrication processing on the microstructure and mechanical properties of oxide dispersion strengthening steels[END_REF][START_REF] Ukai | Development of Oxide Dispersion Strengthened Steels for FBR Core Application[END_REF][START_REF] Ukai | Tube manufacturing and mechanical properties of oxide dispersion strengthened ferritic steel[END_REF], with poor transverse properties compared to longitudinal ones. This is a concern for the cladding applications as the main solicitation is internal pressure creep due to fission gas building up inside the tube, loading it mainly in the hoop direction [START_REF] Michel | A new phenomenological criterion for pelletcladding interaction rupture[END_REF]. However, higher chromium content has been shown to improve corrosion resistance in ODS steels [START_REF] Tanno | Corrosion behavior of ODS steels with several chromium contents in hot nitric acid solutions[END_REF][START_REF] Kimura | Development of Al added high-Cr ODS steels for fuel cladding of next generation nuclear systems[END_REF][START_REF] Cho | Corrosion properties of oxide dispersion strengthened steels in super-critical water environment[END_REF], another significant aspect for cladding applications.

Fewer studies [START_REF] Ukai | Characterization of high temperature creep properties in recrystallized 12Cr-ODS ferritic steel claddings[END_REF][START_REF] Toualbi | Relationships between mechanical behavior and microstructural evolutions in Fe 9Cr-ODS during the fabrication route of {SFR} cladding tubes[END_REF] have been conducted on the mechanical properties of ODS steel tubes than on bulk material, owing in part to the difficulties or the costs surrounding the characterization of the hoop properties on a tube. However, they are necessary to study the material in its final form, having seen a different fabrication route than the bulk material; and to obtain crucial data on the creep lifetime of ODS steel tubes under thermomechanical loading.

In the present paper, results of creep tests performed on the axial and hoop direction of a 9%Cr and a 14%Cr ODS steel tubes are presented. The objective of this study is to evaluate the level of anisotropy in the mechanical behaviour and creep resistance of the tubes and to eventually determine which batch might be the most relevant in the context of cladding applications.

A presentation of the different materials and characterization methods is first proposed, followed by a section in which experimental results, both on microstructure and creep tests, are exposed. A discussion of the results is then presented just before concluding the paper.

Materials & methods

Fabrication route and chemical composition

Metallic powders and yttria powder are mechanically alloyed under hydrogen atmosphere by Plansee, with the nominal chemical compositions: Fe-9Cr-1W-0.3Ti-0.25Y2O3 and Fe-14Cr-1W-0.3Ti-0.25Y2O3. The milled powder is then degassed in a vacuum chamber at 400°C for 2h. Sealed in a soft steel can, the powder is hot extruded into a rod at 1100°C. After drilling and straightening, the raw tube is manufactured into thin cladding tube using cold rolling passes in a HPTR mill, punctuated by intermediate heat treatments to reduce internal stresses. For the 9%Cr ODS steel tube, a austenitizing heat treatment at 1050°C followed by helium quenching (3°C/s) is performed to achieve a martensitic state. Then, a final tempering treatment is performed at 750°C for 30 minutes. The chemical compositions of the final steel tubes (external diameter 10.73mm, thickness 0.5mm, length 2m), as analyzed by Bureau Veritas, are shown in Table 1. Abnormally high Ti apparent content is under investigation.

Microstructural Characterization and fractography

Specimens of the 9%Cr and 14%Cr ODS steel tubes have been prepared to allow the cross-section observation of the microstructure of the materials in both the longitudinal and transverse orientations. These specimens were first mechanically polished and then submitted to electropolishing. Various methods were used to analyze them: electron backscatter diffraction (EBSD) analysis, electron probe micro-analysis (EPMA) and energy dispersive X-ray spectrometry (EDS). Hardness measurements were also conducted on long/cross-sections of the tubes.

Mechanical Characterization

Uniaxial and internal pressure creep tests were carried out. Two types of specimen were used to assess the mechanical properties in the longitudinal (rolling) and circumferential directions, respectively tile and ring specimens. The geometry and specifications of these specimens are shown in Figure 1, as well as the loading conditions for the ring uniaxial creep tests. Circumferential creep tests were also performed by directly imposing constant pressure in tube specimens. The configuration of these internal pressure creep specimens with the holding arrangement used is also presented in Figure 1(c). The specimen is 280mm long, with 10.73mm outer diameter and a wall thickness of 0.5mm. Water flow in the fixtures (welded to the tube) manages the cooling of the specimen prior to its connection to the machine. The welds are located outside the furnace. A mandrel inside the tube reduces the amount of argon gas necessary to maintain high pressure. An argon gas flow around the specimen is also used to reduce external oxidation. Uniaxial creep tests were performed at 650°C (±5). Usual precautions (Teflon coatings, molybdenum disulfide [START_REF] Link | Failure of Zircaloy cladding under transverse planestrain deformation[END_REF][START_REF] Daum | Mechanical Property Testing of Irradiated Zircaloy Cladding Under Reactor Transient Conditions, Small Specimen Test Techniques: Fourth Volume[END_REF]) to limit friction during the ring creep test were prohibited by the high temperature investigated. Internal pressure creep tests have been carried out on both materials at 650°C. Even if both ring and pressure creep specimens are classically used to evaluate the hoop properties of a tube, here they are both tested. Ring specimens, although cheaper in use, suffer in our case of the aforementioned missing precautions, leading to uncertain stress/strain distributions. The data from the ring tests will be presented as is, meaning the strain is obtained by the simple ration of the crosshead displacement (as measured by a LVDT) over the gauge length. Comparisons between the two geometries will tell if the data from the ring tests is relevant, analyzed as well by FE simulations (see appendix). Finally, pressure creep has additional value for its representativity of the cladding thermomechanical loadings.

For these comparisons, the mean hoop stress in the tube has to be evaluated. The method used is based on [START_REF] Onimus | Strain-Path Change Tests and Physically Based Polycrystalline Modeling of the Behavior of Recrystallized Zirconium Alloys[END_REF]. The external pressure being negligible compared with the inner pressure, the mean stress tensor in the tube is expressed as:

𝝈 = 𝑃𝛷 𝑚 2𝑡 ( -𝜂 -𝜂 2 0 0 0 1 -𝜂 0 0 0 1 2 -𝜂 + 𝜂 2 ) ( 1
)
where P is the internal pressure, Φm the mean diameter of the tube, t the thickness and 𝜂 = 𝑡 𝛷 𝑚 ⁄ a geometrical parameter. A second-order approximation in 𝜂 (as opposed to a firstorder approximation, which correspond to the thin wall formula) gives the evaluation for the mean hoop stress in the wall of the tube:

𝜎 𝜃𝜃 = 𝑃𝛷 𝑚 2𝑡 (1 -𝜂) = 9.73 𝑃 (2)
The mean hoop strain is obtained by dividing the evolution of the external diameter 𝛷 𝑒𝑥𝑡 by the initial mean diameter.

𝜀 𝜃𝜃 = ∆𝛷 𝑚 𝛷 𝑚 (0) ≈ ∆𝛷 𝑒𝑥𝑡 𝛷 𝑒𝑥𝑡 (0) -𝑡 (3)
3. Results

Microstructure of the studied 9%Cr and 14%Cr ODS steel tubes

The results of the EBSD analysis are shown in Figure 2. The 9%Cr ODS steel tube exhibits grains of equiaxed shape, the diameter of which is around 5 micrometers, as shown on both the longitudinal section (Figure 2 (c)) and the cross section (Figure 2 (d)) of the tube. RD corresponds to the rolling direction, TD being the transverse direction and ND the corresponding normal direction. The pole figures and orientation maps of both sections also suggest an almost isotropic texture, but with a slight orientation of the <110> direction parallel to RD and the <111> direction to ND in the (RD-TD) plane. This type of microstructure can be seen on other martensitic ODS steels obtained at CEA [START_REF] Toualbi | Relationships between mechanical behavior and microstructural evolutions in Fe 9Cr-ODS during the fabrication route of {SFR} cladding tubes[END_REF][START_REF] Praud | Plasticité d'alliages renforcés par nano-précipitation[END_REF]. Hardness measurements on long and cross sections of the tube give a mean value of 340HV (±12) with no significant differences between samples.

The 14%Cr ODS steel tube exhibits elongated grains in the rolling direction. These grains have an average diameter around 500 nanometers and an average length of 350 micrometers. They show a strong α texture, as the <110> axes of the grains are mostly aligned with RD and the <111> axes with TD, as seen in Figure 2 (a) and (b). Hardness measurements give a mean value of 372HV (±26), again with no significant differences between long and transverse samples. Other studies focus on particle size, composition and distribution in comparable ODS steels [START_REF] Ukai | Alloying design of oxide dispersion strengthened ferritic steel for long life FBRs core materials[END_REF][START_REF] Sakasegawa | Correlation between chemical composition and size of very small oxide particles in the MA957 ODS ferritic alloy[END_REF][START_REF] Brocq | Nanoscale characterization and formation mechanism of nanoclusters in an ODS steel elaborated by reactive-inspired ball-milling and annealing[END_REF][START_REF] Ratti | Influence of titanium on nano-cluster (Y, Ti, O) stability in ODS ferritic materials[END_REF][START_REF] Alinger | On the role of alloy composition and processing parameters in nanocluster formation and dispersion strengthening in nanostuctured ferritic alloys[END_REF]. Here, the only relevant information pertaining to potential anisotropies is the fact that titanium rich particles tend to gather and align along the grains of the 14%Cr steel tube in the rolling direction, as seen with the dark dots in Figure 2. EPMA also reveals this feature, as seen in Figure 3. This is a known feature of ferritic ODS steels [START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF][START_REF] Ukai | Tube manufacturing and mechanical properties of oxide dispersion strengthened ferritic steel[END_REF][START_REF] Steckmeyer | Experimental study and modelling of the high temperature mechanical behaviour of oxide dispersion strengthened ferritic steels[END_REF][START_REF] Klueh | Tensile and creep properties of an oxide dispersion-strengthened ferritic steel[END_REF] and is also observed on 11%Cr martensitic ODS steel with residual ferrite [START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF].

Creep tests

Creep tests have been performed at 650°C (±5) for both ODS steel tubes. Table 2 summarizes the creep properties of these alloys regarding lifetime, final strain and minimal strain rate (as a clear steady state cannot be observed for several tests). Pressure creep tests have been analyzed according to equations ( 2) and ( 3). The data suggests a strong anisotropy between the longitudinal and the transverse direction for the ferritic 14%Cr ODS steel tube concerning creep strength, as also seen in Figure 4. For similar lifetimes, the tile specimens withstand a 150 MPa higher stress. Concerning the 9%Cr ODS steel tube, creep resistance seems less affected by the stress direction, but still with lifetimes for the tile specimens ten to a hundred times longer than that of the ring specimens for similar applied stresses. The results of the internal pressure creep tests are also presented in Figure 4, compared with the uniaxial creep results obtained on the ring specimens. There are obvious similarities between internal pressure and ring creep specimens when looking at creep lifetimes. The results seem to follow the same trend, except the 14%Cr ring at 120MPa again. This ongoing test will be investigated after specimen failure to see if the guiding rods between the crossheads did not generate unwanted friction and thus alter the overall results.

Looking now at creep behaviour of these materials, the minimal strain rate data is plotted in Figure 5. Once again, a significant difference can be observed with the minimal strain rate of ring and tile creep specimens for the 14%Cr alloy. A much higher stress is needed in the rolling direction to obtain the same strain rate as in the transverse direction, and the apparent Norton power-law exponent is higher (n~125) in the long direction than in the transverse direction with ring specimens (n~35). Concerning the 9%Cr ODS steel tube, the data seems to align reasonably well for both directions, with an apparent stress exponent (n<15 and similar in both directions) lower than the one of 14%Cr tube. More similarities between ring and pressure creep data is observed in creep behaviour for both the 9%Cr and 14%Cr steel tubes, with results seemingly aligning, except the 14%Cr ring at 120MPa. Remarkable creep features of ODS steels are low final strains and little or no tertiary creep, especially at low stresses [START_REF] Wilshire | Deformation and damage processes during creep of Incoloy MA957[END_REF][START_REF] Ukai | Hightemperature strength characterization of advanced 9Cr-ODS ferritic steels[END_REF][START_REF] Sakasegawa | Creep constitutive equation of dual phase 9Cr-ODS steel[END_REF][START_REF] Malaplate | Creep behavior of ODS materials: A study of dislocations/precipitates interactions[END_REF]. This is observed here as well in Figure 6 with the creep curves for 9%Cr ODS steel tile specimens. This is particularly true for the pressure creep specimens (see Figure 7), with an apparent lack of tertiary creep stage, during which the strain rate would increase after the traditional steady-state creep. This usually manifests on tubes as a local bulging followed by burst. Here a single crack appears, a few millimeters in length, without significant local diameter change for the tube (total strain at failure around 1%), as seen in Figure 8. However, when comparing the evolution of the external diameter along the tube measured before and after testing, the deformation appears inhomogeneous, as shown in Figure 8a. This is due to temperature inhomogeneity along the tube, creating potential sites for strain localization leading to failure. The water-cooled fixtures on both ends of the tube lowers the temperature in the tube's extremities, where no creep occurs (see Figure 8a). Inside the furnace, temperature is controlled with three thermocouples centered with the three heating elements, yet temperature variations may occur between them. This can explains the lack of apparent tertiary creep stage, since the extensometer is located at the center of the tube specimen, away from the localized deformation that evidently occurs at the end of the test. Figure 9(a) and (c) shows SEM observation of 9%Cr ring specimens (tested at 650°C and 140/120 MPa, lifetimes 70.1/334.9h), both exhibiting a similar pattern. On the outermost part of the ring, dimples can be seen, as well as evidence of tearing and minimal oxidation. Towards the inside, and affecting most of the fracture surface, features characteristic of intergranular decohesion can be seen, as well as much more oxidation.

Figure 10 highlights the intergranular decohesion that can be observed on the fracture surface of 9%Cr ODS steel tube tile specimen tested at 650°C/180MPa, 165.8h. These grains appear highly oxidized. Other regions only exhibit dimples, suggestive of ductile intragranular damage. In Figure 11, the SEM observation of a cross section of a tile creep specimen of the same 9%Cr ODS steel tube tested at 650°C/180MPa is presented. Extensive damage ahead of the main crack can be seen, with no apparent preferred direction for these micrometric cracks. When looking at ring creep specimens of the same alloy tested at 650°C/140MPa/70.1h, we can see in Figure 12 that cracks seem to align on the rolling direction. Figure 14 shows the fracture surface of a 14%Cr tile creep specimen, tested at 650°C and 290MPa, 1662.5h. Again, two types of damage can be seen on the surface, the brighter being the oxidized intergranular area (b), the darker one with ductile damage and minimal oxidation (c). In Figure 15, SEM observations of cross-sections of tile and ring specimens reveal different damage patterns. For the tile specimen, the main fracture and some secondary ones occur perpendicular to the load, and cracks parallel to that loading direction can also be seen. For the ring specimen, only cracks perpendicular to the loading direction can be seen. This is typical of ferritic ODS steels with elongated grains [START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF][START_REF] Ratti | Développement de nouvelles nuances d'aciers ferritiques-martensitiques pour le gainage d'éléments combustibles des réacteurs à neutrons rapides au sodium[END_REF][START_REF] Salmon-Legagneur | Anisotropic intergranular damage development and fracture in a 14Cr ferritic ODS steel under high-temperature tension and creep[END_REF][START_REF] Steckmeyer | Tensile anisotropy and creep properties of a Fe-14CrWTi ODS ferritic steel[END_REF][START_REF] Hervé | Fluage d'aciers renforcés par dispersion nanométrique : caractérisation, modélisation et optimisation de la microstructure[END_REF], for which the grain boundaries, where titanium enriched unwanted particles tend to align as well, facilitate crack nucleation and propagation. Fracture surface observation on the creep ODS steel specimens, both tiles and rings, showed the presence of oxidized intergranular fracture zones and ductile zones. This has been observed on other 14%Cr bulk ferritic ODS steel [START_REF] Salmon-Legagneur | Anisotropic intergranular damage development and fracture in a 14Cr ferritic ODS steel under high-temperature tension and creep[END_REF], with the following proposed two-step crack propagation mechanism: a slow stable intergranular crack propagation (suggested by the strong oxidation of such zones) followed by unstable ductile fracture as soon as a critical stress intensity factor is reached at the tip of the crack. Present observations concur with such fracture phenomena.

Figure 9: SEM observation of the fracture surface on ring creep test specimens of the 9%Cr ODS steel tube (RD=vertical); a) creep test 650°C 140MPa (inner part of the ring on the left); b) intergranular decohesion; c) creep test 650°C 120MPa (inner part of the ring on the right); d) dimples and oxidized intergranular decohesion

Discussion

Creep properties

Two types of mechanical anisotropies are observed on the studied ODS steel tubes, in addition to the microstructural anisotropic features of the 14%Cr ODS steel tube. First, a strong effect of the loading direction is seen in Figure 5 for the stress to strain rate relationship on the 14%Cr ODS steel tube, consistent with bulk ferritic ODS steels with similar chemical composition [START_REF] Salmon-Legagneur | Anisotropic intergranular damage development and fracture in a 14Cr ferritic ODS steel under high-temperature tension and creep[END_REF][START_REF] Steckmeyer | Tensile anisotropy and creep properties of a Fe-14CrWTi ODS ferritic steel[END_REF][START_REF] Fournier | Mechanical behaviour of ferritic ODS steels -Temperature dependancy and anisotropy[END_REF][START_REF] Serrano | Microstructural anisotropy effect on the mechanical properties of a 14Cr ODS steel[END_REF][START_REF] Turba | Anisotropic mechanical properties of the MA956 ODS steel characterized by the small punch testing technique[END_REF][START_REF] Vanegas | Identification of cyclic and anisotropic behaviour of ODS steels tubes[END_REF][START_REF] Salmon Legagneur | Caractérisation de l'endommagement à haute température d'aciers ferritiques renforcés par dispersion de nano-oxydes (ODS)[END_REF]. The 9%Cr ODS steel tube mechanical behaviour seems unaffected by the loading direction. The second anisotropic feature affects both tubes, even the 9%Cr one that both possesses an isotropic microstructure and presents isotropic creep behaviour (see Figure 5): the lifetime at a given applied stress is orders of magnitude lower for the ring specimens than the tile specimens. Tensile tests on a similar 9%Cr ODS steel tube with the same specimen geometries [START_REF] Toualbi | Relationships between mechanical behavior and microstructural evolutions in Fe 9Cr-ODS during the fabrication route of {SFR} cladding tubes[END_REF] also show a loss of ductility when testing the hoop properties with ring specimens. This is a known experimental artifact of the ring specimen: the relationship between the applied crosshead displacement and the effective strain in the gauge section of the ring has indeed been studied for different loading and material configurations [START_REF] Nagase | Optimized Ring Tensile Test Method and Hydrogen Effect on Mechanical Properties of Zircaloy Cladding in Hoop Direction[END_REF][START_REF] Martín-Rengel | Revisiting the method to obtain the mechanical properties of hydrided fuel cladding in the hoop direction[END_REF][START_REF] Arsène | A New Approach to Measuring Transverse Properties of Structural Tubing by a Ring Test-Experimental Investigation, A New Approach to Measuring Transverse Properties of Structural Tubing by a Ring Test-Experimental Investigation[END_REF][START_REF] Dick | Mechanics and full-field deformation study of the Ring Hoop Tension Test[END_REF]. Finite element analysis is an efficient method to determine the mechanical properties from notched ring tests [START_REF] Arsène | A New Approach to Measuring Transverse Properties of Structural Tubing by a Ring Test-Experimental Investigation, A New Approach to Measuring Transverse Properties of Structural Tubing by a Ring Test-Experimental Investigation[END_REF][START_REF] Samal | Chapter 7 -Investigation of failure behavior of thin-walled tubular components and development of a procedure for evaluation of their mechanical and fracture properties[END_REF]. Early FE simulations on the ring specimen (in the chosen experimental set up) conducted in this study are consistent with the aforementioned studies (see Appendix), with the presence of high stress triaxiality on the inner part of the ring, at the center of the gauge section. This leads to early strain and damage localization in this region, resulting in failure of the ring at a misleading crosshead displacement misleading if one is to extract intrinsic material properties. A non-trivial effect of the specimen geometry is therefore suggested and would require further investigations for constitutive modelling and parameter identification for example. The idea would be to use the data from the 9%Cr ODS steel tube, believed to be isotropic. Although martensitic/ferritic ODS steels with seemingly isotropic microstructure can present anisotropic damage mechanics due to the presence of Prior Particle Boundaries (PPB), artifacts of the powder consolidation process [START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF][START_REF] Sakasegawa | Microstructural evolution during creep of 9Cr-ODS steels[END_REF], no evidence of PPB pores could be found either in the as received 9%Cr tube or in the creep samples after rupture. In order to identify a mechanical behaviour and damage model, the data on the tile specimens could be used to obtain material parameters for the model and FE simulation on the ring geometry could then be performed to properly isolate the contribution of geometrical artifacts in the apparent anisotropy for this material. The study of the intrinsic material anisotropies of the 14%Cr tube could then be performed with a high confidence level.

Nevertheless, ring specimens in the chosen loading conditions seem adequate to produce data equivalent to internal pressure creep tests. The latter are the closest way to emulate the thermomechanical load that fuel cladding must withstand. However, they are costly (large specimens), and fraught with technical difficulties: temperature homogeneity and pertinent strain measurement are the main ones encountered in this study. Comparing these results with uniaxial creep tests on notched rings, comparatively easier tests to set up and requiring less specimen volume, confirms that the ring specimens are a good way to assess the hoop creep properties of ODS steel tubes, as the results on both specimens are quite similar (cf. Figure 4 and Figure 5). A possible suggestion is that the multiaxial stress state in the rings somehow emulates the biaxial stress of the pressured tubes, leading to comparable data. This is of particular interest for the characterization of irradiated claddings, where limited material can be put in experimental reactors, and ring specimens are commonly used [START_REF] Yoshitake | Ring-tensile properties of irradiated oxide dispersion strengthened ferritic/martensitic steel claddings[END_REF][START_REF] Yamashita | Irradiation Behavior Evaluation of Oxide Dispersion Strengthened Ferritic Steel Cladding Tubes Irradiated in {JOYO}[END_REF][START_REF] Yano | Effects of neutron irradiation on tensile properties of oxide dispersion strengthened (ODS) steel claddings[END_REF]. Similar complementarity between notched ring tests and burst creep tests on zirconium alloys are documented [START_REF] Seok | High temperature deformation characteristics of Zirlo TM tubing via ring-creep and burst tests[END_REF].

Comparison with commercial or experimental ODS steels

A Larson-Miller plot of the longitudinal creep properties of different ODS steels is presented in Figure 16. Larson-Miller plots are commonly used when comparing creep data obtained at different temperatures. The data on the tile specimen uniaxial creep tests are compared in particular to bulk ODS steels with almost the same chemical composition (Fe-9Cr-1W-0.3Ti-0.3Y2O3 and Fe-14Cr-1W-0.3Ti-0.3Y2O3) as the tubes in this study. The results seem to conform quite well to each other, with the data located well in the scatter of results. [START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF][START_REF] Ukai | Alloying design of oxide dispersion strengthened ferritic steel for long life FBRs core materials[END_REF][START_REF] Praud | Plasticité d'alliages renforcés par nano-précipitation[END_REF][START_REF] Steckmeyer | Experimental study and modelling of the high temperature mechanical behaviour of oxide dispersion strengthened ferritic steels[END_REF][START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF][START_REF] Wilshire | Deformation and damage processes during creep of Incoloy MA957[END_REF][START_REF] Salmon Legagneur | Caractérisation de l'endommagement à haute température d'aciers ferritiques renforcés par dispersion de nano-oxydes (ODS)[END_REF][START_REF] Zakine | Creep behaviour of ODS steels[END_REF][START_REF] Shinozuka | Creep behavior of oxide dispersion strengthened 8Cr-2WVTa and 8Cr-1W steels[END_REF][START_REF] Ohtsuka | Microstructure and high-temperature strength of high Cr ODS tempered martensitic steels[END_REF] We make the same comparison with the hoop creep properties in Figure 17. Data originates from internal pressure creep tests and uniaxial creep tests in the transverse direction (specified by (T)). These latter tests on bulk material in the transverse direction cannot be directly compared to the others. Rather, they are presented here precisely to appreciate the geometric effect of the specimens used to properly assess the hoop properties. In particular, while the 14%Cr tile test results are close to those obtained on CEA Fe-14Cr-1W-0.3Ti-0.25Y2O (bulk), a significant gap exists when looking at the transverse direction compared with the ring specimens.

As mentioned before, data from 'uniaxial' ring tests and internal pressure creep tests align reasonably well. Overall both tubes in this are relatively weaker than other ODS steels, with ODS martensitic steels and high-chromium martensitic with residual ferrite ODS steels performing better. We also compare the results with non-ODS austenitic 316SS, used for cladding/wrapper applications in sodium-cooled fast reactors [START_REF] Jayakumar | High Temperature Materials for Nuclear Fast Fission and Fusion Reactors and Advanced Fossil Power Plants[END_REF][START_REF] Séran | 8 -Irradiation-resistant austenitic steels as core materials for Generation {IV} nuclear reactors[END_REF]. Other ODS steel tubes compare favorably to 316 type steels for long term creep, which can't be established for the ODS steels tubes of this study. The origin of this relative underperformance has not been yet clearly identified. [START_REF] Ukai | 10 -Oxide dispersionstrengthened/ferrite-martensite steels as core materials for Generation {IV} nuclear reactors[END_REF][START_REF] Ukai | Characterization of high temperature creep properties in recrystallized 12Cr-ODS ferritic steel claddings[END_REF][START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF][START_REF] Salmon Legagneur | Caractérisation de l'endommagement à haute température d'aciers ferritiques renforcés par dispersion de nano-oxydes (ODS)[END_REF][START_REF] Shinozuka | Creep behavior of oxide dispersion strengthened 8Cr-2WVTa and 8Cr-1W steels[END_REF][START_REF] Jayakumar | High Temperature Materials for Nuclear Fast Fission and Fusion Reactors and Advanced Fossil Power Plants[END_REF][START_REF] Hamilton | Fabrication Technological Development of the Oxide Dispersion Strengthened Alloy MA957 for Fast Reactor Applications[END_REF]

Conclusion

Uniaxial creep and pressure creep tests carried out at 650°C on a 14%Cr ferritic ODS steel tube with tile and notched ring specimens revealed mechanical behaviour and damage anisotropies, linked with the elongated grains and strong crystallographic texture that result from the fabrication process. To a smaller extent, evidence of damage anisotropy was observed on a 9%Cr martensitic ODS steel tube with a quasi-isotropic microstructure and otherwise isotropic creep properties, identified as geometry-induced anisotropy for the notched ring specimens. Both rings and pressure creep tests gave similar results, validating the use of the former for hoop properties assessment.

The creep results in this study are consistent with the ODS steel literature, yet with slightly below average hoop properties. Most notably, the 14%Cr ODS steel tube performed almost identically to the 9%Cr in the transverse direction, even though it greatly outperforms the martensitic steel in the longitudinal direction.

Further study of the geometry-induced anisotropy is ongoing by way of modeling the mechanical behaviour and damage mechanics of the ODS steel tubes and performing FE simulations of the ring test. Intrinsic material properties will be obtained from this analysis, which will enable a more apt comparison between uniaxial creep tests on rings and tiles, with the aim to properly predict behaviour and failure of the studied 9%Cr and 14%Cr ODS steel tube.

Figure 1 :

 1 Figure 1: specimen geometries. (a) tile specimen; (b) ring specimen; (c) internal pressure specimen

Figure 2 :

 2 Figure 2: pole figures and orientation maps obtained by EBSD analysis. (a) 14%Cr longitudinal section, (b) 14%Cr transverse section, (c) 9%Cr longitudinal section, (d) 9%Cr transverse section

Figure 3 :

 3 Figure 3: Titanium distribution (EPMA) in 14%Cr ODS steel tube. a) (RD-TD) plane; b) (ND-TD) plane

  -10 ± 5E-11 Ongoing test * maximal strain value measured with external diameter profilometry, see Figure 8(a) and 3.2

Figure 4 :

 4 Figure 4: comparison of creep strength for the 9%Cr and 14%Cr steel tubes in the longitudinal and transverse directions performed at 650°C. Arrows denote ongoing tests.

Figure 5 :

 5 Figure 5: Comparison of the stress-strain rate relationship for the 9%Cr and 14%Cr steel tubes in the longitudinal and transverse directions, creep tests performed at 650°C

Figure 6 :

 6 Figure 6: Normalized creep curves of 9%Cr tile specimen tested at 650°C and 140-180MPa

Figure 7 :Figure 8 :

 78 Figure 7: normalized creep curve of some pressure creep specimens tested at 650°C

Figure 10 :

 10 Figure 10: SEM observation of the fracture surface on a tile creep specimen of the 9%Cr ODS steel tube (180MPa, 650°C); magnification (a) x150, (b) x1000

Figure 11 :

 11 Figure 11: SEM observation of the cross section of a 9%Cr tile creep specimen; 650°C 180MPa

Figure 13 :

 13 Figure 13: SEM observation of the fracture surface on a ring creep specimen of the 14%Cr ODS steel tube (RD=vertical); magnification (a) x200, (b) x1000

Figure 14 :

 14 Figure 14: SEM observation of the surface rupture of a 14%Cr ODS tile specimen, creep test at 650°C and 290MPa. (a) x50; (b) dimples (x1000); (c) intergranular damage with oxidation (x1000)

Figure 15 :

 15 Figure 15: SEM observation of cross sections of 14%Cr ODS steel tube creep specimen. (a) tile specimen, 650°C 290MPa; (b) ring specimen 650°C 140MPa

Figure 16 :

 16 Figure 16: Stress vs Larson-Miller plot of the longitudinal creep properties of some ferritic (red), martensitic (blue) and mixed (green) ODS steels[START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF][START_REF] Ukai | Alloying design of oxide dispersion strengthened ferritic steel for long life FBRs core materials[END_REF][START_REF] Praud | Plasticité d'alliages renforcés par nano-précipitation[END_REF][START_REF] Steckmeyer | Experimental study and modelling of the high temperature mechanical behaviour of oxide dispersion strengthened ferritic steels[END_REF][START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF][START_REF] Wilshire | Deformation and damage processes during creep of Incoloy MA957[END_REF][START_REF] Salmon Legagneur | Caractérisation de l'endommagement à haute température d'aciers ferritiques renforcés par dispersion de nano-oxydes (ODS)[END_REF][START_REF] Zakine | Creep behaviour of ODS steels[END_REF][START_REF] Shinozuka | Creep behavior of oxide dispersion strengthened 8Cr-2WVTa and 8Cr-1W steels[END_REF][START_REF] Ohtsuka | Microstructure and high-temperature strength of high Cr ODS tempered martensitic steels[END_REF]

Figure 17 :

 17 Figure 17: Stress vs Larson-Miller plot of the hoop properties of some ferritic (red), martensitic (blue) and mixed (green) ODS steels[START_REF] Ukai | 10 -Oxide dispersionstrengthened/ferrite-martensite steels as core materials for Generation {IV} nuclear reactors[END_REF][START_REF] Ukai | Characterization of high temperature creep properties in recrystallized 12Cr-ODS ferritic steel claddings[END_REF][START_REF] Tanno | Strength anisotropy of rolled 11Cr-ODS steel[END_REF][START_REF] Salmon Legagneur | Caractérisation de l'endommagement à haute température d'aciers ferritiques renforcés par dispersion de nano-oxydes (ODS)[END_REF][START_REF] Shinozuka | Creep behavior of oxide dispersion strengthened 8Cr-2WVTa and 8Cr-1W steels[END_REF][START_REF] Jayakumar | High Temperature Materials for Nuclear Fast Fission and Fusion Reactors and Advanced Fossil Power Plants[END_REF][START_REF] Hamilton | Fabrication Technological Development of the Oxide Dispersion Strengthened Alloy MA957 for Fast Reactor Applications[END_REF]

Table 1 :

 1 chemical composition of the studied ODS steel tubes

	(%wt.)	Cr	W	Ti	Y	O	C	Si	Ni	Mn	Fe
		(OES)	(OES)	(OES)	(PES)	(RMIA)	(CIR)	(OES)	(OES)	(OES)
	9%Cr tube	9.07	0.93	0.53	0.14	0.13	0.104	0.27	0.22	0.28	Bal.
	(T37)	±0.25	±0.03	±0.02	±0.01	±0.01	±0.003	±0.01	±0.01	±0.01
	14%Cr tube	13.93	0.90	0.58	0.15	0.10	0.038	0.27	0.28	0.21	Bal.
	(T31)	±0.25	±0.03	±0.02	±0.01	±0.01	±0.003	±0.01	±0.01	±0.01
	CIR: Combustion Infrared Absorption

Table 2 :

 2 Creep data for tests at 650°C

	Material / specimen	Applied stress (MPa)	Lifetime (h) Final strain (%)	Minimal strain rate (s -1 )	Comments
		300	7.4	0.38	3.30E-08 ± 7E-9
	14%Cr	290	1662.5	0.59	1.54E-10 ± 5E-11
	tile (T31)	290	>14426	0.7867	8.80E-11 ± 3E-11 Ongoing test
		280	>19419	0.687	4.50E-12 ± 5E-13 Ongoing test
	14%Cr ring (T31)	160 140 120	6.3 83.7 >22440	9.1 2.3 1.1	3.30E-07 ± 2E-8 1.00E-08 ± 3E-9 7.00E-11 ± 1E-11 Ongoing test
		200	75.9	2.6	7.90E-08 ± 1E-9
	9%Cr tile	180	165.8	4.4	4.00E-08 ± 3E-9
	(T37)	160	162.9	3.18	3.80E-08 ± 2E-9
		140	>13225	1.335	1.20E-10 ± 1E-11 Ongoing test
	9%Cr ring (T37)	140 120 100	70.1 334.9 2611	1.6 1.1 1.2	2.50E-08 ± 3E-9 1.50E-09 ± 3E-10 1.50E-10 ± 5E-11
	14%Cr	136.2 (140bar)	167.7	0.72*	8.0E-9	± 2E-9
	internal	116.8 (120bar)	699.8	0.46*	1.3E-9	± 3E-10
	pressure	97.3 (100bar)	5418	0.52

* 1.4E-10 ± 4E-11 Ongoing test

Acknowledgments

This study has been supported by CEA. The author gratefully acknowledge Elodie Rouesne for her support with the characterization of the microstructure, be it for the EBSD analysis or the SEM fractographies; Dr. Denis Sornin for the fabrication of the tubes; Elena Vakhitova for her help with the hardness measurements. Thanks finally to Florent Lefebvre, Jean-Philippe Bonthonneau and Véronique Lezaud-Chaillioux, for their support with the mechanical testing.

Appendix: Finite-element simulation of the ring uniaxial creep test

A finite element model of the ring was created using the finite element code Cast3M (http://www-cast3m.cea.fr/). Taking advantage of the specimen symmetries, only an eighth of the ring needs to be simulated. The model of the ring was meshed with solid linear cubic elements with 8 nodes (1472 elements, 2206 nodes with 3 degrees of freedom), as seen in Figure 18. The mandrel permitting the loading of the ring with minimal bending in the gauge section is also modelled. Assumed to be rigid, only the outer shape of the mandrel in contact with the specimen is modelled. Surface contact (no penetration) with friction is simulated. A 0.3 friction coefficient was chosen, deemed conservative enough as a more accurat value could not be directly measured. The material model adopted was an (isotropic) elasto-viscoplastic model (Norton law) with isotropic hardening. Simulations have been performed with parameters identified from the experimental data on 9%Cr ODS steel tile specimens at 650°C. As seen in Figure 19, where symmetries are used in postprocessing to show the whole ring, plastic strain localizes at the center of the gauge section in the inner part of the ring, where stress triaxiality, as defined in Equation (A.4), is higher (most notably >0.33 which corresponds to a uniaxial stress state).

Where 𝑇𝑋 is the triaxiality factor, 𝜎 𝐻 the hydrostatic stress, 𝜎 𝑒𝑞 the equivalent (Von Mises here) stress. The accumulated plastic strain 𝑝 used in Figure 19 is defined in Equation (A.5).

Where 𝜺 𝑝 represents the plastic strain tensor (total strain tensor 𝜺 = 𝜺 𝑒 + 𝜺 𝑝 ).


domain: spi phys

text:
L Pascal
email: lucas.pascal@onera.fr

G Delattre

H Deniau

J Cliquet

New implementation of stability-based transition model by means of transport equations

A new natural laminar-turbulent transition model compatible with Computation Fluid Dynamics is presented. This model accounts for longitudinal transition mechanisms (i.e. Tollmien-Schlichting induced transition) thanks to systematic stability computation on similar boundary profiles from Mach zero to four both on adiabatic and isothermal wall. The model embeds as well the so-called "C1-criterion" for transverse transition mechanisms (i.e. cross-flow waves induced transition). The transition model is written under transport equations formalism and has been implemented in the solver elsA (ONERA-Airbus-Safran property). Comparisons are performed on two-dimensional and three-dimensional configurations against transition database approach.

I. Nomenclature

II. Introduction

Accurate computation of transport aircraft drag strongly relies on natural laminar-turbulent transition prediction capabilities. As computational fluid dynamics (CFD) is now a major component of industrial processes, it is necessary to develop accurate transition prediction techniques for RANS solvers both for aerodynamic performance prediction and design of future laminar transport aircraft concept.

The development of transition prediction methods compatible with Computation Fluid Dynamics (CFD) is a major research topic. A quite recent approach consists in using methods based on Partial Differential Equations (PDE). This approach consists in solving additional transport equations governing the dynamic of quantities that are related to transition. The most famous is probably the "γ -Re θ " of Langtry and Menter [START_REF] Langtry | Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes[END_REF] based on phenomenological reasoning. This method has demonstrated success on many configurations and has been extended to handle as well cross-flow transition [START_REF] Grabe | Transport Modeling for the Prediction of Crossflow Transition[END_REF]. The Amplification Factor Transport (AFT) method was derived more recently by Coder and Maughmer [START_REF] Coder | Computational Fluid Dynamics Compatible Transition Modeling Using an Amplification Factor Transport Equation[END_REF]. This promising method consists in writing under a transport equation the e N method [START_REF] Van Ingen | A suggested semi-empirical method for the calculation of the boundary layer transition region[END_REF][START_REF] Smith | Transition, pressure gradient, and stability theory[END_REF] of Drela and Giles [START_REF] Drela | Viscous-inviscid analysis of transonic and low Reynolds number airfoils[END_REF]. The γ -Re θ and AFT methods are said to be "local" in the sense that the additional transport equations associated to transition only involve values available at RANS computational points. This property reduces much the implementation effort in a RANS solver.

As far as the elsA RANS solver (property of Airbus-Safran-ONERA) is concerned, developments have been conducted to give access to non local variables (for instance integral boundary layer variables) at grid point. This feature has been used to implement the AHD criterion evaluated along mesh lines [START_REF] Perraud | Overview of transition prediction tools in the elsA software[END_REF] and the so-called "parabola method" [START_REF] Perraud | Automatic transition predictions using simplified methods[END_REF] for transition prediction by means of transport equations [START_REF] Bégou | Database Approach for Laminar-Turbulent Transition Prediction: Navier-Stokes Compatible Reformulation[END_REF].

This paper presents the implementation of the AHD and C1 criteria by means of transport equations and their comparison with the parabola method. These criteria are presented in section III while section IV deals with their implementation in a RANS solver. Numerical results are presented in section V.

III. AHD/C1 transition criterion and parabola method for transition prediction A. AHD criterion

The Arnal-Habiballah-Delcourt (AHD) criterion [START_REF] Arnal | Transition prediction in transonic flow[END_REF] is derived by expressing N-factor associated to Tollmien-Schlichting instabilites obtained on similar profiles as curves of the form N = N (R θ -Re θ,cr , Λ 2 ). The method consists then in replacing the Polhausen parameter Λ 2 by its average value Λ2 between the critical point (i.e. the location from which Tollmien-Schlichting instabilites start to grow) of curvilinear abscissa s cr and the current location of curvilinear abscissa s (measured along the streamline at the edge of the boundary layer). Using the Mack's relationship (N T = -2.4 ln(T u /100) -8.43) finally gives the transitional Reynolds number Re θ, t r as:

Re θ, t r = Re θ,cr + A exp(B Λ2 ) ln(CT u ) -D Λ2 (1)
where Re θ,cr is the Reynolds number at the critical point and is given by Re θ,cr = exp G/H 2 i + E/H i -F . This criterion account for receptivity through T u and for flow history through Λ2 .

In its compressible extension [START_REF] Perraud | Stability-Based Mach Zero to Four Longitudinal Transition Prediction Criterion[END_REF], the variables A, B,C, D, E, F, G are function of M e , the Mach number at the edge of the boundary layer. Moreover, this criterion account for effects of wall temperature [11, section V].

B. C1 criterion

The C1 criterion [START_REF] Arnal | Théorie de l'instabilité laminaire et critères de transition en écoulement bi et tridimensionnel[END_REF] is used to predict natural transition induced by cross-flow instabilities. This criterion defines the transition point as the location where the transverse incompressible displacement thickness Reynolds number R δ 2i equals a threshold given by:

Re δ 2 , i, t r =          150 H i ≤ 2.31 300 π arctan 0.106 (H i -2.3) 2.052 1 + γ -1 2 M 2 e 2.31 < H i < 2.65 (2)
(the criterion should not be applied for H i > 2.65).

IV. Implementation in a CFD solver

A. Non local variables Evaluating eqs. (1) or (2) requires the knowledge of boundary layer variables Re θ , Re δ 2 , M e , etc . . . . Contrary to the approaches presented in Ref. [START_REF] Langtry | Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes[END_REF] or [START_REF] Coder | Computational Fluid Dynamics Compatible Transition Modeling Using an Amplification Factor Transport Equation[END_REF], the present method does not rely on correlations between on local variables. Non local variables are evaluated and made available at each cell point in elsA thanks to the fact that it is possible to: i) to get for a cell in the volume the associated wall interface (if there is any) and ii) to know which cells in the volume form the line normal to a given wall interface.

B. "Transition lines method"

An implementation of the AHD criterion has been proposed by Cliquet et al. [START_REF] Cliquet | Application of Laminar-Turbulent Transition Criteria in Navier-Stokes Computations[END_REF]. It consists in assuming that streamlines at the boundary layer edge might be approximated by mesh lines. The implementation is denoted thereafter as "transition lines method". This method has been implemented in the elsA CFD solver and has shown good results on aircraft configuration [START_REF] Hue | Wind-Tunnel and CFD Investigations Focused on Transition and Performance Predictions of Laminar Wings[END_REF], helicopter blades flow [START_REF] Richez | Assessment of laminar-turbulent transition modeling methods for the prediction of helicopter rotor performance[END_REF], etc . . . However, this method requires some effort from the user as the latter is asked to prescribe the starting points of each transition line.

C. Transport equations approach

To alleviate user effort and to account with higher fidelity for three dimensional geometries where streamlines directions might strongly differ from mesh lines, a new implementation of the AHD/C1 criteria has been derived. This implementation is based on transport equations and shares similarities with the method derived by Bégou et al. [START_REF] Bégou | Database Approach for Laminar-Turbulent Transition Prediction: Navier-Stokes Compatible Reformulation[END_REF].

The first prerequisite to estimate the transition threshold Re θ, t r following eq. ( 1) is to know the value of the critical Reynolds number downstream of the critical location. To answer this need, the transported variable Re θ,cr governed by:

∂ t ρ Re θ,cr + ν cr ∇ • ρ Re θ,cr U = (1 -ν cr ) Γ Re θ, c r Re θ,cr -Re θ,cr (3)
is introduced. ν cr equals one where Re θ ≥ Re θ,cr,e and zero elsewhere. As a consequence, as long as the boundary layer is not critical the source term forces Re θ,cr to equal Re θ,cr and if the boundary layer becomes critical Re θ,cr is simply convected. The second prerequisite is to compute the value of Λ2 . To do so, a second transport equation is introduced:

∂ t ρ Λ2 + ∇ • ρ Λ2 U = ν cr ρ||U || s Λ 2 -Λ2 + Γ Λ2 (1 -ν cr ) ρ( Λ2 -Λ 2 ) (4)
where s is an additional transported variable corresponding to the curvilinear abscissa measured from the critical point (upstream of the critical point s equals zero). s is governed by:

∂ t ( ρ s) + ∇ • ρ sU = ρ||U || -Γ s ρ (1 -ν cr ) s . (5)
The transition threshold on Re θ given by (1) can then be evaluated from Λ2,e and Re θ,cr,e obtained by extracting Λ2 and Re θ,cr at the edge of the boundary layer. A last equation is then added to set the intermittency of the transition point given by Re θ > Re θ,cr :

∂ t ρ I + ∇ • ρ IU = ν t r ρ||U || -Γ I ρ (1 -ν t r ) I . (6)
ν t r equals one if Re θ ≥ Re θ, t r or Re δ 2, i reaches Re δ 2, i, t r (given by Eq. ( 2)). As a consequence I corresponds to a curvilinear abscissa measured from the transition point (either induced by longitudinal or transverse instabilities). Its value at the edge of the boundary layer I e is then used to evaluate the intermittency following (in the current implementation, the intermittency is set constant in the whole boundary layer profile):

γ( I e ) = 1 -exp -5 I e l t r 2 . (7)
Eq. ( 7) is derived from Refs. [START_REF] Stock | Navier-Stokes Airfoil Computations with e Transition Prediction Including Transitional Flow Regions[END_REF]Eqs. (1,[START_REF] Drela | Viscous-inviscid analysis of transonic and low Reynolds number airfoils[END_REF]]. Currently, the user is asked to prescribe l t r . Further details will be given in the final paper

V. Validations

Flows over two-dimensional and three-dimensional geometries are considered to validate the prediction method over a wide-range of Mach numbers. Validations are performed against results obtained with the 3C3D solver of ONERA which has be shown to give excellent results for instance in Ref. [START_REF] Hue | Wind-Tunnel and CFD Investigations Focused on Transition and Performance Predictions of Laminar Wings[END_REF]. The latter solves the boundary layer equations on three dimensional geometries. It embeds the AHD/C1 transition criteria and the parabola method.

Computations are performed as well with the method derived by Bégou et al. [START_REF] Bégou | Database Approach for Laminar-Turbulent Transition Prediction: Navier-Stokes Compatible Reformulation[END_REF]. In the final paper, further details will be given about this latter method.

elsA computations are performed with a second order Roe spatial scheme and a backward Euler time scheme. k -ω SST turbulence of Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] is chosen.

Nacelle transition prediction

While numerical validations of transition models in CFD are usually made on wings, numericals results on a generic nacelle configuration are shown in this section. A cut view of the geometry and of the surface mesh is shown Fig. 1. There are about 3 × 10 6 points in the mesh. The turbulence level is set to T u = 0.1%. The computed intermittency is plotted Figs. 2(a,b) at the outer and inner sides of the nacelle. For the sake of visibility, both sides are "unrolled".

Good agreement is obtained against the transition line (defined as the location where γ starts to grow) given by the boundary layer equations solver 3C3D.

Further comparisons and validations cases will be presented in the final paper

VI. Conclusion

A conclusion will be given here in the final paper

γ = Intermittency Λ 2 = 2 ν e dU e ds M = Mach number Re δ 2 =

 222 Polhausen parameter Λ 2 = θ Transverse displacement thickness based Reynolds number Re θ = Momentum thickness based Reynolds number s = Curvilinear abscissa T u = Turbulence level Subscript(s) e = Edge of the boundary layer i = Incompressible cr = Critical point of the boundary layer tr = Transition location

Fig. 1

 1 Fig. 1 Cut view of the nacelle. Pressure boundary condition is imposed on the black surface.

Fig. 2

 2 Fig. 2 Intermittency contours (light and dark corresponds respectively to γ = 0 and γ = 1) at the outer (a) and inner (b) sides of the nacelle. The black line depitcs the transition location predicted by 3C3D.