domain: spi phys

text:
Luis Bernardos
email: luis.bernardos_barreda@onera.fr

François Richez
email: francois.richez@onera.fr

Vincent Gleize
email: vincent.gleize@onera.fr

Rans

RANS modeling of Laminar Separation Bubbles around Airfoils at Low Reynolds conditions

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RANS modeling of Laminar Separation Bubbles around Airfoils at Low Reynolds Number Conditions

L. Bernardos * , F. Richez † and V. Gleize ‡ ONERA, 92190 Meudon, France

In this paper we evaluate the performance of an algebraic non-local laminar separation transition triggering (LSTT) approach that controls the production of turbulence in the transitional region. The LSTT approach is coupled to the k-ω and Spalart-Allmaras turbulence models and the AHD-Gleyzes transition prediction criterion. Diverse low Reynolds number flow configurations featuring laminar separation bubbles are investigated, such as flat-plate case, NACA 0012 airfoil, SD7003 airfoil and T106C turbine airfoil. The influence of chord-based Reynolds number, angle of attack and inflow turbulence level on the prediction capabilities of the LSTT approach are assessed. Good agreement with available DNS, LES and experimental data is obtained.

I. Nomenclature

II. Introduction

L separation bubbles [1, LSBs] appear in numerous aerodynamic applications, such as dynamic stall phenomenon of helicopter blades [START_REF] Richez | Near stall simulation of the flow around an airfoil using zonal RANS/LES coupling method[END_REF][START_REF] Raffel | Micro-PIV and ELDV wind tunnel investigations of the laminar separation bubble above a helicopter blade tip[END_REF], low-Reynolds number micro air-vehicles (MAV) [START_REF] Pines | Challenges Facing Future Micro-Air-Vehicle Development[END_REF] or wind turbines with enhanced laminarity [START_REF] Somers | Design and Experimental Results for the S809 Airfoil[END_REF]. It is well established that the flow initially undergoes laminar separation because of the adverse pressure-gradient [START_REF] Gaster | The Structure and Behaviour of Laminar Separation Bubbles[END_REF], followed by transition onset in the separated shear-layer [START_REF] Malkiel | Transition in a Separation Bubble[END_REF], identified as the amplification of Kelvin-Helmholtz instabilities leading to turbulence [START_REF] Hain | Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils[END_REF] . Increased turbulent mixing in the shear layer may, depending on flow conditions [START_REF] Crabtree | The Formation of Regions of Separated Flow on Wing Surfaces[END_REF], lead to rapid reattachment (short bubble locally modifying wall-pressure distribution [START_REF] Tani | Low-speed flows involving bubble separations[END_REF]) or remain separated for longer distances (long bubble globally modifying wall-pressure distribution [START_REF] Tani | Low-speed flows involving bubble separations[END_REF]). In both cases, airfoil lift, drag and moment are influenced by the presence of the bubble.

RANS (Reynolds-Averaged Navier-Stokes) turbulence models require specific transition information and/or modification to handle the separation-induced transition which dominates the flow [START_REF] Malkiel | Transition in a Separation Bubble[END_REF][START_REF] Sanz | On the Navier-Stokes Calculation of Separation Bubbles With a New Transition Model[END_REF]. Numerous early [START_REF] Tani | Low-speed flows involving bubble separations[END_REF][START_REF] Gaster | The Structure and Behaviour of Laminar Separation Bubbles[END_REF][START_REF] Crabtree | The Formation of Regions of Separated Flow on Wing Surfaces[END_REF][START_REF] Horton | Laminar separation bubbles in two-and three-dimensional incompressible flow[END_REF][START_REF] Carmichael | Low Reynolds number airfoil survey[END_REF] and more recent [START_REF] Hain | Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils[END_REF][START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF][START_REF] Yarusevych | Steady and transient response of a laminar separation bubble to controlled disturbances[END_REF][START_REF] Michelis | On the origin of spanwise vortex deformations in laminar separation bubbles[END_REF] studies, both experimental and computational, highlight the complex physics of the transitional flow inside LSBs, such as pronounced three-dimensionality, high unsteadiness, turbulent breakdown driven by multiple coherent structures and complex oscillator and amplifier instability mechanisms that may lead to notable changes in the flow topology. Several approaches of varying degree of empiricism have been used to predict this difficult flow.

Ultimately, improved transport-equation based transition models [START_REF] Dick | Transition models for turbomachinery boundary layer flows : a review[END_REF] are expected to accurately predict such flows. Correlation-based models [START_REF] Langtry | A Correlation-Based Transition Model Using Local Variables -Part II: Test Cases and Industrial Applications[END_REF][START_REF] Langtry | Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes[END_REF] reformulate widely used transition correlations [START_REF] Abu-Ghannam | Natural transition of boundary-layers -The effects of turbulence, pressure gradient, and flow history[END_REF] into coefficients of transport equations for the intermittency [START_REF] Dhawan | Some properties of boundary layer flow during the transition from laminar to turbulent motion[END_REF][START_REF] Durbin | An intermittency model for bypass transition[END_REF] and other transition parameters, to obtain local, and recently [START_REF] Menter | A One-Equation Local Correlation-Based Transition Model[END_REF] Galilean invariant, transition models. Physics-based models [START_REF] Walters | A New Model for Boundary Layer Transition Using a Single-Point RANS Approach[END_REF][START_REF] Walters | Computational Fluid Dynamics Study of Wake-Induced Transition on a Compressor-Like Flat Plate[END_REF][START_REF] Walters | A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier-Stokes Simulations of Transitional Flow[END_REF] generally rely on the concept [START_REF] Mayle | The path to predicting bypass transition[END_REF] of laminar (pre-transitional) kinetic energy k L ≈ 1 2 u 2 (essentially streamwise, in agreement with transition physics [START_REF] Tani | Boundary-later transition[END_REF] and with the observed rapid increase of the Reynolds-stress tensor anisotropy in very low Reynolds number channel flows [START_REF] Gerolymos | The dissipation tensor ε i j in wall turbulence[END_REF]), which is computed by a specific transport equation, to trigger and control transition in the turbulence model. Transport-equation based transition models are quite successful in mimicking transitional flow effects and in detecting transition [START_REF] Dick | Transition models for turbomachinery boundary layer flows : a review[END_REF]. Given the extreme difficulty of accurately predicting transition, algebraic nonlocal transition models are also developed [START_REF] Kubacki | An algebraic intermittency model for bypass, separation-induced and wake-induced transition[END_REF], including specific adaptations of additional transport equations [START_REF] Minot | Improvement of Laminar-Turbulent Transition Modeling Within a Low-Pressure Turbine[END_REF][START_REF] Colonia | Assessment and Calibration of the γ-Equation Transtion Model at Low Mach[END_REF][START_REF] De Rosa | Validation of Intermittency Model for Transition Prediction in a RANS Flow Solver[END_REF].

Numerous authors have contributed to RANS modeling of LSBs on airfoils, especially at low chord-based Reynolds number Re c . Yuan et al. [START_REF] Yuan | An Investigation of Low-Reynolds-Number Flows past Airfoils[END_REF] and Windte et al. [START_REF] Windte | Validation of the RANS-simulation of laminar separation bubbles on airfoils[END_REF] performed extensive RANS modeling of LSBs for the flow around a SD7003 airfoil at Re c =60,000, with various models, and concluded that Menter's BSL [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] k-ω performed quite well. Source terms were disabled in the laminar regions upstream of transition onset which was determined using an approximate envelope method [START_REF] Windte | Validation of the RANS-simulation of laminar separation bubbles on airfoils[END_REF]. Lian and Shyy [START_REF] Lian | Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil[END_REF] proposed a nonlocal intermittency model for the activation of Wilcox's 1994 k-ω [START_REF] Wilcox | Simulation of Transition with a Two-Equation Turbulence Model[END_REF], through direct weighting of the eddy viscosity ν effective t = ν t γ, and determined transition onset by the e N method [START_REF] Smith | Transition, Pressure Gradient and Stability Theory[END_REF]. Calculations of the SD7003 airfoil at Re c = 60, 000 at various angles of attack gave promising results [START_REF] Lian | Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil[END_REF]. The same flow was studied by Catalano and Tognaccini [START_REF] Catalano | Turbulence Modeling for Low-Reynolds-Number Flows[END_REF] with disabled source terms in the laminar regions upstream of the user-prescribed transition onset. These authors proposed a modification of the F 2 blending function appearing in the expression of the shear limiter of Menter's k-ω SST model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] and of the cut-off value of ω, thus enhancing turbulence production in the transitional region and improving upon the results of the original formulation. An empirical transition onset correlation was added to the method [START_REF] Catalano | A Numerical Method to Detect Laminar Separation Bubbles over Airfoils[END_REF] which was applied to various airfoil calculations (SD7003, NACA 0012, S809 and S827). No specific treatment of the transitional region was performed, in the sense that the activation of the turbulence model was made instantly through the source terms at the transition onset.

Despite the encouraging results of the previously described approaches, accurate prediction of LSB is still challenging. As remarked by Dick and Kubacki [START_REF] Dick | Transition models for turbomachinery boundary layer flows : a review[END_REF], the quality of the predictions of separation-induced transition remains case-dependent. Numerous existing methods focus on correctly predicting the transition onset, but the transition region prediction still relies on the turbulence model's unapprehended ability to correctly produce turbulence at and downstream of the transition onset. In this paper we present and evaluate a recently developed [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] transition modeling formulation that intends to account for the turbulence growth within a LSB and in the downstream relaxation flow region. To this purpose, laminar separation transition triggering (hereafter LSTT) in the RANS model is progressive in the streamwise direction (s), from the criteria-determined [START_REF] Roberts | Calculation of Laminar Separation Bubbles and Their Effect on Airfoil Performance[END_REF][START_REF] Arnal | Laminar Instability Theory And Transition Criteria In Two and Three-Dimensional Flow[END_REF][START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF] transition onset to full turbulence. The assumption is made that the streamwise length between the laminar separation point, which is exactly determined by the solution of the laminar Navier-Stokes equations, and the criteria-determined transition onset is the characteristic lengthscale of the LSTT-process. Therefore, triggering functions of the model are expressed in terms of the aforementioned lengthscale.

Furthermore, wall-normal variations in the LSTT approach are not modeled, in line with many algebraic transition models using intermittency functions [START_REF] Lian | Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil[END_REF][START_REF] Cliquet | Application of Laminar-Turbulent Transition Criteria in Navier-Stokes Computations[END_REF]. The coefficients that define the algebraic form of the triggering functions control the production and destruction of turbulent quantities. They are calibrated to obtain the best possible agreement with available LSB DNS data [START_REF] Laurent | RANS Modeling of a Transitional Laminar Separation Bubble on a Flat Plate with DNS Database[END_REF][START_REF] Laurent | DNS database of a transitional separation bubble on a flat plate and application to RANS modeling validation[END_REF].

Although most methods dealing with the transition region use the concept of intermittency [START_REF] Durbin | An intermittency model for bypass transition[END_REF][START_REF] Menter | A One-Equation Local Correlation-Based Transition Model[END_REF], the transitiontriggering functions used in the present work are not directly related to this concept, but should rather be considered as "empirical weighting coefficients" [START_REF] Arnal | Laminar-turbulent transition problems in supersonic and hypersonic flow[END_REF] designed to reproduce as best as possible the progressive growth of turbulence in LSBs. Therefore they are not necessarily bounded by 1, but may be designed to locally overshoot this bound.

In this work we use linear k-ω models [49, p.124] which are largely used as the underlying RANS framework of transition models [START_REF] Wilcox | Simulation of Transition with a Two-Equation Turbulence Model[END_REF][START_REF] Wilcox | A Half Century Historical Review of the k-ω Model[END_REF]. Additionally, we extend the LSTT approach to the one-equation Spalart-Allmaras [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] turbulence model. This paper is organized as follows. In §III we present the turbulence models used, the transition onset prediction models, and the turbulence triggering approaches. In §IV, the models are evaluated for the prediction of the flow around a NACA 0012 airfoil, SD7003 airfoil and T106C turbine blade.

III. Methodology description A. Turbulence models

In this work we use two different turbulence models, the k-ω Wilcox 2006 [START_REF] Wilcox | Formulation of the k-ω Turbulence Model Revisited[END_REF] model and the one-equation Spalart-Allmaras turbulence model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. Transitional adaptations are performed on the original fully-turbulent formulation of both models. The LSTT approach is implemented in each model through introduction of the dimensionless transition function f tr in the source terms and the dimensionless stress limiter delay function C lim .

The set of equations of the k-ω model is summarized as follows,

∂ t (ρk) + ∂ i (ρu i k) = f tr (-ρu i u j )(∂ j u i ) -β k ρωk + ∂ i [(µ + σ k µ t )∂ i k] (1a) ∂ t (ρω) + ∂ i (ρu i ω) = min( f tr , 1)α ω ω k (-ρu i u j )(∂ j u i ) -β ω ρω 2 + ∂ i [(µ + σ ω µ t )∂ i ω] (1b)
+ σ d ρ ω max((∂ i k)(∂ i ω), 0) (-ρu i u j ) ≈2µ t S i j - 2 3 ρkδ i j (1c) µ t = ρk max ω; C lim 2S i j S i j β k (1d)
Through Eqns. (1a) and (1b) one may observe that the transition function f tr acts on the production terms of both k and ω. As will be presented in the following paragraphs, the purpose of the f tr function is to weight the modeled effective production of turbulence at and downstream of the transition onset, in order to account for the relatively high Reynolds shear stresses identified experimentally in the transitional region. In order to allow for this important production of turbulence, the weighting factor of the production of ω is limited to 1. This is necessary because otherwise the boost of turbulence production expected by the use of f tr > 1 via the production term of k would be mitigated by a boost in the production of the ω equation.

The purpose of the C lim function, which is a constant in the original Wilcox' 2006 model, is to delay the activation of the shear stress limiter, in order to allow for sufficiently high production rates of turbulence in the transitional region. As it can be anticipated, C lim = 0 in the transitional region, thus disabling the shear stress limiter and allowing for important turbulence production, and it yields C lim = 7/8 in the fully turbulent region.

The Spalart-Allmaras [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] model adapted to the LSTT formulation, without tripping term, may be summarized as follows,

∂ t ( ν) + ∂ i (u i ν) = f tr c b1 (1 -f t2 ) S ν -min( f tr , 1) c w1 f w - c b1 κ 2 f t2 ν n 2 (2a) + 1 σ [∂ i ((ν + ν)∂ i ν) + c b2 (∂ i ν) 2 ] (-u i u j ) =2ν t S i j (2b) ν t = ν f v1 (2c)
Through the Eqn. (2a) one may observe that the f tr function acts on both the production and destruction terms of ν. We limit the weighting factor of the destruction of ν to 1. This is done for the same reason as the production of ω is limited in the k-ω formulation, i.e. to allow high rate of turbulence growth in the transitional region where f tr > 1. As one may observe through the equations that constitute the Reynolds shear stresses [Eqns. (2b) and (2c)] it has not been needed to implement the C lim function in the Spalart-Allmaras model using the LSTT approach.

B. Transition onset prediction

Before examining the formulation of the functions f tr and C lim , we note that the LSTT approach requires information on the transition onset location, namely s tr . For this purpose, we employ the Arnal-Habiballah-Delcourt [START_REF] Arnal | Laminar Instability Theory And Transition Criteria In Two and Three-Dimensional Flow[END_REF][START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional Incompressible Flow[END_REF][START_REF] Arnal | Boundary Layer Transition: Predictions Based on Linear Theory[END_REF] (AHD) and the Gleyzes [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF] criteria for the natural attached and separated transition onset computation of the boundarylayer. These criteria are computed at each iteration step of the simulation, and provide the s tr point used by the f tr and C lim functions. The AHD-Gleyzes transition criteria, which are based on linear stability analysis and experimental correlations, are employed to model the amplification of Tollmien-Schlichting and Kelvin-Helmholtz waves. For this reason, these criteria are best suited to natural transition prediction, i.e. at low external turbulence level conditions. The criteria evaluate non-local algebraic expressions employing boundary-layer quantities along the streamwise coordinate s.

The incompressible formulation of the AHD criterion is expressed as follows,

Re θtr = -206 exp(25.7Λ 2 )[ln(16.8T u tr ) -2.77Λ 2 ] + Re θcr (3)
The transition onset is located at the point s = s tr where Re θ(s) = Re θtr is reached. The threshold of transition is determined by the value of the transitional external turbulence level, T u tr . The subscript "cr" denotes the laminar critical point, which is located at s = s cr < s tr when Re θ(s) = Re θcr = exp(52/H (s) -14.8). The mean Pohlhausen parameter Λ 2 is expressed as follows,

Λ 2 = 1 s -s cr ∫ s s cr -θ 2 µ|u e | ∂p wall ∂s ds (4)
Where |u e | denotes the magnitude of the velocity at the boundary-layer edge and p wall the pressure at the wall.

The AHD criterion is used from the laminar critical point up to the point where H = 2.8. If the transition onset has not been reached at such point, named the switch point, then the transition onset prediction is relayed by the Gleyzes criterion, which models the total amplification factor and is expressed as follows,

N(s) = N sw + ∫ Re θ Re θ sw -2.4 B GL dRe θ (5)
Where B GL is the Gleyzes function and is expressed as follows,

B GL =            -162.11093 H 1.1 for 3.36 < H -73[exp [-1.56486(H -3.02)]] for 2.8 < H ≤ 3.36 -103[exp[-4.12633(H -2.8)]] else (6)
The transition onset is located at the critical total amplification factor N cr = N(s = s tr ) of Eqn. [START_REF] Somers | Design and Experimental Results for the S809 Airfoil[END_REF]. This critical value is related to the threshold transitional external turbulence level T u tr via the Mack's [START_REF] Mack | Transition prediction and linear stability theory[END_REF] relationship Eqn. [START_REF] Malkiel | Transition in a Separation Bubble[END_REF],

N cr = -2.4 ln(T u ) -8.43 (7)
The switch total amplification factor N sw expressed in Eqn. ( 5) accounts for the laminar boundary-layer receptivity history up to the switch point. It is determined by replacing Re θ tr by current Re θ (s = s sw ) of Eqn. (3); inject the Mack's relationship Eqn. [START_REF] Malkiel | Transition in a Separation Bubble[END_REF] into Eqn. (3), and finally solving for N(s = s sw ),

N sw = -8.43 -2.4 ln 1 16.8 exp 2.77Λ 2 - Re θ -Re θ cr 206 exp(25.7Λ 2 ) (8)
From the author's experience in using the Gleyzes criterion, we discourage the use of actual boundary-layer values in the neighborhood of a deeply detached boundary-layer. The reason is that massively detached boundary-layers may yield a local decrease of Re θ (s) before the transition onset is reached, thus compromising the growth of N via Eqn. [START_REF] Somers | Design and Experimental Results for the S809 Airfoil[END_REF]. This would provoke a permanent downstream displacement of the transition onset which would translate into numerical instabilities and poor prediction precision. In order to solve this issue, we use an extrapolation approach of H and Re θ values, from a set of points located in the neighborhood of the laminar separation point (cf appendix §A).

Further details on the implementation of the criteria-based approach are presented by Cliquet et al. [START_REF] Cliquet | Application of Laminar-Turbulent Transition Criteria in Navier-Stokes Computations[END_REF] and Perraud et al. [START_REF] Perraud | Overview Of Transition Prediction Tools In The elsA Software[END_REF].

0 ̂ ̂ ̂ = ( -tr )∕( tr -sep ) 0 1 tr + ( -) ̂ -̂ ̂ -̂ - ̂ -̂ ̂ -̂ -̂ ̂ + ̂ ̂ ̂ < ̂ lim =0 ̂ > ̂ lim =1
̂ Fig. 1 Scheme of the transition function f tr (C 1 piecewise cubic polynomials) and shear stress limiter switch C lim used in the LSTT [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] approach with the closure constants s a , s b , s c and a adapted to the turbulence model as shown in Table 1.

C. Turbulence triggering modeling and calibration

In this section we present the algebraic functions f tr and C lim used for triggering the turbulence models [Eqns. ( 1) and ( 2)] at and downstream of the transition onset s tr . As these functions are applied everywhere in the flowfield, they yield specific values at every point of the flow, i.e. f tr (x, y, z) and C lim (x, y, z). In order to determine these values throughout the entire flowfield, a two-step procedure is performed as follows.

In a first step, only the values of f tr and C lim at the wall ought to be determined. This is accomplished by specific algebraic dependence of f tr and C lim upon s, the curvilinear abscissa of the wall, as shown in Fig. 1. In practice, for the airfoil simulations of the current work, s corresponds to the airfoil's wall contour.

For the determination of f tr (s) and C lim (s), the LSTT model makes use of the transition onset location, s tr , provided by the transition criteria presented in §B; and the laminar separation point s sep , retrieved by direct prediction of the Navier-Stokes equations. Hence, the dimensionless curvilinear abscissa s is constructed as presented in Eqn. [START_REF] Crabtree | The Formation of Regions of Separated Flow on Wing Surfaces[END_REF],

s := s -s tr s tr -s sep (9)
Once the wall values f tr (s) and C lim (s) are determined, in a second step we perform a wall-normal extrusion of their values up to the boundary-layer height, namely δ. Outside the boundary-layer, the constant values f tr = 1 and C lim = 1 are set. In this way, f tr and C lim are properly defined at every point in space.

The main role of the LSTT model is to weight the rate at which turbulence is produced from the transition onset. The reason is to account for the smooth and relatively important production of the Reynolds shear stress, -u v , that occurs in the separated shear layer close to the transitional region, as observed in experiments [START_REF] Hain | Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils[END_REF][START_REF] Herbst | Influence of large-scale free-stream turbulence on an SD7003 airfoil at low Reynolds numbers[END_REF] and DNS [START_REF] Laurent | DNS database of a transitional separation bubble on a flat plate and application to RANS modeling validation[END_REF]. This is particularly required in order to avoid the typically low values of skin-friction RANS predictions downstream of the bubble [START_REF] Bernardos | On the k-ω models behavior in the boundary-layer downstream of a short transitional separation bubble[END_REF]. Proper growth of the Reynolds shear stress is accomplished by boosting the triggering function ( f tr > 1), which facilitates a strong imbalance of production-to-dissipation ratio of turbulence in the transitional region. The k-ω model additionally requires a delay on the use of the shear stress limiter up to the region where an established turbulent boundary-layer is expected. This location, governed by the s c closure coefficient, is situated several bubble lengths downstream of the bubble.

As observed in Fig. 1, the triggering region scales with the bubble's laminar length, s trs sep , thanks to the use of the dimensionless abscissa coordinate formulation of Eqn. [START_REF] Crabtree | The Formation of Regions of Separated Flow on Wing Surfaces[END_REF]. This feature is convenient in order to automatically adapt the turbulence triggering to different bubble sizes.

In this work we also present results of a simple off-on triggering of the turbulence model. This commonly used approach, that we denote here as 'Step', consists in disabling the source terms of the turbulence model upstream of the transition onset location. The Step approach is presented with the shear stress limiter enabled. The closure constants of the LSTT model ( s a , s b , s c and a) were calibrated [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] to reproduce the c f distribution of the bubble of a flat-plate DNS reference [START_REF] Laurent | DNS database of a transitional separation bubble on a flat plate and application to RANS modeling validation[END_REF] using k-ω models. In Table 1 we also present the calibration adapted to the Spalart-Allmaras turbulence model, as shown in Fig. 2. As it can be observed, the LSTT calibration for both models is in correct agreement with the DNS skin-friction. Only a slight overestimation of the minimum value of c f is identified for the Spalart-Allmaras model. An interesting observation of Fig. 2 is that not only the c f predictions of the LSTT model are in better agreement with DNS, but also the shape factor H and the velocity profiles.

IV. Results

In the following paragraphs, the LSTT approach is evaluated using the established calibration of Table 1.

A. NACA 0012 Airfoil

In this section we present an assessment of the LSTT approach using k -ω Wilcox (2006) and Spalart-Allmaras turbulence models for two different operating conditions of the NACA 0012 airfoil. These conditions were chosen because high-fidelity simulations are available in literature. The first operating condition comes from the DNS of Jones et al. [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF], which yields α = 5 • and Re c = 50, 000. The second condition is issued from the ILES of Alferez et al. [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF], which yields α = 10.55 • and Re c = 100, 000. DNS Jones et al. [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF] with forced perturbations DNS Jones et al. [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF] In Fig. 3 the RANS predictions of skin-friction using LSTT are compared against the DNS data of Jones et al. [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF]. Two different DNS data sets are presented, resulting from a forced and unforced strategies. The forced DNS performs a velocity perturbation with 0.1% of amplitude of the free-stream velocity at the location (x/c, y/c) = (0.1, 0.129) yielding several frequencies that optimally excite the transition to turbulence within the separated shear-layer [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF]. The unforced DNS yields a self-sustained transition [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF], which is also observed in many unforced high-fidelity simulations [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF][START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF][START_REF] Visbal | High-Fidelity Simulation of Transitional Flows past a Plunging Airfoil[END_REF][START_REF] Breuer | Effect of Inflow Turbulence on an Airfoil Flow with Laminar Separation Bubble: An LES Study[END_REF].

The RANS computations presented in Fig. 3 were performed using automatic transition prediction via AHD-Gleyzes [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF][START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional Incompressible Flow[END_REF] criteria. A sweep of the N cr parameter (which is related to the turbulence intensity via Mack's relationship [START_REF] Mack | Transition prediction and linear stability theory[END_REF]) was performed between N cr = 6 and N cr = 14. The influence of turbulence level upon the displacement of separation and reattachment points is well captured by both turbulence models using LSTT. Indeed, Fig. 3 shows that an increase in turbulence level produces a downstream displacement of the separation point and an upstream displacement of the reattachment point. The RANS predictions using both turbulence models in Fig. 3 show that N cr = 12 curves match quite well the unforced DNS data. Furthermore, RANS predictions at N cr = 6 agree quite well with forced DNS data.

However, it is difficult to evaluate the precision of the transition onset prediction criteria based upon the DNS data. The forced DNS data yield a forced perturbation of 0.1%, which could be assimilated to N cr ≈ 8 following Mack's relationship. For N cr = 8, the RANS simulations predict a bubble which is slightly longer than the forced DNS data. This would mean that the transition criterion tends to predict a slightly downstream transition onset. However, the assumption that N cr = 8 shall be privileged because DNS was forced with a perturbation of 0.1% of amplitude may be overestimated. Nonetheless, from the results of Fig. 3 a,d the available information on the perturbation forcing [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF], we may infer that transition onset prediction is accurate within an uncertainty of about ∆N cr ≈ 2, which we consider quite successful for this case.

In the following we present a comparison of RANS computations against the ILES of Alferez et al. [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF] for the NACA 0012 at α = 10.55 • and Re c = 10 5 . In this case, we compare the predictions of both turbulence models using LSTT and typical Step off-on activation, as well as the predictions using Menter-Langtry γ-Re θt model.

Typical boundary-layer parameters are shown in Fig. 4. The skin-friction and pressure coefficients are computed as

c f ∞ = τ w /(0.5ρ ∞ |u| 2 ∞ ) and C p = (p -p ∞ )/(0.5ρ ∞ |u| 2 ∞ )
, respectively. As the ILES [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF] was unforced, no accurate physics-based information is available on the most appropriate value of N cr to be used. Therefore, preliminary sweeps on the value of N cr were performed (similarly to those presented in Fig. 3) that allowed us to infer that N cr = 12 is an acceptable choice. Note that N cr = 12 yielded also good RANS predictions for the unforced DNS of Jones et al. [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF] shown in Fig. 3.

We can observe in Fig. 4 that the computations of γ -Re θt and Step activations of k -ω and Spalart-Allmaras turbulence models produce a stalled condition. This effect can be seen through the skin-friction evolution. For the three aforementioned RANS computations, once the laminar boundary layer separates at about x/c ≈ 0.01, it remains separated (c f ∞ < 0) up to the trailing edge. The stalled condition predicted by these models can also be seen through the pressure distribution (where they present a lack of suction near the leading edge), the momentum thickness Reynolds number (where they present a too pronounced slope) and the boundary-layer shape factor (where they present very high values all over the airfoil, showing a massive separation). On the contrary, the LSTT activation of the RANS models produce more accurate predictions with respect to the ILES data. The bubble topology is well predicted by the LSTT-activated models, as the reattachment points are very close to the ILES and the characteristic positive peaks of skin-friction right downstream of the bubble are well captured. The Spalart-Allmaras LSTT activation produces a too pronounced negative peak of skin-friction, as it is systematically observed (Figs. 3 and2). Furthermore, the LSTT-predicted evolution of Re θ and H follows the abrupt jumps in the transitional region (0.05 x/c 0.1) that are characteristic of LSBs. Nevertheless, the pressure coefficient suction peak on the bubble region is slightly overestimated by the LSTT predictions.

The velocity profiles predicted by the different RANS computations can be observed in Fig. 5. In line with the boundary-layer predictions presented in the previous paragraph, Fig. 5 shows that γ -Re θt and Step-activation computations predict a stalled condition. The laminar separation (x/c = 0.05) is properly predicted by the aforementioned models, but from x/c 0.075 the velocity profiles begin to deviate from the ILES data, probably due to an insufficient production of turbulence stress. On the contrary, the LSTT-activated RANS models accurately follow the ILES evolution in the separated laminar region (x/c = 0.05), the transitional (x/c = 0.075) and reattachment (x/c = 0.1) regions, and even further downstream. From about x/c ≈ 0.5 the LSTT models predict a more filled velocity profile than the ILES. This is coherent with the slight underestimation of H observed in Fig. 4

B. SD7003 Airfoil

In this section we present the results of the RANS predictions on the flow around the SD7003 [START_REF] Selig | Summary of Low-Speed Airfoil Data[END_REF] airfoil. Several flow conditions are analyzed, as well as different prediction approaches.

In Fig. 6 we compare RANS predictions with available unforced LES data [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF][START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF] at α = 4 • and Re c = 60, 000. Several RANS approaches were employed. The LSTT activation was done using automatic transition prediction criteria [START_REF] Arnal | Laminar Instability Theory And Transition Criteria In Two and Three-Dimensional Flow[END_REF][START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF] using N cr = 11. The choice of N cr = 11 was done following a previous (not shown here) sweep on N cr (similarly to the sweep shown in Fig. 3) and by choosing the value that yielded the closest prediction with respect to the LES data. Remarkably, the chosen value for the present case, N cr = 11, is very close to N cr = 12, which was the value that produced the best LSTT predictions of the transition onset of the flows computed using unforced DNS [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF] and unforced ILES [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF] of Figs. 3 and4, respectively. The Step-activated RANS simulations used in the current study were performed by imposing the activation location exactly at the transition onset location predicted by the LSTT model, which is judged to be accurate enough.

As we observe in Fig. 6, the Menter-Langtry γ -Re θt model (used here with a threshold of N cr = 11 as well) and the Step-activated Spalart-Allmaras model produce very close predictions. The γ -Re θt model properly predicts the transition onset location, based on two observations: (1) the predicted minimum skin-friction peak is located close to the minimum peak of skin-friction of the LES at x/c ≈ 0.6; and (2) the end of the C p plateau agrees reasonably well with the LES one.

However, downstream of the transition onset, the γ -Re θt model predicts a reattachment point that is located 13% of the chord downstream of the LES reattachment point. The skin-friction downstream of the bubble predicted by γ -Re θt model is underestimated by a factor of ≈ 3. These observations downstream of the transition onset are related On the other hand, the LSTT activations using both k -ω and Spalart-Allmaras models produce good agreement with respect to the LES data, in both the laminar, transitional, and turbulent regions. A too pronounced minimum c f peak is observed for the Spalart-Allmaras model, which is also present in the other flows considered in the present study (Figs. 2, (b) k - [START_REF] Wilcox | Formulation of the k-ω Turbulence Model Revisited[END_REF] LSTT [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] (c) k -ω [START_REF] Wilcox | Formulation of the k-ω Turbulence Model Revisited[END_REF] Step (d) Spalart-Allmaras [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] LSTT [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] (e) Spalart-Allmaras [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] Step The authors believe that the reason why the LSTT approach is in better agreement with LES data than γ -Re θt and Step activations, is because LSTT is designed to account for the relatively fast rate of production of Reynolds shear stress in the transitional region, whereas the other approaches underestimate this rate of growth. In order to verify this point, a closer look on the Reynolds shear stress is done in Fig. 7. The Reynolds shear stress -u v is plotted using different RANS approaches, and they are compared against the available PIV experimental data of Hain et al. [START_REF] Hain | Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils[END_REF]. Although the chord-based Reynolds number of the experiment was slightly different than the RANS computations (66,000 versus 60,000), we consider this difference sufficiently low for allowing qualitative comparisons focused on the behavior of the models. In Fig. 7.(b) and 7.(d) we can observe that LSTT approach predicts similar shear stress when using k -ω and Spalart-Allmaras models. The overall shape and location of the produced shear stress predicted by the LSTT computations are in good agreement with the experiment (Fig. 7.(a)), although the maximum value is overestimated. On the contrary, the Step approach (Fig. 7.(c) and (e)) and Menter-Langtry model (Fig. 7.(f)) are in less good agreement with the experimental data, as the predicted values of shear stress are underestimated and the location of its maximum value do not match the location observed in the experiment. Therefore, we may conclude that the reason why γ -Re θt and Step approaches underestimate the skin-friction downstream of the transition onset may be related to a lack of production rate of shear stress in the transitional region, which is the essential modeling issue addressed by the LSTT approach. In Fig. 8 we assess the influence of the angle of attack. With increasing α (Fig. 8), the bubble shrinks from ∼ 55%c at α=2 • (separation at x 0.4c with reattachment at x 0.95c very near the trailing-edge) to ∼ 20%c at α=11 • (separation very near the leading edge with reattachment at x 0.20c).

1.0 = 2 • -1.0 -0.5 0.0 0.5 1.0 ∞ ×10 -2 = 6 • -2 -1 0 1 = 6 • -2 -1 0 1 ∞ ×10 -2 = 8 • -3 -2 -1 0 1 = 8 • 0 0.2 0.4 0.6 0.8 1 ∕ -3 -2 -1 0 1 ∞ ×10 -2 = 11 • 0 0.2 0.4 0.6 0.8 1 ∕ -4 -2 0 = 11 •
Regarding the importance of the triggering approach, the conclusions are quite similar to those drawn for the baseline case (α = 4 • , Re c = 60, 000). Abrupt activation of the turbulence model at the transition onset generally fails, possibly returning stalled flow with an open bubble (Fig. 8) as the prediction using the k -ω model shows.

On the contrary, the LSTT computations are generally in satisfactory agreement (Fig. 8) with LES data [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF][START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF]. At α = 2 • , the agreement with the LES data of Galbraith and Visbal [START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF] is excellent for k -ω and quite good for Spalart-Allmaras. At α = 6 • , there is a slight scatter between the 2 sets of LES data [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF][START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF], the LSTT computations being in close agreement with Catalano and Tognaccini [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF]. At α=8 • , where the 2 sets of LES data are in better agreement one with another, the k -ω + LSTT model suffers from the prediction by the criteria of too early a transition-onset location, which induces an earlier negative c f ∞ peak and earlier reattachment, also resulting in a slight discrepancy of the pressure plateau associated with the LSB. The same remarks apply to the α = 11 • case (Fig. 8) regarding k -ω LSTT. On the contrary, at α = 8 • the Spalart-Allmaras LSTT model does not produce a too upstream transition onset prediction. However, let us highlight that at α = 11 • the Spalart-Allmaras LSTT model predicts a stalled condition. Hence, we believe that the reason why at α = 8 • the k -ω LSTT predicts a too early transition and Spalart-Allmaras LSTT does not, is because Spalart-Allmaras at α = 8 • is very close to bubble bursting.

Despite the fact that Spalart-Allmaras LSTT model does not successfully predict the LES data at this near-stall condition (α = 11 • ), this prediction shows that LSTT activation (which boosts Reynolds stress production) does not inhibit the capacity of the model to predict stall and bubble bursting. Contrarily, by successfully predicting moderate-to-high angles of attack LSBs, the LSTT approach may contribute to enhancing the RANS ability to capture the airfoil's stall threshold. In order to determine the reasons for the observed discrepancies between the k -ω + LSTT predictions and the LES data [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF][START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF] at the higher α ∈ {8 • , 11 • } (Fig. 8) we performed an additional calculation at α=11 • (Fig. 9), where the transition onset was prescribed at x tr = 0.092c further downstream of the criteria-predicted (N cr ≈ 11) location of transition onset at x tr = 0.066c. This ad hoc adjustment of the transition onset location greatly improves the agreement with LES: the location of the negative c f ∞ peak and of reattachment are in excellent agreement with the LES computations, and these results in a substantially improved prediction of the pressure plateau in the C p distributions (Fig. 9).

These results demonstrate that the LSTT approach is quite successful in controlling the turbulence built-up in the transitional region, provided that the correct x tr is obtained by the transition criteria. Inversely, the increased discrepancy of the k -ω + LSTT computations with LES as α increases is principally associated with the inaccurate (early) prediction of transition onset. This would imply that with increasing α, N cr should also increase. Mack's correlation would then suggest that the effective [START_REF] Dullenkopf | An account of free-stream-turbulence lengthscale on laminar heat transfer[END_REF][START_REF] Mayle | The Turbulence That Matters[END_REF] T u tr decreases. An argument in support of this supposition is that as the laminar separation point moves upstream, closer to the stagnation point, with increasing α, the boundary-layer thickness at separation δ sep is modified, and so does the receptivity of the boundary-layer to a given spectrum of external turbulence, in line with theoretical arguments [START_REF] Dullenkopf | An account of free-stream-turbulence lengthscale on laminar heat transfer[END_REF][START_REF] Mayle | The Turbulence That Matters[END_REF] corroborated by numerical stability calculations [START_REF] Vermeersch | Klebanoff-Mode Modeling and Bypass-Transition Prediction[END_REF]. This is naturally incorporated in LES dynamics but not in Mack's correlation (Eq. 7) which assumes a monotonic bijection N cr (T u tr ). With regard to the previous results (Figs. 8,9), it may be surmised that, as α increases and the LSB size shrinks, increased diffusion of momentum by the Reynolds shear-stress -u v is necessary to compensate the high inertial forces of the flow and the increased deceleration associated with the adverse local pressure gradient. It is precisely because the LSTT approach, by boosting production of turbulence and inhibiting the shear limiter of k -ω models until after fully turbulent conditions are reached, achieves a better prediction of the Reynolds shear-stress -u v (Fig. 7) that it performs better than the Step-activated approaches, significantly improving the predictive accuracy of the bubble's topology. The convection effect of the correct prediction of the transitional region in the LSTT approach translates into proper estimation of the c f values downstream of the bubble.

C. T106C turbine blade

The T106C test case is based on the experiments performed by Michalek et al. [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]. They studied the low-pressure high-lift turbine blade T106C on a linear cascade in the VKI S1/C high-speed wind tunnel operating at exit Mach number of M = 0.65. Multiple cases were analyzed, particularly for varying chord-based Reynolds number and free-stream turbulence levels. In this study, two configurations are considered as shown in Fig. 10, the Re c = 120, 000 and Re c = 250, 000 cases, both at T u ∞ = 0.9%. The k -ω [START_REF] Wilcox | Formulation of the k-ω Turbulence Model Revisited[END_REF] LSTT prediction is compared against the γ -Re θt model with Content-Houdeville [START_REF] Content | Méthode innovante pour le calcul de la transition laminaire-turbulent dans les codes Navier-Stokes[END_REF] correlations with T u = 0.9%s. For the LSTT approach, the AHD-Gleyzes [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF][START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional Incompressible Flow[END_REF] criteria were employed, initially using N cr = 2.9 (T u = 0.9%) and then adjusted to N cr = 10 in order to fit the experimental data transition onset for assessing the turbulence model's activation. In Fig. 10 we present the chordwise distribution of isentropic Mach M is of the suction side of the blade. The γ -Re θt model produces good predictions of this flow. We observe only a slightly slow recovery from the end of the plateau of M is , which may translate into an overestimation of the bubble size. On the other hand, the LSTT automatic transition prediction with the prescribed turbulence value of T u = 0.9% (N cr = 2.9) predicts a transition onset located at x tr /c ≈ 0.80, which is quite far upstream of the observed experimental transition onset (x tr /c ≈ 0.85). As a result, LSTT computations using N cr = 2.9 fail to predict the LSB. In order to assess the influence on the value of N cr , a sweep of this parameter was done. It was found that using N cr = 10 the LSTT computations yield very good predictions with respect to experimental data. However, N cr = 10 is associated with very low turbulence level (T u ≈ 0.046) following Mack's relationship. This last aspect is unsatisfactory, as accurate transition criteria should provide correct transition onset prediction for measured experimental turbulence level. This inaccuracy may be caused by one or several of the following points: (a) employed transition criteria may not be precise enough for this flow condition, (b) Mack's relationship may not relate precisely enough the freestream turbulence level to the actual turbulence breakdown threshold and (c) the measured value T u = 0.9% may be much higher than the actual turbulence level value at the laminar boundary-layer edge, T u δ , where the receptivity mechanism takes place.

M is T106C Re c =250,000 T u ∞ =0.9% Experiment γ -Re θt LSTT N cr = 10 LSTT N cr = 2.9
Considering the LSTT computation at N cr = 10, we can evaluate the turbulence model's activation precision with respect to γ -Re θt and the experimental data. Hence, we can observe that LSTT approach yields a steeper recovery after the plateau than γ -Re θt prediction. This results in better agreement with experimental data. A similar effect was observed and discussed upon C p distribution of the SD7003 airfoil in Fig. 6. Thus, in a similar fashion as concluded in the analysis of Fig. 6, the fact that LSTT predicts a steeper and more accurate recovery after the plateau (Fig. 10) may be due to LSTT better accounting for the smooth and important production of Reynolds shear stress in the transitional region.

V. Conclusion

Several transitional RANS approaches applied to a collection of low Reynolds number aerodynamic flows were assessed. The recently proposed non-local Laminar Separation Transition Triggering (LSTT) [START_REF] Bernardos | Algebraic Nonlocal Transition Modeling of Laminar Separation Bubbles using k-ω Turbulence Models[END_REF] approach, originally designed for use with k -ω [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF][START_REF] Wilcox | Formulation of the k-ω Turbulence Model Revisited[END_REF][START_REF] Wilcox | Reassessment of the scale-determining equation for advanced turbulence models[END_REF][START_REF] Kok | Resolving the Dependence on Freestream Values for the k-Turbulence Model[END_REF] turbulence models and transition criteria [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF][START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional Incompressible Flow[END_REF], was extended to the one-equation Spalart-Allmaras [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] turbulence model. The LSTT approach was compared to typical Step off-on [START_REF] Windte | Validation of the RANS-simulation of laminar separation bubbles on airfoils[END_REF][START_REF] Catalano | Turbulence Modeling for Low-Reynolds-Number Flows[END_REF] activation of RANS turbulence models and transport equation-based γ -Re θt [START_REF] Langtry | A Correlation-Based Transition Model Using Local Variables -Part II: Test Cases and Industrial Applications[END_REF][START_REF] Langtry | Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes[END_REF][START_REF] Content | Méthode innovante pour le calcul de la transition laminaire-turbulent dans les codes Navier-Stokes[END_REF][START_REF] Menter | A Correlation-Based Transition Model Using Local Variables -Part I: Model Formulation[END_REF][START_REF] Langtry | Transition Modeling for General CFD Applications in Aeronautics[END_REF] transitional model. RANS computations were systematically compared against available published DNS [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF][START_REF] Laurent | DNS database of a transitional separation bubble on a flat plate and application to RANS modeling validation[END_REF], LES [START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF], ILES [START_REF] Alferez | Study of Stall Development Around an Airfoil by Means of High Fidelity Large Eddy Simulation[END_REF][START_REF] Galbraith | Implicit Large Eddy Simulation of Low-Reynolds-Number Transitional Flow Past the SD7003 Airfoil[END_REF] or experimental [START_REF] Hain | Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils[END_REF][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] data.

Results show that LSTT approach produces an enhancement of the precision of the RANS predictions, particularly for low Reynolds number flows. The present analysis suggests that Step activations and γ -Re θt model tend to underestimate the rate of growth of turbulent shear stress in the transitional region, which provokes a downstream delay of the reattachment point or may even produce an unphysical stalled condition. This scenario is problematic at high angle-of-attack and low Reynolds number flows, where the accurate prediction of the laminar separation bubble determines the RANS model's ability to predict the bubble bursting threshold, which is commonly the main cause of stall. It is inferred that proper modeling of the LSB is also of fundamental interest for accurately predicting the downstream turbulent boundary-layer profile, notwithstanding the natural limits of the original turbulence model's behavior [START_REF] Bernardos | On the k-ω models behavior in the boundary-layer downstream of a short transitional separation bubble[END_REF].

As the LSTT approach requires the use of non-local transition criteria, this means that the accuracy and robustness of the method depends on those of the transition criteria. This may yield several advantages, like the use of accurate physics-based correlations using integral boundary-layer parameters especially suited for capturing each transition mechanism (natural, separation-induced, crossflow, bypass...). But it may also yield several disadvantages, like a more important implementation effort than local transport equation-based approaches [START_REF] Langtry | Transition Modeling for General CFD Applications in Aeronautics[END_REF] particularly for massively parallel unstructured grids. The implementation of non-local transition criteria has proven to be suitable for massively parallel structured codes [START_REF] Cliquet | Application of Laminar-Turbulent Transition Criteria in Navier-Stokes Computations[END_REF], but its extension to unstructured grids is still under work.

This work shows that one of the main assets of the LSTT approach is the use of the non-local length scale s trs sep . This scale, which was also used by Lian and Shyy [START_REF] Lian | Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil[END_REF], seems to be an appropriate choice in order to properly control the non-equilibrium region where the boost of turbulence is promoted. The LSTT approach shows that this scale can be used for appropriately delaying shear limiters that may produce an unfavorable effect in the transitional region. Basing the turbulence activation mechanisms upon this lengthscale allows for proper dependence upon physical conditions like T u , α and Re c . The assumption performed by LSTT which consists in replacing the commonly accepted intermittency factor, γ, as the main mechanism responsible for the turbulence model's activation, by the empirical weighting coefficient f tr , allowed to overcome irrelevant physical constraints inherent to γ. Such constraints are its physical bounds (γ ∈ [0, 1]), its presumed universality as a physical quantity (particularly problematic when used for triggering different turbulence models) or even the controversy on how γ (issued from conditional averaging [START_REF] Dopazo | On conditioned averages for intermittent turbulent flows[END_REF]) interacts with the unconditioned quantities of a RANS framework. On the contrary, this study shows that empirical f tr ( γ) functions designed and calibrated ad-hoc for each considered turbulence model significantly improves the RANS predictions.

A number of perspectives can be inferred from the present work. First, the LSTT approach may be extended to other turbulence models and this may contribute to further understanding the necessity of boosting the turbulent shear stress in the transitional region. Second, modeling work can be performed in order to transform LSTT into a purely local formulation, ready to use with general unstructured grids. Third, a thorough evaluation of the prediction capabilities of the bubble bursting phenomenon, even for dynamic motion of the airfoil, would be desirable as this phenomenon is related to stall. And last, further effort on improving the accuracy and robustness of transition criteria is of great importance in order to produce accurate RANS predictions, as was shown in this study. A reassessment on the Mack's relationship may be desirable in order to better account for the critical total amplification value determining the turbulence breakdown threshold for varying inflow turbulence spectra, including those existing in DNS and ILES.

Where the constants c 1 and c 2 are found by introducing the set { s P 1 , Re θ (s P 1 ) , s P 2 , Re θ (s P 2 ) } into (13), and solving for c 1 and c 2 . We found that extrapolations of the Gleyzes integral function ( 6) is quite insensitive to the value of λ, provided it yields a low enough value. We suggest using λ = 10 -2 .

For concluding the presentation of this extrapolation strategy, we must tell the algorithm how to behave in case one or several requirements cannot be verified or if one or several control points cannot be found. In the event one of the three s P i points cannot be computed (due to, e.g. a mildly detached boundary layer where H max < 5) an attempt can be done to produce a monotonically increasing linear fit with the two remaining control points. In case that a linear fit is not possible or if the resulting parabola fit for φ is convex, the algorithm should use the computed boundary-layer H and Re θ values in [START_REF] Gaster | The Structure and Behaviour of Laminar Separation Bubbles[END_REF]. Following this rationale, the stabilization technique presented in this appendix shall only intervene in highly detached boundary-layers, which actually constitute the problematic scenario where the standard Gleyzes [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF] formulation (5) fails.

  abcissa C lim = shear-stress limiter delay function ŝa-c , a = LSTT closure coefficientsC p = pressure coefficient T u = turbulence intensity, (2k/3) 1/2 /|u| c f = flat-plate's skin-friction coefficient (τ w / 1 2 ρ e |u e | 2 ) ūi = local Reynolds-averaged velocity c f ∞ = airfoil's skin-friction coefficient (τ w / 1 2 ρ ∞ |u ∞ | 2 ) ūs =velocity component projected onto s f tr = Transition function ūse = boundary-layer edge velocity projected onto s H = boundary-layer kinematic shape factor (δ * /θ) |u e | = boundary-layer edge velocity k = turbulence kinetic energy |u ∞ | = inflow freestream velocity k L = laminar kinetic energy α = airfoil's angle of attack M = Mach number γ = Intermittency factor N = total amplification factor δ = boundary-layer thickness N cr = critical total amplification factor δ * = boundary-layer displacement thickness n = wall-normal coordinate θ = boundary-layer momentum thickness p = static pressure Λ 2 = Pohlhausen parameter Re c = chord-based Reynolds number, ρ ∞ c |u ∞ |/µ ∞ µ = molecular dynamic viscosity Re θ = momentum-thickness Reynolds number (ρ e θ |u e |/µ e ) µ t = dynamic eddy viscosity Sij = mean strain rate tensor ν = Spalart-Allmaras' model transported quantity s = wall-tangent curvilinear abscissa ρ = density s sep = laminar separation point τ w = wall shear stress s tr = transition onset ω = Turbulence specific dissipation rate

Fig. 2

 2 Fig. 2 Comparison of predicted skin-friction, shape factor and velocity profiles using the Step and the LSTT approach, coupled with the k-ω [52] and Spalart-Allmaras [51] turbulence models, against the DNS data of Laurent et al. [47] over a flat-plate configuration with adverse pressure gradient.

5 • 4 Fig. 3

 543 Fig. 3 Skin-friction RANS predictions of the influence of the transitional turbulence level threshold, T u (or N cr ), on the suction side of a NACA 0012 at α = 5 • and Re c = 50, 000 using k -ω Wilcox (2006) [52] and Spalart-Allmaras [51] turbulence models with AHD-Gleyzes [44, 53] transition onset prediction criteria and LSTT [41] turbulence activation against DNS data of Jones et al. [13].

Fig. 4

 4 Fig. 4 Boundary-layer parameters on the suction side of the NACA 0012 airfoil at α = 10.55 • and Re c = 10 5 predicted by several RANS models compared against ILES results [59].

8 ILESFig. 5 Fig. 6

 856 Fig. 5 Comparison of velocity profiles for several models for the NACA 0012 at α = 10.55 • and Re c = 10 5 [59]. The airfoil-tangential velocity u s is plotted against the airfoil's wall-normal direction n. The tangential velocity component has been scaled by its value at the boundary layer edge of the ILES data u (ILES) se . The airfoil's distance to the wall n has been scaled with respect to the integral displacement thickness of the ILES data δ * (ILES) .

Fig. 7

 7 Fig. 7 Comparison of measured (Hain et al. [8]) levels of Reynolds shear-stress (α=4 • , Re c =66,000) with computations (α=4 • , Re c =60,000) using different transition triggering strategies (k-ω [52]+LSTT [41], Spalart-Allmaras [51] +LSTT and k-ω [52] with Step activation and shear limiter).

ILES

  Galbraith and Visbal [64] LES Catalano and Tognaccini[START_REF] Catalano | Influence of Free-stream Turbulence on Simulations of Laminar Separation Bubbles[END_REF] k -ω Wilcox (2006) LSTT k -ω Wilcox (

Fig. 8

 8 Fig. 8 Comparison of chordwise distributions of skin-friction (c f ∞ on the suction side) and of pressure coefficient C p obtained from different RANS computations with available LES data [60, 64] at different angles of attack (α=2 • ,6 • ,8 • ,11 • ) at Re c =60,000 and N cr =11 (T u tr ≈0.03%)

Fig. 9

 9 Fig. 9 Comparison of c f ∞ (on the suction side) and C p obtained from LES [64] against RANS computations (k -ω [52] + LSTT [41]) using automatic or prescribed transition onset location.

T106C

  Re c =120,000 T u ∞ =0.9% Experiment γ -Re θt LSTT N cr = 10 LSTT N cr = 2

Fig. 10

 10 Fig. 10 Comparison of isentropic Mach, M is , over the suction side of the T106C turbine blade, using k-ω [52] LSTT model and Menter-Langrty γ -Re θt (with T u = 0.9%) [17, 70] model with Content-Houdeville [69] correlations, against available wind tunnel experimental data.

Table 1 LSTT [41] closure coefficients for the considered turbulence models

 1

	Turbulence model	s a	s b	s c	a
	k -ω Wilcox 2006 [52] 0.182 0.414 5.83 2.15
	Spalart-Allmaras [51] 0.482 3.107	-	7.03

  from about x/c 0.5.

		2	×10 -2								-4 -3			ILES Alferez et al. [13] Wilcox (2006) LSTT Wilcox (2006) Step Spalart-Allmaras LSTT
	∞	0				0.02						-2			Spalart-Allmaras Step γ -Re θt [18]
	-2			-0.02 0.00						-1
	0 2000 -4	0.2	0.4	∕	0.6 0.05	0.8 0.10	1 0.15	0	0	0.2	0.4	∕	0.6	0.8	1
	1500
	1000				400
	500				200
			0	0	0.2	0.4	∕	0.6 0.05 0.10 0.15 0.8 1		0	0.2	0.4	0.6	0.8	1

  3 and 4).

		0	0.012 0.023 0.035	-′ ′ ∕ 2 ∞
	∕ ∕ ∕ ∕ ∕ ∕	0.04 0.06 0.08 0.04 0.06 0.08 0.04 0.06 0.08 0.04 0.06 0.08 0.04 0.06 0.08 0.04 0.06 0.08	∕ 0.35 0.40 0.45 0.50 0.55 0.60 0.65 (a) Wind tunnel [8]

Acknowledgments

The authors would like to acknowledge the great support and useful insights of Prof. G. Gerolymos from Sorbonne Universités and to our colleagues at the DAAA and DMPE Departments at ONERA. Special thanks to Mr. Hain, Mr. Kähler and Mr. Radespiel for sharing their PIV data with us. The simulations run in this study were executed using the software elsA [75], which is issued of the three-party agreement between AIRBUS, SAFRAN, and ONERA. All the preand post-process tasks were performed using the Cassiopee [76] software, and all the figures presented in this work were produced with the Matplotlib [77] library. This work was completely funded by ONERA in the framework of the PhD thesis of the first author.

A. Appendix: Stabilization of Gleyzes Transition Criterion

As the Gleyzes's [START_REF] Gleyzes | A Calculation Method of Leading-Edge Separation Bubbles[END_REF] correlation shows [START_REF] Somers | Design and Experimental Results for the S809 Airfoil[END_REF], the criterion implicitly assumes a monotonically increasing Re θ with respect to the curvilinear abscissa s. However, in general this is not verified, particularly for deeply detached boundary-layers (H 5), as the evolution of Re θ with respect to s may stagnate, or even decrease [START_REF] Horton | Laminar separation bubbles in two-and three-dimensional incompressible flow[END_REF]. Under these circumstances, N(s) value issued from the integration of (5) may stagnate, or even decrease, systematically delaying downstream the transition onset location and compromising the RANS simulation. This is not a physically desired behavior. On the contrary, we would rather expect the inverse behavior, viz. a slight upstream displacement of the transition onset, as under deeply detached conditions the flow is very instable, promoting a more rapid growth of perturbations leading to turbulence.

In order to solve this issue, the authors suggest the use of an algebraic extrapolation technique designed to guarantee the monotonicity of the integrand of [START_REF] Somers | Design and Experimental Results for the S809 Airfoil[END_REF]. This also enhances the numerical convergence of the simulation, as the transition onset prediction becomes less sensitive to the boundary-layer quantities at the transition onset location. The authors suggest that this sensitivity, which is particularly pronounced in recirculating flow, is one of the most common causes of transition onset location oscillations leading to poor convergence.

In the following, the aforementioned algebraic extrapolation technique is presented. The idea consists in replacing the co-processed H(s) and Re θ (s) values by a mathematical approximation φ and ψ constructed from the exact boundary-layer values at several points s P i located in the laminar region, which we denote control points,

The approximation functions φ and ψ shall be monotonically increasing. They shall extrapolate precisely enough the H and Re θ laminar trend from the separation point. The functions shall be reasonably independent on the choice of the control points s P i .

We suggest exactly fitting φ to a concave parabola using three characteristic values, H P 1 = 3, H P 2 = 4 and H P 3 = 5, used to infer the control points locations by assuming local bijection, s P 1 = s(H = H P 1 ) (12a) s P 2 = s(H = H P 2 ) (12b) s P 3 = s(H = H P 3 ) (12c)

We suggest fitting ψ to a monotonically increasing power function, ψ = c 1 s λ + c 2 [START_REF] Jones | Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence[END_REF]