domain: spi phys

text:
M Daroukh
email: majd.daroukh@onera.fr.

C Polacsek
email: cyril.polacsek@onera.fr.

M Carini
email: marco.carini@onera.fr.

Acoustic Assessment of BLI Effects on Airbus Nautilius Engine Integration Concept -Part I: Noise Generation

This paper presents an acoustic evaluation, based on computational fluid dynamics (CFD) simulations, of the Airbus Nautilius engine integration concept applied to the Onera NOVA airplane. This innovative concept is characterized by the integration of two ultra high bypass ratio (UHBR) engines into the fuselage rear end so as to maximize the boundary layer ingestion (BLI). This concept is evaluated against a reference configuration with conventional podded engines. The study is made at take-off conditions representative of the sideline acoustic certification point and is limited to fan tonal noise which is expected to be the most affected contribution to aircraft noise by the BLI. The numerical methodology is based on two steps. First, steady simulations of the whole aircraft with body force modeling for the fan are performed in order to provide the flow characteristics at the inlet and outlet of the engine ducts. Then, unsteady simulations of the engines restricted to the in-duct domain are conducted using the preceding flow characteristics. All numerical simulations are performed with Onera's CFD solver elsA. Steady and unsteady aerodynamic results are evaluated with a focus on the main noise generation mechnisms. Acoustic power estimates are also provided by applying Goldtsein's analogy with the unsteady loadings on fan blades and stator vanes as source terms.

I. Introduction

In order to limit as much as possible the aviation's impact on the environment, the Advisory Council for Aviation Research and Innovation in Europe (ACARE) has set challenging goals in terms of pollutant emissions and noise reduction. In particular, reductions of 75% of CO 2 emissions, 90% of NO x emissions and 65% of noise are to be reached by 2050 [START_REF] Acare | Strategic Research and Innovation Agenda -Executive summary[END_REF]. To reach these ambitious objectives, it seems inevitable to switch from the classic tube-and-wing design that has been optimized over the last decades.

A propulsion system based on boundary layer ingestion (BLI) is one of the disruptive solutions that could help to fulfill the goals related to pollutant emissions. The principle consists in embedding the nacelle within the airframe and thus ingesting the fuselage boundary layer. This results in lower engine inlet and outlet velocities and improves the propulsive efficiency. To be the most efficient, the maximum part of fuselage boundary layer must be ingested while keeping the most uniform inflow in the engines (otherwise, the gains could be counterbalanced by a less efficient engine) [START_REF] Wiart | Exploration of the Airbus 'Nautilius' engine integration concept[END_REF].

Several BLI concepts exist and exhibit different power savings. For example, the MIT D8 concept is a large fuselage aircraft with two engines installed on the flat rear end surface that ingest the upper part of the fuselage boundary layer. The power saving is estimated to 9% when compared to a similar aircraft configuration with podded engines [START_REF] Pandya | Computational Assessment of the Boundary Layer Ingesting Nacelle Design of the D8 Aircraft[END_REF][START_REF] Menegozzo | Boundary Layer Ingestion Propulsion: A Review on Numerical Modeling[END_REF]. The Onera NOVA airplane is of similar size but the BLI is exploited by partially burying the two engines onto the fuselage sides. Comparison with the Onera NOVA airplane equipped with podded engines shows a power saving of 5% [START_REF] Wiart | Exploration of the Airbus 'Nautilius' engine integration concept[END_REF][START_REF] Menegozzo | Boundary Layer Ingestion Propulsion: A Review on Numerical Modeling[END_REF][START_REF] Godard | Fan Design Investigation on the Airbus Nautilius Engine Integration Concept[END_REF]. Both concepts permit only a part of the fuselage boundary layer to be ingested (around 30 to 40%) and makes the fan to operate in a highly distorted flow (because of the important asymetry of the integration).

This paper focuses on the innovative Airbus Nautilius engine integration concept illustrated in Fig. 1(b), with the two engines embedded at the rear end of the fuselage where the tail has been split into two parts that also act as engine hubs [START_REF] Rogero | Ensemble pour aéronef comprenant des moteurs à propulsion par ingestion de la couche limite[END_REF][START_REF] Negulescu | Assembly for aircraft comprising engines with boundary layer propulsion by injection[END_REF]. The idea of this propulsion system is to maximize the portion of ingested fuselage boundary layer while limiting the azimuthal flow distortion at fan face. Recent studies of this concept applied on Onera NOVA aircaft have shown a promising power saving estimate higher than 10% [START_REF] Wiart | Exploration of the Airbus 'Nautilius' engine integration concept[END_REF][START_REF] Godard | Fan Design Investigation on the Airbus Nautilius Engine Integration Concept[END_REF]. The objective of the present study is to provide first acoustic assessment of this configuration, by comparison with a reference one with podded engines. As the BLI is expected to mainly affect the fan tonal noise through the flow distortion, the study is limited to this contribution. The focus is put in this paper on noise generation mechanisms and noise source evaluation as the noise propagation is dealt with in a follow-up paper [START_REF] Lorteau | Acoustic Assessment of BLI Effects on Airbus Nautilius Engine Integration Concept -Part II: Noise Radiation[END_REF].

The paper is structured as follows. The configurations are presented in Sec. II and the numerical methodology is detailed in Sec. III. Aerodynamic and acoustic results are then provided in Secs. IV and V respectively. Finally, conclusions are given in Sec. VI.

II. Presentation of the Configuration

The Nautilius concept applied to the Onera NOVA airplane (called NOVA-Nautilius configuration) is evaluated against the reference NOVA airplane configuration with podded engines (called NOVA-SFN for side fuselage nacelle). Both configurations are shown in Fig. 1. Note that tail plane, rudder and high-lift design were not available for the Nautilius concept and thus they could not be included in the present study. The NOVA-Nautilius configuration considered in the present paper is derived from the one used in the studies that demonstrate the relevance of the concept [START_REF] Wiart | Exploration of the Airbus 'Nautilius' engine integration concept[END_REF][START_REF] Godard | Fan Design Investigation on the Airbus Nautilius Engine Integration Concept[END_REF] but slightly differs after several design improvements realized during the ADEC European project. These are mainly:

• the selection of the appropriate highlight section to remove flow separation occurring inside the nacelle,

• the reduction of high vorticity regions at the fan face produced by the shrinking section of the aft fuselage,

• and the modification of the fan nozzle exhaust area to reach a fan pressure ratio (FPR) closer to the one of the reference configuration. With this last design of the NOVA-Nautilius configuration, a promising power saving of about 11% has been reached. The acoustic study presented in this paper is done at take-off conditions representative of the sideline acoustic certification point (Mach number of 0.25 and angle of attack of 8 degrees). The comparison between the two configurations is performed at iso-rotational speed (around 115% of the nominal speed).

III. Numerical Methodology

A. Generalities

The computational fluid dynamics (CFD) simulations are perfomed using Onera's CFD solver elsA which is based on a cell-centered finite volume approach on a structured multi-block grid [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]. To avoid the prohibitive cost of a unique unsteady simulation, a two-step strategy is adopted:

1) steady simulations of the whole aircraft with body-force modelling of the fan stage are first performed to provide the target engine operating point and the flow field at engine inlet plane, 2) unsteady simulations restricted to the in-duct part of the engine are then performed in order to provide the main acoustic sources. Details on both steps are given below.

B. Airplane Steady Simulations

The airplane simulations are performed by solving the steady Reynolds-averaged Navier-Stokes (RANS) equations using an implicit pseudo-time marching method with backward Euler scheme and a second-order Jameson scheme [START_REF] Jameson | Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes[END_REF] in space. The turbulence closure is dealt with the Spalart-Allmaras model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. To manage the complexity of the configurations, the different parts of the aircraft (fuselage, wing, nacelle) are assembled together with a Cartesian grid using the chimera technique while the fan stage is modelled through a body-force approach [START_REF] Godard | Fan Design Investigation on the Airbus Nautilius Engine Integration Concept[END_REF][START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF]. Far-field boundary conditions are applied on all external boundaries.

Half-part of the domain is computed given the symmetry with respect to the vertical plane and comprises around 35 millions points. The simulations are run over 84 processors and last approximately ten hours.

C. Engine Unsteady Simulations

The engine simulations are performed by solving the unsteady RANS equations in the relative frame with classic 2nd order schemes in space (Roe with van Albada limiter [START_REF] Roe | Approximate Riemann solvers, parameter vectors, and difference schemes[END_REF]) and time (Gear), while the turbulence is modelled by the Wilcox k-ω two-equation model [START_REF] Wilcox | Reassessment of the Scale-Determining Equation for Advanced Turbulence Models[END_REF]. The full-annulus domains shown in Fig. 2 that account for the 18 fan blades and the 40 outlet guide vanes (OGVs) are simulated for both configurations and extend from the engine inlet plane to the engine exit plane. Note that for the Nautilius configuration, the inlet duct is made axisymmetric for simplification (while its original shape is not). The fluid fluxes are transferred between the rotating domain and the fixed one using a sliding no-match interface. A pressure condition that satisfies the radial equilibrium is used at the outlet and a valve law adjusts the value of pressure at a specified point during the iterations to be as close as possible to the targeted operating point. The distortion map extracted from the aircraft simulation is specified at the inlet in terms of total pressure, total enthalpy and flow direction. In order to be expressed in the rotating frame, this distortion map is rotated at each iteration in the opposite direction of the fan rotation (after being made axisymmetric for the Nautilius case). The inlet boundary condition is thus updated at each time step using the so-called trigger capacity of the elsA software.

The meshes are resolved at walls and are designed in order to propagate acoustic waves up to the three times the blade passing frequency (BPF). This leads to a mesh of 565 millions points for the SFN case and of 510 millions points for the Nautilius one (the small difference results from the longer duct in the first configuration). Close to the inlet and outlet planes, the cells are stretched to avoid numerical reflections at the boundaries. Finally, the time step is defined such that one blade passage is described by 200 iterations, which leads to a total number of 3600 iterations per rotation. The simulations are launched over 896 processors and one rotation is computed in about 125 hours. To get a convergence of the acoustic sources (unsteady loadings on fan blades and stator vanes), four rotations have been computed for the SFN case and three rotations were needed for the Nautilius one.

D. Validation of the Chaining Methodology

To ensure the consistency of the proposed methodology, the time-averaged results of the engine uRANS simulations are compared with the ones of the aircraft RANS simulation inside the engine duct. It is especially important to get a reliable distortion map just upstream of the fan as this will be the cause of the penalty noise induced by BLI. In this purpose, the axial velocity results at duct inlet plane are given in Fig. 3 for both configurations. The results have been normalized by the averaged velocity at fan plane of each case. The agreement between both the RANS and uRANS distortion maps is very good for the SFN case. For the Nautilius configuration, minor differences are observed with higher velocities close to the fuselage (right side of the figure) in the uRANS solution, and may be attributed to the axisymmetric assumption. The same comparison made at fan plane for the Nautilius case given in Fig. 4 shows that this higher velocity pocket is attenuated in the duct and does not reach the fan. As a result, the distortion that interacts with the fan is very similar to the one predicted by the aircraft RANS simulation.

IV. Aerodynamic Results

A. Qualitative Flow Visualization

The main flow patterns are first highlighted by qualitative flow visualization. Contour maps of normalized axial velocity extracted from the aircraft RANS simulations are given in Fig. 5 for both SFN and Nautilius cases in the vicinity of the engines. Along the vertical plane (top), the flow fields inside the engines look relatively similar between both configurations. In the horizontal plane however (bottom), the integration of the engines within the fuselage in the Nautilius concept introduces an important asymmetry about the engine axis close to the inlet. Inside the engine duct and relatively far from the inlet plane where the geometry almost match between the two configurations, the flow fields look again similar.

A visualization of the unsteady flow inside the engines is now provided in Fig. 6 in which instantaneous contour maps of normalized velocity magnitude are extracted from the uRANS simulations on a radial slice corresponding to 75% of fan blade height. Note that, for clarity, the slices are unwrapped with the x-axis corresponding to the circumferential position and the y-axis corresponding to the position along the engine axis (flow is going from bottom to top). The results show the acceleration of the flow past the fan blade and outlet guide vane suction sides. The fan blade wakes (and their interaction with the outlet guide vanes) and the outlet guide vane wakes are also highlighted. The velocity profiles along the fan blades depend on the circumferential position in the Nautilius case, which is an effect of important inflow distortion.

B. Engine Performance

The performances obtained from the uRANS simulations after valve law adjustment and convergence are summed up in Table 1. The mass flow rate of air, fan pressure ratio and isentropic efficiency are very similar for both configurations with differences lower than 1%. This is a result of the design of the current NOVA-Nautilius configuration where the nozzle area has been slightly extended in order to get a lower FPR (which is therefore closer to the one of the NOVA-SFN configuration). These close engine performance results make fair the acoustic comparison between both configurations.

C. Distortion Analysis

The distortion is expected to be the main cause of noise modification for BLI engines. Its interaction with the rotating fan indeed causes unsteady loadings on the blades that can contribute to the total tonal noise. It is therefore important to analyze the distortion in such engines. In this purpose, the time-averaged flow field is extracted from the uRANS solutions just upstream of the fan for both configurations. The results in terms of normalized axial velocity are given in Fig. 7. The distortion is severe for the Nautilius configuration, with a low velocity region towards the fuselage axis (fuselage tail) and a high velocity region in the opposite direction. This distortion is caused by both the incidence of the flow and the geometry itself (non axisymmetric duct and full integration within the fuselage). For the SFN configuration, the distortion mainly comes from the angle of attack of the flow despite some effects of the fuselage caused by its proximity. An angular Fourier transform has been performed from these maps to get the amplitude of the distortion harmonics k dist . The process is done at two selected radial stations corresponding to 50% and 75% of blade height. The results are shown in Fig. 8 and have been normalized by the averaged axial velocity value of the corresponding radial station (to highlight deviations from average). It is clearly observed that the harmonic levels are more intense with a broadened content for the Nautilus case. The most significant harmonic orders are within the range 1 ≤ k dist ≤ 5.

The distortion is further analyzed by evaluating its evolution along the inlet duct. This is done by computing at several axial positions x and radial locations r the circumferential distortion coefficient (CDC) based on axial velocity by:

CDC(x, r) = max(V x (x, r)) -min(V x (x, r)) mean(V x (x, r)) . (1)
The results are given in Fig. 9 for three radii corresponding to 25%, 50% and 75% of blade height and for both configurations. The axial position is expressed in terms of the length of inlet duct of the SFN configuration L. These results again highlight the significantly higher levels in the Nautilius case at all positions. The distortion is not only higher at the duct inlet, but it is also less attenuated along the duct which is shorter than the SFN one. The shape of distortion is also modified with the Nautilius concept. While the distortion due to the angle of attack (SFN case) increases with the radius, the evolution is reversed in the Nautilius case with higher levels close to the hub.

D. Fan-Blade Wake Analysis

An analysis of fan-blade wakes is now provided as it is the cause of one of the most important contribution of fan tonal noise when interacting with the OGVs. In this purpose, the time evolution of normalized axial velocity over one blade passage (BP) is shown in Fig. 10 for both configurations. For each configuration, six curves corresponding to six different circumferential positions are given for the three represented radial locations. These circumferential positions are spaced by π/3 which is exactly the space between three fan blades. As a result, in the SFN configuration where the distortion is negligible at this position, the wakes are superimposed for the six different azimuthal positions. On the contrary, for the Nautilius configuration, a difference of both amplitude and phase is observed between these positions. This is to be related to the distortion that modifies the fan-blade wakes through a modification of the flow incidence along the azimuthal direction.

V. Acoustic Results

A. Methodology

The tone noise assessment presented here is restricted to the interaction noise contributions due to the following mechanisms:

1) inlet distorted flow interacting with the fan blades (fan sources), 2) fan blades interacting with the potential flow induced by the OGVs (fan sources), 3) and fan-blade wakes interacting with the stator vanes (OGV sources). Although shocks are present in this take-off configuration, the shock noise contribution is not considered in the present study which is based on a Ffowcs Williams and Hawkings (FWH) analogy and limited to the loading noise term. Calculations are performed using the in-house code FanNoise that solves the Goldstein's formulation (in-duct formulation of the FWH analogy) [START_REF] Goldstein | Aeroacoustics[END_REF]. The acoustic calculations then assume a fully annular duct and a uniform mean flow. Modal analysis is focused on rotor-locked modes (m = nB with m the azimuthal mode order, B the number of fan blades and n the considered BPF harmonic) which are cut-on or nearly cut-on at take-off regime, and giving rise to dominant fan tone noise (due to steady loadings contribution), as well as generalized interaction modes given by the well-known Tyler and Sofrin rule [START_REF] Tyler | Axial Flow Compressor Noise Studies[END_REF], and extended to distorted flow conditions [START_REF] Daroukh | Effect of Distortion on Turbofan Tonal Noise at Cutback with Hybrid Methods[END_REF][START_REF] Garrec | Tone Noise Predictions of a Full-Scale UHBR Engine at Approach Condition with Inflow Distortion Effects[END_REF] as:

m = nB ± kV ± k dist . (2)
V is the number of stator vanes and k can be any integer. The standard Tyler and Sofrin modes (without distortion effects) are summed up in Table 2 for the first five BPFs. The bold values are indicating the cut-on modes m = nB, based on the average flow values extracted at fan plane. In the following acoustic analyses, the tone noise predictions will be split into fan and OGV sources, and by considering respectively intake (sources from fan only) and bypass (sources from OGV only) sound propagation.

B. Blade/Vane Loading Analysis

In order to check and highlight the expected source mechanisms mentioned above, blade/vane loadings issued from uRANS solutions are discussed in this section. Wall-pressure harmonics obtained from a discrete Fourier transform of the pressure signal over one revolution are plotted in Fig. 11 for one rotor blade and in Figs. 12 and 13 for two different stator vanes. Note that all results have been normalized by the same arbitrary value to ease the comparison. It has first been checked that the results are identical for each blade, confirming the blade-to-blade periodicity. In Fig. 11, the intense pressure magnitude on low-order harmonics of rotational frequency (RF), mainly located in the leading edge region, are attributed to the dominant distortion harmonics shown in Fig. 8. The blade interaction with the OGV potential flow is revealed by the RF40 frequency, responsible for visible spots (with significant levels but much lower than the ones due to distortion) mostly located in the trailing edge region. In Figs. 12 and 13, the dominant harmonic levels are linked to the BPF harmonics, with decreasing level when increasing order, and are attributed to the fan wakes impingement on the vane, mostly in the leading edge region. Non-negligible deviations between the two selected vanes in the Nautilius configuration are traducing the asymmetry of the OGV sources caused by the distortion effects on blade wakes.

C. Angular Mode Spectra

Angular mode spectrum analysis is very useful to check the interaction mode rule given by Eq. 2, which gives the Tyler & Sofrin modes given in Table 2 when there is no distortion. For the fan sources (Fig. 14), only the rotor-locked modes m = nB generated by the steady loads emerge in the SFN configuration. As for the Nautilius case, the modes around these rotor-locked modes given by m = nB ± k dist are also highly dominant because of the distortion/fan interaction mechanism. The mode levels associated to the BPF1 are lower (about 10 dB below the higher tones) because the rotor-locked mode m = 18 is theoretically cut-off in the intake at BPF1 (see Table 2). One should note that the present cut-on/cut-off mode condition is very sensitive to the value of the tip Mach number and thus to the axial Mach number adopted in FanNoise, under the restrictive assumptions of annular duct and uniform mean flow. A similar analysis is performed for the OGV sources contribution in Fig. 15. Now, all the interaction modes have significant levels so that they are all visible when using the same level dynamic and they should all contribute to the sound radiation.

Only the standard Tyler and Sofrin modes m = nB ± kV are present in the SFN case, which is a consequence of the quasi-uniform flow. However, for the Nautilius configuration, it can be observed that the modes around them given by m = nB ± kV ± k dist also carry a lot of energy and therefore probably contribute to the total acoustic power. In particular at BPF1, the mode m = 17 is almost as energetic as the mode m = 18 (note that the mode m = 18 is cut-on at BPF1 in the bypass duct).

D. Power Level Spectra

Finally, the power level (PWL) spectra provided by FanNoise are plotted in Fig. 16, focusing on the fan contribution in the intake (left) and the OGV contribution in the bypass duct (right). It can be observed that the overall power level (OAPWL) from the rotor is higher than the one from the OGV in the two configurations. This is due to the steady loadings for both cases and to the distortion-rotor interaction mechanism in the Nautilius concept. At BPF1 where the rotor-locked mode is cut-off in the inlet, the acoustic penalty of the Nautilius concept is high due to the contribution of distortion-rotor interaction modes. For higher BPFs, this penalty is not very clear as the rotor-locked modes from fan sources are more pronounced in the SFN configuration (see Fig. 14). It therefore seems that the energy of the distortion-rotor interaction modes is somehow extracted from the rotor-locked modes for the Nautilius concept. As a result, the penalty of the Nautilius integration on the OAPWL when restricted to fan sources is only of 0.2 dB. Regarding OGV sources, the levels are higher in the Nautilius case except at BPF1. When integrating all frequencies, an acoustic penalty of 2.2 dB is obtained when restricting to OGV sources. It should be kept in mind that these penalties result from the application of Goldstein's analogy which assumes an infinite and constant annular duct and a uniform flow. Duct geometry variations and flow three-dimensional effects are therefore discarded in this noise generation analysis. Those aspects could alter the previous conclusions and they are more specifically adressed in a follow-up paper [START_REF] Lorteau | Acoustic Assessment of BLI Effects on Airbus Nautilius Engine Integration Concept -Part II: Noise Radiation[END_REF].

VI. Conclusion

A first analysis of the noise induced by the Nautilius engine integration has been provided in this paper. The study is limited to fan tonal noise because it is expected to be the most affected contribution to the aircraft noise. Take-off conditions representative of the sideline acoustic certification point are considered. The acoustic results are evaluated by comparison with a reference NOVA aircraft with podded engines. The numerical methodology is based on two steps. First, RANS simulations of the whole aircraft are performed using the elsA solver with body force and chimera techniques. Then, uRANS simulations of the engine, limited to the in-duct domain, are realized by imposing a distortion map (extracted from the preceding simulation) as the inlet boundary condition. Finally, a first acoustic evaluation is done using Goldstein's analogy and the sources of the uRANS simulations.

As expected, the distortion present in the Nautilius configuration is significantly more severe than in the SFN case because of important installation effects and a reduced length of fully-closed inlet duct. The distortion is responsible for acoustic sources on fan blades, essentially located close to the leading edge. This interaction is much stronger than the one with the potential effect with the OGVs. An effect of the distortion on the fan-blade wakes has also been highlighted in the Nautilius case. As a result, a vane-to-vane acoustic source deviation is observed.

The noise estimates based on Goldstein's analogy using rotor and stator loadings give an OAPWL emitted from the rotor about 10 dB higher than the one from the OGVs. This is caused by both the rotor-locked modes generated by the steady loads of fan blades and the neighboring modes generated by distortion-rotor interaction (especially in the Nautilius case for the latter mechanism). These modes are indeed cut-on or nearly cut-on at take-off conditions. The acoustic penalties induced by distortion are then strongly related to the energy distribution between these modes. In the present study, a penalty of only 0.2 dB is obtained in the Nautilius case. This is due to weaker rotor-locked modes (steady loading noise) that counterbalance the much stronger distortion-interaction modes. As Goldstein's analogy assumes an infinite and constant annular duct and a uniform flow, it should however be stressed that this estimate only focuses on the generation of acoustic sources. The propagation effects in the varying duct geometry and heterogeneous flow and the installation effects are specifically adressed in a follow-up paper to complete this first acoustic assessment of the Nautilius engine integration concept.

  (a) NOVA-SFN airplane. (b) NOVA-Nautilius airplane.

Fig. 1

 1 Fig. 1 Overall view of the computed configurations.

  (a) SFN case.(b) Nautilius case.

Fig. 2

 2 Fig. 2 Domains of the uRANS simulations.

  (a) RANS solution -SFN case. (b) Time-averaged uRANS solution -SFN case. (c) RANS solution -Nautilius case. (d) Time-averaged uRANS solution -Nautilius case.

Fig. 3

 3 Fig. 3 Contour maps of normalized axial velocity at duct inlet plane.

Fig. 4

 4 Fig. 4 Contour maps of normalized axial velocity at fan plane.

  (a) SFN case -Vertical plane. (b) Nautilius case -Vertical plane. (c) SFN case -Horizontal plane. (d) Nautilius case -Horizontal plane.

Fig. 5

 5 Fig. 5 Contour maps of normalized axial velocity around the engines extracted from RANS simulations.

Fig. 6

 6 Fig. 6 Contour maps of normalized velocity magnitude extracted from the engine uRANS simulations on a radial slice corresponding to 75% of fan blade height.

  (a) SFN case. (b) Nautilius case.

Fig. 7

 7 Fig. 7 Distortion maps of normalized axial velocity at fan plane location (time-averaged uRANS solution).

  (a) h/H = 0.50. (b) h/H = 0.75.

Fig. 8

 8 Fig. 8 Normalized amplitudes of axial velocity associated to distortion harmonics at fan plane.

Fig. 9

 9 Fig. 9 Evolution of the CDC based on axial velocity along the inlet duct.

  (a) SFN case. (b) Nautilius case.

Fig. 10

 10 Fig. 10 Time evolution of fan-blade wakes extracted just upstream of the OGVs.

  (a) SFN case. (b) Nautilius case.

Fig. 11

 11 Fig. 11 Normalized wall-pressure amplitude on one rotor blade at (from left to right) RF1, RF2 and RF40: suction side (top) and pressure side (bottom).

Fig. 12

 12 Fig. 12 Normalized wall-pressure amplitude on the stator vane #1 at (from left to right) BPF1, BPF2 and BPF3: suction side (top) and pressure side (bottom).

  (a) SFN case. (b) Nautilius case.

Fig. 13

 13 Fig. 13 Normalized wall-pressure amplitude on the stator vane #15 at (from left to right) BPF1, BPF2 and BPF3: suction side (top) and pressure side (bottom).

  (a) SFN case. (b) Nautilius case.

Fig. 14

 14 Fig. 14 Angular mode spectra at the outer wall of the intake due to the fan sources.

Fig. 15

 15 Fig. 15 Angular mode spectra at the outer wall of the bypass duct due to the OGV sources.

( a )

 a Fan sources -Inlet noise. (b) OGV sources -Outlet noise.

Fig. 16 PWL

 16 Fig. 16 PWL spectra in the intake (fan contribution) and in the bypass duct (OGV contribution).

Table 1 Engine performance obtained from the uRANS simulations (with respect to the NOVA-SFN configu- ration).

 1

		SFN Nautilius
	Mass flow rate	-	-0.2 %
	Total pressure ratio	-	+0.6 %
	Isentropic efficiency	-	-0.2 %

Table 2 Expected Tyler and Sofrin modes for NOVA and NAUTILIUS cases at take-off conditions.

 2

		BPF1 BPF2	BPF3	BPF4	BPF5
	Fan	-	36, -4 54, 14, -26 72, 32, -8, -48 90, 50, 10, -30, -70
	OGV	18	36, -4 54, 14, -26 72, 32, -8, -48 90, 50, 10, -30, -70

Acknowledgments

ONERA carried out this study in the framework of ADEC (AdvanceD Engine and aircraft Configurations) and ARTEM (Aircraft noise Reduction Technologies and related Environmental iMpact) projects and in close cooperation with partner Airbus. The ADEC project has received funding from the Clean Sky 2 Joint Undertaking within the European Union's Horizon 2020 research and innovation program under grant agreement No. 945583 -LPA IADP 2020-2021. The ARTEM project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 769350.


text:
Xavier Chanteux

Guillaume Bégou

Hugues Deniau

Olivier Vermeersch

Aerodynamics, Aeroelasticity Ph D Candidate

Construction and Application of Transition Prediction Databased Method for 2 nd Mack Mode on Sharp Cone

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

Construction and Application of Transition Prediction Databased Method for 2 nd Mack Mode on Sharp Cone

Xavier Chanteux * and Guillaume Bégou † DAAA, ONERA, Université Paris Saclay, F-92190 Meudon -France Hugues Deniau ‡ and Olivier Vermeersch §

ONERA-The French Aerospace Lab, Toulouse, France

This study investigates the development of a Reynolds-averaged Navier-Stokes (RANS) model for natural laminar-turbulent transition based on local linear stability theory (LST), and its ability to predict growth rate evolution of Mack's second mode. Unstable growth rates for a given frequency are derived from analytical relations for which coefficients are stored in a database. The beginning of the transitional region is then obtained from the e N -method which consists in integrating the disturbance growth rates along its path of amplification. This approach is an expansion of ONERA's parabolas method to hypersonic laminar-turbulent natural transition and is applied here on a 7°half-angle sharp cone at Mach 8. laminar to turbulent transition is a critical factor in the design of hypersonic vehicles, such as reusable launch vehicles, high-speed interceptor missiles, hypersonic cruise or reentry vehicles: it affects all at once heat transfer, skin friction and boundary-layer properties [START_REF] Reshotko | Boundary Layer Instability, Transition and Control[END_REF]. Thus, it is of practical interest to use already available experiments and theory to evaluate transitional regions.

Nomenclature

In this introduction, the term local refers to quantities accessible in each cell of the mesh, whereas non-local refers to quantities that need integration and/or searching algorithms. From an engineering point of view, Direct Numerical Simulations (DNS) and Large-Eddy Simulations (LES) can provide high-precision results and detailed flow field structures for transitional flows, yet are computationally expensive. With that in mind, efforts have been made to include transition phenomena in modern Reynolds-averaged Navier-Stokes (RANS) solvers via various methods that can roughly be sorted into two categories. On one hand, nonlocal methods based on the Linear Stability Theory (LST) , or Parabolized Stability Equations (PSE), combined with either meta-models [START_REF] Fuller | Neural Network Estimation of Disturbance Growth Using a Linear Stability Numerical Model[END_REF] or databases [START_REF] Arnal | Transition Prediction in Transonic Flow[END_REF] for physics-based applications, or coupled with RANS solvers. The transition onset then is determined by using the e N -method ( [START_REF] Van Ingen | A Suggested Semi-Empirical Method for the Calculation of the Boundary Layer Transition Region[END_REF] and [START_REF] Smith | Transition, Pressure Gradient and Stability Theory[END_REF]). On the other hand, empirical transition criteria (C1 [START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional, Incompressible Flow[END_REF], AHD [START_REF] Arnal | Overview of Laminar-Turbulent Transition Investigations at ONERA Toulouse[END_REF], Drela and Giles' [START_REF] Drela | Viscous-Inviscid Analysis of Transonic and Low Reynolds Number Airfoils[END_REF] ...) used as is or as a basis for correlation-based transition models such as γ -Re θ [START_REF] Menter | A Correlation-Based Transition Model Using Local Variables-Part I: Model Formulation[END_REF] model. This latter solution, originally designed by Menter and Langtry for turbomachine applications at low speed, uses local variables and transport equation for both intermittency and transition onset criteria, in order to evaluate a transition criteria from Re θ , but includes no relevant physical phenomena. It has been extended to supersonic flows for scramjets by Frauholz et al. [START_REF] Frauholz | Transition Prediction for Scramjets Using γ-Re θt Model Coupled to Two Turbulence Models[END_REF] and later to hypersonic flows by Wang et al. [START_REF] Wang | Similarity-Solution-Based Improvement of γ-Re θt Model for Hypersonic Transition Prediction[END_REF] using compressible similarity solutions.

Another local approach designed for turbomachine applications, using transport equations, is the k L -k T -ω model developed by Walters and Leylek [START_REF] Walters | Computational Fluid Dynamics Study of Wake-Induced Transition on a Compressor-Like Flat Plate[END_REF] for bypass and natural transition, where k L and k T respectively describe the laminar and turbulent kinetic energy. On the other hand, Warren and Hassan [START_REF] Warren | Transition Closure Model for Predicting Transition Onset[END_REF] developed a nonlocal method based on LST results, using the time scale of instabilities and taking the contribution of the nonturbulent fluctuations into account. In 2016, inspired by the work of Walter and Leylek, Xu et al. [START_REF] Xu | Fully Local Formulation of a Transition Closure Model for Transitional Flow Simulations[END_REF] established a fully local transition closure model by combining all together Warren and Hassan's modeling idea, incorporating the transport equations for laminar and transitional solutions of Papp and Dash [START_REF] Papp | Rapid Engineering Approach to Modeling Hypersonic Laminar-To-Turbulent Transitional Flows[END_REF], with Menter and Langtry's model for intermittency. It was extended to hypersonic transitional flows dominated by the first and second mode, and laminar separation in 2017 [START_REF] Xu | Parallel Compatible Transition Closure Model for High-Speed Transitional Flow[END_REF]. Wang and Fu [START_REF] Wang | Modelling Flow Transition in a Hypersonic Boundary-Layer with Reynolds-Averaged Navier-Stokes Approach[END_REF] and Wang and Fu [START_REF] Wang | Development of an Intermittency Equation for the Modeling of the Supersonic/Hypersonic Boundary Layer Flow Transition[END_REF] also combined these models between 2009 and 2012 to design their kωγ local model for hypersonic transitional flows, which was improved recently by Zhou et al. ([20], [START_REF] Zhou | Improved K-ω-γ Model for Crossflow-Induced Transition Prediction in Hypersonic Flow[END_REF], [START_REF] Zhou | A Combined Criteria-Based Method for Hypersonic Three-Dimensional Boundary Layer Transition Prediction[END_REF]).

Although all these methods give sufficient results very quickly, they are usually not reliable over a wide range of flow conditions because of the significant number of parameters influencing transition, and give no insight into its mechanism [START_REF] Fu | RANS Modeling of High-Speed Aerodynamic Flow Transition with Consideration of Stability Theory[END_REF]. Comprehensive physicsbased methods are therefore required in RANS solvers, and this means that, at some point, flow stability must be considered either with LST for slightly nonparallel flows, or PSE to account for nonparallel effects. These two methods consider perturbation amplitudes to grow linearly, and estimate the amplification rate for a given base flow, at a specified frequency. However, they do not acount for nonlinear interactions. Amplification rates are usually stored in some database, and the transition onset is evaluated using the e N -method developed in 1956 independently by Van Ingen [START_REF] Van Ingen | A Suggested Semi-Empirical Method for the Calculation of the Boundary Layer Transition Region[END_REF] and Smith and Gamberoni [START_REF] Smith | Transition, Pressure Gradient and Stability Theory[END_REF]: it consists, for a given mode (frequency and amplification direction), in integrating the growth rate along its path of amplification. Later, Van Ingen [START_REF] Van Ingen | The eN Method for Transition Prediction[END_REF] built a database containing stability diagrams. Drela [START_REF] Drela | Implicit Implementation of the Full e N Transition Criterion[END_REF] also built one from stability results of self-similar profiles in order to take account for the varying incompressible shape factor H i . In 1989, Arnal [START_REF] Arnal | Transition Prediction in Transonic Flow[END_REF] used analytical relations to evaluate amplification rates evolution of self-similar profiles with respect to Reynolds number. The different coefficients are then tabulated with respect to some relevant flow parameters (integral thicknesses and M e ). This method is referred to as the parabolas method. It was first developed for two-dimensional incompressible flows over adiabatic walls Arnal [START_REF] Arnal | Description and Prediction of Transition in Two-Dimensional, Incompressible Flow[END_REF], and extended to supersonic flows in 1995 by Arnal. Databases can also be used to train metamodels. Fuller et al. [START_REF] Fuller | Neural Network Estimation of Disturbance Growth Using a Linear Stability Numerical Model[END_REF] were the first ones to implement a neural network to predict growth rate in non reacting jet flows from frequency, wavenumber and momentum thickness. A few years later, Crouch et al. [START_REF] Crouch | Transition Prediction for Three-Dimensional Boundary Layers in Computational Fluid Dynamics Applications[END_REF] developed a similar approach to predict Orr-Sommerfeld results for subsonic and transonic 3D swept-wing boundary layers. This model has been recently improved by Danvin et al. [START_REF] Danvin | Laminar to Turbulent Transition Prediction in Hypersonic Flows with Neural Networks Committee[END_REF] for hypersonic flow.

These physics-based methods are often disparaged in the state of art because of the need to search and integrate non-local flow variables at the edge of boundary layers. In order to tackle this critic, Bégou et al. [START_REF] Bégou | Database Approach for Laminar-Turbulent Transition Prediction: Navier-Stokes Compatible Reformulation[END_REF] recast the integration over a streamline into an advection equation with a source term. This solution allows to evaluate the amplification factor of the e N -method by easily solving an additional transport equation. Combined with the parabolas method up to supersonic flows, this solution allows to evaluate the beginning of the transitional region. This new solution is called the NSP (N-σP) model.

This study focuses on extending the domain of application of the parabolas method to hypersonic flows by taking the evolution of the Mack mode into account.

I. Local Linear Stability Theory and Parabolas Method

A. Local Linear Stability Theory and e N -Method

The linear stability theory, exposed in details by Mack [START_REF] Mack | Boundary-Layer Linear Stability Theory[END_REF], describes perturbations as normal modes propagating in a parallel (or slightly nonparallel) mean flow. The complete unsteady Navier-Stokes equations are linearized about the mean flow by writing each flow variable into a steady mean-flow term and an unsteady small disturbance term. Then the mean-flow solution is dropped from this set of equations, and cross products of disturbances are neglected (linear hypothesis) to obtain the disturbance equations. Mean-flow quantities are supposed to vary slowly in the longitudinal direction, allowing perturbations to be rewritten as normal modes q = q(y) e i(αx+βz-ωt) + C.C.

(

"C.C" denotes the complex conjugate that is needed for the disturbance field to remain real. In the most general case α, β, and ω are complex, but in this study β and ω are supposed real: the amplitude of the wave changes as it propagates downstream in the x direction, which is called spatial amplification theory. The associated wavenumber k is defined by its amplitude k = α 2 r + β 2 r and angular direction relative to the streamwise direction

Ψ = tan -1 (β r /α r ).
With the formulation of equation 1, the wave is amplified when α i is negative. α can be extracted from the complete set of disturbance equations in order to write à q = α B q,

where q = ũ, ṽ, w, p, T, α ũ, α ṽ, α w, α T T , and à and B are two (9N × 9N) matrices (where N is the total number of point of the discretization) containing the mean-flow quantities, and both their first and second derivatives with respect to the normal coordinate.

Solving this equation requires boundary conditions, which, for boundary layer flows, are defined as no-slip condition at wall surface q(0) = 0, and no perturbation in the free-steam q (y → ∞) = 0. The set of boundary conditions and equation 2 define a general eigenvalue problem where α is the eigenvalue and q the eigenvector.

The linear local analyses are performed with an in-house stability code for ideal gas. Inspired by the work of Orszag [START_REF] Orszag | Accurate Solution of the Orr-Sommerfeld Stability Equation[END_REF], derivations are performed with a spectral method using Chebyshev polynomials, on the Gauss-Lobatto collocation points y j = cos (π j/N) (with N being the total number of points in the collocation) recast such that half of the total number of points describe the boundary layer as suggested in Schmid and Henningson [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF] (see Appendix A.5 on chebyshev discretization). Then, the eigenvalue problem is solved using geev function of LAPACK library.

In order to locate the beginning of the transitional region from LST, the e N -method is commonly used. It computes the amplification factor N of an instability, for a given reduced frequency F (see eq. 4 ), expressed as

N(s, F) = ln A A 0 = ∫ s s cr σ( F, s)ds (3)
where s is the path of amplification of σ( F, s), and s cr the abscissa of the initial amplitude A 0 of the instability at the frequency F under consideration. This amplification factor can be regarded as a gain of energy of the disturbance as it propagates, and the transition is expected to occur for a critical value N tr .

B. Parabolas Method for 1 st Mode

In this section, all quantities are made dimensionless using u e and δ 1 .

When investigating the stability of (self-similar) profiles, stability diagrams are often plotted in the ( F, Re δ 1 ) coordinates, where F and Re δ 1 are defined by

F = 2π fν e u e 2 = ω Re δ 1 , Re δ 1 = ρ e u e δ 1 µ e (4)
Thus F evolves linearly with Re δ 1 : iso-values of F are lines with constant slope passing through the origin in the ω, Re δ 1 plan (see Fig. 2 ).

Arnal [START_REF] Arnal | Transition Prediction in Transonic Flow[END_REF] observed that the evolution of the growth rate σ = -α i , for a given reduced frequency and plotted versus Re δ 1 , can be approximated by two half-parabolas for low speed flows. The equation yields, with Figure 1

σ σ M = 1 - Re δ 1 -R M Re k -R M 2 (5)
where The parabolas parameters are then expressed as a function of the reduced frequency F

Re k = R 0 Re δ 1 < R M R 1 Re δ 1 > R M (6) R 0 R M R 1 σ M Re δ 1 σ LST Parabolas
σ M = A M 1 - F F M , (7)
R M = K M F E M , (8)
R 0 = R M 1 -A 0 1 - F F 0 , (9)
R 1 = R M 1 -A 1 1 - F F 1 . (10)
These new coefficients A M , F M , K M , E M , A 0 , F 0 , A 1 and F 1 are stored in a database indexed on the incompressible shape factor H i and Mach number at boundary layer edge M e . Each point of the database corresponds to the stability characteristics of the self-similar profile at specified ( H i , M e ), namely Falkner-Skan-Hartree profiles (which reduces to Blasius profiles in the abscence of pressure gradient) for incompressible flows, and Levy-Lees profiles for compressible ones.

This first model is called the viscous model and is designed for low-speed transitional flows in order to describe instabilities originating from viscous phenomena. Several corrections were added to this model to better describe new phenomena arising at higher velocities: instabilities may come from the presence of an inflection point in the mean profiles [START_REF] Lees | Investigation of the Stability of the Laminar Boundary Layer in a Compressible Fluid[END_REF], which is taken into account through an additional dedicated set of two half-parabolas. A description of the complete parabolas method for the first mode is given in Bégou et al. [START_REF] Bégou | Database Approach for Laminar-Turbulent Transition Prediction: Navier-Stokes Compatible Reformulation[END_REF].

II. Extension of Parabolas Method to 2 nd Mode

In the following of previous studies on the parabolas method, the stability characteristics of Levy-Lees profiles are studied for the prediction of 2 nd mode evolutions in unstable domains. To illustrate the construction method, an example for a baseflow (mean velocity and mean temperature profiles) at Mach number M e =4.5 without any pressure gradient is shown here. The wall is set to adiabatic condition and Sutherland's viscosity law is used with a reference stagnation temperature at boundary-layer edge set to 300K. In the unstable domain of the 2 nd mode, the amplification σ ( F, s) along a reduced frequency F -plotted in red in Figure 2 -is described with two half-parabolas and a linear correction to best fit the LST results. This linear correction is defined by matching first derivatives at ( R, Kσ M ) (see Figure 3).

R 0 R M R R P i R 1 Kσ M σ M

Linear correction

Re δ 1 σ

Figure 3: Definition of all parameters for two halfparabolas and linear correction (iso-F)

This modelization is optimized with a Sequential Least Squares Programming (SLSQP) algorithm. It minimizes, under constraints, the root-mean-squared error between the LST curve and the analytical model. Then, the evolution with F of each coefficient defining the model, namely R 0 , R M , R, R 1 , and σ M , are approximated with a correlation (see equations 11 -15 and associated Figures 4567), and each coefficient of these correlations -namely K 0 , E 0 , etc -are stored in a database. Equations of interpolations of R 0 , R M , R, R 1 , and σ M evolution with reduced frequency F are expressed as

σ M = A m F F 1 + 1 F F 2 + 1 (11) R M = K M 10 5 F E M (12) R 0 = K 0 10 5 F E 0 (13) R 1 = K 1 10 5 F E 1 (14) R R 10 5 F E R (15)
The linear correction is defined by

σ σ M = Re δ 1 -R 1 K R -R 1 ( 16
)
where

K = 2 R 1 -R 2R 1 -R -R M ( 17
)
The parabolas method is used together with the e Nmethod which computes the amplification factor N of an instability, for a given reduced frequency F. To confront the concept of this new model with LST results, a dedicated database has been embedded in ONERA's elsA software -along with the NSP model for the integration of amplification rates -and applied to a sharp cone at Mach 8 at zero angle of attack.

III. Application of Embedded Parabolas method in RANS solver

A. Cone Geometry and Free-stream Conditions

The free-stream conditions and geometry are adapted from Stetson et al. [START_REF] Stetson | Laminar Boundary Layer Stability Experiments on a Cone at Mach 8. II-Blunt Cone[END_REF], which has been reproduced by Malik et al. [START_REF] Malik | Effect of Nose Bluntness on Boundary Layer Stability and Transition[END_REF], Kufner et al. [START_REF] Kufner | Instability of Hypersonic Flow Past Blunt Cones -Effects of Mean Flow Variations[END_REF], and Rosenboom et al. [START_REF] Rosenboom | Influence of Nose Bluntness on Boundary-Layer Instabilities in Hypersonic Cone Flows[END_REF]. The geometry is a 7°half-angle ideally sharp cone at zero angle-of-attack. The gas is considered ideal. The free-stream conditions are as follows: the free-stream Mach number is set to M ∞ = 8.0, the free-stream unit Reynolds number to Re ∞ = 8.202 × 10 6 1/m, and the free-stream stagnation temperature to Ti ∞ = 750K which corresponds to a free-stream static temperature of T ∞ = 54.35K. Schneider [START_REF] Schneider | Hypersonic Laminar-Turbulent Transition on Circular Cones and Scramjet Forebodies[END_REF] specifies that the free-stream Reynolds number used by Rosenboom was computed from the free-stream stagnation pressure Pi ∞ = 4.0MPa by using the Mack-modified form of Sutherland's viscosity law. If normal Sutherland's viscosity law was used, the result would have been Re ∞ = 8.76 × 10 6 1/m, which makes and error about 7%. In his experiment, Stetson had a one-meter-long cone, whereas in the simulations of Rosenboom and Kufner a two-meters-long one was used.

B. Grid Distribution and Numerical Methods

The grid distribution is shown in Figure 8 with the Mach number contour to show the control line following the shock layer. The flow is resolved with 600 points in the freestream direction and 400 points in the wallnormal direction, between the surface and the control line. Grid clustering is applied in the boundary-layer and the shock layer. The grid is composed of 290,000 points in total. The laminar mean flow is computed with ONERA's elsA software using a Roe scheme with Van Albada limiter for space discretization and backward Euler scheme for time integration, with scalar LU-SSOR scheme for implicit time integration. A CFL slope from 0.01 to 10 is used over the 10,000 first iterations and fixed at 10 afterward. The size of the first grid cell is adapted so that the size of the cell at the boundary-layer edge is less than 10% of δ, along with a grid expansion ratio inside the boundary layer set to 1.04 in the wallnormal direction. The simulation is stopped when the boundary-layer profiles are considered to have reached a converged state.

C. Mean-Flow Computation

The mean flow is computed over an adiabatic wall and the accuracy of the simulation is checked by comparing the results with Rosenboom's ones on mean-flow profiles at stations s ∈ [0.1, 0.5, 0.9] (Figures 9 and10), boundary-layer thickness and pressure gradient evolution along the cone arclength s (Figures 11 and12). Three different boundary-layer thicknesses (δ u , δ t , δ h ) were plotted by Rosenboom: they have been computed respectively from velocity (dynamic boundary-layer), temperature (thermal boundary-layer) and total enthalpy (total enthalpy boundary-layer). For enthalpy, the boundarylayer is defined as the wall-normal distance were 99.9% of the total enthalpy of the free stream. In Figure 12 , the pressure gradient is very important at the leading-edge and decreases downstream: the fluid is accelerated until roughly half the cone.

D. Local Linear Stability results comparison

Stability results for the second mode are first compared against those of Rosenboom. In Figure 13 , the current code computes the same evolution of the amplification rates as Rosenboom's except for the end of each parabolas: the nd branch of the second mode goes faster toward zero for the current stability code. It has an impact on the amplification factor maximum of each frequency as shown in Figure 14 . Although, these maxima are not the same between the two stability codes, the limit defined by the critical N-factor N tr is usually crossed before this maxima, where the two stability codes give amplification rates with negligible error.

Higher unstable frequencies are located near the cone leading edge. In this region, the growth rates of the two stability codes are quite different: as plotted in Figure 13, the relative error on the maximum value for frequency 400kHz is about 20%.This could be a critical issue if the transition was to occur here. However, the N-factors are considered low enough to not trigger transition, which is supported by experimental results. Indeed, Stetson et al. [START_REF] Stetson | Laminar Boundary Layer Stability Experiments on a Cone at Mach 8. II-Blunt Cone[END_REF] noticed that the boundary-layer starts transitioning around s = 0.57. Rosenboom deduced from this result that the transition is triggered by the frequency 180kHz. The associated critical N-factor is deduced at N tr ≈ 4.5. Using the same critical N-factor N tr , the stability code used presently predicts a transition triggered by the frequency 175kHz -which makes an error about 3% -and the transition location is found at s = 0.59. The relative error on the transition location between the two codes is about 3%.

These differences between both results can come from mainly two factors. The equations of the current stability code are made dimensionless using boundarylayer quantities and variables at boundary-layer edge. Although the two numerical simulations compute the same mean flow, extracting boundary-layer quantities to compute flow stability requires the determination of the boundary-layer edge. However this velocity is not constant, making this determination tricky. explanation is linked to the stability code itself: Malik [START_REF] Malik | Numerical Methods for Hypersonic Boundary Layer Stability[END_REF] showed that the choice of the scheme for the discretization of the stability equation can have an impact on the growth rate value. However, references in Rosenboom's paper could not give any further information on the stability code. In any case, the current stability code is aimed to be used for the modelization of the second mode growth rate, thus this differences are considered low enough to have negligible impact on the final modelization.

E. Transition prediction with parabolas method

The local linear stability code presented in the previous subsection has been used for the construction of the current database. The comparisation of amplification rate evolutions and N-factors for several frequencies with both methods are presented respectively in Figures 15 and16 to assess the error of the model.

The amplification rates computed from elsA's parabolas method allow to find the same evolution of amplification rates as the LST. For low frequencies, the evolution differs slightly between the two methods, which impacts the maxima of N-factors. As said in the previous subsection, the limit defined by the critical amplification factor N tr is usually crossed before the maximum. Thus, the error of the reconstructed parabolas for low frequencies is considered negligible.

Figure 16 shows the comparison of N-factors between LST and embedded method in elsA. The integration of amplification rates is recasted into a transport equation, using Bégou's method. The N-factors evolutions computed from elsA's parabolas method matches that of the LST for the cone, except for the maxima of low and high frequencies. These discrepancies are the same as the ones for the construction of the model: reconstruction of the LST by the parabolas method at low frequencies is not yet accurate enough to match the second branch. Even tough, at s 0.59, the transition is triggered by a frequency between 170 and 180 kHz, and the associated N t is about 4.5 which is the same results as LST one.

IV. Conclusion

The parabolas method has been extended with a new model for Mack's mode in order to predict natural laminar-turbulent transition for hypersonic flows. This model has been applied to a sharp cone geometry at Mach number 8 following the experiment of Stetson, and has proven its capability to reproduce the amplification rate evolution predicted by the LST. This method, when coupled with the e N -method, is able to give the beginning of the transitional region for RANS code at smaller computational costs and in good agreement with LST results. Moreover, it allows engineers to easily compute transition location because using this method bypasses the resolution of stability equations. This new model still needs to be enriched with isothermal profiles and wider pressure gradient range in order to be used on more complex geometries for hypersonic applications.

δ = Boundary layer thickness δ 1 =

 1 Boundary layer displacement thickness θ = Boundary mayer momentum thickness Re δ 1 = Reynolds number based on δ 1 Re θ = Reynolds number based on θ M e = Mach number at boundary layer edge k

Figure 1 :

 1 Figure 1: Qualitative comparison of LST and approximate growth rates (iso-F)

δ 1 ωFigure 2 :

 12 Figure 2: Stability Diagram at Mach 4.5: Contour lines α i * 10 -3In the unstable domain of the 2 nd mode, the amplification σ ( F, s) along a reduced frequency F -plotted in red in Figure2-is described with two half-parabolas and a linear correction to best fit the LST results. This linear correction is defined by matching first derivatives at ( R, Kσ M ) (see Figure3).

LSTFigure 4 :RFigure 5 :Figure 6 : R 0 interpolation 1 R 1 = K 1 Figure 7 :

 4561117 Figure 4: σ M interpolation

Figure 8 :

 8 Figure 8: Numerical grid: 1 point over 10 is shown Scaling is different in x and y directions

1 FigureFigure 10 :Figure 12 :

 11012 Figure Comparison of velocity profiles along wall-coordinate at stations s = 0.1, 0.5 and 0.9

Figure 13 :Figure 14 :

 1314 Figure 13: Comparison of growth rate of linear local stability code (plain) and Rosenboom's (dashed), for several frequencies propagating along the cone surface

Figure 15 :Figure 16 :

 1516 Figure 15: Comparison of growth rate from elsA (plain) and LST's (dashed) for several frequencies


David R Bickel
email: dbickel@uncg.edu

Derivation of Bayesian and frequentist inference from evidentiary first principles with applications to propagating uncertainty about statistical methods

Keywords: approximate confidence distribution, fiducial inference, foundations of statistics, general principle of maximum entropy, p-hacking, replication crisis

Unquantified uncertainty about probabilistic model assumptions tends to inflate claims of statistical significance, potentially leading to reported results that cannot be replicated. The same bias occurs at a higher level when Bayesian inference or frequentist inference is chosen to achieve significance in the absence of a way to propagate the uncertainty about which statistical paradigm to select.

The objective of this article is to correct that bias at both levels on the basis of evidentiary first principles for statistical inference. It is found that just as Bayesian inference is warranted when a prior distribution is considered alongside the data as evidence, frequentist inference in the form of confidence intervals and their generalization to confidence distributions is warranted when a hypothesis testing procedure is considered as a piece of evidence. Hierarchical evidence in the same framework enables reporting results reflecting uncertainty about which of those pieces of evidence to admit as well as uncertainty about model assumptions. That is illustrated by a method of averaging Bayesian hypothesis testing with frequentist hypothesis testing and by a method of averaging confidence intervals and/or credible intervals.

Figure 1: Each of the first two flowcharts is a different possibility of the body of total evidence admitted for reporting scientific results. The third flowchart is a higher-level body of total evidence for propagating uncertainty about the first two bodies of total evidence to the results. Each rectangle at the top of a flowchart abbreviates a different possibility for background evidence. The other pieces of evidence are less preliminary within the flowchart.

Why unify statistical inference?

The war between the Bayesian and frequentist camps of statistical inference continues [START_REF] Mayo | Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars[END_REF]. The current cease-fire hides unresolved controversies that inflict unintended damage on the sciences. For example, in the special issue of The American Statistician on the replication crisis in science [START_REF] Wasserstein | Moving to a world beyond "p < 0.05[END_REF], the proposed solutions depend on various stances toward the Bayesian and frequentist positions. At two extremes of the associated debate on hypothesis testing, some Bayesians argue for calibrating p-values using posterior probabilities or related quantities, whereas some frequentists argue for the continued use of uncalibrated p-values.

The lack of a consensus among statisticians leaves scientists without clear guidance. Meanwhile, science must carry on. Results and conclusions must be reported, ideally with their uncertainty quantified. But Bayesian and frequentist methods of uncertainty quantification can yield very different results. One approach is to use the statistical paradigm and model that make the conclusions appear the most certain or statistically significant. That is practiced in molecular evolution whenever a Bayesian method is chosen over a frequentist method in order to report a more certain phylogenetic tree [15, p. 431]. The general form of that practice, selecting models and methods of inference to maximize statistical significance, has been condemned as p-hacking and blamed for the inability to replicate research results [START_REF] Andrade | HARKing, cherry-picking, P-hacking, fishing expeditions, and data dredging and mining as questionable research practices[END_REF]. This article proposes a solution in the form of evidentiary first principles for statistical inference; the main special cases are summarized in Figure 1. That results in generally applicable methods of propagating uncertainty about Bayesian and/or frequentist models. 1 2 General theory: Proof distributions from pieces of evidence Let Pr be the joint probability distribution of an observable data set denoted by Data and an unknown quantity or vector of scientific interest denoted by U . The body of total evidence is a sequence of K pieces of evidence, where each piece of evidence is a proposition about Pr. Each piece of evidence is considered more preliminary than the ones after it in the sequence. For example, if a Bayesian prior distribution B is considered more preliminary than an observed data set denoted by data, then the body of total evidence is this sequence of two pieces of evidence:

1. The marginal distribution of U is B , a known prior distribution.

2. The probability that Data = data is 100%.

As a proposition about Pr, each piece of evidence may be represented as a subset of the set of all possible joint probability distributions of Data and U . Then let Ev k be the set representing the k th piece of evidence for k = 1, . . . , K .

Ev 1 , being the most preliminary piece of evidence, is called the background evidence. Each member of the background evidence may be interpreted as practical information encoded in a language [45, chapter 9]. In terms of measure theory, Ev 1 is a set of measures that are not necessarily probability measures. For example, if the marginal distribution of U is uniform on the real line, then it does not integrate to 1. By contrast, Ev k must be a set of probability measures for each k = 2, . . . , K . All measures have the same domain (measure space) for a given application.

Putting the example in the subset representation, the body of total evidence is the ordered pair (Ev 1 , Ev 2 ), where Ev 1 is the set of all joint distributions such that the marginal distribution of U is B , and Ev 2 is the set of all joint distributions such that the probability that Data = data is 100%.

In short,

Ev 1 = {Pr : Pr (U ∈ •) = B } ;
(1)

Ev 2 = {Pr : Pr (Data = data) = 1} . (2)
Two or more pieces of evidence are inconsistent with each other if their intersection is the empty set. For the sake of enough generality to recover frequentist inference as a special case in Section 3, it is not required that the pieces of evidence be consistent with each other. What is required is that each piece of evidence influence the marginal distribution of U to be used for scientific inference as much as possible while respecting which pieces of evidence are more preliminary than others. That requirement is made precise as follows.

The goal for moving from Ev 1 , the most preliminary piece of evidence, to Ev 2 , the next most preliminary piece of evidence, is to identify the distributions in Ev 2 that would not attribute more certainty than warranted by the evidence (Ev

S (Pr 2 || Pr 1 ) =        -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) if Pr 1 Pr 2 -∞ if Pr 1 Pr 2 , (3)
which is short for

Ev 1 Ev 2 = Pr 2 ∈ Ev 2 : sup S Pr 2 || Pr 1 : Pr 1 ∈ Ev 1 = sup {S (Pr 2 || Pr 1 ) : Pr 2 ∈ Ev 2 , Pr 1 ∈ Ev 1 } ,
where sup is the least upper bound.

Similarly, the goal for moving from Ev 1 Ev 2 , the set of the distributions of proof given (Ev 1 , Ev 2 ), to Ev 3 , the next piece of evidence, is to identify Ev 1 Ev 2 Ev 3 , the set of the distributions of proof given (Ev 1 , Ev 2 , Ev 3 ). In analogy with equation ( 4), that is

Ev 1 Ev 2 Ev 3 = arg sup Pr3∈Ev3 sup Pr1,2∈Ev1 Ev2 S (Pr 3 || Pr 1,2 ) .
The process continues in the same way with each successive piece of evidence until achieving

Ev 1 • • • Ev K , the
set of the distributions of proof given the body of total evidence:

Ev 1 • • • Ev K = arg sup PrK ∈EvK sup Pr1,...,K -1∈Ev1••• EvK -1 S (Pr K || Pr 1,...,K -1 ) .
Distributions of proof combine these previous generalizations of the principle of maximum en-tropy [START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Jaynes | Where do we stand on maximum entropy?[END_REF][START_REF] Jaynes | On the rationale of maximum-entropy methods[END_REF][START_REF] Jaynes | Probability Theory: The Logic of Science[END_REF]:

• Maximizing differential entropy over a sequence of linear constraints that need not be mutually consistent [START_REF] Giffin | Updating probabilities with data and moments[END_REF][START_REF] Giffin | Application of the maximum relative entropy method to the physics of ferromagnetic materials[END_REF] • Maximizing over both arguments of S (•||•) rather than only over the first argument [START_REF] Csiszár | An extended maximum entropy principle and a Bayesian justification[END_REF][START_REF] Bickel | Maximum entropy derived and generalized under idempotent probability to address Bayes-frequentist uncertainty and model revision uncertainty: An information-theoretic semantics for possibility theory[END_REF] 3 Bayesian and frequentist special cases

Posterior distributions from prior distributions

With the general framework of Section 2 in place, Bayesian inference will now be derived from the definition of differential entropy (3) and from the example of equations ( 1) and [START_REF] Balch | Mathematical foundations for a theory of confidence structures[END_REF]. By substitution into equation ( 4), the set of the distributions of proof given the body of total evidence is

Ev 1 Ev 2 = arg sup Pr2∈{Pr:Pr(Data=data)=1} sup Pr1∈{Pr:Pr(U ∈•)=B} -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) = arg sup Pr2:Pr2(Data=data)=1 sup Pr1:Pr1(U ∈•)=B -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) = arg sup Pr2:Pr2(Data=data)=1 -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) ,
where ρ 2 is the probability density function of Pr 2 with respect to a dominating measure Pr 1 such that Pr 1 (U ∈ •) = B . Then Ev 1 Ev 2 = {Pr data }; the unique distribution of proof is Pr data , which is the joint probability measure of U and Data such that Pr data (Data = data) = 1 and such that B data , the marginal distribution of U from Pr data , satisfies

B data (•) = Pr data (U ∈ •) = Pr 1 (U ∈ •| Data = data) (5)
[ [START_REF] Williams | Bayesian conditionalisation and the principle of minimum information[END_REF][START_REF] Giffin | Application of the maximum relative entropy method to the physics of ferromagnetic materials[END_REF].

From the right-hand side, it can be seen that B data is the posterior distribution of U indicated by Bayes's theorem. All of the usual Bayesian results can then be derived from B data , including credible intervals, expected-utility-maximizing decisions, expected-loss-minimizing estimates, and the posterior probabilities of hypotheses about U .

Approximate confidence distributions from p-values

Just as Section 3.1 formalizes what evidence is needed to use Bayesian inference to draw scientific conclusions, this subsection clarifies what evidence would be needed to accept or reject scientific hypotheses at various levels of proof using frequentist inference. Instead of admitting a prior distribution as evidence, this section admits a method of generating p-values and confidence intervals.

This frequentist special case of the general framework of Section 2 starts with a p-value testing the null hypothesis that U is equal to a particular scalar or vector u on the basis of data, the observed data set. Such a p-value is written as p data (u) and has the defining property that the conditional distribution of p Data (u) is uniform between 0 and 1 (p Data (u) ∼ Uniform 0,1 ) given

U = u.
Recall that data, typically a vector or matrix of measurements, is a realization of Data, a random data set. An observed 95% confidence set (a confidence interval for scalar u) u data (95%)

is the set of all possible values of U that are not rejected at the 100% -95% significance level, that is,

u data (95%) = {u : p data (u) ≥ 0.05} , (6)
and similarly for confidence levels other than 95%. The frequentist probability that the random confidence set u Data (95%) covers the true value of u is 95%.

For example, if U is the difference between the mean blood pressure of a treatment group and a control group, then the typical null hypothesis to test on the basis of a record data of clinical measurements would be that U = 0, and the p-value testing that hypothesis would be denoted by p data (0). If there were no difference between the treatment and control groups and if the study were repeated enough times, the histogram of such p-values would be roughly flat between 0 and 1, with about 5% of them below 0.05. There is a 95% probability that the 95% confidence interval u Data (95%) includes the true mean difference.

The body of total evidence that recovers much of frequentist inference is this sequence of three pieces of evidence:

1. The background evidence includes every joint measure such that the distribution of p Data (u)

dominates Uniform 0,1 for every value u of the unknown quantity of interest, and such that the marginal distribution of Data is a measure that dominates D, where D is defined in the next piece of evidence in this numbered list. That background evidence is broad enough to represent very little information in itself.

2. In addition to p Data (u) ∼ Uniform 0,1 for every u, the marginal distribution of Data is D, an unknown prior predictive distribution.

3. The probability that Data = data is 100%.

In the mathematical notation, the body of total evidence is an ordered triple (Ev 1 , Ev 2 , Ev 3 ) satisfying these equations: By substitution into equation ( 4), the set of the distributions of proof given the first two pieces of evidence is

Ev 1 Ev 2 = arg sup Pr2:Pr2(Data∈•)=D,Pr2(p Data (u)∈•)=Uniform0,1 for all u sup Pr1:Pr1(Data∈•) D,Pr1(p Data (u)∈•) Uniform0,1 for all u -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) = arg sup Pr2:Pr2(Data∈•)=D,Pr2(p Data (U )∈•)=Uniform0,1 -ρ 1,2 (u, data) log ρ 1,2 (u, data) d Pr 1 (u, data) , (7)
where ρ 1,2 is the probability density function of Pr 2 with respect to any Pr 1 such that Pr sup

Pr ⊥ ∈Ev ⊥ -ρ ⊥,3 (u, data) log ρ ⊥,3 (u, data) d Pr ⊥ (u, data) ,
where ρ ⊥,3 is the probability density function of Pr 3 with respect to Pr ⊥ . By reasoning analogous to that behind equation [START_REF] Bickel | Confidence intervals, significance values, maximum likelihood estimates, etc. sharpened into Occam's razors[END_REF], that results in Pr ⊥,data as a distribution of proof, where

Pr ⊥,data (•) = Pr ⊥ (•| Data = data)
for a Pr ⊥ ∈ Ev ⊥ . Then C data , the marginal distribution of U from Pr ⊥,data , satisfies

C data (•) = Pr ⊥,data (U ∈ •) = Pr ⊥ (U ∈ •| Data = data) . (8)
Since p Data (U ) ∼ Uniform 0,1 and since p Data (U ) and Data are independent under every Pr ⊥ ∈ Ev ⊥ , it follows that p data (U ) ∼ Uniform 0,1 for U ∼ C data . Finally, the probability that U is in the observed 95% confidence set u data (95%) of equation ( 6) is

C data (U ∈ u data (95%)) = C data (U ∈ {u : p data (u) ≥ 0.05}) = C data (p data (U ) ≥ 0.05) = Uniform 0,1 ([0.05, 1]) = 95%.
Since data is fixed rather than random, that probability is a degree of proof, not a frequentist probability.

Since, analogously, the degree of proof that U is in the observed confidence set of any level other than 95% is equal to that level, C data meets the conditions of an approximate confidence distribution [START_REF] Bickel | Confidence intervals, significance values, maximum likelihood estimates, etc. sharpened into Occam's razors[END_REF][START_REF] Bickel | Confidence distributions and empirical Bayes posterior distributions unified as distributions of evidential support[END_REF] such as, in the case of scalar U , an asymptotic confidence distribution [START_REF] Singh | Combining information from independent sources through confidence distributions[END_REF][START_REF] Singh | Confidence distribution (CD) -distribution estimator of a parameter[END_REF]. An advantage of an approximate confidence distribution as opposed to an exact confidence distribution [e.g., [START_REF] Schweder | Confidence and likelihood[END_REF][START_REF] Polansky | Observed Confidence Levels: Theory and Application[END_REF][START_REF] Tian | The highest confidence density region and its usage for joint inferences about constrained parameters[END_REF][START_REF] Bityukov | Confidence distributions in statistical inference[END_REF][START_REF] Kim | Using confidence distribution sampling to visualize confidence sets[END_REF][START_REF] Schweder | Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions[END_REF][START_REF] Taraldsen | Conditional fiducial models[END_REF] is that the former applies not only to continuous data but also to discrete data. For discrete data would require p Data (u) to be only approximately uniform, and thus the 95% frequentist probability of coverage by the confidence set would also be approximate [START_REF] Bickel | Self-consistent confidence sets and tests of composite hypotheses applicable to restricted parameters[END_REF].

For example, suppose u is a scalar and that the function p data is strictly monotonic increasing.

In that case, for any values u 1 and u 2 such that u 1 < u 2 , equation ( 8) then gives

C data (u 1 ≤ U ≤ u 2 ) = C data (p data (u 1 ) ≤ p data (U ) ≤ p data (u 2 )) . = p data (u 2 ) -p data (u 1 ) ,
indicating that p data is the cumulative distribution function of C data . For that reason, p data is called a "confidence distribution" or a "significance function" [START_REF] Singh | Combining information from independent sources through confidence distributions[END_REF][START_REF] Singh | Confidence distribution (CD) -distribution estimator of a parameter[END_REF]. The "confidence" terminology comes from the fact that whenever u 1 and u 2 are the limits of a 95% confidence interval interval,

C data (u 1 ≤ U ≤ u 2 ) = 95%
, and similarly for any level of confidence other than 95%.

The material in this section can be generalized from p-values to what statisticians call "pivots" and "pivotal quantities," which extend Gauss's concept of an error. Steps analogous to those above then lead to the independence of the pivotal quantity and Data. That independence implies that the proof distribution of U is a fiducial distribution [START_REF] Dempster | On direct probabilities[END_REF], thereby establishing an extended principle of maximum entropy as what Evans [20, p. 91] called the lacking "new principle" needed to justify fiducial inference. This century has experienced the development of many fiducial-type distributions, many of which are approximate confidence distributions [25, 24,

4 Evidentiary model averaging and applications to p-hacking

Evidentiary model averaging

Typically, there is uncertainty not only about U , the random variable or random vector of interest, but also about the model assumptions behind the pieces of evidence. In the Bayesian case of Section 3.1, the body of total evidence depends on a prior distribution, whereas in the frequentist case of Section 3.2, the body of total evidence depends on a statistical testing procedure.

In both cases, there is also dependence on the family, the set of possible distributions of Data.

Each value u of the unknown quantity of interest may correspond to multiple distributions in the family. In parametric statistics, u is is a vector of finite dimension and is called the parameter of interest, and the distribution in the family is in general indexed not only by the parameter of interest but also by a nuisance parameter. The nuisance parameter is not explicitly represented in Section 3 since it is integrated out with respect to a prior distribution in the Bayesian case and since the p-values in the frequentist case are valid for all values of the nuisance parameter. In nonparametric statistics, the parameter values are functions rather than finite-dimensional vectors.

In addition to uncertainty about the family of possible data distributions and uncertainty about the prior distribution in the Bayesian case or the testing method in the frequentist case, there is often also uncertainty about whether to use Bayesian inference or frequentist inference. To concisely capture all of the potentially uncertain assumptions, each body of total evidence that is uncertain is called a model of total evidence and is indexed by a possible value of a random variable M . The number of models of total evidence is denoted by N .

That uncertainty about M can be propagated to statements about U by considering a body of total evidence at a hierarchical level above those of the models of total evidence. That is accomplished by applying the framework of Section 2 except with M in place of U and with a higher-level body of total evidence, which is called the body of total evidence about the model. The resulting proof distribution is then equivalent to a probability mass function written as pmf. Then an average proof distribution of U is

Pr {1,...,N } (•) = Prob (U ∈ •) = m=1,...,N Prob (M = m) Prob (U ∈ •| M = m) = m=1,...,N pmf (m) Pr (m) (•) , (9)
where Pr (m) is a distribution of proof from the mth model of total evidence for each m = 1, . . . , N .

If the body of total evidence about the model is itself uncertain enough to matter, then the hierarchy can be analogously extended by adding a yet higher-level body of total evidence. The process may be repeated until the unquantified uncertainty is considered negligible, as is done with hierarchical Bayesian modeling.

In the simplest case, the body of total evidence about the model, not involving Data, is this sequence of pieces of evidence:

1. The only member of the background evidence is the counting measure Pr # on subsets of {1, . . . , N }. That has the effect of putting equal weight on each model of total evidence.

2. This piece of evidence is vacuous in the sense that it is the set Ev * of all distributions on subsets of {1, . . . , N }.

More succinctly, that body of total evidence about the model is the ordered pair (Ev # , Ev * ),

where Ev # = {Pr # }. By substitution into equation ( 4), the set of the distributions of proof given

(Ev # , Ev * ) is Ev # Ev * = arg sup Pr2∈Ev * sup Pr1∈Ev # -ρ 1,2 (m) log ρ 1,2 (m) d Pr 1 (m) = arg sup Pr2∈Ev * -ρ #,2 (m) log ρ #,2 (m) d Pr # (m) ,
where ρ #,2 , as the probability density of Pr 2 with respect to Pr # , is the probability mass function according to Pr 2 . The equation simplifies to

Ev # Ev * = arg sup Pr2∈Ev * - ρ #,2 (m) log ρ #,2 (m) , (10)
which is just maximum Shannon entropy. As is well known, the probability mass function on {1, . . . , N } that maximizes the Shannon entropy is pmf = , the probability mass function assigning equal probability to each member of the set, in this case each model of total evidence. Equation [START_REF] Bickel | Phylogenetic Trees and Molecular Evolution: A Hands-on Introduction with Uncertainty Quantification Corrected[END_REF] then reduces to Ev # Ev * = {pmf = }, which says pmf = is the unique distribution of proof given (Ev # , Ev * ). By equation ( 9),

Pr {1,...,N } (•) = m=1,...,N pmf = (m) Pr (m) (•) = 1 N m=1,...,N Pr (m) (•) , (11)
which means the average proof distribution of U is the equal-weight mixture of the proof distributions of the models of total evidence.

Applications to p-hacking

Equation [START_REF] Bickel | Propagating clade and model uncertainty to confidence intervals of divergence times and branch lengths[END_REF] suggests taking the mean of p-values and/or posterior probabilities over different methods and assumptions instead of p-hacking, the practice of selecting those achieving the most statistical significance. For example, Figure 2 displays probabilities of making a sign error averaged over Bayesian and frequentist methods weighted equally. The more general equation ( 9) can be used to weight the mean as required by the body of total evidence about the model.

Since 95% confidence intervals and 95% credible intervals are special cases of intervals having a 95% degree of proof, they can also be combined using equation [START_REF] Bickel | Maximum entropy derived and generalized under idempotent probability to address Bayes-frequentist uncertainty and model revision uncertainty: An information-theoretic semantics for possibility theory[END_REF]. It can be used to average the approximate confidence distributions behind confidence intervals and/or the posterior distributions behind credible intervals. The result is a 95% proof interval such that there is a 95% degree of proof that the quantity of interest is in that interval, and similarly for degrees of proof higher or lower than 95%. For example, the special case of equation ( 11) is used in Figure 3 to propagate uncertainty about which Bayesian model of molecular evolution to use. Another example, lacking the above evidentiary foundation, was supported by other arguments [START_REF] Bickel | Propagating uncertainty about molecular evolution models and prior distributions to phylogenetic trees URL[END_REF].

Figure 2: Probabilities that either u > u 0 when U ≤ u 0 or u < u 0 when U ≥ u 0 , where u is the observed estimate of U , and u 0 is the value that U would have if the null hypothesis were true, as functions of a two-sided p-value. The frequentist line is the probability of a sign error according to an approximate confidence distribution [START_REF] Bickel | Null hypothesis significance testing interpreted and calibrated by estimating probabilities of sign errors: A Bayes-frequentist continuum[END_REF]. With that distribution as an approximate posterior distribution conditional on U = u 0 [START_REF] Bickel | Null hypothesis significance testing interpreted and calibrated by estimating probabilities of sign errors: A Bayes-frequentist continuum[END_REF], the Bayesian curve is the posterior probability of a sign error using a Bayes factor calibration [START_REF] Sellke | Calibration of p values for testing precise null hypotheses[END_REF] and a 50% (left) or 90.9% [START_REF] Benjamin | Redefine statistical significance[END_REF] (right) prior probability that U = u 0 . Since both probabilities are degrees of proof given their respective pieces of evidence (Section 3) and since there is considerable uncertainty about those pieces of evidence, the Bayesian and frequentist probabilities are "averaged" by taking their mean to obtain the overall degree of proof that there is a sign error (Section 4.1). ; the intervals do not overlap [START_REF] Bromham | Six impossible things before breakfast: Assumptions, models, and belief in molecular dating[END_REF], indicating that neither interval in itself adequately quantifies the uncertainty. The plot on the right displays the corresponding 50% credible intervals assuming the posterior distributions are approximately normal. Under that approximation, the averaged equal-tail 95% and 50% proof intervals from equation ( 11) are also displayed. While the limits of the averaged 95% proof interval are very close to the maximum of the two upper limits and the minimum of the two lower limits, in agreement with a conservative method of uncertainty propagation [START_REF] Bickel | Phylogenetic Trees and Molecular Evolution: A Hands-on Introduction with Uncertainty Quantification Corrected[END_REF], the averaged 50% proof interval is less conservative. By contrast, the method of taking the union of the credible intervals [START_REF] Bickel | Propagating clade and model uncertainty to confidence intervals of divergence times and branch lengths[END_REF] results in sets that are not intervals when the credible intervals do not overlap.

Ev 1 ⊂

 1 {Pr : Pr (Data ∈ •) D, Pr (p Data (u) ∈ •) Uniform 0,1 for all u} ; Ev 2 = {Pr : Pr (Data ∈ •) = D, Pr (p Data (u) ∈ •) = Uniform 0,1 for all u} ; Ev 3 = {Pr : Pr (Data = data) = 1} .

1 Since Ev 1 Ev 1

 111 1 (p Data (u) ∈ •) Uniform 0,1 for all u and such that Pr 1 (Data ∈ •) D. Equation (7) says Ev 1 Ev 2 is the set of joint distributions maximizing the differential entropy while holding the marginal distributions fixed at Uniform 0,1 and D. Since the joint distribution maximizing the entropy while holding the marginal distributions of two random variables fixed is the product distribution corresponding to the independence of those random variables [17, pp. 309-310], it follows that for every Pr ⊥ in Ev Ev 2 , the joint distribution of p Data (U ) and Data is such that p Data (U ) and Data are independent, Pr ⊥ (p Data (U ) ∈ •) = Uniform 0,1 , and Pr ⊥ (Data ∈ •) = D. The set of those distributions is denoted by Ev ⊥ . Ev 2 = Ev ⊥ , it follows that the set of proof distributions given the body of total evidence is Ev 2 Ev 3 = Ev ⊥ Ev 3

Figure 3 :

 3 Figure 3: Intervals of cyanobacteria divergence times in billions of years ago as estimated from molecular and fossil data. The first two intervals on the left are the 95% credible intervals under a Cauchy 50% prior distribution under uncorrelated and autocorrelated models as reported in Betts et al. [4, Fig.1]; the intervals do not overlap[START_REF] Bromham | Six impossible things before breakfast: Assumptions, models, and belief in molecular dating[END_REF], indicating that neither interval in itself adequately quantifies the uncertainty. The plot on the right displays the corresponding 50% credible intervals assuming the posterior distributions are approximately normal. Under that approximation, the averaged equal-tail 95% and 50% proof intervals from equation (11) are also displayed. While the limits of the averaged 95% proof interval are very close to the maximum of the two upper limits and the minimum of the two lower limits, in agreement with a conservative method of uncertainty propagation[START_REF] Bickel | Phylogenetic Trees and Molecular Evolution: A Hands-on Introduction with Uncertainty Quantification Corrected[END_REF], the averaged 50% proof interval is less conservative. By contrast, the method of taking the union of the credible intervals[START_REF] Bickel | Propagating clade and model uncertainty to confidence intervals of divergence times and branch lengths[END_REF] results in sets that are not intervals when the credible intervals do not overlap.

  1 , Ev 2 ). That is accomplished by finding the distributions Pr 2 in Ev 2 and Pr 1 in Ev 1 that maximize the differential entropy of ρ 1,2 , the probability density function of Pr 2 with respect to Pr 1 , assuming that the density exists. The density exists if Pr 1 Pr 2 , which means Pr 1 dominates Pr 2 . That differential entropy is

  which is the additive inverse of the relative entropy or Kullback-Leibler divergence between Pr 2 and Pr 1 . The set of those entropy-maximizing distributions in Ev 2 is symbolized by Ev 1 Ev 2 . They are called the distributions of proof since their probabilities may be interpreted as degrees of proof on the basis of the evidence (Ev 1 , Ev 2 ). More concisely,Ev 1 Ev 2 = arg sup

Pr2∈Ev2 sup Pr1∈Ev1 S (Pr 2 || Pr 1 ) ,

Acknowledgments

The main result of Section 3.2 was derived at the University of Ottawa with partial support from the Natural Sciences and Engineering Research Council of Canada (RGPIN/356018-2009). The rest of this research was supported by the University of North Carolina at Greensboro.