domain: spi phys

text:
Alain Chelius
email: alain.chelius@onera.fr

Itham Salah

El Din

Régis Koch

TOWARD AN ADJOINT BASED AEROACOUSTIC OPTIMISATION FOR PROPELLER NOISE REDUCTION

Keywords:

The objective of the work described in this paper is to set up and validate an adjoint based method for blade shape optimisation aiming at propeller noise reduction. As a first step toward a comprehensive propeller noise optimisation, the present study focuses on the loading noise of a single propeller isolated and without incidence. In this respect, an acoustic objective function is derived, based on the Hanson-Léwy steady loading noise formulation. After defining this function, the first step is to derive its sensitivities to mesh and aerodynamic field parameters. Once implemented in the ONERA adjoint based optimisation procedure, the acoustic function predictions are compared to a classical Ffowcs Williams and Hawkings approach, and the sensitivities are validated by comparison with the finite differences method.

Introduction and motivation

Once the blade shape of a propeller has been designed to comply with targeted aerodynamic performances, reducing its noise generally requires parameter modifications which are antagonistic with the aforementioned performances. For instance, reducing the diameter or the regime mitigates the emitted noise thanks to the tip Mach number reduction, but requires an adaptation of the blades pitch angle to maintain sufficient thrust. Thus, the acoustic mitigation is constrained by the aerodynamic performances. In addition, the acoustic and aerodynamic impacts of local blade shape modifications are difficult to foresee, and a trial-and-error approach based on Computational Fluid Dynamics (CFD) and Computational Aero-Acoustics (CAA) simulations is time consuming. Therefore, blade shape optimisation based on aero-acoustic criteria appears as an alternative approach to this engineer-made design, since modern optimisation techniques may quickly lead to non-contemplated solutions. This is why in this paper we focus on a gradient optimisation with adjoint method applied to propeller noise reduction.

More exactly, this work deals with the case of an isolated single propeller without incidence, since it provides an appropriate framework to validate the method. Indeed, the steady behavior of the flow in the rotating blade frame allows first to rely on steady CFD and adjoint strategies, and then to compute the radiated tonal noise with a frequency domain approach. In this respect, we define an acoustic objective function based on the Hanson-Léwy steady loading noise formulation. Thus, the objective of the present work is to set up and validate this method, based on the RANS adjoint solver of ONERA CFD code elsA [START_REF] Peter | Discrete adjoint method in elsa (part 1): method/theory[END_REF][START_REF] Carrier | Discrete adjoint method in elsa (part 1): Application to aerodynamic design optimisation[END_REF].

The first part of this paper addresses the theoretical aspects of the work, first briefly presenting the adjoint method, and then detailing the formulation of the acoustic objective function and how its sensitivities to the mesh and flow field are obtained. The second part introduces the propeller application case and its numerical set up, and the third section demonstrates the method validity, checking first the acoustic function reliability, and then its sensitivities evaluation.

Aeroacoustic adjoint for propeller noise

The discrete adjoint method

Aerodynamic shape optimisation consists in looking for a set of shape parameters α = (α 1 , ..., α n f ) that minimizes a given objective function J under n c constraint functions G k , with k ∈ [1, n c ]. These objective and constraints functions depend on the geometry and the aerodynamic flow. The latter is described by the discretized equations of fluid mechanics, of residual R(W, X) = 0, where W represents the aerodynamic state variables, and X the mesh of the computational domain. Gradient-based optimisation methods consist in iteratively compute the flow W for a given shape α, evaluate the functions and their gradients with respect to this shape dJ/dα and dG k /dα, define a new shape parameter vector by applying a descent algorithm, and finally mesh a new shape and a new computational domain. There are several methods to compute the above gradients, the discrete adjoint being one of them. It relies on the linearised equation:

dJ dα = ∂J ∂X dX dα + λ T ∂R ∂X dX dα (1)
with the adjoint vector λ such that λ T = -∂J ∂W ∂R ∂W

-1 , i.e. solution of the discrete adjoint equation:

∂J ∂W + λ T ∂R ∂W = 0 (2)
The evaluation of the gradient, based on Eqs. ( 1) and (2), requires the knowledge of the terms ∂J ∂X and ∂J ∂W , the function sensitivities to the mesh and flow respectively. They can be explicitly linearised, as is done for the aeroacoustic function in Section 2.2.3. The mesh sensitivity to the shape parameters ∂X ∂α may be obtained either numerically or with finite differences, depending on the meshing tools used. An extensive description of the discrete adjoint approach can be found in [START_REF] Peter | Numerical sensitivity analysis for aerodynamic optimization: A survey of approaches[END_REF].

The propeller noise function

Aerodynamic sound origins are usually splitted in three terms: thickness noise, related to the volume displacement of moving solid boundaries, loading noise, due to the forces exerted by solid surfaces on the fluid, and quadripolar noise, generated by sources in the flow. In the case of an isolated subsonic propeller, noise is mainly tonal (in opposition to broadband noise) and due to both thickness and loading noise. In the present work, as a first step toward an exhaustive propeller acoustic optimisation, it has been chosen to focus only on the latter noise component. This choice is maintained by the fact that on modern designs, and in most operating conditions, loading noise is the dominant source. Moreover, the linear nature of the three type of sources (see [START_REF] Goldstein | Aeroacoustics[END_REF], Eq. (3.6) for instance) allows to consider the other terms in future works.

Steady loading noise formulation

Let us consider an orthonormal frame Oxyz in translation at uniform velocity U = c ∞ M along the x axis, c ∞ being the speed of sound and M the Mach number. According to [START_REF] Williams | Sound generation by turbulence and surfaces in arbitrary motion[END_REF], the loading noise sound pressure at a point -→ x (x 1 , x 2 , x 3 ) at time t is the following:

p( - → x , t) = S τ F i ∂G ∂y i dτ dS (3)
where S is the propeller blades surface, τ the emission time, F i (i = 1,2,3) the components of the force per surface element exerted on the fluid, -→ y (y 1 , y 2 , y 3 ) the location of the sources, and G( -→ y , τ ; -→ x , t) the Green function in uniform velocity field. This equation may be solved either in the time domain, as in [START_REF] Williams | Sound generation by turbulence and surfaces in arbitrary motion[END_REF], or in the frequency domain. The latter is selected for this study since it allows to relate the loading noise to quantities such as thrust and torque, which have been previously linearised and implemented in the ONERA optimisation chain [START_REF] Dumont | Calculs de gradients pour l'optimisation des performances aérodynamiques d'un rotor d'hélicoptère en vol stationnaire[END_REF][START_REF] Dumont | Aerodynamic shape optimization of hovering rotors using a discrete adjoint of the reynolds-averaged navier-stokes equations[END_REF]. Integration of Eq. ( 3) in the frequency domain has been performed by Hanson [8] and Léwy [START_REF] Léwy | Prévision du bruit de raies émis par un rotor : application à l'aéronautique[END_REF][START_REF] Léwy | Prediction of tone noise radiated by transonic axial flow fans[END_REF], which only differ by their choice of reference frame. In the following we take the Léwy approach, solving the equation in the frame translating with the propeller, though we refer to the Hanson-Léwy formulation.

For a B blades propeller, rotating at the angular velocity Ω = 2πN in the yOz plane, the Fourier transform of Eq. ( 3) leads to the pressure at the harmonic frequencies of the rotation mB, i.e. multiples of order m of the Blade Passing Frequency (BPF). With a far field assumption, i.e. the observation distance is large compared to the source dimensions and the emitted wavelength, it is shown in [START_REF] Léwy | Prévision du bruit de raies émis par un rotor : application à l'aéronautique[END_REF] that the loading noise can be written:

P (mB, - → x ) = ikB 4πS 2 e -ikσ +∞ s=-∞ e -i(mB+s)(φ-π/2) . S M S + x 1 β 2 f (s) 1 + x 2 f (s) 2 + x 3 f (s) 3
.e i(kA 1 -mBψc) J mB+s (kA) dS (4)

In this expression the cylindrical coordinates of -→ x and -→ y are respectivley (d, φ, x 1 ) and (r, ψ, y 1 ), integration is now over the surface S of a single blade, k = 2πmBN c∞ is the wavenumber,

β = √ 1 -M 2 , S = x 2 1 + β 2 (x 2 2 + x 2 3 ), σ = M x 1 +S β 2 , A = rd S , A 1 = y 1 β 2 M + x 1
S , ψ c expresses the chord noncompactness through the relation ψ = 2πN τ + ψ c , J n is the first kind Bessel function of order n, and f (s) i are the forces harmonics of order s from the Fourier series decomposition of F i . As stated in the previous paragraph, it is interesting to replace the latter by introducing the thrust T and torque Q per surface element. In the blade frame, the force locally exerted by the blade on the fluid is decomposed into an axial force along the propeller axis, the thrust, a tangential component equal to Q/r and a radial one. Thus, by decomposing the F i components in the blade frame and neglecting the radial force, one can show that the bracketed term in Eq. ( 4) becomes:

F (s) = -T (s) M S + x 1 β 2 + Q (s) mBS kr 2 (5)
The Sound Pressure Level (SPL) associated to each tone is then L p (mB, -→ x ) = 10.log prms 2 pr 2

, with p r = 2.10 -5 Pa the reference pressure and p rms = √ 2|P (mB, -→ x )| the Root Mean Square amplitude of the tone.

The acoustic objective function

In order to define the acoustic objective function for the optimisation problem, simplifications are made thanks to the fact that the propeller is isolated and without incidence. First, as the blade loads are steady, the Fourier decomposition of F (s) is limited to the mean term F (0) and the summation over s can be removed from Eq. ( 4). Second, knowing that the steady loading noise of a propeller radiates its maximum noise in the propeller plane, it is chosen to focus the objective function on an observation point in this plane, i.e. for which x 1 = 0. Moreover, this radiation is axisymmetrical, therefore we chose φ = π/2 for the sake of simplifications. As a result, x 2 = 0 and x 3 = d. Then, we chose to focus the objective function on one tone only, the fundamental m = 1 being the best candidate for noise reduction since in the case of single isolated propeller it dominates the spectrum. However, the index m is kept in the equations so the optimisation may be tested on the harmonics. Finally, one may notice that once the observation point and the BPF of interest are fixed, the factors in front of the integration term of Eq. ( 4) are constant, so they will not be taken into account in the objective function.

The integration over the blade surface S is obtained from its discretization by the CFD mesh, by summing the contributions of all the surface cells S i . Thus, taking into account the previous simplifications leads to an expression proportional to the acoustic pressure of Eq. (4):

P s (mB, - → x ) = S i    e i   ky 1 M β 2 -mBψc   .J mB (kA)dF i    (6)
where dF i is the elementary steady loading force, related to the elementary steady thrust dT i and torque dQ i by the relation:

dF i = dT i M S β 2 -dQ i mBS kr 2 (7)
The definitions of dT i and dQ i rely on the axial component of the pressure and friction forces exerted on the blade surface element and are detailed in [START_REF] Dumont | Calculs de gradients pour l'optimisation des performances aérodynamiques d'un rotor d'hélicoptère en vol stationnaire[END_REF].

Finally, the goal of the optimisation being to reduce the SPL of the propeller, the acoustic objective function J is defined in ratio of the square of the tone RMS J = |P s (mB, -→ x )| 2 , that we prefer to express with P s the conjugate of P s :

J = P s (mB, - → x ).P s (mB, - → x ) (8)

Mesh and flow sensitivities

As explained in Section 2.1, the adjoint method requires to calculate the objective function sensitivities with respect to the mesh X and flow W . It depends naturally on the analytical expression of J(W, X), but also on the data format used by the adjoint solver. In the case of elsA, the flow is represented by the conservative variables (ρ, ρu, ρv, ρw, ρE) at the center of the cells and at the center of surface boundary cells. Thus, these latter points correspond to the acoustic source locations -→ y . Based on the definition of the acoustic function given by Eqs. [START_REF] Dumont | Calculs de gradients pour l'optimisation des performances aérodynamiques d'un rotor d'hélicoptère en vol stationnaire[END_REF][START_REF] Dumont | Aerodynamic shape optimization of hovering rotors using a discrete adjoint of the reynolds-averaged navier-stokes equations[END_REF][START_REF] Hanson | Helicoidal surface theory for harmonic noise of propellers in the far field[END_REF], one notices that it depends only on the mesh and field adjacent to the blade surface. More precisely, the exponential and Bessel terms of P s depend only on the mesh coordinates at the blade surface, noted X b (x b , y b , z b ), while the elementary force dF i also depends on the first sheet of mesh in the fluid "parallel" to the surface one, noted X p , on the field at the center of the surface boundary cells, noted W b , and the field in the cells adjacent to the blade surface, noted W c . Therefore, the sensitivities of the acoustic function J(W, X) are given by the following differentiations:

∂J ∂X b/p = ∂J ∂x b/p ; ∂J ∂y b/p ; ∂J ∂z b/p T , ∂J ∂W b/c = ∂J ∂ρ b/c ; ∂J ∂ρu b/c ; ∂J ∂ρv b/c ; ∂J ∂ρw b/c ; ∂J ∂ρE b/c T .
These expressions, not detailed here for the sake of brevity, are obtained analytically by hand chain derivation, with the same method and notations as in [START_REF] Dumont | Calculs de gradients pour l'optimisation des performances aérodynamiques d'un rotor d'hélicoptère en vol stationnaire[END_REF], in which ∂dF i /∂(X|W ) and ∂dQ i /∂(X|W ) have already been developped. It is then implemented in the in-house tool HeliOpt. The propeller used in this work cannot be shown and only few details can be given for confidentiality reasons. It is a full scale five blades puller propeller designed for short-range flight aircrafts. In the present case, the operating condition is a cruise flight without incidence, for which some data is provided in Table 1. The propeller is isolated, however the geometry includes the spinner and the hub.

Propeller noise problem

Configuration

Numerical set up

The CFD simulations rely on ONERA's finite volume solver elsA [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]. Since the present case is axisymmetrical, it allows to limit the computational domain to one blade channel of azimuthal extent 2π/B, as illustrated in Fig. 1. As already stated, the flow is steady in the blade frame, therefore the computations are performed in this rotating frame by solving the steady compressible Reynolds-Averaged Navier-Stokes (RANS) equations. The mesh is built with structured full matching multiblocks, extending 10R up and downstream from the blade pitch axis and radially, R being the blade radius. The whole mesh is approximately 6 million points. The RANS equations are closed using the Spalart-Allmaras turbulence model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. The reason for chosing this model is that it is so far the most advanced turbulent model that has been linearised in the elsA adjoint solver, and it is desirable to apply the same numerical schemes in both CFD and adjoint solvers to ensure the optimisation process robustness. The convective fluxes are discretized with a Roe upwind scheme [START_REF] Roe | Approximate riemann solvers, parameters vectors and difference schemes[END_REF] with MUSCL extension to the second order and a van Albada limiter [START_REF] Van Leer | Towards the ultimate conservative difference scheme, v. a second order sequel to godunov's method[END_REF]. The diffusive fluxes are discretized using a second order centered scheme. The time marching scheme is a first order backward Euler.

Method validation 4.1 Acoustic function

As a first step toward validation of the optimisation strategy, the loading noise function implementation has to be validated. For that purpose, the Ffowcs Williams and Hawkings method (FWH) is considered as a reference, using the well validated in-house code KIM [START_REF] Prieur | Aeroacoustic integral methods, formulation and efficient numerical implementation[END_REF], based on blade pressure extracted from the CFD. The acoustic function is not directly comparable to the Sound Pressure Level obtained by FWH, since it is a simplified form of the Hanson-Léwy SPL (cf. Section 2.2.1). Therefore, the latter is considered rather than J in order to perform such comparisons. Directivities along a semi-arc at 73.8R in a plane perpendicular to the propeller one are shown on Fig. 2. In the downstream direction, i.e. over θ = 90 • , both formulations present similar directivities whatever the tone frequency, with an underestimation by Hanson-Léwy between 1 and 2 dB. Below 90 • , it overestimates the FWH predictions up to 17 dB. The far field hypothesis made on the Hanson-Léwy formulation is respected, thus it cannot be the cause of such large differences, and it seems unlikely that the radial forces on the blade, which are neglected, can have such an impact. These discrepancies are still investigated. However, one may note that the BPF levels at θ = 90 • , which has been chosen as the noise level to be evaluated by the J function and reduced through the optimisation, are in satisfying agreeement with a 0.9 dB difference.

The SPL at this particular tone and direction is further investigated. Three shape parameters of blade profile, sweep, chord length and maximum thickness, are modified in order to induce a local blade deformation at 0.75R. Each deformation produces a new blade geometry, which is simulated by CFD and acoustically evaluated with KIM and the acoustic function J. Figure 3 presents the SPL variations compared to the original blade shape for six deformation cases. Although the noise variations are very small (below 0.1 dB), the acoustic function shows trends similar to the FWH results, thus validating the acoustic function ability to reflect the loading noise sensibility to shape variations. In other words, if the optimisation procedure manages to mitigate J, it will mitigate the far field loading noise accordingly.

Sensitivities

The acoustic function sensitivities ∂J ∂X and ∂J ∂W developped in 2.2.3 and computed by HeliOpt are now compared to values obtained with second order finite differences. The latter consist in adding a local small perturbation to the field or flow. For instance, for the sensitivity to x b on the node i, we calculate:

∂J ∂x b,i ≈ J(W (α), X +δx b,i (α)) -J(W (α), X -δx b,i (α)) 2δx b,i (9)
where X ±δx b,i represents the initial mesh X perturbed by ±δx b,i on coordinate x b of node i. The perturbation amplitude is chosen depending on the studied variable local variations. For instance, for x b,i the δx b,i amplitude has to be very small compared to |x b,i±1 -x b,i |. Some results are presented on Fig. 4, for the sensitivites to x b and ρ b . These values are computed on grid points or cell centers on the blade surface along a section close to the tip, where load sources are known to be the strongest. The results by the analytical formulae implemented in HeliOpt are in excellent agreement with those obtained by finite differences. Similar results are obtained for all the variables of X b , X p , W b and W c , as well as for a section at the blade root (0.36R). Therefore, the acoustic function sensitivities to the mesh and flow are validated.

Conclusions and prospects

This study proposes a first approach of adjoint based shape optimisation for propeller noise reduction, and lays the foundation required for that method. The frame of a single subsonic propeller without incidence offers an ideal case for steady adjoint, and allows to focus the acoustic analysis on steady loading noise. Therefore the latter is defined as the objective function that must be minimized, and this acoustic function is derived from the Hanson-Léwy formulation. The sensitivities to the mesh and flow, required for the adjoint solving, have been linearised by hand. Beforehand performing a blade shape optimisation based on this objective, the function is compared to FWH noise simulations on a propeller test case. Although there are discrepancies in the upstream direction, the general trends of the SPL in the propeller plane are correctly predicted, especially the noise variations when changing local shape parameters. Then, the function sensitivities are compared to evaluations realised by finite differences, showing excellent agreement. Therefore, the implementation of the acoustic function and its sensitivities in the optimisation process are validated.

The next step of this work is to chose relevant shape parameters for the optimisation, and validate the gradients calculated by the adjoint solver against finite differences. This exercice is currently on going at ONERA on sweep, chord length and maximum thickness. There are many other approaches for parametrising this optimisation problem, among which the free-form method [START_REF] Dumont | Aerodynamic shape optimization progress on adodg benchmark problems using the elsa software, of Aeronautics[END_REF] is also contemplated for future work. Then, an optimisation may be performed. However, it requires to define a strategy, including the above parametrisation, but also the constraint functions, which in the present case are likely to be the propeller thrust and/or power, since we want to maintain a certain level of aerodynamic performance. The challenge is to foressee what is the most suited strategy that will lead to the best optimisation, i.e. maximum noise mitigation with minimum performance loss.

Figure 1 :

 1 Figure 1: Mesh blocks of the CFD domain

Figure 2 :

 2 Figure 2: Far-field loading noise directivities calculated with KIM ( ) and the Hanson-Léwy formulation (♦), for BPF, 2BPF and 3BPF.

Figure 3 :

 3 Figure 3: Loading noise SPL variations obtained by KIM and the acoustic function J, at the BPF and in the propeller plane, with local blade deformations: 0: no deformation; 1-2: sweep ±2 • ; 3-4: chord ±2%; 5-6: thickness ±2%

Figure 4 :

 4 Figure 4: Sample of acoustic function mesh and flow sensitivites calculated by HeliOpt ( ) and finite differences (♦) on a blade section at 0.82R.

Table 1 :

 1 Propeller cruise condition characteristics.

	Parameter	Value
	Number of blades B	5
	Flight Mach number M	0.50
	Rotation regime [rpm]	1500
	Thrust coefficient C T	0.22
	Static pressure p ∞ [Pa]	26436.3
	Sound velocity c ∞ [m.s -1 ] 299.463

ICSV26, Montreal, 7-11 July 2019 ICSV26, Montreal, 7-11 July 2019

ICSV26, Montreal, 7-11 July 2019

Acknowledgments

Part of this work has been done in the of the project MADELEINE and has received funding from the European Union's Horizon 2020 research and innovation programme under Grant Agreement No 769025.