CONTINUOUS ADJOINT–BASED AEROACOUSTIC SHAPE OPTIMIZATION OF AN AERO–ENGINE INTAKE

. This paper addresses the aeroacoustic shape optimization of a 3D aero-engine intake by means of gradient-based optimization assisted by continuous adjoint. A hybrid aeroacoustic model based on the (U)RANS equations and the permeable version of the Ffowcs Williams-Hawkings (FW-H) acoustic analogy in the frequency domain is used. The ﬂow and adjoint simulations are for compressible ﬂuid ﬂows realized using the in–house GPU–enabled software PUMA. The continuous adjoint formulation includes the differentiation of the Spalart– Allmaras turbulence model. The implementation and differentiation of the 3D FW–H analogy are veriﬁed by comparing the results of the FW–H analogy to both analytical solutions and URANS. In order to save computational cost, periodic boundary conditions are used to reduce the solution domain size together with the use of a moving reference frame which leads to steady ﬂow and adjoint runs. The generatrix of the nacelle lips and the throat area are parameterized using NURBS. The developed tool is used to perform shape optimization to minimize the total energy contained


INTRODUCTION
The design of future aircraft engines and their components is a long, tedious and complicated process, involving many scientific and technical disciplines such as aerodynamics, thermodynamics, structural mechanics and acoustics, to name a few. Among them, the acoustic performance is becoming increasingly important due to ever stricter regulations related to airframe and engine emitted noise. The design trend for future civil aircraft engines is towards high and ultra high by-pass ratio engines, for which the blade tips of the large diameter fan are a very important source of noise generation. In addition, the design of the intake is of paramount importance since it is responsible for providing a uniform air supply to the engine while, at the same time, regulating the fan noise propagation towards far-field. This paper focuses on the latter.
Efficient gradient-based optimization relies upon adjoint methods, as the cost of the sensitivity computation is independent of the number of design variables [1]. Adjoint methods have been a major research topic in aerodynamic shape optimization [8,12], recently also penetrating into the field of acoustic noise reduction [2,3,4,9]. This is mainly due to the unsteady nature of the acoustic problems requiring a time-accurate adjoint solver which, increases memory requirements as the adjoint equations must be solved backwards in time. So far, mostly discrete adjoints are used in aeroacoustic shape optimization [2,3,4]. In one of the first works on the use of adjoint to hybrid acoustic models, [2] turbulent blunt trailing edge noise is reduced by means of the discrete adjoint to the URANS/FW-H analogy. In [9], the continuous adjoint was developed for URANS/FW-H analogy which was tested for pitching airfoils in inviscid flow. With reference to engine-intakes, the continuous adjoint to the linearized Euler equations to optimize the shape of a turbofan inlet duct, can be found in [5].
Herein, an aeroacoustic shape optimization method is developed for an aero-engine intake. This is based on the in-house GPU-enabled CFD software PUMA [6] which solves the (U)RANS equations and extends its continuous adjoint solver initially developed for aerodynamic shape optimizations. The primal and adjoint problems are solved on a cluster of GPUs, exploiting the CUDA environment as well as the MPI protocol for communications among different computational nodes. The sensitivity derivatives computation accounts for variations in turbulence quantities w.r.t. shape changes as it also solves the adjoint to the Spalart-Allmaras turbulence model.
The primal and adjoint solver of PUMA have recently been extended to computational aeroacoustics (CAA) using the FW-H analogy [9]. The hybrid (U)RANS/FW-H model and its adjoint counterpart have been implemented in PUMA based on the permeable version of the FW-H acoustic analogy in the frequency domain. This model provides the sound pressure at the receivers by computing a low-cost surface integral which depends on flow field data.
In this paper, the developed method is used to optimize the shape of an aero-engine intake, Fig. 1, by minimizing the total energy contained in the sound pressure spectrum at a set of receivers. A NURBS model is used to parameterize the shape of the nacelle lips and the throat area of the engine.

CFD Analysis Tool
The in-house GPU-enabled software PUMA is used to numerically solve the 3D (Unsteady) Reynolds-Averaged Navier-Stokes ((U)RANS) equations, [6]. This is based on a vertexcentered finite volume technique for the spatial discretization on unstructured grids and the upwind second-order Roe approximation for the convection terms. The Spalart-Allmaras turbulence model [7] is used. The software implements mixed-precision arithmetics according to which the Jacobian of the residuals is computed in double-but stored in single-precision which speeds up the computations and reduces the required GPU memory, without damaging accuracy.

Noise Propagation Using The FW-H Analogy
Acoustic analogies propagate the flow-generated sound by means of analytical solutions of the wave equation and this leads to a considerable reduction in computational cost [10]. The mathematical formulation of the permeable version of the FW--H analogy, used herein, originates from [11] for cases with a uniform mean flow (as in a wind tunnel), and reads where H(f ) and δ(f ) are the Heaviside and Dirac delta functions, respectively. Values f=0 indicate the user-defined FW-H surface which should include all acoustic sources. Practically, f is the signed distance from this surface, with negative and positive values corresponding to its interior and exterior, respectively. The FW-H surface is located in the part of the grid which is not displaced during the optimization. c ∞ and u ∞ are the sound speed and velocity at far-field. The quadrupole effects have not been included due to their small effect and the disproportionate high computational cost. Only monopole (Q) and dipole (F i ) source terms are retained. ρ = ρ ∞ + ρ , p = p ∞ + p and u i = u ∞i + u i are the density, pressure and velocity components, respectively, and the subscript ∞ indicates far-field quantities. n FWH is the outward unit vector normal to the FW-H surface.
The integral solution to eq. 1 provides the acoustic pressure at a receiver located at x r . For each frequency (ω) value, this yields where the hatˆsymbol indicates quantities after transformation into the frequency domain using Fourier transform, i is the unit complex number and x s denotes position vectors of sources located on the FW-H surface. In subsonic flows, the Green function for 3D problems in the frequency domain isĜ where k = ω/c ∞ is the wave number and

CONTINUOUS ADJOINT TO THE HYBRID ACOUSTIC MODEL
The aeroacoustic objective function J is defined in the frequency domain as the energy contained in the sound spectrum wherep ( x ra , ω) is the outcome of eq. 3 and N r is the number of receivers. |p | = p 2 Re +p 2 Im , where subscripts Re and Im denote the real and imaginary parts of complex variables. For the purpose of this study, integration in eq. 6 degenerates to a single frequency value, namely that of the main (tonal) frequency.
To formulate the adjoint problem, an augmented objective function is defined as J aug = J + Ts Ω Ψ n R n dΩdt, where Ψ n , R n and Ω are the adjoint variable fields, the residuals of the unsteady flow including Spalart-Allmaras turbulence model equations and the CFD domain, respectively. T s is the time window of the unsteady solution. By differentiating J aug w.r.t. design variable b e (e = 1, N with N being the number of design variables) and setting the multipliers of the derivatives of the flow variables w.r.t. b e equal to zero within the field and along the boundaries, the unsteady adjoint equations and their boundary conditions are obtained as (only for the meanflow equation; for the adjoint to the Spalart-Allmaras model, the reader should refer to [13]) where U m , T and f inv nk are the conservative flow variables, temperature and inviscid fluxes, respectively, and The last term in eq. 7 is defined only along the permeable FW-H surface. For the differentiation of this term one should refer to [9]. The adjoint boundary condition along the solid walls is Ψ m+1 n m = 0, where n is the unit normal to the wall. Also, q adj k n k = 0 is imposed along the adiabatic walls. The sensitivities of J w.r.t. design variables can be computed as where f vis nk and τ mk are the viscous fluxes and stresses.

Verification of the FW-H Analogy Implementation
In the past, the implementation of the FW-H analogy in PUMA has been verified for 2D cases [9]. In this section, the implementation is verified in a 3D problem. To do so, results of the FW-H integral, eq. 3, are compared with the analytical solution of the sound field generated by a monopole source in a uniform flow. Additionally, in order to verify the part of the code that differentiates the FW-H integral, derivatives of the objective function, eq. 6, w.r.t. the coordinates of the monopole source, x s , are computed by the code and compared with the outcome of a central second-order finite difference (FD) based on the analytical solution. The monopole source is located at the origin of the coordinate system and is exposed to a uniform flow u ∞ along the +x direction, shown in 2a. The complex velocity potential of the case is where r + and r * are the same as in eq. 5 . The perturbation fields of flow variables are obtained from the real parts of p = −ρ ∞ ( ∂φ ∂t +u ∞1 ∂φ ∂x ), u = ∇φ and ρ = p /c ∞ 2 . In this case, M ∞ = 0.5, A = 0.004 m 2 /s and ω = 3.095 rad/s. The FW-H surface is a cube extending from −0.5m to 0.5m in all three directions (see Fig. 2a), and is discretized by regular surface grids on the six faces with 400 nodes on each face. The time history of p at a receiver located at x r = (10m, 0m, 10m) as computed using the FW-H analogy shown in Fig. 2b perfectly matches the analytical solution. Slight, practically not visible, differences in the plotted values are due to the numerical integration on the FW-H surface. The derivatives w.r.t. to the source coordinates are also compared in Fig. 2c. The agreement between finite differences of the analytical solution and the differentiation of the FW-H is very good. As expected, the derivative w.r.t. x s 2 is zero.

Optimization Framework For the Rotating Frame
Adjoint-based optimization in unsteady flows may become very demanding in terms of time and memory. This is the main reason why the use of adjoint methods is comparatively restricted in aeroacoustic shape optimization. In this particular application, in order to reduce the cost and since the intake geometry is axisymmetric, a CFD domain that corresponds to a single blade passage of the fan is used, with appropriate periodicity conditions. The flow and adjoint equations are solved in a rotating (with the rotational speed of the engine) frame, thus leading to a steady-state solution for both the flow and adjoint problems. A continuous circumferential distribution of receivers at given radius and axial position is used for the computation of the objective function of eq. 6. The workflow of the aeroacoustic optimization is shown in Fig. 3.

SHAPE OPTIMIZATION OF THE AERO-ENGINE INTAKE
The above-described aeroacoustic analysis and optimization framework is applied to optimize an air-engine intake. The aeroacoustic objective function to be minimized, i.e. that of eq. 6, includes only the tonal noise. The geometry of a single blade passage shown in Fig. 4a has about 3.7M nodes arranged on 100 meridional planes. The far-field flow is still and the pressure distribution on the fan-inlet is used as boundary condition. The pressure contours on the nacelle and mid-plane of the engine intake are plotted in Fig. 4b.
An axisymmetric parameterization model is adopted for the intake. The generatrix is firstly reconstructed using NURBS and this gave rise to 15 design parameters controlling the shape of the nacelle lips and the throat area, leaving its outer shape and the part close to the fan intact. NURBS control points (CPs) are allowed to vary in both the axial and radial direction. Since the first and last points are fixed, the optimization has 2×13=26 design variables, Fig. 5.
The FW-H surface, illustrated in Fig. 6a, has 16000 nodes. To perform the integration of eq. 3, the FW-H surface should rotate to cover the full circumference. However, as a cheaper  Nacelle generatrix Active CPs  alternative, the receivers (instead of the FW-H surface) are rotated and the acoustic pressure is retrieved by superimposing pressure signals from each receiver. As both the grid and numerical set-up may affect the acoustic results, the outcome of the FW-H analogy is verified in the current case as well. To do so, the acoustic pressure computed based on the hybrid model is compared with that of the unsteady CFD code at 3 receiver locations, shown in Fig. 6a. The CFD results match reasonably well those obtained by the FW-H analogy, Fig. 6b, given that part of the discrepancies may be due to the neglect of quadrupole terms, representing the noise due to viscous effects and turbulence. The convergence history of the aeroacoustic objective function is presented in Fig. 7a. A clear reduction in the objective value after 13 optimization cycles can be seen. This reduction is also obvious in Fig. 7b that compares the amplitude of the sound pressure at a single receiver in the baseline and the optimized geometries. Changes of the parameterized part of the nacelle are illustrated in Fig. 8. As seen, the biggest geometrical change occurs at the nacelle lip which is pushed forward, while minor changes occur elsewhere. Regarding the aerodynamic performance, the optimized shape increases the total pressure loss by 0.6% compared to the initial one. This could be expected as an acoustic objective was only considered.

CONCLUSIONS
This paper presented recent extensions of the analysis and adjoint-based optimization tool of the in-house CFD solver PUMA in aeroacoustic shape optimization. A previous implementation of the permeable version of the FW-H analogy in the frequency domain, for 2D problems, is extended and verified in 3D, by comparing the results with the analytical solution of a monopole source, including its differentiation. Based on this, a feasible 3D aeroacoustic noise prediction and continuous adjoint-based shape optimization framework is developed for aeroengine intake application which makes use of a steady flow and adjoint solution and builds the unsteady field by rotating steady fields, overcoming the downsides of unsteady adjoint, namely large memory requirement and long solution time.
The developed tool is used to optimize the shape of the nacelle lips and the throat area of an aero-engine intake. The value of the aeroacoustic objective function is noticeably reduced highlighting the functionality of the developed tool. Extensions to consider both aeroacoustic and aerodynamic objectives in a multi-disciplinary framework are straightforward.