As a branch of acoustics,aeroacoustics is to study the noise generated aerodynamically. This kind of aerodynamic noise can be found from either fluid motion or its interaction with solid surfaces. The most prominent of these sources are associated mainly with air vehicles. Other representative sources include wind turbines,high-speed trains,and automobiles. The objective of aeroacoustics is to analyze and reduce noise from these aerodynamic sources.

Aeroacoustics study is pioneered by Lighthill ^[1,2] ,who established the acoustic analogy theory. Lighthill’s theory is considered as the origin of aeroacoustics research. The key idea of this theory is to use an acoustically equivalent source term to represent the complicated noise generation process so that the acoustic field can be described by solving the wave equation with the source term. The successful application of the theory leads to a significant reduction of jet noise and the wide use of high-bypass-ratio aero-engines in the new civil aircrafts. Later,Ffowcs Williams and Hawkings (FW-H equation) ^[3] extended Lighthill’s theory to the prediction of aerodynamic noise from moving surfaces. The FW-H equation is regarded as another milestone in the history of aeroacoustics because it provides the basis of acoustic design for rotating devices,such as helicopter rotors,propeller blades,and fans.

After Lighthill’s theory was established,the prediction of aerodynamic noise has mainly relied on the acoustic analogy theory or its variations. It is no exaggeration to say that the acoustic analogy theory not only provided an indispensable method for the aerodynamic noise prediction,but also acted as a fundamental guide for the development of noise reduction techniques during the past six decades. However,the aerodynamic noise generation is a very complicated process and hence cannot be fully described only by an acoustically equivalent source term. Moreover,the separation of flows and acoustics excludes the effect of flow-sound interaction. For these reasons,a more general aeroacoustics methodology is desirable to reveal the fundamental mechanism of the aerodynamic noise generation.

Since 1980s,the computational fluid dynamics (CFD) advanced rapidly and has been applied extensively in both scientific researches and engineering practices. The success of CFD clearly demonstrated the feasibility and advantage of numerical approach. In this context,it was put forward to solve aeroacoustics problems using computational methods. However,the aeroacoustics problems by nature have significant differences from the traditional fluid dynamics and aerodynamics problems. Accordingly,standard CFD schemes that are dedicated to solving fluid dynamics and aerodynamics problems numerically cannot be directly applied to aeroacoustics problems ^[4,5,6,7,8] . The major reasons are listed as below.

(i) Aerodynamics problems are time independent or only contain low-frequency components, whereas aeroacoustics problems are completely time dependent and usually cover a wide range of frequencies. In particular,accurate resolution of the high frequency wave that has extremely short wavelength poses a great challenge to the numerical schemes ^[6] . Standard CFD schemes are not capable of resolving sound waves in such a wide range of frequencies correctly.

(ii) Sound waves have very small amplitudes,normally five to six orders of magnitude lower than mean flows. Therefore,the numerical scheme is required to have very low numerical noise to avoid the contamination of the simulation results. Standard CFD schemes cannot meet the demand for such high accuracy.

(iii) The quantities of interest in aeroacoustics studies,invariably,are the far-field noise directivity and spectrum so that the solution needs to be accurate throughout the computation domain ^[6] . Therefore,the employed numerical scheme should have a dissipation and dispersion that is as less as possible. To meet such requirement,the standard CFD scheme needs a large number of grid points per wavelength,normally more than twenty. However,it is not acceptable for the aeroacoustics simulation because this means a huge computational resource demand when attempting to solve the high frequency components. A numerical scheme that can give adequate resolution with six to eight grid points per wavelength is needed.

(iv) Numerical simulation is inevitably conducted in a finite domain. Flow disturbances encountered in the aerodynamics or fluid mechanics problems decay very fast. Unlike flow disturbances,acoustic waves decay very slowly during the propagation ^[6] . To make sure the acoustic waves exit the artificial boundary smoothly with little reflection,an adequate nonreflecting boundary condition (NRBC) is needed. However,the existing NRBC used by the standard CFD methodology is not accurate enough for acoustics studies.

Due to the above reasons,standard CFD methodology is not capable of handling aeroacoustics problems. Since 1990s,CAA methodology started to show its advantage in dealing with aeroacoustics problems. With elaborately designed schemes and meshes,accurate flow and acoustic solutions can be obtained by CAA simultaneously and efficiently. This full set of time-dependent data then can be used to analyze the noise generation mechanism as well as the propagation process. Since the foundation of CAA,there have been a number of reviews on the CAA methodology. The representatives include the papers by Tam ^[4,5,6] and Lele ^[8] that were dedicated to reviewing the early advances of CAA,and a book by Tam ^[7] that offered a comprehensive description of CAA methodology. More recently,Li et al. ^[9] reported some advances of CAA in China. The objective of the present paper is to provide a brief review on the advances of CAA in the recent five years.

The rest of the paper is organized as follows. In Section 2,the early advances and applications of CAA are reviewed. This is followed by an introduction to the advances of CAA in the recent five years. Finally,a short discussion of the future prospective of CAA is offered. 2 Early advances and applications of CAA

Two key issues,which set CAA apart from CFD,are high order numerical schemes and nonreflecting boundary conditions. The research interest of CAA algorithms is therefore mainly focused on these two issues. In this section,a review about the early advances in the high order discretization scheme as well as the non-reflecting boundary condition is provided. Several successful applications of CAA methodology are then introduced briefly.

The numerical scheme employed by CAA should be less dispersive and dissipative to make the dispersion relation of the discrete equations almost the same as that of the original partial differential equations. The wave number analysis theory provides an effective way to quantitatively evaluate the dispersion and dissipation error of a numerical scheme. In CAA,this methodology has been substituted by truncated Taylor series method for numerical error analysis and scheme coefficients optimization. Based on this theory,a number of high order finite difference schemes have been proposed since 1990s. The compact scheme ^[10] and the dispersionrelation-preserving (DRP) scheme ^[11] ,which have been widely used in CAA applications,are two representatives. Besides,other high order schemes,such as the optimized compact scheme ^[12] , the high order upwind scheme ^[13] ,the optimized upwind scheme ^[14] ,the optimized compactdifference-base finite-volume scheme ^[15] ,and the compact spectral scheme ^[16] ,were successively proposed. Compared to standard CFD schemes,the major advantage of such high order and high resolution schemes is that the computation resource demand is significantly reduced. Moreover,high order schemes can ensure less phase deviation and amplitude decay of sound waves for long distance propagation.

The discrete partial differential equation by high order scheme supports spurious short waves. These short wave components are not the intrinsic solution of the original partial differential equation and should be damped out to avoid the contamination of the right solutions. Tam and Webb ^[11] proposed an artificial selective damping method to remove these high frequency spurious waves to keep the simulation stable. Bogey and Bailly ^[17] and Berland et al. ^[18] proposed a selective filter to suppress the numerical spurious waves.

The time discretization scheme is another important issue of CAA. To ensure an accurate time-dependent solution,the time integration scheme is also required to be less dispersive and dissipative. Tam and Webb ^[11] optimized the Adam-Bashforth scheme in wave number space. Hu et al. ^[19] proposed the low-dissipation and low-dispersion Runge-Kutta (LDDRK) scheme. Stanescu and Habashi ^[20] later formulated the 2N-storage LDDRK scheme,which requires less memory than the standard LDDRK scheme.

The simulation of aeroacoustics problems requires careful treatment in the artificial boundary. Otherwise,the numerical reflection of the outgoing waves would contaminate the physical solutions. However,it is almost impossible to construct a completely NRBC. The objective of the research on NRBC is to construct a boundary condition that can reduce the numerical reflection as much as possible. Based on different theories,the current boundary conditions can be classified into three categories: the asymptotic boundary condition,the characteristics boundary condition,and the absorbing boundary condition. The NRBC proposed by Bayliss and Turkel ^[21,22] ,Hagstrom and Hariharan ^[23] ,Tam and Webb ^[11] ,and Tam and Dong ^[24] falls into the first category. Extensive numerical experiments have indicated that the asymptotic boundary condition has good performance only when the sound sources are sufficiently far from the boundary. Once the sources move towards the boundary,the effectiveness of the asymptotic boundary condition is degraded. Giles ^[25] ,Thompson ^[26,27] ,and Poinsot and Lele ^[28] proposed to use characteristics to form the NRBC. These characteristics boundary conditions work well for acoustic disturbances incident nearly normally on the boundary. They cannot guarantee satisfactory results at oblique wave incidence or when the tangential component of the mean flow is strong at the boundary. Besides,setting an absorbing layer at the open boundary is another idea of constructing the NRBC. A simple example of absorbing boundary condition is the Newtonian cooling boundary condition. However,Newtonian cooling has not been mathematically proved,and its effectiveness is fully dependent on the user’s experience. Another widely used absorbing boundary condition is the perfectly matched layer (PML). Berenger ^[29] proposed the original PML method for computational electro-magnetics. Hu ^[30] was the first to apply the PML technique to acoustics problems. The PML technique applicable for linearized Euler equations with uniform mean flow was constructed at first and then extended to non-uniform flow,and the stability performance was also improved ^{[31,32,33,34]} . One of the major advantages of the PML technique is that the open boundary can be put very close to the acoustic source. This makes it possible to use a much smaller computational domain.

The objective of CAA is not only to develop high accuracy numerical methods but also to implement these methods to solve aeroacoustics problems. With the advances in numerical scheme and boundary condition,CAA methodology has become a useful approach for the aeroacoustics studies in some model problems,such as the duct fan noise ^[35,36,37] ,jet noise ^{[38,39,40,41,42]} , cavity noise ^[43] ,and airfoil noise ^[44] . However,the application of CAA methodology to realistic engineering problems,that probably involve large spectral bandwidth,length scale disparity of acoustic wave and mean flow,complex geometry,and completely nonlinear flow,is still facing some severe challenges in numerical algorithms. 3 Advances of CAA in recent five years

During the past five years,significant progress has been made in the numerical algorithms of CAA methodology,including spatial discretization schemes,time integration methods,nonreflecting boundary conditions,and the broadband impedance model. The following subsections will give a brief description of the advances in the four research topics. 3.1 Spatial discretization schemes

Complex geometry is one of the major difficulties that limit further application of CAA to realistic aeroacoustics problems. Recently,several advances on the spatial discretization schemes have improved the capability of CAA for problems with complex geometry.

Unstructured grids are often used to deal with complex geometry. However,it is difficult for spatial discretization schemes to achieve high accuracy in unstructured grids. The spectral difference (SD) scheme is a new high-order method for unstructured grids. The SD scheme is efficient and easy to achieve high order accuracy because it is based on the differential form of the governing equations. The SD scheme was first proposed by Liu et al. ^[45] and then extended to 2D Euler equations on triangular grids by Wang et al. ^[46] and 3D Navier-Stokes equations for hexahedral meshes by Sun et al. ^[47] . Gao et al. ^[48] conducted an optimization on the extrapolation and interpolation schemes for the 4th order SD scheme in the waver number space. Through the optimization,the improvements of the SD scheme are in two aspects. One is the reduction of dispersion and dissipation error over the resolvable wave number range. Another is the stability. The original 4th order scheme with Chebyshev-GaussLobatto flux points is weakly unstable. After the optimization,the stability characteristics have been improved. The weak instability even could be eliminated completely if an additional second order filter on selected flux points is used.

The numerical simulation using structured grids puts some requirements on the mesh quality, e.g.,the grid smoothness,orthogonality,and grid-size continuity. This is difficult to be satisfied when the problem involves complex geometry. Gao ^[49] proposed an interface flux reconstruction (IFR) method. In this method,only one coincided grid point is needed at the interface between neighboring blocks. No other requirements on the mesh quality are needed to be satisfied. Therefore,the grid for the IFR method could be discontinuous and non-smooth across the block interface. The common flux at the interface can be computed or reconstructed with the Riemann solver as the way in SD and FR methods. The proposed IFR method is proved to be 4th order accuracy and stable for 7-point DRP scheme ^[11] as well as the 4th order compact scheme ^[10] . Fernando and Hu ^[50] proposed a finite difference scheme based on the discontinuous Galerkin method,i.e.,the DGM-FD. The DGM-FD scheme inherits naturally some features of the DGM,such as the high-order approximations,use of non-uniform grids,and super-accuracy for wave propagations. These features will be helpful in dealing with complex geometry for CAA. 3.2 Time integration methods

The simulation of aeroacoustics problems often involves multi-scale physics. The large disparity in amplitude and length scale between flow and acoustic is a big challenge for the design of computational mesh. More specifically,very fine mesh is needed in the source region to resolve the fine scales,whereas in the region far away from the acoustic source,relatively coarse mesh is sufficient to capture the propagation of sound wave. According to the CFL condition,the maximum allowable time step of the time integration is proportional to the characteristic length scale of the mesh so that the allowable time step of the finest mesh is several orders of magnitude smaller than that of the coarsest mesh. If the uniform time-step integration method is adopted,one has to choose the minimum time step over the entire domain to guarantee a stable computation. Obviously,this strategy is very time consuming.

One way to improve the computational efficiency while retaining the time accuracy is to use the multi-time-step method. In this method,the time marching is conducted with different time steps in different regions. Tam and Kurbatskii proposed the Adam-Bashforth type multitime-step method ^[51] and applied this method in the study of resonant liners ^[43] and jet screech phenomenon ^[52] . Further,Garrec et al. ^[53] extended this method from Cartesian to curvilinear coordinate system. However,there is a common limitation among the previous multi-time-step methods that the grid size ratio of two neighboring blocks is restricted by a fixed value. For the problems with complex geometry,this limitation would make the mesh design very difficult.

Recently,Liu et al. proposed a non-uniform time-step RK method,which was first applied to DG scheme ^[54] and then extended it to a finite difference scheme ^[55] . This non-uniform timestep RK method has been applied by the DLR in its acoustic simulation software DISCIO. The computational efficiency is reported to be increased by 5 times in the simulation of aircraft high lift device noise ^[56] . Lin et al. ^[57] developed an optimized time interpolation scheme and proposed to use time interpolation to achieve multi-time-step integration. This multi-time-step strategy has been successfully implemented in the simulation of flow over NACA0012 airfoil with the spectral difference method ^[58] and the speed-up ratio is up to 2.2 in this case. The proposed two multi-time-step methods do not have any limitation on the grid size ratio across the interface of two adjacent blocks. Multi-time stepping can be more generally applicable for complex geometries and grids,if the implementation is automated. This has been realized by Allampalli ^[59] who developed a new fourth order multi-time-step Adams-Bashforth scheme. This new scheme has been implemented and validated by NASA Glenn Research Center’s broadband aeroacoustic stator simulation (BASS) code. 3.3 Non-reflecting boundary condition

The PML technique has proved its capability of constructing an accurate and efficient nonreflecting boundary condition. However,most of the work done for the PML is only valid when the governing equation is linear or the mean flow is perpendicular to a boundary. The formulation of the PML applicable for nonlinear equations and more general case of mean flow in an arbitrary direction is therefore in need.

The PML technique itself is relatively simple when it is viewed as a complex change of variables in the frequency domain. However,there are still some rules we need to follow. Oth-erwise,we probably cannot derive a stable PML boundary condition. Hu ^[33] has proved that to construct a stable absorbing boundary condition using the PML technique,one needs to make sure that the phase and group velocity of the physical waves supported by the governing equations should be consistent in the same direction. For governing equations that support physical waves of inconsistent phase and group velocities,such as the Euler or Navier-Stokes equations, a space-time transformation is required before applying the PML technique in the derivation process. Following this idea,nonlinear PML boundary conditions applicable for Navier-Stokes equations have been constructed either in a conservative form ^[60] or in a primitive form ^[61] during the past five years. Numerical validations showed that in the case of strongly nonlinear incoming waves,the maximum reflection errors of the nonlinear PML,either in a conservation form or in a primitive form,are reduced by one to two orders compared to the linear PML.

However,there is still a big challenge in the formulation of PML for the case of more general mean flow. When the mean flow is perpendicular to a boundary,the inconsistency of the phase and group velocities only appears in the acoustic waves. A simple space-time transformation is sufficient in eliminating such inconsistency. However,when the mean flow is in an arbitrary direction,the inconsistency of phase and group velocities also arises in the entropy and vorticity waves. The simple space-time transformation is no longer valid. In this case,Parrish and Hu ^[62] proposed to derive the side and corner PML layers independently. The side layers are perfectly matched,and the corner layers are dynamically stable. A series of numerical examples also demonstrated the validity of the proposed PML formulation in the case of oblique mean flow for both linear and nonlinear problems. 3.4 Broadband impedance model

Acoustic liners are widely used in industry for noise reduction. They are indispensable noisesuppression devices for jet engines. Acoustic liners are effective over a wide band of frequencies. To account for such characteristics,broadband impedance models are needed ^[7] .

The acoustic impedance of a surface is originally defined in the frequency domain. Problems involving noise propagation over impedance surfaces may be solved in the time domain. Meanwhile,time-domain methods have obvious advantages over frequency-domain methods for broadband and nonlinear simulation. An equivalent time-domain broadband impedance model is therefore needed. Since 1990s,many broadband impedance models have been proposed to realize time domain simulations. Tam and Auriault ^[63] introduced a three-parameter time-domain impedance model. This model can provide an accurate representation of the liner impedance for single discrete frequency problems. Ozyoruk and Long ^[64] developed a broadband impedance model by a rational function in the z -domain,which can accurately describe the resistance and reactance of the liner impedance obtained by experiments. Rienstra ^[65] proposed an extended Helmholtz resonator model. The impedance of the model is derived by the response of the Helmholtz resonator to periodic acoustic input. Reymen et al. ^[66] developed a new and efficient time-domain impedance boundary condition through the recursive convolution. More recently, Li et al. ^[67] constructed a well-posed multi-freedom broadband impedance model. This model combines the three-parameter model and some features of the multi-pole model.

Since acoustic liners are composed of a large number of micro resonators,the understanding of the absorbing mechanism of the micro resonator is essential for providing an accurate description of the impedance characteristics of acoustic liners. Based on 2D and 3D direct numerical simulations,Xu et al. ^[68] proposed a modified impedance prediction model for resonators under a high sound pressure level. 4 Future prospective of CAA

After the development of more than twenty years,significant progress has been achieved in the numerical schemes of CAA. With elaborately designed meshes,high order finite difference schemes,and high quality artificial boundary conditions,CAA methodology has shown its capability of handling aeroacoustic problems and the potential for more complicated engineering problems. However,the further application of CAA to realistic aeroacoustics problems requires advances in the following aspects.

(i) An advanced turbulence modeling technique. Most of the flow induced noise is associated with the turbulence phenomena. Therefore,the prediction accuracy of the aerodynamic noise is not only dependent on the numerical scheme,but also on the turbulence modeling technique. However,most of the current turbulence models that are mainly developed for energy budget equations are not time-accurate for CAA because the sound generated by turbulence significantly depends on the space-time characteristics of the flow field. The improvement in the accuracy of the space-time correlation is required for the turbulence modeling ^{[69,70,71,72]} . Meanwhile, a large amount of computation resource is still required for the current turbulence modeling technique. An advanced turbulence modeling technique,which can reduce the computation resource,is therefore in urgent need.

(ii) A more accurate artificial boundary condition. The continuous growth in the computation scale now has limited CAA for further application to more complicated engineering problems. One way to resolve this challenge is to develop more accurate artificial boundary condition so that the computation domain could be much smaller than that of the current simulation. This would lead to a significant reduction in computation cost,especially for the three-dimensional simulation.

(iii) A mean-flow-based noise prediction method. From the viewpoint of real application,the noise prediction based on unsteady simulation is still time-consuming. However,the steady simulation technique has become very mature. The typical time duration is very short compared to unsteady simulation. Consequently,noise prediction in conjunction with CAA algorithms based on mean flow properties (e.g.,Reynolds stress) would be very attractive for real applications.

(iv) A high efficiency parallel computing strategy. In recent years,the graphic processing unit (GPU) has found some applications in scientific computation. It is reported that the scientific computation in a single GPU is over 10 times faster than running in the CPU. The hybrid CPU/GPU parallel computing strategy is a promising research topic for CAA.

(v) Reduced-order modeling. The aeroacoustics problems are governed by the full compressible Navier-Stokes equations. The accurate simulation of these governing equations is still very difficult and expensive even with recent progress on the computer hardware and numerical schemes,especially for the acoustic design of engineering problems. A promising approach to overcome this difficulty is to use the reduced-order modeling technique,in which the number of degree of freedom of the governing equations system is reduced and only the dominant behaviors of the original system are extracted ^[73] .