## Abstract

In this paper, an acoustic pressure gradient formula capable of accounting for constant uniform flow effects is suggested. Acoustic pressure gradient calculation is key for acoustic scattering problems, because it may be used to evaluate the hardwall boundary condition. Realistic cases of rotating machines may be evaluated in a moving frame of reference and as such, an acoustic pressure gradient formula capable of accounting for constant uniform flow effects finds significant application. A frequency domain formulation was thus derived for periodic noise source motion located in a moving medium. The suggested formula is mathematically compact and easy to implement. It may offer us significant advantages when tonal noise emissions are dominant, thus finding application potential in acoustic scattering problems in rotating machines in a constant uniform flow. Moreover, the formula contains no Doppler factor, thus facilitating noise prediction for sources in supersonic motion.

## 1. Introduction

The advent of massively parallel computers and recent advances in the field of computational aeroacoustics (CAA) enable accurate prediction of aerodynamic noise from rotating machines. Prediction of the incident sound field generated from the rotor’s blades may be accurately computed by hybrid CAA methods, involving a computational fluid dynamics solution of the flow field coupled to an acoustic analogy [1] solver which propagates noise emissions to the far-field.

Noise prediction of the incident sound field mainly relies on integral methods based on Lighthill’s acoustic analogy [1]. The Ffowcs–Williams and Hawkings (FW–H) equation [2] is among the most commonly used approaches for problems involving a permeable or non-permeable data surface, surrounding the noise sources in the near-field. The solution of the FW–H equation may be obtained in the time or the frequency domain. The most popular time domain solution of the FW–H equation is Farassat’s 1A formulation [3], which has been widely implemented in rotor noise prediction codes [4–9]. In order to avoid some drawbacks of time domain formulae, particularly owing to the Doppler singularity present in supersonic cases, several frequency domain formulae have been suggested [10–15]. Frequency domain methods are useful when tonal noise mechanisms are dominant, and thus noise prediction needs to be realized for specific frequencies only.

However, reflected and scattered sound waves may be of greater amplitude than the incident acoustic field [16], particularly when realistic configurations of turbomachinery or installed aircraft engines are considered [17,18]. Several numerical or analytical approaches have been introduced for calculation of acoustic scattering in aircraft engines and turbomachinery (Mao & Qi [19–22]). These methods require the evaluation of acoustic velocity or the acoustic pressure gradient components on the scattering surface, in order to satisfy the boundary conditions (see [23–27]). Because direct numerical evaluation of the acoustic pressure gradient and acoustic velocity is computationally expensive for realistic cases, Lee *et al.* [28] suggested analytical time domain formulae G1 and G1A for pressure gradient prediction, which provide a way to evaluate the scattering boundary condition, significantly reducing computational time. Recently, Ghorbaniasl *et al.* [29] proposed an analytical acoustic velocity formulation in the time domain, referred to as formulations V1 and V1A, which enables direct treatment of the acoustic scattering boundary condition. A frequency domain version of the acoustic velocity formulae was obtained by Mao *et al.* [30].

In the presence of uniform constant flow, such as a typical wind tunnel case, all the aforementioned formulations require application of the moving observer method to implicitly include effects of uniform constant velocity on the radiated noise. Because within the formulations, the terms associated with the uniform constant flow velocity effects are not present, the observer has to be set in motion along with the data surface, moving with the same constant uniform velocity but in the opposite direction. The moving medium approach provides us an alternative way to solve this problem by explicitly including the terms related to incidence in the acoustic sources and acoustic propagation formulae. The moving medium approach explicitly shows the effect of incidence on the Doppler amplification of radiated noise. In addition, it provides a way of examining the physics of the problem, while processing asymmetric inflow effects. Furthermore, in applications involving rotating machines, the effect of flow incidence can also be equivalent to an unsteady acoustic source, which generates more effective radiation modes [31,32].

In this context, solutions for the convective FW–H equation have been introduced in the time domain (see [32,33]) and the frequency domain [15], thus explicitly accounting for constant uniform flow effects. Despite recent advances in the calculation of acoustic pressure in a moving medium, no formula is available in the literature for direct or indirect calculation of acoustic velocity in the moving medium frame. However, derivation of an acoustic velocity formulation is not straightforward for moving medium problems and leads to a formula which is not mathematically compact. Therefore, an acoustic pressure gradient formulation is preferable for acoustic scattering applications in a moving medium.

The contribution of this paper is the development of an acoustic pressure gradient formulation from the FW–H equation for moving surfaces. The formulation will be derived in the frequency domain for sources in periodic motion, while explicitly accounting for the effects of constant uniform flow and incidence angle on the radiated noise. The suggested formula may be used for prediction of acoustic scattering when a frequency domain solver is preferable and the noise source is located in a moving medium. Such an approach is particularly common when noise from rotating machinery is concerned. The formula will contain no Doppler factor, which will enable fast prediction of noise in supersonic cases, as depicted by Wells & Han [34]. It will be capable of handling supersonic source motion, limiting the data surface to a strictly subsonic uniform constant flow. An important advantage of the derived formula is that the application to a stationary data surface in a constant uniform flow is significantly less costly, thus enabling noise prediction for a wide range of frequencies when realistic cases are concerned. The formulation is verified by computational results for a three-dimensional monopole and dipole in a moving medium for stationary data surface problems. A propeller case with supersonic blade tips is used to prove the validity of the formula for sources in supersonic motion, located in a subsonically moving medium.

The suggested formula can be used to scattering equations adapted accordingly for moving medium problems, which is the future work of the authors.

## 2. Acoustic pressure gradient in the frequency domain

### (a) Sources in motion

The starting point is the convected FW–H equation in the time domain [32] given by
*c*_{0} is the speed of sound. Here, *δ*(*f*) is the Dirac delta function and *H*(*f*) is the Heaviside function of the data surface, *f*(** x**,

*t*), so that

*v*

_{α}the velocity of the data surface. In addition,

*u*

_{α}the local flow velocity components and

*ρ*is the local fluid density, whereas

*ρ*

_{0}denotes the fluid density at rest. In equation (2.3),

*P*

_{αβ}represents the compressive stress tensor, where the flow quantities are given in the coordinates fixed to the medium at rest. Because of the Heaviside function, equation (2.1) is valid in the entire domain so that integral representation of the solution can be readily found using the free space Green’s function. Construction of an integral equation by using the Green’s function combines the effect of sources, propagation, boundary conditions and initial conditions in a simple formula.

In order to derive a solution without Doppler factor and to ease the prediction of the noise from supersonic propeller blades, located in a subsonic flow environment, the Green function can be conveniently expressed as
*k*=*ω*/*c*_{0} is the acoustic wavenumber, *ω* being the angular frequency and *R* and *R** are defined as follows:
*r*=|** x**−

**(**

*y**τ*)| is the distance between the source position

**and the observer position**

*y***. In addition,**

*x*In the following, we concentrate on the dipole source type (second term) and the monopole source type (third term) of equation (2.1), which are efficient in case a permeable data surface is used. Therefore, equation (2.1) can be simplified into the following form
*S* in a reference frame so-called the ** η**-frame, fixed to the data surface, one has

Note that the results before the integrations over *ω* are the spectrum of the acoustic pressure *p*′_{1}(** x**,

*t*) and

*p*′

_{2}(

**,**

*x**t*). Therefore,

Equation (2.15) gives the solution for the convected three-dimensional FW–H equation, for surface sources in motion. The influence of constant uniform flow velocity and acoustic effects of incidence are explicitly taken into account. This formulation is derived without Doppler factor, so it may be applied more conveniently to supersonic data surfaces [34], located in a strictly subsonic flow environment.

It is worth noting that the present formulation is mathematically equivalent to the formulation suggested by Xu *et al.* [15]. The present formulation is mathematically more compact than the one derived in [15], whereas the latter separates the contributions of monopole and dipole sources. Equation (2.15) may be implemented in frequency domain solvers for noise prediction in the cases of rotating machinery in a constant uniform flow.

Based on the solution formulation of the convected three-dimensional FW–H equations, the acoustic pressure gradient can be found by taking the gradient of the terms ** x**,

**and**

*η***are independent variables, ∂/∂**

*τ**x*

_{α}was taken inside the integrals.

One can use the following relations

It should be pointed out that when sources in motion are considered, the data surface can either enclose the sources, thus being stationary, or can be chosen to coincide with the moving body surface (i.e. data surface chosen as the surface of a rotating blade). In the latter case, the data surface will also be in motion. Although the derived formula are mathematically valid to arbitrarily moving sources, it is not practically preferable for non-periodic noise source motion. In this case, time domain approaches such as formulation G1A are clearly more advantageous.

This formula may be used for evaluation of the boundary conditions on the scattering surface, for acoustic scattering prediction. In addition, the derived acoustic pressure gradient formula should be used in a system of equations [22] modified accordingly for scattering surfaces located in a uniform constant flow.

Equation (2.20) may thus be used alongside equation (2.15) to predict the scattered and incident acoustic field, in a frequency domain solver, when a moving medium approach is needed. The suggested formula is mathematically compact and may be used efficiently for noise prediction of rotating machinery in uniform constant flow, where tonal noise is dominant.

### (b) Stationary data surface

In most applications, the data surface is considered to be stationary, i.e. *v*_{β}=0, which is a valid hypothesis when simulation of most realistic applications is concerned. This leads to simpler formulations, because *R* and *R** are time-independent variables, thus allowing us to take *R* and *R** outside the retarded time integrand. Consequently, by using the definition of the Fourier transform, equations (2.15) and (2.20) can be reduced to
*Q* and *F*_{β}, respectively.

Frequency domain formula containing two integrals (one in time and one in space) are significantly costly to compute, when a large range of frequencies is required. Consequently, frequency domain formula are limited to noise calculation of specific tones, thus being more efficient than time domain formulations for such cases. On the other hand, equations (2.21) and (2.22) enable efficient calculation of the acoustic pressure and the acoustic pressure gradient, respectively, for the case of a stationary data surface. The need of evaluating one extra integrand in time, as seen in equations (2.15) and (2.20), has thus been obviated, and the input flow data are acquired in the frequency domain by a FFT.

The improved efficiency of equations (2.21) and (2.22) allows their application in frequency domain solvers for incident and scattered acoustic field prediction, even for a much broader range of frequencies.

## 3. Numerical results

Three test cases are presented in the following sections to validate the derived frequency domain solver formulation for a moving medium.

The two first test cases consist of stationary monopole and dipole sources located in a moving medium. A permeable stationary data surface is used for source calculation, because such cases are indicative of practical and industrial applications, where the moving sources are usually surrounded by a stationary data surface. In addition, the use of a stationary data surface obviates the need to calculate the time integral in the formulation, leading to efficient calculation. Far-field noise radiated from the sources is predicted by the proposed formulation. The acoustic pressure and pressure gradient time history and directivities are then evaluated and compared against the analytical solution.

The third case is the Isom consistency test and was chosen based on the need to confirm the validity of the derived formula for moving data surfaces. Noise generated from a two-bladed rotor is predicted using the developed formulations, and the data surface is solid, coinciding and rotating with the blades. The validation procedure is based on the comparison of the calculated acoustic pressure gradient components and those generated by the monopole noise source. Supersonic source motion is also considered, in a subsonic moving medium.

### (a) TC1: monopole source

The first validation case is a stationary monopole point source located in a moving medium with constant uniform flow. A single-frequency monopole source placed at the origin of a Cartesian coordinate system is considered.

As already explained in [34], the extended formulation of the complex potential for a single-frequency monopole source positioned at the origin and in a uniform flow with arbitrary configuration, is given as follows
*R* and *R** are obtained from equation (2.5). The particle velocity can be calculated by the gradient of the complex potential given in equation (3.1)
*R* and *R** required in equation (3.2) can be obtained from equation (2.14). The acoustic pressure of the monopole source in a moving medium of arbitrary flow orientation is then computed by the unsteady Bernoulli equation

The acoustic pressure gradient in the *x*_{α}-direction is given by
*A* to 1 m^{2} s^{−1}. The ambient speed of sound *c*_{0} is chosen 340 m s^{−1}, whereas the free stream flow density is assumed to be 1.234 kg m^{−3}. The emission frequency of the monopole is ** ω**=10

*π*rad s

^{−1}. The observers are located on a circle of radius

*r*

_{ob}=20 m. The source terms are calculated over one emission period

*T*=2

*π*/

*ω*. Sixty-four observer time points are used per time period, ensuring fine enough temporal resolution. The data surface is a sphere of radius

*r*

_{s}=1 m, and a mesh study was realized to ensure sufficient spatial resolution. The spherical coordinate system (

*r*,

*θ*,

*φ*) is introduced to describe the penetrable data surface, with 80 and 50 uniformly spaced points used to discretize the azimuthal angle

*θ*and the polar angle

*φ*, respectively. The total number of computational grid points is 4000.

Figure 1 shows the root mean square of the acoustic pressure at *r*_{ob}=20 m, and figure 2 depicts the acoustic pressure time history for an observer located at *r*_{ob}=20 m, *θ*=0° and *φ*=90°. In both graphs, the results obtained from the derived formulations are plotted against the analytical solution. Evidently, the predicted results are in very good agreement with the analytical solution for all the investigated inflow Mach numbers. The excellent agreement between results demonstrates that the developed frequency domain formulation of acoustic pressure is valid for the problems in the moving medium with arbitrary orientation.

Figure 1 further indicates that a constant uniform flow in different directions defined in the Cartesian coordinate system produces various effects on acoustic pressure directivity. It is worth mentioning that there is no convective effect if the monopole source is moving in a plane perpendicular to the direction of moving medium, which can be seen in figure 1*b*. Furthermore, if the flow Mach number in the source plane is increasing, then the convective effect on the noise propagation will increase simultaneously. Therefore, the level of acoustic pressure at a point upstream of the source will be greater than a corresponding point downstream, as seen in figure 1*c*.

In figure 3, the acoustic pressure gradient time history for an observer located at *r*_{ob}=20 m, *θ*=0° and *φ*=90° is compared against the analytical solution for different Mach numbers. The predicted results accurately match the solution obtained analytically. This result confirms the accuracy of the proposed frequency domain formulation for prediction of the acoustic pressure gradient.

### (b) TC2: dipole source

The second test case is a stationary dipole source in a moving medium with constant uniform flow of arbitrary orientation. The dipole source is equivalent to two very closely positioned monopole sources with opposite fluctuation phases. A dipole is described by an axis which is the line connecting the centres of the two nearby monopoles. In this article, we consider that the dipole axis is aligned with the *x*_{2}-axis in a Cartesian coordinate system and has a distance of *d*=0.01 m between the centres of two monopoles. The monopoles are equidistant from the origin. Applying the aforementioned velocity potential for the monopole source, the velocity potential for the dipole source can be conveniently obtained
*c*_{0} is chosen 340 m s^{−1}, whereas the free stream flow density is assumed to be 1.234 kg m^{−3}. The emission frequency of the monopole is *ω*=10*π* rad s^{−1}, and the velocity potential amplitude is *A*=100 m^{2} s^{−1}. The observers are located on a circle of radius *r*_{ob}=100 m. The source terms are calculated over one emission period *T*=2*π*/*ω*, whereas 64 observer time points are used per time period, ensuring fine enough temporal resolution.

Figure 4 depicts the comparison of the acoustic pressure directivity at *r*_{ob}=100 m, whereas figure 5 presents the acoustic pressure time history for an observer located at *r*_{ob}=100 m, *θ*=45° and *φ*=90°, for three different flow configurations defined in the Cartesian coordinate system. The predicted results very well duplicate the analytical solution, thus proving the validity of the formulation for acoustic pressure prediction.

Furthermore, figure 6 depicts the acoustic pressure gradient time history for an observer located at *r*_{ob}=100 m, *θ*=45° and *φ*=90°. The agreement between the predicted results with the analytical solution is also excellent, further validating the reliability and accuracy of the developed frequency domain formulation of acoustic pressure gradient.

### (c) TC3: Isom noise consistency

In order to show the applicability of the derived formula for a moving source surrounded by a moving data surface, the Isom thickness noise for propeller blades is used. Isom thickness noise property was described in the review paper of Brentner & Farassat [35]. It is shown that if a constant aerodynamic load, *v*_{n} over the surface.

This consistency was initially tested for a conventional helicopter blade in [36]. It was shown that inclusion of the blade tip source and refinement of the grid at the inner and outer radii of the blade, led to a good matching of the calculated Isom noise and monopole noise results [36]. It was also indicated that by decreasing the thickness ratio at the extremities of the blade the calculated results agree with the theory. This study dealt with evaluating the Isom thickness noise property for rotating blades in a medium at rest. This test case was then successfully tested for rotating blades in the presence of a uniform constant flow with an arbitrary orientation [32]. Recently, the Isom consistency has also been shown for the acoustic velocity field simulated in a medium at rest [29]. The next step in the research of the Isom consistency test is the evaluation of its validity for acoustic velocity or acoustic pressure gradient in a moving medium frame. This paper will provide us numerical confirmation of this consistency for the acoustic pressure gradient predicted in a subsonically moving medium. It will be assumed that the source is in supersonic motion. To the best knowledge of the authors, it is the first time that this consistency is shown for supersonic source motion.

A rudimentary rotor consisting of two equally spaced blades is considered. The blades have a constant chord of 0.4 m, and the thickness ratio is 10%, which vanishes at the inner and outer radii of the blade [29]. The diameter of the rotating blades is 10 m. The blades rotate around the *z*-axis, on the *x*–*y* plane of a Cartesian coordinate system with a rotational frequency of 13 Hz, corresponding to a supersonic blade tip Mach number of 1.2. The blade surface is discretized into 75 cells around the blade aerofoil and 100 divisions along the radial direction. Considering the importance of noise sources located on the blade edges, a non-uniform mesh density is used in both directions to refine the grids near blade edges. A calculation is performed for 120 sampling time points per period to resolve the pressure frequency spectrum.

The calculations are performed for one observer point located at *r*_{ob}=50 m, *θ*=0° and *φ*=90°. Two flow configurations are considered, corresponding to a medium at rest and a subsonically moving medium of

## 4. Conclusion

In this paper, an analytical formulation is suggested for the prediction of the acoustic pressure gradient in the frequency domain. The formula is derived for sources in periodic or rotating motion, located in a moving medium, thus explicitly accounting for the acoustic effects of incidence and angle of attack on the radiated noise.

A verification study was realized through two analytical test cases of a monopole and dipole source located in a moving medium. The obtained results were in exceptionally good agreement with the analytical solution, thus providing verification of the derived formula for the acoustic pressure gradient for stationary data surfaces. In order to extend the validation to data surfaces moving with sources, a two-bladed rotor with supersonic blade tips was chosen and was located in constant uniform subsonic flow. The calculated results well matched the theory even for sources in supersonic motion.

A frequency domain approach was preferred to a time domain approach, in order to obtain a mathematically compact formula. The frequency domain solution provides us further benefits when there is a need to evaluate the acoustic field of only few dominant tonal modes. The derived formulations present an important advantage, because their application to a stationary data surface in a constant uniform flow can be realized at very low computational cost, thus allowing application to realistic cases and investigation of a wide range of frequencies. Additionally, the developed formula contains no Doppler factor, which will enable fast noise prediction, when cases of supersonic data surface motion are concerned.

The suggested formula may be used for evaluation of acoustic scattering effects, when the source is located in a moving medium. The approach finds significant application potential in rotating machines and aircraft, where an angle of attack may introduce additional noise sources, whereas acoustic scattering in the vicinity of the noise source cannot be neglected. Application of the suggested formulations to problems of moving data surfaces will be illustrated in future work of the authors.

## Ethics

This work did not involve any experiments on humans or animals.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

G.G. conceived of the suggested formulations and the mathematical methodology required, and then carried out the derivation procedure; Z.H. coded the mathematical formulations and implemented them in the test cases for validation purposes; L.S.-R. set up the test cases and drafted the manuscript; C.L. coordinated the work and helped draft the manuscript. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The research performed by the first and fourth author was funded by Vrije Universiteit Brussel (VUB). The research performed by the second author was funded by a CSC-VUB scholarship (grant no. [2014]3026), while the third author was funded by a PhD grant from the Agency for Innovation by Science and Technology in Flanders (IWT).

## Acknowledgements

The research was supported by PhD grants from the CSC-VUB scholarship (grant no. [2014]3026) and the Agency for Innovation by Science and Technology in Flanders (IWT). The authors gratefully acknowledge this support.

- Received May 26, 2015.
- Accepted October 28, 2015.

- © 2015 The Author(s)