## Abstract

Kirchhoff’s integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff’s formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.

## 1. Introduction

The Kirchhoff formula was suggested by Gustav Kirchhoff in 1883 [1] and was primarily used in the theory of light diffraction and other electromagnetic problems. It is one of the most useful results in wave propagation theory and particularly in acoustics. As an integral method applicable to noise prediction problems in modern computational aeroacoustics (CAA), the Kirchhoff method allows flow-generated noise to be propagated and predicted based on the pressure and its derivatives in time and space, on an arbitrary control surface (data surface). The linear wave equation is assumed to be valid outside the data surface. The data surface should thus be placed in the nonlinear flow region, including all nonlinear flow effects. Acoustic pressures predicted by the Kirchhoff formula can be substantially in error if the Kirchhoff surface is located inside the nonlinear region. This is the main shortcoming of the method, compared with the Ffowcs Williams & Hawkings (FW-H) [2] equation, which may predict noise emissions fairly accurately, even when the integration surface is close to the noise-generating surface, thus neglecting external quadrupoles. The two methods are therefore equivalent when the Kirchhoff surface is placed in the linear flow region [3]. On the other hand, the FW-H equation is based on the conservation laws of fluid mechanics rather than the wave equation, which is the case for the Kirchhoff formula. Consequently, the FW-H equation is not appropriate for all types of wave propagation. In practical CAA problems, the Kirchhoff approach finds wide application to aerodynamic noise generation involving high-lift devices [4,5], rotors [3,6–9] and jet flows [10–14].

Initially, the Kirchhoff formula was limited to a stationary data surface [15]. In 1930, the original Kirchhoff formula was extended to moving surface applications by Morgans [16], using the Green’s function approach. Morino also used the Green’s function method and developed a time-domain and frequency-domain formulation for a uniformly translating surface [17,18], which he then extended to a rotating surface [19]. This approach however was significantly cumbersome and complicated. In order to achieve a simpler derivation of the Kirchhoff formula, Farassat & Myers in 1988 [20] suggested a fast and modern method to obtain Morgans’ result by using generalized functions theory in the time domain. They also provided an extension of this formula to moving surfaces, which has been widely used for high-speed helicopter rotor noise prediction. More recently, Farassat & Posey [21] presented a faster and simpler approach for the derivation of the aforementioned Kirchhoff formulation for moving surfaces [20]. An extensive review of the use of Kirchhoff’s method in CAA and its applications was provided by Lyrintzis [22].

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 emitted noise. The terms associated with the mean flow velocity effects are not present in the formulations and, thus, the observer must be put in motion along with the data surface, with the velocity of the flow but in the opposite direction. The same problem, however, can be alternatively solved by the moving medium approach. The main advantage of moving medium methods is that terms related to incidence are explicitly included in the acoustic sources and acoustic propagation formulae, thus showing the effects 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. When rotating noise sources such as propeller blades are concerned, the effects of mean flow incidence can be equivalent to an unsteady acoustic source, resulting in generation of more effective radiation modes [23,24]. Recently, Lee & Lee [25] proposed a convective Kirchhoff formula for boundary integral problems, by considering only streamwise flow motion.

The Kirchhoff approach has thus not been implemented for aerodynamic noise prediction for sources in arbitrary motion located in a moving medium with an arbitrary orientation. The contribution of this paper is the development of a convected Kirchhoff formulation for moving surfaces that explicitly takes into account the effects of mean flow and angle of incidence on radiated noise. The formulation will be derived in the time domain for moving surfaces, with respect to a moving medium frame and will be an extension of Farassat & Myers’ formula [20] to moving medium problems.

The derivation method of the formulation will follow the fast and modern procedure depicted by Farassat & Posey [21], by using generalized functions theory [26,27]. This formulation consists of the partial derivatives with respect to the source coordinates and time, thus making the formulation easy to use when studied numerically. The extension of the formula to moving medium problems will be realized according to the procedure described by Ghorbaniasl & Lacor [24], who suggested a moving medium solution of the FW-H equation [2]. A formula with the Doppler factor will be developed because it is more easily interpreted and is more helpful in examining the physics of systems. For subsonic cases, the present formulation explicitly shows the mean flow effect and its physical meaning. The formulation will be verified through two analytical subsonic test cases for stationary and moving Kirchhoff data surfaces.

As already mentioned, supersonic jet flows is an area of significant application potential for the Kirchhoff approach. The presented technique may thus be used to derive a formula which contains no Doppler factor, enabling fast prediction of noise in supersonic cases, as depicted by Wells & Han [28]. Another approach without the Doppler factor and its mathematical explanation can be found in [29], leading to formulation 4 of Farassat.

## 2. Governing equation in a moving medium

### (a) Derivation of the equation

Let us assume that *f*(** x**,

*t*)=0 (table 1) is the moving Kirchhoff data surface with velocity

**defined in such a way that the unit outward normal is**

*v**f*, then one can redefine it as

*f*/|

**∇**

*f*|, which does satisfy this condition on the surface. Furthermore, the definition of the unit normal in terms of the function defining the surface relies on a normalization of

*f*, which is always possible. The aim is to show how to derive the Kirchhoff formula for acoustic wave prediction radiated from sources on the data surface in a moving frame, using generalized functions. Assume the medium moves in an arbitrary direction with a constant velocity of

*x*

_{i}being the

*i*th component of

**. Equation (2.1) is valid for the region outside the data surface**

*x**f*(

**,**

*x**t*)=0. In order to extend the domain of the problem to inside the surface, one defines the function

*p*′(

**,**

*x**t*).

The extended governing equation is given by
*f*=0, then the jump is defined as
*δ*( *f*), by using the definition of equation (2.2), one has
*i*th component of

Similarly, generalized time derivatives *D*/*Dt* as follows:
*M*_{n}=*v*_{n}/*c*_{0} is the local normal Mach number of the surface, and *D*/*Dt*, one has
*Φ*_{t}=∂*Φ*/∂*t* and *y* and *τ*, i.e. *Φ*_{t}=*Φ*_{t}(** y**,

*τ*) and

As will be shown in appendix A, the derived convective wave equation (2.15) is consistent with the convective FW-H equation [24], when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3].

### (b) Solution of the equation

The governing wave equation obtained for the Kirchhoff formula is valid in the open space. We can now solve this equation using the Green’s function for the wave equation in the open three-dimensional space. One may use the Green’s function for a moving medium approach [24], i.e.
*τ*≤*t*, where *g*=*τ*−*t*+*R*/*c*_{0}. The quantities *R** and *R* are defined as
** y**(

*τ*) and the observer position

**(**

*x**t*), i.e.

**=|**

*r***(**

*x**t*)−

**(**

*y**τ*)|. Based on this Green’s function, the solution of equation (2.15) is given by

*S*:

*f*=0 be defined in a reference frame, called the

**-frame, fixed to the data surface, where the variable**

*η***is transformed to the variable**

*y***. For the sake of simplicity, we assume that the surface is non-deformable. It can, though, be easily extended to deformable surfaces. Therefore, equation (2.19) leads to**

*η***and**

*x***are independent variables, equation (2.20) can be rewritten in the following form:**

*y*The integration over d*τ* can be performed by using the methods of generalized functions theory [27]. It can be seen that [24]
*e* denotes that quantities inside the bracket are calculated in the emission time *τ*_{e}=*t*−*R*/*c*_{0}. This relation can be applied to equation (2.24), leading to
*Φ* and its time derivative *Φ*_{t} and space derivative **∇***Φ* as inputs for the wave propagation. It explicitly takes into account the convection effect on the noise prediction.

The formulation is equivalent to the Kirchhoff formula derived by Farassat & Myers [20] if the medium is set to be at rest, i.e. *θ**=*θ* and *R**=*R*=*r*.

In general, the Kirchhoff data surface *S* can deform. However, for the sake of simplicity, the derivative in this paper used a non-deformable data surface. In addition, in many applications of the Kirchhoff formula in aeroacoustics, the data surface is assumed to be rigid.

The derived formulation is based on the retarded time solution and its stability will depend on the relative position of the source and observer, the data surface velocity and the time step when it is studied numerically.

The above equation is a noise prediction formula based on the classical retarded time algorithms. This formula explicitly takes into account the mean flow influence and the aerodynamic and acoustic effects of incidence and is easy to implement in noise prediction codes. This formulation is derived with the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems.

### (c) An alternative form of the solution

Writing the formulation in terms of geometry and available kinematic quantities makes the formulation practically useful. In addition, in order to improve the accuracy of the results, it is also essential that the time derivative in the integrand of the second term on the right-hand side of equation (2.26) can be evaluated analytically. As demonstrated by Farassat & Succi [31], this can be realized by using the rule

Substituting equations (2.27), (2.29) and (2.33) into equation (2.26) gives the following formulation:

It can be checked that this formulation leads to the Kirchhoff formula for non-deformable moving surfaces derived by Farassat & Myers [20], under no mean flow influence.

#### (i) Simplified form for a stationary data surface

The simple case of a stationary data surface is suitable for practical applications and is commonly used in the hybrid computational fluid dynamics and acoustic analogy approach. The derived formulation (2.34) along with (2.35) and (2.36) can be simplified for stationary data surfaces as follows:
*r* not being dependent on the retarded time.

## 3. Numerical results

Here, the suggested convected Kirchhoff formulation for moving surfaces is validated through two analytical test cases. The first test case is a stationary monopole in mean flow with a stationary data surface, whereas the second case consists of a monopole translating along with the data surface, in a moving medium. Descriptions of the test cases and the calculated results are given in §3a,b.

### (a) TC1: monopole in uniform constant flow

In order to verify the present formulations, the analytical solution of a three-dimensional monopole source in uniform constant flow has been used, as described by Lyrintzis & Mankbadi [10]. This case is equivalent to the problem of a monopole in motion with a constant velocity in the *x*-direction, located in an ambient quiescent fluid. The analytical solution for the acoustic pressure at the data surface resulting from the monopole [10] is given by
*ω* is the angular frequency of the source, whereas the monopole source is located at (*x*_{s},*y*_{s},*z*_{s}) and the observer at (*x*,*y*,*z*).

In this study a spherical Kirchhoff surface of radius *r*_{s}=0.1 m is used as the integration (data) surface (a two-dimensional cross section of the domain is depicted in figure 2). The data surface is discretized into 4095 points, well distributed on the spherical surface, with angle increments equal to 4°. This resolution for the data surface was considered to be fine enough. The ambient speed of sound and the density of the undisturbed medium are chosen as *c*_{0}=1 m s^{−1} and *ρ*_{0}=1 kg m^{−3}, respectively. The monopole source is located at the origin (0, 0, 0) and its angular frequency *ω* is 1 rad s^{−1}. For discretization in time, 256 time steps are used. The inputs required for the Kirchhoff formula, namely the pressure and its derivatives in time and space, are calculated from equation (3.1).

The acoustic pressure fluctuations *p*′ obtained by the developed Kirchhoff formulation are plotted versus the analytical results for different observer locations. Three observers are positioned in the *x*−*y* plane (*θ*=0°), on a circle of radius *r*_{obs}=10 m and at angle increments of *α*_{1}=0°, *α*_{2}=90° and *α*_{3}=180°, as defined in a spherical coordinate system by
*x*−*y* plane on a circle of radius *r*_{obs}=10 m from the origin.

Two cases corresponding to mean flow Mach numbers of

### (b) TC2: monopole in uniform constant flow with a moving data surface

In order to achieve validation of the developed formulation for cases involving a data surface in motion, a second test case was considered. Because, to the best of our knowledge, an analytical test case of a moving data surface is not available in the literature, the aforementioned three-dimensional monopole was set in translating motion along with the Kirchhoff data surface at a constant velocity of *M*=0.2, while located in a uniform constant flow of

The noise prediction was realized by the moving medium formulation developed here. The input for the formulation, and thus the sources on the data surface, was calculated for a mean flow of *M*=0.2, towards the direction of the mean flow.

The reference for comparison was obtained by computing the emitted noise for the exact same case, using the Kirchhoff formula derived by Farassat for a medium at rest [20], applied with a moving observer approach. The inputs used for the reference solution were the noise sources calculated on the data surface, as described above. The reference solution was then obtained by imposing translational velocity, equal to *M*=0.2, on the source, whereas the source is also translating along with the observer with velocity

The acoustic pressure time history obtained by the developed Kirchhoff formulation is plotted versus the results from the moving observer method for the three aforementioned observer locations. Moreover, noise directivity obtained by the moving medium and moving observer approaches were calculated and plotted for 36 observers equally distributed in the *x*−*y* plane on a circle of radius *r*_{obs}=10 m from the origin. All required parameters, as well as the time and space discretization, are defined as described in §3a.

The outcome of the comparison of the present formulation against the moving observer method is depicted in figures 7 and 8, for the acoustic pressure time history and directivity, respectively. Apparently, the results of the two methods are in excellent agreement. In this case, more significant than the actual input to the methods is the final outcome, which proves that both methods provide identical results, thus validating the developed formulation for a moving Kirchhoff data surface.

## 4. Conclusion

A convected Kirchhoff formulation was suggested for arbitrarily moving surfaces in a uniform moving medium of arbitrary orientation. The convective wave equation was obtained using generalized functions theory, and the Kirchhoff formula was derived in the time domain with respect to a moving medium frame. The equation explicitly takes into account mean flow and incidence effects on the radiated noise and is easy to implement in noise prediction codes. It was shown that the suggested formula can be regarded as an extension of Farassat and Myers’ formula [20] to moving medium problems.

The derived formula consisted of the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems. The same methodology can be used to obtain a formula which contains no Doppler factor, thus facilitating noise prediction for supersonic cases.

A verification study was conducted based on two test cases involving a stationary and a moving data surface for an observer located in a moving medium. In the former case, the calculated results were compared with an analytical solution, whereas for the latter case results from the moving observer approach served as a reference. The comparisons verified the accuracy of the developed formula.

The suggested formulations may be applied to codes based on the Kirchhoff method for prediction of aerodynamic noise. Specific cases include jet noise, rotors and high-lift devices and in general any problems where the influence of mean flow and angle of attack is significant and should be taken into account.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

G.G. conceived the formulations and the required mathematical methodology, then carried out and drafted the derivation procedure; L.S.-R. validated the formulations through the analytical test cases and drafted the manuscript; C.L. coordinated the research and helped draft the manuscript. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This research is funded by the Research Foundation - Flanders (FWO) and a PhD grant from the Agency for Innovation by Science and Technology in Flanders (IWT).

## Acknowledgements

This research is supported by the Research Foundation - Flanders (FWO) and the Agency for Innovation by Science and Technology in Flanders (IWT). The authors gratefully acknowledge this support.

## Appendix A. Comparison of the Kirchhoff and FW-H equations in the moving medium

It has already been mentioned that the Kirchhoff formulation is consistent with the FW-H equation, when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3]. We will show here the equivalence of the moving medium FW-H equation and the moving medium Kirchhoff formula, equation (2.14), in the linear region, as shown by Brentner & Farassat [3] for a medium at rest. The parameter under consideration here is acoustic pressure, because this equivalence is of interest for aeroacoustic applications.

The FW-H equation in a moving medium can be given as [24]

From (A 1) and (A 2), one can obtain the following equation:

In order to show the relation between equations (2.14) and (A 3), let the Kirchhoff source terms in equation (2.14) be denoted by *Q*_{Kir}, where *φ* corresponds to the pressure fluctuation *p*′. Subsequently, the Kirchhoff source terms are added to and subtracted from equation (A 3), resulting in

Having

We can finally conclude that in the linear flow region, where

Evidently, the only remaining source term in equation (A 8) is second order in the perturbation quantity *u*_{i} and would be negligible in the derivation of the wave equation from the fluid conservation laws. Therefore, it is shown that the Kirchhoff and the FW-H formulations for a moving medium are equivalent, when the data surface is placed in the linear propagation region.

- Received October 1, 2015.
- Accepted February 18, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.