# Experimental Prediction of Performance and Broadband Noise of a Wind Turbine by Blade Element Theory

Soichi Sasaki, Shinsuke Mukae, Ken Tsushima

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Soichi Sasaki, Shinsuke Mukae, Ken Tsushima. Experimental Prediction of Performance and Broadband Noise of a Wind Turbine by Blade Element Theory. 17th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC2017), Dec 2017, Maui, United States. hal-03004966

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# Experimental Prediction of Performance and Broadband Noise of a Wind Turbine by Blade Element Theory

Soichi Sasaki<fnref n="1" />*, Shinsuke Mukae<fnref n="2" />, Ken Tsushima<fnref n="3" />

<figure><img src="image_2.png" /></figure>

ISROMAC 2017

International
Symposium on
Transport
Phenomena and

Dynamics of
Rotating Machinery

Hawaii, Maui

December 16-21
2017

**Abstract**

In order to provide a theory that can be used at the initial design stage for the high performance wind turbine, this work predicts not only the aerodynamic characteristics of the wind turbine but its broadband noise based on blade element momentum theory (BEM). The BEM represented the output coefficient of the measured value of the actual wind turbine. When the scaling law of the trailing edge noise model is applied, the predicted value by the analytical flow of the BEM could represent the measured broadband noise level generated from the actual wind turbine. The results of BEM indicate that hump noise in the high frequency domain is generated by the von Karman Vortex shedding in the hub side of the impeller. The broadband noise in the high frequency domain of the wind turbine was induced by the relative velocity in which the high speed at the tip side. On the other, the broadband noise in the low frequency domain is induced by the separated flow until the full stall condition.

**Keywords**

Aerodynamic Noise, Wind Tunnel Experiment, Renewable Energy

## 1. INTRODUCTION

A micro wind turbine can be used in various situations which are the backup power at the disaster, the auxiliary power supply in the public place, the power supply in an isolated place, and so on. Moreover, in Japan, restrictions for the corporation in the energy business have been relaxed by the liberalization of electricity (1) since April 2016, and the operation in the renewable energy business by the micro wind turbines has already started. However, the aerodynamic noise of the wind turbine which is driven by a combination of various mechanisms becomes 80 dB or more. To solve this noise problem is one of the important technical issues for the development of this kind of machine.

Aerodynamic noise generated from a solid boundary such as an impeller of a wind turbine is described by the acoustic analogy of N. Curle (2). In the case of aerodynamic noise, because the representative velocity is much slower than the speed of sound, the dipole sound source becomes the major factor. Moreover, in the noise generated from the wind turbine, the broadband noise becomes the major factor except for the periodical mechanical vibration noise sources such as the bearing, the motor, the resonance of the structure, and so on. The broadband noise resembles the characteristics of trailing edge noise. The primary model for this noise has been studied by R. K. Amiet or M. S. Howe in the seventies (3-4). The trailing edge noise is the broadband noise generated by perturbation of the convection flow released to the wake in the vicinity of the trailing edge of the blade. Roger & Moreau extended Amiet's trailing edge noise model to the limited chord length model (5-6); this principle has been applied in a lot of studies on the aerodynamic noise of turbomachinery (7-10). On the other

hand, a detailed review on the aerodynamics of the wind turbine rotor is described by Herman Snel (11-12). In the reference (11), the two-dimensional blade element momentum theory (BEM) has been explained. BEM is the method for the analysis on the fluidic properties of the blade in which the annular momentum is used for the global flow region. In fact, the application of the two momentum equations results in analytical relations between the induced velocity components and the aerodynamic forces exerted by the elements of the blade in this region. In the two dimensional BEM, it is assumed that these forces are equal to the forces on the same aerodynamic profile taken from two dimensional wind tunnel test. Relating the annulus aerodynamic momentum to the force in the wind tunnel, the closed set of a non-linear algebraic equations is obtained which can be solved numerically. This feature of the BEM is computationally very efficient. The BEM has been actively used for analysis of the initial development of the wind turbine (13-15), whereas commercial CFD codes are used for a lot of predictions on the aerodynamic noise problem (16). However, studies based on the BEM very seldom address the aerodynamic noise, more especially the broadband noise.

Therefore, in this study, we predict not only the aerodynamic characteristics of the wind turbine but its broadband noise based on the BEM in order to provide a theory that can be used at the initial design stage for the high performance wind turbines. First of all, the characteristics on the momentum and the aerodynamic noise of NACA00 series symmetric isolated airfoils are clarified by the wind tunnel test. In the next, the aerodynamic characteristics and noise of the actual wind turbine are measured by the wind tunnel test. Finally, while comparing the actual measured broadband noise

<footnote n="1">¹Graduate School of Engineering, Nagasaki University, Japan</footnote>

<footnote n="2">²Graduate School of Engineering, Nagasaki University, Japan</footnote>

<footnote n="3">³Nagasaki Industrial Promotion Foundation, Japan</footnote>

<footnote>*Corresponding author: souichi@nagasaki-u.ac.jp</footnote>
generated by the wind turbine with the predicted value by the BEM, we discuss the characteristics of the broadband noise generated from the propeller wind turbine.

# 2. EXPERIMENTAL SETUP

## 2.1 Wind tunnel

The wind tunnel for the measurement of the properties of the isolated blade is shown in Fig.1. The main dimensions of the blade are listed in Table 1. The angle of attack is defined as the angle formed by the center line of the blade and the stream line of the main flow. Two kinds of wind tunnels with the different nozzle scales are used for the performance test of the isolated blade. The aerodynamic noise generated from the blade is measured in the wind tunnel with the nozzle size 100 mm in Fig. 1 (a). When the Reynolds number defined by the nozzle size is approximately 2.0 × 10⁵, the turbulence of the main flow within the measurement region is less than 0.5%. The leading edge of the blade is set at 100 mm downstream from the nozzle exit.

<figure><img src="image_3.png" /><figcaption>(a) Nozzle size 100mm×100mm</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(a) Nozzle size 400mm×400mm</figcaption></figure>

<caption>Fig. 1 Exhaust type wind tunnel.</caption>

<table><caption>Table 1 Main dimensions of the blade</caption><thead><tr><th></th><th>Chord<br>C (mm)</th><th>Thickness<br>T (mm)</th><th>Span<br>L (mm)</th><th>t/C*100<br>(%)</th></tr></thead><tbody><tr><td>NACA0008</td><td>30</td><td>2.4</td><td>100</td><td>8</td></tr><tr><td>NACA0012</td><td>100</td><td>12</td><td>400</td><td>12</td></tr><tr><td rowspan="2">NACA0018</td><td>30</td><td>5.4</td><td>100</td><td rowspan="2">18</td></tr><tr><td>100</td><td>18</td><td>400</td></tr></tbody></table>

A 1/2-inch microphone (Ono-Sokki, LA-4350) is set at a distance of 0.5 m from the trailing edge; the angle between the main flow and the microphone is 90°. The signal from the noise level meter is sent to the FFT analyzer (Ono-Sokki, CF5210), and the frequency response characteristics of the noise are analyzed in the 1/3 octave band frequency. In this measurement method, we discuss the far field noise properties than the wave length of the point sound source without take into account of the directivity. The lift coefficient and drag coefficient of the blade are measured in a wind tunnel with the nozzle size of 400 mm in Fig. 1 (b). The force acted on the blade is measured by the load cell which has the capacity of 25 N (Tech - Gihan, TL2B09-25N). One end of the blade that is supported at both ends is affected by the momentum of the half of the fluid. In the two-component load cell, the lift and drag can be measured directly. Lift coefficient and drag coefficient are defined as;

$$C_L = \frac{2L}{\rho V^2 A}, C_D = \frac{2D}{\rho V^2 A} \qquad (1)$$

where, L is the lift force, D is the drag force, V is the main flow velocity, and A is the projected area of the blade.

In Fig. 2, the schematic view of the suction type wind tunnel is shown. The wind tunnel is composed of the collector, the measurement duct, the nozzle, the deformed duct and axial flow fan. Two-stage honeycombs of the two stages are attached to the collector; the turbulence of the incoming flow is removed by the honeycombs. The section of the measurement duct is 1800mm square, and the entire length is 5000 mm. The choke-mechanism of diameter 1200mm is provided to obtain enough uniform velocity to turn the impeller of the wind turbine. The mainstream velocity in the wind tunnel is controlled by the command of the axial flow fan.

<figure><img src="image_5.png" /><figcaption>Fig. 2 Suction type wind tunnel for turbine test.</figcaption></figure>

The experimental apparatus of the wind turbine is shown in Fig. 3. The diameter of the impeller is 1170 mm, the number of blades is three and the rating output is 400W. The experimental apparatus is equipped with the three-phase induction motor (Mitsubishi Electric Co., Ltd., SF-JR 400 W) for giving the reaction torque and the torque-meter (Ono-Sokki, SS-020) are equipped. This experimental setup can measure the mechanical output of the turbine directly. The rotational speed of the impeller is measured by the photoelectric rotation speed meter. In order to estimate the aerodynamic characteristics of the wind turbine, the output coefficient C<sub>p</sub> and the tip speed ratio λ is defined as;
$$ \lambda = \frac{R\omega}{V}, \quad C_p = \frac{2T\omega}{\rho AV^3} \qquad (2) $$

where, R is the radius of the impeller, ω is the angular speed, V is the mainstream velocity, T is the torque, ρ is the density of air and A is the projection area of the impeller. A 1/2-inch microphone (Ono-Sokki. LA-4350) is set at a distance of 1.0 m from the impeller inlet on the rotational axis of the impeller. The frequency response signal is sent to the FFT analyser (Ono-Sokki, CF5210).

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3 Experimental apparatus of the wind turbine.</figcaption></figure>

## 2.2 Blade Element Momentum Theory

Figure 4 illustrates the flow model around the impeller of the wind turbine. It is assumed that the flow passing thorough the impeller of the wind turbine is as an annular flow that does not interfere with other annuli. In this figure, Ω is the angular velocity of the impeller, α is the angle of attack, θ is the pitch angle, φ is the yaw angle. As shown in Fig. 4 (b), the velocity component of the relative velocity W is given by the axial induced velocity coefficient "a" and the rotational induced velocity coefficient "a'". At this time, the relationship between the angle of attack and the pitch angle is expressed by Eq. (3);

$$ \alpha = \phi - \theta \qquad (3) $$

Here,

$$ \tan \phi = \frac{1}{\lambda(r)} \frac{1-a}{1+a'}, \quad \lambda(r) = \frac{r \omega}{V} $$

where, r is the distance from the rotation center of the impeller to the blade element. The fluid forces C<sub>x</sub> and C<sub>y</sub> in the x and y directions are given by Eq. (4);

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 4 Flow model in the blade and impeller frames.</figcaption></figure>

$$ \begin{aligned} C_x &= C_L \sin \phi - C_D \cos \phi \\ C_y &= C_L \cos \phi + C_D \sin \phi \end{aligned} \qquad (4) $$

These two coefficients "a" and "a'" are determined analytically by successive approximation.

$$ \begin{aligned} a &= \frac{\sigma C_y}{4 \sin^2 \phi + \sigma C_y} \\ a' &= \frac{\sigma C_x}{4 \sin \phi \cos \phi - \sigma C_x} \end{aligned} \qquad (5) $$

Here,

$$ \sigma = \frac{B c}{2 \pi r} $$

where B is the number of blades, σ is the local solidity at the position r. If these coefficients are analytically obtained, the relative velocity and the angle of attack for the blade element can be given as Eq. (6) and Eq. (2).

$$ W = \sqrt{(r \Omega (1+a'))^2 + (V(1-a))^2} \qquad (6) $$

The output coefficient of the blade element is given as;

$$ C_p = 4a'(1-a) \lambda(r)^2 \qquad (7) $$

When the coefficient is integrated from the hub to the blade tip, the output of the wind turbine is obtained as;

$$ P = \frac{1}{2} \overline{C_p} \rho V^3 \pi R^2 \qquad (8) $$

with
$$ \overline{C_p} = \frac{1}{\pi R^2} \int C_p \cdot 2 \pi r dr $$

## 2.3 Prediction of the broadband noise

Figure 5 illustrates the flow regime around a two-dimensional blade. The broadband noise of the wind turbine is approximated by multiplying the trailing edge noise of one blade by the number of blades. According to Ref. (5), the power spectrum density (PSD) of the far field sound pressure $S_{pp}$ ($\omega$) generated from one blade element is given by;

$$ S_{pp}(\omega) = \left( \frac{\omega c}{2\pi a_0 S^2} \right)^2 L \Phi_{pp}(\omega) l_y(\omega) |I|^2 \quad (9) $$

<figure><img src="image_10.png" /><figcaption>(a) Boundary layer and perturbation on the blade</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) Back-scattering model at the trailing edge</figcaption></figure>

<caption>Fig. 5 Turbulent flow past a trailing edge.</caption>

In this analysis, all quantities are estimated by the BEM and the boundary layer displacement thickness by semi-empirical equation. The PSD of the wall pressure suggested by P. Gliebe et al. (17) is given as;

$$ \Psi_{pp} = \frac{10^{-4}}{(1 + 0.5 \tilde{\omega}^2)^{5/2}}, \quad \tilde{\omega} = \frac{\omega \delta_1}{W} \quad (10) $$

with

$$ \Phi_{pp} = \Psi_{pp} \rho^2 \delta_1 W^3 $$

Here, $\delta_1$ is the boundary layer displacement thickness, and $W$ is the relative velocity. The boundary layer displacement thickness between the pressure surface side and the suction side is evaluated by the exponential law.

$$ 2\delta_1 = 0.093 c R_e^{-0.2} \quad (11) $$

Corcos' model (18) has been chosen for the spanwise

correlation length.

$$ l_y(K_2, \omega) = \frac{\omega/(b_c W_c)}{K_2^2 + \omega^2/(b_c W_c)^2} \quad (12) $$

In present model, the spanwise wave number $K_2$ is assumed to be zero for simplicity. According to Ref. [6], the square radiation integral in the limit of very high frequency can be read as (see Appendix);

$$ |I|^2 \approx \frac{(1+M_0)(1-M_0-M_c)}{\bar{\mu}^2 M_c (1+x_1/S_0)} \left[ \frac{\beta^2}{M_c} + M_0 - \frac{x_1}{S_0} \right] \quad (13) $$

When the frequency is normalized by the chord length $c$ and the relative speed $W$, the equation (14) is established.

$$ \frac{\omega_o}{\omega_m} = \frac{c_m W_o}{c_0 W_m} \quad (14) $$

The ratio of the $S_{pp}$ of the measured value for the model in the wind tunnel test to the target value of the blade element in the actual wind turbine is given by;

$$ \frac{S_{pp}^o}{S_{pp}^m} = \left( \frac{\omega_o}{\omega_m} \frac{c_o}{c_m} \frac{R_m}{R_o} \right)^2 \frac{L_o}{L_m} \frac{\Phi_{pp}(\omega)_o}{\Phi_{pp}(\omega)_m} \frac{l_y(\omega)_o}{l_y(\omega)_m} \left| \frac{I_o}{I_m} \right|^2 \quad (15) $$

where, the superscript $m$ is the model value, and $o$ is the objective value in the wind turbine. In the case of the noise level, since decibel notation is common, we express equation (16) as logarithm;

$$ \begin{aligned} SPL_{o, j} = SPL_{m, j} &+ 20 \log \left( \frac{W_{o, j}}{W_{m, j}} \right) + 20 \log \left( \frac{R_{m, j}}{R_{o, j}} \right) \\ &+ 10 \log \left( \frac{L_{o, j}}{L_{m, j}} \right) + 10 \log \left( \frac{\Phi_{pp}(\omega)_{o, j}}{\Phi_{pp}(\omega)_{m, j}} \right) \\ &+ 10 \log \left( \frac{c_{o, j}}{c_{m, j}} \right) + 20 \log \left| \frac{I_{o, j}}{I_{m, j}} \right| \end{aligned} \quad (16) $$

where, superscript $j$ is the counter on the segment of the blade element. In the experimental analysis of the broadband noise using the scaling law, the target noise level of the wind turbine can be predicted by giving the model size of the isolated blade and the scale of the actual wind turbine to equation (16). Finally, we can predict the overall noise spectra generated from the wind turbine as;

$$ L_A = 10 \log \sum_{j=1}^{n} 10^{\frac{SPL_{o,j}}{10}} \quad (17) $$

where, $n$ is the number of segments. We transpose wind tunnel experiment made on an isolated airfoil to rotating blade segments. This transposition may be accompanied with a Doppler-effect correction. The basis for this has been proposed by Paterson & Amiet in the seventies (25) and re-addressed recently by Sinayoko et al, (26). In the present case of broadband noise at low Mach number
ignoring the correction is not expected to have a significant effect, especially in view of the other approximations made in the paper.

### 3. RESULTS AND DISCUSSION

In Fig. 6, the lift coefficient and drag coefficient of the NACA0012 are shown. The main flow velocity is 20 m/s, the Reynolds number based on chord length is $1.3 \times 10^5$. The thick solid line in Fig. 7 (a) is the lift coefficient of NACA0012 as predicted by Xfoil (19). The measured lift coefficient is almost the same as the predicted value by Xfoil up to the vicinity of the stall point. Because the lift coefficient becomes smaller from the angle of attack 14°, the stall point of this blade is in the vicinity of 14°. The measured drag coefficient becomes slightly larger than the theoretical value of Xfoil due to actual flow disturbance up to the vicinity of the stall point.

<figure><img src="image_12.png" /><figcaption>Fig. 6 Aerodynamic characteristics of the NACA0012.</figcaption></figure>

Fig. 7 is the relationship between the angle of attack and the overall noise level of the NACA0008. The main flow velocity is 30 m/s, the Reynolds number based on chord length is $5.9 \times 10^4$. The noise level becomes maximum at the angle of attack 3°, then the noise level decreases sharply as the angle of attack increases. The noise level rises from the angle of attack of 6° to the stall point, and as the angle of attack becomes larger than the stall point, the noise level drops again. In the Fig. 8, the spectral distributions of aerodynamic noise generated from the NACA0008 with different angles of attack are compared. The broken line is the background noise of the wind tunnel. The discrete frequency noise (DFN) is generated in the vicinity of 4000 Hz from the blade with the angle of attack 3°. This is the noise generated synchronized with the vortex shedding frequency (20). In reference [20], it was clarified that DFN was generated when the boundary layer on the blade surface was laminar and separated at the location upstream of the trailing edge. Meanwhile, they could not observe a close relationship between the existence of a separation

bubble and the generation of DFN. That is, they pointed out the DFN was generated even when the attack angle of the flow was nearly zero and then no separation bubble was observed. At the angle of attack of the stall point, a hump is seen in the noise spectrum below 1000 Hz. This noise is also induced by large scale vortex shedding; on the other hand, in the high frequency region above 2000 Hz, aerodynamic noise over a wide frequency band is generated from the blade.

<figure><img src="image_13.png" /><figcaption>Fig. 7 Noise characteristics of the NACA0008</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 8 Spectral distribution of the aerodynamic noise of the NACA0008.</figcaption></figure>

The relationship between the tip speed ratio and the output coefficient is compared in Fig. 9. Empty symbols stand for the wind tunnel test, and the filled symbols for the BEM. The wind turbine performance of the BEM near the tip speed ratio 6.5 is able to reproduce the measured values. In the present work, we have not measured sufficient blade properties of the negative angle of attack. When the rotation speed of the impeller is much higher than that of the design point ($\lambda$=6.5), the relative flow separates on the front surface of the impeller disc with the negative angle of attack because the flow cannot passing through the impeller. Therefore, in the scope of this study, we could not analyze the performance in the domain of the high tip speed ratio. In the following discussion, the characteristics around the tip speed ratio 6.5 are selected for comparing with the analytical results.
<figure><img src="image_15.png" /><figcaption>Fig. 9 Performance of the wind turbine.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 10 Spectral distribution of the aerodynamic noise of the wind turbine.</figcaption></figure>

In Fig. 10, the spectral distributions of the noise generated from the wind turbine with different main flow velocities are compared. The open symbols are the predicted noise by the BEM. In the case of main flow velocity 6.3 m/s, the broadband noise is large near 5000 Hz. The predicted value is able to represent this increase. The frequency of the predicted value is approximately the same as the center frequency of the hump noise in the high frequency domain of the actual wind turbine. In the case of the main flow velocity 8.7 m/s, the measured noise spectrum exhibits the discrete frequency noise (DFN) in the vicinity of 576 Hz, whereas in the predicted noise has not the DFN. Therefore, this result indicates that caused by mechanical vibration due to the resonance between the rotation of the impeller and the natural frequency of the experimental apparatus. In such a case which generates the strong mechanical vibration noise, we assumed that the von Karman vortex noise in the high frequency domain is not generated because the wake flow cannot make the stable staggered array of vortices due to the vibration.

The spanwise distribution of the sound pressure level contributes as predicted by the BEM and the estimated angle of attack are shown in Fig. 11. The noise level at the blade tip is the strongest. At the position, the angle of attack is -1 °, then the isolated NACA blade generated the broadband noise (see Fig. 8). The tip side mainly contributes to the increase of the broadband noise. According to Eq. (16), the increase in relative velocity at the tip is responsible for the increase in broadband noise.

<figure><img src="image_17.png" /><figcaption>Fig. 11 Distribution of the span direction on the sound pressure level and angle of attack estimated by the BEM.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 12 The noise spectra of the blade segments are compared at different span positions.</figcaption></figure>

In Fig. 12, the noise spectra of the blade segments are compared at different span positions. The broadband noise in the high frequency domain of the blade tip is the largest due to the influence of the relative velocity, whereas it can be seen that the humped noise in the high frequency domain is generated at 225 mm. The angle of attack of this span position was 2 °. This results indicates that the weak separated flow at the span position
generates the discrete frequency noise associated with the vortex shedding.

## 4. CONCLUSION

(1) From the results of the wind tunnel test, the NACA0008 airfoil generates discrete frequency noise in the vicinity of the angle of attack 3°, then the overall noise level at the angle of attack became maxima.

(2) The blade element momentum theory (BEM) predicts the output coefficient in accordance the measured value of the actual wind turbine. When the scaling law of the trailing edge noise model is applied, the predicted value based on the analytical flow by the BEM reproduces the measured broadband noise level generated from the actual wind turbine.

(3) It is clarified experimentally that the hump noise in the high frequency domain is the broadband noise generated from the von Karman Vortex shedding. The results of the BEM indicated that the hump noise was generated at the hub side of the impeller.

(4) The broadband noise in the high frequency domain of the wind turbine is related to the high relative velocity at the blade tip. On the other hand, the analysis of the BEM indicated that the broadband noise in the low frequency domain was induced by the separated flow until the full stall condition.

## APPENDIX

Both equations (2) and (4) in Ref. [6] can be compared in the limit of very high frequencies, for which the radiation integral involved in Amiet's solution reduces to the main trailing-edge scattering term (21).

$$I = -\frac{e^{2iC}}{iC} \times \left\{ (1+i)e^{-2iC} \sqrt{\frac{B}{B-C}} E^*[2(B-C)] - (1+i)E^*[2B] + 1 \right\}$$

with

$$B = \bar{K}_1 + \bar{\mu}(1 + M_0), \quad C = \bar{K}_1 - \bar{\mu} \left( \frac{x_1}{S_0} - M_0 \right)$$

The high-frequency trend for an observer in the mid-span plane follows by making $kc = 2 \beta^2 \bar{\mu} \to \infty$. According to Ref. [22],

$$E^*(x) = \int_0^x \frac{e^{-it}}{\sqrt{2\pi t}} dt = C_2(x) - iS_2(x)$$

with

$$S_2\left(\frac{\pi}{2}x^2\right) = \frac{1}{2} - f(x)\cos\left(\frac{\pi}{2}x^2\right) - g(x)\sin\left(\frac{\pi}{2}x^2\right)$$

$$C_2\left(\frac{\pi}{2}x^2\right) = \frac{1}{2} + f(x)\sin\left(\frac{\pi}{2}x^2\right) - g(x)\cos\left(\frac{\pi}{2}x^2\right)$$

and, for large values of x,

$$f(x) \approx \frac{1}{\pi x}, g(x) \approx \frac{1}{\pi^2 x^3}$$

Therefore

$$E^*(x) \approx \frac{1+i}{2} - \frac{e^{-i\xi}}{\sqrt{2\pi\xi}}$$

and only the constant term needs to be retained. The limit behavior of the squared radiation integral reads

$$|I|^2 \approx \frac{(1+M_0)(1-M_0-M_c)}{\bar{\mu}^2 M_c (1+x_1/S_0)} \left[ \frac{\beta^2}{M_c} + M_0 - \frac{x_1}{S_0} \right]$$

Let further transform the expressions using

$$\frac{x_1}{S_0} = \frac{\cos \theta_e + M_0}{1 - M \cos \theta_e}$$

in terms of the emission coordinates (Re, θe), that are defined as the true emission angle and propagating distance with respect to the moving fluid (23,24). These are related to the reception time coordinates by the transformation formulae

$$\cos \theta = \frac{R_e}{R} (\cos \theta_e + M_0), \quad \frac{R}{R_e} = \sqrt{1 + M_0^2 + 2M_0 \cos \theta_e}$$

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