A Consistency Test of Thickness and Loading Noise Codes Using Ffowcs Williams and Hawkings Equation

Abstract

The thickness noise predicted by the Ffowcs Williams and Hawkings (FW&H) equation depends on the normal velocity 𝑣_𝑛 which is very sensitive to the meshing size. Isom showed that in far field a monopolar source is equivalent to a dipolar source induced by a uniform distribution of the load on the entire moving surface. The main objective of this paper is to determine a specific expression of Isom's thickness noise in time and frequency domains for subsonic fans. The scope of the proposed expression of Isom's thickness noise is to define a benchmark test of consistency for thickness and loading noise codes in both time and frequency domains for subsonic fans when using the free field solution of FW&H's equation.

1. Introduction

In free field, Isom showed in [1] that the acoustic pressure generated by a mopole is equivalent to these generated by dipole induced by a uniform load equal to 𝜌₀𝑐² over a moving surface. Consequently, the calculation of the thickness noise becomes completely independent of the normal velocity 𝑣_𝑛 over the surface.

Farassat shows in [2, 3] that it is more suitable to determine the thickness noise using Isom's formulation than methods based on the resolution of Ffowcs Williams and Hawkings (FW&H) equation [4] using fluctuating velocity. The latter presents an important variation on the surface which makes the numerical integration very sensitive to the meshing resolution. On the other hand, Isom's thickness noise formula can present some difficulties for rotating machinery because of its sensitivity to the geometry of the blade tips [5].

Isom's thickness noise formula was used by Ghorbaniasl and Hirsch [6] as a consistency benchmark to show that it should not exist discrepancy between FW&H's thickness noise and Isom's noise when Farassat 1A formulation presented by Farassat and Succi in [7] is numerically solved in time domain. Khelladi et al. presented in [8] a specific formulas of thickness noise in time and frequency domains for subsonic fans based on initial Isom's formulation.

In this paper, we will present a validation of the formulas proposed in [8]. In Section 2 a brief remainder of the Isom's thickness noise formulas is presented. To illustrate the potential of the proposed approach, two examples are discussed in Section 3 to review a code using FW&H equation. In example 1, a two-bladed rotor will be studied. The effect of each part of the thickness noise as well as the effect of the tip Mach number on the overall noise will be shown. In example 2, thickness noise obtained by the FW&H equation for a high-rotational speed centrifugal fan [9] will be compared to that obtained by the proposed formulas. The method is used as a consistency test in time and frequency domains for the two examples. Finally, conclusions are drawn in Section 4.

2. Isom's Thickness Noise

Let 𝑓(𝐱,𝑡)=0 be a function taking into account the geometry and the kinematics of a moving body surface, defined such as 𝑓>0 outside the body, see Figure 1. Let 𝐻(𝑓) be the Heaviside function.

Figure 1 

Calculation domain.

According to [3], the wave equation corresponding to the trivial function 𝜌₀𝑐²[1−𝐻(𝑓)] is given by[1𝑐²𝜕²𝜕𝑡²−∇²]{𝜌₀𝑐²𝜕[1−𝐻(𝑓)]}=𝜕𝑡[𝜌₀𝑣_𝑛+𝜕𝛿(𝑓)]𝜕𝑥_𝑖[𝜌₀𝑐²𝑛_𝑖𝛿(𝑓)], (1) where 𝑛_𝑖=𝜕𝑓/𝜕𝑥_𝑖 and 𝑣_𝑛=𝐯⋅𝐧 are, respectively, the local unit normal vector and the local normal velocity on the body surface.

2.1. Time Domain Solution

The far and free field solution of (1) is given by𝜌₀𝑐²1[1−𝐻(𝑓)]=−𝜕4𝜋𝜕𝑡_𝑆[𝜌₀𝑣_𝑛]𝑟𝐷_ret+1𝑑𝑆𝜕4𝜋𝜕𝑥_𝑖𝑆[𝜌₀𝑐²𝑛_𝑖]𝑟𝐷_ret𝑑𝑆, (2) where 𝐷=|1−𝑀_𝑟| is the Doppler factor.

Outside the body 𝐻(𝑓)=1, then,𝜌₀𝑐²[1−𝐻(𝑓)]=0. (3) Equation (2) gives then,1𝜕4𝜋𝜕𝑡_𝑆[𝜌₀𝑣_𝑛]𝑟𝐷_ret1𝑑𝑆=𝜕4𝜋𝜕𝑥_𝑖𝑆[𝜌₀𝑐²𝑛_𝑖]𝑟𝐷_ret𝑑𝑆. (4) This equality shows that the monopole source solution is not unique. It is, also, equal to a dipole source induced by a steady and uniform load 𝜌₀𝑐² over the moving body surface.

Isom's thickness noise is then given by𝑝^′_Isom1(𝐱,𝑡)=𝜕4𝜋𝜕𝑥_𝑖𝑆[𝜌₀𝑐²𝑛_𝑖]𝑟𝐷_ret𝑑𝑆. (5) In far field on can show that𝑝^′_Isom1(𝐱,𝑡)=−4𝜋_𝑆[𝐫{1𝑐𝐷𝑟𝑟𝜕(𝐅𝜕𝜏𝐷𝐅)+𝑟³𝐷𝑐𝐫⋅𝐌}]_ret𝑑𝑆, (6) where 𝑡=𝜏+𝑟/𝑐₀ is the reception time and 𝐅=𝜌₀𝑐²𝐧 is the force induced by the uniform load 𝜌₀𝑐².

In far field, the term in 1/𝑟³ is neglected, (6) becomes after expansion𝑝^′_Isom1(𝐱,𝑡)=4𝜋_𝑆[𝐫⋅𝐅𝑐𝐷³𝑟²𝜕𝐷𝜕𝜏_I−𝐫⋅(𝜕𝐅/𝜕𝜏)𝑐𝐷²𝑟²_II]_ret𝑑𝑆. (7)𝐷 is given as function of 𝑟, by: 𝐷=|1+(1/𝑐)(𝜕𝑟/𝜕𝜏)|.

Part (I) of (7) expresses the unsteadiness of the source motion while part (II) expresses the effect of the rotating load (𝜌₀𝑐²) source.

Equation (7) can, then, be written as𝑝^′_Isom(𝐱,𝑡)=𝑝^′_I(𝐱,𝑡)+𝑝^′_II(𝐱,𝑡), (8) where𝑝^′_I1(𝐱,𝑡)=4𝜋_𝑆[𝐫⋅𝐅𝑐²𝑟²|1+(1/𝑐)(𝜕𝑟/𝜕𝜏)|³𝜕²𝑟𝜕𝜏²]_ret𝑝𝑑𝑆,_′II1(𝐱,𝑡)=−4𝜋_𝑆[𝐫⋅(𝜕𝐅/𝜕𝜏)𝑐𝑟²|1+(1/𝑐)(𝜕𝑟/𝜕𝜏)|²]_ret𝑑𝑆. (9)

Equations (9) are nothing more than the loading noise parts of Formulation 1A of Farassat. As it is written, because of the existence of partial derivatives in the denominators, the formulas are ambiguous to interpretation. This ambiguity is a mathematical subtlety discussed in the NASA publication of Farassat [10]. In our case, this form of writing will be very useful for the continuation of the development and the ambiguity will be circumvented. An exact expression of 𝑟 will be given below and partial derivatives will be fully defined.

2.2. Isom's Thickness Noise for Fans

Let us consider a fan turning at a velocity Ω. The angular position of a point on the blade is related to the moment of noise emission 𝜏 by Ψ=Ω𝜏+Ψ₀ where Ψ₀ is the initial position at 𝜏=0. Suppose that Ψ₀=0 at 𝜏=0. On Figure 2, 𝑆 is a noise source rotating around 𝑒₃ with an angular velocity Ω and a distance 𝑟_𝑠. F, a force applied by the fluid on the surface on 𝑆, defined by its radial, tangential, and axial components (𝐹_𝑟,𝐹_𝑡,𝐹_𝑎). 𝑂 is an observer defined by (𝑟₀,𝜑,𝜃) and r the distance between 𝑆 and 𝑂.

(a)

(b)

Figure 2 

Reference frame: (a) Axial fan (𝐹𝑟=0,𝐹𝑡,𝐹𝑎), (b) Centrifugal fan (𝐹𝑟,𝐹𝑡,𝐹𝑎=0).

In paper [8] Khelladi et al. demonstrated that the two parts of the thickness noise for fans can be given by𝑝^′_IΩ(𝐱,𝑡)=²sin(𝜃)4𝜋𝑟₀𝑐²×_𝑆𝑟_𝑠cos(Ω𝜏−𝜑){𝔄}|1+𝐴sin(Ω𝜏−𝜑)|³𝑑𝑆, (10) where 𝔄 denotes sin(𝜃)cos(Ω𝜏−𝜑)𝐹_𝑟+sin(𝜃)sin(Ω𝜏−𝜑)𝐹_𝑡+cos(𝜃)𝐹_𝑎.𝑝^′_II(𝐱,𝑡)=Ωsin(𝜃)4𝜋𝑟₀𝑐_𝑆sin(Ω𝜏−𝜑)𝐹_𝑟+cos(Ω𝜏+𝜑)𝐹_𝑡|1+𝐴sin(Ω𝜏−𝜑)|²𝑑𝑆, (11) where 𝐴=(𝑟_𝑠Ω/𝑐)sin(𝜃).

If 𝐧 is local unit normal vector on the blade surface in the moving reference frame, then, the force three components are given by 𝐹_𝑟=𝜌₀𝑐²(𝐧⋅𝐧_𝑟), 𝐹_𝑡=𝜌₀𝑐²(𝐧⋅𝐧_𝑡), and 𝐹_𝑎=𝜌₀𝑐²(𝐧⋅𝐧_𝑎), where 𝐧_𝑟, 𝐧_𝑡 and 𝐧_𝑎 are, respectively, the local unit vectors in radial, tangential, and axial directions.

The direction of F components depends on the direction of the source rotation and position.

It is noticed that the axial component 𝐹_𝑎 of the load 𝜌₀𝑐²𝑛_𝑖 does not exist in (11). So, for centrifugal fans, part (II) of thickness noise is influenced by both radial and tangential components of the load 𝜌₀𝑐²𝐧, whereas, for axial fans, only tangential component has influence on it.

Let us consider a 𝑍 blades fan, the overall acoustic pressure generated by the fan is given by𝑃^′_Isom(𝐱,𝑡)=^𝑍_𝑛=1𝑝^′_Isom𝛼(𝐱,𝑡+^𝑛Ω), (12) where 𝑝^′ is the acoustic pressure of one blade and 𝛼_𝑛 is the angle between two successive blades.

2.3. Frequency Domain Solution

In frequency domain it was demonstrated in [8] that the two components of the thickness for a Z-blades fan are given by𝑃^′(𝑚)_I=(𝐱)𝑍Ω²sin(𝜃)8𝜋²𝑐²𝑟₀𝑒^{𝑖𝑚𝑍(Ω(𝑟0/𝑐)+𝜑)}×_𝑆^2𝜋−𝜑_−𝜑𝑟_𝑠cos(𝜉){𝔛}(1+𝐴sin(𝜉))²𝑒^{𝑖𝑚𝑍(𝜉−𝐴cos(𝜉))}𝑃𝑑𝜉𝑑𝑆,_′(𝑚)II=(𝐱)𝑍Ωsin(𝜃)8𝜋²𝑐𝑟₀𝑒^{𝑖𝑚𝑍(Ω(𝑟0/𝑐)+𝜑)}×_𝑆^2𝜋−𝜑_−𝜑sin(𝜉)𝐹_𝑟+cos(𝜉+2𝜑)𝐹_𝑡𝑒|1+𝐴sin(𝜉)|^{𝑖𝑚𝑍(𝜉−𝐴cos(𝜉))}𝑑𝜉𝑑𝑆, (13)(1)(2)(3)(4)(5) where 𝔛 denotes sin(𝜃)cos(𝜉)𝐹_𝑟+sin(𝜃)sin(𝜉)𝐹_𝑡+cos(𝜃)𝐹_𝑎 and 𝜉=Ω𝜏−𝜑 and 𝐴=(𝑟_𝑠Ω/𝑐)sin(𝜃).

Contrary to (10) and (11), (13) needs a surface integral over only one blade, which constitute a consequent time and numerical storage reduction.

Hawkings and Lowson showed in [11] that for high-Mach numbers and high-sound levels, the effects of nonlinear acoustic propagation must be taken into account. They stated that nonlinear propagation causes some noticeable changes in the observed acoustic field, especially in its spectral characteristics. Consequently, the formulas described above in frequency domain will be more appropriate for low-Mach numbers and linear acoustic propagation cases.

3. Examples and Discussion

3.1. Example  1: A Two-Bladed Rotor

A NACA 0010 two-bladed rotor at several tip Mach numbers in time domain is studied in this example. The blades are 5 m in outer radius and 1 m in inner radius with a constant chord of 0.4 m and 10% of thickness ratio along the entire span of the blade. The observer is in the rotor plane and 50 m from the center. The surface expressed by 𝑓(𝐱,𝑡)=0 is assumed to be the upper, the lower, and the tip surfaces of the blade.

Figure 3 shows the acoustic pressure given by (12) for 0.2, 0.5, and 0.9 tip Mach numbers. To review the effect of each part of the acoustic pressure, expressions given by (10) and (11) in time domain are also included in this figure. Recall that 𝑝^′_I given by (10) expresses the effect of the kinematics and the geometry of the source and 𝑝^′_II given by (11) expresses the effect of the moving load 𝜌₀𝑐².

(a)

(b)

(c)

Figure 3 

Acoustic pressure signal: (a) Tip Mach number 0.2, (b) Tip Mach number 0.5, (c) Tip Mach number 0.9.

For low-tip Mach number (Figure 3(a)), the amplitude of 𝑝^′_I is very small compared to 𝑝^′_II and the overall acoustic pressure 𝑝^′_Isom results mainly from the effect of the moving load 𝜌₀𝑐². When the tip Mach number increases (Figures 3(b) and 3(c)), the effect of the kinematics and the geometry of the source become more important on the overall acoustic pressure 𝑝^′_Isom. If we compare now the acoustic pressure obtained by Isom and by FW&H, we observe a good agreement between the two formulation. That means that the numerical implementation is correct and the meshing of tip and root surfaces of the blade was correctly performed. For more details, an assessment of the causes of errors on the time-based formulation is discussed in [12] in which the same example was treated but using a decreasing thickness ratio at the extremities of the blade. The influence of the tip surfaces grid refinement can be found in [6].

3.2. Example  2: A High Rotational Velocity Centrifugal Fan

Centrifugal fans, made up of an impeller, a diffuser and a return channel (Figure 4), are widely used in vacuum cleaners. These fans turn at a relatively high rotational speeds of about 35000 rpm. Their low efficiency of about 30% and their high acoustic levels are mainly due to the flow disorganization in the impeller-diffuser interface, the junction diffuser-return channel, and the exit of the return channel [9, 13]. The unsteady flow is strong and can reache 250 m/s. The aerodynamic and geometrical characteristics at the operating point are given, respectively, in Tables 1 and 2.

Table 1 

Aerodynamic characteristics at operating point.

Table 2 

Geometrical characteristics of the centrifugal fan.

Figure 4 

Centrifugal fan: Impeller, diffuser, and return channel.

In [9], Khelladi et al. showed that the thickness noise of this centrifugal fan calculated by the Ffowcs Willams and Hawkings equation is not negligible compared to the dipole source. It was also shown that the corresponding acoustic pressure in frequency domain is given by𝑝^′(𝑚)_thickness(𝑥)=𝑖𝑚𝑍²Ω𝜌₀4𝜋𝑟₀𝑒^{𝑖𝑚𝑍Ω(𝑟0/𝑐)}×_𝑆^+∞_𝑘=−∞𝑒^{𝑖(𝑚𝑍−𝑘)(𝜑−𝜋/2)}𝐽_{𝑚𝑍−𝑘}(𝐴)𝑣_𝑛^(𝑘)𝑑𝑆, (14) where

𝐴=𝑚𝑍Ω(𝑟_𝑠/𝑐₀)sin(𝜃) and 𝑀_𝑟𝑠=𝑟_𝑠Ω/𝑐₀  

𝐽_{𝑚𝑍−𝑘}(𝐴): first kind Bessel function.

For both FW&H and Isom thickness noise formulations, the observer is placed in the rotor plane and 1 m from the center. The surface expressed by 𝑓(𝐱,𝑡)=0 is assumed to be the pressure and the suction sides of the impeller blade.

Figure 5 presents the spectra of the thickness noise given by both FW&H and Isom equations as a function of harmonics of the Blade Passing Frequency (BPF).

Figure 5 

SPL versus blade passing frequency harmonics (𝑀=0.56) at 1 m from the source.

It was observed that the calculation of the Isom's thickness noise is very sensitive to the numerical integration model mainly for frequencies corresponding to high harmonics. For the three first harmonics the variation of the SPL between FW&H and Isom is about 2% for harmonic 1 and about 14% for harmonics 2 and 3. This large variation is probably caused by the absence of the effect of the other parts of the fan in Isom's modeling. For this case, only harmonic 1 calculated by the FW&H equation can be validated by Isom. The overall acoustic level (OASPL) is dominated by the first harmonic: 81.20 dB given by Isom and 82.81 dB given by FW&H (a variation of about 2%). The proposed Isom's formulas give then a good estimation of the OASPL.

4. Conclusion

In this paper, specific formulas of thickness noise in time and frequency domains for subsonic fans based on the initial Isom's formulation were defined and used as a benchmark test of consistency when using the free-field solution of the FW&H's equation. The proposed formulas highlight also the effect of each geometrical parameter of the fan on the overall thickness noise and present a fast computational mean and low memory storage capability since the acoustic pressure in frequency domain is calculated for only one blade. The overall acoustic pressure was decomposed into two parts; the first one highlighted the effect of the kinematics and the geometry of blades, and the second one highlighted the effect of the moving load over blades expressed by 𝜌₀𝑐².

To illustrate the use of the proposed formulas, two examples were treated. The first one concerns a two-bladed rotor at several tip Mach numbers. Thickness noise was calculated using Isom's and FW&H's formulations. Concerning Isom's formulation, the results showed a different effect of the two parts of the acoustic pressure, depending on whether we used low or high Mach number. A good agreement between Isom's and FW&H's formulations was observed. That proves that the numerical implementation was correctly set up, and the meshing of tip and root surfaces of the blade was correctly performed.

The second example concerns a high rotational speed centrifugal fan. For this case, the thickness noise was calculated by the FW&H equation and compared to the proposed formulas at frequency domain. The result showed that only harmonic 1 calculated by the FW&H equation was correctly estimated by Isom. It was observed also as a good estimation of the overall SPL (only 2% of variation). For this case, one can conclude that the proposed consistency test is at least applicable to validate the first harmonic SPL and the overall SPL.

Nomenclature

𝑐:	Velocity of sound in a medium at rest
𝐷:	Doppler amplification factor due to the moving source
𝑓:	A function taking into account the geometry and the kinematics of the moving surface when 𝑓(𝐱,𝑡)=0
𝐅:	Force induced by the constant load 𝜌0𝑐2 on the moving surface
𝐻:	Heaviside function
𝑚:	Rank of harmonic
𝐌:	Mach number associated to the absolute velocity of the source
𝑀𝑟:	Mach number of the sources in the direction of listening point given by 𝑀𝑟=(𝐫/𝑟)⋅𝐌
𝐧:	A unit vector normal to the moving surface
𝑝′:	Acoustic pressure
𝑃′:	Overall acoustic pressure
𝐫:	Distance between the observer and the source given by 𝐫=𝐱−𝐲
𝑠:	Rank of harmonic
𝑡:	Reception time
𝑣𝑛:	Velocity normal to the moving surface
𝑥:	Position of the observer
𝑦:	Position of the source
𝛼𝑛:	Angle between two successive fan blades
𝛿:	Dirac function
Ω:	Rotational velocity in rad/s
𝜙:	Observer angular position
𝜌0:	Density of the propagation medium
𝜏:	Emission time
𝜃:	Observer angular position.