16.2 Time domain prediction methods for tone noise

16.2.1 Loading noise

Loading noise is caused by the time variation of the compressive stress tensor p_ij on a blade surface as it rotates and may be predicted by the second term in the Ffowcs-Williams and Hawkings equation (5.2.13), which gives the radiated acoustic pressure as

${\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{loading}}=-\frac{\partial }{\partial x_i}{\displaystyle \underset{S_o}{\int }{\left[\frac{p_{ij}{n}_j}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}dS\left(\mathbf{z}\right)$

The surface integral in Eq. (16.2.1) should be carried out over the complete surface of the rotor blade, but in many instances it can be simplified to an integral over the blade planform. The planform is the projection of the rotor geometry into the rotor disc plane, as shown in Fig. 16.10. This approximation is valid if the blades are thin and the acoustic wavelength at the maximum frequency of interest is much larger than the blade thickness. We can then ignore the displacement of the upper surface from the lower surface and define the force per unit area applied to the fluid by the difference between the values of p_ijn_j evaluated on the upper and lower surfaces for a given point on the blade planform (see Fig. 16.10). We define

$f_{i}d\Sigma =\left({\left[p_{ij}{n}_j\right]}_{\mathrm{upper}}-{\left[p_{ij}{n}_j\right]}_{\mathrm{lower}}\right)dS\mathrm{and}\left|n_1\right|dS=d\Sigma$

where dΣ is the planform area in the disc plane (see Fig. 16.10) of the surface element dS. Substituting into Eq. (16.2.1) we obtain the thin airfoil approximation

${\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{loading}}=-\frac{\partial }{\partial x_i}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{f_i\left(\mathbf{z},\tau \right)}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)$

For blades with lean and sweep there may also be a considerable displacement of the mean camber line of the blade from the rotor disc plane. This displacement should be included in the retarded time calculation that takes place inside the integral in Eq. (16.2.2).

If the variation of the surface loading f_i repeats itself during each blade rotation, then tone noise results. To calculate the acoustic field, we need to know the precise variation of f_i over the complete blade surface, as a function of emission time τ. However, there are some inherent difficulties with this requirement because the surface integral must be evaluated at a fixed observer time, and so the emission time τ will vary over the surface of the blade.

To address these difficulties, we use the approach pioneered by Farassat [1] and shift the spatial derivatives to source time, as explained in Section 5.2. The result was given by Eq. (5.5.2) for a far-field observer. If we project the blade loading onto the surface planform as described above, we obtain

${\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{loading}}\approx \frac{1}{c_{\infty }}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{x_i}{4\pi \left|\mathbf{x}\right|{}^2{\left(1-M_r\right)}^2}\left\{\frac{\partial f_i}{\partial \tau }+\frac{f_i}{\left(1-M_r\right)}\frac{\partial M_r}{\partial \tau }\right\}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)$

To evaluate Eq. (16.2.3) we need to calculate the location of each surface element defined in the integrand at a given observer time.

To proceed we need to define the coordinate system of the blade and the observer. There are several different choices for the coordinates, and care must be used in specifying the convention that will be used. In aircraft and marine propeller applications the convention [2] is to define coordinates (x,y,z) with x pointing in the direction of thrust. In helicopter applications the z coordinate is chosen in the direction of thrust. In actuator disc theory the x coordinate is chosen in the direction of flow through the propeller. In this chapter we choose the aircraft propeller convention with x or x₁ in the direction of thrust as shown in Fig. 16.11. Furthermore, we need to choose the direction of blade rotation. A propeller rotating anticlockwise when viewed from upstream is defined as having “right-hand rotation.” If it rotates in a clockwise direction it is said to have “left-hand rotation.” We will choose right-hand rotation as shown in Fig. 16.11.

To evaluate the integrand in Eq. (16.2.3) we solve the equation t−τ−r(τ)/c_∞=0, so we can specify the azimuthal location ϕ=ϕ₁+Ωτ of each surface element as a function of observer time t, radius R, and azimuthal location ϕ₁ in blade-based coordinates. For example, consider a rotor which is in the y₂y₃ plane at τ=0 and is moving with linear velocity (U_o,0,0) as shown in Fig. 16.11. In blade-based coordinates the location of each surface element is z=(0,R cos ϕ₁, R sin ϕ₁), and in fixed coordinates the surface element is located at y=(U_oτ, R cos(ϕ₁+Ωτ), R sin(ϕ₁+Ωτ)). The retarded time equation is given by t−τ−|x−y(τ)|/c_∞=0, and to solve this equation we use an interpolation method. First we evaluate t using a uniformly spaced set of points τ_m=mΔτ in source time and then interpolate the results to obtain values of τ at fixed intervals in observer time t_j=jΔt. Fig. 16.12 shows a plot of source location vs observer time for a stationary propeller rotating with a tip Mach number of 0.8 for source located at 30%, 60%, and 90% of the tip radius R_tip and an observer in the acoustic far field at x=(40R_tip, 30R_tip, 0). Notice how the curves cross at different observer times because the source at the outer radius is sometimes closer, and sometimes further from the observer as the blade rotates.

For blades with a subsonic tip speeds it is a relatively simple task to use this curve to compute the location of each blade element for uniformly spaced steps in observer time. However, for supersonic propellers this process becomes more difficult because the curve is no longer monotonically increasing, and we will return to this issue in Section 16.2.3.

The noise from steady loading is readily computed by defining the blade thrust F_L and the blade drag F_D for blade element of span ΔR located at R. (Note the sign convention requires that f_i is the force per unit area applied to the fluid and is equal and opposite to the force per unit area applied to the blade.) For steady loading f₁dΣ=−F_L is constant, but the direction of the drag force varies with blade location, so f₂dΣ=F_Dsin(ϕ₁+Ωτ) and f₃dΣ=−F_Dcos(ϕ₁+Ωτ). Typically the drag force is about 10% of the thrust. The acoustic far field for each surface element can be computed using Eq. (16.2.3).

To illustrate a typical calculation, consider an observer at x=(40R_tip, 30R_tip, 0) for surface elements at 30%, 60%, and 90% of the blade radius. To evaluate Eq. (16.2.3) we need to define both the blade loading and the relative Mach number M_r, which is given, for x₃=0 and U_o=0, by

$M_r=-\frac{x_2\Omega Rsin\left({\varphi }_1+\Omega \tau \right)}{\left|\mathbf{x}\right|c_{\infty }}$

We then obtain a nondimensional acoustic signal defined as

$\begin{array}{c}{C}_p\left(t\right)=\frac{4\pi \left|\mathbf{x}\right|c_{\infty }{\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{loading}}}{\Omega F_L}\approx -{\left[\frac{x_1}{\left|\mathbf{x}\right|{\left(1-M_r\right)}^3}\frac{\partial M_r}{\partial \left(\Omega \tau \right)}\right]}_{\tau =\tau ⁎}\\ +{\left[\frac{x_2\left(D/L\right)}{\left|\mathbf{x}\right|{\left(1-M_r\right)}^2}\left\{cos\left({\varphi }_1+\Omega \tau \right)+\frac{sin\left({\varphi }_1+\Omega \tau \right)}{\left(1-M_r\right)}\frac{\partial M_r}{\partial \left(\Omega \tau \right)}\right\}\right]}_{\tau =\tau ⁎}\end{array}$

with

$\begin{array}{c}\frac{\partial M_r}{\partial \left(\Omega \tau \right)}=-\left\{\frac{x_2\Omega R}{\left|\mathbf{x}\right|c_{\infty }}cos\left({\varphi }_1+\Omega \tau \right)\right\}\hfill \end{array}$

There are several important features to this result. First notice how the loading closest to the tip generates the most significant part of the signature because the rate of change of M_r is largest at this radius. Second note that if drag to lift ratio is small then the lift force dominates the calculation. In this case F_D/F_L=0.1 and the peak of the pressure pulse is only marginally increased by including the drag term (Fig. 16.13).

The magnitude of the signature in this example is small because the thrust force is constant during the rotation of the blade. However, if it has a harmonic time dependence, so f₁dΣ=−F_Lf(τ), where f(τ)=cos(mΩτ), then Eq. (16.2.5) becomes

$C_p\left(t\right)\approx -{\left[\frac{x_1}{\left|\mathbf{x}\right|{\left(1-M_r\right)}^2}\left\{\frac{\partial f\left(\tau \right)}{\partial \left(\Omega \tau \right)}+\frac{f\left(\tau \right)}{\left(1-M_r\right)}\frac{\partial M_r}{\partial \left(\Omega \tau \right)}\right\}\right]}_{\tau =\tau ⁎}$

For large values of m the rate of change of f(τ) is much greater than the rate of change of the relative Mach number M_r, and so the radiated levels increase dramatically as shown in Fig. 16.14 for the case when m=10.

The peak signal level in Fig. 16.14 is approximately 30 times larger than in Fig. 16.13, showing the importance of the unsteady loading in these calculations. Also note how the frequency of the signature varies during the blade rotation. During the first part of the cycle the blade is moving away from the observer, and the frequency is lower than the source frequency because of a Doppler frequency shift. In the second part of the cycle the blade is moving toward the observer, and the Doppler shift increases the observed frequency so that it is higher than the source frequency. This effect is dependent on the observer position and is more dramatic in the plane of the rotor than along the rotor axis. The importance of unsteady loading is even more significant when the blade encounters a sudden change in angle of attack caused by an inflow distortion, an encounter with a wake, or a vortex in a BVI. To illustrate this, consider a fluctuation in thrust on the blade segment of span ΔR, located at R, given by

$f_{1}d\Sigma =F_L\left(\left(\tau -{\tau }_o\right)/T_v\right)exp\left(-{\left(\tau -{\tau }_o\right)}^2/{T_v}^2\right)$

as illustrated in Fig. 16.15. The choice of T_v=0.02T_p gives a pulse which lasts for approximately one-tenth of the period of rotation in source time. The observed signature from this pulse at the observer location is given in Fig. 16.16 and is much shorter than the source signature as shown in Fig. 16.15. The source is at 90% of the blade span, and so the Doppler frequency shift can be large, and the observed signature becomes very impulsive, with a peak level that is much greater than for a harmonic variation in loading as shown in Fig. 16.14.

In the examples given above the signature from individual blade elements has been shown. To obtain a complete prediction of rotor noise the surface integral must be evaluated, and this can be achieved by numerical integration or summing the calculations for each blade element across the span at the correct retarded time. This is a relatively straightforward extension of the procedures described above providing that the blade surface loading is known as both a function of blade radius and azimuth. In low-frequency applications it is often reasonable to ignore the distribution of loading in the chordwise direction and replace f_i with the sectional blade lift and drag per unit span for each blade radius. This assumes that the blade chord is small compared to the acoustic wavelength and is not always a good assumption, but it simplifies the calculations considerably and is a valid approach to obtain first estimates of blade noise signatures at low frequencies.

16.2.2 Thickness noise

Thickness noise is described by the third term in the Ffowcs-Williams and Hawkings equation (5.2.13) which gives the acoustic field as

${\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{thickness}}=\frac{\partial }{\partial t}{\displaystyle \underset{S_o}{\int }{\left[\frac{{\rho }_{\infty }{V}_j{n}_j}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}dS\left(\mathbf{z}\right)$

The evaluation of this source term for a propeller can be achieved using the method that was described in Section 16.2.1. The same procedure for evaluating the retarded time solution can be employed with the same limitations on numerical accuracy. The main difference is the appropriate evaluation of the source strength on the blade surface, which is easier in this case because it is defined exactly by the blade geometry.

As before we can replace the surface integral for thin blades by an integral over the blade planform in the rotor disk plane. We denote the upper and lower surface coordinates, displaced axially (in the y₁ direction from the rotor disk plane) as y₊(R,ϕ) and y₋(R,ϕ). As shown in Fig. 16.10 we define the blade volume in terms of the upper surface function f₊(R,ϕ)=y₁−y₊(R,ϕ)=0 and the lower surface function f₋(R,ϕ)=y₋(R,ϕ)−y₁=0. The blade thickness is then defined by h=y₊−y₋. The normal to the blade surface is then given by n=∇f₊/|∇f₊| on the upper surface and n=∇f₋/|∇f₋| on the lower surface. It follows that on the upper surface

$V_j{n}_j=\frac{1}{\left|\nabla f_{+}\right|}\left(V_1-V_2\frac{\partial y_{+}}{\partial y_2}-V_3\frac{\partial y_{+}}{\partial y_3}\right)$

Similarly, on the lower surface

$V_j{n}_j=\frac{1}{\left|\nabla f_{-}\right|}\left(-V_1+V_2\frac{\partial y_{-}}{\partial y_2}+V_3\frac{\partial y_{-}}{\partial y_3}\right)$

Since dS=dΣ/|n₁|, $\nabla h=\nabla y_{+}-\nabla y_{-}$, and |n₁|=1/|∇f₊| or 1/|∇f₋|, we find that the contributions from the upper and lower surfaces can be combined to give

$V_j{n}_{j}dS=-\left(V_2\frac{\partial h}{\partial y_2}+V_3\frac{\partial h}{\partial y_3}\right)d\Sigma =-\mathbf{V}\cdot \nabla hd\Sigma$

Thickness noise can then be defined as an integral over the blade planform as

${\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{thickness}}=-\frac{\partial }{\partial t}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{{\rho }_{\infty }\mathbf{V}\cdot \nabla h}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)$

This shows how the blade thickness contributes to the source strength, and the result is independent of the blade angle of attack or camber. Consequently, this source is defined as “thickness” noise.

Evaluation of the source term in this equation at the correct retarded time can be challenging numerically, but an elegant alternative [1] simplifies this considerably. To derive this alternate form, we return to the Ffowcs-Williams and Hawkings equation (5.2.4) and only retain the thickness noise terms on the right-hand side. For an impermeable surface for which V_jn_j=v_jn_j we obtain

${\left[\frac{1}{c_{\infty }^2}\frac{{\partial }^2\left(H_s{p}^{\prime }\right)}{\partial t^2}-\frac{{\partial }^2\left(H_s{p}^{\prime }\right)}{\partial x_i^2}\right]}_{\mathrm{thickness}}=\frac{\partial }{\partial t}\left[{\rho }_{\infty }{V}_j{n}_j\delta \left(f\right)|\nabla f|\right]$

However, for a body that does not deform as it moves we have, using Eqs. (5.1.12), (5.1.3),

$\frac{\partial }{\partial t}\left(1-H_s\left(f\right)\right)=-\frac{\partial f}{\partial t}\delta \left(f\right)=V_j{n}_j\delta \left(f\right)|\nabla f|$

so the wave equation for thickness noise sources can be given in an alternate form as

${\left[\frac{1}{c_{\infty }^2}\frac{{\partial }^2\left(H_s{p}^{\prime }\right)}{\partial t^2}-\frac{{\partial }^2\left(H_s{p}^{\prime }\right)}{\partial x_i^2}\right]}_{\mathrm{thickness}}=\frac{{\partial }^2}{\partial t^2}\left[{\rho }_{\infty }\left(1-H_s\left(f\right)\right)\right]$

The term (1−H_s(f)) represents the volume inside the moving surface, and so thickness noise can be considered as being completely equivalent to the sound from a displaced mass of fluid moving through a stationary medium, and this is referred to as Isom's result [1]. The solution to Eq. (16.2.15) is obtained following the procedures given in Section 5.2, with the double derivative with respect to time requiring integration by parts twice, followed by the integration (5.2.12), giving

${\left[p^{\prime }\left(\mathbf{x},t\right)\right]}_{\mathrm{thickness}}=\frac{{\partial }^2}{\partial t^2}{\displaystyle \underset{V_{\infty }}{\int }{\left[\frac{{\rho }_{\infty }\left(1-H_s\left(f\right)\right)}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}dV\left(\mathbf{z}\right)$

This result is relatively straightforward to evaluate numerically by breaking the volume of the blade down into acoustically compact volume elements of volume ΔV_k, centered on y^(k), and using $\partial /\partial t=\left(1-M_r\right)\partial /\partial \tau$, so

${\left[p^{\prime }\left(\mathbf{x},t\right)\right]}_{\mathrm{thickness}}=\sum _k{\rho }_{\infty }\mathrm{\Delta }{V}_k{\left[\frac{1}{\left(1-M_r\right)}\frac{\partial }{\partial \tau }\left(\frac{1}{\left(1-M_r\right)}\frac{\partial }{\partial \tau }\left(\frac{1}{4\pi r|1-M_r|}\right)\right)\right]}_{\begin{array}{l}\mathbf{y}={\mathbf{y}}^{\left(k\right)}\\ \tau =\tau ⁎\end{array}}$

This result shows that thickness noise can be computed directly from volume elements within the blade, with the correct allowance for the propagation distance from the source to the observer. The signature from different parts of the blade planform will arrive at the observer at the same time, so the calculation is nontrivial, but it is tractable using numerical methods.

For an observer in the far field terms of order r⁻² can be dropped, and this result simplifies to

${\left[p^{\prime }\left(\mathbf{x},t\right)\right]}_{\mathrm{thickness}}=\sum _k{\rho }_{\infty }\mathrm{\Delta }{V}_k{\left[\frac{1}{4\pi r}\left(\frac{1}{{\left(1-M_r\right)}^4}\frac{{\partial }^2{M}_r}{\partial {\tau }^2}+\frac{3}{{\left(1-M_r\right)}^5}{\left(\frac{\partial M_r}{\partial \tau }\right)}^2\right)\right]}_{\begin{array}{l}\mathbf{y}={\mathbf{y}}^{\left(k\right)}\\ \tau =\tau ⁎\end{array}}$

We can normalize this result to obtain a nondimensional thickness noise pulse for each volume element as

$C_q\left(t\right)=\frac{4\pi \left|\mathbf{x}\right|c_{{}_{\infty }}{\left(p^{\prime }\left(\mathbf{x},t\right)\right)}_{\mathrm{thickness}}}{\Omega \left({\Omega }^{2}R{\rho }_{\infty }\mathrm{\Delta }{V}_k\right)}={\left[\frac{c_{\infty }}{\Omega R}\left(\frac{1}{{\left(1-M_r\right)}^4}\frac{{\partial }^2{M}_r}{\partial {\left(\Omega \tau \right)}^2}+\frac{3}{{\left(1-M_r\right)}^5}{\left(\frac{\partial M_r}{\partial \left(\Omega \tau \right)}\right)}^2\right)\right]}_{\begin{array}{l}\mathbf{y}={\mathbf{y}}^{\left(k\right)}\\ \tau =\tau ⁎\end{array}}$

Using this normalization, we see that the thickness noise is scaled on an equivalent force equal to Ω²Rρ_oΔV_k, and so will depend on the blade volume. As an example the normalized signature for an observer at (40R_tip, 30R_tip, 0) and a source at 90% of the blade span, with tip Mach number M_tip=0.8, is shown in Fig. 16.17. The thickness noise gives a clearly defined pulse as the blade moves toward the observer, and its level exceeds that of a steady loading signature for high tip Mach numbers for the same equivalent blade loading. It is important to appreciate, however, that this calculation does not include all the effects of blade shape and retarded time across the blade chord, and more detailed calculations are required to define the exact thickness noise pulse.

16.2.3 Supersonic tip speeds

The calculations that have been described above are for blades with subsonic tip speed. When the blade tip Mach number exceeds one then a number of effects occur. These include the presence of shock surfaces in the flow and a singularity in the integrals (16.2.1), (16.2.2), and (16.2.14) when M_r=1. Another important issue for supersonic rotors is that there is no longer a one-to-one relationship between the emission time and the observer time. Fig. 16.18 shows the calculation of blade position for a given observer time for a supersonic rotor when the observer is in the plane of the rotor. Close to the blade tip it is seen that there are in fact three solutions for ϕ for an observer time of t/T_p=0.75. Consequently, the source strength from many parts of the rotor disc is concentrated at this instant of observer time, and a large impulsive peak in the sound signature is expected. Numerical calculations in and around this instant are clearly complex, and the reader is referred to the papers by Farassat [1] on the details of how to address this problem. Providing the correct asymptotic numerical coding is used, the time domain methods described above can be extended to supersonic rotors. However, as will be shown in the next section, many of these numerical difficulties with time domain methods are overcome if frequency domain methods are employed, at the expense, unfortunately, of other numerical issues.

16.3.1 Harmonic analysis of loading and thickness noise

In Section 16.2 we showed that the loading and thickness noise from a rotor could be defined as an integral over the blade planform, which was given by the combination of Eqs. (16.2.2), (16.2.14) as

$p^{\prime }\left(\mathbf{x},t\right)=-\frac{\partial }{\partial x_i}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{f_i}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)-\frac{\partial }{\partial t}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{{\rho }_{\infty }\mathbf{V}.\nabla h}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)$

For tone noise the resulting acoustic signature will be periodic, and the contribution from each blade will be repeated after each revolution. In this section we make use of this characteristic to evaluate the integrals in Eq. (16.3.1). We limit consideration to stationary propellers, but note that the theory can be readily extended to propellers and rotors in flight.

The signature from each blade is identical, and so if p₁(x,t) is the signature from one blade in isolation, the signature from a rotor with B blades is

$p^{\prime }\left(\mathbf{x},t\right)={\displaystyle \sum _{n=1}^B{p}_1\left(\mathbf{x},t+n{T}_p/B\right)}$

where T_p is the period of one rotation. Because the acoustic signature is periodic we can expand the signature from a single blade as a Fourier series of the type

$p_1\left(\mathbf{x},t\right)={\displaystyle \sum _{j=-\infty }^{\infty }{c}_j{e}^{-2\mathit{\pi ijt}/T_p}}={\displaystyle \sum _{j=-\infty }^{\infty }{c}_j{e}^{-\mathit{ij\Omega t}}}$

where Ω=2π/T_p is the rotational frequency. Combining Eqs. (16.3.2), (16.3.3) then gives

$p^{\prime }\left(\mathbf{x},t\right)={\displaystyle \sum _{j=-\infty }^{\infty }{c}_j}{\displaystyle \sum _{n=1}^B{e}^{-ij\left(\Omega t-2\pi n/B\right)}}=B{\displaystyle \sum _{m=-\infty }^{\infty }{c}_{mB}{e}^{-\mathit{imB\Omega t}}}$

where we have made use of the fact that the sum over n is only nonzero when j=mB. The important feature of this result is that we can define the time history from a rotor with B blades from the Fourier coefficients of the time history from a single blade, but only the coefficients of order mB will contribute.

To determine the Fourier coefficients required in Eq. (16.3.3) we evaluate the integral

$c_n\left(\mathbf{x}\right)=\frac{\Omega }{2\pi }{\displaystyle \underset{0}{\overset{T_p}{\int }}{p}_1\left(\mathbf{x},t\right)e^{\mathit{in\Omega t}}}dt$

Combining this with Eq. (16.3.1) then gives

$c_n\left(\mathbf{x}\right)=-\frac{\Omega }{2\pi }{\displaystyle \underset{0}{\overset{T_p}{\int }}\left\{\frac{\partial }{\partial x_i}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{f_i}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)\right\}{e}^{\mathit{in\Omega t}}}dt-\frac{\Omega }{2\pi }{\displaystyle \underset{0}{\overset{T_p}{\int }}\left\{\frac{\partial }{\partial t}{\displaystyle \underset{{\Sigma }_o}{\int }{\left[\frac{{\rho }_{\infty }\mathbf{V}\cdot \nabla h}{4\pi r|1-M_r|}\right]}_{\tau =\tau ⁎}}d\Sigma \left(\mathbf{z}\right)\right\}{e}^{\mathit{in\Omega t}}}dt$

In the far field the differentiation of the first integral can be changed to a differential over time (see Section 5.2) using $\partial /\partial x_i~\left(-x_i/r_o{c}_o\right)\partial /\partial t$, where r_o=|x| so that by using the properties of Fourier series we obtain

$\begin{array}{c}{c}_n\left(\mathbf{x}\right)=\frac{\mathit{in}{\Omega }^2}{2\pi }\left({\displaystyle \underset{{\Sigma }_o}{\int }{\displaystyle \underset{0}{\overset{T_p}{\int }}{\left[\frac{-x_i{f}_i}{4\pi r_o^2{c}_o|1-M_r|}\right]}_{\tau =\tau ⁎}{e}^{\mathit{in\Omega t}}}}\mathit{dtd\Sigma }\left(\mathbf{z}\right)\right.\\ \left.+{\displaystyle \underset{{\Sigma }_o}{\int }{\displaystyle \underset{0}{\overset{T_p}{\int }}{\left[\frac{{\rho }_{\infty }\mathbf{V}.\nabla h}{4\pi r_o|1-M_r|}\right]}_{\tau =\tau ⁎}{e}^{\mathit{in\Omega t}}}}\mathit{dtd\Sigma }\left(\mathbf{z}\right)\right)\end{array}$

The integral over observer time can be changed to an integral over emission time because t=τ+r(τ)/c_o, and so the time differentials are related by dt=|1−M_r|dτ. It follows that since the source terms are also periodic with the same time scale, Eq. (16.3.7) reduces to

$\begin{array}{c}{c}_n\left(\mathbf{x}\right)=\frac{\mathit{in}{\Omega }^2}{2\pi }\left({\displaystyle \underset{{\Sigma }_o}{\int }{\displaystyle \underset{0}{\overset{T_p}{\int }}\frac{-x_i{f}_i\left(\mathbf{z},\tau \right)}{4\pi r_o^2{c}_o}{e}^{in\Omega \left(\tau +r\left(\tau \right)/c_o\right)}}}\mathit{d\tau d\Sigma }\left(\mathbf{z}\right)\right.\\ \left.+{\displaystyle \underset{{\Sigma }_o}{\int }{\displaystyle \underset{0}{\overset{T_p}{\int }}\frac{{\rho }_{\infty }\mathbf{V}\left(\mathbf{z},\tau \right).\nabla h\left(\mathbf{z}\right)}{4\pi r_o}{e}^{in\Omega \left(\tau +r\left(\tau \right)/c_o\right)}}}\mathit{d\tau d\Sigma }\left(\mathbf{z}\right)\right)\end{array}$

One of the most significant aspects of this transformation is that it eliminates the singularity that occurs when M_r=1, so the integrals are harmonic and well behaved for all blade speeds.

For an observer in the acoustic far field the integral over time in Eq. (16.3.8) can be evaluated directly. Consider the blade element at radius R and azimuthal location ϕ₁ in blade-fixed coordinates, which is at y=(0, Rcos(ϕ₁+Ωτ), Rsin(ϕ₁+Ωτ)). Using the approximation r~r_o−x·y/|x| the propagation distance from this element to the far-field observer is given by (see Fig. 16.11).

$r\left(\tau \right)\approx r_o-x_{2}Rcos\left({\varphi }_1+\Omega \tau \right)/\left|\mathbf{x}\right|-x_{3}Rsin\left({\varphi }_1+\Omega \tau \right)/\left|\mathbf{x}\right|$

It is advantageous to specify the observer location in spherical coordinates so that

$x_1=r_{o}cos{\theta }_o{x}_2=r_{o}sin{\theta }_{o}cos{\varphi }_o{x}_3=r_{o}sin{\theta }_{o}sin{\varphi }_o$

then

$r\left(\tau \right)\approx r_o-Rsin{\theta }_{o}cos\left({\varphi }_1-{\varphi }_o+\Omega \tau \right)$

The final step in the analysis is to make use of the Fourier series expansion

$e^{-i\alpha cos\theta }={\displaystyle \sum _{m=-\infty }^{\infty }{J}_m\left(\alpha \right)e^{-im\left(\theta +\pi /2\right)}}$

where J_m(α) is a Bessel function of the first kind of order m. This function has well-known properties and is readily available on most computational systems, so making use of it to simplify the analysis is advantageous. However, it is difficult to compute accurately for large orders, and so asymptotic formulae for the Bessel function are sometimes required. We can now write

$e^{\mathit{in\Omega r}\left(\tau \right)/c_{\infty }}=e^{in\Omega r_o/c_{\infty }}{\displaystyle \sum _{m=-\infty }^{\infty }{J}_m\left(\frac{n\Omega Rsin{\theta }_o}{c_o}\right)e^{-im\left({\varphi }_1-{\varphi }_o+\Omega \tau +\pi /2\right)}}$

Combining all these results into Eq. (16.3.8) gives the Fourier series coefficients in a convenient form as

$c_n\left(\mathbf{x}\right)=\frac{in\Omega e^{in\Omega r_o/c_o}}{4\pi r_o{c}_o}{\displaystyle \sum _{m=-\infty }^{\infty }{\displaystyle \underset{R_i}{\overset{R_{\mathrm{tip}}}{\int }}{Q}_{m,n}\left(R,\mathbf{x}\right)J_m\left(\frac{n\Omega Rsin{\theta }_o}{c_o}\right)}}dR$

with the blade planform surface element defined as dΣ=Rdϕ₁dR and the blade surface given by ϕ_TE(R)<ϕ₁<ϕ_LE(R) and R_i<R<R_tip. The source term is given as

$Q_{m,n}\left(R,\mathbf{x}\right)=\frac{\Omega }{2\pi }{\displaystyle \underset{{\varphi }_{TE}}{\overset{{\varphi }_{LE}}{\int }}{\displaystyle \underset{0}{\overset{T_p}{\int }}\left\{\frac{-x_i{f}_i\left(\mathbf{z},\tau \right)}{r_o}+{\rho }_{\infty }{c}_o\mathbf{V}\left(\mathbf{z},\tau \right)\cdot \nabla h\left(\mathbf{z}\right)\right\}{e}^{i\left(n-m\right)\Omega \tau -im\left({\varphi }_1-{\varphi }_o+i\pi /2\right)}Rd{\varphi }_1}}d\tau$

We have now reduced the problem to two relatively straightforward integrals. The first integral in Eq. (16.3.14) is over the blade span, and the second, in Eq. (16.3.15), gives the source term and requires integrals over the blade chord and the period of rotation. We will start by considering the case in which we can ignore the unsteady loading terms and limit consideration to only the steady loading and thickness noise terms. If the thrust per unit area on the blade surface is f_L(R,ϕ₁) and the drag is f_D(R,ϕ₁) then we have f_i=(−f_L,−f_Dsin(Ωτ+ϕ₁), f_Dcos(Ωτ+ϕ₁)), and we can define

$\frac{x_i{f}_i}{r_o}=-f_{L}cos{\theta }_o-f_{D}sin{\theta }_{o}sin\left(\Omega \tau +{\varphi }_1-{\varphi }_o\right)$

Similarly, the thickness noise term may be simplified as in Section 16.2.2, as

${\rho }_{\infty }\mathbf{V}\left(\mathbf{z},\tau \right)\cdot \nabla h\left(\mathbf{z}\right)={\rho }_{\infty }\left(\Omega R\right)\frac{1}{R}\frac{\partial h}{\partial {\varphi }_1}$

which is independent of time. Using these results in Eq. (16.3.15) then gives

$\begin{array}{c}{Q}_{m,n}\left(R,\mathbf{x}\right)=\frac{\Omega }{2\pi }\underset{{\varphi }_{TE}}{\overset{{\varphi }_{LE}}{\int }}\underset{0}{\overset{T_p}{\int }}\left\{f_L\left(R,{\varphi }_1\right)cos{\theta }_o+f_D\left(R,{\varphi }_1\right)sin{\theta }_{o}sin\left(\Omega \tau +{\varphi }_1-{\varphi }_o\right)\right.\\ \left.+{\rho }_{\infty }{c}_o\Omega \frac{\partial h\left(R,{\varphi }_1\right)}{\partial {\varphi }_1}\right\}{e}^{i\left(n-m\right)\Omega \tau -im\left({\varphi }_1-{\varphi }_o+\pi /2\right)}Rd{\varphi }_{1}d\tau \end{array}$

We have now completely specified the time dependence of the sources, and so the integral over time may be evaluated analytically. We note that both the thrust term and the thickness noise term are independent of time, and so their integral is only nonzero when n=m. Similarly, for the drag term the sine function may be split into the sum of two exponentials, and so the integral over time will only be nonzero when n=m±1. Finally, we note that the integral of the thickness noise term over ϕ₁ can be carried out by parts, and providing h=0 at the leading and trailing edges we obtain

$Q_{m,n}\left(R,\mathbf{x}\right)={\displaystyle \underset{{\varphi }_{TE}}{\overset{{\varphi }_{LE}}{\int }}\left\{f_L\left(R,{\varphi }_1\right)cos{\theta }_o{\delta }_{mn}+f_D\left(R,{\varphi }_1\right)sin{\theta }_o\left(\frac{{\delta }_{n,m-1}-{\delta }_{n,m+1}}{2i}\right)+im\Omega {\rho }_{\infty }{c}_{o}h\left(R,{\varphi }_1\right){\delta }_{mn}\right\}{e}^{-im\left({\varphi }_1-{\varphi }_o+\pi /2\right)}R}d{\varphi }_1$

This provides a simple formula for the source term that can be readily evaluated. We still need to know the distribution of the thrust, drag, and thickness on the blade surface, but this can be obtained from the blade design characteristics. As a first approximation we can use a point-loading approximation, but this is inaccurate for the higher harmonics. One of the most important features of this result is that the integral over time has reduced the summation in Eq. (16.3.14) to only three values of m for a given harmonic n. The summation in Eq. (16.3.14) is therefore limited to only three terms (and can be further simplified by using properties of Bessel functions if required). This would not have been the case for unsteady loading because f_L would then be a function of time. To show how unsteady loading can be incorporated into the calculation, we can assume that the loading is periodic in source time and can therefore be expanded as a Fourier series, so

$f_L\left(R,{\varphi }_1,\tau \right)={\displaystyle \sum _{k=-\infty }^{\infty }{f}_L^{\left(k\right)}\left(R,{\varphi }_1\right)e^{-\mathit{ik\Omega \tau }}}$

and similarly for the drag term. Using this result in Eq. (16.3.18) and integrating over time shows that the integral will only be nonzero when n−m=k (or n−m=k±1 for the drag term), so we can rewrite Eq. (16.3.19) to include unsteady loading as

$Q_{m,n}\left(R,\mathbf{x}\right)={\displaystyle \underset{{\varphi }_{TE}}{\overset{{\varphi }_{LE}}{\int }}\left\{f_L^{\left(n-m\right)}\left(R,{\varphi }_1\right)cos{\theta }_o+f_D^{\left(k\right)}\left(R,{\varphi }_1\right)sin{\theta }_o\left(\frac{{\delta }_{n,m+k+1}-{\delta }_{n,m+k-1}}{2i}\right)+im\Omega {\rho }_{\infty }{c}_{o}h\left(R,{\varphi }_1\right){\delta }_{mn}\right\}\times e^{-im\left({\varphi }_1-{\varphi }_o+\pi /2\right)}R}d{\varphi }_1$

In this result the number of terms required in the summation over m in Eq. (16.3.14) has been significantly increased, and the expansion Eq. (16.3.20) may converge only slowly. So, the evaluation of Eq. (16.3.21) may be time-consuming, but the computational procedure is relatively straightforward. One of the advantages of this approach is that it allows the distribution of blade loading and thickness over the chord to be readily included in the calculation. It should also be noted that the radial integral in Eq. (16.3.14) includes a dependence on the Bessel function J_m(nΩRsinθ_o/c_o) which has a strong impact on the directionality of the acoustic field. These functions can be cumbersome to compute, but fortunately there are a number of asymptotic approximations, which simplify the task of computing the radial integral.

As an illustrative example we will calculate the Fourier series coefficients c_n(x) for the Gaussian derivative impulsive blade loading given by Eq. (16.2.7). The loading coefficients in Eq. (16.3.20) can be calculated analytically in the limit that T_v≪T_p and are

$f_L^{\left(k\right)}\left(R,{\varphi }_1\right)=\frac{ik{F}_L{\left(\Omega T_v\right)}^2\sqrt{\pi }}{4\pi }exp\left(ik\Omega {\tau }_o-{\left(k\Omega T_v/2\right)}^2\right)\left\{\frac{\delta \left({\varphi }_1\right)\delta \left(R\right)}{R}\right\}$

where the terms in {} are required so that the force acts at a point. It follows then that

$Q_{m,n}\left(R,\mathbf{x}\right)=\frac{i\left(n-m\right)F_L{\left(\Omega T_v\right)}^2\sqrt{\pi }}{4\pi }{e}^{i\left(n-m\right)\Omega {\tau }_o-{\left(\left(n-m\right)\Omega T_v/2\right)}^2}\delta \left(R\right)cos{\theta }_o{e}^{im\left({\varphi }_o-\pi /2\right)}$

A contour plot of these coefficients is shown in Fig. 16.19 and clearly shows the importance of the terms with increasing m and n.

The nondimensional Fourier series coefficients of the acoustic signature are then obtained as

$C_p^{\left(n\right)}=\frac{4\pi r_o{c}_{\infty }{c}_n\left(\mathbf{x}\right)}{\Omega F_L}=in\Omega e^{in\Omega r_o/c_o}{\displaystyle \sum _{m=-\infty }^{\infty }{J}_m\left(\frac{n\Omega Rsin{\theta }_o}{c_o}\right)cos{\theta }_o{e}^{\mathrm{i}\mathrm{m}\left({\varphi }_o-\pi /2\right)}}\times \left\{\frac{i\left(n-m\right){\left(\Omega T_v\right)}^2}{4\sqrt{\pi }}{e}^{i\left(n-m\right)\Omega {\tau }_o-{\left(\left(n-m\right)\Omega T_v/2\right)}^2}\right\}$

and the amplitude of these are plotted on a dB scale in Fig. 16.20. Note that the scale has a very large range, and the harmonics that contribute significantly are only those with indices <32. Evaluating the pulse observed in the far field from these harmonics reproduces the signature calculated directly using the time series approach, as shown in Fig. 16.16.

We can also evaluate the harmonics of the thickness noise pulse that was given in Fig. 16.17. The nondimensional form of these harmonics is given by

$C_q^{\left(n\right)}=\frac{4\pi r_o{c}_{\infty }{c}_n\left(\mathbf{x}\right)}{\Omega \left({\rho }_o{\Omega }^{2}R\mathrm{\Delta }{V}_k\right)}=-\left(\frac{c_{\infty }}{\Omega R}\right)n^2{J}_n\left(\frac{n\Omega Rsin{\theta }_o}{c_{\infty }}\right)e^{\mathrm{i}\mathrm{m}\left({\varphi }_o-\pi /2\right)+in\Omega r_o/c_{\infty }}$

and these are also shown in Fig. 16.20. From this result we see that thickness noise tends to dominate for the lower harmonics, and unsteady loading noise dominates for the higher harmonics. Note also that for a rotor or propeller with B blades only the harmonics n=mB will contribute, so the contribution of thickness noise may be limited to the first two blade-passing frequencies.

16.5.1 Amplitude modulation

Amplitude modulation occurs when the source level on a rotating blade varies significantly with position. An example is a rotor operating in a very uneven inflow, such as the rotor operating near a wall that was discussed in Chapter 10. In this case the rotor blades pass through a turbulent boundary layer that extends over about one-fourth of the rotor disc plane. The noise levels from a particular blade are low when it is in the free stream flow outside the boundary layer, and all the leading-edge noise is generated when the blade passes through the high levels of turbulence in the boundary layer. The signal from each blade is therefore strongly modulated, but the modulation is partially mitigated by the number of blades in the rotor. If the blade count is low, then the effect of modulation is significant because there are times when no blade is in the region of high-level turbulence. However, if the blade count is high then there are always a number of blades in the region where noise is produced and the signal has very little variation with time. Another example is the effect of a nonuniform mean flow on trailing edge noise. If the nonuniform mean flow causes a significant change in blade angle of attack, then the trailing edge noise will be locally increased (or decreased as the case may be) and the far-field sound will have a signal that is modulated. In each case we will assume that the signal from each blade is uncorrelated, which will be the case for trailing edge noise but not always the case for leading-edge noise, and we will discuss the impact of blade-to-blade correlation in the next section.

To illustrate the effect that amplitude modulation has on the measured spectrum consider a source signal from each blade that can be modeled as f_s(t), where s is the blade number and f_s(t) is uncorrelated for different blades. If the signal is modulated as the blade rotates by a mean flow effect, then the modulation can be represented by the function g(t−sT_p/B) for blade number s, where T_p is the time for one blade rotation and B is the blade count. The total signal is then

$a\left(t\right)={\displaystyle \sum _{s=1}^B{f}_s\left(t\right)g\left(t-s{T}_p/B\right)}$

An example of the signal from one blade is shown in Fig. 16.23. This signal is only nonzero for one-fifth of the period of blade rotation, and the signals from successive blades should be added to this. Fig. 16.23B shows the signal for seven blades summed with the correct time delay. The signal appears to be continuous, and no periodic character is apparent from the overall signature, which is misleading.

The modulating function is periodic and so can be expressed as a Fourier series

$g\left(t-s{T}_p/B\right)={\displaystyle \sum _{n=-\infty }^{\infty }{g}_n{e}^{2\pi in\left(t-s{T}_p/B\right)/T_p}}$

and so the Fourier transform of the signal is given by

$\tilde{a}\left(\omega \right)={\displaystyle \sum _{s=1}^B{\displaystyle \sum _{n=-\infty }^{\infty }{\tilde{f}}_s\left(\omega -2\pi n/T_p\right)g_n{e}^{2\mathit{\pi ins}/B}}}$

The power spectrum of the signal is given by

$S_{\mathit{aa}}\left(\omega \right)={\displaystyle \sum _{s=1}^B{\displaystyle \sum _{r=1}^B{\displaystyle \sum _{n=-\infty }^{\infty }{\displaystyle \sum _{m=-\infty }^{\infty }\frac{\pi }{T}\mathit{Ex}\left[{\tilde{f}}_s\left(\omega -2\pi n/T_p\right){\tilde{f}}_r^{⁎}\left(\omega -2\pi m/T_p\right)\right]g_n{g}_m^{⁎}{e}^{2\pi i\left(\mathit{ns}-mr\right)/B}}}}}$

If we assume that the loading on each blade is statistically independent the double summation is reduced to a single summation, and if the base signal is statistically stationary and independent of blade number, then only those terms for which n=m will be nonzero, so

$S_{\mathit{aa}}\left(\omega \right)=B{\displaystyle \sum _{n=-\infty }^{\infty }{S}_{\mathit{ff}}\left(\omega -2\pi n/T_p\right)\left|g_n\right|{}^2}$

As an example we can consider a signal that is modulated by a periodic rectangular window of length T_g, where T_g=T_p/5 as shown in Fig. 16.23A, and assume that the source spectrum is given by

$S_{\mathit{ff}}\left(\omega \right)=\frac{{\left(\omega T_s\right)}^2}{1+{\left(\omega T_s\right)}^4}$

as shown in Fig. 16.23C. The signal has a time scale T_s=0.3T_p and the spectrum peaks at the frequency ω=Ω/2. When the signal is modulated and summed as in Eq. (16.5.1) the resulting spectrum is quite different and shows a series of peaks and an oscillatory level sometimes referred to as scalloping. The peaks do not occur at exact multiples of the rotation frequency or blade passage frequency because S_ff peaks at a frequency that is nonzero. The spectrum, however, is quite different from that of a single blade as a direct result of the periodic modulation of the signal.

16.5.2 Blade-to-blade correlation

In Amiet's method it was assumed that the signals generated by each blade are statistically independent, and so there is no blade-to-blade correlation. This is a reasonable approximation for trailing edge noise that depends on the individual blade boundary layers and for situations where the length scale of the incoming turbulence is much smaller than the distance between the blades. However, it is known that when turbulence enters a rotor it can be stretched in the direction of the flow, and this extends its axial length scale. This stretching can be substantial and alone can result in a single turbulent structure being cut multiple times by successive blades so that the blade loading is correlated between the blades. This results in a quasiperiodic signal in the acoustic far field that includes bursts of pulses at the BPF and a sound spectrum with broadband peaks at near multiples of the BPF, referred to as haystacks. The criterion for this effect to occur is that the BPF should be significantly higher than the axial inflow speed U_o divided by the axial turbulence length scale BΩL/U_o ≫1. If this is a large parameter then blade-to-blade correlation needs to be considered, if not then Amiet's approximation can be applied.

This feature was originally identified by Sevik [4] for a propeller operating in grid-generated isotropic homogeneous turbulence. It was expected that the spectrum from this interaction would be a smooth function of frequency determined by the wavenumber content of the inflow turbulence. However, Sevik found that the spectrum included a series of humps that peaked at frequencies slightly above the BPF. It was later shown by Martinez [5] that the shifting of the humps from the BPF was caused by the pitch angle of the blades to the axial flow.

To illustrate the characteristics of blade-to-blade correlation, consider a series of pulses that persist for a limited period of time such as would be generated by a rotor with B blades cutting through an eddy at the blade-passing interval. The signature from each blade passage is the same, but the amplitude is modulated by the variation in the strength of the eddy as it passes though the rotor. A model for the time history of the sound produced by one eddy is

$p\left(t\right)={\displaystyle \sum _{n=-\infty }^{\infty }f\left(t-2\pi n/B\Omega \right)e^{-{\left(U_{o}t/L\right)}^2}}$

Here the envelope exp(−(U_ot/L)²) defines the time variation in the strength of the eddy as it is convected through the rotor with axial velocity U_o. We model the observed signature of a single-blade passage through the eddy as f(t)=(Ut/L_o)exp(−(Ut/L_o)²) which is the same shape as the signal shown in Fig. 16.15. In this model L represents the axial lengthscale of the turbulence, L_o is the transverse lengthscale, and U is the flow speed relative to the blade.

Fig. 16.24A shows the time history modeled by Eq. (16.5.2). The spectrum of these pulses is shown in Fig. 16.24B for BΩL/U_o=Bπ/4 and BΩL_o/U=2. The time scale of the interaction is relatively small in this case indicating that the axial flow speed U_o is large and the axial lengthscale L is small, so the blades only chop the eddy once, and the spectrum of the pressure time history is comparatively smooth. However, when the axial flow speed is reduced, and the lengthscale L is increased, the criterion for haystacking BΩL/U_o is increased. Fig. 16.25A gives the time history for the case when BΩL/U_o=Bπ. Because several blades interact with a single eddy the time history has multiple pulses at the blade-passing interval and the spectrum (Fig. 16.25B) shows clearly defined peaks at the blade-passing frequencies. This is the haystacking phenomenon.

Details of a time domain approach to this problem can be found in Glegg et al. [6].