2.1. Method for flow simulation

As observed in the experiment of Alexander et al. (Reference Alexander, Molinaro, Hickling, Murray, Devenport and Glegg2016), the presence of the rotor has a negligible effect on the generation and development of the upstream cylinder wake because of the large separation distance between the rotor and the cylinder. This allows the simulations of wake generation and wake-rotor interaction to be carried out separately. The wake-generation simulation, discussed in detail in § 3, is performed in the stationary frame of reference and provides inflow data for the rotor simulation. The latter is conducted in the rotor frame of reference in which the spatial coordinates are fixed on the rotor. This decoupled approach is computationally efficient because, aside from avoiding the need for using a sliding-mesh technique and the associated computational overhead, the cylinder wake only needs to be computed once and the same inflow data saved can be used repeatedly for rotor simulations with different rotor advance ratios and wake-strike positions.

The rotor-frame simulations are performed using the conservative formulation (Beddhu, Taylor & Whitfield Reference Beddhu, Taylor and Whitfield1996) of the spatially filtered (denoted by an overbar) Navier–Stokes equations and continuity equation,

(2.1)\begin{equation} \left. \frac{\partial\bar{u}^r_i}{\partial t} \right|_r + \frac{\partial}{\partial x_j^r} \bar{u}^r_i \bar{u}^r_j - \frac{\partial}{\partial x_m^r} \epsilon_{mjk} \varOmega_j x_k^r \bar{u}^r_i + \epsilon_{ijk} \varOmega_j \bar{u}^r_k = - \frac{1}{\rho} \frac{\partial\bar{p}}{\partial x_i^r} + \nu \frac{\partial^2 \bar{u}^r_i}{\partial x_j^r \partial x_j^r} + \frac{\partial}{\partial x_j^r} \tau_{ij}^r, \end{equation}

(2.2)\begin{equation} \frac{\partial\bar{u}^r_i}{\partial x_i^r} = 0, \end{equation}

where $\bar {u}^r_i$ are components of the absolute velocity in the rotating frame of reference $x^r_i$, $\varOmega _i$ are components of the angular velocity of the reference frame, $p$ and $\rho$ are the fluid pressure and density, respectively, $\nu$ is the kinematic viscosity, $\epsilon _{ijk}$ is the alternating tensor, $\tau _{ij}^r = \bar {u}^r_i\bar {u}^r_j-\overline {u_i^r u_j^r}$ is the subgrid-scale stress, and ${\partial }/{\partial {t}}|_r$ is the time derivative with respect to the rotating frame of reference. The equations solved in the stationary frame of reference are those in (2.1) and (2.2) with $\varOmega _j$ set to zero and the superscript and subscript ‘$r$’ dropped.

A finite volume, unstructured-mesh LES code for incompressible flow developed at Stanford University (You, Ham & Moin Reference You, Ham and Moin2008) is enhanced with the rotating-frame formulation (2.1) and used for flow simulations. It employs a fully implicit, fractional-step time-advancement method and an algebraic multigrid Poisson solver for pressure, and is second-order accurate in both time and space. The spatial discretization is energy conserving and low dissipative, and thus can accurately capture a wide range of flow scales relevant to sound generation. The effect of subgrid-scale motions is modelled using the dynamic Smagorinsky model (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991). The accuracy of this code for flow-noise studies has been established in a number of previous configurations including rough-wall boundary layers (Yang & Wang Reference Yang and Wang2013), tandem cylinders and airfoil in a cylinder wake (Eltaweel et al. Reference Eltaweel, Wang, Kim, Thomas and Kozlov2014).

2.2. Method for acoustic calculation

The FW–H integral formulation of the Lighthill equation, which allows surfaces in arbitrary motion (Ffowcs Williams & Hawkings Reference Ffowcs Williams and Hawkings1969; Goldstein Reference Goldstein1976) is employed,

(2.3)\begin{align} p({\boldsymbol{x}},t) = \iint_{S(\tau)}n_jp_{ij}\frac{\partial{G}}{\partial{y_i}}\,\mathrm{d}S\,\mathrm{d}\tau-\iint_{S(\tau)}\rho_{\infty} n_j V_j \frac{\partial{G}}{\partial{\tau}}\,\mathrm{d}S\,\mathrm{d}\tau+\iint_{V(\tau)}T_{ij}\frac{\partial^2{G}}{\partial{y_i}\partial{y_j}}\,\mathrm{d}V\,\mathrm{d}\tau, \end{align}

where $p({\boldsymbol {x}},t)$ is the fluctuating pressure, which is equal to the acoustic pressure outside the nonlinear flow region, at observer location ${\boldsymbol {x}}$ and observer time $t$; $S(\tau )$ is the integration surface, taken to be the solid surface, at source emission time $\tau$; $n_j$ are components of the unit normal to $S$ pointing into the fluid; $p_{ij}$ is the compressive stress tensor dominated by pressure; $G = G({\boldsymbol {x}}, t; {\boldsymbol {y}}, \tau )$ is the free-space Green's function; ${\boldsymbol {y}}$ is the source position vector with $y_i$ being its components; $\rho _{\infty }$ is the ambient density; $V_j$ are components of the surface velocity; and $T_{ij}$ is the Lighthill stress tensor dominated by $\rho _{\infty } u_i u_j$ and distributed in the source region $V(\tau )$. The three terms on the right-hand side represent loading noise, thickness noise and quadrupole contributions, respectively. At low Mach numbers unsteady loading is the dominant source of noise. As a result, (2.3) can be simplified in the acoustic far-field as (Brentner & Farassat Reference Brentner and Farassat2003)

(2.4)\begin{equation} p({\boldsymbol{x}},t) \approx \frac{1}{4{\rm \pi} c_{\infty}}\frac{\partial}{\partial t}\int_S \left[\frac{r_{d_i}}{r_d^2}\frac{p_{ij}n_j}{|1-M_r|}\right]_{\tau^{\ast}}\,\mathrm{d}S, \end{equation}

where $r_d=|{\boldsymbol {x}}-{\boldsymbol {y}}(\tau )|$ is the distance between the observer and the source, $r_{d_i} = x_i-y_i$, and $M_r = (r_{d_i}/r_d)M_i$ is the Mach number of the source in the radiation direction with $M_i$ being components of the source Mach number vector. The integrand is evaluated at the retarded time $\tau ^{\ast }$ which is the root of the equation $\tau = t - |{\boldsymbol {x}}-{\boldsymbol {y}}(\tau )|/c_{\infty }$. For the rotor geometry and Mach numbers considered in this study, the rotor blades can be assumed acoustically compact in the chordwise direction but not necessarily in the radial direction. To facilitate evaluation of (2.4), each blade is divided into acoustically compact strips stacked in the radial direction, and the far-field sound is then the sum of contributions from all strips,

(2.5)\begin{equation} p({\boldsymbol{x}},t)\approx\frac{1}{4{\rm \pi} c_{\infty}}\frac{\partial}{\partial t}\sum_{n=1}^{N_b}\sum_{m=1}^{N_s}\left[ \frac{r_{d_i}}{r_d^2}\frac{F_i}{|1-M_r|} \right]^{m,n}_{\tau^{\ast}}, \end{equation}

where $N_b$ is the number of blades, $N_s$ is the number of strips on a blade and

(2.6)\begin{equation} F^{m,n}_i = \int_{S_{mn}} p_{ij} n_j\,\mathrm{d}S \end{equation}

is the net unsteady force on each strip.

Alternatively, the time derivatives in (2.4) and (2.5) can be moved from the observer time to the source time using ${\partial }/{\partial t} = ({1}/({1-M_r}))({\partial }/{\partial \tau })$, leading to (see, for example, Brentner & Farassat Reference Brentner and Farassat2003; Glegg & Devenport Reference Glegg and Devenport2017)

(2.7)\begin{equation} p({\boldsymbol{x}},t) \approx \frac{1}{4{\rm \pi} c_{\infty}} \int_S \left[ \frac{r_{d_i}}{r_d^2(1-M_r)^2} \left( \frac{\partial (p_{ij} n_j)}{\partial \tau}+\frac{p_{ij} n_j}{|1-M_r|}\frac{\partial M_r}{\partial \tau} \right) \right]_{\tau^{\ast}}\,\mathrm{d}S \end{equation}

and

(2.8)\begin{equation} p({\boldsymbol{x}},t)\approx\frac{1}{4{\rm \pi} c_{\infty}} \sum_{n=1}^{N_b}\sum_{m=1}^{N_s} \left[ \frac{r_{d_i}}{r_d^2(1-M_r)^2} \left( \frac{\mathrm{d} F_i}{\mathrm{d} \tau} + \frac{F_i}{|1-M_r|} \frac{\mathrm{d} M_r}{\mathrm{d} \tau} \right) \right]^{m,n}_{\tau^{\ast}} \end{equation}

in the acoustic far-field. In this formulation the acoustic dipole sources ${\partial (p_{ij} n_j)}/{\partial \tau }$ and $\mathrm {d} F_i/\mathrm {d} \tau$ are revealed explicitly, and the terms proportional to ${\partial M_r}/{\partial \tau }$ are relatively small if the Mach number of the blade tip is small. Calculations performed with both formulations (2.5) and (2.8) have produced negligible differences in results.

4.1. Simulation set-up

The simulation set-up for the rotor flow is shown schematically in figure 5(a,b). For convenience the hub radius $R \equiv R_{{hub}}$ is used as the length scale for normalization. For the modified Sevik rotor, the rotor diameter is $7.2R$, and the blade chord length is uniform from the root to the tip, and equal to $0.9R$. The Reynolds number based on the free stream velocity and hub radius is $Re_R = 81\,000$. Simulations are conducted in a cylindrical domain with dimensions of $20R$ in the axial direction and $14.4R$ in the radial direction. The radius of the domain, being four times the rotor radius, is relatively small to save computational cost, but is not expected to cause a significant confinement effect given that the rotor is at zero- or low-thrust. The rotor centre, chosen as the origin of the coordinates, is $8R$ from the inlet. The cylinder wake is parallel to the $x$–$z$ plane. The computational mesh from two different perspectives is illustrated in figure 5(c,d). The simulation employs approximately $100$ million mesh cells in total. There are $53$ million hexahedral and tetrahedral cells in the mesh block around the blades with the first off-wall cell height of $0.004R$ and surface grid spacings ranging from $0.0028R$ (leading and trailing edges) to $0.013R$ (midchord) in the blade circumferential direction and from $0.008R$ (hub and tip) to $0.017R$ (midspan) in the spanwise direction. This grid is designed to capture the turbulence-ingestion noise, but is insufficient to resolve the blade turbulent boundary layers accurately and thus the rotor self-noise computed is unreliable. There are $32$ million prism, hexahedral and tetrahedral cells used in the region upstream of the blade block with a grid spacing of approximately $0.04R$ in all directions to provide adequate resolution for the incoming turbulent wake, and $15$ million cells distributed downstream of the rotor with gradual stretching towards the outlet. Stress-free conditions with radial velocity $u_r = 0$ are imposed on the outer boundary, no-slip and no-penetration conditions are imposed on the solid surfaces, and convective boundary conditions are employed at the exit. At the inlet the inflow data containing time series of the cylinder-wake turbulence profiles are fed into the computational domain through bilinear interpolation in space and linear interpolation in time. After each time step $\Delta t$, the inflow profiles are rotated by $-\varOmega \Delta t$ relative to the rotor mesh, where $\varOmega$ is the rotor angular velocity. The mesh near the inlet plane is refined to reduce the interpolation error.

Figure 5.

Rotor simulation set-up (a,b) and computational mesh (c,d) from two different perspectives.

4.2. Flow characteristics

In this section the basic flow characteristics obtained from the LES are presented for both the zero thrust ($J = 1.44$) and thrusting ($J = 1.05$) rotors. To generate these results, the simulations for both cases were started with an initially uniform flow field and run for approximately 40 time units in terms of $R/U_{\infty }$, which is equivalent to two flow-through times based on the computational-domain length, or $3.9$ and $5.3$ rotations for $J=1.44$ and $1.05$, respectively, to wash out the transients. The next 88 time units ($4.4$ flow-through times, or $8.5$ rotations for $J=1.44$ and $11.6$ rotations for $J=1.05$) were used to collect data for statistical and acoustic analysis. The time steps were determined based on maximum Courant–Friedrichs–Lewy numbers of 1.5 for the $J = 1.44$ case and 1.0 for the $J = 1.05$ case. Simulations were typically run on 1152 Intel Xeon E5-2698v3 cores, and cost approximately 63 000 and 85 000 core-hours per flow-through time for $J=1.44$ and $1.05$, respectively.

An instantaneous flow field for the zero-thrust case is illustrated in figure 6 from three different perspectives: a three-dimensional view and two plane views that are parallel and perpendicular to the wake, respectively. The isosurfaces of the Q-criterion show a wide range of flow structures interacting with the rotor. Tip vortices are also clearly visible in regions where they are not disrupted by the turbulent wake. The level of details of the flow structures captured by the simulation is well illustrated in this figure, and allows a detailed analysis of the broadband and tonal noise generation processes. The vortical structures around the thrusting rotor (not illustrated) show similar characteristics as in the zero-thrust case but are more elongated as they are drawn into the rotor.

Figure 6.

Isosurfaces of $Q$-criterion at $Q(R/U_{\infty })^2=1.2$ from different perspectives coloured by the velocity component perpendicular to the cylinder wake for the zero-thrust case ($J = 1.44$).

Figure 7 shows isocontours of the instantaneous radial vorticity on two cylindrical surfaces at $r/R=1.26$ and $3.34$ for the zero-thrust case. The smaller surface is entirely immersed in the cylinder wake, and thus turbulence structures are observed around all blades. Due to the small radius, the blade chord Reynolds number based on the relative velocity $\sqrt {U^2_{\infty } + (\varOmega r)^2}$ is relatively low, and no trailing-edge vortex shedding is observed. In contrast, on the larger surface, which is close to the blade tip, not all blades are immersed in the wake at the same time, and the chord Reynolds number is significantly higher, leading to trailing-edge vortex shedding and associated noise as will be shown in the acoustic results.

Figure 7.

Isocontours of instantaneous radial vorticity $\omega _r R/U_{\infty }$ on two cylindrical surfaces for the zero-thrust case ($J = 1.44$). (a) $r/R=1.26$; (b) $r/R=3.34$.

Phase-averaged axial velocities at three different streamwise locations $x/R=-0.38$, 0 and 0.38, which correspond to the blade leading edge at the root, rotor midplane and blade trailing edge at the root, respectively, are shown in figure 8 for both zero thrust and thrusting cases. The phase averaging is performed over 8.4 rotor rotations, or 84 blade passages for the zero-thrust rotor and $11.6$ rotor rotations, or $116$ blade passages for the thrusting rotor. The phase is selected such that two blades are aligned with the cylinder wake centreline in the rotor midplane. It can be seen that the thrusting rotor accelerates the flow mildly in the blade passage, whereas the zero-thrust rotor has a much smaller effect on the flow. Figure 9 depicts the phase-averaged turbulent kinetic energy at the same three streamwise locations. In contrast to the mean axial velocity, the turbulent kinetic energy is little affected by the rotor advance ratio except in the blade wake (figure 9c,f), where the thrusting rotor produces more turbulent kinetic energy due to its faster rotational speed.

Figure 8.

Phase-averaged axial velocity $U/U_{\infty }$ for the zero thrust (a–c) and thrusting (d–f) cases at different streamwise locations: (a,d) $x/R=-0.38$; (b,e) $x/R=0$; (c,f) $x/R=0.38$.

Figure 9.

Phase-averaged turbulent kinetic energy $\overline {u'_i u'_i}/(2U_{\infty }^2)$ for the zero thrust (a–c) and thrusting (d–f) cases at different streamwise locations: (a,d) $x/R=-0.38$; (b,e) $x/R=0$; (c,f) $x/R=0.38$.

5.2. Radiation from different blade regions

To examine the relative contributions to the radiated acoustic field from different blade regions, each blade is divided into five segments of equal width, numbered 1–5 from the root to the tip, along the radial direction. Figure 14 shows a comparison of the acoustic pressure spectra produced by these five segments for the zero thrust and thrusting cases at two observer locations $z=0$, $r_o=200$, $\theta _o = 30^{\circ }$ and $75^{\circ }$. At both locations, the spectral level increases with the radial coordinate of the source region at frequencies higher than the BPF ($\,f_{{BP}} \approx 0.96 U_{\infty }/R$ for $J = 1.44$ and $1.32 U_{\infty }/R$ for $J = 1.05$). This increase is significantly larger at the shallower observer angle where the axial dipoles are more dominant, and slightly larger in the thrusting case compared with the zero-thrust case. A prominent spectral peak is seen at $f\kern0.05em R/U_{\infty }\approx 16.4$ and $21.8$ for $J = 1.44$ and $1.05$, respectively, for segments $4$ and $5$ but not others, indicating that the outer region is the dominant contributor to blade vortex shedding, which is consistent with the observation made in figure 7. In both cases at frequencies below the BPF, particularly at the cylinder shedding frequency, the spectral variation with segment position is less consistent and highly dependent on the observer angle. At $\theta _o=30^{\circ }$ the shedding peak varies only mildly and non-monotonically with the segment position, whereas at $\theta _o=75^{\circ }$ the shedding peak varies more significantly, showing mostly a decrease as the segment position varies from the root to the tip.

Figure 14.

Contributions of different blade segments to rotor noise spectra for the zero thrust (a,b) and thrusting (c,d) cases in the $z=0$ plane at $r_o/R=200$ and two observer angles. Segment 1 (, red); segment 2 (, green); segment 3 (, blue); segment 4 (, black); segment 5 (, orange); entire rotor (, magenta). (a) $J = 1.44$, $\theta _o=30^{\circ }$; (b) $J = 1.44$, $\theta _o=75^{\circ }$; (c) $J = 1.05$, $\theta _o=30^{\circ }$; (d) $J = 1.05$, $\theta _o=75^{\circ }$.

6.1. Accuracy of Sears theory for rotor noise

As a classical model for airfoil-turbulence interaction noise, the Sears theory (Sears Reference Sears1941) provides a relationship between the unsteady lift of a thin airfoil and the upwash velocity encountered by the airfoil. It is of interest to evaluate its accuracy for predicting the noise of rotor ingesting the turbulent wake using the LES data. For a general two-dimensional gust the unsteady sectional lift $L$ can be calculated from

(6.1)\begin{equation} L=\sum_{n=1}^N {\rm \pi}\rho_{\infty} C U_c \hat{v}_n \textrm{e}^{-\textrm{i}\omega_n t}S(\varOmega_n), \end{equation}

where $C$ is the airfoil chord length, $U_c$ is the convection velocity, $\hat {v}_n$ is the upwash velocity component at angular frequency $\omega _n$, and $\varOmega _n=\omega _n C/(2U_c)$ is the reduced frequency. The Sears function is defined as

(6.2)\begin{equation} S(\varOmega_n)=\frac{2}{{\rm \pi} \varOmega_n [H_0^{(1)}(\varOmega_n)+i H_1^{(1)}(\varOmega_n)]}, \end{equation}

where $H_0^{(1)}$ and $H_1^{(1)}$ are Hankel functions of the first kind. The summation in (6.1) is over all relevant Fourier modes, and the Fourier component of upwash velocity $\hat {v}_n$ is related to the time-domain upwash velocity $v_n(x,t)$ through

(6.3)\begin{equation} v_n(x,t)=\sum_{n=1}^{N} \hat{v}_n \exp\left({-\textrm{i}\omega_n(t-x/U_c)}\right). \end{equation}

In the current implementation, the upwash velocity is defined as the component of the relative velocity perpendicular to the blade chord just upstream of the rotor and the convection velocity is defined as the velocity component parallel to the chord. During the flow simulation, the velocities in the rotor inlet plane, taken as $x/R = -0.38$, are saved at every 16th time step. For each strip on the rotor blades, the convection velocity $U_c$ is estimated by $\sqrt {U_{\infty }^2+(\varOmega r)^2}$ in the middle of the strip, which is a good approximation of the spanwise-averaged, chord-parallel velocity at low angles of attack, and the upwash velocity $v_n$ at a given time is taken as the spanwise average of the upwash velocity. The unsteady lift for each strip is calculated using (6.1)–(6.3), summing up all the available Fourier modes, and the acoustic pressure is calculated using (2.5) and (2.6). Calculations have been performed using 10 strips and 118 strips along the blade span, and the resulting acoustic spectra are essentially the same.

Figure 15 shows, for both the zero thrust and thrusting cases, comparisons of the acoustic spectra predicted based on the unsteady loading from the Sears theory and that directly from the LES at $r_o/R = 200$, $\theta _o = 45^{\circ }$. It is found that the Sears theory predicts the rotor noise reasonably well in the mid-frequency range of $0.7 \lesssim f\kern0.05em R/U_{\infty } \lesssim 12$ for $J = 1.44$ and $0.2 \lesssim f\kern0.05em R/U_{\infty } \lesssim 16$ for $J = 1.05$. The high-frequency bounds for the two cases are comparable in terms of $f/f_{BP}$ (${\approx}12$), beyond which the Sears theory predictions fall off more quickly because they do not include the rotor self-noise. Large discrepancies are also seen between the Sears theory and LES predictions at low frequencies, particularly for the zero-thrust rotor. Similar comparisons are found at other observer angles.

Figure 15.

Comparison of sound pressure spectra at $r_o/R = 200$, $\theta _o = 45^{\circ }$ computed using blade unsteady loading from the Sears theory and LES for both advance ratios: Sears theory (, red); LES (, green). (a) $J=1.44$; (b) $J=1.05$.

6.2. Contribution from wake mean velocity defect

Since the Sears theory is linear, it allows an easy separation of the noise generated by the mean and fluctuating velocities in the wake. To this end the velocity in the rotor inlet plane is decomposed into a mean component and a fluctuating component, and their contributions to the sound pressure spectra are evaluated. The upwash velocity and its components caused by the mean and fluctuating wake velocities for the zero-thrust case are shown in figure 16 as a function of time at $x/R = -0.38$, $r/R=2.3$ and 3. Since the outer region of the rotor blade enters and leaves the wake periodically, the mean component appears as a periodic signal in the rotor frame of reference. Comparing the velocities experienced by blades at the two radial locations, it can be seen that the mean velocity curve has longer bottoms at the outer position because the blade section spends more time outside the finite-thickness wake, and consequently, turbulent velocity fluctuations are seen during shorter time intervals. Velocity fluctuations are small when the blade is outside the wake at both locations.

Figure 16.

Time history of the upwash velocity and its components caused by the mean and fluctuating wake velocities at $x/R = -0.38$ and two radial positions: mean velocity (, red); fluctuating velocity (, blue); total velocity (, green). The rotor advance ratio $J=1.44$. (a) $r/R = 2.3$; (b) $r/R = 3$.

The sound pressure spectra based on Sears theory predictions using the total upwash velocity and its mean and fluctuation components are compared in figure 17 for $r_o/R = 200$, $\theta _o = 45^{\circ }$. The mean velocity defect in the wake is a major contributor to the BPF peak but produces negligible sound at other frequencies. This suggests the potential for using RANS simulations to predict the BPF tone (e.g. Moreau Reference Moreau2019a). The turbulent velocity fluctuations are responsible for the broadband sound at all frequencies.

Figure 17.

Sound pressure spectra for $J = 1.44$ predicted by the Sears theory at $r_o/R = 200$, $\theta _o = 45^{\circ }$ using the upwash velocity based on: total velocity (, red); mean velocity (, green); fluctuating velocity (, blue).

6.3. Mach number scaling

The Sears theory can be employed to derive the appropriate Mach number scaling of the acoustic pressure. Based on (6.1), the unsteady sectional lift $L \sim \rho _{\infty } C U_{c} v_n$, and from (2.8), for $M_r \ll 1$,

(6.4)\begin{equation} p \sim \frac{1}{c_{\infty} r_d} \frac{\mathrm{d} F}{\mathrm{d} \tau} \sim \frac{1}{c_{\infty}} \frac{R}{r_d} \rho_{\infty} U^2_{{c}} v_n = \gamma p_{\infty} \frac{R}{r_d} \frac{U^2_{{c}}}{c^2_{\infty}} \frac{v_n}{c_{\infty}} \end{equation}

noting that $F \sim LR$ (assuming the blade span $\sim R$), ${\mathrm {d}}/{\mathrm {d} \tau } \sim U_{{c}}/C$ and $c^2_{\infty } = \gamma p_{\infty }/\rho _{\infty }$. In (6.4) the convection velocity $U_{{c}} \approx \sqrt {U^2_{\infty } + (\varOmega r)^2}$, and the upwash velocity is dominated by turbulent fluctuations in the wake, $v_n \sim U_{\infty }$. Consequently, $p \sim (R/r_d) \gamma p_{\infty } M_{\infty } M^2_{{c}}$, where $M_{{c}} = U_{{c}}/c_{\infty }$. The acoustic pressure spectrum $\phi _{pp} \sim p^2/f$ with $f \sim U_{\infty }/R$ or $f_{{BP}}$, and therefore should scale with $M^2_{\infty } M^4_{{c}}$.

In figure 18 the rotor acoustic pressure spectra for $J = 1.44$ and 1.05 are compared at microphone position 1 with different Mach number and frequency scalings. The Mach number $M_{{c}}$ is evaluated based on the convection velocity at the blade tip, $U_{{c}} = \sqrt {U^2_{\infty } + (\varOmega R_{{tip}})^2}$, since noise emission is dominated by the outer region of the rotor. It can be seen that unlike the $M^6_{\infty }$ scaling (figure 18a,c), the $M^2_{\infty } M^4_{{c}}$ scaling (figure 18b,d) successfully collapses the spectral curves for the two advance ratios in the frequency range between the BPF and the trailing-edge vortex shedding frequency, in which the Sears theory is accurate. The frequency scaling ($R/U_{\infty }$ vs. $f_{{BP}}$) also has an effect on the spectral level but it is less significant than the effect of Mach number scaling.

Figure 18.

Sound pressure spectra for the zero thrust and thrusting cases at microphone position 1 with different Mach number and frequency scalings. $J = 1.44$ (, red); $J = 1.05$ (, black). (a) $M_{\infty }^6$ versus $U_{\infty }/R$ scaling; (b) $M_{\infty }^2 M_{{c}}^4$ versus $U_{\infty }/R$ scaling; (c) $M_{\infty }^6$ versus $f_{{BP}}$ scaling; (d) $M_{\infty }^2 M_{{c}}^4$ versus $f_{{BP}}$ scaling.

Figure 19 shows the $M^2_{\infty } M^4_{{c}}$ scaled sound pressure spectra from three of the five blade segments shown in figure 14 for both rotor advance ratios. The local convection velocity, $U_{{c}} = \sqrt {U^2_{\infty } + (\varOmega r)^2}$ where $r$ is evaluated in the middle of the blade segment, is used to calculate $M_{{c}}$. At both observer locations illustrated in figure 19(a,b), there is a frequency range of approximately one decade ($0.8\text {--}8f_{{BP}}$ at $\theta _0 = 30^{\circ }$ and $1\text {--}10f_{{BP}}$ at $\theta _0 = 75^{\circ }$) within which all spectral curves for different blade regions (from the root to the tip) and different advance ratios show a reasonable collapse, in contrast to the spectral curves with $M^6_{\infty }$ scaling shown in figure 14. The collapse of the BPF peaks and broadband spectra in this frequency range also verifies that they are produced by turbulence ingestion and not by self-noise sources such as tip vortices and trailing-edge vortex shedding. The $M^2_{\infty } M^4_{{c}}$ scaling is invalid at high frequencies where the spectra are dominated by blade self-noise, and at low frequencies where the Sears theory prediction is inconsistent with simulation results.

Figure 19.

Sound pressure spectra with $M^2_{\infty } M^4_{{c}}$ scaling of the noise emitted from different blade segments for both $J = 1.44$ and 1.05, evaluated at $z = 0$, $r_o/R = 200$ and two observer angles. Segment 1, $J = 1.44$ (, black); segment 3, $J = 1.44$ (, black); segment 5, $J = 1.44$ (, black); segment 1, $J = 1.05$ (, red); segment 3, $J = 1.05$ (, red); segment 5, $J = 1.05$ (, red). (a) $\theta _o = 30^{\circ }$; (b) $\theta _o=75^{\circ }$.

It should be mentioned that although in figures 18 and 19 the $M^2_{\infty } M^4_{{c}}$ scaling of rotor turbulence-ingestion noise is only verified for the $J = 1.44$ and 1.05 cases based on numerical simulation data, the VT experimental data (Alexander et al. Reference Alexander, Molinaro, Hickling, Murray, Devenport and Glegg2016, private communication 2016) contain a wider range of advance ratios from $J = 0.58$ to 1.44. An examination of the measured acoustic spectra indicates that this Mach number scaling is valid for rotor turbulence-ingestion noise at all the advance ratios tested in the experiment.