On the differences in noise predictions based on solid and permeable surface Ffowcs Williams

Abstract

The Ffowcs Williams–Hawkings equation is widely used in computational aeroacoustics to post-process unsteady simulations and provide the sound at distances beyond the accurate range of the grid. It distinguishes the monopole/dipole contributions from solid surfaces Σ and that from quadrupoles present in the volume of the fluid. Curle showed that at low Mach numbers, the solid-surface terms are stronger than the volume term. Very few studies have included the volume term itself, but many have used a permeable Ffowcs Williams–Hawkings surface Σ, which in principle surrounds the quadrupoles, giving valid results independent of Mach number; in reality for external flows, the surface cannot surround all the quadrupoles. However, in spite of doubts over the solid-surface approach even at low Mach numbers, it is widely used because of its simplicity and the difficulties associated with turbulence crossing the permeable surface. We consider Mach numbers M up to 0.25, which challenges approximations based on the property that “M ≪1.” An additional attraction of the solid-surface approach is the idea of identifying the “true” source of the sound by computing separately the Ffowcs Williams–Hawkings integrals for different components. We wish to determine whether this “self-evident” argument gives an effective approach, and in general to assess Curle’s approximations, using a sphere-dipole problem and three model problems related to landing-gear noise, namely an isolated rectangular body, a fuselage with a cavity, and one with a bluff body under it. One key test is the shielding of sound toward various directions; any approach that misses this shielding is suspect. The overall conclusion is, again, that Curle’s approximations succeed in some cases but are quite inaccurate in the audible range for an airliner in approach and for high-speed trains, and also that separating the components’ contributions is often misleading. In contrast we verify that, for our model problems and with adequate grid resolution and surface placement, the permeable-surface results are as accurate as the simulation itself is, irrespective of distance, direction, or Mach number.

Keywords

Airframe noise computational aeroacoustics Ffowcs Williams–Hawkings method solid and permeable control surfaces Curle’s approach

Introduction

Overview and Ffowcs Williams–Hawkings (FWH) formula

Our purpose is an improved understanding based on quantitative results of the issues covered in a primarily theoretical paper by one of the present authors on Acoustic Analogies,¹ hereafter JSV13 (with equation numbers such as JSV13-6). Its motivation was to clarify the community’s thinking and reporting for aerodynamic noise at low but industrially relevant Mach numbers such as 0.25, and an outcome was a challenge to some of Curle’s arguments,² which very much guide the interpretation of many results in this field. For instance, Curle’s statement that “quadrupole sound” scales with M⁸ while “dipole noise” scales with M⁶ was shown to be incorrect (there is a cross-term of order M⁷ in JSV13-16, and thus the approximation is of lower order than claimed, namely M instead of M², which is a potent difference when M = 0.25). Yet, in the literature, the attraction of the M⁶ law is powerful. A practical consideration when conducting simulations, also related to Curle’s theories, is the choice between solid and permeable Ffowcs Williams–Hawkings surfaces,³ which we discussed at length in JSV13, in Spalart et al.,⁴ and other papers. However, no clear guidelines on this choice are yet available in the literature.

Below we present a literature survey starting in 2013 and weight it toward 2017 and 2018, much of it based on the LAGOON model,⁵ hoping this sums up the state of the art.

For reference, we present the FWH formula as we generally use it, in its far-field approximation, although a near-field version will also be used

\begin{array}{l} 4 π | x | (1 - M_{r}) p^{'} (x, t) = \frac{x_{j} x_{l}}{{| x |}^{2} c_{0}^{2}} \frac{\partial^{2}}{\partial t^{2}} \int_{V} {T_{jl}}_{ret} d V \\ + \frac{x_{j}}{| x | c_{0}} \frac{\partial}{\partial t} \int_{Σ} {p^{'} n_{j} + ρ u_{j} (u_{n} - U_{n})}_{ret} d Σ \\ + \frac{\partial}{\partial t} \int_{Σ} {ρ_{0} u_{n} + ρ^{'} (u_{n} - U_{n})}_{ret} d Σ \end{array}

(1)

Here, $x$ is the vector to the observer, where the pressure fluctuation $p^{'}$ is desired, and $M_{r}$ the relative Mach number between observer and body, to which the solid or permeable surface is attached. c₀ is the speed of sound in the far-field, T the Lighthill stress tensor, ρ the density, u and U the velocity vectors of the fluid and the body, respectively. Primes denote fluctuations, and {}_ret that a quantity is taken at the retarded time, earlier than the reception time by the time sound takes to travel at c₀ from the emitter to the observer. The principal option is between a surface Σ that is on the body, and one that is permeable and in principle surrounds all the turbulence, so that T is negligible outside Σ, and the first term can be neglected without approximation.

Curle² correctly argued that, for aerodynamic sound at low Mach numbers, the volume term, the first one on the right-hand-side of equation (1), is of order M⁴, whereas the other two if evaluated on the solid surface are of order M³. He failed to recognize the cross-term of order M⁷ which appears when forming the intensity ${p^{'}}^{2}$ , having given Lighthill’s finding of M⁸ scaling for quadrupoles in free space too much weight (JSV13-16). To our knowledge, this is the only part of his work that contains an error.

Permeable surfaces and end-caps

A known issue with permeable Σ surfaces is that, for external flows, the turbulence and therefore the quadrupoles in the wake of the body inevitably cross the surface, thus negating the assumption that Σ “surrounds all the turbulence” so that the volume integral can be omitted with rigor. To combat this error, the simulation grid can be extended, but not with a resolution fine enough to sustain the turbulent eddies to the same standards as in the near-field; this has been called the “Departure Region,” as opposed to the “Focus Region.”⁶ The simulation gradually removes the small-scale kinetic energy (faster than physical dissipation does, unless the Reynolds number is low), either via a subgrid-scale model, or numerical dissipation brought about by upwind-differencing or filtering. In either case, these are substantial viscous-like effects which to be rigorous would be added to the quadrupole tensor; this is far from simple and has not been done. We know of no rigorous estimate of how close to the body it is tolerable to initiate “departure-region errors.” The danger of stretching the Σ surface far downstream is of calculating “the accurate integral for turbulence that has become inaccurate,” due to numerical dissipation.

Somehow, the spurious sound caused by this “hydrodynamic” effect is often stronger for upstream observers and at low frequencies. Quite a few remedies have been proposed, which allow weakening the spurious sound. A common one is to omit the integral on the downstream “end-cap,” i.e. to use “open” control surfaces. Another is to extend the surface so far that the turbulence has essentially decayed, a decay possibly forced by the inevitable coarsening of the grid as explained above. Along with this, some approximate corrections based on “frozen turbulence” being convected through the end-cap have been tried with some successes, but we find the necessary assumptions crude, considering that the turbulence is certainly not frozen, especially for static jets.

Lockard and Casper⁷ clearly based their correction on a “frozen gust” assumption. They report success for a 2D vortex, but uncertain findings for fully 3D unsteady RANS and detached-eddy simulations. We have not found extensions of this work, but it was influential.

Rahier et al.⁸ in order to create an end-cap correction appear to invoke the absence of a noise source through a surface $S^{'}$ moving with the fluid velocity, which is not clear to us physically. Results for a jet without the correction show an over-prediction relative to experiment at low frequencies for open and simply closed Σ surfaces. The corrected formula has an under-estimation, but a milder one, and is less dependent on the surface length which was encouraging. In this and other studies, agreement with experiment is not the final proof of the accuracy of the FWH component of the complete noise-prediction system, because separate sources of error can partly compensate each other.

The proposal by Ikeda et al.⁹ is similar, again invoking frozen turbulence and a hypothetical FWH surface which would travel with the mean-flow velocity. They also had partial success, reducing the sensitivity to surface length, but the agreement with experiment depended on direction and frequency. They tested in addition open surfaces which failed at low frequencies (for a jet) and multiple end-caps which had little effect. Overall, the semi-analytical end-cap corrections do not appear fully understood, as of 2019.

An idea which may have more basis in the equations is to use multiple permeable surfaces with different end-caps, and to average the sound calculated from each of them (in the time-domain, or the complex signal in the Fourier domain).¹⁰ The motivation is that the retarded times in equation (1) combine the contributions to the integral that are “traveling” toward the observer at a velocity c₀. The hydrodynamic content that injects spurious contributions into the second and third terms of equation (1) travel essentially at the fluid velocity, so that the contribution of a large eddy, for instance, will approximately cancel thanks to phase differences over the multiple end-caps, if the volume they define is larger than the volume of the eddy. With M < 0.25, the disparity of velocities is substantial, even for an observer in the downstream direction. This approach is by no means exact, but it is considered effective and used by a significant part of the community.^11,12 Some authors regret the additional cost and decision load (number and location of the caps), but both are moderate, and any approach that allows a shorter domain for the simulation (or more to the point a shorter focus region⁶) will probably be quite advantageous in terms of overall cost. Recall that the placement of the lateral boundaries is itself delicate, with the need to avoid either contact with turbulence, or excessive grid coarsening.

The elaborate efforts to improve the permeable-surface approach illustrate how it is in theory the correct one, but applying it flawlessly is not possible for practical flows. We also note that several careful studies obtained satisfactory results from the solid-surface terms for such practical flows; in some cases, permeable-surface results came out close also. These include work of Vatsa et al.,¹³ Casalino et al.,¹⁴ and Khorrami et al.¹⁵ Overall, in the LAGOON summary of Manoha and Caruelle,⁵ the permeable-surface results were often disappointing. It is possible the permeable approach was partly misused in some cases, but we cannot presume this was the case, and we must consider the solid-permeable question to be controversial, before and after the present paper appears.

Mach-number scaling

Turning to the Mach-number scaling, a fair number of papers were discussed in JSV13, many of which arrived at exponents well below 6 and therefore in conflict with Curle’s or any other theory. A more recent example of the “attraction” mentioned is in the interesting experimental paper of Rego et al.¹⁶ for a simplified landing gear. In Figure 17(b) of their paper, they present the Mach-number exponent as a function of frequency; they are comparing spectra at fixed frequency (called “Strouhal number based on c”) rather than fixed Strouhal number (based on flow velocity U₀). They appear to detect for ${p^{'}}^{2}$ a power close to 6 at low frequencies and to 8 at high frequencies, which fits the theory of dipoles and quadrupoles, and give an empirical fit which rises from 6.22 to 8. However, at low frequencies, their measured exponent is well below 6, and if visually averaged through the fluctuations it is not far from the 4 which a monopole would justify. Their very brief discussion gives no reason for this mismatch or for the different exponents in different directions (sideline or flyover). The OASPL in Figure 18 of Rego et al.¹⁶ closely follows M⁶.

The paper by Reger and Cattafesta¹⁷ on measurements over the rudimentary landing gear (RLG) is ambitious as it covers a wide range of Mach numbers from 0.09 to 0.18, and configurations with and without a plate representing the body of the airplane. It follows JSV13 for analysis in terms of powers of M. Some results strongly support M⁶ scaling, as predicted by Curle. In fact, this seems to extend to values of M×St close to 1, with the Strouhal number St based on wheel diameter, so that Curle’s compactness approximations are not well justified. The data have a curious step between 2000 and 2500 Hz, which is mentioned only in passing in the paper, and fairly predictable behavior on either side of it. With the plate in place, the results at different M do not collapse, which is explained by the reflection; a Strouhal number based on the distance between the wheel and its mirror image is justified when seeking a compactness argument, and is almost 4 times as large as the St based on wheel diameter. The results without the plate have even better collapse when plotted versus frequency and with M⁷ scaling. If both this scaling and that versus St and M⁶ apply, the spectrum must be proportional to 1/St². The figures satisfy this property quite well, again separately on both sides of the step (and of course not toward zero frequency, because the integral of 1/St² would be infinite). We do not know of a theoretical reason for a 1/St² variation over two octaves and more; note that the RLG is free of small components, being composed of smooth wheels and posts of similar size but with rectangular cross-sections. Therefore, the eddies produced by the flow separation are of a rather homogeneous size, and the turbulent energy cascade proceeds from them. Such a 1/St² behavior shifts the bulk of the sound intensity to low frequencies, where the simple M scaling stops working at fixed frequency; as a result the M scaling of the overall sound level is probably much closer to M⁶ than M⁷. The fixed-frequency figures in Reger and Cattafesta¹⁷ also illustrate how the interference effects due to the plate are set by the time to travel from body to image at the speed of sound. Finally, there is no indication of M⁸ scaling, even at high frequencies.

The solid FWH formula is often justified by it giving good agreement with experiment. This is the case for Ricciardi et al.¹⁸ who use it partly because of the (questionable) M⁸ argument, partly because of the permeable difficulties downstream, and partly because of better previous results. Curiously, here and in other papers the results for far-field noise on LAGOON appear significantly better than the wall pressures, which were the source of data for the noise; error cancellation must be taking place in the integrals. Many of the results in Figure 28 of Ricciardi et al.’s paper are excellent, especially at high frequencies, even though the Mach number was 0.23. There were open questions about the application of FWH in an open wind tunnel.

Hajczak et al.¹⁹ specifically addressed the issue of solid and permeable surfaces (possibly open at the end), using a single wheel body and M = 0.23. Unfortunately, they did not have a reference result to compare with (something we believe we achieved in the present work). After seeing agreement between solid results and permeable results when that surface was resolved properly, they concluded in favor of the simpler solid formulation.

A recent full-scale simulation is that of a Gulfstream full aircraft with main landing gear down by Appelbaum et al.²⁰ This computationally massive and thorough study used the PowerFLOW lattice-Boltzmann approach, with up to 17 billion voxels and a finest spacing in contact with the surface of 0.32 mm; five million time steps were taken. The volume of the permeable FWH surface is of the order of 100 m³ (based on Figure 4 and assuming symmetry), providing in the average on the order of 2 × 10⁸ voxels per m³, leading to a typical voxel size under 2 mm, which we assume was maintained up to the permeable surface. This appears to resolve the audible range, assuming on the order of seven points per wavelength; they present results up to 5 kHz, a wavelength of 7 cm. This estimate however ignores the near-body refinement, and in fact Figure 3 suggests a voxel size at the permeable surface closer to 12 mm; this may not be a figure of the finest grid (which was 2.25 times finer than the coarse grid). The flow visualization in Figure 8 is consistent with a voxel size of a few millimeters, with typical LES aspect for the smallest eddies. In the FWH results, they found the permeable surface to be noticeably more accurate when comparing with the sound extracted from the simulation, while surmising that the sound in the simulation suffered from “departure effects” damping the sound at higher frequencies, even though they extended a fine-grid region to the observer. This paper did not provide comparisons with measurements. Those in the paper of Khorrami et al.,¹² on the same aircraft and from essentially the same team, were ambiguous in that the raw spectra in Figure 12 did not agree, but spectra after beam-forming did. This latter paper was also ambivalent between solid and permeable surfaces.

Yu and Lele²¹ extensively studied the quadrupole issue, in particular introducing the concept of “true physical quadrupole noise,” and actually calculating volume integrals. They also used permeable surfaces. The flow was over an airfoil, with a circular rod nearby causing vortex shedding over an already turbulent boundary layer, itself resolved by LES (unlike those in our fuselage simulations below). They found that the solid-surface formula was not accurate enough for M = 0.3, and still far from perfect at M = 0.1; the relative error certainly did not come down by a factor of M² (the peak pressure fluctuation $p^{'}$ changed by a factor of about 67, which is slightly less than M⁴, therefore definitely higher than Curle’s approach would predict, but the directivity changed too much to expect a simple exponent). They established that volume sources in equation (1) are important (if Σ is solid), but also that results with permeable surfaces were inaccurate; they did not try corrections, open surfaces, or multiple end-caps, and recognized that volume integrals are impractical outside relatively simple model problems.

To summarize, the measured Mach-number scaling of the sound can be far from Curle’s prediction, which many authors fail to recognize; solid FWH surfaces are often preferred but primarily for convenience; the use of permeable FWH surfaces while it is better justified is delicate and exposed to numerical errors and other little-known effects in the simulation itself.

Partial surface integrations

As mentioned above, the idea of partial solid FWH (PSFWH) integrals has currency in some groups, and would appear to have much potential for technological purposes, if it allows one to identify which parts of a system, especially one as complex as landing gear and its cavity, are to blame for more of the noise.^13,19,22 This would compare well with the beam-forming technique, which from measured sound identifies “hot spots” using the phase relationship detected by a phased array of microphones. It is not always very precise in locating the sources. The two concepts have rather similar purposes, and both can be applied over selected frequency ranges. Naturally, sound can be “made” by one component, but due to turbulence generated by another component impinging on it, so that engineering judgment is needed. In the present paper, the PSFWH idea is tested.

We suspect that, for instance if the landing-gear cavity is treated as a partial solid surface, the formula could fail to reflect the key fact that the pressure fluctuations derive from turbulence active inside the cavity, as opposed to turbulence outside it (i.e. we imagine that the inside of the cuboid is the solid body, and the flow surrounds the cuboid, and argue that the same pressure fluctuations would be interpreted as radiating sound upwards). If so, there could be little difference between the sound predicted upwards and downwards. In fact, if the cavity ceiling is considered by itself, the argument is exact since only the $p^{'}$ term subsists in the solid terms of equation (1).

De Azevedo and Wolf²³ in their extensive landing-gear study separated not only the contributions of various parts of the solid surface, but also parts of the permeable surfaces, as well as many of the terms in equation (1). The permeable end-cap was dominant, which was a serious concern. When its contribution was removed, the predictions went from too high to much too low, compared with solid-surface calculations and with experiment. The authors left a solution of this problem for future work, possibly helped by multiple end-caps, but viewed the solid-surface approach as satisfactory.

Giret²² explicitly applied the PSFWH approach to a two-wheel LAGOON landing gear. He did not use any permeable surfaces. A remarkable observation was the strong cancellation between apparent contributions from the left and right wheel. Giret concluded that they together formed a cavity, so that separating their contributions was artificial. These results were not in an earlier paper by Giret et al.,²⁴ which focused on grid-refinement studies.

Numerical simulation approach

All the computations were performed with the use of the in-house general-purpose code NTS²⁵ which is capable of simulating turbulent flows in the framework of a full range of turbulence-modeling and turbulence-resolving approaches. This is a structured finite-volume high-order CFD/CAA code accepting multi-block overlapping grids of Chimera type. It has been validated on numerous aerodynamic (in the framework of DNS and hybrid RANS-LES approaches) and aeroacoustic problems and shown to be rather reliable and accurate (see, e.g. Shur et al.,^26–28 Spalart et al.,²⁹ Suzuki et al.³⁰). For numerical solutions of the compressible Navier–Stokes equations, the code employs an implicit flux-difference splitting method of Roe type.³¹ For the approximation of the viscous fluxes in all the governing equations the second-order centered scheme is used, while the inviscid fluxes are approximated with the use of the weighted fifth-order upwind-biased/fourth-order centered scheme with a solution-dependent weight function.³² In the considered flows computed with the use of a hybrid RANS-LES approach delayed detached-eddy simulation (DDES³³) this resulted in a low-dissipation scheme with the weight of the upwind part $σ_{upwind}$ =0.5 in the inviscid and RANS regions and $σ_{upwind} \approx$ 0.1 in the LES region. For the time integration, the code employs a three-layer backward second-order scheme with dual time-stepping and sub-iterations in pseudo time with “infinite” pseudo-time step. The set of discrete equations for the iterations is solved with the use of Gauss–Seidel relaxation by planes.

Calculations of the FWH integrals were performed with two different acoustic post-processors. Namely, the far-field noise was computed with a cost-efficient method^10,34 implemented in Fourier space, whereas the FWH-based computations of the acoustic pressure in the near-field were performed with a general FWH utility operating in the time-domain,³⁵ which was generously provided by Drs A. Duben and T. Kozubskaya of Keldysh Institute of Applied Mathematics (Moscow). It was verified that in the far-field the two utilities give virtually identical results.

Overview of the study

The first problem we consider is inviscid and without flow, making the modeling choices simple. In contrast, the other problems, as already mentioned, are treated with DDES.³³ The reason not to use the preferable direct numerical simulation (DNS), besides a high Reynolds number, is that the problems have a large fuselage over which the flow is attached and quasi-steady, while the flow is highly unsteady in the region of interest. This is similar to the practical problem of the most interest, namely landing gear, both for the external components and cavities. In addition, the bodies here impose separation at sharp edges, which lessens the impact of the boundary-layer turbulence, or the Reynolds number for that matter. Motivation came from results, not presented in Ricciardi et al.,¹⁸ by the team of University of Campinas and Boeing, which had the symptom mentioned above, namely that the solid-surface sound predictions failed to have the expected directivity; this is discussed in detail below.

Another obstacle to DNS, mentioned above is the impracticality of applying DNS resolution over enough of the domain. Conceivably, the entire focus region would have DNS resolution, and contain the permeable FWH surface Σ, but the cost of this would be very high. Admittedly, in the present study and in industrial ones, the resolution in various regions is assessed mostly visually; grid-refinement studies have been conducted only in isolated instances, one of them here.

The paper proceeds with the four model problems: the dipole-sphere in still air, isolated bluff body, fuselage-cavity, and fuselage-bluff-body, followed by conclusions. We believe the questions we pose in the present paper remain delicate and will not have a simple answer, but the scientific and industrial motivation to properly process simulation flow fields is very high. New attempts at a reliable solution will be made in the near-future.

Dipole placed under a sphere

Problem statement and sound field

Our first problem does not involve quadrupoles, but leads to the identification of sound sources, and shielding. As shown in Figure 1(a), a dipole is placed under a sphere. This is a generic model for an aircraft component such as a wheel applying an unsteady force to the fluid. The grid used has 200, 220, and 204 cells in the radial, azimuthal, and polar directions, respectively, and is fine enough for the simulation results to be the reference at all the observer positions considered; the departure region begins beyond those. This contrasts with the other problems, which cannot avoid the uncertainties due to modeling of small turbulent eddies, or to departure effects.

Figure 1.

Dipole placed under sphere. (a) Geometry and (b) pressure field. Coordinates are normalized with the wavelength λ = R/3.

The “straightforward” identification approach we called PSFWH would calculate the solid-FWH sound field, and declare that “this is the sound of this part of the airplane.” Here, the sound field of the dipole is known analytically. The dipole is injected into the flow field via source terms $f$ (in the momentum equation) and $V \cdot f$ (in the energy equation in conservation form). The spatial dependence of the source term cannot be a point, for numerical reasons, and has the following form, which is similar to the form of the monopole source term proposed in Spalart et al.³⁶

f = A_{f} \frac{Ψ (r)}{V_{f}} φ (t)

Here $A_{f}$ is the amplitude, $φ (t) = \sin (2 π f t)$ gives the harmonic time-dependence, and $V_{f} = \int \int \int Ψ dx dy dz$ is the effective source volume for the spatial distribution $Ψ (r)$ . The latter has the bell shape $Ψ (r) = \cos^{2} [(π / 2) \min {| r - r_{0} | / Δ r_{0}, 1}]$ , where the size $Δ r_{0}$ should be chosen large enough to be well-resolved by the grid and sufficiently small to ensure the source compactness. In the present work it was set $0.12 λ$ , where $λ = c_{0} / f$ is the wavelength. Note that with such definition of the source, the corresponding point source in the wave equation for the acoustic pressure is $- \nabla \cdot F δ (r - r_{0})$ with $F = A_{f} φ (t)$ , independently of the chosen value of $Δ r_{0}$ .

Below we only show cases with a vertical dipole (F_y in the figure), but results with a horizontal dipole led to the same conclusions.

There is no flow, and the walls are treated as free-slip surfaces. The parameters are as follows: the wavelength is 1/3 times the sphere radius R, and the dipole is 1/2 a wavelength away from the surface. This creates interference patterns with approximately 45° for the dominant direction, as seen in Figure 1(b); other cases, with different offsets for the dipole, produced different patterns. As could be expected, the shielding by the sphere in the upward direction is very definite.

Results

We now turn to quantitative results, comparing the sound at a distance of 11 wavelengths λ from the origin, i.e. in the near-field, which allows comparisons with the direct output of the simulation, where the grid is still fine. The same conclusions would be reached for far-field sound, but the simulation cannot provide that, since the grid coarsens for large r and a sponge layer was introduced. In Figure 2(a) the simulation result, solid line, confirms the interference pattern and the shielding seen in Figure 1(b). The peak level at ±45° downwards is 30 dB higher than the lowest level, close to the straight-up direction. This result is accurately reproduced by the permeable FWH formula, independently of the radius of the FWH sphere (dotted and dash-dot curves). The sound field of the isolated dipole itself (long-dashed), in contrast, does not reproduce shielding; the only difference between the up and down direction is due to the difference in distance (14.5λ versus 7.5λ). It also misses the interference patterns. The sound field of the dipole with an opposite image dipole inside the sphere (not shown), which would be appropriate for a plane wall, does capture interference, to leading order, but it misses the upwards shielding and in any case does not provide a very practical method, with non-trivial geometries.

Figure 2.

Sound extracted from simulation, and sound produced by integral formulas (a, b), and FWH surface term on the sphere (c). In frames (a, b), observers are located at $x = 0 and \sqrt{y^{2} + z^{2}} = 11 λ$ . OASPL: overall sound pressure level.

Of more interest are the results of the sphere’s solid-surface contribution, dashed curve in Figure 2(b), and its combination with the dipole, dash-dot-dot. In the upwards direction, the two almost cancel. This combination gives a very accurate answer, almost as good as the permeable formula. On a theoretical basis, this may be unsurprising, since this problem has no quadrupoles. However, in practice, this shows that while the dipole is the true source of sound, the calculation of the sound field must include the sphere’s contribution, which in aeronautical practice would mean the airplane’s fuselage and wing. Figure 2(c) shows the FWH integrand on the sphere, in other words the “footprint” of the dipole. Its extent appears to be a few times the distance from the dipole to the surface. This will be a guide in practice when designing FWH surfaces, solid or permeable, and even grids, that is here the upper hemisphere contributes very little.

Isolated bluff body over a range of Mach numbers

Problem statement and flow field

The body is a right parallelepiped, with unit height h, width 1.5h, and length 3h. The Reynolds number based on h is 10⁵, and four Mach numbers from 0.0625 to 0.5 were considered. The flow is treated by DDES; the grid is seen in Figure 3 and has 340 by 245 by 245 cells in the (x, y, z) directions, and the time step is 5 × 10⁻³h/U. The grid is clustered at all the walls, with the near-wall step equal to 10⁻⁴h, which corresponds to not more than 1.5 in the law of wall units, except for the vicinity of the forward stagnation point. The lateral steps in the “wakes” of the refined grid near the walls are equal to 5 × 10⁻³h. Away from the wall, the wall-normal grid steps increase with the ratio of 1.3, until reaching Δ = 5 × 10⁻²h in all the three spatial directions, and then is kept constant up to the sides and the front face of the control FWH surface. The streamwise grid step at the closest end-cap is twice larger, 0.1h. At the highest considered Mach number M = 0.5, this grid ensures accurate resolution of the sound waves propagation inside the FWH surface up to Strouhal number based on the body height around 6.

Figure 3.

Fragment of grid, FWH surfaces, and vorticity contours.

The duration of the simulations was ≈80 time units, based on U and h, the last 40 of which were used for the FWH-based noise computations. Naturally, this leaves noticeable statistical noise in the spectra especially for Strouhal numbers of O(1), and scientific judgment is essential when interpreting the results. A decisive improvement in this respect would require orders of magnitude more computing effort. The figure also displays the FWH surfaces, vorticity contours, and the position of two near-field observers (white circles). The separation at the front corner naturally pours turbulence along the lateral faces and into the wake; the range of scales is rather wide. The usual gradual grid coarsening in the departure region is visible, and the eddy size adapts to the grid. Organized vortex shedding is not present due to the three-dimensionality of the shape.

Figure 4 displays the overall sound power level spectra, in raw form and then rescaled from different Mach numbers to M = 0.0625 using Curle’s M⁶ law. The power level is calculated relative to the reference power W_ref = 10⁻¹²W (PWL( f ) = 10log₁₀(W( f )/W_ref)), for the upward direction in the xy-plane (as if the acoustic pressure were independent of the azimuthal angle $φ$ ). To obtain W( f ), the integration of the surface-normal component of the mean sound intensity is performed in the far-field over a wave-front, which is a surface of a constant emission distance R_e, i.e. the sphere of radius R_e with the center shifted downstream of the origin by the distance R_eM, with the use of the following formula

W (f) = 2 π ​ R_{e} \int_{θ_{e} = 0}^{π} \frac{PSD (f, θ_{e})}{ρ_{0} c_{0}} {(1 - M \cos θ_{e})}^{2} \sin θ_{e} d θ_{e}

where

PSD (f, θ_{e})

is the power spectral density of the acoustic pressure and

θ_{e}

is the polar emission angle.

Figure 4.

Sound power level: (a) raw results, and (b) and (c) corrected using M⁶ scaling law. PWL: power level.

The spectra in Figure 4 reflect the broad-band turbulence. Curle’s theory is here quite successful, in two respects. First, the M⁶ law renders the Mach-number effect well, although the statistical noise on the spectra is too large for precise conclusions. Second, the solid-surface result is quite close to the permeable result, at least up to Mach 0.25 and more so at low frequencies, which is highly consistent with the key role of the product M×St. Still, the agreement appears to apply well outside the compact-body range, since M×St is well above 1. At M = 0.5, the permeable result is consistently higher, indicating a positive correlation between the dipole and quadrupole contributions. The apparent Mach-number power is between 6 and 7. Note that comparisons between FWH approaches in the same simulation are less hampered by statistical noise than comparisons between separate runs at different Mach numbers. We recognize that the “true” spectra would be significantly smoother, and judgment is needed for simulations of manageable cost when dealing with massive separation.

Figure 5 qualifies this finding about Curle’s theory, by examining results in specific observer directions rather than the sound power level. The far-field results at an angle of 90° repeat the findings of Figure 4, but those at 0° (directly upstream) do not: solid results are up to 10 dB lower than permeable ones. The latter results come from averaging the three end-caps, which we consider the most reliable approach. However, results from single surfaces were very close to the averaged results, suggesting that there is little contamination by end-cap effects. A factor that is more difficult to rule out is the effect of coarsening grid on the eddies, mentioned in the “Introduction” section, but this coarsening is most visible precisely between the three end-caps, so that it is reasonable to expect that it would force a difference between the results from the different end-caps. An end-cap at x = 4 would move outside the major coarsening region, but also begin to eliminate part of the focus region, and therefore well-resolved eddies and quadrupoles.

Figure 5.

Spectra in different directions ( $θ_{e}$ is the polar emission angle). Upper graphs, far-field formula over range of Mach numbers reduced to emission distance 1.0h. Lower graphs, comparison with simulation in near-field (white circles in Figure 3), M = 0.25. FWH: Ffowcs Williams–Hawkings; SPL: sound pressure level.

The lower frames of Figure 5 finally present direct comparisons between the two FWH predictions and the true sound of the simulation in the near field at M = 0.25; this is the “litmus test.” These show that the permeable-surface FWH formula reproduces the near-field sound in the simulation quite accurately for both considered near-field observers. Other than that, the near-field results are consistent with the far-field ones in the sense that at θ = 90° the solid and permeable FWH results are very close and at θ = 0° the noise from the solid surface is tangibly lower than from the permeable one. However, the difference between the solid and permeable predictions at θ = 0° in the near field is larger at the high frequencies, whereas in the far-field (top right frame in Figure 5) it is larger at the low frequencies. This reflects a significant input of high powers of 1/r in the near-field terms, which are absent in equation (1).

Overall, the results for a single bluff body producing massive separation and broad-band turbulence although free of a wide range of scales in the geometry tend to support Curle’s theory, even at Mach numbers which are not very small. There are however mild qualifications, particularly about the solid formula in the upstream direction. We now turn to more complex geometries.

Generic fuselage with a simplified landing-gear cavity

Problem statement and flow field

Next, a generic fuselage with a simplified landing-gear cavity is considered, as illustrated in Figure 6.

Figure 6.

Simplified landing-gear cavity. Top left, geometry; top right, vorticity contours; middle, pressure field in XY and YZ planes (time derivative in the acoustic range normalized with $ρ_{0} c_{0}^{3} / D$ ); bottom, permeable surfaces and location of observers in the near field.

It has some key ingredients of the industrial problem, while being far less costly to compute, and separating cavity effects from bluff-body effects. This flow having quadrupoles, eddies crossing the permeable surface, and a departure region, the results will not be as close to perfection as in the first case, but we expect strong effects of shielding, with the PSFWH approach separating various parts of the surface in the FWH integral possibly leading to paradoxes.

The Mach number is 0.25 and the Reynolds number based on fuselage diameter D is 10⁷. The cavity has length L_x = 0.4D, width L_z = 0.2D, and depth L_y = 0.2D. The flow is treated with DDES,³³ with most of the fuselage boundary layer in quasi-steady RANS mode. The grid used in the simulation (see Figure 7) has $N_{r} = 140$ , $N_{φ} = 200$ , and $N_{θ} = 530$ cells in the fuselage grid block, and $N_{x} \times N_{y} \times N_{z}$ = $80 \times 60 \times 80$ cells in the cavity grid block with strong clustering of the grid in the cavity shear layer (30–40 cells across the shear layer, depending on the streamwise position), resulting in ∼15 million cells total. The time-averaged pressure field has significant variations along the body, but the time derivative in Figure 6 is fully dominated by the sound generated at the cavity. The simulation produces the expected highly unsteady shear layer bounding the cavity, and weaker but fine-scale vorticity inside it. The evolution from the attached, fully modeled boundary layer to the shear layer with resolved turbulence is rapid enough, suggesting a satisfactory grid resolution. Turbulence then propagates along the body, being gradually damped by the sub-grid-scale model and grid coarsening. It is rather devoid of small eddies in the region where it crosses the permeable surface, but the acoustic content, having a much larger length scale, is very visible with no obvious signs of damping, at least near the dominant frequency (this could be different for high frequencies). Multiple end-caps were not considered. The pressure field is marked by waves with a wavelength slightly longer than D (somewhat controlled by grid spacing). This corresponds to a Strouhal number based on D of about 4.

Figure 7.

Some details of computational grid used for simplified landing-gear cavity flow.

Acoustic results

Figure 8 shows that unlike the solid-surface formula the permeable-surface FWH formula reproduces the near-field sound in the simulation very accurately, suggesting that the departure, end-cap, and similar effects are very moderate. This conclusion is however qualified when observing that the observer points are close to the permeable surface, so that the sound is primarily produced by local contributions to the integrals; this could conceal end-cap errors, for instance. The agreement would in a sense be “tautological.” Simulation results free of dissipation at larger distances are not expected with the grids we were able to run on. Vatsa et al.¹³ to overcome this limitation extended a funnel-shaped fine-grid region all the way to the microphones, at a distance of the order of 2D, and the permeable results were only slightly more accurate. This was in the downward direction, like Point 14 is; it would be most valuable for them to repeat the exercise in the upward direction, emulating Point 5. The present comparison, at least, strongly suggests that the FWH utility is free of errors. Appelbaum et al.²⁰ conducted a very similar test, already reporting the same success only for the permeable surface. Far-field permeable predictions are therefore also considered reliable. This includes the strong sheltering, since the sound level at Point 5, above the body, is about 25 dB lower than at Point 14, below it; this is very consistent with Figure 6. This result, observed at a distance of D/2 from the fuselage, may not apply for large distances. The peak is at a Strouhal number close to 3.5; compare with the rough visual estimate of 4. On the other hand, the spectrum shape for low frequencies up to St ∼ 5 is similar although by no means identical at those two points, which we have not explained yet. The plateau over St=(8, 15) for Point 14 is also difficult to predict.

Figure 8.

Spectra from simulation and from solid- and permeable-surface integrals at two points above and below fuselage (see Figure 6). FWH: Ffowcs Williams–Hawkings; SPL: sound pressure level.

Figure 9 presents another test of the near-field permeable utility of Abalakin et al.³⁵ In theory, the FWH formula returns zero for the pressure field inside the surface. This can be viewed as a “compatibility condition” for the distribution of all the terms on Σ in equation (1). The figure shows that the level drops by at least 26 dB when crossing the surface inwards, which is quite satisfactory and strongly suggests that the quadrupoles external to the permeable surface have a very small impact. It also suggests that the simulated flow field is free of strong “viscous-like” effects outside the surface, which were discussed in the “Introduction” section, and that the grid is fine enough near the surface for an accurate integration. The fact that the integrand contains wavenumbers twice as large as the flow field does presents a challenge to accurate numerical integration (this phenomenon depends on the exact formulation, and in equation (1) it is essentially driven by steep variations of the retarded time, for high frequencies).

Figure 9.

Distribution of near-field overall sound-pressure level along vertical axis (x = z = 0), as produced by permeable FWH integration. OASPL: overall sound pressure level.

Figure 10 presents far-field results of different approaches, which are not as simple as for the sphere-dipole case. Namely, it compares sound power spectra in the upwards and downwards direction computed with the use of solid FWH and permeable FWH approaches, the latter with two nested surfaces which give virtually identical spectra. The figure shows that according to permeable FWH the difference between directions is weaker than in Figure 8, around 10 dB in terms of the peak values, due to this applying in the far-field. However, this is still a large difference.

Figure 10.

Sound power levels in the vertical direction with different FWH approaches. Far-field results. Left: upwards; right: downwards. PWL: power level.

Other than that, the solid and permeable results in Figure 10 are almost identical up to a Strouhal number of 4, and in particular the peak at St ∼ 3.5 is identical. For St larger than 4, the two results diverge, and the solid-FWH calculations essentially miss the shielding effect, even though these are results with the full surface (both cavity walls and fuselage skin), an approach which was successful for the sphere-dipole case. This appears to reflect the quadrupole contribution, which would have constructive interference with the surface terms in one direction, and destructive interference in the other direction. The relative success of the solid formula for low frequencies (which would not be relevant in airline practice) followed by an abrupt divergence has not been explained yet; in particular, a simple argument such as the source being compact does not seem to apply since λ ∼ D. This is if the entire fuselage is viewed as the source, as opposed to the cavity, of depth D/5, not to mention the eddies in the shear layer, of smaller size yet.

Figure 11 gives a more detailed idea on the relationship between the sound levels predicted with the use of different approaches including the PSFWH with partial surfaces being the fuselage skin w/o the cavity and the cavity walls only. Strouhal numbers below 4, which are too tainted by statistical imperfections, were not included. In addition to the considerable difference between the solid and permeable results mentioned above, the figure strongly confirms our conjecture in the “Partial surface integrations” section about the apparent contribution of the cavity walls: it has almost no directivity in the transverse vertical plane, while its directivity in the longitudinal plane is almost the same upwards and downwards, but curiously includes a much higher level in the upstream direction. The directivity for the ceiling alone (not shown) exactly confirmed our conjecture in the “Partial surface integrations” section. The directivity of the permeable result, peaking at 50° downwards, is fully consistent with Figure 6.

Figure 11.

Far-field sound pressure level directivity curves for two symmetry planes of the configuration reduced to emission distance 1.0D. The SPL is computed by integration of spectra over the Strouhal number range St = [4, 20]. Upper graphs: XY-plane; lower graph: YZ-plane. SPL: sound pressure level.

The dominant frequency is of interest. If we assume that the large eddies in the mixing layer propagate at half the freestream velocity, St = 3.5 corresponds to an eddy spacing of D/7, which is much larger than the visual spacing in the simulation, based on Figure 6. The Strouhal number based on cavity length L is 1.4, which would approximately correspond with the third Rossiter mode (using St_L = (n −γ )/(M + 1/κ) with γ = 0.25 and κ = 0.57). Finally, the cavity length and depth are 0.4D and 0.2D, respectively, so that conjectures based on half- or quarter-wavelength modes are not satisfied either. In other words, various plausible estimates based on the geometry or the flow visualization fail to predict the observed frequency.

Overall, our results for the generic fuselage with a simplified landing-gear cavity configuration strongly support the permeable-surface FWH formula, and indicate that the solid-surface formula can be highly misleading regarding the actual sources of sound, and in particular fail to capture directional effects. This conclusion applies only for frequencies above the primary peak, but in a typical airliner, St = 3.5 would be below 50 Hz, and therefore of little interest; St ∼ 50 is far more relevant in flight.

Generic fuselage with a simple bluff body

The next problem aims at turbulence generated by massive separation in the vicinity of a large body. The shape is again simple, a right parallelepiped of thickness h, length 3h, and width 1.5h with h = 0.1D, free of disparate scales (unlike real landing gear), and with sharp corners which minimize the effects of Reynolds number and turbulence model. The turbulence will originate away from the fuselage, but still interact with it. The flow Mach number and the Reynolds number based on D are the same as in the fuselage-cavity problem considered in the previous section.

Figure 12 shows the geometry and the grid used in the bluff body block with size $N_{x} \times N_{y} \times N_{z}$ = $290 \times 135 \times 140$ . The grid in the fuselage block is similar to that for the fuselage-cavity configuration but has a wider region of fine radial grid aimed at better resolution of the flow past the bluff body so that the number of cells is $N_{r} = 215$ versus 140 resulting in around 23 million cells total. The permeable surfaces are very similar to those for the cavity problem. The separation at the front corner again naturally pours turbulence along the lateral faces and into the wake (see Figure 13).

Figure 12.

Geometry of the fuselage with bluff body configuration and grid in the bluff body block.

Figure 13.

Instantaneous vorticity contours.

Figure 14 shows that the pressure perturbations generated by the bluff body interact intimately with the fuselage, even though the turbulence itself almost does not make contact with the fuselage surface (based on the vorticity contours in Figure 13). The figure also makes the shielding in the upwards direction again very visible. This flow has a much wider range of wavelengths than the previous one; the waves which reach above the fuselage have a wavelength close to 0.6D. The damping by grid coarsening appears definite around 3D below the fuselage, but is not relevant to the permeable surfaces.

Figure 14.

Time-derivative of the pressure in the acoustic range.

The far-field spectra in Figure 15 reveal much about all approaches. First, similar to the fuselage-cavity configuration, the permeable approach captures the shielding, which is of the order of 8 dB, but the solid approaches essentially fail to. Second, the solid predictions are unnatural for the upwards direction. Namely, the contribution of the fuselage in this direction is about 5 dB larger than that of the bluff body although physically, the latter is a major sound source. Overall, considering this and, also, based on the results obtained with the use of the permeable FWH surface, the PSFWH answers are quite different and misleading.

Figure 15.

Sound power levels in upwards direction (left column) and downwards direction (right column). PWL: power level.

Note finally, that the simulation results presented above were repeated on a grid with the fuselage block refined by a factor of 2 in the azimuthal and polar directions (this Grid 2 had around 62 million cells). This improved both the resolution of the simulation and of the FWH integration. This however did not lead to any systematic alteration of the noise predictions discussed above. See Figure 16 comparing the SPL spectra obtained on the two grids at θ_e = 90° for the upwards and downwards direction with the use of both solid and permeable FWH surfaces. The differences in Figure 16(a) for St < 6 are noted, but may be statistical. This supports the accuracy of the computations, at least, in terms of the sound waves’ propagation and their scattering by the fuselage surface.

Figure 16.

Effect of grid on the far-field SPL spectra at $θ_{e}$ = 90°. FWH: Ffowcs Williams–Hawkings; SPL: sound pressure level.

Finally, Figure 17 addresses the impact of the cross-term between dipoles and quadrupoles, discussed in the “Introduction” section (JSV13-16). The quadrupole term ${p^{'}}_{Q}$ was defined as the difference between the permeable and the full-solid output (denoted by ${p^{'}}_{D}$ ); it did not come from a volume integral. Results are shown in the far-field, in the time-domain, and are conclusive. First, the cross-term is far from negligible, contrary to what some colleagues had predicted, and the correlation between ${p^{'}}_{D}$ and ${p^{'}}_{Q}$ is strongly negative. The correlation coefficient reaches −0.96 in the upwards direction (y = +100D); in other words the two contributions cancel each other, which reflects the shielding by the body. The effect is roughly 10 dB, which is stronger than that seen in Figure 15, presumably because of larger differences above St = 20. In the downwards direction, the correlation coefficient is about −0.375, which is much more moderate. Physically, a correlation between the quadrupole contribution and that from dipoles in their relatively close physical proximity is not surprising at all, but the fact that it is a negative correlation in both directions was not anything we would have predicted. It may be loosely explained by the sound that would be emitted by the “mirror image” of the turbulence with respect to the solid surface, although the argument would be more convincing with a flat surface than the cylinder. Based on Figure 13, the ratio between the distance from the turbulent eddies to the wall and the radius of curvature of the latter is not very small.

Figure 17.

Time-domain traces of the dipole, quadrupole, and cross-term contributions to the square of pressure, ${p^{'}}^{2}$ , obtained with the use of permeable FWH. Top, upwards observer; bottom, downwards observer. In each graph, the numbers show the time average of the terms over full time-sample equal to 50 $D / U_{0}$ .

Conclusions

We presented simulations and acoustic results via different versions of the FWH integral formula for a set of four model problems relevant to airframe noise, including in some cases wide variations of the Mach number, and moderate variations of the grid spacing. Curle’s widespread low-Mach-number theory is emphasized in two respects: his prediction that the solid-surface terms dominate, and that aerodynamic sound scales like M⁶ (for ${p^{'}}^{2}$ ). An additional issue was the (post-Curle) idea of identifying sound sources by separating the solid-surface integrals from various components of the aircraft.

Few of our questions had simple answers, and the success of Curle’s theory is case-dependent. This paper adds knowledge to the JSV13 line of thinking, without exhausting the subject. Sharp answers are made more difficult by the statistical noise in the spectra; presuming that this effect scales with $1 / \sqrt{T}$ where T is the duration of the simulation, decisive reductions of it would be onerous. Still, we consider it established that solid-surface integrals are not accurate enough for industrial purposes at the Mach numbers of high-speed trains and airliners in approach. Curiously, the accuracy of Curle’s theory does not seem to clearly hinge on the product of Mach and Strouhal number, as the equations predict (in some cases, the length scale used to define the Strouhal number is not unique). Regarding the M⁶ scaling, it was quite successful here, although our analysis of the literature, especially experimental, was not supportive and suggested much over-confidence in the community.

Conclusive results for the near-field sound showed that the permeable-surface FWH formula gives results that are as accurate as those of the simulation. The design of that surface is based on flow visualization and “common sense,” but careful grid generation with very gradual stretching is needed. By extension, the same formula gives far-field results that are as accurate as the simulation is, whereas the solid-surface formula fails often, especially when shielding effects are powerful. In addition, the partial surface FWH approach is close to illusory.

Progress in both research and industrial computational aero-acoustics will depend on the present results being accepted by the community, and confirmed in different enough flow solvers, FWH utilities, and without excessive sensitivity to the placement of the permeable surface. This will take time. The scientific community very much values independent findings, and for good reason. A favorable development would be the selection of one of the present test cases for a future workshop, in the BANC or other series. The range of scales in the geometries was not very wide, and it must be remembered that pantographs, on trains, and brake lines, on airplanes, are very thin and generate high frequencies. This will severely challenge the creation of grids that are fine enough to resolve such phenomena over a large-enough volume to reach a global permeable FWH surface. The accurate simulation of landing gear, 5 m tall and with 1 cm tubes shedding vortices (well within the audible range), is clearly demanding of many billions of grid points and tens of thousands of time steps; this is even before attempting to calculate the radiated sound. The “brute force approach” is not practical yet, recent claims at conferences notwithstanding.

Footnotes

Acknowledgements

The calculations were performed at the Computer Center “Polytechnichesky” of St Petersburg Polytechnic University. We enjoyed excellent discussions with Dr U. Michel, Prof. S. Lele, Dr T. Ricciardi, and Dr W. Wolf. Finally, we thank Drs A. Duben and T. Kozubskaya for sharing their near-field FWH utility.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Spalart

PR.

On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers. J Sound Vib 2013; 332: 2808–2815.

Curle

The influence of solid boundaries upon aerodynamic sound. Proc Roy Soc 1955; A231: 505–514.

Ffowcs Williams

Hawkings

DL.

Sound generation by turbulence and surfaces in arbitrary motion. Phil Trans Roy Soc 1969; A264: 321–342.

Spalart

Shur

Strelets

, et al. Initial noise predictions for rudimentary landing gear. J Sound Vib 2011; 330: 4180–4195.

Manoha

Caruelle

Summary of the LAGOON solutions from the benchmark problems for airframe noise computations-III workshop. In: The 21st AIAA/CEAS aeroacoustics conf., Dallas, TX, 22–26 June 2015, AIAA 2015-2846.

Spalart

PR.

Young person’s guide to detached-eddy simulation grids. NASA CR-2001-211032. Hampton, VA: Langley Research Center, 2001.

Lockard

Casper

JH.

Permeable surface corrections for Ffowcs Williams and Hawkings integrals. In: The 11th AIAA/CEAS aeroacoustics conf., Monterey, CA, 23–25 May 2005, AIAA 2005-2995.

Rahier

Huet

Prieur

Additional terms for the use of Ffowcs Williams and Hawkings surface integrals in turbulent flows. Comput Fluids 2015; 120: 158–172.

Ikeda

Enomoto

Yamamoto

, et al. Quadrupole corrections for the permeable-surface Ffowcs Williams–Hawkings equation. AIAA J 2017; 55: 2307–2320.

10.

Shur

Spalart

Strelets

MK.

Noise prediction for increasingly complex jets. Part I: methods and tests. Int J Aeroacoust 2005; 4: 213–246.

11.

Mendez

Shoeybi

Lele

, et al. On the use of the Ffowcs Williams-Hawkings equation to predict far-field jet noise from large-eddy simulations. Int J Aeroacoust 2013; 12: 1–20.

12.

Khorrami

Ravetta

Lockard

, et al. Comparison of measured and simulated acoustic signatures for a full-scale aircraft with and without airframe noise abatement. In: 2018 AIAA/CEAS aeroacoustics conf., Atlanta, GA, 25–29 June 2018, AIAA 2018-2975.

13.

Vatsa

Lockard

Khorrami

, et al. Aeroacoustic simulation of a nose landing gear in an open jet facility using FUN3D. In: The 18th AIAA/CEAS aeroacoustics conf., Colorado Springs, CO, 4–6 June 2012, AIAA 2012-2280.

14.

Casalino

Ribiero

AFP

Fares

, et al. Lattice-Boltzmann aeroacoustic analysis of the LAGOON landing-gear configuration. AIAA J 2014; 52: 1232–1248.

15.

Khorrami

Ravetta

Lockard

, et al. Measured and simulated acoustic signature of a full-scale aircraft with airframe noise reduction technology installed. In: The 25th AIAA/CEAS aeroacoustics conf., Delft, The Netherlands, 20–23 May 2019, AIAA 2019-2477.

16.

Rego

Pereira

LTL

Catalano

, et al. Aerodynamic and aeroacoustic experimental investigation of a simplified nose landing gear. In: AIAA SciTech Forum, Kissimmee, FL, 8–12 January 2018, AIAA 2018-0759.

17.

Reger

Cattafesta

LN.

Experimental study of rudimentary landing gear acoustics. AIAA J 2015; 53: 1715–1720.

18.

Ricciardi

Wolf

Speth

Acoustic prediction of LAGOON landing gear: cavity noise and coherent structures. AIAA J 2018; 56: 4379–4399.

19.

Hajczak

Sanders

Vuillot

, et al. Investigation of the Ffowcs-Williams and Hawkings analogy on an isolated landing gear wheel. In: AIAA SciTech Forum, Kissimmee, FL, 8–12 January 2018, AIAA 2018-3301.

20.

Appelbaum

Duda

Fares

, et al. Airframe noise simulations of a full-scale aircraft. In: 2018 AIAA/CEAS aeroacoustics conf., Atlanta, GA, 25–29 June 2018, AIAA 2018-2973.

21.

Lele

SK.

Volume noise sources in turbulent wake interaction problems: true quadrupole noise? In: The 20th AIAA/CEAS aeroacoustics conf., Atlanta, GA, 16–20 June 2014, AIAA 2014–3042.

22.

Giret

JC.

Simulations aux grandes échelles des écoulements instationnaires turbulents autour des trains d’atterrissage pour la prédiction du bruit aérodynamique. PhD Thesis, National Polytechnic Institute of Toulouse, France, 2014 (large parts of this document are in English).

23.

De Azevedo

PRG

Wolf

WR.

Noise prediction of the LAGOON landing gear using acoustic analogy and proper orthogonal decomposition. In: The 22nd AIAA/CEAS aeroacoustics conf., Lyon, France, 30 May–1 June 2016, AIAA 2016-2768.

24.

Giret

Sengissen

Moreau

, et al. Prediction of LAGOON landing-gear noise using an unstructured LES solver, In: The 19th AIAA/CEAS aeroacoustics conf., Berlin, Germany, 27–29 May 2013, AIAA 2013-2113.

25.

Shur

Spalart

Strelets

, et al. High-order implicit multi-block Navier-Stokes code: ten-years experience of application to RANS/DES/LES/DNS of turbulent flows. In: The 7th Symposium on overset composite grids and solution technology, Huntington Beach, CA, 2004.

26.

Shur

Spalart

Strelets

MK.

LES-based evaluation of a microjet noise reduction concept in static and flight conditions. J Sound Vib 2011; 330: 4083–4097.

27.

Shur

Spalart

Strelets

MK.

Noise prediction for underexpanded jets in static and flight conditions. AIAA J 2011; 49: 2000–2017.

28.

Shur

Spalart

Strelets

, et al. Synthetic turbulence generators for RANS-LES interfaces in zonal simulations of aerodynamic and aeroacoustic problems. Flow Turbulence Combust 2014; 93: 63–92.

29.

Spalart

Belyaev

Garbaruk

, et al. Large-eddy and direct numerical simulations of the Bachalo-Johnson flow with shock-induced separation. Flow Turbulence Combust 2017; 99: 865–885.

30.

Suzuki

Spalart

Shur

, et al. Unsteady simulations of a fan/outlet-guide-vane system: broadband-noise computation. AIAA J 2019. Epub ahead of print 31 July 2019. DOI: 10.2514/1.J058177.

31.

Roe

PL.

Approximate Rieman solvers, parameter vectors and difference schemes. J Comput Phys 1981; 43: 357–372.

32.

Travin

Shur

Strelets

, et al. Physical and numerical upgrades in the detached-eddy simulation of complex turbulent flows. In: Friedrich

Rodi

(eds) Advances in LES of complex flows. Vol. 65. Munich: Springer, 2002, pp.239–254.

33.

Spalart

Deck

Shur

, et al. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoret Comput Fluid Dyn 2006; 20: 181–195.

34.

Spalart

Shur

ML.

Variants of the Ffowcs Williams–Hawkings equation and their coupling with simulations of hot jets. Int J Aeroacoust 2009; 8: 477–492.

35.

Abalakin

Bakhvalov

Gorobets

, et al. Parallel research code NOISEtte for large-scale CFD and CAA simulations. Vychislitelnye Metody i Programmirovanie 2012; 13: 110–125 (in Russian).

36.

Spalart

Shur

Strelets

MK.

Added sound sources in jets; theory and simulation. Int J Aeroacoust 2009; 8: 511–534.

On the differences in noise predictions based on solid and permeable surface Ffowcs Williams–Hawkings integral solutions

Abstract

Keywords

Introduction

Overview and Ffowcs Williams–Hawkings (FWH) formula

Permeable surfaces and end-caps

Mach-number scaling

Partial surface integrations

Numerical simulation approach

Overview of the study

Dipole placed under a sphere

Problem statement and sound field

Results

Isolated bluff body over a range of Mach numbers

Problem statement and flow field

Generic fuselage with a simplified landing-gear cavity

Problem statement and flow field

Acoustic results

Generic fuselage with a simple bluff body

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References