© The Author(s), Published by EDP Sciences, 2024

1 Introduction

The Heating Ventilation and Air Conditioning (HVAC) system of a car has to provide airflow to ensure the passengers comfort regarding the temperature inside the vehicle cabin without damaging the acoustic environment. To obtain a satisfactory indoor environment, the acoustical design of a ventilation system is as important as its thermal design and becomes even more significant since the development of the electrical and hybrid electric vehicles. In a HVAC, the noise is caused by the complicated acoustic and turbulent flow interactions with multiple discontinuities located in a compact housing duct at a low mean velocity interacting mainly with a moving element (the blower) but also with fixed elements such as the heat exchanger, the mixing and distributing flaps, the filter, bends, constrictions and finally with the outlet grille before radiating into the passenger compartment. Because of the compactness of the HVAC housing required by the car manufacturers, high amplification and whistling can sometimes be produced [1, 2] and there is a need for an appropriate description of the physical phenomenon which causes these amplifications. The prediction of the aero-acoustic sound power spectra (SWL) of single elements in ducts [3, 4] such as spoilers, bends, orifices is usually based on the Nelson-Morfey’s theory [5]. This formulation assumes a dipole source distribution resulting of the drag fluctuating forces arising from the turbulent flow near the element and that the ratio between these forces and the static drag force depends on the Strouhal number. This approach is very convenient from an industrial point of view because the SWL can be easily predicted for a wide range of configurations thanks to similarity considerations [6]. The prediction of noise due to the presence of multiple elements in air ducts have also been the subject of research in the past. The theory, which is also based on Nelson and Morfey’s formulation, relies on the cross-spectra between the fluctuating forces between each element [7, 8]. These predictive models are however based on the fluctuations of the aerodynamic forces on the obstacles and the scattering of acoustic waves as well as possible resonance effects are usually ignored in the analysis. Depending on the situation, these resonances can be purely acoustical [9] or can result from a more complex flow-acoustic phenomena whereby a vortex shedding mechanism and the resonant acoustic frequencies are locked together [10]. Although this is not the place for a complete survey on this difficult matter, one can cite the work of Ziada and Lafon [11] in the field of industrial aero-acoustics. The authors identified three different mechanisms which generate upstream feedback of disturbance: (1) structural vibrations, (2) fluid-resonant mechanism triggered when the frequency of vortex shedding becomes close to that of an acoustic resonator, and (3) the fluid-dynamic mechanism caused by flow impingement on a downstream object.

The aim of this paper is twofold: to show that mechanism (3) can be responsible for the production of high level sound power around specific frequencies because of the closeness of the HVAC elements and also to give all necessary details allowing to capture these fluid-acoustic interaction correctly via the use of a commercial Computational Fluid Dynamics (CFD) software. The academic study consists of two identical diaphragms with rectangular openings inserted in a duct of rectangular cross-section and separated by three different distances which are comparable with the transverse dimensions of the duct. It is interesting to note that similar configurations, mainly conducted with circular geometries both for the duct and the orifice, have been the subject of many papers in the recent years and we can cite the work of Sovardi et al. [12], Karban and Schram [13, 14], Sack et al. [15, 16], and also [17]. In a somewhat more industrial context, other research work investigated the ability of different CFD software to predict broadband noise and the presence of tonal noise, see [18–20] for instance. Both experimental and numerical aspects are presented here in a rectangular duct. The article is organized as follows: in Section 2, experimental results are shown in terms of power spectra for the sound power radiated downstream by the obstacle which consists either of a single diaphragm or two diaphragms in tandem. It is shown that the latter configuration can result in a significant increase of the noise production with the presence of peaks which are typical of flow-acoustics resonance or acoustic feedback-loop phenomena. This is evidenced by matching the frequencies of the peaks with a classical feedback-loop equation as found for cavity or airfoil tonal noise. In Section 3, numerical calculations using LES simulations are performed, analyzed and compared with experimental results. The modal pattern observed at the resonant peaks, signature of large coherent structures formed downstream of the diaphragms, is a further evidence of the flow-acoustics resonance. The paper ends with a conclusion and some discussion.

2 Experimental results

2.1 Experimental setup and measurement procedure

Although the detailed description of the test set up can be found in [21], it is worth recalling its main characteristics. The flow generation part is composed of a fan, mufflers, a plenum chamber and a hot wire air flow meter. Two mufflers connected with an elbow located upstream of the blower are designed to reduce fan and flow noise. The air passes through a flow tranquilizing box which minimizes the turbulence and a flow nozzle equipped with a differential pressure gauge to measure the volumetric flow rate and sensors have been used to measure the humidity as well as the temperature of the environment. The measurement part (visible in Fig. 1) includes a total of 96 flush mounted microphones located upstream and downstream the test section. Figure 1 also shows two anechoic terminations placed on both sides of the duct.

Figure 1 Sketch of the Aero-Acoustics test rig.

Acoustic measurements are based on the 2-N port method [22]. The formulation takes the form of a matrix system (here ports 1/2 correspond to the left/right part of the test section)

$$\left(\begin{array}{l}{\mathsf{P}}^{1-}\\ {\mathsf{P}}^{2+}\end{array}\right)=\left(\begin{array}{ll}{\mathsf{S}}^{11}& {\mathsf{S}}^{12}\\ {\mathsf{S}}^{21}& {\mathsf{S}}^{22}\\ & \end{array}\right)\left(\begin{array}{l}{\mathsf{P}}^{1+}\\ {\mathsf{P}}^{2-}\end{array}\right)+\left(\begin{array}{l}{\mathsf{P}}_s^{1-}\\ {\mathsf{P}}_s^{2+}\end{array}\right),$$(1)

where $\mathsf{S}$ is the scattering matrix and the source vectors ${\mathsf{P}}_s^{1-}$ and ${\mathsf{P}}_s^{2+}$ contain the amplitudes of the duct acoustic modes propagating downstream (+) and upstream (−).

Pressure signals are recorded by a 96 channels LMS Scadas acquisition with a bandwidth of 6400 Hz and 4096 spectral lines. The measurement procedure is performed using an averaging process over 600 realizations to achieve statistical convergence (the acquisition time is about 2 min long for each configuration). The acoustic scattering matrix of the obstructed duct and the impedance of the surrounding environment are first identified by measuring the acoustic response to given external excitations. The test rig was designed to perform modal decomposition up to the 8th propagating mode (N = 8). Therefore, the computation of the scattering matrix must require at least 2N configurations which must be linearly independent. This is achieved by using 10 loudspeaker positions both upstream and downstream the test section (see Fig. 1). In the case of a single diaphragm inserted in the duct, measured scattering matrix coefficients have been validated by comparison with numerical simulations [21].

In a second step, the aeroacoustic noise produced by the flow-obstacle interaction can be measured. In order to obtain a formulation for both random and periodic types of signals, the sound power from values of the source vectors is obtained via the modal cross-spectrum matrix [15, 22]. The formulation extracts the source strength using cross-spectrum densities between the four different sections S₁, S₄, S₇, S₈ according to the procedure first described in [21] and also applied in [23] (see Appendix B in [23] for all necessary details). The procedure allows us to suppress the hydrodynamic fluctuations, particularly on the downstream side where the flow is expected to be turbulent.

The convection of the propagating waves due to the flow can be neglected here and the sound power radiated downstream by the obstacle is estimated via

$$W^{2+}=\frac{1}{2{\rho }_0\omega }\sum _{m\ge 0}\mathrm{}\sum _{n\ge 0}\mathrm{}\mathrm{Re}\left(k_{\mathrm{mn}}\right)\langle {|P_{s,\mathrm{mn}}^{2+}|}^2\rangle =\sum _{m\ge 0}\mathrm{}\sum _{n\ge 0}\mathrm{}{W}_{\mathrm{mn}}^{2+},$$(2)

where k_mn are the longitudinal wavenumbers associated with mode (m, n), i.e.

$$k_{\mathrm{mn}}=\sqrt{k^2-(\mathrm{m\pi }/a{)}^2-(\mathrm{n\pi }/b{)}^2},$$(3)

where k = ω/c₀ (note the summation is restricted to propagating modes only, i.e. k_mn must be a real quantity). Here, $P_{s,\mathrm{mn}}^{2+}$ are the amplitudes of the duct acoustic modes from the source vector. The dimensions of the cross-section of the duct are a = 20 cm and b = 10 cm. The geometry of the diaphragm inserted in the test section consists of a constriction of rectangular shape with dimension 10 cm × 5 cm × 0.8 cm centered in the rectangular duct (Fig. 2). In the case of two identical diaphragms in tandem, three separation distances with L = 6 cm, 13 cm, and 21 cm are considered.

Figure 2 Two diaphragms in tandem and dimensions of a single diaphragm.

2.2 Case of a single diaphragm and scaling law for the fundamental mode

The hypothesis used in Nelson and Morphey paper is that the root mean square fluctuating drag force on the obstacle is proportional to the steady drag force $\bar{F}$ at a certain frequency band and the expression for the sound power carried by the fundamental mode travelling in one direction is given by

$$W_{00}^{2+}=\frac{S_{\mathrm{FF}}}{4{\rho }_0{c}_{0}A} \mathrm{with} S_{\mathrm{FF}}={\bar{F}}^2{K}^2.$$(4)

In the case of a diaphragm, we follow the recent development of Kårekull et al. [24] who revisited the Nelson-Morfey semi-empirical scaling law by suggesting that the dynamic force is assumed to scale with the momentum flux: $\bar{F}={\rho }_0\mathrm{AU}{U}_d$ where U_d is a characteristic velocity taken as the velocity in the vena contracta evaluated from the formula presented by Durrieu et al. [25] and given by U_d = U/σ and

$$\sigma =\frac{\frac{A_{\mathrm{o}}}{A}}{1+\sqrt{0.5\left(1-\frac{A_{\mathrm{o}}}{A}\right)}}.$$(5)

The key result in the analysis is to view the source strength spectrum K²(St) as function of the Strouhal number

$$\mathrm{St}=\frac{f{D}_h}{U_d}.$$(6)

Here, the characteristic length is chosen to be $D_h=\sqrt{4A/\pi }$ which corresponds to the hydraulic diameter of the rectangular duct. Experimental results are scaled following the previous model and are given Figure 3 for a diaphragm whose area orifice is A_o = 0.005 m² and a vena contracta ratio σ = 0.16 (evaluated using Eq. (5)). A good collapse is found for all velocities over the majority of the frequency range (up to around 3000 Hz). Note that these results are restricted to the contribution of the plane wave mode only and the change in inclination which appears when St ≈ 2.2 correspond to the cut-off frequency of the first transverse mode (around 860 Hz) of the duct. Generalization to the multimodal context could be made following the original work [5] but this is not investigated here. It should be noted that if results are somewhat consistent with experimental investigations reviewed in [4], they are in excellent agreement with those reported in Figure 14 in [5]. In particular below cut-off where it is shown that the curve of 20log(K(St)) exhibits a decay of 20 dB per decade with increasing Strouhal number.

Figure 3 Dimensionless source strength spectra for a single diaphragm (only the contribution of the plane wav mode is considered).

2.3 Case of two diaphragms in tandem

The prediction of noise due to the presence of multiple elements in air ducts have been the subject of research in the past and we can cite [7, 8]. The theory relies on the interaction factor which is a function of frequency, distance between the various duct elements and geometries. Depending on the cross-spectra between the fluctuating forces on both diaphragms, it is anticipated that the radiated acoustic power should increase by 3 dB up to a maximum of 6 dB (assuming that forces on both diaphragms are identical). A series of measurements has been conducted for three separation distances. Results are shown in Figure 4 where the sound power levels are compared with the case of a single diaphragm for a mean flow velocity U = 7 m/s.

Figure 4 Total sound power measured downstream for two diaphragms in tandem with U = 7 m/s.

Two main observations can be made: first, the broadband noise is increased by 10–20 dB due to impinging turbulence at the downstream orifice. The increase in power decreases with frequency and also when the distance separation is larger. These results are in line with those published in [16] in the case of two diaphragms inserted in a circular duct (see Fig. 14 in [16]) where it is shown that the increase in power becomes marginal when the distance exceeds at least 10 times the hydraulic diameter. Second, a series of consecutive and clearly distinguishable peaks at low frequencies below the first cut-off frequency. Two additional peaks can be identified, the first one appears at the first cut-off frequency of the duct and the second one, around 1500 Hz, corresponds to an acoustic resonance due to the cavity made by the two diaphragms. The modulation of the sound power spectra due to resonance between the orifice plates is also observed and discussed in [16]. The peaks at low frequencies are due to high amplitude acoustic waves and this suggests the presence of flow-acoustic feedback mechanisms. This is further investigated in Figure 5 where the effect of the flow velocity is now shown. Again, a series of low frequency peaks can be identified, except when the flow velocity is too low. At this stage, it is instructive to report these resonant frequencies with respect to the mode number n corresponding to the nth identified peak in Figures 4 and 5. This is illustrated in Figure 6. For each configuration, it is observed that peak frequencies behaves like an affine function with respect to mode number. This is typical of the Rossiter’s formula [26, 27] which originates from the study of feedback mechanisms of flows past open cavities. The frequential peaks depends upon the flow velocity and the separation distance as

$$f_n=\frac{n+\gamma }{L\left(\frac{1}{U_c}+\frac{1}{c_0}\right)}.$$(7)

Figure 5 Total sound power measured downstream for two diaphragms in tandem with L = 13 cm.

Figure 6 Peak frequencies for all configurations.

The physical interpretation for this formula is that these tones are the result of a feedback effect produced by the vortices impingement on the second diaphragm causing acoustic waves propagating upstream and triggering a new set of vortices. Here n is the number of vortices located between both diaphragms and γ is an empirical constant. In the original paper [26], the convection velocity of the vortices is usually estimated as a fraction of the external upstream flow velocity U_∞ and U_c = κU_∞ and in our situation, it is more relevant to consider the flow velocity in the vena contracta so we put U_∞ = U_d. In Table 1 are reported the values of the empirical constants γ and κ which yield a fairly good collapse with the experimental data. Despite the fact that the present configuration notably differs from the more classical cavity flow, the values are somewhat in line with those reported in the scientific literature and formula (7) is representative of the flow mechanisms. We may notice that the empirical constant γ is found to be positive here (as in the experimental data in [28]) though most studies report a negative value [27].

3 Numerical results

The aim of this section is to offer a better understanding of the flow oscillations and the radiated acoustic power by means of numerical simulations. It also shows some useful guidelines which permits to capture the physical mechanisms correctly taking place in the duct.

3.1 Numerical set-up and parameters

The numerical results are obtained for the separation distance L = 13 cm with an inlet velocity of U = 7 m/s. To prevent acoustic waves to bounce off the boundaries, two Large Eddy Simulations have been tested: the first one, called LES1, utilizes acoustic damping zones at both ends. These zones must stretch over a sufficient distance in order to cover a full wavelength. The lowest frequency of interest is 200 Hz which means that the damping zone is 1.7 m long. In a second setup, named LES2, the acoustic damping is used only in the upstream direction and the non-reflective boundary condition, via a waveTransmissive pressure boundary condition, is prescribed downstream. In both configurations, the same computational domain has been used and the duct length on both sides of the two diaphragms is 3 m long. All surfaces, which includes the duct walls and the obstacles are considered perfectly rigid and a no slip wall condition is applied. The boundary conditions at both end of the duct are: a constant velocity is imposed at the inlet and a constant pressure is imposed at the outlet. Note that planes of symmetry have not been exploited here and the full domain has been simulated.

The computation has been performed with the compressible solver rhoPimpleFoam from OpenFOAM CFD software. In order to assess the convergence, two different meshes are tested. The first mesh G1 consists of a structured hexahedral grid containing 14 million cells. The size of the elements is chosen to ensure that the propagation of acoustic waves is well simulated over a sufficient large distance which must include the position of the virtual microphones, placed on the duct wall as in the experimental setup. A simple rule of thumb shows there are at least 30 cells per wavelength at 1 kHz. Cells are much finer in the vicinity of the shear layer, and are stretched away from the obstacles and in the damping zones. The size of the smallest cells around the edges of the diaphragm where vortices are created is 5 × 10⁻⁴ m. A finer mesh, G2, is built with the same refinement strategy and contains 33 millions hexahedral cells, the smallest cell size around the obstacle is 3.25 × 10⁻⁴ m. Figure 7 shows a cross-sectional close-up view of both meshes in the plane x₃ = 0 (the axes are defined in Fig. 8).

Figure 7 Cross-sectional view (x3 = 0 plane) of the mesh G1 (top) and G2 (bottom), centered around the edges of the two diaphragms.

Figure 8 Instantaneous velocity and pressure fields in the planes x3 = 0 (top) and x2 = 0 (bottom).

LES computations are performed with a given initial state obtained with a RANS k–ω SST (Shear Stress transport) computation. The WALE subgrid-scale model is used, and second order schemes both in space and time are employed, using respectively linear and backwards schemes. Since the time scheme is implicit, the maximum Courant number is around 5 for both meshes and this corresponds to a time step of 3 × 10⁻⁵ s and 2.3 × 10⁻⁵ s respectively. In all results shown here, the simulation time is around 1 s. As a comparison, i.e. the flow-through time, the amount of time required for the fluid to cross the gap between the obstacles is around 0.05 s.

3.2 Convergence check

Results are post-processed after a transient period of 0.54 s. This value, which corresponds to 11 flow-through times, has been determined using the procedure proposed by Mockett et al. [29] to detect the presence of an initial transient content. This procedure is based on the hypothesis that the initial transient of a signal may be reflected through a distortion of its mean or standard deviation. In Figure 9 (top) is shown the time history of the drag force exerted on the second obstacle F₂(t). To illustrate the presence of this initial transient, Figure 9 (bottom) shows the evolution of the product of a statistical random error on the mean μ and standard deviation σ of F₂, as the start time t₀ is shifted. Then, the detection of the minimum of this product indicates the end of the initial transient t_t.

Figure 9 Time history of the force exerted on the second obstacle (top), and variation of the product $\sigma \left[{\mu }_{F_2}\right]\times \sigma \left[{\sigma }_{F_2}\right]$ with t0, illustrating the detection of tt (bottom).

3.3 Aerodynamic predictions

Figure 8 shows the instantaneous velocity (left) and pressure (right) fields in the planes x₃ = 0 and x₂ = 0. While the geometry and boundary conditions possess symmetry properties, the computed flow is not, as shown in the planar streamlines representation in Figure 10 (left). These streamlines have been obtained from a velocity field averaged on a period of Δt ≈ 0.3 s (approximately 6 flow-through times). Furthermore, these planar representations do not give insight on the normal velocity component: the 3D streamlines (Fig. 10 (right)) show a strong swirl movement inside the cavity. Isosurfaces of the Q-criterion (Q = 2 × 10⁷) of the instantaneous flow colored by velocity magnitude is shown in Figure 11. In the side view (Fig. 11 (left)), ring vortices (flagged with black arrows) are clearly identifiable after the first obstacle. Another type of large coherent structure is more visible when the Q-criterion is computed from the mean velocity, see Figure 11 (right) with Q = 2 × 10⁶. These streamwise coherent structures, flagged with red arrows and clearly seen in Figure 11 (right), seem to break apart the ring vortices as they travel past the first obstacle.

Figure 10 Planar (left) and three dimensional (right) streamlines.

Figure 11 Q-criterion, top view (left) and axial view (right).

In order to distinguish the flow features at different frequencies, Figure 12 shows the instantaneous dilatation field, defined as ρ₀∂_tρ, filtered at different frequencies. The top picture shows the unfiltered field, and the pictures below show the same field filtered at the peak frequencies identified in Figure 13. The filtered fields exhibit a noticeable pattern as a succession of white and black spots (modal structure), which are present both in the cavity and downstream of the second diaphragm. It needs to be pointed out that (i) the same filtering procedure at another arbitrary frequency would lead to a uniform grey zone in the cavity with no particular pattern and (ii) the fields shown here correspond to LES2 results and using LES1 results would lead to the same observations. From these observations, it emerges that each frequency f_n is associated with a number n of vortices in the cavity, and

$$n{\lambda }_a\le L\le \left(n+1\right){\lambda }_a,$$(8)

where λ_a = U_c/f_n is the aerodynamic wavelength, that is the spacing between two vortices, and this is in line with the empirical equation (7).

Figure 12 Dilatation field at an arbitrary time step: unfiltered field (top) and filtered at the mode frequency fn.

Figure 13 Acoustic power radiated downstream (computed with Eq. (2) where modal amplitudes are extracted using the 2-N port formulation): influence of the mesh (left) and influence of the boundary condition (right).

3.4 Acoustic predictions

In the compressible simulation, virtual microphones have been placed in the computational domain at the same location, i.e. at sections S₅, S₆, S₇, S₈ as in the experimental setup (see Fig. 1). Amplitudes of the duct acoustic modes are extracted using the 2-N port method as described earlier. Contrary to the experimental procedure, radiation conditions are assumed to be exact at both ends of the duct and no acoustic wave travel toward the test section (i.e. ${\mathsf{P}}^{1+}={\mathsf{P}}^{2-}=0$). Thus the acoustic power radiated downstream is straightforward without the need for computing the scattering matrix. Note that this greatly simplifies the analysis and we can refer to a complete computational procedure developed in [15] which includes both the aero-acoustic source strength and the scattering matrix of the in-duct components.

The sound power spectra obtained with LES1 for both meshes are compared with the experimental one in Figure 13 (left). The grey curves correspond to the numerical results: the solid one has been obtained with the coarsest mesh G1 and the dotted one with the finest mesh G2. Both numerical results are seen to agree, and this implies that the mesh convergence has been reached. Furthermore, the numerical results show similar trend with the experimental ones, which suggests that the feedback phenomenon is indeed captured by the simulation. However, two main differences with the experimental results may be observed: (i) the numerical results exhibit an overall underestimation of the acoustic level and (ii) peak frequencies do not coincide perfectly, especially the second and third peak.

The influence of the downstream boundary condition is shown in Figure 13 (right). While both LES computed results show the same trend, the acoustic power level computed with LES2 is higher by a few dB up to 600 Hz, and peaks seem slightly sharper with a better agreement with experimental data. Even though the exit section is located approximately 3 m downstream, the numerical treatment of the radiation condition can have a discernible effect on the radiated sound power.

In order to highlight the importance of the role played by the acoustic waves on the feedback-loop mechanism, an incompressible computation (LES3) is carried out with a mesh similar to G1 and by using the same schemes and subgrid-scale model. Here the length of the simulated domain is reduced to remove the damping zones and the incompressible solver pimpleFoam is used. Since acoustic waves can not be captured by the numerical simulation, the approach utilized earlier can not be followed. Instead, the formulation of Nelson and Morfey [5], which allows one to compute the acoustic power from the sole knowledge of the pressure fluctuations on the obstacles can be used. Since all peaks appear below the first cut-off frequency, the expression for the radiated acoustic power can be simplified with the plane wave hypothesis, as in [30]. As two obstacles, assumed acoustically compact, are present here, three different terms appear in the final expression: the first two terms relate to the drag fluctuations on both obstacles and the last one takes into account the cross-spectra between the fluctuating forces (see a derivation of the formula in Appendix)

$$W_{00}^{2+}=\frac{1}{4A{\rho }_0{c}_0}\left(S_{F_1{F}_1}+S_{F_2{F}_2}+2\mathrm{Re}\left(S_{F_1{F}_2}{\mathrm{e}}^{-i\mathrm{kL}}\right)\right).$$(9)

In the present case, most of the noise is generated by the vortices impinging on the second obstacle and the corresponding term is predominant over the whole bandwidth. This is illustrated in Figure 14 (left) where the three components of (9) are plotted separately. The contribution of the first obstacle increases with frequency but remains below the one of the second obstacle. The correlation term is particularly high at the peak frequencies and rapidly decreases with frequency. It is worth observing that the total radiated power, computed with LES2 compressible simulation, is in good agreement with results of Figure 13 (recall that Fig. 14 is obtained with equation (9) while equation (2) is used in Fig. 13)). This means that formula (9) is very reliable, at least for the plane wave mode, and the source term which normally appears in the Lighthill’s acoustic analogy can be discarded in the integral formulation (A2). The reasons for this is that solid walls, which includes the duct and the obstacles, act as perfect sound reflectors and are therefore responsible for most of the radiated power [31].

Figure 14 SWL for the plane wave mode only computed via (9) from the fluctuating forces exerted on the two diaphragms. Contribution of the different terms of (9) computed with LES2 (left). Comparison between LES2 (compressible) and LES3 (incompressible) simulations (right).

Results obtained with and without compressibility effects (LES2 and LES3 respectively) are now shown in Figure 14 (right). It can be seen that, although broadband noise spectral density are in good agreement, no peaks are identifiable if the flow is assumed incompressible. This point is not as trivial as it may appears since, depending on the configuration, incompressible flow solvers are sometimes sufficient to capture flow oscillation effects, see for instance [32]. Finally, it is worth noting that other numerical setups have also been tried and they all failed to capture the acoustic feedback loop. In particular when symmetries of the geometry are used and only one fourth of the domain is simulated (the fact that symmetry of the flow is broken is confirmed by the simulation in Fig. 10 for instance) and also when limited schemes such as the vanLeer and linearUpwind are used.

4 Conclusions

In this paper, the fluid-acoustic interactions caused by flow impingement on two identical diaphragms with rectangular openings inserted in a duct of rectangular cross-section are investigated both experimentally and numerically. The former relies on the the 2-N port method, while the latter compares results for the first time from both compressible and incompressible large eddy simulations with the open-source software OpenFOAM. Good overall agreement is achieved between all approaches, providing new insight in the noise mechanisms.

It is first shown that the presence of the second diaphragm can increase the broadband noise by 10–20 dB with the presence of additional peaks at low frequencies. In fact, most of the noise is generated by the vortices impinging on the second obstacle and the corresponding term in the equation of the acoustic power solely obtained from the pressure fluctuations on the diaphragms is predominant over the whole bandwidth.

The location of the peak frequencies are then observed to fit well with Rossiter’s formula which originates from the study of acoustic feedback-loop mechanisms of flows past open cavities. This phenomenon is highlighted using Direct Noise Computations. In particular, the dilatation field filtered at the resonant frequencies shows a distinct pattern corresponding to the Rossiter modes. It is also shown that, although broadband noise spectral density are in good agreement, no resonance peaks are identifiable if the flow is assumed incompressible. Although the Rossiter model has been shown to be relevant in order to give some useful interpretation of the feedback mechanisms, the authors of the present paper are aware that the studied configuration departs noticeably from the more academic case of a flow past a rectangular cavity and there is room for further studies. In particular a more thorough analysis could be conducted and this could be done in the spirit of [33] for instance.

Acknowledgments

The computations were made on the supercomputers Cedar from Simon Fraser University and Niagara from University of Toronto, both parts of Compute Canada’s national platform of Advanced Research Computing resources.

Conflict of interest

The authors declare that they have no conflicts of interest in relation to this article.

Data availability statement

Data are available on request from the authors.

Appendix

Lighthill’s equation for the pressure fluctuations in a turbulent flow takes the form of the inhomogeneous Helmholtz equation

$$\left(\Delta +k^2\right)p=-\frac{{\partial }^2{T}_{\mathrm{ij}}}{\partial x_i\partial x_j}.$$(A1)

Under the hypothesis of an isentropic low-Mach number flow at high Reynolds number, sound sources due to density, viscous stress and entropy changes are neglected and Lighthill’s stress tensor becomes T_ij ≈ ρ₀u_iu_j, where scalar quantities u_i can be provided either by compressible or incompressible flow simulations. Note that (A1) is written in the frequency domain with the convention e^−iωt. Now, using the Taylored Green’s function G_d for the rectangular duct [30], the acoustic pressure radiated away has the explicit closed form expression:

$$p\left(\mathbf{x},\omega \right)=\int \mathrm{}{T}_{\mathrm{ij}}\left(\mathbf{y},\omega \right)\frac{{\partial }^2{G}_d}{\partial y_i\partial y_j}{\mathrm{d}}^3\mathbf{y}+{\int }_{\mathrm{\Gamma }}\mathrm{}p\left(\mathbf{y},\omega \right)\frac{\partial G_d}{\partial n_y}{\mathrm{d}}^2\mathbf{y},$$(A2)

where p(y, ω) stands for the pressure fluctuations on the rigid obstacle(s) Γ and n_y is the normal unit vector to the surface. Because we are interested in the low frequency propagation, i.e. below the first cut-off duct acoustic mode, the first term of the modal series is predominant [30] and

$$G_d\approx -\frac{{\mathrm{e}}^{\mathrm{i}k\left|x_1-y_1\right|}}{2\mathrm{i}\mathrm{kA}}.$$(A3)

The theory of Nelson and Morfey [5] relies on the assumption that the volume integral in (A2) is negligible compared to the surface integral. This yields (we use the same notation as in (4)):

$$W_{00}^{2+}=\frac{1}{4A{\rho }_0{c}_0}{\int }_{\mathrm{\Gamma }}\mathrm{}{\int }_{\mathrm{\Gamma }}\mathrm{}{S}_{\mathrm{pp}}\left(\mathbf{y},{\mathbf{y}}^{\mathrm{\prime }},\omega \right){\mathrm{e}}^{-\mathrm{i}k\left(y_{1^{\text{'}}}-y_1\right)}{n}_{1y}{n}_{1y^{\text{'}}}{\mathrm{d}}^2\mathbf{y}{\mathrm{d}}^2{\mathbf{y}}^{\mathrm{\prime }},$$(A4)

where

$$S_{\mathrm{pp}}\left(\mathbf{y},{\mathbf{y}}^{\mathrm{\prime }},\omega \right)=\langle p\left(\mathbf{y},\omega \right)p^{\mathrm{*}}\left({\mathbf{y}}^{\mathrm{\prime }},\omega \right)\rangle ,$$(A5)

is the cross power spectral density for the pressure fluctuations. The formula can be simplified in the presence of two acoustically compact obstacles Γ_j(j = 1,2) separated by distance L and by calling

$$F_j\left(\omega \right)={\int }_{{\mathrm{\Gamma }}_j}\mathrm{}p\left(\mathbf{y},\omega \right)n_{1y}{\mathrm{d}}^2\mathbf{y},$$(A6)

the drag force on each obstacle, we end up with formula (9) or (4) in the presence of a single obstacle.

References

All Tables

All Figures

	Figure 1 Sketch of the Aero-Acoustics test rig.
In the text

	Figure 2 Two diaphragms in tandem and dimensions of a single diaphragm.
In the text

	Figure 3 Dimensionless source strength spectra for a single diaphragm (only the contribution of the plane wav mode is considered).
In the text

	Figure 4 Total sound power measured downstream for two diaphragms in tandem with U = 7 m/s.
In the text

	Figure 5 Total sound power measured downstream for two diaphragms in tandem with L = 13 cm.
In the text

	Figure 6 Peak frequencies for all configurations.
In the text

	Figure 7 Cross-sectional view (x3 = 0 plane) of the mesh G1 (top) and G2 (bottom), centered around the edges of the two diaphragms.
In the text

	Figure 8 Instantaneous velocity and pressure fields in the planes x3 = 0 (top) and x2 = 0 (bottom).
In the text

	Figure 9 Time history of the force exerted on the second obstacle (top), and variation of the product $\sigma \left[{\mu }_{F_2}\right]\times \sigma \left[{\sigma }_{F_2}\right]$ with t0, illustrating the detection of tt (bottom).
In the text

	Figure 10 Planar (left) and three dimensional (right) streamlines.
In the text

	Figure 11 Q-criterion, top view (left) and axial view (right).
In the text

	Figure 12 Dilatation field at an arbitrary time step: unfiltered field (top) and filtered at the mode frequency fn.
In the text

	Figure 13 Acoustic power radiated downstream (computed with Eq. (2) where modal amplitudes are extracted using the 2-N port formulation): influence of the mesh (left) and influence of the boundary condition (right).
In the text

	Figure 14 SWL for the plane wave mode only computed via (9) from the fluctuating forces exerted on the two diaphragms. Contribution of the different terms of (9) computed with LES2 (left). Comparison between LES2 (compressible) and LES3 (incompressible) simulations (right).
In the text