Collapsible tubes can be employed to study the sound generation mechanism in the human respiratory system. The goals of this work are (a) to determine the airflow characteristics connected to three different collapse states of a physiological tube and (b) to find a relation between the sound power radiated by the tube and its collapse state. The methodology is based on the implementation of computational fluid dynamics simulation on experimentally validated geometries. The flow is characterized by a radical change of behavior before and after the contact of the lumen. The maximum of the sound power radiated corresponds to the postbuckling configuration. The idea of an acoustic tube law is proposed. The presented results are relevant to the study of selfexcited oscillations and wheezing sounds in the lungs.
I. INTRODUCTION
In daily medical practice, auscultation is among the most costeffective and timely diagnostic tools for diseases of the respiratory and cardiovascular system (Bohadana , 2014). Different pathologies are characterized by specific sounds (sometimes called adventitious sounds) that can guide physicians toward identifying and assessing the disease. From an acoustic perspective, the problem of sound generation in the human body (although one should be careful when considering blood, as for its incompressible nature, it cannot directly create sound) and its propagation represents a challenging question that is far from being completely understood (Hassan , 2019; Palnitkar , 2020).
From a biomechanical perspective, for respiratory and cardiovascular pathologies, the generation of adventitious sounds is often correlated to the interaction between the biological fluid (air or blood) and the human vessel (airways, or veins and arteries) (İçer and Gengeç, 2014). The interest in quantitatively modeling the fluidstructure interaction (FSI) phenomenology and the corresponding sound generation mechanisms in the context of different diseases is justified by the ambition to develop and improve soundbased diagnostic tools. In this perspective, thanks to more accessible largescale highperformance computational resources, computational fluid dynamics (CFD) represents a powerful tool able to furnish a 4D description of mass transportation in the human body and the corresponding FSI dynamics (Sikkandar , 2019).
The current computational power allows to study cardiovascular (Itatani , 2022) and respiratory (Lasota , 2023; Schoder , 2020) diseases in both patientspecific (Kraxberger , 2023; Mylavarapu , 2013) and idealized geometries (Li , 2023; Shabbir , 2023). One remarkable example of an idealized geometry model of a human vessel is a collapsible tube. Collapsible tubes can reproduce all the relevant physical behavior of a human vessel and are widely employed in both numerical models (Babilio , 2023) and clinical practice (Wellman , 2014). The advantage of idealized geometries for studying pathophysiological flows is not only found in the lesser computational cost. Numerical models based on CFD simulation with the ambition to provide a reliable quantitative analysis need to be experimentally validated. The possibility to access the large amount of independent highquality experimental data (Bertram, 2003; Gregory , 2021; Zarandi , 2021) targeting collapsible tubes under different conditions makes the validation step possible and reliable. Moreover, by reducing the complexity of a biological system to its very fundamental parts, it is possible to identify the main physical mechanisms driving its dynamics. For these reasons, collapsible tubes play a major role in analyzing many different pathologies of the cardiovascular and respiratory systems.
One application example of collapsible tubes in biomedical acoustics is the study of respiratory wheezes. Wheezing is one of the most common adventitious sounds generated in the lungs. It is connected to several different pathologies of human lungs, such as asthma (Gern, 2010) and chronic obstructive pulmonary disease (Barnes , 2009). Tonal features characterize it and can happen during both inhalation and expiration. The sound generation mechanism of wheezing sounds is associated with the onset of selfexcited oscillations in the airways due to the interaction between air and the conduct (Heil and Boyle, 2010). It is possible to study such FSIinduced oscillations by implementing numerical (Whittaker , 2010a) and experimental (Oruç , 2007) studies involving collapsible tubes. However, the connection between the FSI and the corresponding generated acoustic signal is still to be completely understood (Zhang and Liu, 2023).
In most of the studies in the literature, collapsible tubes have been employed to study biological fluid flows, selfexcited oscillations, collapse critical pressure, or to produce acoustic reducedorder models. However, the nature of the fluidborne sound generation mechanisms in collapsible tubes relevant for biomedical applications and their relation with fluid mechanics structures is still an open question. This work addresses these problems by studying the fluid behavior in three distinguished collapse phases: prebuckling, postbuckling, and postcontact. The methodology is based on CFD and computational aeroacoustics (CAA) modeling. The two goals of this work are: 1) to determine the main fluid characteristics related to sound generation under different collapse conditions; and 2) to relate the overall sound power level generated by the flow with the collapsible state of the tube. This last point is also motivated by a similar experimental investigation on urethral obstruction modeled as a collapsible tube (Teriö, 1991).
The geometry of a collapsible tube is described by three nondimensional parameters (Fig. 1): the lengthtodiameter ratio d = l_{0}/D, the thicknesstodiameter ratio γ = h/D, and the axial prestretch ratio l = L/l_{0}.
The values of these parameters have been chosen to be relevant for biomedical applications (Hoppin , 1977; Horsfield and Cumming, 1968) and are listed in Table I.
d [–] .  γ [–] .  l [–] . 

3 (2.5–6)  0.06 (0.03–0.1)  1.25 (1–1.6) 
d [–] .  γ [–] .  l [–] . 

3 (2.5–6)  0.06 (0.03–0.1)  1.25 (1–1.6) 
When the flow is present, p_{int} ≠ 0, and consequently the plot in Fig. 2 is no longer valid. Although for larger values of the external pressure (given a fixed internal pressure p_{int}), the tube still undergoes the transitions from prebuckling to postbuckling to postcontact configurations, the value of the intramural pressure does not follow the tube law anymore. The relation between the intramural pressure and the area of the section of the tube is called, in this case, fluid law (Gregory , 2017). The main qualitative difference with the tube law happens in the postcontact configuration, where the geometry of the constriction induces a large pressure loss across the tube. As the area is further reduced, the flow rate drops, and the intramural pressure increases. An important remark is related to the shape of the cross section in the postbuckling configuration. In general, this shape is characterized by the presence of n ≥ 2 lobes. The number of lobes is determined either by geometric imperfections in the structure or by instabilities caused by the flow (Shamass , 2017). In the present work, the focus will be on the n = 2 cases. In a future investigation, the effects of multiple lobes will be investigated. The transition between the three phases of the collapse (prebuckling, postbuckling, and postcontact) is marked by two particular values of the intramural pressure called buckling critical pressures and contact critical pressure. The value of such critical pressures depends on both geometric and elastic parameters of the system (Kozlovsky , 2014; Zarandi , 2021). In Laudato (2023), the authors have shown that the buckling critical pressure can be estimated by treating the buckling as a secondorder phase transition (see also Turzi, 2020, for a rigorous proof for a 1D ring). In Laudato and Mihaescu (2023), a treatment for estimating the contact critical pressure is derived.
The fluid behavior in a collapsible tube under different collapse scenarios has been extensively studied from theoretical (Shapiro, 1977), numerical (Heil, 1997), and experimental (Gregory , 2021) points of view. However, understanding the relation between the tube's characteristic flow structures and the corresponding acoustic features remains an open query. Some relevant studies have been conducted (Alenius , 2015; Åbom , 2006), in which the authors have studied the aeroacoustics of an orifice plate in a duct using the twoport scattering method. This method, however, targets the characterization of the scattering matrix of the system for an already existing acoustic signal (Boden and Åbom, 1995). Moreover, the system's geometry under analysis does not reflect the 3D behavior of a human vessel.
The present work lies in the larger project of establishing a relation between the acoustics of a human vessel and its collapse level, with the potential development of new diagnostic tools for respiratory and cardiovascular diseases. Section II presents the numerical model implemented in this study. The goal is to perform an indepth analysis of the fluid behavior under the three collapse phases (prebuckling, postbuckling, and postcontact) of a collapsible tube. The flow features responsible for the onset of acoustic waves propagating in the tube are studied. Finally, the relation between the acoustic power level generated in the system and the tube's collapse level is presented, supporting the existence of an acoustic tube law.
The structure of the paper is as follows. In Sec. II, the details of the numerical model implemented in this work are discussed. Section III concerns the sound generation mechanisms due to the interaction of the fluid flow with the geometries corresponding to three different stages of the collapse. In Sec. IV, the analysis of the acoustic power radiated by the collapsible tube under the three different collapse phases is discussed. The onset of acoustic waves in the tube domain is investigated in Sec. V. Finally, discussion and conclusions can be found in Sec. VI.
II. NUMERICAL MODEL
It is possible to study the airflow under the three collapse states of a physiological tube and the corresponding acoustic field by employing numerical methods. The numerical model is implemented in the commercial software Siemens StarCCM+ (version 2210). The simulation strategy consists of two steps:

Solidonly simulation of a collapsible tube under the effect of an isotropic external pressure. The goal is to compute the resulting deformed configurations of the tube corresponding to the prebuckling, postbuckling, and postcontact phases of the collapse. These will be compared for validation with experimental results and used as fluid domain geometries in the second step.

Simulations of the velocity and pressure field of air flowing in the three fixed collapsed configurations obtained in the previous step. Since air is treated as a compressible ideal gas, the resulting acoustic pressure fluctuations are predicted at the same time.
A. Solid elastic model
At each time step (Δt = 0.1 s), the value of the intramural pressure and the area of the central cross section are registered. It is then possible to determine the corresponding tube law. The validation of the numerical model has been performed by comparison with a public experimental dataset (Gregory , 2017, 2021) obtained via 3D camera measurements of the area of the central cross section of the tube. A digital replica of the experimental specimen has been implemented in StarCCM+, and the value of the maximum pressure P has been set to match the experimental conditions. The comparison between the numerical and experimental tube laws shows a fair match (Fig. 3).
The output of these solidonly simulations consists of three experimentally validated deformed configurations of a collapsible tube, corresponding to the prebuckling, postbuckling, and postcontact phase (Fig. 4). Such deformed configurations will be employed as numerical domains for the subsequent fluid flow and acoustic field simulations.
B. Fluid model
The three geometrical computational domains depicted in Fig. 4 are used to simulate the air velocity and pressure fields. The domain is extruded from the collapsed tube in the upstream and downstream directions to prevent any influence of the imposed boundary conditions on the region of interest. This results in a division of the computational domain into three regions (Fig. 5): inlet, sound generation and propagation region, and acoustic suppression zone (ASZ) outlet. The sound generation and propagation region defines the region of interest. The inlet and outlet are treated as free stream boundaries, which impose a flow velocity of 10 m/s and a static temperature of 308 K. One advantage of free stream boundaries is that they are perfect nonreflective boundaries for plane waves. As discussed in the following, the geometry and boundary conditions of the problem allow only plane waves to propagate in the acoustic frequency range. The flow pressure fluctuations, characterized by a much smaller length scale, are dissipated by the mesh stretching in the ASZ and, therefore, will not be reflected. All the other walls are treated with noslip boundary conditions.
The simulation solver and mesh grid are tuned to perform an unsteady compressible flow direct noise calculation (DNC) (Bailly , 2010), which allows the whole aeroacoustics characterization of the system to be performed within one simulation. Both the unsteady flow, which causes the noise generation, and the resulting radiated sound waves are solved simultaneously. This approach imposes precise requirements on the mesh grid, which is built in three steps:

A coarse unstructured mesh of polyhedral elements is defined to discretize the system. It allows us to obtain a wellconverged preliminary solution based on the Reynolds Averaged NavierStokes approach that will be used as the initial condition for the following unsteady simulation.

Since the time evolution of the fluid features is resolved via a segregated large eddy simulation (LES) model with a WALE subgrid scale model, a first mesh refinement is implemented in the region of interest to resolve the Taylor microscale. Moreover, 10 prism layers are implemented on the walls to ensure wellresolved boundary layers.

To ensure that all the waves in the acoustic frequency range can propagate in the domain, the mesh is further refined to ensure that at least 20 grid cells resolve the shortest acoustic wavelength.
Outside the region of interest, the mesh is coarser and is stretched in the ASZ. In this way, the smallerscale aerodynamic fluctuations are numerically dissipated away from the region of interest and cannot reflect into the domain.
The choice of the time step can also influence the numerical dissipation of acoustic waves. An implicit secondorder time marching scheme is implemented in the model. A minimum of 15 time steps per period of the highest frequency sound wave are recommended for a DNC simulation. These requirements have been extensively tested in previous numerical (Pietroniro , 2022) and experimental (Schickhofer , 2019) works by the authors. In particular, it has been proved that boundary layer phenomena, wave propagation, and wave dispersion can be accurately modeled by StarCCM+ if the mesh satisfies these requirements. For this simulation, the time step has been set to Δt = 3.1 × 10^{–6} s, corresponding to about 16 time steps for a 20 000 Hz acoustic wave.
The presented fluid model has been implemented in the commercial software StarCCM+ by Siemens. The fluid flow LESbased solver has been previously validated in similar geometries by comparing the pressure predictions to corresponding experimental midline pressure data (Schickhofer and Mihaescu, 2020).
Velocity and pressure fields are extracted to calculate timeaveraged and fluctuating components on relevant cross sections for the following postprocessing. The full time history of pressure and velocity fields on the domain surfaces are stored at each time step to compute the surface Fourier transform. Several monitoring points for both velocity and pressure fields are placed in the domain.
III. SOUND GENERATION MECHANISMS
The maximum velocity magnitude of the airflow under the three different collapse states of the tube is on the order of 40 m/s, corresponding to a low Mach number regimen ( $ Ma \u2243 0.12$). Under such conditions, sound is generated mainly by the unsteady pressure loads caused by the flow of vortical structures washing the walls of the domain. Therefore, analyzing such interaction in the three geometries under investigation is interesting. To this end, the distribution of the mean axial component of the velocity is analyzed on two orthogonal longitudinal cross sections (Figs. 6 and 7), corresponding to the lateral view (y–z plane) and the top view (x–z plane).
The Poiseuillelike flow in the prebuckling configuration of the tube does not show any interesting flow feature. It is irrelevant for the sound generation as it shows negligible timedependent interactions with the walls. Consequently, it will not be further discussed in this section. The flow in the postbuckling configuration (Fig. 6) is characterized by a clear Coandă effect. This effect describes the tendency of a jet flowing from an orifice to attach to one side of the domain, developing a recirculation region on the other side. As shown in the y–z plane cross section (lateral view, top panel in Fig. 6), the buckled configuration acts on the fluid like a sudden contractionsudden expansion zone. The resulting jet stays attached to the lower part of the domain (due to the Coandă effect), producing a large recirculation area in the y–z plane. As discussed in Sec. V, the Coandă effect is responsible for the onset of acoustic waves in the domain. The cross section in the x–z plane (top view, bottom panel in Fig. 6) shows the symmetric behavior of the airflow through the two lobes of the buckled configuration.
In the postcontact configuration of the tube, the lumen is partially closed. Such a critical change in the geometry corresponds to a clear change in the fluid behavior. Due to the contact of the internal walls of the tube, the Coandă changes its direction (lateral view, top panel of Fig. 7). As shown in the cross section in the x–z plane (top view, bottom panel in Fig. 7), the contact region acts like an adverse pressure gradient region, causing the separation of the boundary layers from the wall. Consequently, a large recirculation area is produced, constraining the flow toward the lateral walls of the domain and, in turn, generating sound.
Interestingly, the topology change from the postbuckling to the postcontact configuration of the collapsible tube induces a rotation of 90° of the main recirculation area. The periodic flow separations in the postbuckling and postcontact configurations cause an oscillating force on the fluid in the perpendicular direction to the flow. Such unsteady force is the primary sound generation mechanism in a collapsible tube under physiological conditions. It is possible to estimate the unsteady pressure loads developed on the walls in the three collapse configurations by analyzing the pressure root mean square (RMS) on a probe line located on one boundary of the domain (bottom panel in Fig. 8). Remarkably, the postbuckling configuration shows much higher values for the pressure fluctuations on the wall, suggesting that this configuration generates a higher sound power level with respect to the postcontact configuration. To prove this point, an indepth sound power analysis is presented in the next section.
Due to the quasistatic assumption, the pressure fluctuations induced in the fluid by the motion of the walls are not included in the analysis. These effects can potentially contribute to the sound generation mechanisms and will be investigated in a followup investigation.
IV. SOUND POWER ANALYSIS
The analysis of the velocity field discussed in Sec. III shows that the flow characteristics and the corresponding sound generation mechanisms depend dramatically on the collapse state of the tube. The sound power level can be computed by processing the results of the unsteady compressible CFD simulations, analysis carried out for all three different collapsed geometries under analysis. The analysis is based on the employment of the source field Lighthill's acoustic analogy (Lighthill, 1952). This entails that any conversion of acoustic energy into flow kinetic energy is neglected, as well as the sound field interference on the fluid flow. It is also assumed that the effects of the fluid motion on the sound propagation in the source region are not considered.
The present section contains an analysis of the sound power level generated by the fluid flow in the nearfield under the different collapse phases. The investigated domain can be interpreted as the acoustic nearfield. Indeed, as will be shown in Sec. V, the wavelength corresponding to the acoustic pressure fluctuations is of the same order as the longitudinal dimension of the domain. Section V will present the study of the subsequent propagation of acoustic waves in the domain. It will exploit the full power of the DNC simulations, which describe the coupling between the sound and the flow fields.
Interestingly, the maximum of the sound power level corresponds to the postbuckling configuration (at least, in the quasistatic regimen of the model). The explanation is related to the stronger interaction between the flow and the walls in the postbuckling configuration. Indeed, as discussed in Sec. III, the numerical model predicts larger RMS pressure, with respect to the postcontact configuration, in the region located downstream of the minimum crosssectional area where the developed shear layers are interacting with the walls (Fig. 8). Larger pressure fluctuations result in higher values of $ F \u0307 2 \xaf$ in Eq. (7) and, in turn, in larger sound power levels. Such observation implies an optimal collapsed tube configuration, which yields maximum sound power between the postbuckling and the postcontact phase. This suggests the existence of an acoustic tube law, relating the intramural pressure (or the cross section area) to the sound power radiated by the collapsible tube. The derivation of the acoustic tube law will be the object of future work, in which the system will be studied in an FSI setting with a continuous collapse. It is important to remark that the value computed using Eq. (7) represents the total acoustic energy generated by the flow. However, it does not specify how such an energy will be radiated. Partially, it will generate acoustic waves, partially it will excite the domain walls, resulting in aeroelastic perturbations, and partially will be dissipated. The farfield propagation of the sound, both within the lumen and outside, will be investigated in future work. A final consideration is that the effects of the quadrupole sources can be further investigated, although their effects in the physiological low Mach number regimen, should not change the qualitative results shown in Fig. 9.
V. ONSET OF ACOUSTIC WAVES
The sound generation mechanisms described in Sec. III induce pressure fluctuations in the collapsible tube. The DNC setup implemented for the numerical simulations allows us to analyze at the same time the aerodynamic and acoustic components of such pressure fluctuations. This section analyzes the spectral properties of the pressure field in the tube under different collapse states. Moreover, the onset of acoustic waves is observed.
Probing points for the pressure field are positioned in sensible positions in the domain to capture the fluctuations associated with the vortical structures developed in the shear layer (Figs. 10–12). From the time history of the pressure signals, it is possible to obtain the corresponding power spectral density (PSD) for the three different collapse states of the tube.
A first remark is that the PSD for the prebuckling configuration has negligible amplitudes compared to the more collapsed states of the tube, as expected from the negligible unsteadiness of the flow behavior in this configuration. The second observation is that all the PSD plots show narrowband peaks. The corresponding frequencies are related to the smallscale fluctuations in the shear layers associated with the separation mechanisms described in Sec. III. The shear layer fluctuations are visible in correspondence of the probe points in the axial velocity snapshot plots in Figs. 11 and 12.
The PSD for the postbuckling configuration shows clear asymmetric amplitudes at the peak frequency f = 5630 Hz for the two nearfield probes, which is explained by the asymmetric flow induced by the Coandă effect. Such asymmetry is not detected in the PSD levels for the postcontact configuration due to the disappearance of the Coandă effect, which induces a more symmetric flow behavior. It is interesting to notice that, although the power density at the peak frequency in the postcontact configuration (f = 6250 Hz) is higher than the corresponding one in the postbuckling configuration, the analysis in Sec. IV shows that the overall energy of the pressure fluctuation in the postbuckling configuration is larger.
Both configurations show aerodynamic and acoustic pressure fluctuations, indicating that acoustic waves are generated at the frequency corresponding to the flow shear layer fluctuations. The short wavelength pressure features downstream of the constriction have a clear aerodynamic origin. Indeed, since Eq. (9) shows the time evolution of one frequency component of the signal, it is possible to estimate the phase velocity c_{f}, as λƒ = c_{f}, where λ is the wavelength of the feature. At the same time, it is possible to identify pressure fluctuations characterized by waves with larger wavelengths moving at the speed of sound. These waves are most noticeable upstream of the constriction, where the aerodynamic fluctuations are absent. Hence, it is possible to conclude that the fluid dynamics features described in Sec. III, characterized by frequencies f_{PB} and f_{PC}, induce the onset of acoustic waves within the domain.
VI. DISCUSSION AND CONCLUSION
The analysis of the flowinduced sound generation mechanisms in a collapsible tube has been performed using CFD and CAA methods. The different flow features associated with the three collapse states of the tube have been identified and related to the generation of acoustic power. The analysis of the pressure fluctuation on the walls of the tube and the corresponding sound power analysis has shown that the maximum sound power is produced in the postbuckling configuration. This suggests the existence of a nontrivial acoustic tube law, with a maximum sound power level located in the range of pressures corresponding to the nonbuckling configuration. Clearly, the effects of moving boundaries might affect the results obtained using the present model, and additional investigations are in order. Moreover, the compressible LES model has shown the onset of acoustic plane waves at the frequencies of the shear layer fluctuations. The results obtained in this analysis open to new perspective research questions.
The relation between the airflow peak frequencies indicated in Figs. 10–12 and the frequency of possible selfexcited oscillations of the airways' wall, in turn related to wheezing (Grotberg and Gavriely, 1989), is not straightforward. Heil and Boyle (2010) have shown that a collapsible tube in the postbuckling configuration shows selfexcited oscillations for large enough Reynolds numbers. However, understanding the underlying physical mechanisms for the onset of such oscillations is currently an open question (Heil and Hazel, 2011). A possible research question is how the energy content of the flow pressure fluctuations due to the unsteady flow interacting with one side of the collapsible tube in the postbuckling configuration is coupled with the solid structure. In other words, it is relevant to determine the frequency response function of a collapsible tube and how it is related to the collapsible state of the system. The role of the flowinduced acoustic waves on the onset of selfexcited oscillations can be also investigated. The analysis of the flow and acoustic features presented in this work represents a first step in this direction, and it will be addressed in a future investigation involving a twoway coupled FSI numerical model. Moreover, possible refining of the solid models can be considered to produce more realistic simulations. In particular, as in many biological systems, it is possible to consider the walls of the domain as including a microstructure of fibers that can have a nonnegligible effect on the elastic dynamics of the system (Ciallella and Steigmann, 2023; Giorgio , 2018). Finally, another approach could be to employ the perturbed convective wave equation method to analyze the acoustic features in a collapsed geometry using a perturbative method (Maurerlehner , 2022).
Another interesting perspective is related to the analysis of the nondimensional frequency associated with the peaks in the PSD plots 10–12 and the physics of the fluid features in the postbuckling and postcontact configuration of the tube. A more indepth analysis can be designed to prove this thesis. The strategy would be to simulate the fluid flow in several instances of postbuckling and postcontact configurations and to compute the corresponding Strouhal number. The thesis would be proven if the Strouhal number is consistently equal to $ f \u0303 P B$ for all the postbuckling simulations and to $ f \u0303 P C$ for all the postcontact simulations. This analysis will be the object of a future investigation.
Acknowledgments
This work was supported by KTH Engineering Mechanics in the thematic area of Biomechanics, Health, and Biotechnology. M.L. was supported by Swedish Research Council Grant No. 2022–03032. E.Z. was supported in part by Swedish Research Council Grant No. 2020–04668. The authors acknowledge PRACE for awarding access to the Fenix Infrastructure resources at CINECA, which are partially funded by the European Union's Horizon 2020 research and innovation programme through the ICEI project under Grant Agreement No. 800858. The simulations were partly run on the Swedish National Infrastructure for Computing resources at the PDC Centre for High Performance Computing.
Author Declarations
Conflict of Interest
The authors declare no conflict of interest.
DATA AVAILABILITY
The simulation data can be provided under reasonable request.