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 post-buckling configuration. The idea of an acoustic tube law is proposed. The presented results are relevant to the study of self-excited oscillations and wheezing sounds in the lungs.

In daily medical practice, auscultation is among the most cost-effective 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 fluid-structure interaction (FSI) phenomenology and the corresponding sound generation mechanisms in the context of different diseases is justified by the ambition to develop and improve sound-based diagnostic tools. In this perspective, thanks to more accessible large-scale high-performance 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 patient-specific (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 high-quality 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 self-excited oscillations in the airways due to the interaction between air and the conduct (Heil and Boyle, 2010). It is possible to study such FSI-induced 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, self-excited oscillations, collapse critical pressure, or to produce acoustic reduced-order models. However, the nature of the fluid-borne 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: pre-buckling, post-buckling, and post-contact. 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 non-dimensional parameters (Fig. 1): the length-to-diameter ratio d = l0/D, the thickness-to-diameter ratio γ = h/D, and the axial pre-stretch ratio l = L/l0.

FIG. 1.

(Color online) Top, Radial cross-sectional view of the domain pictured in its post-buckling configuration. The (half) diameter D/2 and the thickness h are sketched. Bottom, Side view of the actual configuration of the domain under the action of the boundary conditions pext (red dashed arrows) and l (light blues arrow). Two black solid arrows indicate the clamped boundary. Blue box corresponds to the initial configuration of the system.

FIG. 1.

(Color online) Top, Radial cross-sectional view of the domain pictured in its post-buckling configuration. The (half) diameter D/2 and the thickness h are sketched. Bottom, Side view of the actual configuration of the domain under the action of the boundary conditions pext (red dashed arrows) and l (light blues arrow). Two black solid arrows indicate the clamped boundary. Blue box corresponds to the initial configuration of the system.

Close modal

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.

TABLE I.

Values of the geometric parameters of the system. The values in parentheses correspond to the physiological range in the human airways.

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) 
In this study, one short side of the tube is clamped while the other is stretched by an amount indicated by the pre-stretch parameter l and then clamped. Under these assumptions, the collapse is only driven by the so-called intramural pressure, i.e., the pressure difference between the interior and the exterior of the tube:
p intr = p int p ext .
(1)
When this difference is negative, the tube starts to collapse. In the absence of flow, pint = 0, and the collapse is described by the so-called tube law (Whittaker , 2010b), which relates the area of the central cross section of the tube and the intramural pressure (Fig. 2). The tube collapses for small negative values of the intramural pressure by maintaining its axisymmetric shape (pre-buckling phase). When the intramural pressure assumes larger negative values, the tube undergoes a buckling phenomenon, resulting in a non-axisymmetric shape (post-buckling phase). Finally, for even larger negative values of the intramural pressure, the internal walls of the tube touch each other (post-contact phase) (Flaherty , 1972). At first, the contact region is one point, and it becomes a line for more negative intramural pressures (Fig. 2).
FIG. 2.

(Color online) The tube law for (d, γ, l) =  (4, 0.06, 1.1). Right axis indicates the intramural pressure. Top axis indicates the area of the central cross section normalized on the initial area. Green areas represent the regions of the transition from the pre-buckling to the post-buckling phase and to the post-contact configuration (Laudato , 2023).

FIG. 2.

(Color online) The tube law for (d, γ, l) =  (4, 0.06, 1.1). Right axis indicates the intramural pressure. Top axis indicates the area of the central cross section normalized on the initial area. Green areas represent the regions of the transition from the pre-buckling to the post-buckling phase and to the post-contact configuration (Laudato , 2023).

Close modal

When the flow is present, pint ≠ 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 pint), the tube still undergoes the transitions from pre-buckling to post-buckling to post-contact 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 post-contact 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 post-buckling 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 (pre-buckling, post-buckling, and post-contact) 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 second-order 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 two-port 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 in-depth analysis of the fluid behavior under the three collapse phases (pre-buckling, post-buckling, and post-contact) 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.

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 Star-CCM+ (version 2210). The simulation strategy consists of two steps:

  1. Solid-only 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 pre-buckling, post-buckling, and post-contact phases of the collapse. These will be compared for validation with experimental results and used as fluid domain geometries in the second step.

  2. 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.

The main assumption of the model is to treat the collapse of the walls using a quasi-static assumption. The limitations of this assumption on the sound generation mechanism will be discussed in detail in Secs. IV and V.

The reference configuration geometry is a 3D cylindrical flexible tube with finite thickness, and it is implemented in the built-in CAD software in Star-CCM+. The values of the geometric parameters (d, γ, l) are listed in Table I and are typical for human airways. The system is subject to the following boundary conditions (Fig. 1). One short side of the tube is clamped, whereas the other one is first stretched by a quantity related to the parameter l and then clamped. The external walls of the domain are under the effect of an isotropic inward pressure pext, which linearly increases in time. Due to the absence of flow, the internal pressure vanishes, pint = 0, and the intramural pressure depends only on the external pressure, pintr = – pext(t). The following relation describes the time dependence of the external pressure:
p ext ( t ) = P τ t ,
(2)
where P is the maximum external pressure and t [ 0 , τ ] are the time steps. By changing the value of P, it is possible to obtain, at time t = τ, the desired collapse state of the tube. If P < P c r b, i.e., smaller than the buckling critical pressure, the tube will be in its pre-buckling phase. Analogously, if P c r b < P < P c r c or P c r c < P, the resulting collapse state will correspond to the post-buckling and post-contact phase, respectively, where P c r c is the contact critical pressure. The values corresponding to the three different collapse states are 1000, 2000, and 3000 Pa for the pre-buckling, post-buckling, and post-contact configuration, respectively (also Fig. 3) and are compatible with typical physiological conditions of the airways. Indeed, under normal breathing conditions, the intramural pressure is approximately 2000 Pa (Lai-Fook and Rodarte, 1991). However, for asthma patients, for which the collapse of the airways is often manifested, these values can reach up to 10 000 Pa (Evans and Whitelaw, 2009), depending on the size of the patient (Lausted , 2006). For values of p ext > P c r c, the internal walls of the tube (the lumen) touch each other. The contact is handled employing a repulsive virtual plane (Laudato and Mihaescu, 2023), which prevents the intersection of the solid domain. A sensitivity study of the ratio P/τ has been performed by Laudato (2023), and it is not reported here for brevity. For all the following analyses, τ = 2.5 s is used.
FIG. 3.

(Color online) Comparison between the tube law obtained via the presented numerical model (red dashed line) versus the corresponding experimental data from Gregory (2021).

FIG. 3.

(Color online) Comparison between the tube law obtained via the presented numerical model (red dashed line) versus the corresponding experimental data from Gregory (2021).

Close modal
Under such boundary conditions, the system undergoes large deformation that cannot be adequately treated with linear elasticity (Laudato , 2023). Therefore, the tube is modeled as a Neo-Hookean material whose strain energy potential is defined as
U = Ψ ( I 3 ) + Φ 2 ( I ̃ 1 ) 2 ,
(3)
where I and I ̃ are the first and second invariants of the right Cauchy-Green tensor, respectively. The coefficients ψ and Φ are defined as
Ψ = E 4 ( 1 + ν ) , Φ = E ν ( 1 2 ν ) ( 1 + ν ) ,
(4)
where E is the Young's modulus and ν is the Poisson ratio. The values of these elastic parameters are the same as the samples used in Gregory (2021), whose experimental data will be used to validate the numerical results (E = 106 Pa, ν = 0.49, ρ = 1000 kg/m3).

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 Star-CCM+, 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 solid-only simulations consists of three experimentally validated deformed configurations of a collapsible tube, corresponding to the pre-buckling, post-buckling, and post-contact phase (Fig. 4). Such deformed configurations will be employed as numerical domains for the subsequent fluid flow and acoustic field simulations.

FIG. 4.

(Color online) From left to right: the cross sections of the fluid domains obtained from the solid model simulations for the pre-buckling, post-buckling, and post-contact configurations.

FIG. 4.

(Color online) From left to right: the cross sections of the fluid domains obtained from the solid model simulations for the pre-buckling, post-buckling, and post-contact configurations.

Close modal

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 non-reflective 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 no-slip boundary conditions.

FIG. 5.

(Color online) Lateral view of the computational domain of the fluid numerical model. Green represents the inlet region. Blue rectangle shows the region of interest and represents the sound generation and propagation region. Red represents the acoustic suppression zone.

FIG. 5.

(Color online) Lateral view of the computational domain of the fluid numerical model. Green represents the inlet region. Blue rectangle shows the region of interest and represents the sound generation and propagation region. Red represents the acoustic suppression zone.

Close modal

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:

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

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

  3. 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 smaller-scale 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 second-order 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 Star-CCM+ 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 Star-CCM+ by Siemens. The fluid flow LES-based solver has been previously validated in similar geometries by comparing the pressure predictions to corresponding experimental mid-line pressure data (Schickhofer and Mihaescu, 2020).

Velocity and pressure fields are extracted to calculate time-averaged and fluctuating components on relevant cross sections for the following post-processing. 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.

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 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 (yz plane) and the top view (xz plane).

FIG. 6.

(Color online) Mean axial velocity in the two main longitudinal cross sections (yz plane and xy plane, respectively) of the domain in post-buckling configuration. The flow is characterized by the presence of Coandă effect.

FIG. 6.

(Color online) Mean axial velocity in the two main longitudinal cross sections (yz plane and xy plane, respectively) of the domain in post-buckling configuration. The flow is characterized by the presence of Coandă effect.

Close modal
FIG. 7.

(Color online) Mean axial velocity in the two main longitudinal cross sections (yz plane and xy plane, respectively) of the domain in post-contact configuration. Empty regions correspond to the location of the contact of the lumen.

FIG. 7.

(Color online) Mean axial velocity in the two main longitudinal cross sections (yz plane and xy plane, respectively) of the domain in post-contact configuration. Empty regions correspond to the location of the contact of the lumen.

Close modal

The Poiseuille-like flow in the pre-buckling configuration of the tube does not show any interesting flow feature. It is irrelevant for the sound generation as it shows negligible time-dependent interactions with the walls. Consequently, it will not be further discussed in this section. The flow in the post-buckling 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 yz plane cross section (lateral view, top panel in Fig. 6), the buckled configuration acts on the fluid like a sudden contraction-sudden 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 yz 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 xz 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 post-contact 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 xz 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 post-buckling to the post-contact configuration of the collapsible tube induces a rotation of 90° of the main recirculation area. The periodic flow separations in the post-buckling and post-contact 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 post-buckling 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 post-contact configuration. To prove this point, an in-depth sound power analysis is presented in the next section.

FIG. 8.

(Color online) Top, Pressure RMS comparison between the three different geometries. The post-buckling configuration shows the largest pressure fluctuations on the wall. Bottom, Position of the probe line where the pressure RMS has been calculated.

FIG. 8.

(Color online) Top, Pressure RMS comparison between the three different geometries. The post-buckling configuration shows the largest pressure fluctuations on the wall. Bottom, Position of the probe line where the pressure RMS has been calculated.

Close modal

Due to the quasi-static 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 follow-up investigation.

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 near-field under the different collapse phases. The investigated domain can be interpreted as the acoustic near-field. 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.

Lighthill's equation is a reformulation of the compressible Navier-Stokes equations in the form of an inhomogeneous wave equation expressed in terms of acoustic pressure fluctuation:
1 c 0 2 2 p t 2 2 p x i x i = t ( 1 c 0 2 t ( p c 0 2 ρ ) ) g i x i + 2 x i x j ( ρ u i u j τ i j ) .
(5)
Here, the prime variables ρ′ and p′ are acoustic fluctuations of the density and pressure mean-field, respectively, c0 is the speed of sound, gi represents the forces acting on the fluid, and τij is the deviatoric stress tensor. The source term in Eq. (5) can be decomposed into the following terms:
s m = t ( 1 c 0 2 t ( p c 0 2 ρ ) ) , s d = g i x i , s q = 2 x i x j ( ρ u i u j τ i j ) .
(6)
These terms correspond to the n = 0, 1, 2 terms of a multi-pole expansion of order 2n, i.e., monopole, dipole, and quadrupole source terms. The monopole term sm describes sound source mechanisms due to deviation from adiabatic assumption, and it vanishes in the present analysis due to quasi-static assumption. The low Mach number regimen of the system implies that the contribution of the quadrupole forces (which describe the sound produced by turbulence) is negligible compared to the dipole term. Indeed, the ratio of sound power generated by a dipole source to the one generated by a quadrupole source scales as Ma– 2, which implies that the contribution due to quadrupole sources to the total sound power level is in this case of about 2%. Consequently, the main contribution to the sound power generated by the flow in a collapsible tube under physiological conditions comes from the dipole source sd.
The dipole source term describes the sound produced by the fluctuating forces due to flow separation from a rigid wall. As anticipated in Sec. III, the periodic character of the developed flow structures in the generated shear layer, as well as the flapping nature of the developed shear layer washing the wall, causes an oscillating force on the walls in the perpendicular direction with respect to the flow. Another consequence of the low Mach number regimen is that the collapsible tube acts as a compact acoustic source. Indeed, if T is the typical temporal scale for sound production and U is the typical mean flow velocity, the typical acoustic frequency is ƒ = 1/T ∼ U/D, where D is the diameter of the tube. The corresponding wavelength is λ = c0/ƒ ∼ D/Ma, which implies that the source region is much smaller than the typical wavelength for low Mach numbers. Under these assumptions, the sound power radiated by a dipole source (Åbom, 2006) can be written as follows:
W d = F ̇ 2 ¯ 12 π ρ 0 c 0 3 ,
(7)
where the dot represents the time derivative, and F 2 = | F | 2, i.e., the (squared) magnitude of the total force acting on the fluid, F i = V g i d V, where V is the volume of the fluid domain. It is possible to compute this term by extracting the total force acting on the fluid from the CFD simulations for the pre-buckling, post-buckling, and post-contact configurations. The resulting sound power levels are plotted in Fig. 9.
FIG. 9.

(Color online) Comparison of sound power level (SWL) defined as 10log(Wd), where Wd is defined in Eq. (7) in the three collapse states under analysis. The value for the pre-buckling is smaller than the reference and has been set to zero.

FIG. 9.

(Color online) Comparison of sound power level (SWL) defined as 10log(Wd), where Wd is defined in Eq. (7) in the three collapse states under analysis. The value for the pre-buckling is smaller than the reference and has been set to zero.

Close modal

Interestingly, the maximum of the sound power level corresponds to the post-buckling configuration (at least, in the quasi-static regimen of the model). The explanation is related to the stronger interaction between the flow and the walls in the post-buckling configuration. Indeed, as discussed in Sec. III, the numerical model predicts larger RMS pressure, with respect to the post-contact configuration, in the region located downstream of the minimum cross-sectional area where the developed shear layers are interacting with the walls (Fig. 8). Larger pressure fluctuations result in higher values of F ̇ 2 ¯ 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 post-buckling and the post-contact 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 far-field 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.

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.

FIG. 10.

(Color online) Top, Pressure PSD magnitude for the pre-buckling configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

FIG. 10.

(Color online) Top, Pressure PSD magnitude for the pre-buckling configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

Close modal
FIG. 11.

(Color online) Top, Pressure PSD magnitude for the post-buckling configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

FIG. 11.

(Color online) Top, Pressure PSD magnitude for the post-buckling configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

Close modal
FIG. 12.

(Color online) Top, Pressure PSD magnitude for the post-contact configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

FIG. 12.

(Color online) Top, Pressure PSD magnitude for the post-contact configuration. Bottom, Instantaneous axial velocity and probe positions in the domain. The probes' colors correspond to the colors in the PSD plots.

Close modal

A first remark is that the PSD for the pre-buckling 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 narrow-band peaks. The corresponding frequencies are related to the small-scale 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 post-buckling configuration shows clear asymmetric amplitudes at the peak frequency f = 5630 Hz for the two near-field probes, which is explained by the asymmetric flow induced by the Coandă effect. Such asymmetry is not detected in the PSD levels for the post-contact 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 post-contact configuration (f = 6250 Hz) is higher than the corresponding one in the post-buckling configuration, the analysis in Sec. IV shows that the overall energy of the pressure fluctuation in the post-buckling configuration is larger.

The value of the peak frequencies in the post-buckling and post-contact configuration can be compared by means of the corresponding dimensionless frequency (Strouhal number) defined as
f ̃ = f D U ,
(8)
where D = 2 A min / π is the effective diameter, Amin is the area of the central cross section of the tube, and U is the bulk flow velocity (U = 22 m/s). The values of Strouhal number for the post-buckling and post-contact cases are f ̃ P B = 0.68 and f ̃ P C = 0.52, respectively. This difference is likely attributed to the different flow mechanisms related to the post-buckling and post-contact collapse state. Indeed, the flow behavior changes completely as the tube transitions from the post-buckling to the post-contact configuration.
To study the onset of acoustic waves within the domain induced by the flow features presented in Sec. III, it is possible to compute the Fourier transform of the pressure on the two main longitudinal cross sections. The computation is performed using a built-in implementation in Star-CCM+. The pressure data are windowed with a Hann function, and the number of analysis blocks is 6 with an overlap factor of 0.25. Once the Fourier transform is obtained, it is possible to compute the corresponding real and imaginary parts evaluated at the two peak frequencies for the post-buckling and post-contact configurations, fPB = 5630 Hz and fPC = 6250 Hz, respectively. These can be used as the coefficients of the components of the Fourier series of the pressure time evolution p(t) at the frequency fPB and fPC. In formulae:
p ( t ) = Re [ p ̂ ( f i ) ] cos ( 2 π t T ) + Im [ p ̂ ( f i ) ] sin ( 2 π t T ) ,
(9)
where p ̂ ( f i ) is the Fourier transform of the pressure signal evaluated at the frequency fi, where i = {PB, PC}. The corresponding pressure fluctuations at specific peak frequencies on the two main longitudinal cross sections are displayed in Figs. 13 and 14 for the post-buckling (ƒi = ƒPB) and the post-contact (ƒi = ƒPC) configurations, respectively.
FIG. 13.

(Color online) Snapshot of the pressure fluctuations in the post-buckling configuration at ƒ = ƒPC = 5630 Hz. The larger wavelength fluctuations propagate at the speed of sound.

FIG. 13.

(Color online) Snapshot of the pressure fluctuations in the post-buckling configuration at ƒ = ƒPC = 5630 Hz. The larger wavelength fluctuations propagate at the speed of sound.

Close modal
FIG. 14.

(Color online) Snapshot of the pressure fluctuations in the post-contact configuration at ƒ = ƒPC = 6250 Hz. The larger wavelength fluctuations propagate at the speed of sound.

FIG. 14.

(Color online) Snapshot of the pressure fluctuations in the post-contact configuration at ƒ = ƒPC = 6250 Hz. The larger wavelength fluctuations propagate at the speed of sound.

Close modal

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 cf, as λƒ = cf, 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 fPB and fPC, induce the onset of acoustic waves within the domain.

It is possible to further determine the nature of these acoustic waves by analyzing the cut-off frequency f n c of the system, i.e., the minimum frequency that allows the propagation of higher-order modes of order n in the collapsible tube. The corresponding formula is (Åbom, 2006) as follows:
f n c = c 0 k n 2 π 1 Ma 2 ,
(10)
where k n represents the n-th eigenvalues of the Helmholtz equation obtained by assuming a harmonic time-dependent solution in the homogeneous wave equation. For a circular duct, the first eigenvalue is k 1 = 1.841 / r, where r = 0.003 m is the radius of the non-deformed part of the collapsible tube, where the wave propagation happens. Consequently, the corresponding cut-off frequency is on the order of 30 kHz, which implies that only acoustic plane waves can propagate in the domain for the frequency range considered in this study.

The analysis of the flow-induced 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 post-buckling configuration. This suggests the existence of a non-trivial acoustic tube law, with a maximum sound power level located in the range of pressures corresponding to the non-buckling 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 self-excited 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 post-buckling configuration shows self-excited 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 post-buckling 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 flow-induced acoustic waves on the onset of self-excited 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 two-way 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 non-negligible 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 non-dimensional frequency associated with the peaks in the PSD plots 10–12 and the physics of the fluid features in the post-buckling and post-contact configuration of the tube. A more in-depth analysis can be designed to prove this thesis. The strategy would be to simulate the fluid flow in several instances of post-buckling and post-contact configurations and to compute the corresponding Strouhal number. The thesis would be proven if the Strouhal number is consistently equal to f ̃ P B for all the post-buckling simulations and to f ̃ P C for all the post-contact simulations. This analysis will be the object of a future investigation.

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.

The authors declare no conflict of interest.

The simulation data can be provided under reasonable request.

1.
Åbom
,
M.
(
2006
).
An Introduction to Flow Acoustics
, 4th ed. (
KTH Royal Institute of Technology
,
Stockholm, Sweden
).
2.
Åbom
,
M.
,
Allam
,
S.
, and
Boij
,
S.
(
2006
). “
Aero-acoustics of flow duct singularities at low mach numbers
,” in
12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference)
, p.
2687
.
3.
Alenius
,
E.
,
Åbom
,
M.
, and
Fuchs
,
L.
(
2015
). “
Large eddy simulations of acoustic-flow interaction at an orifice plate
,”
J. Sound Vib.
345
,
162
177
.
4.
Babilio
,
E.
,
Mascolo
,
I.
, and
Guarracino
,
F.
(
2023
). “
From static buckling to nonlinear dynamics of circular rings
,”
Front. Appl. Math. Stat.
9
,
1115227
.
5.
Bailly
,
C.
,
Bogey
,
C.
, and
Marsden
,
O.
(
2010
). “
Progress in direct noise computation
,”
Int. J. Aeroacoust.
9
(
1
),
123
143
.
6.
Barnes
,
P. J.
,
Drazen
,
J. M.
,
Rennard
,
S. I.
, and
Thomson
,
N. C.
(
2009
).
Asthma and COPD: Basic mechanisms and clinical management
, 2nd ed. (
Elsevier
,
Amsterdam
).
7.
Bertram
,
C.
(
2003
). “
Experimental studies of collapsible tubes
,” in
Flow Past Highly Compliant Boundaries and in Collapsible Tubes: Proceedings of the IUTAM Symposium, University of Warwick, UK
, pp.
51
65
.
8.
Boden
,
H.
, and
Åbom
,
M.
(
1995
). “
Modelling of fluid machines as sources of sound in duct and pipe systems
,”
Acta Acust.
3
,
549
560
.
9.
Bohadana
,
A.
,
Izbicki
,
G.
, and
Kraman
,
S. S.
(
2014
). “
Fundamentals of lung auscultation
,”
N. Engl. J. Med.
370
(
8
),
744
751
.
10.
Ciallella
,
A.
, and
Steigmann
,
D. J.
(
2023
). “
Unusual deformation patterns in a second-gradient cylindrical lattice shell: Numerical experiments
,”
Math. Mech. Solids
28
(
1
),
141
153
.
11.
Evans
,
J. A.
, and
Whitelaw
,
W. A.
(
2009
). “
The assessment of maximal respiratory mouth pressures in adults
,”
Respir. Care
54
(
10
),
1348
1359
.
12.
Flaherty
,
J. E.
,
Keller
,
J. B.
, and
Rubinow
,
S.
(
1972
). “
Post buckling behavior of elastic tubes and rings with opposite sides in contact
,”
SIAM J. Appl. Math.
23
(
4
),
446
455
.
13.
Gern
,
J. E.
(
2010
). “
The ABCs of rhinoviruses, wheezing, and asthma
,”
J. Virol.
84
(
15
),
7418
7426
.
14.
Giorgio
,
I.
,
Dell'Isola
,
F.
, and
Steigmann
,
D.
(
2018
). “
Axisymmetric deformations of a 2nd grade elastic cylinder
,”
Mech. Res. Commun.
94
,
45
48
.
15.
Gregory
,
A.
,
Agarwal
,
A.
, and
Lasenby
,
J.
(
2017
). “
Collapse of flexible tubes—The tube law
,”
Apollo - University of Cambridge Repository
.
16.
Gregory
,
A.
,
Agarwal
,
A.
, and
Lasenby
,
J.
(
2021
). “
An experimental investigation to model wheezing in lungs
,”
R. Soc. Open Sci.
8
(
2
),
201951
.
17.
Grotberg
,
J. B.
, and
Gavriely
,
N.
(
1989
). “
Flutter in collapsible tubes: A theoretical model of wheezes
,”
J. Appl. Physiol.
66
(
5
),
2262
2273
.
18.
Hassan
,
T.
,
McKinney
,
L.
,
Sandler
,
R. H.
,
Kassab
,
A.
,
Price
,
C.
,
Moslehy
,
F.
, and
Mansy
,
H. A.
(
2019
). “
A system for measuring sound transmission through joints
,” in
2019 IEEE Signal Processing in Medicine and Biology Symposium (SPMB)
(IEEE, New York)
, pp.
1
4
.
19.
Heil
,
M.
(
1997
). “
Stokes flow in collapsible tubes: Computation and experiment
,”
J. Fluid Mech.
353
,
285
312
.
20.
Heil
,
M.
, and
Boyle
,
J.
(
2010
). “
Self-excited oscillations in three-dimensional collapsible tubes: Simulating their onset and large-amplitude oscillations
,”
J. Fluid Mech.
652
,
405
426
.
21.
Heil
,
M.
, and
Hazel
,
A. L.
(
2011
). “
Fluid-structure interaction in internal physiological flows
,”
Annu. Rev. Fluid Mech.
43
,
141
162
.
22.
Hoppin
,
F. G.
, Jr.
,
Hughes
,
J.
, and
Mead
,
J.
(
1977
). “
Axial forces in the bronchial tree
,”
J. Appl. Physiol.
42
(
5
),
773
781
.
23.
Horsfield
,
K.
, and
Cumming
,
G.
(
1968
). “
Morphology of the bronchial tree in man
,”
J. Appl. Physiol.
24
(
3
),
373
383
.
24.
İçer
,
S.
, and
Gengeç
,
Ş.
(
2014
). “
Classification and analysis of non-stationary characteristics of crackle and rhonchus lung adventitious sounds
,”
Digital Signal Process.
28
,
18
27
.
25.
Itatani
,
K.
,
Sekine
,
T.
,
Yamagishi
,
M.
,
Maeda
,
Y.
,
Higashitani
,
N.
,
Miyazaki
,
S.
,
Matsuda
,
J.
, and
Takehara
,
Y.
(
2022
). “
Hemodynamic parameters for cardiovascular system in 4D flow MRI: Mathematical definition and clinical applications
,”
Magn. Reson. Med. Sci.
21
(
2
),
380
399
.
26.
Kozlovsky
,
P.
,
Zaretsky
,
U.
,
Jaffa
,
A. J.
, and
Elad
,
D.
(
2014
). “
General tube law for collapsible thin and thick-wall tubes
,”
J. Biomech.
47
(
10
),
2378
2384
.
27.
Kraxberger
,
F.
,
Näger
,
C.
,
Laudato
,
M.
,
Sundström
,
E.
,
Becker
,
S.
,
Mihaescu
,
M.
,
Kniesburges
,
S.
, and
Schoder
,
S.
(
2023
). “
On the alignment of acoustic and coupled mechanic-acoustic eigenmodes in phonation by supraglottal duct variations
,”
Bioengineering
10
,
1369
.
28.
Lai-Fook
,
S. J.
, and
Rodarte
,
J. R.
(
1991
). “
Pleural pressure distribution and its relationship to lung volume and interstitial pressure
,”
J. Appl. Physiol.
70
(
3
),
967
978
.
29.
Lasota
,
M.
,
Šidlof
,
P.
,
Maurerlehner
,
P.
,
Kaltenbacher
,
M.
, and
Schoder
,
S.
(
2023
). “
Anisotropic minimum dissipation subgrid-scale model in hybrid aeroacoustic simulations of human phonation
,”
J. Acoust. Soc. Am.
153
(
2
),
1052
1063
.
30.
Laudato
,
M.
, and
Mihaescu
,
M.
(
2023
). “
Analysis of the contact critical pressure of collapsible tubes for biomedical applications
,”
Continuum Mech. Thermodyn.
36
,
217
228
.
31.
Laudato
,
M.
,
Mosca
,
R.
, and
Mihaescu
,
M.
(
2023
). “
Buckling critical pressures in collapsible tubes relevant for biomedical flows
,”
Sci. Rep.
13
(
1
),
9298
.
32.
Lausted
,
C. G.
,
Johnson
,
A. T.
,
Scott
,
W. H.
,
Johnson
,
M. M.
,
Coyne
,
K. M.
, and
Coursey
,
D. C.
(
2006
). “
Maximum static inspiratory and expiratory pressures with different lung volumes
,”
BioMed. Eng. Online
5
(
1
),
29
.
33.
Li
,
P.
,
Laudato
,
M.
, and
Mihaescu
,
M.
(
2023
). “
Time-dependent fluid-structure interaction simulations of a simplified human soft palate
,”
Bioengineering
10
(
11
),
1313
.
34.
Lighthill
,
M. J.
(
1952
). “
On sound generated aerodynamically: I. General theory
,”
Proc. R. Soc. London, Ser. A: Math. Phys. Sci.
211
(
1107
),
564
587
.
35.
Maurerlehner
,
P.
,
Schoder
,
S.
,
Tieber
,
J.
,
Freidhager
,
C.
,
Steiner
,
H.
,
Brenn
,
G.
,
Schäfer
,
K.-H.
,
Ennemoser
,
A.
, and
Kaltenbacher
,
M.
(
2022
). “
Aeroacoustic formulations for confined flows based on incompressible flow data
,”
Acta Acust.
6
,
45
.
36.
Mylavarapu
,
G.
,
Mihaescu
,
M.
,
Fuchs
,
L.
,
Papatziamos
,
G.
, and
Gutmark
,
E.
(
2013
). “
Planning human upper airway surgery using computational fluid dynamics
,”
J. Biomech.
46
(
12
),
1979
1986
.
37.
Oruç
,
V.
, and
Çarp
,
M.
(
2007
). “
An experimental investigation of collapsible tube flows at the onset of self-excited oscillations
,” in
WIT Transactions on the Built Environment
, p.
92
.
38.
Palnitkar
,
H.
,
Henry
,
B. M.
,
Dai
,
Z.
,
Peng
,
Y.
,
Mansy
,
H. A.
,
Sandler
,
R. H.
,
Balk
,
R. A.
, and
Royston
,
T. J.
(
2020
). “
Sound transmission in human thorax through airway insonification: An experimental and computational study with diagnostic applications
,”
Med. Biol. Eng. Comput.
58
,
2239
2258
.
39.
Pietroniro
,
A. G.
,
Trigell
,
E.
,
Jacob
,
S.
,
Mihaescu
,
M.
,
Abom
,
M.
, and
Knutsson
,
M.
(
2022
). “
Effects of boundary layer and local volumetric cells refinements on compressor direct noise computation
,”
SAE Int. J. Adv. Curr. Prac. Mobility
5
,
786
802
.
40.
Schickhofer
,
L.
,
Malinen
,
J.
, and
Mihaescu
,
M.
(
2019
). “
Compressible flow simulations of voiced speech using rigid vocal tract geometries acquired by MRI
,”
J. Acoust. Soc. Am.
145
(
4
),
2049
2061
.
41.
Schickhofer
,
L.
, and
Mihaescu
,
M.
(
2020
). “
Analysis of the aerodynamic sound of speech through static vocal tract models of various glottal shapes
,”
J. Biomech.
99
,
109484
.
42.
Schoder
,
S.
,
Weitz
,
M.
,
Maurerlehner
,
P.
,
Hauser
,
A.
,
Falk
,
S.
,
Kniesburges
,
S.
,
Döllinger
,
M.
, and
Kaltenbacher
,
M.
(
2020
). “
Hybrid aeroacoustic approach for the efficient numerical simulation of human phonation
,”
J. Acoust. Soc. Am.
147
(
2
),
1179
1194
.
43.
Shabbir
,
M. S.
,
Ali
,
N.
, and
Abbas
,
Z.
(
2023
). “
Impact of unsteadiness on the non-Newtonian flow of blood in a vascular tube with stenosis and aneurysm: Analytical solution
,”
Waves Random Complex Media
2023
,
1
15
.
44.
Shamass
,
R.
,
Alfano
,
G.
, and
Guarracino
,
F.
(
2017
). “
On elastoplastic buckling analysis of cylinders under nonproportional loading by differential quadrature method
,”
Int. J. Struct. Stab. Dyn.
17
(
7
),
1750072
.
45.
Shapiro
,
A. H.
(
1977
). “
Steady flow in collapsible tubes
,”
J. Biomech. Eng.
99
(
3
),
126
147
.
46.
Sikkandar
,
M. Y.
,
Sudharsan
,
N. M.
,
Begum
,
S. S.
, and
Ng
,
E.
(
2019
). “
Computational fluid dynamics: A technique to solve complex biomedical engineering problems-a review
,”
WSEAS Trans. Biol. Biomed.
16
,
121
137
.
47.
Teriö
,
H.
(
1991
). “
Acoustic method for assessment of urethral obstruction: A model study
,”
Med. Biol. Eng. Comput.
29
,
450
456
.
48.
Turzi
,
S. S.
(
2020
). “
Landau-like theory for buckling phenomena and its application to the elastica hypoarealis
,”
Nonlinearity
33
(
12
),
7114
.
49.
Wellman
,
A.
,
Genta
,
P. R.
,
Owens
,
R. L.
,
Edwards
,
B. A.
,
Sands
,
S. A.
,
Loring
,
S. H.
,
White
,
D. P.
,
Jackson
,
A. C.
,
Pedersen
,
O. F.
, and
Butler
,
J. P.
(
2014
). “
Test of the starling resistor model in the human upper airway during sleep
,”
J. Appl. Physiol.
117
(
12
),
1478
1485
.
50.
Whittaker
,
R. J.
,
Heil
,
M.
,
Jensen
,
O. E.
, and
Waters
,
S. L.
(
2010a
). “
Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes
,”
Proc. R. Soc. A
466
(
2124
),
3635
3657
.
51.
Whittaker
,
R. J.
,
Heil
,
M.
,
Jensen
,
O. E.
, and
Waters
,
S. L.
(
2010b
). “
A rational derivation of a tube law from shell theory
,”
Q. J. Mech. Appl. Math.
63
(
4
),
465
496
.
52.
Zarandi
,
M. A. F.
,
Garman
,
K.
,
Rhee
,
J. S.
,
Woodson
,
B. T.
, and
Garcia
,
G. J.
(
2021
). “
Effect of tube length on the buckling pressure of collapsible tubes
,”
Comput. Biol. Med.
136
,
104693
.
53.
Zhang
,
S.
, and
Liu
,
H.
(
2023
). “
A fully-coupled three-dimensional fluid-structure interaction study on the externally-pressurized collapsible tube and the internal flow
,”
J. Theor. Appl. Mech.
61
(
2
),
395
406
.