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The article delves into the intricate flow patterns within human airways, emphasizing the significance of understanding these dynamics for effective drug delivery and protective ventilation. It introduces Correlation Velocimetry (CV), a novel measurement technique that enables high-temporal-resolution and long-duration measurements, capturing over 1000 breathing cycles. This approach reveals deterministic velocity fluctuations and superimposed oscillations, challenging existing explanations of turbulent events. The study highlights the potential of the human airways as fluidic oscillators, driven by the fundamental breathing frequency, and suggests that these oscillations are organized flow structures rather than purely turbulent phenomena. The findings promise to advance respiratory physiology and improve therapeutic interventions.
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Abstract
Normal breathing induces significant variations in Reynolds numbers throughout the human airways, resulting in distinct flow regimes. However, the onset of instabilities and the development of flow structures in this context are not yet fully understood. This study presents the application of a novel point-wise measurement technique, Correlation Velocimetry (CV), to investigate unsteady velocity variations within breathing cycles at very high temporal resolution over a strongly extended measurement duration. Our approach enabled the evaluation of velocity data from more than 1000 successive breathing cycles in a realistic airway model, providing unprecedented statistical robustness. We observed both high- and low-frequency oscillating structures with spatial and temporal coherence across all investigated breathing regimes, ranging from Reynolds numbers of 274 to 4382. The cycle-to-cycle repeatability of these structures suggests the presence of defined physical mechanisms. Contrary to previous interpretations attributing similar fluctuations to turbulence or transitional states, our analysis indicates that these oscillations likely arise from geometric features forming systems of harmonic oscillators driven by the fundamental breathing frequency. This study provides new insights into the complex, multi-scale nature of respiratory airflow dynamics, challenging existing models and offering implications for improving computational simulations and our understanding of respiratory physiology.
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1 Introduction
The flow inside the human airways is a complex phenomenon, depending on the airway geometry and breathing conditions. A detailed understanding of this system is essential for effective drug delivery and a protective artificial ventilation. The gas and particle transport strongly depends on the local flow conditions Koullapis (2018), which in turn depend on the local geometry, Janke et al. (2017); Bauer et al. (2012).
During normal breathing, a tidal volume of 500 ml is typically inhaled and exhaled at a frequency of about 15 breaths per minute. This approximately sinusoidal breathing pattern leads to a maximum Reynolds number of about \(Re = 2000\) in the trachea, where \(Re = u\cdot D / \nu\). Here, u denotes the area-averaged velocity, D is the diameter of the trachea (typically about 20 mm), and \(\nu\) is the kinematic viscosity of air (\(\nu = {16.3e-6\,\mathrm{\square \text {m}\text {s}^{-1}}}\) at 30 \(^{\circ }\text {C}\)). This Reynolds number suggests a transitional regime. Previous literature does not reach a clear consensus on whether the flow is laminar or turbulent. Numerical simulations typically assume turbulent flow and apply either turbulence models or Large Eddy Simulations (LES) (Sommerfeld (2021); Koullapis (2018); Reid and Hayatdavoodi (2024)), as particle transport and deposition are most often investigated for constant inflow conditions at a maximum Reynolds number of at least 2000 or higher. Real breathing conditions are only rarely considered (such as in Adler and Brücker (2007); Bauer et al. (2012); Janke et al. (2017); Jedelsky et al. (2024, 2012); Elcner et al. (2016); Cui et al. (2020)). Moreover, when and how turbulence initiates in this oscillatory flow remains unclear. In contrast to constant flow velocity, the acceleration and deceleration of the flow, as well as changes in geometry due to flow reversal, create a complex interplay of stabilizing and destabilizing effects.
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Most frequently, Particle Image Velocimetry (PIV) measurements have been employed to characterize typical flow patterns (Adler and Brücker (2007); Große et al. (2007); Janke et al. (2019); Bauer et al. (2019)) in the human airways. However, the results were typically evaluated phase-averaged and moreover, the temporal resolution is usually restricted to about 8 phase angles within one breathing cycle. Achieving both, very high temporal resolution and statistically reliable results based on the repetition of many breathing cycles, is almost impossible for PIV measurements in the human airways due to the very high amount of data to evaluate.
Point measurements offer a promising approach to investigate unsteady and transitional behavior with higher temporal resolution. Researchers have employed various measurement methods in the literature, including Hot Wire (Johnstone et al. (2004)), Phase Doppler Anemometry (PDA) (Elcner et al. (2016); Jedelsky et al. (2012, 2024)) and Laser Doppler Anemometry (LDA) (Liu et al. (2024); Kerekes (2016)). A comprehensive review of measurement techniques applied to study the flow in human airways is provided by Lizal (2018). While these studies have provided valuable insights, they typically analyze a relatively small number of breathing cycles (often in the single digits or lower double digits). Although this limited number of cycles allows for the calculation of statistical quantities, it may not fully capture the complexity of the overall process. Furthermore, frequency spectra derived from such short samples may lack resolution in the lower-frequency range, which is crucial for relatively slow processes like breathing. This limitation could potentially obscure relevant effects and lead to incomplete conclusions. Therefore, expanding the scope of these measurements could yield more robust and representative results.
Here, we present a new point-wise measurement technique called Correlation Velocimetry (CV) derived from pioneering work of Chaves et al. (1993), Chaves et al. (2006), Chaves (1998), Chaves et al. (2004) to investigate the onset of possible transitional and turbulent events with very high temporal resolution and long measurement duration (hours), allowing for statistical evaluation of more than 1000 subsequent breathing cycles. As a result, the averaged values are statistically highly reliable.
The principle of CV is derived from Laser-Two-Focus (L2F) Velocimetry Schodl and Forster (1991). In L2F, two focused laser beams create distinct measurement points in space. When tracer particles pass through these focal points, they temporarily block the laser light, and these interruptions are detected by a sensor. The evaluation of the detector signals in L2F is performed through time-of-flight analysis of individual particles, yielding their individual velocities. In contrast, CV uses a single-LED light source to illuminate a measurement volume, which is projected onto receiving optics with two photodetectors spaced a few micrometers apart and aligned with the flow’s main velocity. A passing particle will first shadow one detector and, after a slight delay, likely shadow the second detector as well. This results in two detector signals, offset in time. The time offset is determined by applying cross-correlation of signal windows. If the distance between the detectors is known, the average flow velocity within the considered signal window can be calculated.
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The optical system for CV is far less complex compared to similar methods, such as L2F, Laser Doppler Anemometry (LDA) or Phase Doppler Anemometry (PDA). It is also robust to variations in the refractive index within the optical system. The cross-correlation of the two detector signals introduces a statistical component that provides robust results and makes the method applicable in environments with high particle densities. The details of this specific setup are described in the methods section. In the results section, we explain the nature of velocity fluctuations during the breathing cycles and provide further insight into the ongoing discussion about whether turbulent events occur in the upper airways.
2 Material and methods
2.1 Experimental setup
The velocity measurements were carried out at a CT scan of the human airways including the mouth and throat region as well as the lower airways with trachea down to approximately the 7th bifurcation generation (Fig. 1 a). This model corresponds to the SimInhale Benchmark model (Lizal 2018) and was originally developed and described in more detail by Lizal et al. (2012). From the CT scan a transparent silicone model has been manufactured which allowed optical access under refractive index matching. Thus, as working fluid a mixture of glycerine and water (mass ratio of 58/42) was used instead of air. Flow similarity is achieved by adapting the flow parameters to achieve Reynolds and Womersley number similarity. As oscillatory flow is applied, a peak Reynolds number is used here as reference which is calculated with
$$\begin{aligned} Re = \frac{4 V f_0}{\nu D}, \end{aligned}$$
(1)
where V denotes the tidal volume, \(f_0\) is the cycle frequency, D is the diameter of the trachea with \(D = {16.3}{mm}\), and \(\nu\) denotes the kinematic viscosity of the liquid with \(\nu = {8.4e-6\,\mathrm{\square \text {m}\text {s}^{-1}}}\) at 20 \(^{\circ }\text {C}\). The Womersley number \(\alpha\) denotes the ratio of unsteady (oscillatory) to viscous forces and is determined by
All investigated flow parameters are summarized in Table 1. Case No. 8 with TV = 500 ml and a frequency of \(f_0\) = 0.15 Hz in water & glycerine corresponds to quiet breathing in air based on the same Reynolds and Womersley number. The complete setup and oscillatory motion generation by a linear actuator is described in more detail in Janke et al. (2019).
Table 1
Summary of experimental parameter variations
Case no.
Tidal volume V (mL)
Frequency \(f_0\) (Hz)
Reynolds number Re
Womersley number \(\alpha\)
1
125
0.075
274
1.93
2
250
0.075
548
1.93
3
375
0.075
822
1.93
4
500
0.075
1096
1.93
5
125
0.15
548
2.73
6
250
0.15
1096
2.73
7
375
0.15
1643
2.73
8\(^*\)
500
0.15
2191
2.73
9
125
0.30
1096
3.86
10
250
0.30
2191
3.86
11
375
0.30
3287
3.86
12
500
0.30
4382
3.86
*Conditions for quiet breathing in air
Fig. 1
a Airway geometry, b detailed view of the measurement position. Distances are given in mm
For flow visualization tracer particles are added. The tracer particles have a mean diameter \(d_{50} = {50}\upmu {\hbox {m}}\) (Vestosint 1174, Evonik Degussa) and a density \(\rho _p\) of \({1060}{\hbox {kg}\, \hbox {m}^{-3}}\), closely matching the density of the surrounding fluid (\(\rho _\textrm{fl} = {1150}{\hbox {kg}\, \hbox {m}^{-3}}\) at 20\(^\circ\)). The Stokes number defined through
is \(St \approx 0.002\) for the maximum velocity present in the experiments with \(u = {2.4\,\mathrm{\text {m}\text {s}^{-1}}}\). Consequently, the velocity error due to particle drag is below 1 % (Dring 1982).
Fig. 2
Sketch of the measurement setup consisting of: linear motor generating the velocity profile, trigger for syncing LED with linear motor, optical lenses, receiving optics and data recorder (sound card)
The optical axis of the CV setup is positioned perpendicular to the flow. Initially, 6 measurement points were considered, as marked in Fig. 1b. However, this study will focus on points 2 and 4 only, with the other locations to be examined in a future study.
The measurement volume is illuminated using an LED light source (OSRAM Oslon SSL 80, \(\lambda = {470}{\hbox {nm}}\)) as shown in Fig. 2. Light is focused in front of the lung model using a Fresnel lens (\(f={219}{mm}\)) with an aperture diameter of 255mm, creating an image of the LED’s emitting surface within the lung model. The Fresnel design is chosen to collect the maximum amount of light and achieve the required focal length within the limited assembly space. An area of approximately 400\(\upmu {\hbox {m}}\) is illuminated in the measurement volume. On the back side of the model, the transmitted light is collected using an achromatic lens (\(f={50}{mm}\)) and projected onto the receiving optics, which contains two optical fibers (diameter 250\(\upmu {\hbox {m}}\)) that guide the light to the photodetectors. The photodetectors used are OSRAM BPE 21 silicon photodiodes, suited for light in the wavelength range of 350–820 nm. The entire setup is illustrated in Fig. 2.
Fig. 3
Schematic of the receiving optic’s projection into the measurement volume. Physical dimensions are shown on the right, while the corresponding imaged dimensions on the left demonstrate the \(6.55\times\) magnification
The optical setup imposes a magnification of the measurement volume on the receiving optics by a factor of \(M = 6.55\) (see Fig. 3). Consequently, the cylindrical measurement volume observed by each optical fiber has a size of 38.2 \(\upmu\)m, which is slightly smaller than the size of the tracer particles used. The real distance between both light guides is \(\Delta s_{real} = {1.32}{\hbox {mm}}\). For evaluation in the measurement volume, this distance must be divided by the magnification factor, resulting in \(\Delta s = \Delta s_{real} / M = {272}\upmu {\hbox {m}}\).
Since the setup is calibrated in situ and the measurement locations are defined within the calibrated model, optical aberrations—such as those caused by curved surfaces, differences in refractive indices, and misalignments—are far less critical compared to LDA or PDA. This advantage, combined with the use of an LED instead of a laser, makes the approach highly robust and suitable for systems with less defined optical properties.
2.3 Measurement method
The measurement technique relies on the sequential shadowing of two detectors by tracer particles as they traverse the measurement volume. The optical fibers are aligned with the principal flow component, which significantly increases the probability that a single particle will shadow both detectors. This shadowing creates a deflection in each detector signal, with the deflections separated by a measurable time delay.
The responses of the photodetector diodes are recorded using a standard external USB stereo PC sound card with a sampling rate (\(f_s\)) of 44.1 kHz. The photodiodes operate in photoelectric mode, as the investigated flow, with a maximum velocity of 2.4 ms\(^{-1}\), is relatively slow. For faster flows, it is recommended to use transimpedance amplifiers to increase the response time of the photodiodes, along with faster analog–digital converters with higher sampling rates.
The minimum detectable velocity is limited by the sample length used for signal correlation, the recording frequency, and the distance between the detectors. For the present setup configuration, velocities below 0.012 ms\(^{-1}\) cannot be reliably measured. This limitation primarily affects measurements close to the flow reversal points (\(\varphi = 0\), \(\varphi = \pi\)), where velocities approach zero. The maximum size of this interval is \(\pm \approx 0.1\) rad for \(V = 125\) ml at \(f_0 = 0.075\) Hz and decreases with \(Re\). Data in these regions should be interpreted with appropriate caution. To mitigate this limitation, the correlation window size can be increased, though at the cost of reduced temporal resolution. This trade-off between minimum detectable velocity and temporal resolution is inherent to correlation-based measurement techniques.
The present setup allows for extended recordings, capturing more than 1000 breathing cycles in each run. The length of the recorded stereo signals ranges from 1 h to over 2 h, depending on the breathing frequency (\(f_0\)). Each measurement run was started by a trigger signal, ensuring synchronization between individual measurements.
An exemplary signal window containing 1024 samples is shown in Fig. 4 for each detector. By applying cross-correlation, we can determine the mean temporal displacement \(\Delta t\) between both signals. The average velocity within the signal window is then calculated by dividing the known distance \(\Delta s\) between the detectors by \(\Delta t\). Cross-correlation enhances performance under low signal-to-noise ratios, as it considers multiple passing particles in the evaluation. However, there is a trade-off between robustness (achieved with larger windows) and temporal accuracy (obtained with smaller windows).
Fig. 4
Example sections of the two detector signals consisting of 1024 samples and the corresponding correlation function. The distinct peak of the correlation function at a time lag of \(-\)0.16 ms is marked by an arrow
The raw data consist of a stereo audio file containing the electrical response of the photodetector diodes. Particle shadowing of the detector produces a spike in the measurement signal. Due to statistical fluctuations in particle detection across both detectors, accurate identification of individual particle transits is not feasible. Consequently, signal windows encompassing multiple particle transits are utilized.
The cross-correlation algorithm employed in this study is based on fast Fourier transform (FFT) and is implemented using the SciPy programming library. This approach ensures robust and accurate computation of cross-correlation values.
In the first step, the signals are normalized by subtracting the mean value and by dividing them by the standard deviation. Window sizes N of 2048 samples (46.44 ms at 44.1 kHz) for peak velocities below \({0.5}\,{\hbox {m}}\,{\hbox {s}}^{-1}\) and 1024 samples (23.22 ms at 44.1 kHz) for higher velocities have been found to deliver reasonable results. A representative signal window for each detector containing 1024 samples is shown in Fig. 4, along with the respective cross-correlation function. The time lag between the two signals is identified at the maximum of the correlation function. To achieve higher accuracy in determining the time lag \(\Delta t\), a three-point peak fit using a Gaussian kernel is applied. The cross-correlation reduces the temporal resolution of the signal depending on the window size N. The resulting sampling rate for the time lags is defined by \(f_s / N\) for non-overlapping windows, which yields 43.01 Hz for \(f_s = {44.1\,\mathrm{\text {k}\text {Hz}}}\) and \(N = 1024\).
By introducing an overlap \(o_w\) between the windows, the temporal resolution can be increased. However, this comes at the cost of increased computational effort for the evaluation. The effect of the overlap on the temporal resolution can be calculated using the formula \(f_s / (N (1 - o_w))\). In this study, an overlap of 80 % is chosen, which has proven to be a good compromise between accuracy and computational costs. The resulting temporal resolution of the time lags, and consequently velocity, is 215.33 Hz for \(N=1024\). At a breathing frequency of \(f_0 = {0.15\,\mathrm{\text {Hz}}}\), this results in approximately 1433 samples per period.
2.5 Velocity signal processing
To convert the derived time lags \(\Delta t (t)\) to velocity, they must be multiplied reciprocally with the imaged distance between the detectors \(\Delta s\), yielding \(u(t) = \Delta s / \Delta t(t)\). This calculation produces a sinusoidal velocity signal. The frequency of this signal can be determined by performing a FFT. For all experiments conducted, the difference between the frequency of the velocity signal and the imposed breathing frequency \(f_0\) is below 0.1 mHz, which underlines the validity of the measurements.
Since the sampling frequency of the velocity signal is not guaranteed to be an integer multiple of the flow frequency \(f_0\), phase shift effects can occur. This phenomenon arises from the mathematical formulation of a sampled periodic sinusoidal signal:
$$\begin{aligned} u_s[n] = A \sin \left( 2 \pi \frac{f_0}{f_s/(N (1-o_w))} n + \varphi _0\right) \end{aligned}$$
(4)
where \(u_s[n]\) represents the discretely sampled velocity signal after cross-correlation processing, A is the amplitude and n is an integer representing the individual samples. If the term \(f_0/(f_s/(N (1-o_w)))\) is not an integer, a phase shift is introduced that increases with n. Considering the case of \(f_0 = {0.15\,\mathrm{\text {Hz}}}\), the phase shift per period is \({\Delta _p \varphi \approx {0.00438}}\,\mathrm{\text{rad}}\), which accumulates to \(\Delta \varphi \approx 1.4\pi\) after 1000 periods. A graphical representation of this effect is provided in Fig. 5 (top), where individual periods with length \(1/f_0\) are superposed.
To correct the phase shift, a specific procedure is applied to each individual period. The phase angle \(\varphi _0\) of the respective period is obtained by fitting a sinusoidal function:
where the amplitude \(\hat{u}\) and \(\varphi _0\) are the fitting parameters. However, only \(\varphi _0\) is needed for the phase correction. Once \(\varphi _0\) is known, each period can be aligned to a reference phase angle, such as \(\varphi _0 = 0\), allowing for precise results in subsequent processing. The effect of the phase correction is clearly visible in Fig. 5 when comparing the uncorrected (top) and corrected (bottom) superimposed periods. At this stage, the individual periods are also resampled to 1024 samples each, facilitating subsequent evaluations such as averaging. Additionally, any outliers with a velocity magnitude higher than 3 ms\(^{-1}\) and correlation values below 0.3 are replaced by linear interpolation using neighboring data points. This procedure primarily affects the regions at \(\varphi = 0\) and \(\varphi = \pi\), where no directed flow is present, and must be considered during the evaluation and discussion.
Fig. 5
1024 transparent overlaid periods of the velocity signal. Top: Signal without phase correction. Bottom: Phase-corrected signal with the mean value over all periods shown in red
The measured velocity signals contain more than 1000 periods. After the initial processing, the individual phases can be superimposed and statistical quantities like mean value \(\bar{u}\) and standard deviation \(\sigma _u\) can be derived. To validate the statistical robustness of the measurements and to determine the minimum number of periods required for reliable results, a convergence analysis is performed. Specifically, the convergence of the mean velocity at position 4 under typical breathing conditions (\(V={500\,\mathrm{\text {m}\text {L}}}\) and \(f_0 = {0.15\,\mathrm{\text {Hz}}}\)) at peak inspiration (\(\varphi = \pi /2\)) is examined. For progressively increasing subsample sizes, the deviation between the subsample mean and the mean value computed from all available periods is calculated. The analysis showed that the deviation falls below 0.1 % for a subsample size of 207 periods. Since the size of the complete sample is five times larger, it can be safely assumed that the statistical quantities of the sample are representative for the overall process. A detailed description is provided in the Appendix (5).
Fig. 6
Phases-averaged mean velocity for different tidal volumes V and \(f_0\) at position 2. In addition, the standard deviation \(\sigma _u\) of the velocity as well as the turbulent intensity TI defined in Fig. 6 is shown in green and blue, respectively
Phases-averaged mean velocity for different V and \(f_0\) at position 4. In addition, the standard deviation \(\sigma _u\) of the velocity as well as the turbulent intensity TI defined in Fig. 6 is shown in green and blue, respectively
The phase-averaged velocity profiles \(u(\varphi )\) for different combinations of the tidal volume V and \(f_0\) are shown in Figs. 6 and 7 for positions 2 and 4. The parameter combinations investigated are summarized in Table 1. The imposed sinusoidal oscillation is clearly visible. At position 2, a pronounced asymmetry between inhalation and exhalation exists, where the peak velocity during inhalation exceeds the peak velocity during exhalation by 20% to 30 %. This can be caused by the flow detachment at the larynx, which is present during inhalation but not during exhalation. The detachment constricts the flow, which results in higher local velocities at equal volume flow rates. In contrast, position 4, which is located further away from geometrical discontinuities, shows a stronger symmetry between inhalation and exhalation.
The amplitude of the sinusoidal velocity increases with \(Re\). For the same \(Re\), the observed velocities have similar magnitudes, as expected. A prominent feature emerging from all phase-averaged profiles is the presence of superimposed oscillations. These are especially distinct after the reversal from inhalation to exhalation at \(\varphi = \pi\). Since they are apparent in the averaged images, they must be deterministic, reproducible and coupled to the fundamental frequency. The frequency of these oscillations is higher than \(f_0\), and their amplitude is significantly smaller and varies among different parameter configurations. Jedelsky et al. (2024), Jedelsky et al. (2012) and Elcner et al. (2016) observed similar fluctuations, albeit over substantially fewer periods. Jedelsky et al. (2024) attributed them to Taylor-Görtler vortices based on simulation results by Lin et al. (2007), while Elcner et al. (2016) explained them with turbulent transition states in the laryngeal jet. However, as these characteristic fluctuations already occur at very low \(Re\) , they can be neither of turbulent nor transitional nature. Moreover, Taylor-Görtler vortices typically appear as longitudinal vortices at concave surfaces and except in the study of Lin et al. (2007) they have never again been confirmed for the human airways. Hence, a more detailed analysis of these fluctuations will be provided in Sect. 3.2, where we will propose an alternative explanation for these phenomena.
The phase-averaged standard deviation of the velocity is shown in green in Figs. 6 and 7. Due to the previously discussed limitations of the LCV method near the reversal points, the standard deviation near \(\varphi =0\) and \(\varphi = \pi\) should be interpreted with caution.
The magnitude of \(\sigma _u\) scales with Reynolds number (Re), exhibiting comparable magnitudes during both inhalation and exhalation phases.
During inhalation (\(0 \le \varphi \le \pi\)), both measurement positions initially show small values of \(\sigma _u\). At position 2, for \(Re < 2000\) and at position 4 for \(Re> 1000\), this low-magnitude regime persists until the early acceleration phase. At \(\varphi = \pi /4\), corresponding to the point of maximum acceleration, different behaviors can be observed at both positions. At position 4 (Fig. 7), a distinct jump in \(\sigma _u\) occurs for \(Re \ge 1643\), while at position 2 (Fig. 6), the increase is more gradual. Following this increase, \(\sigma _u\) maintains at a relatively constant level until \(\varphi \approx 3\pi /4\). In the deceleration phase (\(3\pi /4 \le \varphi \le \pi\)), when the flow approaches reversal, \(\sigma _u\) increases again at both positions (e.g., Fig. 6g and k or Fig. 7f and j).
During exhalation (\(\pi \le \varphi \le 2\pi\)), \(\sigma _u\) initially decreases for all cases, which can be attributed to the stabilizing effect of flow acceleration (e.g., Fig. 6h or Fig. 7f). For \(Re \le 548\), \(\sigma _u\) remains at this low level throughout the exhalation phase. For higher Re, an increase can be observed at \(\varphi = 5\pi /4\), corresponding to the point of maximum acceleration. This increase manifests differently at the two measurement positions: At position 2, it occurs abruptly (see Fig. 6h), while at position 4, for \(Re \ge 1643\), the increase is more gradual, as shown in Fig. 7h.
In the final deceleration phase (\(3\pi /2 \le \varphi \le 2\pi\)), position-specific behavior is observed. At position 2, for \(Re \ge 2191\), a further increase in \(\sigma _u\) occurs, visible in Fig. 6k, while for lower Re, \(\sigma _u\) remains approximately constant. At position 4, \(\sigma _u\) maintains a relatively constant level across all Re.
Following Jedelsky et al. (2024), we introduce the phase-related turbulent intensity TI. However, we modify the definition by normalizing TI with the mean flow velocity originating from the imposed boundary conditions, defined as \(\dot{V}(\varphi ) / A\), where A is the cross-sectional area of the trachea (\(D = {16.3}\,\,{\hbox {mm}}\)), instead of the phase-averaged velocity. This separates the normalization from the actual measurement, which improves the comparability between different measurement series and prevents unwanted effects from fluctuations in phase-averaged mean velocities \(\bar{u}\). The resulting equation for TI is:
which essentially represents a normalized standard deviation. TI is additionally shown in Figs. 6 and 7 as a blue area. Since TI will diverge around the point of flow reversal, the regions close to \(\varphi = 0\) and \(\varphi = \pi\) are not shown in Figs. 6 and 7 and are not considered in the further discussion.
The values range from 0.05 to 0.1, which aligns with the findings reported by Jedelsky et al. (2024), Jedelsky et al. (2012). The behavior of TI exhibits distinct patterns that depend on three key factors: Reynolds number, spatial location and phase angle within the breathing cycle. At position 2, TI initially decreases during flow acceleration, remains nearly constant after peak acceleration and increases again during flow deceleration during both half-cycles. For \(Re \le 822\), larger TI values are observed during inhalation compared to exhalation. For higher \(Re\), TI has similar magnitudes during inhalation and exhalation. The previously described shape behavior is preserved. At position 4, the magnitude of TI and the shape of the curves are similar between inhalation and exhalation for all investigated \(Re\).
3.2 Deterministic structures
The dominant component of the oscillating velocity profile closely follows the sinusoidal wave imposed by the linear actuator at frequency \(f_0\). While this frequency exhibits the highest amplitude in the velocity profiles, additional higher-frequency oscillations are superimposed upon it. To analyze the deviation from the underlying sine, a Notch filter centered at \(f_0\) is used. The filter was applied in both forward and backward directions to maintain phase preservation. The phase-averaged, filtered velocity profiles for both measurement positions are presented in Fig. 8.
Fig. 8
Notch-filtered velocity profiles at both measurement positions with the filter centered at \(f_0\)
The fluctuations, previously observed in Figs. 6 and 7, are more pronounced in Fig. 8. Both high-frequency (\(\approx 30 f_0\)) and low-frequency (\(\approx 2 f_0\)) modulations can be identified. During inhalation, for \(Re \le 1096\), the two measurement positions exhibit opposing behavior. The velocity modulation at position 2 is initially oriented opposite to the main flow direction, while at position 4, it aligns with the flow. At \(\varphi \approx \pi /4\), corresponding to peak acceleration, this pattern reverses, with position 2 aligning with the flow direction, while position 4 shows opposition. A subsequent reversal occurs at \(\varphi \approx 3 \pi /4\), coinciding with peak deceleration. At higher Reynolds numbers (\(Re \ge 1643\)), position 4 exhibits behavior which is opposed to both acceleration and deceleration, which can be attributed to inertial effects. The velocity modulation at position 2 displays similar characteristics but is offset toward negative values (in the direction of flow).
During exhalation, larger-scale fluctuations are observed. Beginning at \(Re = 274\) (Fig. 8a), multiple distinct deflections are evident. Both amplitude and frequency increase with \(Re\) up to \(Re = 1096\). Beyond this point, while the frequency remains constant, the relative amplitude diminishes. At approximately \(Re = 2191\), the behavior becomes similar to that observed during inhalation, where the velocity modulation opposes the fluid acceleration, which can be attributed to inertial effects.
The origin of the low-frequency velocity modulation during inhalation can be explained by examining spatial information of the velocity profile. Janke et al. (2019) provided velocity profiles (D1–D2) near the measurement locations for inhalation with constant flow rates at \(Re = 1231\), \(Re = 2460\), and \(Re = 4286\). Despite the transient nature of the presented results, the overall spatial profile of the curve is expected to be comparable to the PIV results of Janke et al. (2019) during the inspiration phase. Their measurements showed a maximum velocity in the region of position 2. From this peak, the velocity decreased rapidly toward the posterior wall, indicating a recirculation zone (Fig. 10b III), and decreased slowly toward position 4. The velocity modulation during inhalation is hypothesized to originate from the growth of the recirculation zone, which constricts the flow near position 2 and results in increased velocities. Similar effects have been documented in the field of osci-jet nozzles, where changes in the recirculation zone size also result in velocity modulation (Woszidlo et al. (2019); Ostermann et al. (2018)). The opposite behavior, observed at position 4, may be attributed to entrainment of the laryngeal jet.
The higher-frequency effects, which are particularly prominent during the exhalation phase, are superimposed by lower-frequency inertial effects. To obtain a clearer visualization of these higher-frequency components, the notch-filtered velocities shown in Fig. 8 were subjected to additional processing using a fifth-order high-pass Butterworth filter. The cutoff frequency was set to \(8f_0\), ensuring complete removal of the inertial effects occurring at approximately \(4f_0\). The resulting filtered velocity signals are denoted \(\tilde{u}\) and presented in Fig. 9, allowing for a detailed analysis of the higher-frequency phenomena.
Fig. 9
High-pass-filtered velocity fluctuations \(\tilde{u}\) filtered by a Butterworth filter of order five with a cut off frequency of \(8f_0\)
In all measurements, the most prominent high-frequency oscillations are observed following the reversal from inhalation to exhalation at \(\varphi> \pi\). These oscillations exhibit their maximum amplitude at \(Re = 822\) and persist throughout the entire exhalation phase. For both lower and higher \(Re\), the amplitude decreases with increasing distance from \(Re = 822\), and the oscillations shift toward the flow reversal point at \(\varphi = \pi\). The oscillations occurring immediately before and after flow reversal, including those after \(\varphi = 0\), can be attributed to vortices generated during the preceding flow deceleration and subsequently convected through the measurement volume during acceleration. However, the oscillations observed during the middle of a phase must originate from different mechanisms.
Measurements at \(Re = 1096\) were conducted across all three \(f_0\) values. During inhalation, distinct fluctuations were observed. At \(f_0 = {0.3\,\mathrm{\text {Hz}}}\), approximately 5 periods were present (Fig. 9i). The number of oscillations increased with decreasing frequency: approximately 9 at \({0.15\,\mathrm{\text {Hz}}}\) and approximately 16 at \({0.075\,\mathrm{\text {Hz}}}\). The corresponding physical frequency was determined to be in the range of 1.2–1.5 Hz for all cases. The presence of deterministic fluctuations at both measurement positions indicates an upstream source, e.g., geometrical features, within the respiratory system that modulates the flow. Flow visualizations by Johnstone et al. (2004), extensively presented in Ball et al. (2008), clearly show recirculation zones in the glottal region as well as the detachment due to the laryngeal jet during inhalation, as shown in Fig. 10b I and III, respectively. The recirculation zone \(\textrm{I}\) particularly resembles an oscillating structure known as wedge edge resonator oscillator, as described in the review of fluidic oscillators by Campagnuolo and Lee (1969) and depicted schematically in Fig. 10a.
Fig. 10
Schematic of a wedge resonator oscillator with a closed cavity \({\textbf {a}}\), adapted from Campagnuolo and Lee (1969), and sketch of the airway model showing potential recirculation zones and flow detachments during inhalation \({\textbf {b}}\) and exhalation \({\textbf {c}}\)
According to Campagnuolo and Lee (1969), the frequency of such a resonating system is determined by its geometric properties and flow velocity, which aligns with the observation of similar physical modulation frequencies at identical \(Re\), as described above. While the current literature on fluidic oscillators predominantly focuses on feedback channel oscillators (Woszidlo et al. 2015, 2019; Ostermann et al. 2018; Schmidt et al. 2017), cavity oscillators have received comparatively little attention. However, Luo et al. (2017) conducted experimental investigations of a cavity resonating oscillator without a wedge, demonstrating clear jet modulation. Similar effects are expected in the glottal region, though the complex interplay between geometric and fluid dynamic aspects necessitates more targeted investigations.
No comparable explanation has been identified for the exhalation phase thus far. However, since distinct fluctuations are evident in the phase-averaged data, deterministic effects must also be the source in this case. Downstream separation regions, as sketched in Fig. 10c IV and V, may induce upstream pressure fluctuations and flow modulations. Alternatively, the merging of flows from both bronchi could modulate the flow field. The underlying mechanism appears to be dynamic in nature, as evidenced by effects such as the fanning of fluctuations observed at \(Re = 822\) (Fig. 9c).
3.3 Frequency analysis
To further analyze the coherent structures observed in the phase-averaged velocity profiles, the fast Fourier transform (FFT) of the velocity signal is calculated using Welch’s method (Welch 1967). To prevent any overlap of sampling and fluid dynamic effects, the raw velocity signal after cross-correlation, sampled at 215.33 Hz for a window size of 1024 and 107.66 Hz for a window size of 2048, is used. These sampling rates are not an integer multiple of any of the investigated frequencies: \(0.075\) Hz, \(0.15\) Hz and \(0.3\) Hz. The window size used in Welch's method corresponds to 64 periods, with no overlap used. The resulting frequency resolution is \(1.15\) mHz, \(2.31\) mHz and \(4.62\) mHz for \(f_0 = { 0.075\,\text {Hz} ,\ 0.15\,\text {Hz} \ and\ 0.3\,\text {Hz}}\), respectively. The present resolution significantly surpasses that of previous numerical and experimental studies (Calmet 2016; Jedelsky et al. 2024; Cui et al. 2020; Elcner et al. 2016; Reid and Hayatdavoodi 2024; Jedelsky et al. 2012), which typically analyzed only a few cycles or even fractions of a cycle, potentially leading to inaccuracies, particularly in the low-frequency regime.
Fig. 11
Power density spectra calculated from the velocity signal by using Welch’s method. The harmonics of the respective breathing frequency \(f_0\) are shown as vertical gray dashed lines. The black dashed line indicates a trend of \(1/f^2\)
The calculated power density spectra are shown in Fig. 11. The frequency is normalized by the respective \(f_0\) to facilitate direct comparison. All spectra exhibit a distinct peak at their respective breathing frequency \(f_0\). This peak is the largest for all investigated cases, indicating that it contains the most energy, as expected. Additionally, all spectra display harmonic overtones at integer multiples of the base frequency, which is consistent with the high temporal coherence observed. The number and characteristics of these harmonics vary among different cases and locations.
The FFT spectra support the hypothesis of a harmonic oscillatory system driven by the base breathing frequency. The first harmonics exhibit decreasing amplitudes, which is characteristic for sawtooth-like modulations. Such modulations are well-documented in the field of fluidic oscillators and are induced by periodic changes in the size of recirculation zones restricting the flow (Woszidlo et al. 2019; Ostermann et al. 2018; Schmidt et al. 2017; Funaki et al. 2007). It is hypothesized that the growth and reduction of the recirculation zone in the laryngeal jet, shown in Fig. 10b \(\textrm{III}\), induces similar effects. Upon examination of the low-frequency modulations presented in Fig. 8, a degenerated sawtooth-like pattern can be observed during inhalation.
The amplitudes in the frequency spectrum of a sawtooth wave with finite falling slope decrease proportionally to \(1/n^2\), where n represents the harmonic number. Under these conditions, certain harmonics may be weakened or vanish entirely. This behavior is clearly observable in the spectra presented in Fig. 11. To illustrate this relationship, a reference line with a \(1/f^2 \propto 1/n^2\) slope has been added to each diagram in Fig. 11, demonstrating close agreement with the experimental data. Since the first harmonic contains the energy of the fundamental oscillation, the initial amplitude of the reference line has been fitted to match the mean value of both measurement positions at the second harmonic. This reference line approximates the contribution of the low-frequency sawtooth oscillation, thereby providing a basis for distinguishing between peaks associated with the sawtooth pattern (those below the line) and those arising from other phenomena. Comparable frequency spectra have been reported by Schmidt et al. (2017) for feedback channel oscillators, emphasizing the similarities in velocity modulation induced by variations in recirculation zone size.
The frequency spectra for \(Re = 822\) (Fig. 11c) show the strongest deviation from the \(1/f^2\) trend. This observation aligns with the pronounced oscillations during exhalation shown in Fig. 9c. The peaks at higher frequencies remain harmonics of the breathing frequency, suggesting the presence of resonance effects. For \(Re = 1096\), these high-frequency peaks appear with reduced amplitude and continue to diminish with increasing distance from \(Re = 822\). For larger \(Re\), this attenuation can be attributed to increasing inertial effects, which progressively dominate the smaller-scale oscillations. At lower \(Re\), the dynamic forces may be insufficient to fully excite the oscillatory system.
The interpretation of the spectrum is challenging due to the complexity of the system and the multiple origins of various modulations. The system appears to be fundamentally harmonic, with all frequencies based on the breathing frequency. Several factors complicate the analysis:
(i)
The presence of sawtooth-shaped modulations contributes to all harmonics, potentially obscuring relevant effects.
(ii)
Constructive interference may explain amplitudes larger than expected.
(iii)
Destructive interference can potentially mask relevant frequencies.
Consequently, the provided interpretation of the spectrum should be considered more qualitative than quantitative. The complex interactions within the system necessitate caution in drawing definitive conclusions from the spectral analysis alone.
4 Conclusion
In this study, point velocity measurements were conducted in the upper respiratory tract using an innovative method. The number of recorded cycles, exceeding 1000, significantly surpasses all measurements previously known to the authors. This results in robust statistics that allow for detailed insights into mean velocities and fluctuation quantities. Particularly with regard to mean velocities, spatially and temporally coherent structures were identified, which can easily be obscured by random fluctuations in shorter measurements.
This provided comprehensive analysis offers novel insights into the complex fluid dynamics of oscillatory breathing. Our key findings can be summarized as follows:
(i)
Asymmetry in flow patterns: We observed a clear asymmetry between inhalation and exhalation, particularly at position 2 near the larynx. This aligns with previous investigations. The asymmetry is induced by larger flow detachment during inhalation, which constricts the flow and results in higher velocities.
(ii)
Velocity fluctuations: The velocity fluctuations quantified by \(\sigma _u\) and TI change over the breathing cycle and show jumps, gradual increase and oscillations, depending on whether the flow is accelerating or decelerating. While \(\sigma _u\) slightly increases with \(Re\), the TI remains nearly constant over all investigated parameter configurations.
(iii)
Superimposed oscillations: Our phase-averaged velocity profiles revealed coherent, reproducible oscillations superimposed on the fundamental breathing frequency. These oscillations were particularly prominent after the reversal from inhalation to exhalation. Due to the remarkable coherence the origin of these fluctuations seems to be systematic rather than turbulent.
(iv)
Frequency analysis: Fast Fourier transform (FFT) analysis revealed a complex harmonic structure, with distinct peaks at integer multiples of the breathing frequency. The presence and characteristics of these harmonics varied with Re and measurement location.
(v)
Low-frequency modulations: Low-frequency modulations were observed to correlate with changes in the mean flow velocity. While several potential mechanisms including flow restrictions, jet entrainment and variations in the laryngeal jet width may contribute to these modulations, their relative importance remains to be determined. The amplitude of these modulations shows systematic variation with Reynolds number, suggesting a coupling to the overall flow dynamics.
(vi)
Oscillatory mechanisms: The observed oscillatory patterns share characteristics with known fluid dynamic oscillator systems, particularly in their frequency response and phase coupling, while the geometric features of the airways, such as the glottis, may act as flow-induced oscillators. Additional research is needed to conclusively establish these mechanisms. The systematic nature of the oscillations and their Reynolds number dependence suggests organized flow structures rather than purely turbulent phenomena.
These findings provide an alternative interpretation to previous studies that attributed similar fluctuations to Taylor-Görtler vortices or transitional states in the laryngeal jet. The observed frequency spectra and coherent behavior suggest the presence of organized harmonic oscillators driven by the fundamental breathing frequency and shaped by the airway geometry. The experimental approach presented here contributes to the understanding of airflow dynamics in human airways by providing statistically robust, high-resolution temporal measurements. These insights may inform future developments in:
(i)
Detailed mapping of flow structures using advanced flow visualization techniques.
(ii)
Investigation of the relationship between specific anatomical features and observed oscillatory patterns.
(iii)
Exploration of how these flow dynamics may change in diseased or artificially altered airways.
(iv)
Development of computational models that can accurately capture the complex oscillatory behavior observed in this study.
In conclusion, this work demonstrates that fluid flow in the human airways is a rich, multi-scale phenomenon with deterministic structures that warrant further investigation. Understanding these complex dynamics is crucial for advancing our knowledge of respiratory physiology and improving respiratory therapies.
Acknowledgements
This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number 257981040.
Declarations
Conflict of interest
The authors declare no competing interests.
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To demonstrate that the sample size of the recordings is sufficient to ensure stable mean values, the convergence of means is investigated. According to the Law of Large Numbers, the mean of a sample approaches the population mean as the number of samples increases. An arbitrary phase angle is selected, in this case \(\varphi =\pi / 2\), and the average velocity \(\bar{u}(\pi / 2)\) is calculated for an increasing number of periods. The resulting curve, shown in Fig. 12, exhibits exponential behavior.
Fig. 12
Convergence of mean value for \(f_0\) = 0.15 Hz and \(V = {500}\) ml at \(\varphi = \pi / 2\). The horizontal line describes the population mean at \({-1.204}\,\) ms\(^{-1}\)
This can be described by the equation for an exponential decay
$$\begin{aligned} y(x) = y_0 + A e^{-x/\tau } \end{aligned}$$
(7)
where \(y_0\), A and \(\tau\) are free parameters. Fitting the data shown in Fig. 12 with Eq. 7, \(\tau = 29.5\) is obtained. For \(7 \tau\) the deviation between the sample mean and the population mean is below 0.1 %. In the present case, \(7 \tau\) correspond to approximately 207 samples. Since more than 1000 samples are used in the statistical evaluation, it can be assumed with certainty that the considered number of periods is sufficiently large and representative for the whole population.
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