VORTICITY TRANSPORT IN HUMAN AIRWAY MODEL

Abstract

Many previous studies concerned with respiratory fluid mechanics have simplistically assumed steady and laminar flow. Above a certain ventilation frequency, the unsteady nature of the respiratory flow becomes apparent, and inhalation and exhalation cannot be approximated as quasi-stationary processes. Moreover, due to the geometrical structure of the bronchial tree, flow unsteadiness and transition to turbulence can incept even at Reynolds numbers usually considered laminar in parallel flows. Here we investigate the primary features of the oscillatory flow through a 3D printed double bifurcation model that reproduces, in an idealized manner, the self-similar branching of the human bronchial tree. We consider Reynolds and Womersley numbers relevant to physiological conditions between the trachea and the lobar bronchi. Three-component, three-dimensional velocity fields are acquired at multiple phases of the ventilation cycle using Magnetic Resonance Velocimetry (MRV). The phase-averaged volumetric data provide a detailed description of the flow topology, characterizing the main secondary flow structures and their spatio-temporal evolution. We also perform twodimensional by Particle Image Velocimetry (PIV) for the steady inhalation case at a Reynolds number Re = 2000. PIV is carried out by matching the refractive index of the 3D printing resin with a novel combination of anise oil and mineral oil. The instantaneous measurements reveal unsteadiness of the separating unsteady flow in the bifurcation, and the ensemble averages show a clear Reynolds stress pattern indicating that the flow is turbulent at the first bronchial bifurcation already at this relatively low Reynolds number. INTRODUCTION The human respiratory system is structured as a network of branching airways. The trachea splits in the two main bronchioles, which successively bifurcate about sixteen more times down to the terminal bronchi, followed by roughly six generations of alveolar ducts involved in the gas exchange (Kleinstreuer & Zhang 2010). While the actual anatomy is fairly complex and varies between different subjects, general features have been long identified which are remarkably consistent: at the i bronchial generation, the daughter-tomother branch diameter ratio is h = Di+1/Di ≈ 0.8, the length-todiameter ratio of each branch is Li/Di ≈ 3.5, and the bifurcation angle is θ ≈ 60-70° (Weibel 1997). Such proportions minimize the through-flow time during the respiration process and the energy expenditure in bifurcating flow systems, and therefore canonical airway models with such characteristics have been extensively studied. The most classic case is the planar version of the Weibel A model (Weibel 1963), in which the branches consist of circular tubes that bifurcate symmetrically and lay on the same plane. Despite the simplicity of such representation, this has been shown to capture many key aspects of the respiratory airflow both in inspiratory and expiratory mode. Above a certain ventilation frequency, the unsteady nature of the respiration becomes apparent, and inhalation and exhalation cannot be approximated as quasi-stationary processes. This is especially important in the upper and central airways, where length and velocity scales are larger, making inertia and acceleration effects dominant over viscous dissipation. In this study we experimentally investigate the primary features of the oscillatory flow through a symmetric double bifurcation, which models the self-similar branching of the human bronchial tree. The focus is on the generation and transport of vorticity during the respiratory cycle/ This process is at the root of the strong secondary flows and characterizes the instantaneous shear layers in this complex, unsteady flow. EXPERIMENTAL METHODOLOGY The airway geometry we investigate is a planar Weibel A double bifurcation, and it replicates the model studied numerically by Comer et al. (2001). A schematic of the geometry with dimensions is given in Fig. 1, along with a 3D rendering. Similar (when not identical) geometries have been used in several previous experimental and numerical studies (Zhang and Kleinstreuer 2002, Longest and Vinchurkar 2007, Fresconi and Prasad 2007, among others). This specific model was chosen because of the full description of the geometry and velocity fields reported by Comer et al. (2001). Their study suggests that fine details as the rounding radius at the carina have a moderate effect as compared to more macroscopic features such as the bifurcation angle. We label the three branching generation G0 (mother branch, receiving the inflow), G1 (daughter branches, after the first bifurcation), and G2 (granddaughter branches, after the second bifurcation). Typical airway labeling schemes use G0 for the trachea, G1 for the main bronchi, etc., but here we do not strictly associate the mother branch with the trachea. Rather, assuming a quasi10 International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017 2 2D-1 homothetic airway model, this geometry may represent a different location in the airway tree depending on the Reynolds number. We investigate a range of Reynolds and Womersley numbers relevant to physiological conditions between the trachea and the lobar bronchi. We define the Reynolds number defined as Re = U0D0/ν, where D0 and U0 are the diameter and peak bulk velocity at G0, and ν is the kinematic viscosity. The Womersley number is defined as Wo = D0/2√(ω/ν), where ω is the angular frequency of oscillation. Here we present measurements at Wo = 3 and peak Reynolds number Re = 2000. The airway geometry we investigate is a planar Weibel A double bifurcation, and it replicates the model studied numerically by Comer et al. (2001), see Fig. 1. We label the three branching generation G0 (mother branch, receiving the inflow), G1 (daughter branches, after the first bifurcation), and G2 (granddaughter branches, after the second bifurcation). The model is 3D printed by stereolithography with a layer thickness of 25 micron, which warrants high geometric precision and hydrodynamic smoothness. We investigate a range of Reynolds and Womersley numbers relevant to physiological conditions between the trachea and the lobar bronchi. We define the Reynolds number defined as Re = U0D0/ν, where D0 and U0 are the diameter and peak bulk velocity at G0, and ν is the kinematic viscosity. The Womersley number is defined as Wo = D0/2√(ω/ν), where ω is the angular frequency of oscillation. To this end we use a in-house built oscillatory pump. For the present study we impose a peak Reynolds number Re = 2000 and vary the Womersely number in the range Wo = 1 12, spanning conditions from slow breathing to high frequency ventilation. Here we show results for the Wo = 3 case. Fig. 1. Schematic of the geometry of the double bifurcation. A physical model of the bifurcation is 3D printed using a clear resin (Waterhsed XC 11122), and is hermetically sealed to plastic tubing that connect it to an oscillatory pump. The pump consists of a piston sliding through a 8 cm diameter cylinder, and driven by a numerically controlled stepper motor which imposes the desired sinusoidal waveform. Threecomponent, three-dimensional velocity fields are acquired at multiple phases of the ventilation cycle using Magnetic Resonance Velocimetry (MRV). For maximum signal in the MRV measurements, the working fluid is water with 0.06 mol/L of CuSO4. A 3 Tesla Siemens MRI scanner (Fig. 2) is employed. Velocity data are obtained using the method described by Elkins et al. (2007), with the signaling and data acquisition sequence from Markl. et al. (2012). Threedimensional, three-component (3D-3C) velocities are obtained on a uniform Cartesian grid at a resolution of 0.6 mm. The field of view is 153.6 by 307.2 by 26.4 mm and includes both the fluid and the solid walls of the test section. By gating the MRI signal, 10 phased-averaged velocity fields are obtained within the respiration cycle. The procedure is similar to the one used by Banko et al. (2016) who studied the oscillatory flow in a subject-specific airway model. Figure 3 shows the excellent agreement between the imposed waveform and the flow rate measured by MRV at a cross-section in generation G0. Fig. 2. The 3 Tesla Siemens MRI scanner used for the MRV measurements, with the model inserted in the head coil. Fig. 3 Flow rate measured by MRV at the trachea (symbols), compared to the ideal sinusoidal waveform imposed by the oscillatory piston pump. Three-component, three-dimensional velocity fields are acquired at multiple phases of the ventilation cycle using Magnetic Resonance Velocimetry (MRV). For maximum signal in the MRV measurements, the working fluid is water with 0.06 mol/L of CuSO4. A 3 Tesla Siemens MRI scanner is employed. Velocity data are obtained using the method described by Elkins et al. (2007), with the signaling and data acquisition sequence from Markl. et al. (2012). Threedimensional, three-component (3D-3C) velocities are obtained on a uniform Cartesian grid at a resolution of 0.6 mm. The field of view is 153.6 by 307.2 by 26.4 mm and includes both the 10 International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017 3 2D-1 fluid and the solid walls of the test section. By gating the MRI signal, 10 phased-averaged velocity fields are obtained within the respiration cycle. The procedure is similar to the one used by Banko et al. (2016) who studied the oscillatory flow in a subject-specific airway model. Excellent agreement is found between the imposed waveform and the flow rate measured by MRV through the mother branch G0. Additionally, 2D velocity fields are obtained by Particle Image Velocimetry (PIV) along the symmetry plane of the bifurcation. The PIV system includes an Nd:YAG laser operated at 3 Hz and a 4 Megapixel CCD camera. Refractive index matching between the 3D printed model and the working fluid is achie