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Oscillatory flow facilitates gas exchange in human respiration system. In the present study, both numerical calculation and PIV (Particle Image Velocimetry) measurement indicate that, under the application of HFOV (High Frequency Oscillatory Ventilation), apparent steady streaming is caused and augmented in distal airways by the continuous oscillation, i.e., the core air moves downwards and the peripheral air evacuates upwards within bronchioles. The net flow of steady streaming serves to overcome the lack of tidal volume in HFOV and delivers fresh air into deeper lung region. Also, numerical calculations reveal that the intensity of steady streaming is mainly influenced by the geometry of airways with provided oscillatory frequency and tidal volume, and it rises with Re and Wo up to a Re of about 124 and Wo of about 5. Steady streaming is considered as an important factor for the ventilation efficiency of HFOV.

Oscillatory flow is a widespread phenomenon and plays an important role in many fields, e.g. pneumatic propulsion, piston-drived flow, and acoustic oscillation are commonly used in mechanical engineering; pulsatile blood circulation, respiratory flow in lung, and capillary waves are of much interest in bio-mechanics; seasonal reversing wind, ocean circulations as well as tide flow are of high concern in meteorology, etc. More than mere oscillation or repetition, mass, momentum, and energy may be transferred via these reciprocating movements. The present study focuses on the effect of reciprocal motion in peripheral human lung airways that is much more complex than in a cylindrical channel. If Poiseuille flow is oscillated in a uniform duct, no net flux will occur when the flow restores, although the Stokes layer may vary unsteadily and the flow is in sinusoidal motion during the oscillation.

However, for the oscillatory flow in non-uniform channels, fluid elements would not in general return to original locations at the end of oscillation. This displacement is the integrated result from steady component of oscillation, and often referred to as steady streaming that can be defined as

where

Recently, this time-averaged effect (steady streaming) has gradually been found important and functional in many fields. Steady streaming generated by no slip boundary conditions was surveyed with respect to flows in blood vessels by Padmanabhan and Pedley [

The flow phenomena induced by oscillatory respiration are quite complex in human lung, due to the complicated pulmonary structure, different respiration scenarios, mechanical properties of lung tissue, and the interaction with organs or body parts, etc. The Reynolds number varies from thousands at the trachea drastically to lower than 0.1 in the alveoli sacs during our rest breathing, and the flows may differ more when the gas is oscillated by faster frequency even with lower tidal volume because the Reynolds number could be increased by higher-frequency vibration in airways, which indicates the flow is more turbulent in the upper lung channels, and if the oscillation is fast enough, out-of-phase flow would be induced mainly in the intermediate airways. Currently, the studies of airflow under HFOV (High-Frequency Oscillatory Ventilation) basically center on the upper or intermediate lung region due to the flow particularity there.

Anatomically, an adult human lung bifurcates from trachea to alveoli 23 times and thus forms a multi-branching structure with 24 generations (G0 - G23) according to Weibel’s lung model [

Therefore, we assume that there might be progressive or cumulative net flow, aside from molecular diffusion, to overcome the shortage of tidal volume in HFOV and bring fresh air to the alveoli indirectly. Whether or not steady streaming acts in the transitional bronchioles where the low tidal volume cannot reach needs to be confirmed, which is the first purpose of the present study. If the steady streaming is found working within this zone, its importance and reason need to be analyzed and illustrated, which is the second purpose of this study. The relevant investigations implemented by Haselton and Scherer [

For the oscillatory flow in channels of uniform cross section, it is relatively convenient to figure out the exact solution of velocity distribution. One of the classic oscillatory problems is the Stokes’ second problem which deals with the velocity distribution of viscous flow over an oscillatory plate, as shown in

Another noted oscillatory flow is the unsteady oscillatory flow in cylindrical channel, in a cylindrical coordinate, the governing equation can be expressed by

where

poiseuille,

While for Wo

K represents the acceleration per unit mass,

gas as shown in

In the case of oscillatory flow in pipe of uniform rectangular cross section, similarly, we assume the central axis in X direction, the flow velocity

solved by substituting into the governing equation

boundary conditions for the flow are

proximation of poiseuille velocity distribution is

where

strated in

These cases share one characteristic in common that the flows are oscillated in uniform channel. Velocity distribution varies at different instant of a cycle, however, the net flow is zero after integral cycles of oscillation due to the symmetry of velocity distribution at incoming and outgoing phases, which also means the resultant steady streaming is zero. While in a non-uniform channel, the steady streaming is generally nonzero after the oscillations, a coefficient [

where

In non-uniform channels, it is not convenient to find the exact solution of velocity distribution and the coefficient

with respect to the coefficient

As foresaid, a progressive mechanism is expected to act under the application of HFOV, and further to deliver fresh gas into the transitional zone. Therefore, a cluster model of transitional bronchioles airways needs to be built numerically or realistically, for clarifying the flow feature during oscillations.

In numerical model, the top inlet is fed with oscillatory gas, the four outlets are connected to the pressure boundary conditions. The calculations are implemented by Star-ccm^{+}® which is based on the FVM (finite volume method) algorithm. The governing equations for the gas flow in airway model are:

where

convective terms are discretized by MARS method in second order. Initial temperature and pressure are set at 293.10 K and 101.3 kPa, respectively. Because the flow velocity is lower than 0.1 m/s, and local Reynolds number is less than 10 in this distal region, incompressible Newtonian fluid and laminar flow are selected as the flow properties.

For the numerical inlet boundary, gas velocity is used based on the feature of oscillatory Poiseuille flow, Equation (5) is employed to guarantee 50 mL tidal volume for sinusoidal oscillation with different frequencies in HFOV, where u_{z} and 2.5 × 10^{−}^{4} (m) indicate velocity in z direction and the radius of G18 airway.

For the four outlets at the bottom, gas pressure is assigned as the boundary condition, which comprises lung compliance C, laminar resistance in airway R_{j} and volumetric flow rate in bronchi for a given generation q as demonstrated in Equation (6). And lung compliance C is defined as the ratio of volume difference

where the suffix i indicates the generation number of interest, and j counts from i+1 to the terminal.

Both Lagrangian method and VOF (Volume of Fluid) scheme are employed in calculation to demonstrate the internal flow. VOF method is normally used to distinguish the interface between different species of fluid. The gas property in the airways is homogeneous without species difference, however, VOF can be employed here for judging the net flow which is based on the deformation of fictive interface. In VOF setting, the effect of molecular diffusion is neglected to clearly show the interface between different fluids, and all the fluids share identical physical properties.

Similarly, in the experimental setup, as illustrated in

four truncated elastic tubes, to simulate the compliances of following airways. Additionally, in G18 - G20, the Reynolds number is less than 10, and Womersley number is lower than 1, the Peclet number is about 1 in the HFOV application, which indicates that viscous laminar flow and parabolic quasi-steady flow are dominant in this region, also the advective transport rate and diffusive transport rate are generally in the same order of magnitude.

Our rest breathing normally takes 5 seconds for one cycle of respiration, while the HFOV achieves 50 oscillations within the same duration. As shown in

the gas particles returned to their original locations after one cycle of rest breathing. While in the case of HFOV, a regular redistribution of gas particles is presented after 50 fast and shallow oscillations-the core particles move downwards while the peripheral particles evacuate upwards, and the ones extremely close to the wall do not relocate apparently due to the no-slip condition, which obviously is a time-averaged mechanism.

Similar phenomenon is revealed by means of VOF as demonstrated in

To confirm this phenomenon of net flow induced by HFOV, PIV measurement is thereafter carried out by using the realistic airway model, the gas-feeding pattern is set identical to the numerical calculation-sinusoidal oscillation with 10 Hz frequency and 50 mL tidal volume. After obtaining velocity distribution at different phases in a cycle, the particle dislocations can be produced by integrating the velocity distribution at different instants of the cycle as shown in

HFOV has been confirmed by PIV experiment. According to the particle tracks obtained in experiment, it can be found that the down-coming and up-going routes do not superpose each other for every particle, which implies the oscillatory flow is remarkably irreversible in distal airways. And the irreversible pattern gives rise to the net flow or steady streaming.

The steady streaming phenomenon which is caused by irreversible flow has been confirmed by both CFD calculation and PIV measurement. In bifurcation geometry, the steady streaming is assumed to be mainly affected by the asymmetric geometry in airway, according to the assumption of F. E. Fresconi et al. [_{X} will be investigated under sinusoidal oscillation with 10 Hz frequency and 50 mL tidal volume in one-second HFOV application. Also, the molecular diffusion is neglected for clearly observing the interface between the nominal fresh air and used air.

As illustrated in

Cylinder

Cone

Bifurcation (2 generations)

Branch (3 generations)

Shapes in volume mesh

Moreover, the volume of net flow (V_{nf}) which denotes the volume of fresh gas left in the following region after oscillation has been calculated along with the coefficient CE_{X}, as listed in _{nf} and higher CE_{X}. The CE_{X} of branch model is as high as 0.164, which means 16.4% of the tidal volume has been thrusted into the lower region after 10 oscillations, this lung-like airways apparently is more suitable to be a steady-streaming generator. In addition, the tidal volume at G18 is 50 mL/2^{18} = 1.91 × 10^{−4} mL, because the tidal volume is normally applied to trachea in clinical application of HFOV. For the distal airways, the volume of oscillatory flow is quite limited due to the huge number of branches.

Both bifurcation model and branch model consist of bifurcating channels, however, the volume of net flow and CE_{X} are much higher in the branch model. Therefore, it is considered that more following bifurcations bring greater net flow and stronger steady streaming, this cumulative effect implies that steady streaming may be much more considerable in HFOV application due to the multi-bifurcating structure in the real human lung which bifurcates 23 times from trachea all the way to alveoli.

In clinical field, the airway geometry has been defined already, a series of Wo and corresponding Re are then selected and applied to check their influences on CE_{X}, which may be instructive to HFOV application. The oscillatory frequency is changed to regulate the magnitude of Wo, and the changed velocity then determines Re, while the tidal volume is kept at 50 mL at trachea.

The variation of penetration volume for three-cycle oscillation (sinusoidal, 50 mL tidal volume) under different Womersley numbers (from 0.1 to 10) are illustrated in

Shape | Volume of net flow | CE_{X} |
---|---|---|

Cylinder | 1.36E−6 mL | 0.00713 |

Cone | 3.40E−6 mL | 0.0178 |

Bifurcation | 6.71E−6 mL | 0.0352 |

Branch | 3.12E−5 mL | 0.164 |

tidal volume, while at the end of each oscillation, the values of penetration volume diverge from each other because of the different magnitudes of streaming net flow in airways, which means different amounts of fresh air are left in the following region after the oscillation. The curves indicate that the net flows are relatively low when Wo < 1, while for Wo > 1, the magnitudes of net flow rise drastically as Womersley number grows up to about 5, then fall down gradually when Wo further increases. This trend can be noticed clearly in _{nf} and Wo in three cycles. For all the three cycles, net flow maximizes at about Wo = 5 as well.

The detailed values of oscillatory frequency, Wo, Re, V_{nf}, and CE_{X} after three cycles are listed in the following _{X} reaches a maximum 0.345 when Wo = 5, Re = 123.6, the corresponding frequency is as high as 955.0 Hz. The net flows are compared for different frequencies in same cycles here, if it is compared in an identical duration, the difference will be much greater because the higher frequency accomplishes more cycles in a certain duration. This result indicates that the steady streaming in lung can be enhanced drastically by increasing the oscillatory frequency up to nearly one thousand, which may further give an improvement direction for current HFOV.

Haselton and Scherer investigated streaming magnitude in a single bifurcation channel whose sizes are much greater than our airway models, they found the maximum streaming displacement increases with Re and Wo up to a Re of about 100 and Wo of about 5, after which a gradual decline occurs [

Frequency (Hz) | Wo | Re | Volume of net flow (mL) | Three cycles CE_{X} |
---|---|---|---|---|

0.382 | 0.1 | 0.049 | 1.81E−06 | 0.0094 |

3.438 | 0.3 | 0.445 | 3.19E−06 | 0.0167 |

9.550 | 0.5 | 1.236 | 4.69E−06 | 0.0246 |

18.72 | 0.7 | 2.423 | 5.21E−06 | 0.0273 |

38.20 | 1 | 4.944 | 7.23E−06 | 0.0379 |

152.8 | 2 | 19.778 | 2.38E−05 | 0.1246 |

343.8 | 3 | 44.500 | 4.77E−05 | 0.2497 |

611.2 | 4 | 79.110 | 6.49E−05 | 0.3398 |

955.0 | 5 | 123.61 | 6.59E−05 | 0.3450 |

1375.1 | 6 | 177.991 | 6.16E−05 | 0.3225 |

1871.7 | 7 | 242.262 | 5.83E−05 | 0.3052 |

2444.6 | 8 | 316.418 | 5.53E−05 | 0.2899 |

3094.0 | 9 | 400.467 | 5.19E−05 | 0.2721 |

3820.0 | 10 | 494.440 | 4.81E−05 | 0.2522 |

and Grotberg [

Obviously, the Womersley number can be changed by changing either frequency or viscosity within a provided geometry. But only oscillatory frequency has been altered to change Wo in this study, one reason is in clinical field, the viscosity of ventilating gas is almost constant, another reason is that we anticipate faster oscillation could incur better ventilation efficiency by causing more turbulence in upper lung region as well as more steady streaming. From

A common scenario of HFOV (sinusoidal with 10 Hz frequency and 50 mL tidal volume) has been investigated by both CFD calculation and PIV measurement in the present study. An apparent time-averaged net flow phenomenon has been found in peripheral lung airways which help to deliver the fresh air into deeper region centrally and discharge the used air peripherally. It demonstrates that even though the tidal volume is smaller than dead space of lung in HFOV, steady streaming works to thrust the fresh gas deeply and thereby overcome the lack of tidal volume. The distal lung region is therefore more convective than previously expected. In view of steady streaming found in upper lung airways by other researchers, it is considered that steady streaming may exist in the whole lung tract and is an important factor of HFOV effect.

A series of airway geometry have been numerically surveyed to clarify the influence of geometry on steady streaming. It is found that more divergent channels bring stronger steady streaming, and the multi-bifurcated channel causes greater steady streaming than the single bifurcation does, which indicates the magnitude of steady streaming in real lung may be much greater than that in the truncated branch model under HFOV application.

By changing the oscillatory frequency, different Womersley numbers and corresponding Reynolds numbers have been obtained to testify their influences on steady streaming. It has been found that the streaming magnitude rises with Re and Wo up to a Re of about 124 and Wo of about 5, which is in a good agreement with the experimental results brought out by other researchers. Moreover, the volume of net flow maximizes when the frequency reaches nearly 1000, which may imply the feasibility of super-fast HFOV in the future.

The authors are grateful to Kohei Morita and Tomonori Yamamoto for assistance in numerical calculation and PIV measurement.

Han, B., Hirahara, H. and Yoshizaki, S. (2016) Streaming Cau- sed by Oscillatory Flow in Peripheral Airways of Human Lung. Open Journal of Fluid Dynamics, 6, 242-261. http://dx.doi.org/10.4236/ojfd.2016.63019

C: Compliance [m^{3}/Pa]

f: Frequency [Hz]

G: external force [N]

p: Pressure [Pa]

Q: Flow rate [m^{3}/s]

Pe: Peclet number

r: radial position [m]

Re: Reynolds number

t: time [s]

T: flow cycle period [s]

u: oscillatory velocity [m/s]

U: velocity amplitude [m/s]

^{3}/s]

u_{z}: velocity in z direction [m/s]

V_{T}: tidal volume [m^{3}]

Wo: Womersley number

υ: kinematic viscosity [m^{2}/s]

ω: angular frequency [rad/s]

ρ: fluid density [kg/m^{3}]

i: ith generation of lung

j: jth generation of lung

z: z-direction