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Research Papers

# Contributions of Kinetic Energy and Viscous Dissipation to Airway Resistance in Pulmonary Inspiratory and Expiratory Airflows in Successive Symmetric Airway Models With Various Bifurcation AnglesOPEN ACCESS

[+] Author and Article Information
Sanghun Choi

Department of Mechanical Engineering,
Kyungpook National University,
Daegu 41566, South Korea
e-mail: s-choi@knu.ac.kr

Jiwoong Choi

IIHR-Hydroscience & Engineering,
Iowa City, IA 52242;
Department of Mechanical and
Industrial Engineering,
The University of Iowa,
Iowa City, IA 52242
e-mail: jiwoong-choi@uiowa.edu

Ching-Long Lin

IIHR-Hydroscience & Engineering,
Iowa City, IA 52242;
Department of Mechanical and
Industrial Engineering,
3131 Seamans Center for the Engineering
Arts and Sciences Iowa City,
The University of Iowa,
Iowa City, IA 52242
e-mail: ching-long-lin@uiowa.edu

1Corresponding author.

Manuscript received April 27, 2017; final manuscript received September 26, 2017; published online November 9, 2017. Assoc. Editor: Alison Marsden.

J Biomech Eng 140(1), 011010 (Nov 09, 2017) (13 pages) Paper No: BIO-17-1180; doi: 10.1115/1.4038163 History: Received April 27, 2017; Revised September 26, 2017

## Abstract

The aim of this study was to investigate and quantify contributions of kinetic energy and viscous dissipation to airway resistance during inspiration and expiration at various flow rates in airway models of different bifurcation angles. We employed symmetric airway models up to the 20th generation with the following five different bifurcation angles at a tracheal flow rate of 20 L/min: 15 deg, 25 deg, 35 deg, 45 deg, and 55 deg. Thus, a total of ten computational fluid dynamics (CFD) simulations for both inspiration and expiration were conducted. Furthermore, we performed additional four simulations with tracheal flow rate values of 10 and 40 L/min for a bifurcation angle of 35 deg to study the effect of flow rate on inspiration and expiration. Using an energy balance equation, we quantified contributions of the pressure drop associated with kinetic energy and viscous dissipation. Kinetic energy was found to be a key variable that explained the differences in airway resistance on inspiration and expiration. The total pressure drop and airway resistance were larger during expiration than inspiration, whereas wall shear stress and viscous dissipation were larger during inspiration than expiration. The dimensional analysis demonstrated that the coefficients of kinetic energy and viscous dissipation were strongly correlated with generation number. In addition, the viscous dissipation coefficient was significantly correlated with bifurcation angle and tracheal flow rate. We performed multiple linear regressions to determine the coefficients of kinetic energy and viscous dissipation, which could be utilized to better estimate the pressure drop in broader ranges of successive bifurcation structures.

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## Introduction

Investigating flow physics in a successive branching tubular structure is crucial for understanding human lung physiology, as a human lung consists of airway structures up to the 23rd generation that range from ∼20 mm (trachea) to ∼100 μm (terminal bronchioles) [1]. Furthermore, in order to better understand global lung function estimated by pulmonary function tests, especially of diseased lungs [2,3], estimating airway resistance in human airways is important. For instance, in asthma, airway narrowing [4,5] and air trapping [6,7] are known to be distinct phenotypes that may cause an increase in the pressure drop between mouth and terminal bronchioles by elevating airway resistance. Despite previous efforts to understand flow physics associated with airway resistance in human airways, the effect of the bifurcation angle (θ) and flow rate (Q) on the airway resistance at multiple generations needs further investigation. As air trapping may occur during expiration, especially in severe asthma [6,7], understanding flow physics on expiration is crucial. However, most of the existing studies on airway resistance are limited to cases of inspiration. In this context, the main interest of this study is to investigate the relationship between variables that can potentially contribute to airway resistance, including kinetic energy and viscous dissipation, under different conditions of θ (airway structure) and Q (lung function) for both cases of inspiration and expiration.

About four decades ago, Pedley et al. [812] made a seminal contribution to the study of airway resistance for inspiratory flows. Employing experiment data, an airway resistance model was empirically derived using a laminar boundary layer theory. The major contribution of their study was the establishment of a relationship between the viscous pressure drop (ΔPVis) and Q. The existing Poiseuille flow assumption [13] relies on the linear relationship between the two variables, which means that the ratio of ΔPVis to Q is only a function of airway structural variables such as length (L) and diameter (D). However, Pedley et al. [9] demonstrated that ΔPVis is proportional to Q1.5, implying that the ratio of ΔPVis to Q is a function of L, D, and Reynolds number (Re). In addition to viscous dissipation, kinetic energy was parameterized with a constant coefficient using experimental data that can be used to estimate the change in kinetic energy due to bifurcation between generations [9]. Using a theoretical analysis, Hyatt and Wilcox [14] explained that kinetic energy only accounts for 10% of the total pressure drop in an entire lung and, thus, the effect of kinetic energy was negligible in the sense of global lung function. As a consequence, the effect of kinetic energy was ignored in the existing network resistance models [15,16]. However, the effect of kinetic energy should be considered at the local segmental level when one performs branch-by-branch analysis or develops a network one-dimensional (1D) model for entire lung simulations [15,16].

Owing to advances in its computational power, computational fluid dynamics (CFD) has been used to validate or improve existing viscous dissipation models. Comer and Zhang [17] analyzed CFD-predicted flow structures using tubular bifurcation structures with a three-generation geometry, representing airways from the third to fifth generation, at three different Re and compared them with the model of Pedley et al. [812], concluding that their model can be useful when the viscous dissipation coefficient is improved. However, the study was limited by the small generation number on inspiration and the highly simplified airway model it utilized. Later, CFD simulations using realistic healthy human airway models became capable of resolving complex flow physics under various breathing conditions [1821]. Employing realistic human airways, van Ertbruggen et al. [22] performed CFD simulations, and predicted pressure drops at each bronchial segment. They derived the coefficients of viscous dissipation and compared them by generation with the model of Pedley et al. [812]. On the other hand, Katz et al. [23] employed a model [13] to estimate major and minor losses, where the major loss was the same as the viscous dissipation assumed by Poiseuille flows and the minor loss was predicted by CFD simulations. Unfortunately, both models [22,23] neglected the effect of kinetic energy, assuming that the pressure drop due to viscous dissipation was the same as the total pressure drop. More recently, Borojeni et al. [24] compared the above three models by testing their applicability to both adult and child lungs, demonstrating that the model of van Ertbruggen et al. [22] is a good option for adults, whereas the models of Pedley et al. [812] and Katz et al. [23] work better for children.

Although symmetric and realistic human airway models have been used to characterize airway resistance, all of the previous studies have focused on inspiratory flows. Besides, there have been only a limited number of reports on the relationship between kinetic energy and viscous dissipation. Although viscous dissipation has been regarded as a major contributor to airway resistance in previous studies, it has yet to be precisely analyzed for inspiratory flows and expiratory flows under various flow rate conditions. Furthermore, from our previous population-based analysis [4], the bifurcation angle between two daughter branches varied from 40 deg to 100 deg. A CFD study [25] based on a single bifurcation model concluded that θ has a marginal effect on the airway resistance. Both θ (a structural variable) and Q (a functional variable) can be altered in diseased lungs [4]. Thus, this study aims to differentiate the contributions of kinetic energy and viscous dissipation to airway resistance, and further investigate the effects of θ and Q on the airway resistance during both inspiration and expiration. Symmetric airway models are adopted for this study to control other factors associated with airway resistance, such as branching asymmetry between daughter branches, homothetic diameter, and length ratios.

## Methods

###### Generation of Symmetric Airway Models.

Figure 1(a) shows symmetric airways up to the 20th generation, which was employed by Weibel-type model considering trachea as the zeroth generation [26]. Based on the existing literature by Pedley et al. [9], the diameter (D) of the zeroth generation was set to 18 mm, the successive diameter ratio of the daughter to parent branch (homothetic ratio) was imposed as 0.78, and the length-to-diameter (L/D) ratio was 3.5 (Fig. 1(b)). Diameter ratios of the glottis and the cross section above the glottis to the trachea were empirically imposed as 0.677 and 1.24, respectively [27]. The following θ values were used to examine the effect of the bifurcation angle on the airway resistance: 15 deg, 25 deg, 35 deg, 45 deg, and 55 deg (Table 1, Fig. 1(c)). In this study, θ was defined as the angle between the current branch and its parent branches (Fig. 1(c)). Unlike right airways from trachea located in-plane, left airways were generated out-of-plane to avoid an interference between left and right airways (see Supplemental material for various views of the airway model which is available under “Supplemental Data” tab for this paper on the ASME Digital Collection). We confirmed that the effect of being out-of-plane on the right airways compared to the left ones is negligible in terms of airway resistance, although it is not presented here. Between the models, L, D, and Re were the same for a given Qtrachea. To examine the effect of Qtrachea, low, moderate, and high Qtrachea of 10 L/min, 20 L/min, and 40 L/min were imposed on the airway model with a θ of 35 deg (Table 1), being equivalent to Re of 719, 1438, and 2875 at the zeroth generation (trachea), respectively. We used tracheal diameter in calculation of Re. In our analysis, we defined the parent branch of the first bifurcation as the zeroth generation (Fig. 1(b)). Note that the region with glottal constriction above the zeroth generation was not investigated in this analysis.

The computational domain (Fig. 1(d)) was partitioned to create surface and volume meshes for each subdomain. Two representative paths that extended to the 20th generation were extracted and analyzed (Fig. 1(b)), as they represented two extreme paths: a branch was successively generated toward the left direction from right main bronchus and another was successively generated toward the right direction. To create triangular surface and tetrahedral volume meshes for branch-by-branch domains, we employed the software gmsh [28]. The grid size of each subdomain was determined from the branch diameter and flow rate for creating similar wall units [27]. As for the mesh size, the average distance from the wall to the first grid for the case with Qtrachea = 20 L/min was 4.7 in wall units (y+), which was smaller than the viscous sublayer thickness (y+ = 5). A more specific description for mesh generation can be found in Ref. [27]. This mesh size was validated in the previous lung airflows simulation [29]. The computational meshes for the five different θ cases contained about four million tetrahedral elements (Table 1).

###### CFD Simulations and Boundary Conditions.

A finite element method [30] with a large eddy simulation technique [31] was adopted to solve filtered incompressible Navier–Stokes equations, as follows: 1Display Formula

(1a)$∇⋅u=0$
Display Formula
(1b)$ρ∂u∂t+ρ(u⋅∇)u=−∇p+∇⋅[(μ+μT)(∇u+∇uT)]$

where u, ρ, p, μ, and μT are filtered velocity vector, fluid density, pressure, fluid dynamic viscosity, and subgrid-scale (SGS) turbulent eddy viscosity, respectively. The ρ and μ values were set to 1.12 kg/m3 and 1.84 kg/(m s), respectively. The effect of gravity was neglected because this was an airflow simulation. Boundary conditions for these CFD simulations are described as follows: On the airway walls, no-slip boundary conditions (u = 0) were imposed. On inspiration, at the cross section above the glottal constriction, i.e., the inlet, a synthetic turbulence model [27,32] was imposed to feed synthetic turbulence, and a parabolic velocity profile was imposed at the openings of the ending branches, i.e., the outlet. On expiration, a parabolic velocity profile was imposed at the openings of the ending branches, i.e., the inlet, and a uniform velocity profile was imposed at the cross section above the glottal constriction, i.e., the outlet. The flow rates in two daughter branches were the same and equaled half of the flow rate in the parent branch for both inspiration and expiration because of the symmetry. With the equal flow distribution at the same generation, we could compute flow rates and impose velocity profiles (with known cross-sectional areas) at ending branches as Dirichlet boundary conditions. After reaching a quasi-steady state, the simulation was run for 2.4 s corresponding to about three through-flow times, and the resulting flow variables were used to calculate the ensemble-averaged turbulence statistics.

###### Pressure Drops Due to Change in Kinetic Energy and Viscous Dissipation.

To analyze the pressure drop in a partitioned subdomain (Fig. 1(d)), we employed an energy balance equation based on Eq. (1)Display Formula

(2)$∂∂t(ρu22)=−∇⋅[pu+ρu22u]+u⋅[(μ+μt)∇2u]$

where u is the magnitude of the velocity. As we analyzed ensemble-averaged velocities with quasi-steady-state simulation data, the steady-state assumption was acceptable. In addition, when the Gauss–divergence theorem is applied, the weak form of Eq. (2) in a control volume becomes Display Formula

(3)$∫Spu⋅ndA︸(a)+∫Sρu22u⋅ndA︸(b)=∫Su⋅[(μ+μT)∇2u]dV︸(c)$

where S and Ω denote control surface and control volume, respectively, and n is a unit outward normal vector. Terms (a), (b), and (c) in Eq. (3) were associated with the total pressure drop, the change in kinetic energy, and the viscous dissipation (ΦVis), respectively. With the no-slip boundary condition (u = 0) applied to the walls, the total pressure drop (ΔPTot) and the kinetic energy-driven pressure drop (ΔPKE) in a partitioned domain (Fig. 1(d)) were expressed as follows: Display Formula

(4)$ΔPTot=∫S,i−1pu⋅ndAQi−1−∫S,ipu⋅ndAQi, ΔPKE=∫S,i−1ρu22u⋅ndAQi−1−∫S,iρu22u⋅ndAQi$

where Qi is the flow rate at the ith control surface. For both the inspiration and expiration cases, the proximal and distal surfaces were defined as the (i−1)th and ith control surface respectively, so that the sign of the pressure drop was likely to be positive on inspiration and negative on expiration. By continuity Eq. (1a), Qi−1 shall be equal to Qi. The energy balance equation (3) was recast as Display Formula

(5)$ΔPTot≡ΔPVis−ΔPKE$

where ΔPVis denotes the pressure drop due to viscous dissipation. ΔPVis consists of grid-resolved and SGS turbulent eddy viscous dissipations because the large eddy simulation model was used. The pressure drop due to the grid-resolved flow structures (ΔPVis,Resolved) was computed as follows: Display Formula

(6)$ΔPVis, Resolved=∫Ωu⋅[μ(∇u+∇uT)]dVQ(i)≈12∫Ωμ(∇u+∇uT)2dVQ(i)$

The difference between ΔPVis and ΔPVis,Resolved was the pressure drop due to the SGS turbulent eddy viscous dissipation.

###### Dimensional Analysis for Kinetic Energy and Viscous Dissipation.

In the 1D energy balance equation [13], PKE is estimated as KρU2, where U is an average velocity on a cross-sectional area, and K is a parameter depending on velocity distribution. In each ith distal control surface (Fig. 1(d)), we calculated K as follows: Display Formula

(7)$Ki=PKEiρUi2=∫S,iρu22u⋅ndAρUi2Qi$

If velocity is nonuniform on a cross-sectional area, K becomes larger. Thus, K is larger in parabolic flows than in turbulent flows, e.g., K ∼ 0.5 for fully developed turbulent flows and K = 1.0 for fully developed laminar flows [13]. Pedley et al. [812] empirically estimated the value of K as 0.85 in their experimental airway model. In this study, we evaluated K on inspiration and expiration associated with different θ and Qtrachea.

Viscous dissipation models for human airways have been developed in several previous studies [9,22,23]. In a representative study conducted by Pedley et al. [812], a model was derived from a symmetric bifurcation angle of 70 deg between daughter branches, which is equivalent to a θ of 35 deg in this study. Employing the laminar boundary-layer theory, they derived a relationship between ΔPVis and the pressure drop due to viscous dissipation in fully developed laminar flows in a straight tube (ΔPPoiseuille) as follows: Display Formula

(8)$γ(i)=(ReddL)−0.5ΔPVis(i)ΔPPoiseuille(i)$

where γ was empirically determined as 0.327 in the case of inspiration [812]. With Eq. (8) and ΔPVis from Eq. (5), we computed the coefficient γ(i) in the (i)th control volume up to the tenth generation because ΔPVis was similar to ΔPPoiseuille in the airways beyond the tenth generation. Finally, airway resistance was calculated as follows: Display Formula

(9)$resistance=ΔPTot(i)Qtrachea$

Note that ΔPTot in Eq. (9) consists of both ΔPKE and ΔPVis, as expressed in Eq. (5).

## Results

###### Pressure Drop Due to Change in Kinetic Energy and Viscous Dissipation.

With θ = 35 deg, the total pressure drop (ΔPTot), the pressure drop due to change in kinetic energy (ΔPKE), the pressure drop due to viscous dissipation (ΔPVis), and the pressure drop due to grid-resolved viscous dissipation (ΔPVis,Resolved) were computed for each control volume (Fig. 1(d)), for both the inspiration and expiration cases (Fig. 2). The inspiration case with low Qtrachea (Fig. 2(a)) and the expiration cases with three different Qtrachea (Figs. 2(b), 2(d), and 2(f)) demonstrate that ΔPVis is similar to ΔPVis,Resolved. This indicates that the SGS effect is negligible for the pressure drop due to viscous dissipation. For the inspiration cases with moderate and high Qtrachea (20 L/min and 40 L/min) (Figs. 2(c) and 2(e)), ΔPVis was larger than ΔPVis,Resolved up to the third generation, implying that turbulent flows were active in these regions. Figure 3 also shows that the effects of a turbulent laryngeal jet emanating from the glottal constriction on inspiration, specifically the effects of its turbulent kinetic energy, could reach the third generation. Besides, for both the inspiration and expiration cases, ΔPKE was not negligible up to approximately the ninth generation, indicating non-negligible contribution of flow developments emanating from the parent branch and the two child branches [21]. For example, for the inspiration case with high Qtrachea (Fig. 2(e)), ΔPKE at the first and second generations was similar to ΔPVis, resulting in almost zero ΔPTot. Overall ΔPTot on inspiration was smaller than that on expiration, although the absolute value of ΔPVis on inspiration was larger than that on expiration. Thus, the effect of kinetic energy shall be taken into consideration in predicting the pressure drop up to about the ninth generation.

###### Wall Shear Stress and Airway Resistance.

Figure 4(a) shows the mean and standard deviation (SD) of average wall shear stress (τw) at the wall surface of each generation (Fig. 1(d)) for the inspiration and expiration cases. The mean τw and its SD were calculated with the results from five different θ at the same Qtrachea of 20 L/min. Note that the effect of θ is shown in Sec. 3.4. For both the inspiration and expiration cases, the minimum τw was observed at around the sixth–ninth generations, and its maximum was observed at either lower or higher generations (Fig. 4(a)). As expected, τw increased with an increase in Qtrachea for both the inspiration (Fig. 4(c)) and expiration (Fig. 4(e)) cases. For both cases, we computed log–log ratios as follows: [log(τwhigh) − log(τwlow)]/[log(Qtracheahigh) − log(Qtrachealow)]. The log–log ratio between wall shear stress and Qtrachea was about 1.2 up to the seventh generation, and about one after the eighth generation, indicating nonlinear and linear relationships between τw and Qtrachea depending on the generation number. In contrast to the average wall shear stress, airway resistance was much larger on expiration than that on inspiration (Fig. 4(b)), because ΔPKE plays a critical role in decreasing ΔPTot on inspiration and in increasing ΔPTot on expiration. The resistance varied depending on the Qtrachea up to the ninth generation. Within the ninth generation, the resistance did not show any consistent patterns on inspiration, whereas it increased with increasing Qtrachea on expiration. Resistance became fairly constant for both the inspiration and expiration cases beyond the ninth generation, where the effect of kinetic energy is negligible and the Poiseuille flow assumption is valid. Figure 5 further demonstrates the difference between ΔPVis and ΔPPoiseuille for three different Qtrachea. Consistent with airway resistance in Fig. 3, the difference between ΔPVis and ΔPPoiseuille was negligible in the airways beyond the ninth generation. Therefore, we focused on the branches up to the tenth generation when conducting the analyses in Secs. 3.3 and 3.4.

###### Dimensional Analysis for Kinetic Energy.

Employing Eq. (7), the coefficient of kinetic energy (K), a measure of velocity uniformity on a cross section, was computed at the distal control surface of a control volume (Fig. 1(d)) with different θ and Qtrachea for both the inspiration and expiration cases (Fig. 6). For both cases, K ranged from 0.8 to 1.35, and it increased with increasing generation number. Refer that K = 0.5 and 1.0 for fully developed turbulent flows and fully developed laminar flows, respectively. In addition, K was larger on inspiration than that on expiration. This may be because of the nonuniform velocity distribution that is generated on inspiration. Inspiratory flows and kinetic energy (Figs. 7(a) and 7(c)) were nonuniform owing to the inertial effect from the parent branch, whereas expiratory flows and kinetic energy (Figs. 7(b) and 7(d)) were fairly uniform owing to the merger of downstream in the parent branch. While a pair of counter rotating vortices was observed on inspiration, two pairs of counter-rotating vortices were observed on expiration. The strong secondary motions on expiration possibly affect uniform distribution of kinetic energy. Furthermore, for the same Qtrachea, K was insensitive to θ (Figs. 6(a) and 6(b)). K appeared to vary with Qtrachea only in inspiratory flows (Fig. 6(c)), whereas K was less dependent on Qtrachea in expiratory flows (Fig. 6(d)). On inspiration, a pair of counter rotating vortices associated with the effect of turbulence (Fig. 7(c)) may lead to momentum redistribution, as Qtrachea increases. The significant variation of K at the zeroth to second generation on inspiration was possibly due to the laminar- or transitional-flow distribution at the low Qtrachea of 10 L/min, which led to the nonuniform flow distribution on the cross section. Figure 2(a) shows that ΔPVis and ΔPVis,Resolved were essentially overlapped, indicating that turbulent flows are inactive in the case of low Qtrachea.

###### Dimensional Analysis of Viscous Dissipation.

Employing Eq. (8), the viscous dissipation coefficient (γ) was computed for each control volume (Fig. 1(d)) for different θ and Qtrachea during both inspiration and expiration (Fig. 8). For both cases, γ ranged from 0.1 to 0.7, and it increased with increasing generation number. Note that Pedley et al. [812] derived an averaged coefficient of 0.327 with the aid of experimental data. Overall, γ was smaller for the expiration cases than for the inspiration cases. Figures 9(a) and 9(b) illustrate the elevated wall shear stress on inspiration as compared with that on expiration. Figures 9(c) and 9(d) further depict the grid-resolved viscous dissipation (ΦVis,Resolved). These plots reveal the regions of large dissipation for inspiration and expiration, respectively. For the inspiration cases, flows from a parent branch impinge on the carina and develop a boundary layer on the inner walls of the following bifurcation (Fig. 9(c)). For the expiration cases, larger dissipation was mainly found in the shear layer formed by the merger of two fluid streams emanating from daughter branches. At any rate, the summation of dissipation on expiration was smaller than that on inspiration (Figs. 8(c) and 8(d)). Different from K, γ was somewhat sensitive to θ. As θ increased, γ increased. This is because more inclined walls on the downstream could further reduce boundary layer thickness, yielding an increase in viscous dissipation (shear stress). In addition, Figs. 8(c) and 8(d) show that γ significantly depends on Qtrachea for both the inspiration and expiration cases, except for the branches of the zeroth to second generation. Thus, the effects of θ and Qtrachea shall be taken into consideration in modeling airway resistance.

###### Improved Modeling of Dimensionless Parameters.

Although the kinetic energy coefficient K was less sensitive to θ, it was sensitive to generation number (Gen) and Q, especially on inspiration (Fig. 6). To control the flow rate effect when obtaining K, with the data between the zeroth and tenth generation for the inspiration and expiration cases, we performed multiple linear regressions to determine the relationship between K and Re, and K and Gen. As the low Qtrachea case on inspiration (Fig. 6(c)) exhibits a relatively large deviation, we excluded the two data points at the zeroth and first generation in the following analysis. The ninth and tenth generations were also excluded in the case of inspiration because K reached a plateau beyond the eighth generation, where the mean (SD) of K was 1.33 (0.04). Thus, 61 (nine generations × seven cases − two points) and 77 (11 generations × 7 cases) observations were used to derive the linear regressions for inspiration and expiration, respectively. The linear regression models were derived as follows: 10Display Formula

(10a)$K=0.8519+6.42×10−2Gen-5.456×10−5(Gen⋅Re),for 0≤Gen≤8K=1.33,for 9≤Gen≤10 on inspiration$
Display Formula
(10b)$K=0.8277+3.696×10−2Gen-2.671×10−5(Gen⋅Re), for 0≤Gen≤10 on expiration$

The adjusted R2 values for the inspiration and expiration linear regression models were 0.94 and 0.91, respectively, and the Akaike information criterions (AIC) were −210 and −276, respectively. Employing all the data points between the zeroth and tenth generations, the model for K defined by Eq. (10) was compared with the CFD-predicted K, as shown in Figs. 10(a) and 10(b). More significant correlations were observed on expiration than on inspiration because several outliers exist at the first and second generations at low Qtrachea on inspiration, which were not included in the linear regression model defined by Eq. (10a). Beyond the tenth generation, K could be assumed as 1.0, because of parabolic velocity profiles.

Next, we performed multiple linear regressions to determine the relation between γ and θ, and γ and Q. As illustrated in Fig. 8, for inspiration, γ was more sensitive to the θ up to the third generation, whereas it was sensitive to both the Q and θ from the third to tenth generation; thus, we divided the generations into two regions in order to obtain better regressions. From the seven inspiration cases (Table 1), the mean (SD) of γ at the zeroth generation on inspiration was 0.0834 (SD = 0.002). We obtained linear regressions between γ and θ on inspiration (the first and third generation, 21 data points: three generations × seven cases) as well as on expiration (the zeroth and third generation, 28 data points: four generations × seven cases) Display Formula

(11a)$γ=0.0834, for generation=0γ=0.0979+0.2039θ, for 1≤generation≤3 on inspiration$
Display Formula
(11b)$γ=0.0429+0.1964θ, for 0≤generation≤3,on expiration$

where θ is in radians. These models demonstrated significant correlations, i.e., the adjusted R2 values for both inspiration and expiration cases were 0.979 and 0.988, respectively, while the AICs were 31 and 14, respectively.

As the variability due to θ and Qtrachea was significant between the fourth and tenth generation, we used a variable η instead of the constant 0.5 from Eq. (8) to control the effect of Qtrachea. Forty-nine observations (seven generations × seven cases) were employed for the linear regressions for both inspiration and expiration. For linear fitting, Eq. (8) was transformed into the log scale as follows: Display Formula

(12)$log (ΔPVisΔPPoiseuille)=log [γ(ReddL)η]=log γ+η log (ReddL)$

Based on Figs. 8(c) and 8(d), we assumed that γ was a linear function of θ and Gen, and η was a linear function of Gen. The multiple linear regressions provided the following equations between the fourth and tenth generations: 13Display Formula

(13a)$γ=exp(−1.322+0.093 Gen+0.537 θ) η=0.3671−2.737×10−2 Genfor 4≤generation≤10;on inspiration$
Display Formula
(13b)$γ=exp(−1.030+0.076 Gen+0.447 θ) η=0.2631−2.313×10−2 Gen for 4≤generation≤10.on expiration$

For both inspiration and expiration, the adjusted R2 values for these models were 0.914 and 0.885, respectively, and the AICs were −133 and −154, respectively. Beyond the tenth generation, it is reasonable to invoke the Poiseuille flow assumption in calculating the pressure drop, as shown in Figs. 2 and 4. Figure 10 demonstrates that the new viscous dissipation model using variables γ and η provided significantly improved correlations, and the trends aligned with the identity line.

## Discussion

Investigating flow physics in the context of airway resistance is key to understanding the global mechanism of lung function in healthy airways, which further helps us to understand the lung pathophysiology in asthma and chronic obstructive pulmonary disease patients [2,3]. Asthma and chronic obstructive pulmonary disease may be characterized by luminal narrowing of central airways and/or air trapping in the peripheral airways, which lead to a significant increase in airway resistance and pressure drop. However, there are few studies that compare airway resistance in inspiratory and expiratory flows with successive airway bifurcations and various flow conditions. Although airway resistance is determined by both kinetic energy and viscous dissipation, most of the previous studies [22,23] focused on viscous dissipation only. Besides, airflow obstruction may be more associated with expiration than inspiration [33]; therefore, it is important to understand airway resistance during expiration. Using a CFD technique and an energy balance analysis (Eq. (2)), we differentiated contributions of pressure drops due to kinetic energy and grid-resolved and SGS turbulent eddy viscous dissipations. Furthermore, we performed dimensional analyses for both kinetic energy and viscous dissipation to examine the effects of θ and Qtrachea on the airway resistance during inspiration and expiration.

The effect of kinetic energy tends to be overlooked when estimating the total pressure drop because it is only dominant in several large airways [14]. This may be true at the global organ scale, but it may not be negligible at the local segmental scale. It has been typically omitted for the sake of simplicity [15,16]. In regard to flow physics, bulk velocity should decelerate on inspiration owing to an increase in the total cross-sectional area by generation, whereas it should accelerate on expiration owing to a decrease in the total cross-sectional area [1]. Consequently, it contributes to the increase (or decrease) of the total pressure drop on expiration (or inspiration). In addition, with increasing Qtrachea, the effect of kinetic energy on the airway resistance becomes more dominant, as shown in Figs. 2(e) and 4(d). This is because kinetic energy is a function of Q2 when determining pressure drop, whereas viscous dissipation is empirically a function of Q1.5 (for lower generations) or Q (for higher generations). Therefore, besides viscous dissipation, kinetic energy may play an important role in lung hysteresis during breathing [29,34]. This may have a larger effect on asthmatic lungs because their homothetic ratio is not uniform, and therefore, abrupt constriction from parent to daughter branches may increase local velocity, leading to an increase in kinetic energy. Thus, this study suggests that kinetic energy should not be neglected for the sake of simplifying computational models.

We computed wall shear stress and airway resistance for both inspiration and expiration (Fig. 4). We found that both parameters of wall shear stress and airway resistance behaved differently during inspiration and expiration. Specifically, airway wall shear stress (or viscous dissipation) was greater on inspiration than on expiration, whereas airway resistance was greater on expiration than on inspiration. The different characteristics of the two variables on inspiration and expiration can be explained by taking into consideration the contribution of kinetic energy to the total pressure drop, i.e., deceleration on inspiration and acceleration on expiration. This finding also implies that caution must be taken when referring to either viscous dissipation or airway resistance. We further found that the minimum wall shear stress occurred at about the sixth–ninth generation, whereas larger wall shear stress occurred within regions of either lower or higher generation. While large wall shear stress is expected due to high Re in larger airways, relatively larger wall shear stress is noticeably observed in regions of smaller airways (>the ninth generation) due in part to the variation of total cross-sectional area from generation to generation. Small airways are usually classified with the diameters smaller than 2 mm [35], which corresponds to the diameter of branch in the ninth generation in this study. Since smaller airways tend to have thinner airway walls, we speculate that these regions are more susceptible to external stimulation, indicating the importance of peripheral airway resistance [36].

To parameterize the effects of θ and Qtrachea on kinetic energy and viscous dissipation, we performed dimensional analyses. In human airways [4], the bifurcation angle between daughter branches ranged from 40 deg to 100 deg, being equivalent to a θ of 20–50 deg in this analysis. Thus, the cases where θ is equal to 15 deg and 55 deg could be considered as two extreme cases. The kinetic energy coefficient, K, was less sensitive to θ for both inspiration and expiration cases, but it was sensitive to the generation number. Linear regression models were used to predict kinetic energy. Thus, a constant coefficient, e.g., 0.85, suggested by Pedley et al. may not be sensitive enough to capture local variations [812]. The increase in K with increasing generation number indicates the nonuniformity of flow distribution due to a successive branching structure. K also tends to drop after the eighth generation, implying that flow distribution returns to a uniform state. The deviation of low Qtrachea from the moderate and high Qtrachea cases at the first and second generations (Fig. 6(c)) was possibly due to the dominance of laminar or transitional flows in the case of low Qtrachea. The result implies that three-dimensional (3D) CFD simulation should be used to predict these regions at least when using 3D and 1D coupled simulation [21]. Furthermore, as Qtrachea increases on inspiration, K tends to be reduced, implying more uniform flow distribution (Fig. 6(c)). This may be because turbulent flows and one pair of counter rotating vortices induce momentum redistribution on a cross section, similarly found in the study of Comer and Zhang [17]. Jalal et al. [37] investigated flow uniformity, using a different measure [38] for various Re, and demonstrated the similar feature of momentum redistribution with the current study. We also found that overall K is larger on inspiration than on expiration, because, as a result of the inertial effect, inspiration flows are more nonuniform than expiratory flows (Fig. 7), and two pairs of counter rotating vortices strongly affect momentum redistribution on expiration.

As demonstrated in the Falkner–Skan solution using a similarity analysis [39], larger θ may decrease boundary layer thickness and consequently increase shear stress. In our analysis, the increase in γ with increasing θ was fairly aligned with the Falkner–Skan solution. Besides, during both inspiration and expiration, the relationship ΔPVis ∼ Q1.5 from Pedley et al. [812] seemed to relax the effect of Q up to the third generation. However, it is speculated that the model of Pedley et al. may not work for airways beyond the fourth generation for both inspiratory and expiratory flows because γ is still a function of Q between the fourth and tenth generations. Therefore, to improve the relationship between ΔPVis and Q, we performed multiple linear regressions. Since the derived γ is a dimensionless parameter, it allows to predict pressure drop, if the information including flow rate, flow direction, diameter, length, generation is available.

As mentioned earlier, for the in-plane branching patterns, we used symmetric branching structures to control other factors that may affect airway resistance, such as branching asymmetry between daughter branches, homothetic diameter, and length ratios. Thus, this study focused on airflow resistance under well-controlled conditions. It is critical to understand underlying flow physics before performing simulations in complicated structures. However, the proposed new resistance models need further validation in realistic human airways models that have more realistic branching patterns such as nonplanarity of branching angle and smooth corners of bifurcation radius. In addition, we used a circular airway shape, but the actual shape of the airway becomes elliptic depending on disease severity [4], so hydraulic diameter should be considered. We did not include the quantitative analysis of the glottis region above the zeroth generation where turbulence is strongly dominant, because the interests of the current study were successive bifurcation structures. Further studies are necessary to parameterize airway resistance in this glottal-constricted region. Moreover, airways deform during breathing, adding uncertainties to the estimation of viscous dissipation. Besides, in the extreme breathing cases [40], the viscous dissipation may induce an increase in temperature, leading to a change in fluid properties, such as density and viscosity. The heterogeneity of diameter in a segment [41] needs further consideration to improve the estimation of viscous dissipation. In addition, we imposed a boundary condition with equally divided flow rates at the same generation based on a symmetric model, but lobar flow distribution in real lungs is variable depending on lobes. For example, it has been reported that lower lobar ventilation is larger than upper and middle lobar ventilation in healthy subjects [42]. In order to reflect regional ventilation, Yin et al. [43,44] and Miyawaki et al. [29] developed subject-specific boundary condition models using static/dynamic images. The current resistance model could be potentially utilized to improve existing boundary conditions only if static images are available. Future studies based on the present study are strongly required to better understand the effects of different factors, e.g., airway shape, heterogeneity, fluid properties, ventilation distribution, waveform, and branching patterns.

In conclusion, using symmetric airways, we performed a total of 14 3D CFD simulations to examine the effects of θ and Qtrachea on kinetic energy and viscous dissipation for both inspiration and expiration cases. Using an energy balance equation, we examined the contributions of different variables to the total pressure drop, including kinetic energy and grid-resolved and SGS turbulent eddy viscous dissipations. We found that kinetic energy is a key component of flow physics that discriminates airway resistance on inspiration and expiration. The overall pressure drop and airway resistance were larger on expiration than on inspiration, but wall shear stress and viscous dissipation were smaller on expiration than on inspiration. Dimensional analysis demonstrated that both the kinetic energy coefficients and viscous dissipation coefficients were correlated with generation number, and the viscous dissipation coefficients were further correlated with θ and Q. Based upon physical observations, we performed multiple linear regressions to propose new resistance models for kinetic energy and viscous dissipation that may be useful for accurate and efficient multiscale 3D–1D coupled simulations of pulmonary airflows.

## Acknowledgements

We thank Shinjiro Miyawaki for assisting with data analysis. Computational time was supported by XSEDE.

## Funding Data

• National Institutes of Health (Grant Nos. R01 HL094315, S10 RR022421, and U01 HL114494).

• Basic Science Research Program through the National Research Foundation of Korea (Grant No. NRF-2017R1D1A1B03034157).

## References

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Choi, S. , Hoffman, E. A. , Wenzel, S. E. , Castro, M. , and Lin, C.-L. , 2014, “ Improved CT-Based Estimate of Pulmonary Gas Trapping Accounting for Scanner and Lung Volume Variations in a Multi-Center Study,” J. Appl. Physiol., 117(6), pp. 593–603. [PubMed]
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Pedley, T. J. , Schroter, R. C. , and Sudlow, M. F. , 1971, “ Flow and Pressure Drop in Systems of Repeatedly Branching Tubes,” J. Fluid Mech., 46(2), pp. 365–383.
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Lin, C.-L. , Tawhai, M. H. , McLennan, G. , and Hoffman, E. A. , 2007, “ Characteristics of the Turbulent Laryngeal Jet and Its Effect on Airflow in the Human Intra-Thoracic Airways,” Respir. Physiol. Neurobiol., 157(2–3), pp. 295–309. [PubMed]
Choi, J. , Tawhai, M. H. , Hoffman, E. A. , and Lin, C.-L. , 2009, “ On Intra-and Intersubject Variabilities of Airflow in the Human Lungs,” Phys. Fluids, 21(10), p. 101901.
Choi, J. , Xia, G. , Tawhai, M. , Hoffman, E. A. , and Lin, C.-L. , 2010, “ Numerical Study of High-Frequency Oscillatory Air Flow and Convective Mixing in a CT-Based Human Airway Model,” Ann. Biomed. Eng., 38(12), pp. 3550–3571. [PubMed]
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van Ertbruggen, C. , Hirsch, C. , and Paiva, M. , 2005, “ Anatomically Based Three-Dimensional Model of Airways to Simulate Flow and Particle Transport Using Computational Fluid Dynamics,” J. Appl. Physiol., 98(3), pp. 970–980. [PubMed]
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## References

West, J. B. , 2008, Respiratory Physiology: The Essentials, Lippincott Williams & Wilkins, Baltimore, MD.
Sorkness, R. L. , Bleecker, E. R. , Busse, W. W. , Calhoun, W. J. , Castro, M. , Chung, K. F. , Curran-Everett, D. , Erzurum, S. C. , Gaston, B. M. , Israel, E. , Jarjour, N. N. , Moore, W. C. , Peters, S. P. , Teague, W. G. , and Wenzel, S. E. , and for the National Heart Lung and Blood Institute's Severe Asthma Research Program, 2008, “ Lung Function in Adults With Stable but Severe Asthma: Air Trapping and Incomplete Reversal of Obstruction With Bronchodilation,” J. Appl. Physiol., 104(2), pp. 394–403. [PubMed]
Make, B. J. , and Martinez, F. J. , 2008, “ Assessment of Patients With Chronic Obstructive Pulmonary Disease,” Proc. Am. Thorac. Soc., 5(9), pp. 884–890. [PubMed]
Choi, S. , Hoffman, E. A. , Wenzel, S. E. , Castro, M. , Fain, S. B. , Jarjour, N. N. , Schiebler, M. L. , Chen, K. , and Lin, C. L. , 2015, “ Quantitative Assessment of Multiscale Structural and Functional Alterations in Asthmatic Populations,” J. Appl. Physiol., 118(10), pp. 1286–1298. [PubMed]
Montaudon, M. , Lederlin, M. , Reich, S. , Begueret, H. , Tunon-de-Lara, J. M. , Marthan, R. , Berger, P. , and Laurent, F. , 2009, “ Bronchial Measurements in Patients With Asthma: Comparison of Quantitative Thin-Section CT Findings With Those in Healthy Subjects and Correlation With Pathologic Findings,” Radiology, 253(3), pp. 844–853. [PubMed]
Choi, S. , Hoffman, E. A. , Wenzel, S. E. , Castro, M. , and Lin, C.-L. , 2014, “ Improved CT-Based Estimate of Pulmonary Gas Trapping Accounting for Scanner and Lung Volume Variations in a Multi-Center Study,” J. Appl. Physiol., 117(6), pp. 593–603. [PubMed]
Busacker, A. , Newell , J. D., Jr. , Keefe, T. , Hoffman, E. A. , Granroth, J. C. , Castro, M. , Fain, S. , and Wenzel, S. , 2009, “ A Multivariate Analysis of Risk Factors for the Air-Trapping Asthmatic Phenotype as Measured by Quantitative CT Analysis,” Chest, 135(1), pp. 48–56. [PubMed]
Pedley, T. J. , Schroter, R. C. , and Sudlow, M. F. , 1970, “ Energy Losses and Pressure Drop in Models of Human Airways,” Respir Physiol., 9(3), pp. 371–386. [PubMed]
Pedley, T. J. , Schroter, R. C. , and Sudlow, M. F. , 1970, “ The Prediction of Pressure Drop and Variation of Resistance Within the Human Bronchial Airways,” Respir. Physiol., 9(3), pp. 387–405. [PubMed]
Pedley, T. J. , Schroter, R. C. , and Sudlow, M. F. , 1971, “ Flow and Pressure Drop in Systems of Repeatedly Branching Tubes,” J. Fluid Mech., 46(2), pp. 365–383.
Pedley, T. J. , Sudlow, M. F. , and Milic-Emili, J. , 1972, “ A Non-Linear Theory of the Distribution of Pulmonary Ventilation,” Respir. Physiol., 15(1), pp. 1–38. [PubMed]
Pedley, T. J. , 1977, “ Pulmonary Fluid Dynamics,” Annu. Rev. Fluid Mech., 9(1), pp. 229–274.
White, F. M. , 2011, Fluid Mechanics, McGraw-Hill, New York.
Hyatt, R. E. , and Wilcox, R. E. , 1963, “ The Pressure-Flow Relationships of the Intrathoracic Airway in Man,” J. Clin. Invest., 42, pp. 29–39. [PubMed]
Ismail, M. , Comerford, A. , and Wall, W. A. , 2013, “ Coupled and Reduced Dimensional Modeling of Respiratory Mechanics During Spontaneous Breathing,” Int. J. Numer. Methods Biomed. Eng., 29(11), pp. 1285–1305.
Kim, M. , Bordas, R. , Vos, W. , Hartley, R. A. , Brightling, C. E. , Kay, D. , Grau, V. , and Burrowes, K. S. , 2015, “ Dynamic Flow Characteristics in Normal and Asthmatic Lungs,” Int. J. Numer. Methods Biomed. Eng., 31(12), p. e02730.
Comer, J. K. , Kleinstreuer, C. , and Zhang, Z. , 2001, “ Flow Structures and Particle Deposition Patterns in Double Bifurcation Airway Models,” J. Fluid Mech., 435, pp. 25–54.
Lin, C.-L. , Tawhai, M. H. , McLennan, G. , and Hoffman, E. A. , 2007, “ Characteristics of the Turbulent Laryngeal Jet and Its Effect on Airflow in the Human Intra-Thoracic Airways,” Respir. Physiol. Neurobiol., 157(2–3), pp. 295–309. [PubMed]
Choi, J. , Tawhai, M. H. , Hoffman, E. A. , and Lin, C.-L. , 2009, “ On Intra-and Intersubject Variabilities of Airflow in the Human Lungs,” Phys. Fluids, 21(10), p. 101901.
Choi, J. , Xia, G. , Tawhai, M. , Hoffman, E. A. , and Lin, C.-L. , 2010, “ Numerical Study of High-Frequency Oscillatory Air Flow and Convective Mixing in a CT-Based Human Airway Model,” Ann. Biomed. Eng., 38(12), pp. 3550–3571. [PubMed]
Choi, J. , 2011, “Multiscale Numerical Analysis of Airflow in CT-Based Subject Specific Breathing Human Lungs,” Ph.D. dissertation, University of Iowa, Iowa City, IA.
van Ertbruggen, C. , Hirsch, C. , and Paiva, M. , 2005, “ Anatomically Based Three-Dimensional Model of Airways to Simulate Flow and Particle Transport Using Computational Fluid Dynamics,” J. Appl. Physiol., 98(3), pp. 970–980. [PubMed]
Katz, I. M. , Martin, A. R. , Muller, P. A. , Terzibachi, K. , Feng, C. H. , Caillibotte, G. , Sandeau, J. , and Texereau, J. , 2011, “ The Ventilation Distribution of Helium-Oxygen Mixtures and the Role of Inertial Losses in the Presence of Heterogeneous Airway Obstructions,” J. Biomech., 44(6), pp. 1137–1143. [PubMed]
Borojeni, A. A. , Noga, M. L. , Martin, A. R. , and Finlay, W. H. , 2015, “ Validation of Airway Resistance Models for Predicting Pressure Loss Through Anatomically Realistic Conducting Airway Replicas of Adults and Children,” J. Biomech., 48(10), pp. 1988–1996. [PubMed]
Kang, M. Y. , Hwang, J. , and Lee, J. W. , 2011, “ Effect of Geometric Variations on Pressure Loss for a Model Bifurcation of the Human Lung Airway,” J. Biomech., 44(6), pp. 1196–1199. [PubMed]
Weibel, E. R. , 1963, Morphometry of the Human Lung, Springer-Verlag, Berlin.
Miyawaki, S. , Hoffman, E. A. , and Lin, C.-L. , 2017, “ Numerical Simulations of Aerosol Delivery to the Human Lung With an Idealized Laryngeal Model, Image-Based Airway Model, and Automatic Meshing Algorithm,” Comput Fluids, 148, pp. 1–9. [PubMed]
Geuzaine, C. , and Remacle, J. F. , 2009, “ Gmsh: A 3-D Finite Element Mesh Generator With Built-In Pre‐ and Post-Processing Facilities,” Int. J. Numer. Methods Eng., 79(11), pp. 1309–1331.
Miyawaki, S. , Choi, S. , Hoffman, E. A. , and Lin, C.-L. , 2016, “ A 4DCT Imaging-Based Breathing Lung Model With Relative Hysteresis,” J. Comput. Phys., 326, pp. 76–90. [PubMed]
Lin, C. L. , Lee, H. , Lee, T. , and Weber, L. J. , 2005, “ A Level Set Characteristic Galerkin Finite Element Method for Free Surface Flows,” Int. J. Numer. Methods Fluids, 49(5), pp. 521–547.
Vreman, A. W. , 2004, “ An Eddy-Viscosity Subgrid-Scale Model for Turbulent Shear Flow: Algebraic Theory and Applications,” Phys. Fluids, 16(10), pp. 3670–3681.
Jarrin, N. , Benhamadouche, S. , Laurence, D. , and Prosser, R. , 2006, “ A Synthetic-Eddy-Method for Generating Inflow Conditions for Large-Eddy Simulations,” Int. J. Heat Fluid Flow, 27(4), pp. 585–593.
Weinberger, S. E. , Cockrill, B. A. , and Mandel, J. , 2008, Principles of Pulmonary Medicine, 5th eds., Elsevier Health Sciences, Philadelphia, PA. [PubMed] [PubMed]
Jahani, N. , Choi, S. , Choi, J. , Iyer, K. , Hoffman, E. A. , and Lin, C. L. , 2015, “ Assessment of Regional Ventilation and Deformation Using 4D-CT Imaging for Healthy Human Lungs During Tidal Breathing,” J. Appl. Physiol., 119(10), pp. 1064–1074. [PubMed]
Pare, P. D. , Wiggs, B. R. , James, A. , Hogg, J. C. , and Bosken, C. , 1991, “ The Comparative Mechanics and Morphology of Airways in Asthma and in Chronic Obstructive Pulmonary Disease,” Am. J. Respir. Crit. Care Med., 143(5 Pt. 1), pp. 1189–1193.
Wongviriyawong, C. , Harris, R. S. , Greenblatt, E. , Winkler, T. , and Venegas, J. G. , 2013, “ Peripheral Resistance: A Link Between Global Airflow Obstruction and Regional Ventilation Distribution,” J. Appl. Physiol., 114(4), pp. 504–514. [PubMed]
Jalal, S. , Nemes, A. , Van de Moortele, T. , Schmitter, S. , and Coletti, F. , 2016, “ Three-Dimensional Inspiratory Flow in a Double Bifurcation Airway Model,” Exp. Fluids, 57(9), p. 148.
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## Figures

Fig. 1

Schematics of (a) symmetric airway geometries with five different bifurcation angles (15 deg, 25 deg, 35 deg, 45 deg, and 55 deg), (b) two major branch paths, (c) bifurcation angle between the current branch and its parent branch, and (d) the control surface and volume defined for quantitative analysis of the pressure drop

Fig. 2

Pressure drop averaged over two major paths (Fig. 1(b)) with the bifurcation angle 35 deg. Left [(a), (c), (e)] and right [(b), (d), (f)] columns show inspiration and expiration cases, respectively. Top [(a), (b)], middle [(c), (d)], and bottom [(e), (f)] rows show results when low (10 L/min), moderate (20 L/min) and high (40 L/min) flow rates were applied, respectively.

Fig. 3

Contour on a sliced plane and isosurface at 0.25 of turbulence kinetic energy: (a) on inspiration and (b) on expiration at the high flow rate (Qtrachea = 40 L/min)

Fig. 4

(a) Average wall shear stress and (b) airway resistance averaged for five different bifurcation angles for inspiration and expiration at the moderate flow rate (Qtrachea = 20 L/min). (c) Average wall shear stress and (d) airway resistance for inspiration at three different flow rates with the bifurcation angle 35 deg. (e) Average wall shear stress and (f) airway resistance for expiration at three different flow rates with the bifurcation angle 35 deg.

Fig. 5

Comparison of the pressure drop due to viscous dissipation ΔPVis and the pressure drop due to the Poiseuille flow assumption (ΔPPoiseuille) at three different flow rates with the same bifurcation angle of 35 deg: (a) inspiration and (b) expiration

Fig. 6

Kinetic energy coefficient (K) according to generation number of (a) moderate inspiration (Qtrachea = 20 L/min) with five different bifurcation angles, (b) moderate expiration (Qtrachea = 20 L/min) with five different bifurcation angles, (c) inspiration with three different flow rates at the bifurcation angle 35 deg, and (d) expiration with three different flow rates at the bifurcation angle 35 deg

Fig. 7

Color-coded velocity vectors [(a) and (b)] and kinetic energy distribution [(c) and (d)] on the distal control surface between inspiration [(a) and (c)] and expiration [(b) and (d)]. The first fluctuating eigenmode (POD-derived coherent vortical structure) identified by the isosurface of λ2 = −0.02, and velocity vector tangent on the distal control surface between inspiration (e) and expiration (f).

Fig. 8

Viscous dissipation coefficient (γ) according to generation number of (a) inspiration with five different bifurcation angles, (b) expiration with five different bifurcation angles, (c) inspiration with three different flow rates, and (d) expiration with three different flow rates

Fig. 9

Average wall shear stress projected on surface meshes [(a) and (b)] and grid-resolved viscous dissipation on a perpendicular plane inside the volume meshes [(c) and (d)]. Left figures [(a) and (c)] are for inspiration and right figures are for expiration [(b) and (d)] at the moderate flow rate with a bifurcation angle 35 deg.

Fig. 10

Model comparisons using constant and variable coefficients of kinetic energy (K) and viscous dissipation (γ and η) for inspiration [(a) and (c)] and expiration cases [(b) and (d)]. (a) and (c) Use the constant and variable coefficients of kinetic energy (K), while (c) and (d) use the constant and variable coefficients of viscous dissipation (γ and η). All data points between the zeroth and tenth generations were employed in this analysis.

## Tables

Table 1 Simulation cases and the number of nodes and elements for each case: A series of 14 CFD simulations (seven for inspiration and seven for expiration) marked with “Ο” were performed

## Errata

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