Evaluation of the Polyhedral Mesh Style for Predicting Aerosol Deposition in Representative Models of the Conducting Airways

Abstract

A critical factor affecting the accuracy of Computational Fluid Dynamic (CFD) simulations and the time required to conduct them is construction of the computational mesh. This study aimed to evaluate the relatively new polyhedral mesh style for simulating aerosol deposition in the upper conducting airways compared with established meshing techniques and experimental data. Hexahedral and polyhedral mesh solutions were compared in two benchmark geometries: 1) a 90°-bend with flow characteristics similar to the extrathoracic airways of an adolescent child, and 2) a double bifurcation representing bifurcations B3-B5 in an adult. Both 4-block and 5-block hexahedral meshes were used in the 90°-bend to capture the potential of fully-structured hexahedral meshes. In the 90°-bend, polyhedral elements matched polydisperse in vitro deposition data with 20% relative error (RE; averaged across the particle sizes considered), which is an improvement on the accuracy of the 4-block hexahedral mesh (35% RE) and is similar to the accuracy of the 5-block hexahedral mesh (19% RE). In the double bifurcation, deposition fraction relative differences evaluated between polyhedral and hexahedral meshes ranged from 0.3% to 28.6% for the different particle sizes assessed, which is an order of magnitude improvement compared with previous studies that considered hexahedral vs. hybrid tetrahedral-prism meshes for the same flow field. Solution convergence time with polyhedral elements was found to be 50% to 140% higher than with hexahedral meshes of comparable size. While application dependent, the increase in simulation time observed with polyhedral meshes will likely be outweighed by the ease and convenience of polyhedral mesh construction. It was concluded that the polyhedral mesh style, with sufficient resolution especially near the walls, is an excellent alternative to the highly regarded hexahedral mesh style for predicting upper airway aerosol transport and deposition and provides a powerful new tool in the assessment of respiratory aerosol dosimetry.

Graphical Abstract

1. Introduction

Computational fluid dynamics (CFD) simulations of aerosol transport and deposition in airway models have become a widely used tool that significantly contributes to the understanding of respiratory dosimetry of inhaled particles (Longest et al., 2018). Primary applications of these simulations include analysis of inhaled pollutants for computational toxicology assessments (Chen, Lee, Mutuku, & Hwang, 2018), as well as understanding the drug delivery characteristics of current medical inhalers and designing the next generation of medical aerosol delivery systems (Longest et al., 2018; Ruzycki, Javaheri, & Finlay, 2013; Wong, Fletcher, Traini, Chan, & Young, 2012). Recent advances in the use of CFD to simulate airway particle deposition and drug delivery include the analysis of diseased airways (Bos et al., 2017; Sul et al., 2018), airway wall motion (Bates et al., 2019), and the development of CFD-based complete airway models (Kolanjiyil & Kleinstreuer, 2017; Longest, Tian, Khajeh-Hosseini-Dalasm, & Hindle, 2016). CFD models have also recently focused on aerosol drug delivery in different age groups including adults (Golshahi, Walenga, Longest, & Hindle, 2014; Longest, Tian, Walenga, & Hindle, 2012; Wong et al., 2010), children (Bass & Longest, 2020), and infants (Bass, Boc, Hindle, Dodson, & Longest, 2019). Compared with in vitro experiments and in vivo analysis of aerosol particle dosimetry, advantages of CFD predictions (once validated) include the ability to understand and visualize underlying transport physics (potentially leading to new discoveries), rapidly conduct parameter sensitivity analysis, and reduce the size of expensive and time-consuming human subject trials.

Considering the numerical methods required for pulmonary aerosol research, one source of significant difficulty is the need to tailor computational meshes for both the airway region being analyzed and the underlying continuous and discrete phase transport physics within that region. CFD model predictions cannot be accurate unless a mesh is properly designed to resolve the physics of each airway region, regardless of the solver model selections. An obvious challenging factor in airway mesh development is the range of length scales that must be addressed, which includes both the span of the airway diameter and the micrometer boundary layers in the near-wall region of the upper and midlevel airways. Excellent mesh quality across these scales is required both to resolve the flow field (Wilcox, 1998) and to adequately determine particle transport, often including microscale particle-wall interactions.

Compared with turbulence model selection (Ball, Uddin, & Pollard, 2008; Jayaraju, Brouns, Lacor, Belkassem, & Verbanck, 2008; Kleinstreuer & Zhang, 2003; Longest, Tian, Delvadia, & Hindle, 2012; Matida, Finlay, & Grgic, 2004; Stapleton, Guentsch, Hoskinson, & Finlay, 2000; Tian, Longest, Su, Walenga, & Hindle, 2011; Wilcox, 1998) and other components of aerosol transport prediction in the lungs, relatively less attention has been paid to mesh development and the impact of mesh style on solution accuracy and quality. Of the available studies that focus on mesh style effects, it is commonly held that properly resolved hexahedral elements provide a gold-standard for CFD accuracy. This meshing style has been extensively demonstrated to accurately capture aerosol deposition when compared to in vitro (Holbrook & Longest, 2013; Walenga et al., 2014) and in vivo (Kolanjiyil & Kleinstreuer, 2017; Longest et al., 2016; Tian, Hindle, Lee, & Longest, 2015) experimental data. Several studies (Bass & Longest, 2018; Longest & Vinchurkar, 2007a; Vinchurkar & Longest, 2008) have provided insight concerning hexahedral (or hex) cell performance in airway models. Theoretically, a hex cell’s unique alignment with the flow field (Ferziger & Peric, 1999), despite a relatively large aspect ratio, combined with its ability to minimize artificial (or numerical) diffusion or dispersion errors make it an ideal numerical structure for retaining accuracy while minimizing mesh size. Similarly, it is known that other mesh cell types – such as tetrahedral, prismatic, and hybrid – are useful for respiratory simulation in complex geometries because they are easy to generate and can be selectively refined in regions of interest. While convenient, Longest and Vinchurkar (2007a) discovered that tetrahedral meshes were less able to resolve secondary flow features in bifurcating airway flows compared with hex cells, resulting in less accurate comparisons with in vitro particle deposition data. This discrepancy was especially large in small particle (1 μm – 5 μm) deposition, for which unstructured meshes predicted up to five times higher values than structured hex cells. Vinchurkar and Longest (2008) found that tetrahedral cells offer poorer grid convergence and solution accuracy than their hexahedral counterparts (even at very high resolutions) but also concluded that this effect might be mitigated by constructing a mesh with near-wall prisms to attain better convergence and more accurate deposition profiles, as has been implemented by other researchers as well (J. D. Schroeter, Garcia, & Kimbell, 2011; Jeffry D. Schroeter, Kimbell, Asgharian, Tewksbury, & Singal, 2012; Jeffry D. Schroeter, Tewksbury, Wong, & Kimbell, 2015; J. Xi & Longest, 2008, 2009; J. Xi, Longest, & Martonen, 2008; J. X. Xi, Berlinski, Zhou, Greenberg, & Ou, 2012).

Bass and Longest (2018) evaluated flow field and particle deposition accuracy for tetrahedral meshes with near-wall prism layers and compared results to hexahedral mesh predictions. This study provided recommendations for appropriate near-wall mesh spacing when using the tetrahedral with prism layer hybrid approach as well as efficient turbulent model selections and solver settings for upper airway respiratory dynamics. It was determined that tetrahedra are not as effective as hexahedra for predicting particle transport, but that adequate results can be attained with good mesh resolution in the near-wall prism region.

Despite the known and expected limitations of tetrahedral cells, this mesh style is still often required for pulmonary research, especially in complex geometries such as the nasal cavity. For example, Walenga, Longest, Kaviratna, and Hindle (2017) found good matches to in vitro data using tetrahedral cells with pentahedral near-wall layers in multiple nose-throat geometries while evaluating the effects of inter-subject variability in nose-to-lung pharmaceutical aerosol delivery. This study and many others (Frank-Ito, Wofford, Schroeter, & Kimbell, 2016; Gemci, Ponyavin, Chen, Chen, & Collins, 2008; Golshahi et al., 2013; Rygg, Hindle, & Longest, 2016; Rygg & Longest, 2016) support the standard approach of using tetrahedral meshes with near-wall prism layers to evaluate nose-mouth-throat (NMT) and other airway geometries that are difficult to mesh with hexahedral cells.

In choosing a mesh style, a tradeoff has traditionally been required between accuracy and mesh construction time. Hexahedral elements are associated with high accuracy but difficult construction, whereas tetrahedral and hybrid meshes typically exhibit lower accuracy (even at very high resolutions) while offering fast and convenient construction. Hexahedral mesh assembly requires dividing irregular volumes into regular blocks, assigning boundary conditions to the block surfaces, creating an efficient and quality-assuring method of placing nodes along the block edges, and using an interpolation scheme (such as Cooper) to mesh the volume. For every variation in airway or device geometry – whether to model patient-specific airways or explore the effects of characteristic airway damage such as mucus plugging – this process must be repeated. In contrast, unstructured meshes may be generated automatically based on a few sizing inputs, allowing for accelerated research at the expense of some accuracy, high cell counts, and extended run times.

The relatively new development of polyhedral cells for use in CFD presents a unique potential solution to the current accuracy vs. convenience dilemma of mesh type selection. Polyhedral cells offer the advantages of rapid unstructured mesh construction but reportedly have superior accuracy and computational efficiency when compared with traditional tetrahedral and hybrid unstructured cells. Theoretically, the structure of these polyhedral elements lends itself well to accurate simulation; polyhedral cells typically have more faces per cell than tetrahedra or hexahedra, which makes it easier for the mesh to resolve gradients in and out of the cell and helps to maintain nearly right angles between the primary flow direction and the control volume faces.

While currently popular in the CFD industry, few scientific studies have documented the potential advantages of polyhedral meshes. A relevant study was recently performed by Bass et al. (2019) where tetrahedral and polyhedral cells were directly compared in an infant nasal airway geometry. They found that between polyhedral and tetrahedral meshes of comparable density, polyhedral elements were able to converge more quickly and closely match in vitro data. Bass and Longest (2020) used polyhedral elements with near-wall prisms in the development of an infant DPI interface that minimizes device and extra-thoracic aerosol deposition losses. The study iteratively examined multiple design features, so the comparatively short polyhedral mesh generation process was a considerable asset in assessing multiple designs. Polyhedral cells have been used to model other cardiovascular and respiratory flows (Bates et al., 2019; Kabilan et al., 2016; Spiegel et al., 2011), but these studies have largely relied on the single cardiovascular study from Spiegel et al. (2011) for evidence that the polyhedral style is comparable to tetrahedral for the modeling of bifurcating flows and have not evaluated the effects of the mesh style on particle transport. Considering the significant differences in the transport physics of particles in the cardiovascular and pulmonary systems, as well as the potential benefit of identifying a meshing technique that is both easy to construct and highly accurate, additional assessment of polyhedral cell use in respiratory aerosol dynamics is needed.

The objective of this study is to evaluate performance of the relatively new polyhedral mesh style for simulating aerosol deposition in the upper conducting airways compared with established meshing techniques and experimental data. While the polyhedral mesh style is being incorporated by industry and academic CFD research and initial results are positive, a number of questions have not been answered in a rigorous way. For example: can polyhedral meshes of the upper conducting airways accurately predict regional and local aerosol deposition in comparison to benchmark experimental deposition data for varying particle sizes and flow rates? How do polyhedral meshes compare with hexahedral cells in terms of grid convergence and solution time? Do polyhedral meshes introduce numerical dispersion errors into secondary flow patterns that are sufficient to alter particle deposition? If so, is the effect great enough to impair the accuracy of local deposition profiles as observed with other unstructured mesh styles (Vinchurkar & Longest, 2008)? This study aims to provide data that may be used to address these questions by comparing the results of polyhedral and hexahedral elements in characteristic flows that represent the extrathoracic and upper tracheobronchial (TB) regions. Ultimately, this study seeks to quantify and document the performance of polyhedral meshing techniques compared with well-established hexahedral meshing in a way that can justify the expanded use of polyhedral cells as a new numerical tool in the field of pulmonary aerosol research.

2. Methods

2.1. Study Overview

To directly compare the results of hexahedral and polyhedral mesh styles, flow features and aerosol deposition are evaluated in two characteristic geometries: a 90°-bend (Bass & Longest, 2018; Pui, Romay-Novas, & Liu, 1987) and a double bifurcation airway model of airway bifurcations B3 to B5 constructed of Physiologically Realistic Bifurcation (PRB) units (Heistracher & Hofmann, 1995; Longest & Oldham, 2006; Oldham, Phalen, & Heistracher, 2000; Weibel, 1963). Heistracher and Hofmann (1995) introduced the PRB units, which are equation-driven models constructed to mimic key features of bifurcation geometries (Horsfield, Dart, Olson, Filley, & Cumming, 1971; Weibel, 1963). Based on a Reynolds Number of 6,000 and a flow rate of 21 LPM, the 90°-bend model may be considered representative of deposition in the extrathoracic airways of a 10–15yo adolescent undergoing light exercise. The double bifurcation model represents airway generations in an adult based on the lung model of Weibel (1963). With a Reynolds Number of 1,940 and a flow rate of 7.5 LPM, the bifurcation model is expected to represent aerosol deposition in bifurcations B3 through B5 (where the trachea splits at B1 and subsequent bifurcations are B2, B3, etc.) of an adult undergoing heavy exertion with a tracheal inspiratory flow rate of 60 LPM. The polyhedral mesh is compared to both 4-block and 5-block (“O-grid”) hexahedral meshes in the 90°-bend but only to a 4-block mesh in the double bifurcation.

Oldham et al. (2000) introduced benchmark in vitro experimental deposition data for the double bifurcation that has been critical to the development of CFD models of the bifurcating airways and has been essential for this study as well. They used the PRB method outlined by Heistracher and Hofmann (1995) to generate a double bifurcation geometry from B3 to B5 using dimensions for an adult based on data from Weibel (1963). Note that there are differences between airway dimensions when comparing the Weibel (1963) morphometric data to more recent ICRP (1994) data and the result is that the double bifurcation geometry used in this study is somewhat smaller than other adult models. Oldham et al. (2000) injected 10 μm fluorescent latex particles into a physical prototype of the geometry at a flow rate of 7.5 LPM, which represents a heavy exertion state with a 60 LPM inspiratory flow through the trachea. In vitro outlet conditions were not specified, but a subsequent study from Longest and Oldham (2008) elaborates that the mass flow was equally divided at the outlets. In the current study, deposition patterns from hexahedral and polyhedral meshes of the double bifurcation are compared to the localized deposition profiles provided by Oldham et al. (2000).

Pui et al. (1987) analyzed and created detailed benchmark data of aerosol deposition for the 90°-bend geometry. They considered curved pipe sections with varying bend ratios and cross-sectional diameters. Liquid droplets were generated and injected in flows with Reynolds numbers of 100, 1,000, 6,000, and 10,000 resulting in Stokes numbers that ranged from 0.03 to 1.46. Total deposition data was recorded for the varying particle sizes, flow rates, and geometries, which allows for verification of CFD models in the laminar through turbulent range in curved tubes. The current study utilizes the 6,000 Reynolds number flow, which consists of a tube diameter of 5.03 mm and bend ratio of 5.7, to compare the hexahedral and polyhedral mesh styles. Seven particle sizes were tested at Stokes numbers of 0.03, 0.13, 0.19, 0.36, 0.60, 0.72, and 1.00, so that the size-dependent deposition predictions of the hexahedral and polyhedral styles could be directly compared.

2.2. Geometry and Mesh Construction

The 90°-bend was generated in Solidworks and is shown in Figure 1a, where the bend outlet is highlighted for future reference. Extensions of 60 mm were attached to the inlet and outlet to match the experimental setup and to prevent recirculation through the outlet boundary.

Figure 1:

a) The 90°-bend including extensions. b) The double bifurcation geometry of region B4 to B5 composed of two physiologically realistic bifurcation (PRB) units and including extensions on the outlets. Slice A is the inlet of the B5 region.

The double bifurcation geometry (Figure 1b) was generated using the PRB equations (Heistracher & Hofmann, 1995) with parameters specified in Table 1. A code was written in MATLAB R2019a (MathWorks, Inc., Natick, MA) to interpret those parameters and generate guide curves that were processed with Solidworks to create the model surface. Extensions with a length of 34 mm (roughly ten diameters) were added to the outlets to ensure nearly fully-developed flow at the exit boundaries. The resultant geometry is shown in Figure 1b, with Slice A being the inlet to the second bifurcating region. Note that the reported PRB parameters in the Oldham et al. (2000) study did not include curvature radius or carina radius. The values shown in Table 1 provided the closest geometric match to published images of the experimental model based on visual comparison of the two geometries.

Table 1:

Parameters used to generate the PRB-based double bifurcation model of airway generations B3 to B5 in MATLAB.

Section	Diameter [mm]	Length [mm]	Curvature Radius [mm]	Carina Radius [mm]
B3	5.6	8.6	15	0.45
B4	4.5	9.2	10.8	0.36
B5	3.6	12	-	-

Gambit v2.4.6 was used to generate hexahedral meshes due to its strengths with this meshing style. Parasolid files were imported from Solidworks, then boundary conditions were set and nodes placed along the appropriate edges before meshing with the Cooper approach. Gambit was also used to generate a tetrahedral surface mesh, which was processed with the Fluent 2019 R1 (ANSYS Inc., Canonsburg, PA) mesh generation tool to create polyhedral meshes. Figure 2 and Figure 3 show the inlet and surface meshes for the 90°-bend and double bifurcation, respectively. Only half of the inlet mesh in each case is shown so that the structures may be seen in greater detail. Symmetry planes were not utilized in the simulations. Per best meshing practices established by Bass and Longest (2018), mesh in the near wall region was made with five layers, a growth ratio of 1.2, and a first cell height that yielded an average y⁺ = 1. The y⁺ metric is a non-dimensional length defined as: y⁺ = yu*/v, where y is the distance from the wall to the first near-wall cell centroid, v is the fluid kinematic viscosity, and u^∗ is the fiction velocity, defined as: $u*=\sqrt{{\tau }_w/\rho }$, where τ_w is the wall shear stress and ρ is the fluid density. Creating a mesh that results in a y⁺ of 1 ensures that the mesh is sufficiently resolved to create accurate turbulence and velocity profiles in the near-wall region.

Figure 2:

Inlet meshes for the a) 4-block, b) polyhedral, and c) 5-block cases of the 90°-bend geometry. d) The 90°-bend surface mesh from the 4-block style.

Figure 3:

Inlet meshes for the a) hexahedral and b) polyhedral cases of the double bifurcation geometry. c) The double bifurcation surface mesh from the polyhedral style.

Two hexahedral mesh styles were utilized in the 90° bend – one where the cross-section was divided into 5 blocks (structured hexahedral) and another with 4 blocks (unstructured hexahedral). Figure 2a and Figure 2c illustrate these mesh styles. Both block structures were applied to the double bifurcation flow by Vinchurkar and Longest (2008), who found that the meshes predicted similar deposition for 10 μm particles but that the 5-block predicted significantly less deposition for small (≤ 5μm) particles. It is generally more difficult to generate a 5-block hex structure, but this advantage for solving small particle deposition should be considered when evaluating a new cell type against the hexahedral style.

2.3. Solver Settings, Boundary Conditions, and Particle Injections

FLUENT 2019 R1 (ANSYS Inc., Canonsburg, PA) was used to solve the transport equations. The k-ω turbulence model was selected based on its strong record of accurate solutions in upper airway geometries (Longest & Vinchurkar, 2007b). Low Reynolds number (LRN) and shear flow corrections were used in the double bifurcation, but the fully-turbulent flow and lack of a jet or recirculation region in the 90°-bend case reduces the need for the LRN model, so it was not included in the 90°-bend case. Per the recommendations of Bass and Longest (2018), the solver settings included the SIMPLEC scheme with a Green-Gauss Node-Based discretization for space and second order discretization of pressure, momentum, turbulent kinetic energy (k), and specific dissipation rate (ω). Flow solutions were discontinued once residuals were no longer reduced and had decreased by ten or more orders of magnitude from initial values.

Boundary conditions were applied based on the values specified by benchmark data (Oldham et al., 2000; Pui et al., 1987) as well as by the best practices established from previous studies. For the 90°-bend, a constant velocity inlet with a magnitude of 17.4 m/s (Bass & Longest, 2018) was used at the extension inlet to attain the desired Reynolds number of 6,000 and produced a nearly fully-developed turbulent velocity profile at the bend inlet to mimic the experimental setup of Pui et al. (1987). For the double bifurcation, a turbulent inlet velocity profile was generated using an iterative approach based on a fully developed shear-stress dependent model (Kays & Crawford, 1993; Longest & Vinchurkar, 2007b) for an inlet Reynolds number of 1,945 and a flow rate of 7.5 LPM, which corresponds to a 60 LPM inspiratory flow. In both systems, wall boundaries were set to the no slip condition and the outlets to the outflow boundary type. Consistent with the experiments of Oldham et al. (2000), which were further described by Longest and Oldham (2008), mass flow was equally divided among the four outlets of the double bifurcation. Effects of wall roughness and upstream flow were neglected.

Blunt particle injection profiles were generated with a random spatial distribution in the core flow and a decrease in particle-count near the walls that is governed by the 1/7^th power law, which ensures the decline is approximately proportional to the velocity profile in the near-wall region. Injections were placed at the inlet of each core flow domain rather than at the inlet of the extension. Particles were assumed to deposit when they contacted the wall with no bounce or rebound.

2.4. Particle Transport and Near-Wall Corrections

Particle trajectories were determined using the Discrete Phase Model (DPM) in FLUENT v19.0 (ANSYS Inc., Canonsburg, PA) and the particle transport equations were integrated with the Runge-Kutta scheme with default accuracy control (tolerance set to 1E-05 with 20 max refinements) and up to 1E+06 timesteps per particle path. The discrete random walk model was used to incorporate turbulent stochastic particle motion. Key transport factors included the flow field and turbulent dispersion. These settings have been well tested by our group (Bass et al., 2019; Bass & Longest, 2018; Longest & Xi, 2007; Walenga & Longest, 2016) and have been consistently proven effective in matching in vitro data for microparticles considered in upper airway geometries. When a DPM is used with the LRN k-ω turbulence model, it is typical for the system to predict higher microparticle deposition than concurrent in vitro datasets because the LRN k-ω model treats turbulence as isotropic when it determines the fluctuating velocity components. This assumption results in wall-directed fluctuations near the wall that are unrealistically high and cause an increase in the deposition of small particles, which are influenced strongly by turbulent dispersion. To address this, Longest and Xi (2007) and others (Matida et al., 2004) have developed near-wall corrections, applied with user-defined functions (UDFs), to introduce more physically correct anisotropic turbulence in the near-wall region. Using the approach of Bass and Longest (2018), corrections first interpolate velocity between cell centroids to find the velocity at the particle location and then calculate anisotropic velocity fluctuations for use in calculating the particle trajectory. These corrections also include a wall-normal damping function to capture the effects of particle-wall hydrodynamic interaction. A key variable in this approach is a near-wall limit, which is used to tune the damping function to in vitro data by establishing the wall-normal distance at which the damping takes effect (Bass & Longest, 2018). This limit has been effective for accurate particle deposition in the upper airways when set to around 1 μm to 2 μm (Bass et al., 2019; Bass & Longest, 2018), which is on the order of magnitude of the particle diameter. A range of values for this limit are assessed in the 90°-bend simulations of this study. The y+ limit was set to 60 and not varied in these simulations.

2.5. Mesh Independence and Computational Performance

Mesh independence is determined based on convergence of turbulent kinetic energy (k), wall shear stress (WSS), localized vorticity, and aerosol total deposition fraction (DF). Turbulent kinetic energy is useful for its direct proportionality to turbulent dispersion and was monitored by mass average over the entire flow domain. WSS is a derivative quantity that is expected to be highly sensitive to changes in mesh style and is monitored with an area-weighted average on the domain walls. Localized vorticity is used to quantify the magnitude of secondary flows that are produced in these representative airways. For the 90°-bend, an area-weighted average of vorticity magnitude is taken at the bend outlet (see Figure 1) and in the double bifurcation the same quantity is calculated at Slice A (see Figure 1), which is the inlet of the second bifurcating region. DF is an important quantity for this study and is taken as the number of particles that deposit in each case divided by the number injected. It is expected that validated particle deposition in addition to convergence of the three flow-related metrics (k, WSS, and vorticity, which are indicative of whole-geometry, derivative, and local flow features) should be sufficient to indicate convergence of the mesh.

The 4-block, polyhedral, and 5-block mesh styles were evaluated at resolutions of approximately 85, 210, 360, 660, and 1,300 thousand cells (not including the extensions) to determine mesh independence in the 90°-bend. The near-wall layers for all meshes were constructed with a first layer height of 20 μm, a growth ratio of 1.2, and a total of five layers, which was shown to be adequate for modeling particle transport in the study of Bass and Longest (2018).

The hexahedral and polyhedral mesh styles were evaluated at resolutions of approximately 45, 120, 220, 515, and 1,080 thousand cells to determine mesh independence in the double bifurcation. The near-wall layers for all meshes were constructed with a first layer height of 40 μm, a growth ratio of 1.2, and a total of five layers.

Computational performance is evaluated two ways. First, the mass-averages of velocity and k are monitored for the first 1,000 iterations in the 90°-bend and 600 iterations in the double bifurcation. These iteration ranges were selected to best visualize the convergence history profiles. The plots are compared and the mesh that converges with fewer iterations is considered to be more efficient per iteration. Second, the time and iteration count required for the residuals to stabilize while simulating with each mesh are recorded and compared to assess which mesh arrives at a solution more quickly in real time. It is noted that computation times are dependent on the computational system and for this study an Intel® Core™ i9–9960X CPU @ 3.10 GHz processor with 128 GB of RAM was used.

2.6. Comparison Criteria

2.6.1. Relative Difference and Error

Relative difference used to evaluate mesh convergence, was calculated as the absolute difference between the selected variables for higher and lower resolution meshes divided by the value from the highest resolution mesh. In the context of comparison with experimental results, relative error was calculated as the absolute difference between the experimental value and CFD prediction, divided by the experimental value.

2.6.2. Spread Error

Spread error, used to evaluate solution agreement of mesh styles, was calculated as the difference of the largest and smallest converged values divided by the mean of all converged values for a specific flow variable, such as velocity magnitude.

3. Results

3.1. 90°-bend Results

3.1.1. Mesh Independence and Solution Performance

The solution variables shown in Figure 4 were evaluated to determine the resolution required for mesh independence. The three flow-related metrics (k, WSS, vorticity) indicate that the 4-block and polyhedral meshes produce highly similar results once a sufficient mesh resolution is reached. The 5-block solutions show the same convergence profile as the other two meshes, but at notably higher flow metric magnitudes. For flow convergence of the three mesh types, a mesh resolution of around 660k cells calculated the flow-related metrics with less than 3% relative error compared to the 1,300k mesh. The 360k meshes calculated the flow-related metrics with relative errors less than 7% compared to the 1,300k mesh.

Figure 4:

Mesh convergence plots in the 90°-bend for a) mass-averaged turbulent kinetic energy (k), b) wall shear stress(WSS), c) vorticity at the outlet of the bend, and d) deposition fraction of 3 μm particles with experimental “Target” based on data from Pui, Romay-Novas et al. (1987). Mesh count does not include the extensions.

Mesh convergence is also evaluated based on particle deposition. Across the seven particle sizes considered, the average relative differences between the 360k cell meshes and the 1,300k cell meshes were −0.37%, −0.99%, and 1.89% for the 4-block, polyhedral, and 5-block styles, respectively. However, smaller particle sizes displayed larger changes between mesh resolutions. Figure 4d shows the convergence of 2.97 μm particle deposition. For this particle size, the relative differences in DF between the 360k and 1,300k meshes were −19.6%, 6.4%, and 28.3% for the 4-block, polyhedral, and 5-block cases. The 660k meshes yielded values of −28.3%, −15.6%, and 8.4% for the same comparison. As the 660k meshes did not demonstrate a clear improvement in mesh convergence even with the consideration of single particle sizes, the 360k cell meshes were considered sufficiently converged, based on the particle-averaged values presented above.

The experimental target deposition fraction for 2.97 μm particles is 30% (Pui et al., 1987). At a resolution of 360k elements, the relative (and absolute) errors between the in vitro deposition and the deposition predicted by the 4-block, 5-block, and polyhedral mesh types are 63% (19%), 18% (6%), and 23% (7%), respectively. After considering the convergence for flow-related metrics and particle deposition in addition to the error between predicted and in vitro deposition for 2.97 μm particles, the 360k meshes were selected for comparison of mesh styles in the 90°-bend.

Figures 5a and 5b show that for all three mesh styles at the selected resolution of 360k, velocity converges in roughly 600 iterations and k converges in roughly 500 iterations. The velocities converged within 0.1 m/s of one another for a spread error of 0.4% and the turbulent kinetic energies within 0.25 m²/s², resulting in a spread error of 10%. Figure 5c and Figure 5d reveal that, though the number of iterations is not strongly dependent on mesh style, it does take longer for a polyhedral mesh to converge compared to a hexahedral mesh. For the recommended mesh resolution, polyhedral elements (1.80 hrs) required 140% more time to converge than the 4-block hexahedral mesh (0.75 hrs).

Figure 5:

a) Mass-averaged velocity and b) mass-averaged turbulent kinetic energy (k) for the first thousand iterations in the 90°-bend geometry. c) Hours taken to solve and d) iterations taken to solve vs. mesh count in the 90°-bend geometry.

3.1.2. Secondary Flow

Secondary flow comparisons were evaluated qualitatively at the outlet of the 90°-bend using velocity vectors and contours. Figure 6 shows the secondary flow takes the form of a vortex pair which causes air at the perimeter of the tube to flow towards the inner radius and merge along the center plane with peak velocities of about 4 m/s. The average primary velocity (normal to the flow shown) has a magnitude of 17.4 m/s, which means that the peak secondary flow (depicted) occurs at about 23% of the primary flow velocity. The three mesh styles produced very similar secondary flow patterns, although the hexahedral meshes indicate a higher peak velocity where the vortexes merge compared with the polyhedral mesh.

Figure 6:

Vectors and contours of secondary velocity at the outlet of the 90°-bend for the a) 4-block, b) polyhedral, and c) 5-block cases.

3.1.3. Particle Deposition

To mimic the experimental conditions of Pui et al. (1987), the particle sizes evaluated in the 90°-bend range from 1.18 μm to 6.82 μm with Stokes numbers that range from 0.03 to 1.00. Figure 7 shows the deposition of particles on the 90°-bend for the three mesh styles. The patterns are similar, and they show that deposition of particles >3 μm (upper panel) primarily occurred along the outer radius of the bend while particles <3 μm (lower panels) that deposited generally did so along the center and inner radii, likely as a result of the secondary flow shown in Figure 6.

Figure 7:

Particle deposition locations of larger (> 3μm) particles in the a) 4-block, b) polyhedral, c) 5-block cases; and particle deposition locations of smaller (< 3μm) particles in the d) 4-block, e) polyhedral, and f) 5-block cases.

Figure 8 compares the in vitro data from Pui et al. (1987) to deposition fractions calculated with varying near-wall limit (see Section 2.4) across the range of particle sizes. It is clear across mesh styles and ranges in near-wall limit that the “nw_limit = 0” cases provide the best match to the in vitro profile. Increases in the limit resulted in an unrealistic reduction of small particle deposition for the specific conditions considered. Figure 8d (with nw_limit = 0 for the three mesh styles) emphasizes the key finding that the polyhedral mesh predicts a better match to the in vitro data than the 4-block hexahedral mesh and closely approximates the accuracy of the 5-block mesh. Average relative errors (averaged across the different particle sizes considered) between the deposition found by Pui et al. (1987) and the deposition predicted by the 4-block, 5-block, and polyhedral mesh styles for the nw_limit = 0 cases were 35%, 19%, and 20%, respectively (Figure 8d).

Figure 8:

Curved tube deposition with varying nw_limits for a) polyhedral, b) 4-block hexahedral, and c) 5-block hexahedral meshes. d) Comparison of the nw_limit = 0 deposition profile for each case.

3.2. Double Bifurcation Results

3.2.1. Mesh Independence and Solution Performance

The plots in Figure 9 show the dependence of particle deposition and the three flow-related metrics on mesh style and resolution in the double bifurcation geometry. Flow convergence for the 4-block and polyhedral meshes are very similar across the range of mesh resolutions with the exception of k (Figure 9a). Turbulent kinetic energy in this low Reynolds number flow is assumed to be largely dependent on wall effects. Therefore, the difference in k between the two mesh styles is assumed to reveal a difference between the two mesh styles in how wall-induced turbulence is generated. Excluding k, meshes with 515k cells calculated the flow-related metrics with less than 2% relative difference compared to the 1,080k mesh. For the same flow related metrics, the 220k meshes produced relative differences of less than 4% compared with the 1,080k case.

Figure 9:

Mesh convergence plots in the double bifurcation geometry for a) mass-averaged turbulent kinetic energy (k), b) area-weighted average wall shear stress (WSS), c) area-weighted average vorticity at Slice A, and d) particle deposition fraction of 10 μm particles. Mesh count does not include the extensions.

Figure 9d shows the convergence of 10 μm particle deposition across the range of mesh resolutions considered. The relative differences between the 220k meshes and 1,080k meshes were 0.6% and 0.8% for the hexahedral and polyhedral styles, respectively. The 515k meshes yielded 0.1% and 0.5% for the same comparison. The target in vitro deposition fraction is 81% (Oldham et al., 2000). For the 515k meshes, relative (absolute) errors between the in vitro deposition and the deposition predicted by the hexahedral and polyhedral styles were 4.0% (3.2%) and 3.6% (3.0%), respectively. After considering the mesh resolutions based on differences in flow-related metrics and 10 μm particle deposition as well as the error between predicted deposition and in vitro data, the 515k cell meshes were selected for comparison of mesh styles in the double bifurcation.

Figure 10a and Figure 10b show that for both mesh styles, velocity converges in roughly 200 iterations and k converges in roughly 300. The velocities for the different styles converged within 0.02 m/s of one another for a spread error of 0.4% and k values converged within 0.05 m²/s², resulting in a spread error 40%, which represents the difference in wall-induced turbulence generation between the two mesh styles. Figure 10c and Figure 10d reveal that the number of iterations and the amount of time required are not strongly dependent on mesh style except at very high resolution (1080k cells). However, the polyhedral mesh (0.58 hrs) requires approximately 53% more time to converge at the selected resolution than the hexahedral mesh (0.38 hrs).

Figure 10:

a) Mass-averaged velocity and b) mass-averaged turbulent kinetic energy for the first 600 iterations in the double bifurcation geometry. c) Hours taken to solve and d) iterations taken to solve vs. mesh count.

3.2.2. Secondary Flow

Secondary flow comparisons were qualitatively evaluated with velocity vectors and contours at Slice A (see Figure 1), the inlet of the second bifurcating region. Figure 11 shows vectors and contours of the secondary flow formed in the wake of the first bifurcation. This flow takes the form of a vortex pair which causes air at the perimeter of the slice to flow away from the inside (or carinal) wall and align at the midplane with a peak velocity of nearly 1 m/s. The average primary velocity (normal to the flow shown) has a magnitude of 3.9 m/s, such that the peak secondary flow velocity is roughly 26% of the primary flow velocity. Both mesh styles produced similar secondary flow patterns with only minor visible differences. The peak secondary velocity of the polyhedral mesh is only marginally smaller than that of the hexahedral style.

Figure 11:

Contours and vectors of secondary velocity at Slice A for the a) hexahedral and b) polyhedral cases in the double bifurcation geometry.

3.2.3. Particle Deposition

Figure 12 illustrates the deposition of particles on the walls of the double bifurcation for both mesh styles. The patterns are qualitatively similar, and confirm that, as expected, deposition of large particles primarily occurs through impaction in the bifurcating regions and along the downstream carinal walls. Small particles deposit primarily through impaction on the carinal ridges but also via the secondary flow around the inlet of the second bifurcation.

Figure 12:

Particle deposition locations in the double bifurcation for the a) hexahedral and b) polyhedral meshes.

Figure 13 illustrates localized deposition patterns for the double bifurcation by plotting cumulative deposition fraction against the Y coordinate (see orientation in Figure 3; aligns with inlet branch) for four different particle sizes considered. For 10 μm particles (Figure 13a), the in vitro profile from Oldham et al. (2000) is shown alongside the CFD predictions. The average relative errors across the Y coordinate between the in vitro profile and the hexahedral and polyhedral styles are 9.1% and 9.6%, respectively. Though there is no in vitro data for the localized deposition of small particles in this geometry, the two mesh styles show close agreement even for the 1 μm particles. This reveals a key improvement over results from Vinchurkar and Longest (2008), who analyzed the same flow and showed that hybrid tetrahedral elements predicted five times more deposition of 1 μm particles than hexahedral elements.

Figure 13:

Cumulative deposition fraction vs. Y-coordinate for a) 10 μm, b) 5 μm, c) 3 μm, and d) 1 μm particles in the double bifurcation geometry.

4. Discussion

A primary finding of this study is that a properly constructed polyhedral mesh can predict regional aerosol deposition equivalent to hexahedral meshes with minimal disadvantages. In the 90°-bend, the 4-block mesh tended to under-predict deposition of small particles, but the 5-block and polyhedral styles minimized this issue despite having similar near-wall layer height construction. Small particle deposition is of high importance in key pharmaceutical aerosol simulations and continues to play a pivotal role in the development of advanced drug delivery techniques such as Excipient Enhanced Growth (EEG) (Bass et al., 2019). Hence, the use of 5-block hexahedral meshes over the 4-block configuration remains preferred when possible despite even longer mesh generation times (Vinchurkar & Longest, 2008). The polyhedral mesh was able to match the 2.97 μm regional deposition found by Pui et al. (1987) with similar accuracy to the 5-block hexahedral mesh, with a relative (absolute) difference of 5% (1%) between the two styles. Further, the average relative difference between the two styles across the range of particle sizes varied by only 1%. Therefore, this study finds that the regional predictive ability of a polyhedral mesh is equivalent to that of a 5-block hexahedral style with the added benefit of significantly lower mesh generation time.

A secondary finding of this study is that polyhedral meshes predict local deposition profiles that match experimental data with the accuracy of hexahedral meshes. Localized deposition is more difficult to predict than regional deposition, but both mesh styles provide a close match to the in vitro profile found by Oldham et al. (2000) for 10 μm particles. For small particles, mesh style did impact the local deposition profiles in the double bifurcation geometry as observed by Vinchurkar and Longest (2008) for hexahedral vs. hybrid tetrahedral meshes. However, the difference in small particle deposition between hex and poly mesh predictions is relatively small compared with the hex and hybrid tet mesh predictions observed by Vinchurkar and Longest (2008). Moreover, regional deposition between the hex and poly meshes across the measured particle sizes were acceptably small with relative differences of 0.3% to 28.6%, which are significantly less than the difference observed by Vinchurkar and Longest (2008) when comparing hexes with tet/prisms (up to 400% relative difference). While it is not clear which mesh is more accurate for predicting the local deposit of small particles, the polyhedral mesh appears to be an acceptable substitute to the hexahedral case based on the low average relative difference observed. This finding for polyhedral meshes is a reversal of the finding of Vinchurkar and Longest (2008) where it appeared that unstructured meshes could not accurately predict the local deposition profiles captured by hexahedral cells.

As described in the Introduction, a known disadvantage of non-hexahedral mesh styles is increased numerical dispersion, which affects predictions of kinetic energy and turbulence (Longest & Vinchurkar, 2007a). This study finds that the error applies to polyhedral elements as well. The mesh independence and computational performance analyses indicate that the polyhedral mesh converges with a slightly lower volume-averaged velocity than a hexahedral mesh does for the same flow. This is likely because the poly mesh predicts secondary velocities of lower magnitude than the hex mesh. Figure 14 shows how secondary velocities – a primary cause of small particle deposition – calculated by the 4-block and poly meshes differ from one another (max relative difference ~20%) especially in areas of high velocity gradient. In addition to deviations in velocity profile, polyhedral elements seem to predict less wall-induced turbulence in the transitional flow of the double bifurcation than their hexahedral counterparts (see Figure 10b). While it is suspected that numerical dispersion is a cause for error in aerosol transport (Longest & Vinchurkar, 2007a) for other non-hex meshes, this study finds that it does not largely impact local or regional deposition in polyhedral mesh solutions for micrometer scale particles. Based on these observations it can be concluded that while polyhedral elements exhibit more numerical dispersion than their hexahedral counterparts, their predictive accuracy for micrometer aerosol transport and deposition in the airways is negligibly impacted.

Figure 14:

Velocity contours of the absolute velocity difference between polyhedral and 4-block solutions at a) the outlet of the 90°-bend and b) Slice A of the double bifurcation.

Another important finding of this study is that, for comparable mesh size, it takes longer for a polyhedral mesh to converge than a hexahedral mesh. For both airway geometries the time difference is dependent on the mesh resolution (see Figure 5 and Figure 10), where high mesh resolution exacerbates the time-discrepancy. An increase in convergence time requires consideration by CFD researchers when choosing between hexahedral and polyhedral elements for their studies. In many CFD research areas, large-scale transient simulations are common and require days or weeks to converge. Increasing this period may slow the speed at which research can progress, but it should be noted that the increased solution time does not require additional researcher input. Shortening mesh development time may be a significant advantage in whole-lung research, evaluations of patient-specific or diseased airways, development of aerosol delivery devices, and other CFD applications that require extensive and repetitive development of flow geometry.

The primary limitation of this study was the relatively narrow scope of the two flow geometries that were evaluated relative to the breadth and complexity of the airways. The 90°-bend produced flow that was similar to conditions of light exertion in a 10–15yo adolescent’s extrathoracic airways and the double bifurcation represented flow in the upper airways of an adult under heavy exertion. The bifurcation geometry was based on the morphometric data from Weibel (1963), which differs from the more recent ICRP (1994) data. These specific and idealized models were useful for comparing mesh solutions especially with the aid of validation studies from Pui et al. (1987) and Oldham et al. (2000). However, they may not show mesh style effects that could arise with varying inhalation conditions in the same domains or in flows that are difficult to model with hexahedral elements such as the laryngeal jet or flow through the nasal cavity (J. Xi et al., 2008; J. X. Xi, Si, Kim, & Berlinski, 2011). Transient effects such as breathing pattern and time-dependent flow features were not considered, but could conceivably influence the study findings. Another limitation was the lack of localized validation data for small particles in the double bifurcation. Oldham et al. (2000) did evaluate regional deposition of 1 μm particles and found a total deposition of 0.01%, but this has only been matched with the addition of upstream effects from the in vitro setup (Longest & Oldham, 2006). The result is that although the hex and poly meshes produced similar local deposition profiles even for small (1–3 μm) particles, those profiles have not been validated with experimental deposition results beyond the case of 10 μm monodisperse solid particles. Further, the increased numeric dispersion associated with the polyhedral mesh could have an influence on submicron particle deposition in either of the models, but particles in this range were not evaluated in the current study.

5. Conclusions

This study has demonstrated that the polyhedral mesh style, with sufficient near-wall and core mesh construction, is an excellent alternative to the highly regarded hexahedral mesh style in the prediction of respiratory aerosol transport and deposition. The polyhedral mesh was observed to minimize previously observed shortcomings of other unstructured meshing techniques such as unacceptably high differences in local particle deposition and numerical dispersion. The primary disadvantage of the polydisperse mesh style was a 50% to 140% increase in solution time. Nevertheless, it is expected that in most respiratory aerosol applications, savings in mesh construction time associated with polyhedral meshes will result in a net increase in CFD research efficiency. Furthermore, based on findings of this study, the useful contribution to CFD simulations provided by polyhedral meshes should be considered for other biological particle-based systems and aerosol dynamics applications.

Highlights:

• Hexahedral meshes offer high accuracy but are difficult to construct

• Polyhedral mesh building may be automated, but hexahedral mesh comparison is needed

• Polyhedral meshes can match total and local aerosol deposition of hexahedral meshes

• Solution convergence time is 50%–140% longer for polyhedral vs. hexahedral meshes

• Polyhedral meshes are an acceptable approach in computational pulmonary aerosol research