Computational Rhinology: Unraveling Discrepancies between In Silico and In Vivo Nasal Airflow Assessments for Enhanced Clinical Decision Support

Introduction

Part I - Computational Rhinology

I.4 Nasal Flow Resistance

Flow resistance, $R$, is an intrinsic flow channel-specific attribute which correlates the pressure drop over the channel, $\Delta P$, and the volumetric flowrate, $Q$,

$$\Delta P=RQ.$$

(1)

The flow resistance can vary with the flowrate, and its characteristics can be determined by measuring the pressure drop for known flowrates, or vice versa. For pressure-driven flows such as respiratory airflow, Equation (1) offers insight into how the breathing effort (quantified as pressure drop) must increase to maintain a consistent flowrate when encountering heightened flow resistance. For instance, to achieve an adequate inspiratory flowrate in an obstructed airway, a greater intrathoracic negative pressure is required than in an unobstructed airway. This offers a simple explanation why oral breathing may become the favored mode of respiration in the presence of NAO. In general, heightened flow resistance serves as an indicator of diminished patency.

The nature of (internal) fluid flow heavily depends on the geometry of the flow channel. E.g., for fully developed, steady, laminar, single-phase flow in a straight channel, the flow resistance is inversely proportional to the hydraulic diameter, $D_h\equiv 4A/O$, raised to the 4^th power (Hagen–Poiseuille equation);

$$R=\frac{128L\mu }{\pi D_h^4},$$

(2)

where $A$, $O$, and $L$ are the cross-sectional area, perimeter, and length of the flow channel, respectively, and $R=\Delta P/Q,$ is the dynamic viscosity of the fluid.

In complex flow channels such as the nasal cavities, slight changes in the geometrical features may cause unpredictable airflow response. E.g., it can be imagined that in parts of the nasal cavity featuring narrow passages such as in the nasal vestibule or the olfactory slit, a slight modification may reduce the perimeter considerably without affecting the cross-sectional area notably. This may severely impact the local hydraulic diameter. Moreover, flow instabilities or recirculation zones triggered by geometrical features (e.g. abrupt changes in flow direction or cross-sectional area) may result in effective cross-sections smaller than the actual cross-section. It is not obvious that the concept of hydraulic diameter and associated standard flow resistance correlations are applicable for nasal airflow [70,71,72,73].

A peculiar feature of the nasal passages, known as the nasal cycle, is caused by temporal, asymmetric swelling and deswelling of the nasal mucosa. This effect causes an intermittent variation in nasal resistance which can be observed through the sensation of unilateral nasal obstruction. It is believed that this effect is important for the removal of deposited dust particles etc. Not all humans have it, however, and the periodicity of the phenomenon is neither the same between different individuals nor constant in the same individual [74,75]. The nasal cycle is expected to affect unilateral nasal resistance measurements.

Nasal cavity compliance may permit local expansion or contraction of the nasal cavity cross-section due to periods of over and under pressure occurring during the respiratory cycle. This may cause local pressure dependency in the hydraulic diameter, hence the nasal resistance, which may introduce asymmetry with respect to exhalation/inhalation and temporal effects such as hysteresis in the pressure-flow relationship (Equation (1)).

I.1.1. Rhinomanometry

Figure 1. Rhinomanometry output for the current patient. Red corresponds to right side and blue corresponds to left side RMM, and light/dark colors indicate before/after administration of decongestive nasal spray. Black curves with 10% error bars show the selected "measured data" used in the current paper.

Figure 1. Rhinomanometry output for the current patient. Red corresponds to right side and blue corresponds to left side RMM, and light/dark colors indicate before/after administration of decongestive nasal spray. Black curves with 10% error bars show the selected "measured data" used in the current paper.

Rhinomanometry (RMM) is the only technique in clinical use, that allows quantitative assessment of the respiratory function of the nose [76], and it is thus of significant importance for the calibration and validation of patient-specific mathematical models of nasal air flow. The theory and background has been thoroughly covered by Vogt et al. [77]. RMM is a method that measures the pressure drop in the nose as a function of volumetric air flowrate. The resulting pressure-flow curves relate the volumetric flowrate to the pressure drop. The measured volumetric flow and pressure drops form the basis for calculation of the nasal resistance and estimation of representative hydraulic diameters [78]. Figure 1 shows an example of a pressure-flow curve.

Whereas bilateral RMM considers the simultaneous measurement of both nasal passages, unilateral RMM considers only one nasal passage at the time, by occluding one nostril while assessing the airflow in the open passage. It follows from the definition (Equation (1)) that the reciprocal bilateral resistance is the sum of the reciprocals of the individual unilateral resistances of the two nasal passages,

$$\frac{1}{R_{bi}}=\frac{1}{R_{uni,left}}+\frac{1}{R_{uni,right}}.$$

(3)

In posterior RMM the pressure is measured directly in or close to the nasopharynx by a pressure probe typically inserted via the oral cavity. In anterior RMM, one nostril is closed, and the pressure probe is inserted into the nasal vestibule behind the occlusion to give an indirect measurement of the choanae pressure. It has been shown that posterior and anterior RMM are equivalent with respect to (unilateral) pressure measurement [79]. Passive RMM is performed by enforcing external nasal airflow. More commonly employed is active RMM, where the patient's own physiological airflow is utilized. The RMM procedure combining active breathing with anterior measurement is denoted active anterior RMM (AAR).

Because the nasal passages are lined with a mucosal membrane subject to unpredictable, temporal variations in swelling, RMM is typically performed twice, before and after decongestion. Application of topical nasal decongestant (e.g. xylometazoline) or physical exercise serves to eliminate the vascular component of nasal obstruction caused by swelling of the turbinates, allowing quantification of the anatomical component of NAO [80]. The anatomical component is determined solely by the rigid tissue of the nose (e.g. bone and cartilage), and it is thus related to the maximal nasal volume, independent of the nasal cycle [45]. In clinical rhinology, such decongestion tests are performed to quantify the different roles of the skeleton and mucosa in NAO. For the mathematical modelling of airflow in the nasal cavities, it is convenient to disregard the complexities associated with soft tissue and temporal variations in unilateral nasal resistance.

I.1.2 Mathematical Illustration of the Principles of Rhinomanometry and Inherent Hysteresis using Bernoulli's equation

The mathematical background of rhinomanometry has been thoroughly covered by others [77]. Here, a brief illustration of the concept of rhinomanometry is provided, based on the mathematical description of simple pipe flow.

In pipe and duct flow engineering, it is common practice to estimate pressure drops by utilizing Bernoulli's equation [81]. For unsteady, fully developed, horizontal, incompressible flow through a straight, rigid duct of constant cross-sectional area, the pressure drop per unit length results from the contributions of a friction term and an unsteady, inertial term,

$$\frac{\Delta P}{L}=\underset{\mathrm{flow}\mathrm{resistance}\mathrm{per}\mathrm{unit}\mathrm{length},R/L}{\underbrace{\frac{8\rho }{{\pi }^2{D}_h^5}\left[\underset{\mathrm{friction}\mathrm{term}}{\underbrace{f_{D}Q}}+\underset{\mathrm{unsteady}\mathrm{term}}{\underbrace{\frac{\pi D_h^3}{2}\frac{\partial \ln Q}{\partial t}}}\right]}}Q,$$

(4)

where $\rho$ is the fluid mass density and $f_D$ is the Darcy friction factor,

$$f_D=\left\{\begin{array}{lll}{f}_{D,lam}=64/\mathrm{Re}\hfill & \mathrm{for}\hfill & \mathrm{Re}\le 2000\hfill \\ f_{D,turb}(\mathrm{Re},{\epsilon }_r)\hfill & \mathrm{for}\hfill & \mathrm{Re}\ge 4000\hfill \end{array}\right.,$$

(5)

where $\mathrm{Re}$ is the Reynolds number, which for internal duct flow, can be expressed as

$$\mathrm{Re}=4Q/\pi \nu D_h,$$

(6)

where $\nu =\mu /\rho$ is the fluid kinematic viscosity, and ${\epsilon }_r$ is the relative wall roughness. Equation (4) reduces to Equation (2) for laminar flow ($\mathrm{Re}\le 2000$) with $\partial Q/\partial t=0$. Several alternative formulations exist for the turbulent friction factor, e.g. the Haaland equation [82],

$$\frac{1}{\sqrt{f_{D,turb}}}=-1.8{\log }_{10}\left[{\left(\frac{{\epsilon }_r}{3.7}\right)}^{1.11}+\frac{6.9}{\mathrm{Re}}\right].$$

(7)

In laminar flow, the flow is characterized by smooth, locally parallel streamlines, and flow variables behave deterministically. In the case of turbulent flow, however, flow variables behave stochastically. In duct flow engineering, the flow is typically considered laminar for Reynolds numbers below 2000 and turbulent for Reynolds numbers above 4000, but these thresholds may vary depending on the specific flow configuration. In the intermediate range, the flow is transitional, which is a generally poorly understood, complex and dynamic flow state. For transitional flow, the friction factor can be approximated by a smooth, weighted average of the laminar and turbulent friction factors, $f_{D,tran}\approx wt\cdot f_{D,lam}+(1-wt)\cdot f_{D,turb}$, where the weight, $wt$, varies smoothly between 1 and 0 in the laminar and turbulent regimes, respectively.

To improve the general understanding of RMM, a simplified model is used to represent nasal airflow, based on a straight, smooth (${\epsilon }_r=0$) duct of a given hydraulic diameter and fully developed, sinusoidal (respiratory) flow,

$$Q=Q_{\max }\sin \left(\omega t\right),$$

(8)

where $t$ is the time variable, $\omega =2\pi /T$ is the angular frequency, and $T$ is the period of the respiratory cycle. Employing this model, synthetic RMM pressure-flow curves can be generated to study the impact of the various parameters. In Figure 2, data are shown for $Q_{\max }=600ml/s$, $T=5s$, and $D_h=10mm$, to compare the effects of assuming laminar or turbulent flow and to illustrate the effect of the unsteady term. Figure 2a shows that there is a phase shift between the flowrate and pressure drop due to the unsteady term. In Figure 2b, it can be seen that this causes a hysteresis effect such that the pressure-flow curve does not pass through the origin.

The hysteresis width is found by evaluating Equation (4) at $Q=0$,

$$W_{\Delta P}=\frac{8\rho L\omega Q_{\max }}{\pi D_h^2},$$

(9)

where Equation (8) was used. It follows that the relative hysteresis width, defined as the width of the pressure-flow curve hysteresis at the level of $Q=0$ divided by the maximum laminar pressure drop, and can be expressed as

$$W_{\Delta P,rel}=\omega D_h^2/16\nu .$$

(10)

Figure 3 illustrates how the relative hysteresis width increases for increasing hydraulic diameter while the flow resistance per unit length decreases.

In general, it can be observed through analysis of Equation (4) that:

• Steady, laminar pressure-flow curves are straight lines passing through the origin.
• The slope of the pressure-flow curve decreases with increasing flow resistance. That is, laminar pressure-flow curves are steeper than turbulent ones due to the higher friction factor in turbulent flow. Distinct change in slope in in vivo RMM pressure-flow curves may thus be an indication of transition to turbulence.
• The unsteady flow resistance term causes a hysteresis effect, such that the pressure-flow curve becomes a closed loop not passing through the origin.
• The relative hysteresis width is determined by the hydraulic diameter and the period of the respiratory cycle.

The complex shape of the nasal cavity as well as effects of lateral wall movement may complicate this picture considerably for nasal airflow. Blevins [81] provides an overview of applicable methods to approximate friction factors and pressured drop for channels with noncircular cross-sections, bends, changes in flow area, etc.

Part II: Overview of Published Literature on in vitro and in silico Nasal Airflow Studies

II.2 Summary of Part II

• An increasing number of scientific publications are being published in the cross-disciplinary field of computational rhinology. As of 2018, the publication rate was approximately 80 papers per year after several years of near-exponential increase.
• The rapid increase in the publication rate is mainly attributed to the advancements made in automatic segmentation of medical imaging data, streamlining the process of 3D geometry generation, and automatic, unstructured meshing of complex geometries.
• There is no consensus regarding the choice of turbulence model in nasal airflow simulations. Laminar flow modelling appears to be the most popular approach, by far, followed by RANS models. Few publications have reported from LES or DNS modelling.
• Most studies have investigated steady, inspiratory flow. Relatively few publications have reported from expiratory or transient/respiratory flow.

Table 1. A non-exhaustive overview of experimental and numerical studies of human nasal airflow employing steady and transitional flow boundary conditions.

Table 1. A non-exhaustive overview of experimental and numerical studies of human nasal airflow employing steady and transitional flow boundary conditions.

		Volumetric Flowrate
		Steady	Transient
Experiments in physical replicas (in vitro)		[42,83,117,151,173,203,204,205,214,215,216,219,220,221]	[42,127,129,130,149,151,157,197,200,201,202,203,204,206,222,223,224]
Navier-Stokes-based models (in silico)
Laminar		[5,50,57,58,59,61,62,63,65,66,67,68,83,86,88,90,100,101,106,116,117,123,126,129,131,148,152,158,172,175,176,180,192,199,205,207,208,209,211,219,225,226,227,228,229,230,231,232,233,234,235,236,237] The present study	[126,129,153,158,172]
RANS	$k\epsilon$	[3,4,42,83,170,204,210,220,226,231,238,239,240] The present study	[224]
	$k\omega$	[83,117,121,122,173,204,205,207,226,230,238,241,242,243]	[173,197]
	$k\omega$ SST	[83,101,124,148,172,175,199,204,205,212,213,220,244,245,246] The present study	[118,160,172,194,223]
	Spalart-Allmaras	[204,226,247]
	Reynolds Stress Model	[83]
LES and RANS-LES hybrid models		[83,148,204,206,217,238,248]	[159,249,250,251] The present study
DNS		[83]
Lattice-Boltzmann-based methods (in silico)		[42,125,195,252,253,254]	[127]

Part III. In silico RMM Simulation Results

III.1 Methods

The present study used patient-specific clinical data for creation of and comparison with CFD simulations. Computed tomography (CT) images were utilized to create a digital 3D geometry model of the patient's nasal cavity, which was used as basis for CFD simulations of active anterior rhinomanometry (AAR). In silico pressure-flow curves were obtained from the CFD simulations and compared with the patient's AAR data.

Figure 4. Computed tomography images of the current patient. The modelled airway is colored red, and the blue lines in each cross-section indicate the position of the other cross-sections.

Figure 4. Computed tomography images of the current patient. The modelled airway is colored red, and the blue lines in each cross-section indicate the position of the other cross-sections.

III.1.3 In silico Rhinomanometry (CFD modelling)

In silico RMM was performed by CFD simulation of unilateral nasal airflow in the patient-specific 3D geometry. During the simulations, both nasal cavities were included, but one nostril was closed such that there was only airflow through one nasal cavity at the time. The transnasal pressure drop was thus approximated by the pressure difference between the open and the closed nostril, similar to in vivo AAR. See Section I.3.2 for additional discussion of AAR.

CFD modelling was performed in ANSYS Fluent 2019 R2 [257]. Steady state simulations were performed to produce quasi-steady RMM curves. These simulations utilized laminar, $k\epsilon$ realizable, and $k\omega$ SST models. Transient simulations were performed with the Large Eddy Simulation (LES) to produce the entire RMM curves including potential effects of unsteady, respiratory flow. Three breathing cycles were simulated on each side.

All simulations were isothermal with constant air properties ($\rho =1.225kg/m^3$, $\mu =1.7894\cdot {10}^{-5}Pas$). Additional details regarding the simulations are summarized in Table 2. Most simulation settings were kept at default values as proposed by ANSYS Fluent. Refer to ANSYS Fluent user and theory guides for more details regarding the model settings and theory background [113,258]. In all simulations, convergence was assumed when residuals dropped below 0.001. In the transient simulations, this required between 1 and 7 iterations per time step, depending on the flowrate.

The simulation case-files and simulation results from the LES simulations have been made available to the public under the Creative Commons Attribution-Noncommercial (CC BY-NC) license [14].

Figure 5. Flow geometry obtained from segmentation of CT images. Paranasal sinuses and the oral cavity are omitted from the model. Computational boundaries are indicated by color: Air-tissue wall boundary (gray), pharyngeal flow boundary (red); right nostril (green); left nostril (yellow).

Figure 5. Flow geometry obtained from segmentation of CT images. Paranasal sinuses and the oral cavity are omitted from the model. Computational boundaries are indicated by color: Air-tissue wall boundary (gray), pharyngeal flow boundary (red); right nostril (green); left nostril (yellow).

Turbulence Modelling

For laminar flow, eqs. (11) and (12) (p. 20) are solved directly. For turbulent flow employing Reynolds averaged Navier-Stokes (RANS) turbulence models, such as the $k\epsilon$ and $k\omega$ model families, the velocity is considered a sum of a mean velocity and a fluctuating component whose ensemble average is zero, $u\to \overline{u}+u^{\prime }$ with $〈u^{\prime }〉=0$, such that $〈u〉=\overline{u}$ , where the brackets indicate ensemble averaging. Inserting for the fluctuating velocity in eqs. (11) and (12) and performing ensemble averaging results in the RANS equations, in which the Reynolds stress term, $〈\left(u^{\prime }\cdot \mathsf{\nabla }\right)u^{\prime }〉$, represents the effect of turbulence. In order to close Equation (12), these stress terms are modelled by the Boussinesq hypothesis, where the Reynolds stress terms are assumed to be related to the mean velocity gradients such that the effect of turbulence can be modelled through an increased effective viscosity (eddy viscosity model). Eqs. (11) and (12) are thus written as the RANS equations,

$$\mathsf{\nabla }\cdot \overline{u}=0,$$

(18)

and

$${\partial }_t\overline{u}+\left(\overline{u}\cdot \mathsf{\nabla }\right)\overline{u}=-\left(1/\rho \right)\mathsf{\nabla }\overline{P}+\left(\nu +{\nu }_t\right){\mathsf{\nabla }}^2\overline{u},$$

(19)

where the ${\nu }_t$ is the isotropic turbulent kinematic viscosity. In the $k\epsilon$ and $k\omega$ model families, additional equations are solved for the turbulent kinetic energy, $k$, dissipation rate, $\epsilon$, and specific dissipation rate, $\omega$, which are used to compute the turbulent kinematic viscosity. $k\epsilon$ realizable [259] is an improved variant of the standard $k\epsilon$ model which has been shown to have superior performance in modelling complex flows with e.g. separation and secondary flow features [113]. $k\omega$ SST is a hybrid model combining the strengths of the $k\omega$ model, which is considered more accurate in the near-wall region, and the $k\epsilon$ model, which is considered better in the outer parts of the turbulent boundary layer. The Enhanced wall treatment (EWT) option in ANSYS Fluent, enables near-wall modelling capabilities in the $k\epsilon$ models, also. Thus, both $k\epsilon$ with EWT and $k\omega$ SST are y⁺ insensitive models [112].

For LES turbulence models, the governing equations (eqs. (11) and (12)) are filtered through the finite-volume discretization. The LES Continuity equation becomes identical to Equation (18), but the Reynolds stress term in the Momentum equation is replaced by the subgrid-scale stress tensor ,${\tau }_{SGS}$,

$${\partial }_t\overline{u}+\left(\overline{u}\cdot \mathsf{\nabla }\right)\overline{u}=-\left(1/\rho \right)\left[\mathsf{\nabla }\overline{P}+\mathsf{\nabla }\cdot \overline{{\tau }_{SGS}}\right]+\nu {\mathsf{\nabla }}^2\overline{u}.$$

(20)

The RANS and LES momentum equations are, however, formally identical. In ANSYS Fluent, the subgrid-scale stress models employ the Boussinesq hypothesis, as the RANS models, and the main difference between the available LES variants are in the calculation of the subgrid-scale turbulent viscosity. Here, the Wall-Adapting Local Eddy-Viscosity (WALE) subgrid-scale stress model [260] was employed. This model has the trait that it predicts zero turbulent viscosity in laminar flow, and it is therefore considered suitable for modelling wall bounded flows with laminar turbulent transition [111,261].

The core idea of the LES modelling strategy is that the majority of the energy carrying turbulent eddies are modelled directly on a sufficiently fine grid, while the remaining (small) fraction of eddies are modelled via the subgrid-scale model. LES is widely accepted as the most accurate turbulence model, second to Direct Numerical Simulation (DNS), which resolves even the smallest turbulent eddies. It is assumed that LES approaches DNS in the limit of high spatial resolution (small grid cells). Additional discussion of turbulence models was provided in Section I.4.5, and the turbulence models available in ANSYS Fluent are well documented in the ANSYS Fluent User and Theory guides [113,258].

Boundary and Initial Conditions

The boundaries of the flow geometry consist of the nasal cavity walls, the left and right nostrils, and the pharyngeal flow boundary (see Figure 5).

CFD simulations were set up to replicate unilateral RMM, where one nostril is occluded while measuring the volumetric flowrate and pressure drop in the contralateral nasal cavity. Hence, all simulations were performed unilaterally, with one nostril closed and the other open, leading to two specific scenarios: left open-right closed (LO-RC) and left closed-right open (LC-RO).

The nasal cavity walls and closed nostril were treated as standard smooth, non-slip wall boundaries. The open nostril was specified as a pressure inlet with a specified, constant pressure of 0 Pa(g). In ANSYS Fluent, the pressure formulation at a pressure inlet depends on the flow-direction across the boundary. Thus, the total and static pressures at the nostril boundary were alternatingly equal to the specified pressure, during inhalation and exhalation, respectively (see the Ansys Fluent user guide for details [258]). This choice of boundary condition was aimed at mimicking the difference between inhalation from and exhalation into a stagnant reservoir of air. During inhalation, it is expected that air will experience loss free acceleration from the far field to the nostril, conserving the total pressure. This implies a negative static gauge pressure of approximately $\rho U^2/2$ during inhalation, where $U$ is the mean velocity at the nostril [81]. During exhalation, however, it is expected that the jet of air exiting the nostril will dissipate and lose all its kinetic energy to the surrounding air, and the pressure at the nostril is approximately equal to the far field static pressure.

The pharyngeal flow boundary was specified as an inlet velocity boundary condition, with a uniform velocity corresponding to the specified volumetric flowrate. Negative and positive flowrates indicated inhalation and exhalation, respectively. For steady state simulations, the constant, volumetric flowrate was specified for each simulation, but for the transient simulations, tidal respiration was approximated by

$$Q(t)=-Q_{\max }\cos \left(\frac{2\pi (t-t_0)}{\tau }\right),$$

(21)

where $t$ was the flow-time, $t_0=1s$ was the duration of the pseudo-steady initialization described below, $\tau =5s$ was the duration of one breathing cycle, and the peak volumetric flowrate was $Q_{\max }=600\mathrm{ml}/\mathrm{s}$. This corresponds to a tidal volume of $955\mathrm{ml}$, which is above what would be expected in normal breathing [161,262].

The same turbulence boundary conditions were employed for the open nostril and the pharyngeal flow boundary. Default values were used; 5% turbulent intensity, turbulent viscosity ratio of 10, and intermittency of 1 ($k\omega$ SST, only). For the LES simulations, the synthetic turbulence generator option was selected. See the ANSYS Fluent User and Theory guides for details [113,258].

Initial conditions for the transient LES-based simulations, were obtained by first performing a steady state fine-mesh $k\omega$ SST simulation with peak inspiratory flow, $Q=-Q_{\max }$. Next, the resulting flow fields were converted by employing the text-user-interface command solve/initialize/init-instantaneous-vel before pseudo-steady LES simulation with constant peak inspiratory flow, was run for 1 second flow-time ($t_0=1s$). This initialization procedure allowed the development of the flow fields, and no pronounced start-up effects were observed, in contrast to the report by Bradshaw et al. [159].

Spatial and Temporal Resolution

Ansys Meshing [257] was employed to create coarse and fine computational meshes. Default software settings were used, but the capture proximity and capture curvature options were enabled. Inflation layers consisting of five layers of pentahedral prismatic cells and a growth ratio of 1.2 were established along all wall surfaces and the closed nostril, and tetrahedral cells were used for the remaining geometry. The coarse and fine meshes used element sizes of 1 and 0.2 mm, respectively, which resulted in 876 thousand and 44.7 million grid cells. Figure 6 showcases coarse and fine surface meshes. Figure 7 illustrates the computational meshes at selected cross-sections throughout the nasal cavity. For the coarse mesh, the gap between opposing inflation layers was approximately the width of 2-3 grid cells, in the narrowest passages. On the fine mesh, however, even the narrowest passages were finely resolved by many grid cells.

Time steps of 10 microseconds were used for the transient LES simulations.

III.2 Results

The main content of the CFD study presented in this part of the paper was the generation of in silico RMM pressure-flow curves from high-fidelity CFD modelling and their comparison to in silico curves obtained from simpler CFD models as well as patient-specific, clinical in vivo measurements.

Most of the simulation results presented here were obtained from the fine-mesh CFD models described earlier. Unless explicitly stated otherwise, the presented data were obtained from the fine-mesh $k\omega$ SST simulations at peak inspiratory flow. Results from the coarse-mesh steady state $k\epsilon$ realizable and $k\omega$ SST models are limited to the in silico RMM pressure-flow curves. The coarse-mesh laminar simulations were, in addition, used to compare pressure profiles to the LES simulations. Animations of showcasing the transient behavior of selected flow field variables, obtained from the LES simulations, are available in the supplementary material.

During the transient simulations, data was systematically saved at regular intervals. For instance, the differential pressure difference between the right and left nostrils, along with pressure and velocity data at selected cross-sections throughout the geometry, was saved at every time step. Additionally, simulation data files containing instantaneous values of all flow variables as well as recorded turbulence statistics, were saved every ten thousandth time step (every tenth second). The entire data set accessible in the public domain for non-commercial use [14]. A separate document describing the contents of the data set is available in the data repository.

Despite capturing turbulence statistics during the transient LES simulations, they are not incorporated or discussed in the current report. This decision was rooted in the continuous variation of mean flow rate in accordance with its sinusoidal boundary condition, making it unclear how turbulence statistics could contribute to the present analysis. Nevertheless, this data is provided in the published dataset, inviting other researchers to scrutinize and engage in discussions regarding its potential applications.

III.2.1 In silico RMM-results

In the transient LES-based simulations, three breathing cycles were simulated on each side of the nose. Throughout these simulations, the differential pressure between the right and left nostrils were monitored throughout the simulations. Figure 8 shows the transient behavior of the differential pressure. There were no evident differences between the three breathing cycles on either side. For steady state simulations, repeated simulations with different pharyngeal flow rate conditions were performed.

In generating in silico RMM pressure-flow curves, the volumetric flowrate was plotted against the differential pressure data. During inhalation, the pressure at the closed nostril is lower than at the open nostril, while during exhalation, the opposite situation occurs. Consequently, the differential pressure between the right and left nostril will be positive during inhalation and negative during exhalation on the right side, and vice versa on the left side. It is noted that in the clinical in vivo RMM pressure-flow curves depicted in Figure 1, both differential pressure and volumetric flowrate are non-negative. In the in silico RMM pressure-flow curves presenting the current simulation data, the volumetric flowrate remains non-negative throughout the respiratory cycle. However, the sign of the differential pressure was adjusted to align with the convention where inhalation corresponds to a positive pressure difference between the open nostril and the nasopharynx, and exhalation corresponds to a negative pressure difference. The resulting in silico RMM pressure-flow curves thus emulate clinical RMM curves, ensuring that pressure-flow curves for inhalatory and exhalatory breathing phases on the left and right sides are appropriately plotted in the corresponding quadrants of the pressure-flow chart. Specifically, pressure-flow curves for the inhalation phase on the right and left sides are represented in the upper and lower right quadrants, respectively. Conversely, pressure-flow curves for the exhalation phase on the right and left sides are plotted in the lower and upper left quadrants, respectively.

Figure 9 illustrates how the various steady state coarse-mesh simulations (laminar, $k\epsilon$ realizable, and $k\omega$ SST models) produced almost identical RMM pressure-flow curves, on both sides of the nose. These curves closely agree with the fine-mesh $k\omega$ SST model, at peak inspiratory flow. During exhalation, the agreement among the models is somewhat diminished, with the laminar model predicting the highest nasal resistance and the $k\epsilon$ realizable model predicting the lowest. Figure 10 reveals that the transient fine-mesh LES yields RMM pressure-flow curves nearly identical to those generated by the steady state coarse-mesh laminar simulations. Figure 11 provides a comparison between in silico RMM pressure-flow curves and representative, decongested in vivo RMM pressure-flow curves for the specific patient

In this study. It is evident that the CFD simulations severely underpredicted the pressure drop, hence nasal resistance, displaying a discrepancy exceeding one order of magnitude.

Assuming a defined channel length and curve-fitting Equation (4) (p. 14) to RMM data, representative hydraulic diameters for the nasal cavities can be determined. In Figure 12, best-fit curves and best-fit hydraulic diameters are presented for current in vivo and in silico RMM data utilizing a channel length of $L=0.1m$. The nasal resistance on the left side was higher than on the right, resulting in a smaller hydraulic diameter on the left side. The hydraulic diameters that best capture the in silico data are notably higher than those representing the in vivo data. Additionally, a discernible difference between inhalation and exhalation, particularly on the right side, is observed in the in silico data. Examination of the in vivo data reveals that on the left side, the curves flatten out for high pressure drops. The inability of the volumetric flow rate to increase with rising pressure drop indicates a proportional relationship between hydraulic resistance and pressure drop, leading to a constant volumetric flow rate (refer to Equation (1), p. 12). This phenomenon may be attributed to a shrinking cross-sectional area caused by the collapse of the nasal cavity. It is interesting to note that despite the observed effect on the left side, the in vivo RMM curves exhibit symmetry with respect to inhalation and exhalation. This symmetry suggests that nasal compliance occurs due to strong Venturi effect, which remains insensitive to the flow direction.

Figure 13 illustrates hysteresis loop resulting from the by the unsteady flow in the transient LES simulation, as discussed in Section I.3.3. The hysteresis width introduces a fitting restraint on the channel length and hydraulic diameter, and the channel length $L$ is eliminated. The combination of eqs. (4) and (9) results in

$$\Delta P=\frac{W_{\Delta P}}{\omega Q_{\max }}\left[\frac{f_D}{\pi D_h^3}{Q}^2+\frac{1}{2}\frac{\partial Q}{\partial t}\right],$$

(22)

where $W_{\Delta P}$ is the hysteresis width. By applying Equation (21) to describe the unsteady volumetric flowrate and employing the hysteresis widths observed in the LES-based RMM curves, the hydraulic diameter was adjusted to achieve the best-fit curves shown in Figure 13. The curve fitting process was conducted separately for the inspiratory and expiratory phases on both sides. To optimize the fit, the Haaland equation (Equation (7), p. 14) for the (turbulent) friction factor was employed across all flow-rates. While the best-fit curve accurately represented the hysteresis loop on the right side, the fit on the left side was less precise. The resulting channel lengths derived by inserting the best-fit hydraulic diameter, measured hysteresis width, and air properties into Equation (9) were $L=3.3cm$ and $2.6cm$ for inhalation and exhalation on the left side, respectively. On the right side, the corresponding lengths were $L=4.2cm$ and $2.9cm$. These numbers are significantly lower than the physical length of the nasal cavity, underscoring a limitation in the comprehensiveness of Equation (22) in capturing essential physical aspects of the intricacies of nasal air flow. Hydraulic diameters derived from curve-fits of the laminar simulation data demonstrated good agreement with those representing the LES hysteresis when similar channel lengths were employed. This consistency suggests that the laminar simulation effectively captures pertinent physics.

III.2.2 Spatial and temporal resolution

Based on peak inspiratory and expiratory flowrates, the coarse-mesh simulations yielded wall $y^{+}$ values below 5, almost everywhere. The fine-mesh simulations yielded wall $y^{+}$ values below 1. Hence, the near wall grid cells were situated in the viscous sublayer almost everywhere, for all flowrates (see Equation (16), p. 32), in all the simulations. Wall $y^{+}$ contour plots obtained from the fine-mesh $k\omega$ SST model are shown in figures 14 and 15 for the LO-RC and LC-RO peak inspiratory flow simulations, respectively.

Figure 14. Estimated wall $y^{+}$ values based on the LO-RC steady state fine-mesh kω SST simulation of peak inspiratory flow

Figure 14. Estimated wall $y^{+}$ values based on the LO-RC steady state fine-mesh kω SST simulation of peak inspiratory flow

Figure 15. Estimated wall $y^{+}$ values based on the LC-RO steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 15. Estimated wall $y^{+}$ values based on the LC-RO steady state fine-mesh kω SST simulation of peak inspiratory flow.

Due to the heightened velocities resulting from the narrower nasal cavity, $y^{+}$ values on the left side were slightly higher than on the right side. The narrowest sections exhibited the highest wall shear stress, hence $y^{+}$ values, due to the relatively elevated flow velocities in these regions.

Figure 16. Kolmogorov micro scales: a) length scale; and b) time scale computed from the LO-RC and LC-RO fine-mesh kω SST peak inspiratory flow simulations.

Figure 16. Kolmogorov micro scales: a) length scale; and b) time scale computed from the LO-RC and LC-RO fine-mesh kω SST peak inspiratory flow simulations.

The subgrid-scale turbulent viscosity in the WALE LES model, as implemented in ANSYS Fluent and used in the current simulations, is proportional to the square of the mixing length for subgrid scales [113],

$$l_{LES}=\min (\kappa y,0.325\Delta ),$$

(23)

where $\kappa =0.4187$is the von Kármán constant, $y$ represents the distance to the nearest wall, $\kappa y$ is thus the length scale of the largest eddies, and $\Delta$ is the cube root of the grid cell volume. On the fine mesh, it was only in the two first layers of prismatic grid cells along the walls, that $\Delta >\kappa y/0.325$. Thus, the subgrid-scale turbulent viscosity was predominantly determined by the computational grid cell size. The two first layers of near-wall grid cells were inside the viscous sublayer for all flowrates, which aligns with the capabilities of the WALE LES model to correctly represent the behavior of wall bounded flows [113].

The Kolmogorov length and time scales, representing the smallest scales of turbulence, can be expressed as

$$\eta ={\left({\nu }^3/\epsilon \right)}^{1/4},$$

(24)

and

$${\tau }_{\eta }={\left(\nu /\epsilon \right)}^{1/2},$$

(25)

respectively, where $\nu$ is the kinematic viscosity, $\epsilon =-dk/dt$ is the turbulent dissipation rate, and $k$ is the turbulent kinetic energy [109]. Utilizing results from the fine-mesh $k\omega$ SST simulations at peak inspiratory flow, the Kolmogorov micro scales were computed from eqs. (24) and (25). Contour plots at selected cross-sections are depicted in Figure 16 for LO-RC and LC-RO simulations.

To assess the spatial and temporal resolution of the fine-mesh simulations, local subgrid scale mixing lengths determined from Equation (23) and time step size were compared to the Kolmogorov length and time scales determined by eqs. (24) and (25), respectively. The turbulent dissipation rates needed to compute the Kolmogorov micro scales were obtained from the fine-mesh $k\omega$ SST simulations of peak inspiratory flow. As the turbulent dissipation rate varies with location and flow velocity, the Kolmogorov micro scales also exhibit corresponding variations. The contour plots in Figure 17 depict the ratio of a) the LES and Kolmogorov length scales, $l_{LES}/\eta$ ; and b) the time step size and the Kolmogorov time scale,$\Delta t/{\tau }_{\eta }$, at selected cross sections for LO-RC and LC-RO simulations. Figure 18 presents the distributions of length and time scale ratios through histogram plots with a bin size of 0.1. It is evident that the vast majority of grid cells exhibited subgrid scale mixing lengths smaller than the local Kolmogorov length scale ($l_{LES}/\eta <1$) and had time steps shorter than the Kolmogorov time scale ($\Delta t/{\tau }_{\eta }<1$). However, the LO-RC simulation generally demonstrated higher ratios compared to the LC-RO simulation, attributed to the overall higher flow velocities in the former. in the LO-RC and LC-RO simulations, 1.2% and 0.2% of the grid cells, respectively featured subgrid scale mixing lengths exceeding the Kolmogorov length scale. Only the LO-RC simulation exhibited a minor fraction of grid cells (0.01%) where the Kolmogorov time scale was smaller than the time step size. Grid cells with elevated length scale ratios were generally situated within the wall boundary layers, albeit not directly at the wall, in regions characterized by relatively high wall shear stress. The grid cells with the highest time scale ratios were found at the anterior of the middle meatus.

The subgrid-scale turbulent viscosity ratio was close to zero almost everywhere in the nasal cavity, as shown in Figure 19. In the nasopharynx, however, the turbulent viscosity ratio is substantially higher. The grid cells displaying the poorest spatial and temporal resolution were in concordance with those featuring the highest sub-grid scale turbulent viscosity.

$CFL$ numbers (see Equation (17), p. 33), as calculated by ANSYS Fluent, were obtained from the fine-mesh $k\omega$ SST simulations of peak inspiratory flow. Contour plots presented in Figure 20 reveal values below 2 on the right side and below 4 on the left side. Hence, the chosen time steps exceeded the recommended range for LES ($CFL\sim 1$). It has, however, been shown that $CFL\sim 5$ can work in some cases [111]. Given the time-varying flowrates during the respiratory cycle, the $CFL$ number varied periodically between 0 and these maximum values. The implicit formulation of the CFD simulation is robust with respect to high $CFL$ numbers, and no issues regarding numerical stability were experienced.

III.2.3 Flow velocity fields.

In Figure 21, flow velocity contours on selected cross sections for the LO-RC and LC-RO fine-mesh $k\omega$SST simulations of peak inspiratory flow are presented. Notable variability is evident within each cross-section, reflecting a complex flow pattern. The lower part of the nasal cavity, adjacent to the inferior turbinate, emerges as the preferred flow path. Qualitatively, it was noted that the favored flow path on the left side was slightly higher than on the right side, closer to the middle turbinate.

Figure 22 displays velocity contours and vectors on the sagittal symmetry plane, illustrating distinct flow patterns in the nasopharynx at peak inspiratory and expiratory flow. During inspiration, significant mixing occurs, dominated by relatively large unsteady vortical structures. During expiration, however, the nasopharyngeal flow is dominated by the nasopharyngeal jet, as highlighted by Bradshaw et al. [159]. Airflow velocities were generally low enough that the assumption of incompressible flow was valid.

Using unilateral hydraulic diameters from the open side of the cross-sections in Figure 23, Reynolds numbers based on the peak flow-rate (600 ml/s) were computed (see Equation (6), p. 14). Figure 24 depicts these results for unilateral flow on the left and right sides, respectively. In the anterior part of the nose (nasal gateway), the Reynolds number is notably higher on the left side, whereas in the posterior nasal cavity, it is slightly higher on the right side. Throughout the nasal cavity, the estimated peak Reynolds numbers are in the range which would be expected to be turbulent, for fully developed flow in straight channels (>4000).

Figure 18. Histograms illustrating the distribution of ratios of a) LES simulations mixing lengths (Equation (23)) to the Kolmogorov length scales (Equation (24)); and (b) LES simulations time step size (10 μs) to Kolmogorov time scales (Equation (25)), based on the LO-RC and LC-RO fine-mesh kω SST peak inspiratory flow simulations. Bin sizes are 0.01.

Figure 18. Histograms illustrating the distribution of ratios of a) LES simulations mixing lengths (Equation (23)) to the Kolmogorov length scales (Equation (24)); and (b) LES simulations time step size (10 μs) to Kolmogorov time scales (Equation (25)), based on the LO-RC and LC-RO fine-mesh kω SST peak inspiratory flow simulations. Bin sizes are 0.01.

Figure 19. Estimated turbulent viscosity ratio values based on the fine-mesh kω SST peak inspiratory flow simulations.

Figure 19. Estimated turbulent viscosity ratio values based on the fine-mesh kω SST peak inspiratory flow simulations.

Figure 20. CFL numbers based on the fine-mesh kω SST peak inspiratory flow simulations and time step size of $\Delta t=10\mu s$.

Figure 20. CFL numbers based on the fine-mesh kω SST peak inspiratory flow simulations and time step size of $\Delta t=10\mu s$.

Figure 21. Flow velocity contour plots at selected cross-sections for a) LO-RC and b) LC-RO, fine-mesh kω SST peak inspiratory flow simulations.

Figure 21. Flow velocity contour plots at selected cross-sections for a) LO-RC and b) LC-RO, fine-mesh kω SST peak inspiratory flow simulations.

Figure 22. Nasopharyngeal velocity contours and vectors drawn at the sagittal symmetry plane for peak inspiratory and expiratory flow for the transient LES-based LC-RO simulation on the fine mesh.

Figure 22. Nasopharyngeal velocity contours and vectors drawn at the sagittal symmetry plane for peak inspiratory and expiratory flow for the transient LES-based LC-RO simulation on the fine mesh.

In figures 25 and 26, Q-criterion isosurfaces colored by velocity provide a visual representation of vortical structure generation in the transient LES simulations. Notably, during exhalation, there is minimal production of vortexes in the straight pharyngeal inlet section. Conversely, the nasal cavity exhibits substantial vortex production and mixing. During inhalation, the vortexes produced in the nasal cavity are transported downstream through the pharyngeal section. Intriguingly, some vortexes penetrate deeply into the closed nasal cavity. Although their velocities are low, vortexes facilitate the movement of air toward the olfactory region at the top of the nasal cavity, where the sense of smell is located.

Figure 23. Cross section positions used in figures 24, 30, and 31. The cross-sections 1-13 were positioned at regular intervals of 0.5mm. The two nasal cavities merge into the nasopharynx between cross-sections 13 and 14. Nasopharynx is thus represented by cross-sections 14-19. The cross-sections 20 and 21 are identical in shape and size to the pharyngeal outflow boundary.

Figure 23. Cross section positions used in figures 24, 30, and 31. The cross-sections 1-13 were positioned at regular intervals of 0.5mm. The two nasal cavities merge into the nasopharynx between cross-sections 13 and 14. Nasopharynx is thus represented by cross-sections 14-19. The cross-sections 20 and 21 are identical in shape and size to the pharyngeal outflow boundary.

Figure 24. Estimated Reynolds number based on unilateral peak flow (600 ml/s) and the unilateral hydraulic diameter of the open side (Equation (6)). Cross-section numbers refer to positions given in Figure 23.

Figure 24. Estimated Reynolds number based on unilateral peak flow (600 ml/s) and the unilateral hydraulic diameter of the open side (Equation (6)). Cross-section numbers refer to positions given in Figure 23.

Figure 25. Q-criterion iso surfaces (value of $3\cdot {10}^{6}1/s^2$) colored by velocity magnitude, at peak inspiration (left) and expiration (right) during the third respiratory cycle in in silico unilateral rhinomanometry, based on transient LES simulations on the fine mesh.

Figure 25. Q-criterion iso surfaces (value of $3\cdot {10}^{6}1/s^2$) colored by velocity magnitude, at peak inspiration (left) and expiration (right) during the third respiratory cycle in in silico unilateral rhinomanometry, based on transient LES simulations on the fine mesh.

Figure 26. Q-criterion iso surfaces (value of $3\cdot {10}^{6}1/s^2$) colored by velocity magnitude, at peak inspiration (left) and expiration (right) during the third respiratory cycle in in silico unilateral rhinomanometry, based on transient LES simulations on the fine mesh.

Figure 26. Q-criterion iso surfaces (value of $3\cdot {10}^{6}1/s^2$) colored by velocity magnitude, at peak inspiration (left) and expiration (right) during the third respiratory cycle in in silico unilateral rhinomanometry, based on transient LES simulations on the fine mesh.

III.2.4 Stress fields

Figure 27 displays static pressure contours on selected cross sections, for the LO-RC and LC-RO fine-mesh $k\omega$ SST simulations of peak inspiratory flow. In figures 28 and 29, the distributions of normal stress (static pressure) and wall shear stress on the nasal cavity wall are presented. The most notable pressure drops occur at the nasal gateways, coinciding with the smallest cross-sectional areas. These regions also exhibit the highest wall shear stresses, consistent with the identified $y^{+}$ hot spots in figures 14 and 15, due to the elevated flow velocity. The significantly higher pressure drop observed in the left nasal gateway, compared to the right, is attributed to the narrower passage on the left due to the deviating septum (see Figure 7).

Figure 30 and Figure 31 show the area-averaged static and total pressures at the cross-sections indicated in Figure 23 for peak inspiratory and expiratory flow in the fine-mesh LES and coarse-mesh laminar simulations, respectively. The averaging was exclusively based on the open nasal cavity. The figures reveal a generally good agreement between the laminar and LES-based models. Additionally, two notable observations emerge from these curves:

• Comparison of the total pressure development along the length of the nasal cavity indicates varying pressure losses and local flow resistance in different parts of the nose, depending on the flow direction. This suggests that the total unilateral nasal resistance, derived from the integral of the local resistances along the nasal cavity passage, may exhibit dependence on the flow direction.
• During inhalation, the stagnation pressure at the occluded nostril corresponds well to the static pressure in the nasopharynx for both sides of the nose. Conversely, during exhalation, a closer correspondence is observed between the stagnation pressure at the occluded nostril and the total pressure in the nasopharynx.

III.3 Discussion

The simulation of nasal airflow using CFD poses a challenging task owing to the intricate geometry of the nasal cavity, the dynamic nature of respiratory flow, and limited availability of validation measurements. Despite the growing popularity of CFD in computational rhinology, there remains a shortage of high-fidelity studies assessing the validity of prevalent modeling approaches, such as steady state laminar flow or RANS turbulence modeling, in nasal airflow.

This paper presents highly detailed nasal airflow simulations, representing some of the most comprehensive work published in this field to date. The aim is to contribute significantly to ongoing debates on optimal simulation set-up. Specifically, focus has been on establishing a benchmark for evaluating laminar and RANS-based models, addressing observed gross discrepancies between in vivo and in silico RMM. A key finding is the minimal variation between in silico RMM pressure-flow curves resulting from different turbulence models, including coarse-mesh steady state laminar, $k\epsilon$ realizable, and $k\omega$ SST and fine-mesh steady state $k\omega$ SST and transient LES. This challenges the study's initial hypotheses that the disagreement between in vivo and in silico RMM may be explained by poor choice of turbulence model, inadequate spatial/temporal resolution, or unsteady effects.

The following discussion delves into detailed considerations for each element of the initial hypothesis, followed by exploration of factors related to the nasal cavity geometry model. The section is concluded with some final remarks regarding the patient-specific RMM pressure-flow curves and the curve-fitting

Figure 27. Static pressure contour plots at selected cross-sections for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 27. Static pressure contour plots at selected cross-sections for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 28. Static pressure contour plots at the nasal cavity wall for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 28. Static pressure contour plots at the nasal cavity wall for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 29. Wall shear stress contour plots at the nasal cavity wall for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 29. Wall shear stress contour plots at the nasal cavity wall for a) LO-RC and b) LC-RO, steady state fine-mesh kω SST simulation of peak inspiratory flow.

Figure 30. Area-averaged static pressures computed at cross-sections indicated in Figure 23. The data were obtained from fine-mesh LES (solid line) and coarse-mesh laminar (X) simulations, at peak expiratory (black) and inspiratory (red) unilateral flow. The dashed, horizontal lines indicate the area-averaged static pressure at the occluded nostril, obtained from the LES simulation. The hatched area highlights the nasopharyngeal cross-sections.

Figure 30. Area-averaged static pressures computed at cross-sections indicated in Figure 23. The data were obtained from fine-mesh LES (solid line) and coarse-mesh laminar (X) simulations, at peak expiratory (black) and inspiratory (red) unilateral flow. The dashed, horizontal lines indicate the area-averaged static pressure at the occluded nostril, obtained from the LES simulation. The hatched area highlights the nasopharyngeal cross-sections.

Figure 31. Area-averaged total pressures computed at cross-sections indicated in Figure 23. The data were obtained from fine-mesh LES (solid line) and coarse-mesh laminar (X) simulations, at peak expiratory (black) and inspiratory (red) unilateral flow. The dashed, horizontal lines indicate the area-averaged static pressure at the occluded nostril, obtained from the LES simulation. The hatched area highlights the nasopharyngeal cross-sections.

Figure 31. Area-averaged total pressures computed at cross-sections indicated in Figure 23. The data were obtained from fine-mesh LES (solid line) and coarse-mesh laminar (X) simulations, at peak expiratory (black) and inspiratory (red) unilateral flow. The dashed, horizontal lines indicate the area-averaged static pressure at the occluded nostril, obtained from the LES simulation. The hatched area highlights the nasopharyngeal cross-sections.

of the Bernoulli equation (Equation (4), p. 14). The discussion should be understood in the context of the broader discussion provided in Section I.5.

III.3.1 Turbulence modelling

While the assumption of laminar flow has traditionally dominated CFD modelling of nasal airflow (refer to Part II), a consensus on whether its characteristic behavior is laminar, transitional, or turbulent remains elusive [83,84,85]. Published comparisons between laminar and turbulent modelling approaches have typically employed steady state RANS-based turbulence models relying on the Boussinesq hypothesis. A big disadvantage with this approach is its assumption that turbulent viscosity is an isotropic scalar quantity, which is not generally true. More sophisticated modelling approaches that account for transient phenomena or employ turbulence models such as Reynolds Stress Models (RSM), Large Eddy Simulation (LES), or Direct Numerical Simulation (DNS) tend to be computationally expensive. Few publications have investigated the effects of imposing unsteady flow boundary conditions, such as respiratory tidal flow, or applied these turbulence models to simulate nasal airflow. In addition to the present study, recent investigations utilizing finely resolved LES models with time-varying flow have been conducted by Lu et al. [249], Calmet et al. [250,251], Bradshaw et al. [159], and Hebbink et al. [127]. Other publications have also utilized high fidelity CFD models under steady flow conditions; refer to Table 1 (p. 41) for an overview.

The present paper's findings, asserting the similarity between laminar CFD models and more complex turbulence modeling approaches in predicting gross flow features, find support in several other studies. Li et al. [83] demonstrated that their laminar model exhibited good agreement with LES and DNS results at a volumetric flowrate of 180 ml/s. Calmet et al. [251] similarly concluded that a laminar flow model accurately predicted gross flow features, albeit with some underestimation of local flow fluctuations and turbulence intensity. Calmet et al. [251] and Bradshaw et al. [159] observed transitional and dominantly laminar nasal airflow, respectively. In the current study, the unilateral pressure drop exhibited low sensitivity to the choice of computational mesh or turbulence modelling approach on both sides of the nose across a wide range of volumetric flow rates.

Considering the calculated Reynolds numbers for the current patient-specific model alone, turbulent flow would be anticipated in both nasal cavities at peak flow-rates (refer to Figure 24). However, when factoring in the respiratory frequency of $f=0.2Hz$, the kinematic viscosity of air, and a hydraulic diameter of less than 1 cm into Equation (14) (p. 28), it becomes evident that the Womersley number for the current simulations was of the order of 1. This falls within the low range, as discussed by Xu et al. [168], suggesting a delayed transition to turbulence. These findings, coupled with the simulation results indicating near-laminar flow, illustrate that the critical Reynolds number for turbulent pipe flow is not a reliable indicator for selecting an appropriate turbulence model for respiratory nasal airflow.

In wall-bounded flows, there is a risk of underestimating turbulence intensity in RANS and LES-based CFD models if the wall boundary layer is not treated appropriately. The nasal cavity is characterized by intricate flow cross-sections featuring high aspect ratios between the axial, spanwise, and wall-normal length-scales. Hence, geometrical complexity alone makes it challenging to achieve a computational mesh that satisfies the wall requirements of conventional wall-function-based turbulence models. For varying volumetric flow rates, the challenge amplifies. It is, therefore, recommended to employ a computational grid that resolves the viscous sublayer and utilize $y^{+}$ insensitive turbulence models capable of accurate representation of the wall shear stresses. In the present study, these criteria were met for all turbulence models employed, indicating the attainment of accurate and reliable simulation results. A discussion of spatial and temporal resolution is provided below.

Lastly, it is acknowledged that turbulence requires initiation. This can occur through boundary effects such as wall roughness, turbulent fluctuations e.g. at inlets, geometric irregularities such as bends, edges, or steps, or flow obstacles such as nasal hair. If crucial turbulence triggers are missing from the model, predicted laminar-like flow might not accurately reflect the physiological situation. As shown in Figure 25, during exhalation, little vortical flow existed in the pharyngeal inlet section. The vortices generated at the inlet dissipated before reaching the nasopharynx. Nevertheless, the significant vortex production in the nasopharynx and nasal cavities implies that there were sufficient triggers present to generate turbulence in the nasal cavity.

III.3.2 Spatial and temporal resolution

The wall $y^{+}$-insensitive RANS models provided in ANSYS Fluent are expected to perform well in the wall $y^{+}$ ranges seen in the current simulations, both on the coarse and fine meshes. Using these RANS models avoids the problems associated with employing wall functions [112]. Wall $y^{+}<1$, which was seen almost everywhere at peak flow in the fine-mesh simulations, generally satisfies the requirement of fully resolved LES models [111], where the wall shear stress is computed directly from the laminar stress-train relationship [113].

The Kolmogorov length and time scales (eqs. (24) and (25)) offer a valuable metric for evaluating the spatial and temporal resolution in numerical models of turbulent flow [108,109,110]. At the Kolmogorov scale, viscous forces dominate, leading to dissipation of turbulent energy into thermal energy. These scales characterize the smallest turbulent eddies, and detailed models such as DNS must therefore resolve the Kolmogorov micro scales [263]. Kolmogorov's hypotheses [264] posit that turbulence is isotropic at the smallest scales, implying that the Boussinesq hypothesis can be reasonably accurate. Consequently, $y^{+}$ insensitive RANS- and LES-based models exhibit high accuracy, approaching DNS, when the spatial and temporal resolution is of the order of the Kolmogorov micro scales.

Figure 18 indicates that the majority of fine-mesh grid cells featured LES mixing lengths and timesteps smaller than the Kolmogorov length and time scales, respectively. The spatial and temporal resolution, comparable to the Kolmogorov micro scales, along with a finely resolved wall boundary layer, suggests that the fine-mesh LES simulations were finely resolved, albeit not fully, during peak flow. Because the Kolmogorov micro scales increase with decreasing velocities, the LES was most likely fully resolved at lower volumetric flow rates, however. As the computational mesh exhibited sufficient resolution to resolve the smallest turbulent eddies, and the temporal resolution effectively resolved the turbulent time scale, the LES can be considered near-DNS, within the nasal cavity. This view is further supported by the relatively small turbulent viscosity ratios depicted in Figure 19. Consequently, errors attributable to spatial or temporal resolution are deemed negligible in the fine-mesh LES.

It is noted, however, that the turbulent dissipation rate is not readily available from the present LES simulations, since it relies on statistical analysis of fully developed turbulent flow. In the current simulations, the volumetric flowrate was continuously varying due to the oscillatory nature of respiratory flow, and pseudo steady statistics were thus not available. Hence, the formulae for calculating the Kolmogorov micro scales (eqs. (24) and (25)) are not directly applicable for assessing the LES simulations. To address this limitation, the Kolmogorov micro scales used in figures 16-18 were derived from the steady state fine-mesh $k\omega$ SST simulations at peak inspiratory flow. These simulations assumed constant volumetric flowrate, hence pseudo-steady turbulence statistics. This approach, while a necessary adaptation, provides valuable insights despite the dynamic nature of the respiratory flow in the current simulations.

Rigorous statistical analysis of the LES data was not performed in the current study, but the simulation data has been made freely available to the public [14]. Other investigators are encouraged to examine and analyze the dataset. It is worth noting that in the continuously developing flow fields due to the oscillatory nature of respiratory flow, the steady state statistics of classical turbulence theory might not hold. The temporal (and spatial) development of the out-of-equilibrium turbulent energy spectrum, is a topic for further study [265].

III.3.3 Transient effects

The physiological respiratory cycle deviates from a perfectly sinusoidal function of time. Notably, there exists an inherent asymmetry between inhalation and exhalation, leading to a marginally higher peak inspiratory flow rate compared to the peak expiratory rate. Consequently, during exhalation, a plateau emerges, characterized by an almost constant flow rate. Realistic breathing cycles have been presented by e.g., Van Hove et al. [223], Calmet et al. [251], and Pawade [129]. In transient simulations, obtaining turbulence statistics benefits from prolonged intervals of near-constant flow rate, enabling a broader time-averaging window. With a continuously varying mean flow velocity, as in the current LES, turbulence statistics sampling becomes challenging with respect to computational cost, since it must rely on a high number of breathing cycles.

Turbulence evolves distinctively in continuously accelerating/decelerating flow compared to steady flow [167]. A period of near-steady flow, as seen in physiological expiration curves, could potentially promote turbulence development if the Reynolds number is sufficiently high. Nevertheless, in the current simulations, the RANS-based models, representing fully developed steady flow, contradict this notion concerning peak flow in the specific patient-specific model.

Inthavong et al. [105] proposed that eliminating start-up effects in transient simulations, starting from a quiescent state, required two to three breathing cycles, whereas Bradshaw et al. [159] dismissed the first cycle. In the current simulations, a fine-mesh steady state $k\omega$ SST model was employed to establish initial peak inspiratory flow conditions for the transient LES. Notably, no start-up effects were observed, and there were no discernible differences between successive breathing cycles. Consequently, it can be inferred that the start-up effects were either enduring or entirely absent.

The nasal airflow simulation results presented in this paper show that a series of steady state simulations closely replicated the transient simulation of the entire (sinusoidal) breathing cycle. Thus, for the specific patient considered, the respiratory nasal airflow could be characterized as quasi-steady. This observation aligns with expectations, given the low Womersley number (Equation (14), p. 28). Quasi-steadiness implies that the shape of the respiratory cycle is non-substantial. However, it is important to note that the width and shape of the unsteady hysteresis affecting RMM pressure-flow curves are influenced by the time-derivative of the volumetric flow rate, hence the slope of the breathing profile (refer to Section I.3.3 and Figure 13). It is underlined that the hysteresis loops affecting the pressure-flow curves resulting from the transient LES were solely due to the non-zero time derivative of the flow rate, not start-up effects or nasal compliance.

To sum up, while the present simulation data revealed some transient effects, their influence on the in silico RMM results was found to be minimal. Therefore, transient effects are deemed highly improbable as an explanation for the significant disparity observed between in silico and in vivo RMM results.

III.3.4 Geometry

One potential source of error in the comparison of in silico and in vivo RMM data, not investigated in this study, pertains to the consistency between the digital geometry model used in CFD simulations and the actual nasal cavity geometry during RMM. For instance, Aasgrav [4] highlighted the substantial impact of segmentation settings on nasal resistance by contrasting CFD models based on different HU threshold settings. Beyond this, there are additional points that warrant discussion in relation to patient-specific geometry models. This section aims to underscore key aspects to be mindful of when evaluating the congruence between the digital representation and the real anatomical features. Refer also to the broader discussions in sections I.4.3, I.5.2, and I.5.4.

A notable deviation from realistic airway geometry in the current digital patient-specific model is the truncation of the geometry at the nostrils and the subsequent truncation and extrusion of the pharyngeal tract. The former restricts the faithful representation of airflow distribution across the open nostril during inhalation and exhalation, as well as the turbulent intensity during inhalation. However, Taylor et al. [180] observed that overall flow patterns and measures were insensitive to the detailed prescription of inflow conditions at the nares. The latter predominantly affects the turbulent intensity during exhalation, as illustrated in Figure 25, where the flow featured minimal vorticity when it entered the nasopharynx, during exhalation. This indicates that production of vorticity and turbulent structures was delayed until the nasopharynx and nasal cavity. While the precise influence on simulation results remains unclear, this observation aligns with Bradshaw et al.'s findings [159], underscoring the importance of accurately describing and including the pharyngeal tract to model vortex generation and transport during both inhalation and exhalation. Neglecting the paranasal sinuses is consistent with the perspective of several other researchers [126,159,170,171].

When exploring additional factors that could influence the comparability of the patient-specific geometry employed in in silico RMM and the actual airway geometry during in vivo RMM, two primary considerations emerge: 1) the condition of the nasal cavity during CT image acquisition, and 2) the impact of nasal compliance.

• During in vivo RMM, nasal cavity decongestion was achieved through the application of xylometazoline. This is expected to maximize the nasal cavity volume, through reduction of turbinate swelling, and minimize the nasal resistance. However, the CT image acquisition, basis for the digital patient-specific geometry model, took place in the natural, non-decongested state. As a result, it was subject to the nasal cycle and various factors affecting spontaneous turbinate and nasal mucosa swelling. For instance, in the natural non-decongested state, postural effect on the nasal resistance and nasal cycle may be anticipated [33,133,134]. The nasal resistance is expected to be higher in the supine position, in which CT images were obtained, than in the sitting position, in which the in vivo RMM data were obtained. The lack of decongestion and the postural effect are both expected to increase the nasal resistance. Hence, the correction of these sources of error would presumably reduce the nasal resistance predicted by the in silico RMM, further increasing the disagreement with the in vivo RMM data.
• It has been proposed that nasal compliance may affect RMM curves (see Section I.5.1), and that rigid CFD geometries will fail to reproduce in vivo RMM curves due to this. However, for the current patient, it is observed that the patient-specific in vivo RMM curves are almost symmetrical with respect to in/exhalation (see Figure 12a). At the same time, particularly on the left side, the in vivo RMM pressure-flow curve plateaus, suggesting a significant increase in nasal resistance beyond a critical flow rate. To explain these observations by nasal compliance, a collapsible constriction is required, where the Venturi effect dominates over the static pressure in such a way that the collapse is independent of the flow direction. Without such a constriction, asymmetrical collapse would be expected due to the under/over pressures in the nasal cavity during the inspiratory and expiratory phases, respectively. For instance, the phenomenon of nasal gateway collapse exemplifies this, where the collapse occurs exclusively during inhalation [266,267].

The first point suggests that the discordance between in silico and in vivo RMM may be more prominent than indicated by the current findings. On the other hand, the second point provides a plausible rationale for the significant misalignment between simulations and measurements. This contradicts the conventional notion that a decongested nasal cavity retains rigidity, calling for robust evidence to demonstrate the (partial) symmetrical collapse of certain nasal cavity parts during both inhalation and exhalation.

For future studies comparing in silico and in vivo RMM, it is recommended to exercise caution in ensuring that the patient-specific geometry model accurately represents the airway geometry during in vivo RMM. This can be achieved by minimizing the time gap between clinical RMM and CT data acquisition and ensuring that both are obtained in a decongested state, preferably in the same position (either supine or sitting). This approach will contribute to reducing a major source of error in the analysis of in silico nasal airflow data.

As a final point, the role of nasal hair should be considered. Hahn et al. [151] and Stoddard et al. [174] presented differing conclusions regarding the significance of nasal hair. Hahn et al. found little impact on overall airflow, while Stoddard et al. observed a positive effect on both subjective and objective measures of nasal obstruction following the removal of nasal hair, suggesting its noteworthy contribution to nasal resistance. A recent CFD study by Haghnegahdar et al. [268] indicated that nasal hair significantly influenced nasal airflow patterns, although they did not report nasal resistance. The potential impact of nasal hair on nasal resistance appears to warrant increased attention.

III.3.5 Final Remarks and Observations Regarding the Analysis of Rhinomanometry

Pressure measurements

In the course of anterior Rhinomanometry (RMM), the assessment of nasopharyngeal pressure involves measuring the stagnation pressure at the occluded nostril. This pressure is then used as an approximation of the nasopharyngeal pressure. The differential pressure between the open nostril and the occluded nostril thus serves as an indicator of the pressure drop across the nasal cavity, facilitating the calculation of nasal resistance (refer to Section I.3 for more details). However, the simulation findings discussed in the preceding section (see figures 30 and 31) indicate that interpreting these measurements may not be straightforward. Firstly, the results indicate that nasal resistance could be influenced by the direction of airflow (inhalation versus exhalation). Additionally, it is noted that the measured stagnation pressure corresponds primarily to the static nasopharyngeal pressure during inhalation, while during exhalation, it aligns more closely with the total pressure.

Anticipation of the former aligns with classical potential theory, where symmetry in flow direction reversal only occurs in ideal situations. As illustrated in figures 25 and 26, the vorticity of nasal airflow is substantial, and non-recoverable pressure losses, such as those caused by vortex shedding, are expected to vary based on the flow direction, particularly when there are abrupt changes in cross-sectional areas or flow direction [81].

The latter phenomenon can be elucidated using principles akin to those governing airspeed measurements with Pitot tubes in, for example, aviation (see

Figure 32). In this context, when the flow of air is directed towards the cavity with its opening facing the flow, it is deflected due to a force exerted by the fluid inside the cavity. This force is counteracted by the stagnation pressure at the cavity's bottom. Conversely, for the cavity with its opening facing the opposite direction, the wake behind the cavity may induce a suction force, reducing the pressure at the cavity's bottom. The observed disparity in simulation results between the static pressure at the occluded nostril and the total pressure in the nasopharynx during exhalation can be attributed to flow curvature. As the airflow enters the nasopharynx, it "bends" towards the open nasal cavity, and vortical structures penetrate into the occluded nasal cavity (see figures 25 and 26). These factors necessitate the application of a force that influences the stagnation pressure. In the context of RMM, this implies a potential overestimation of nasopharyngeal pressure during exhalation, leading to an underestimation of pressure drop and nasal resistance.

In standard RMM, there is unfortunately insufficient information to rectify the inherent overprediction of nasopharyngeal pressure during exhalation. However, if the cross-sectional area of the open nostril were measured, it would allow for the estimation of total pressure at the nostril, enabling the calculation of pressure drop as a change in total pressure. In channels with varying cross-sectional areas, recoverable changes in static pressure may result from the acceleration or deceleration of the flow. Conversely, changes in total pressure are solely due to non-recoverable losses, suggesting that total pressure measurements may provide a more reliable basis for estimating nasal resistance compared to static pressure measurements.

From this discussion, it becomes evident that interpretation and application of standard clinical anterior RMM examinations should consider two key factors: 1) the physical nasal resistance may be influenced by the direction of airflow, and 2) exhalatory pressure measurements are likely to underpredict the pressure drop and unilateral nasal resistance.

The use of Bernoulli's equation

The patient-specific clinical in vivo RMM pressure-flow data presented in Figure 1 shows that there was significant effect on the nasal resistance by decongesting the nose, on both sides. However, the data exhibits significant variability, with evident large hysteresis loops affecting each breathing cycle. Particularly on the left side, where the nasal resistance was highest, the scatter in differential pressure values span a wide range, for given volumetric flowrates. The characteristic loops observed in the in vivo RMM pressure-flow curves have been previously discussed by Vogt et al. [77,139], who proposed that nasal compliance contributes to the hysteresis, and Groß and Peters [140], who suggested that measurement techniques in RMM may be a contributing factor. In Section I.3.3, it was illustrated how hysteresis might result from unsteady flow, alone.

The representative in vivo RMM curves presented in Figure 1 were created by manual placement of pressure-flow points, to represent the minimal nasal resistance observed in the data. It is evident that this representative nasal resistance significantly underestimates the actual nasal resistance during certain recorded breathing cycles.

The discrepancy between in silico and in vivo RMM is evident not only in terms of nasal resistance but also in other aspects. First, there is much less scatter in the in silico pressure data. Although pressure fluctuations are present in the LES-based pressure-flow curves (see figures 8 and 10), they manifest as a noisy signal typical of turbulent fluctuations. Additionally, the tight hysteresis envelopes inherent in the LES data appear quite different from the large hysteresis loops observed in the in vivo data (compare figures 1 and 13).

Excellent curve-fits of Bernoulli's equation (Equation (4), p. 14) were obtained for both in silico and in vivo RMM pressure-flow curves (refer to figures 12 and 13). Deviation from the straight lines associated with laminar flow in a straight tube, have been attributed to the transition to turbulent flow [78]. Such behavior was observed both in in vivo and in silico RMM curves presented here. The present CFD simulations have, however, provided compelling evidence that the flow was laminar. Conversely, the optimal curve-fit was obtained using the (turbulent) friction factor obtained from the Haaland equation (Equation (7)) regardless of Reynolds number. These findings suggest that considerations of turbulence alone may not be sufficient for assessing RMM pressure-flow curves. Without delving into an exhaustive analysis, it is suggested that the apparent turbulent behavior, when assessing the Bernoulli equation curve-fit in isolation, may be influenced by other factors related to the deviation between the nasal cavity and a straight pipe of constant diameter. For instance, the non-circular shape of the nasal cavity, numerous abrupt changes in cross-sectional area, bends, and curves, along with the presence of nasal hair and mucosal lining, may introduce minor friction losses that should be accounted for [81]. Additionally, the impact of unsteady flow, such as significant vortex shedding, as illustrated in figures 25 and 26, and the possibility of nasal compliance should be considered.

Nomenclature

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