Fluid Mechanics Based Classification of the Respiratory Efficiency of Several Nasal Cavities

Transcription

1 Fluid Mechanics Based Classification of the Respiratory Efficiency of Several Nasal Cavities Andreas Lintermann a,,matthiasmeinke a,wolfgangschröder a a Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 5262 Aachen, Germany Abstract The flow in the human nasal cavity is of great importance to understand rhinologic pathologies like impaired respiration or heating capabilities, a diminished sense of taste and smell, andthepresenceofdrymucousmembranes. Tonumericallyanalyze this flow problem a highly efficient and scalable Thermal Lattice-BGK (TLBGK) solver is used, which is very well suited for flows in intricate geometries. The generation of the computational mesh is completely automatic and highly parallelized such that it can be executed efficiently on High Performance Computers (HPC). Anevaluationofthefunctionalityofnasal cavities is based on an analysis of pressure drop, secondary flow structures, wall-shearstressdistributions, andtemperature variations from the nostrils to the. The results of the flow fields of three completely different nasal cavities allow their classification into ability groups and support the aprioridecision process on surgical interventions. Keywords: Lattice Boltzmann, Thermal Lattice Boltzmann, Respiration capability,heatingcapability,nasal cavity flows. Introduction The functionality of the human nasal cavity is crucial for the comfort of the individual person, since it not only plays aroleinolfactionanddegustation,butitalsofilterstheair from particles and moisturizes and heats it to achieve optimal fluid conditions in the lung. As a consequence, a reduction of these functions causes subjective pathological complaints by the individual patient. Functional degradations are not only generated by simple coryza or allergic reactions, but can also occur due to malformations or deformations of the geometric shape of the nasal cavity and as such reduce the inspiration and heating capabilities. In some cases surgical interventions are required to reestablish certain functionalities by geometric optimization of the human nasal cavity. Surgery planning is a challenging task in rhinology and is usually performed based on expert knowledge. To support this decision process, the flow characteristics in the nasal cavity can be numerically analyzed in detail. This development has led to the field of Computer Assisted Surgery [], which allows to better understand the intricate flow physics in a human nasal cavity. First experimental investigations based on a model of the nasal cavity were performed by Masing [2] and later on complemented by Brücker et al. [3] by measuring the spatial velocity distribution. Highly resolved numerical simulations in the same model of the nasal cavity as that in [3] were conducted by Hörschler et al. [4, 5] using an Advection Upstream Splitting Method (AUSM)-based Finite-Volume Method (FVM) of second-order accuracy on a multi-block structured grid. Recent simulations with commercial flow solvers were presented in [6 ]. Naftali et al. [6] and Elad et al. [7] use FLUENT to simulate the unsteady pulsatile three-dimensional inspiratory flow in a nose-like and in an anatomical model obtained from medical images. Assuming laminar flow they investigate the air-conditioning capacity [6] and the influence of wall-shear stress [7] during inhalation. The results in [6] show that the models can provide up to 9% of the heat flux and moisturization required to achieve sufficient breathing conditions and that the endonasal structural components play a major role in this process. From the investigations by Elad et al. [7] it is known that the wall-shear stress in the simulations reaches values on the order of Pa and that locally even higher stresses can occur, which could effect the functional performance of endothelial and epithelial cells [, 2]. Zachow et al. assumed turbulent flow [8] and use ANSYS CFX with an elementbased FVM and the Shear-Stress Transport (SST) turbulence model [3] for the Reynolds-averaged Navier-Stokes (RANS) equations to predict the flow in a nasal cavity extracted from Computer Tomography (CT) images. This study delivers a proof of concept without discussing the respiration and heating capability in detail. Riazuddin et al. [] used FLUENT to solve the RANS-equations closed by the SST turbulence model for a nasal cavity. Under a variation of the Reynolds number, the velocity, the resistance, the wall-shear stress, vortex formations, and turbulence intensities were investigated for the in- and expiration phase. The results of this study show that the resistance is greater in the inspiratory phase than in the expiratory phase, where the mixing due to turbulence is higher. Hörschler et al. [4] addressed the question of laminar or turbulent flow and performed a highly resolved simulation of the flow through a model of the human nasal cavity without incorporating any kind of turbulence model and obtained a good agreement with experimental findings for the primary flow structures. That is the nasal cavity flow which Corresponding author. Tel.: ; fax: addresses: A.Lintermann@aia.rwth-aachen.de (Andreas Lintermann), M.Meinke@aia.rwth-aachen.de (Matthias Meinke), office@aia.rwth-aachen.de (Wolfgang Schröder) Preprint submitted to Computers in Biology and Medicine December 5, 26

2 is spatially and temporally in a transitional state, could be well described by a sufficient resolution without influencing the numerical solution by turbulence modeling. The aforementioned investigations focused on model-like geometries or single real nose configurations. They do lack the comparison of the air-conditioning functionality and wallshear stress distribution for a variety of anatomically correct nasal cavities and hence, can be considered individual solutions. Yu et al. [9], however, simulate unsteady in- and expiratory flows for 24 three-dimensional models obtained from medical images, three solutions of which are discussed in great detail. They use ANSYS to solve the RANS-equations and determine the pressure drop and the volume flux depending on the pressure gradient. The simulations are based on a fully turbulent flow assumption, whereas the flow in the nasal cavity is mostly in the laminar or transitional regime [4, 5]. Nevertheless, it is fair to state that RANS-based simulations seem to provide reasonable integral results only under certain conditions since turbulence models were developed for different flow regimes. That is regarding the intricacy of human nasal cavities and a detailed analysis of the resulting flow structure either a higher fidelity turbulence model that represents the local variance of the flow structure is necessary or a higher resolved direct-numerical-simulation-like approach is to be used to capture the time-dependent flow physics inside the nasal cavity. Therefore, following the findings of Hörschler et al. [4] highly resolved numerical simulations using a Lattice- Boltzmann Method (LBM) are performed in this investigation. Furthermore, the volume flux will be in a range such that the Reynolds number Re which is proportional to the volume flux will be approximately Re 9 and only inspiratory flow will be analyzed, i.e., the transition phase from inspiration to expiration or vice versa will not be considered. Hence, considering the results discussed in detail in [5] only steady inhalation will be computed since in the Reynolds number range of Re 9 unsteady effects can be neglected on an integral scale. Unlike in former studies [5 8,, 4, 5] the flow structures of three completely different nasal cavities are investigated and a classification of the respiratory efficiency based on the flow and the temperature fields is suggested. This article is organized as follows. The numerical method, the boundary conditions, the grid generation process, and the quantities to analyze the flow in the nasal cavity are discussed in Sec. 2. In Sec. 3 the three different nasal cavities are presented and subsequently, the flow fields of the varying configurations are juxtaposed. That is the pressure drop, the overall three-dimensional flow characteristics, the wall-shear stress, and the heat transfer are analyzed. Based on an evaluation of the various distributions a tentative classification of the respiratory efficiency is given. Finally, some conclusions are drawn in Sec Numerical method To simulate the flow in the nasal cavity a Lattice Boltzmann Method (LBM) is used since it is well known that such an approach has some major advantages over Finite-Element (FE) or Finite Volume (FV) methods when highly intricate geometries are considered [6]. Among other applications [7] the LB method has been proven to be well suited for biomechanical flow problems in the moderate Reynolds number (,, ) ; (,, ) ; (,, ) α =...5 ξ i = ξ (,, ) ; (,, ) ; (,, ) α =6...7 (,, ) α =8 Figure : D3Q9 model for phase space discretization. In three dimensions 9 directions are specified to model molecular collision and propagation processes. Table : Computation of macroscopic variables from moments of the PPDFs. variable discrete moment density ρ = 8 i= fi = 8 i= f eq i momentum ρv α = 8 i= ξi,αfi = 8 eq i= ξi,αfi temperature T = 8 i= gi = 8 i= geq i regime, i.e., for flow simulations in the human lower respiratory system [8, 9]. In addition, a good parallel scale-up and straightforward boundary treatment makes the LB method attractive to simulate nasal cavity flows as it has been shown by Finck et al. [5] and Eitel et al. [2]. The LB code used in this study has been previously used and validated by Freitas et al. [8] and Eitel-Amor et al. [7]. In the following, a brief description of the LB method is given, the mesh generation process is described, and the numerical approaches to analyze and categorize nasal cavity flows is presented. 2.. Governing equations and notations The Lattice Boltzmann Method is a gas-kinetic stochastical approach to simulate continuum flows by numerically solving the Boltzmann equation, i.e., by solving for the particle probability functions f i (PPDFs) in the Lattice-BGK equation f i ( x + ξ iδt, t + δt) =f i ( x, t )+ωδt (f eq i ( x, t ) f i ( x, t )) () in three dimensions with the D3Q9 model [2] shown in Fig.. A detailed description of the quasi-incompressible LBM can be found in [7, 8, 22]. To determine the temperature field a multi-distribution function approach (MDF) described in [23 26] is used, where in addition to Eq. a passive scalar transport equation for the temperature is solved by coupling the Thermal Lattice-BGK equation g i ( x + ξ iδt, t + δt) =g i ( x, t )+Ω T (g eq i ( x, t ) g i ( x, t )), (2) to Eq.. The macroscopic variables are obtained by the moments of the PPDFs f i and g i as listed in Tab.. 2

3 2.2. Boundary conditions To determine the respiration and heating capability of the nasal cavity at inspiration, the following boundary conditions at the inlets, i.e., at the nostrils of the nasal cavity, at the outlet, i.e., at the, and at the walls are defined. Note that in the following discussion variables labeled with an overbar, e.g. v, areconsideredtobedimensional,whilevariables without an overbar are considered to be non-dimensional. At the inlets a formulation introduced in [5, 4, 27] is used. It is based on the well-known equation of Saint-Vernant and Wantzel [28] to determine the isentropic expansion speed v [ ( ) γ ] v = 2γ p p γ, (3) γ ρ p where the subscript denotes the stagnation state, γ = c p/c v is the isentropic exponent defined by the ratio of the specific heats, p is the pressure, and p and ρ are the reference pressure and density. Rearranging Eq. 3 leads to Using the equation of state ( p = p γ ) γ ρ v 2 γ. (4) 2γ p p = ρr T = ρ c s 2, (5) with R being the specific gas constant, T the temperature, and c s the speed of sound, and introducing dimensionless parameters ρ = ρ/ ρ, c s = c s/ ξ and v = v/ ξ,where ξ is the molecular velocity, results in ρ = ( γ 2γ ( v c s ) 2 ) γ γ. (6) Applying c s =/ 3 to Eq. 6 yields the iterative formulation for the dimensionless density ( ρ ι = γ 2γ 3 ρ 2 ι (ρ ι v ι ) 2 ) γ γ for iteration step ι. Notethatthemomentumρ ι v ι is linearly extrapolated from the next inner cell layer to the outer layer. Thus, using Eqs. 3 and 7 the velocity and the density are known at the inflow boundary. Furthermore, the dimensionless temperature at the inlets is set T =, i.e., the equilibrium distribution functions are determined for a given temperature of T =293.5K, which also defines the reference temperature in the simulation. At the outlet the pressure is prescribed by using the isentropic pressure-density relation (7) p = c 2 sρ = ρ. (8) 3 The flow field is initialized by setting ρ =suchthatimposing ρ<attheoutletleadstoareductionofthepressureand to an increase of the velocity based on the difference of the PPDFs in neighboring cell layers. Note that the LB method is a low Mach number method such that small density variations can occur. The velocity distribution, however, has to match the volume flux that is relevant for the respiratory flow problem. Considering the proportionality between the volume ρ ι e+4 4e+4 6e+4 iteration step ι ρ(n g ) ρ(n m ) ρ(n b ) Re l (N g ) Re l (N m ) Re l (N b ) 8e+4 e Figure 2: Time-dependent adaptation of the density ρ and the local Reynolds number Re l in the using Eqs. 9 and 2. The quantities N g, N m,andn p denote different nasal cavity geometries. flux and the Reynolds number, which is defined by the mean velocity v avg, the hydraulic diameter D h, and the kinematic viscosity ν, i.e.,re = v avg D h / ν, aniterativeprocedureis imposed to determine the pressure in the or via Eq. 8 the density that fits the volume flux and the Reynolds number, respectively. The following algorithm varies the density as a function of the differences Re = Re Re l,ι,wherere is the physically relevant Reynolds number and Re l,ι is the local Reynolds number at iteration step ι { Re >,ρ ι δ ρ ρ ι =, (9) Re <,ρ ι + δ ˆρ using ( δ ρ = min ( δ ˆρ = min ρ max, ρ max, 5 Re l ) Re ρ ι Re Re l,ι () ) Re ρ ι Re l,ι Re l,ι, () where ρ ι = ρ ι ρ ι 2 is the density iteration difference from the previous iteration. The iteration is repeated until the density distribution leads to the defined Reynolds number and volume flux, respectively. Note that for stability reasons the step sizes δ ρ and δ ˆρ are bound by the term ρ max such that a smooth transition to the target Reynolds number is achieved. Furthermore, the Reynolds number is increased as a function of time via a scaling function λ(t) Re = Re + λ(t) (Re f Re ), (2) where Re is an initial Reynolds number, Re f is the target Reynolds number, which is defined by the hydraulic diameter D h,p of the, the kinematic viscosity of air, and a volume flux of 25 ml/s, which corresponds to an average flow rate for adults during inhalation at rest [6, 29]. The temporal scaling function λ(t) isdefinedby λ(t) = 2 [ ( tanh ( ) tanh ( t t n t ) 5 ) ] (3) The parameters t and t define the initial and current time level and n t represents the number of iteration steps the temporal scaling function λ(t) isapplied. Fig.2showsthatthe 3

4 function λ(t) allowsasmoothtransferfromre to Re f and also for the density in the as a function of time. The no-slip wall boundary condition is based on the interpolated bounce-back modeling for inclined walls [3] for the velocities, while for the temperature the equilibrium PPDFs for a specified body temperature of T b =39.5K and the dimensionless notation T b = T b /T =.546 is used Mesh generation Complex geometries like the human nasal cavity are extracted from Computer Tomography (CT) images and are available in Standard Tessellation Language (STL). Toobtain asurfacefromct-data,theregionofinterest,i.e.,thevolume of air specified in the CT-image via a range of density values, is marked in a segmentation step using a recursive region growing algorithm [3]. The two-fold partitioned image is contoured using the Marching Cubes algorithm [32], which generates a two-manifold closed surface of the volume of air. A realistic surface is obtained by using a Windowed Sinc Smoother as proposed by Taubin et al. [33] to smooth high frequent surface fluctuations generated by the cubic structure of the voxels in the partitioned image. The Cartesian Meshes are based on an octree structure implied by the hierarchical, iterative subdivision of an initial cube surrounding the geometry of interest [34], i.e, the surface of the human nasal cavity. The generation process is parallelized and is performed completely automatically. In an initial step, each processor initially refines a minimal surrounding cube around the geometry to a certain level l min by continuously decomposing each cube into eight sub-cubes defining parent-child relations. This process constitutes the octree structure of the grid. Cells outside the fluid domain are deleted and do not participate in the subdivision process anymore. Then, domain decompositioning is performed by a Hilbert decompositioning method using space filling curves as described by Sagan et al. in [35]. By keeping only its corresponding cells each process continues the subdivision process and the removal of outliers until a starting-grid level l start is reached. At this stage of the algorithm all the active computational cells, i.e., the leaf nodes of the octree, are on the same level l start. Furtherrefinementtoafinallevell final introduces alocallyrefinedgridthatenablestheresolutionofhighshear regions such as wall-bounded or free-shear layers. In the case of refinement a bandwidth b defines the wall-distance in cell units in which the level l final should be reached. A subsequent smoothing step guarantees a level difference l of at most between a cell c on level L(c) atalevelinterfaceand its neighboring cell N (c) onadifferentlevel l = L (c) L(N (c)). (4) Smoothing across domain boundaries is controlled by communicating local changes to the neighboring domains, in which the smoothing process continues. The resulting grid is stored by efficient parallel I/O routines using the NetCDF format [36]. Grid cells on the order of O ( ) can be generated by this method in less than one minute of computing time on O ( 4) cores Flow field analysis The discussion of the flow fields of various human nasal cavities will be primarily based on the pressure loss, the heating capability, vortical flow structures, and the wall-shear stress. Consequently, these distributions will be also used to evaluate the respiration efficiency of the configurations. To clearly define how these quantities are determined the essential equations are given next. Pressure loss. The inspiration capability is evaluated by the dimensionless total pressure loss between the left l (δp l ), respectively the r (δp r), and the at the peak volume flux. Considering the pressure-density relation in Eq. 8 the static pressure difference is defined by p x = p x p p = ρx ρp, (5) 3 where x [l, r] andthesubscriptp denotes the. According to a generalized form of the Bernoulli equation, the total pressure p t is determined by p t,x =(p x + p d,x )+δp x = p f,x + δp x, (6) where p f,x consists of the static pressure p x and the dynamic pressure p d,x =(ρ/2) v 2 and the total integral pressure loss is denoted by δp x.thetotalintegralpressurelossfortheleft and right nasal cavity is defined by δp t,l = Ā l δp t,r = p f,l dā p f,p dā Ā l Ā (7) p Ā p p f,r dā p f,p dā, (8) Ā r Ā p Ā p Ā r where Āl and Ār are the surface areas of the left and right nostril and Āp is the surface area of the. The specific total pressure loss on a characteristic streamline between the nostril cross section s/s t =andthearbitraryposition s/s t on the streamline, where s t is the total arc length of the streamline, reads p t(s/s t)=p f () p f (s/s t). (9) Vortical flow structures. The vortical structures are determined by the -criterion [37] ( ) 3 [ ] 2 Q det ( v ) c = + >, (2) 3 2 which is based on the Q-criterion [38] 2 Q = ( Ω 2 S 2) >, (2) 2 [ ] v +( v ) T is the strain tensor, Ω = where S = 2 [ ] v ( v ) T is the vorticity tensor and denotes the Frobenius norm. Notethatthedimensionlessvelocityvector v = ( v / ξ, v / ξ, v 2/ ξ ) is used to obtain c. Contoursof c are either colored by using the definition of the turbulent kinetic energy ( ) v 2 + v 2 + v 2 2 k = (22) 2 to highlight fluctuation related structures or by the vorticity v. To determine the impact of eddies generated in the nasal cavity on the downstream flow, the discrete crosscorrelation ˆR xy(m) = { N m n= x n+my n, m ˆR yx( m), m < (23) 4

5 computational grid nasal cavity surface computational grid nasal cavity surface τ w,l q l v α main flow direction T α = T b T v β T β normal n l v γ normal n l T γ x x velocity profile temperature profile Figure 3: Computation of the local wall-shear stress and the local heat flux based on a polynomial representation of the primitive variables. Wall-shear stress computation at vertex l using a polynomial representation of the velocity distribution. Heat flux computation at vertex l using a polynomial representation of the temperature distribution. of the time signals x and y of the dimensionless velocity v at distinct locations is evaluated over N time steps. In this equation m denotes the time-lag between the signals. Additionally, the rms-values of ˆR xy ˆR xy,rms = ˆR xy 2 (24) are used to compare the results at different locations. Considering normalized autocorrelations { N m R xx(m) N n= x n+mx n, m = (25) R xx( m), m < eddy turnaround times τ e are determined by integrating R xx in the time interval [,t ], where t is the first root of R xx τ e,x = t R xx(m)dm. (26) Furthermore, unsteady secondary flow structures are analyzed by Welch s method [39] for the power spectral density PSD of the change of the velocity-components of the dimensionless velocity vector v over time. Wall-shear stress. To analyze local separation regions and pronounced shear load on the walls the wall-shear stress distribution is determined. The computation of the wall-shear stress is based on a polynomial representation of the velocity profile. To reduce the number of cells in the original mesh, a layer possessing a thickness of of three cell diameters normal to the nasal cavity surface is defined. As indicated in Fig. 3 the velocities v k,k {α, β, γ} are sampled at three equidistant locations along the inward facing normal n l of vertex l of the triangular representation of the nasal cavity surface at a distance of δx. Thevelocitycomponentsaretrilinearlyinterpolated to second-order accuracy using the values at the cell centers. At the surface the no-slip condition, i.e., v α =,is satisfied and a second-order polynomial v t(x) =αx 2 + βx + γ is used to describe the tangential velocity distribution at the sampling position x on the wall normal n l such that the wallshear stress τ w,l = η v l n l (27) x= can be determined. The wall-shear stress per triangle j, averaged by its number of vertices and its area A j is given by τ w(j) = Āj 3 τ w,l. (28) The average wall-shear stress in the nasal cavities is computed by τ w,avg = τ w(j), (29) L λ where L λ denotes the total number of triangles for the various configurations λ. Tocomparetheresultsforthedifferent nasal cavities the local derivatives are scaled by the grid distance δx. Additionally,thedistributionofthewall-shearstress over the surface is discussed using a binning c τ at an interval difference of ϖ τ and an index ς τ normalized by the number of surface vertices P λ for the various configurations λ P λ l= χ τ,l c τ =, (3) P λ where χ τ,l = {, τ w,l [(ς τ ) ϖ τ,ς τ ϖ τ ], otherwise l j. (3) Heating capability. The heating capability is determined by the dimensionless temperature increase δt = T p,da dā Ti = Tp T (32) Ā p Ā p from the nostril given by the left and areas A l, A r with a total area of Ā t = Āl + Ār and the local temperature of T i = T = to the at an area Āp and the local temperature T p. To analyze the influence of the surface area on the heating capability the heat flux at the wall, i.e., x =,iscomputedby using the spatial derivative of the temperature in the wall normal direction n l for triangle j defined by its vertices l weighted by the triangle area Āj q Ā,j = Āj 3 l=2 l= q l, (33) 5

6 where q l = κ T n l (34) x= is the heat flux at vertex l. Similartothecomputationofthe wall-shear stress, the heat flux distribution at sampling points x along the surface normal n l is represented by a secondorder polynomial (see Fig. 3) and at the iso-thermal wall T α =.546 is satisfied. To show the dependence of the heating capability on the surface area, the surface-area-averaged heat flux in the nasal cavities is computed by q avg = q A Ā,j. (35) Again, the distribution of the heat flux over the surface is discussed using a binning c q over all vertices l at an interval difference of ϖ q and an index ς q normalized by the total number of vertices P λ for the different configurations λ P λ l= χ q,l c q =, (36) P λ j where χ q,l = {, q l [(ς q ) ϖ q,ς q ϖ q], otherwise 3. Results and discussion. (37) The numerical simulation of the respiratory flow in a nasal cavity allows the analysis of fundamental flow properties like the pressure loss, the heating capability, the wall-shear stress, and vortical flow structures. The evaluation of the flow field of single solutions of solely one specific nasal cavity configuration delivers only a limited insight on the overall problem and no fluid-mechanics based classification can be given. Therefore, an analysis and comparison of three different nasal cavities is performed in this study. A juxtaposition of the fundamental Table 2: Structural features and categorization of the nasal cavity geometries. The notation left (l) and right (r) refers to the patient s point of view. The subscripts n and p define the surface area Ā of the in- and outlet geometry (nostril and ) and the hydraulic diameter D h. label category structural features Ā[m 2 ] Ā l n[mm 2 ] Ā r n[mm 2 ] Ā p[mm 2 ] Dh,p [mm] (pers. eval.) N g good - slightly converging left nasal cavity N m medium - extreme deviation swollenlowerturbinateintheright nasal cavity N p poor - small hydraulic diameter of the missinglowerturbinateintheright nasal cavity -missingcenterturbinateintheleft nasal cavity -widechanneltotheleftparanasalsinus -perforation - notch in area (c) slightly narrowed left channel swollen niche like regions missing center turbinate perforation wide channel to paranasal sinus swollen deviation missing lower turbinate Figure 4: Frontal view on CT-slices of the nasal cavities N g, N m,andn p. The cavity N g has only a slightly narrowed left channel. The cavity N m suffers from a deviation and swollen turbinates, while cavity N p (c) has a perforation and missing center and s, which have been removed in surgery. 6

7 left paranasal sinus s left nostril right paranasal sinus left paranasal sinus right nostril right paranasal sinus (c) (d) niche like areas niche like areas left paranasal sinus swollen lower turbinate right paranasal sinus swollen lower turbinate deviation deviation left paranasal sinus right paranasal sinus (e) (f ) left lower turbinate left paranasal sinus missing center turbinate missing lower turbinate right paranasal sinus left paranasal sinus notch right paranasal sinus perforation Figure 5: Surfaces of the nasal cavities extracted from the CT-images shown in Fig. 4. Profile view of nasal cavity Ng. Frontal view of nasal cavity Ng. (c) Profile view of nasal cavity Nm. (d) Frontal view of nasal cavity Nm. (e) Profile view of nasal cavity Np. (f) Frontal view of nasal cavity Np. flow properties leads to the classification of the geometries. The analysis of the flow properties is based on time-averaged solutions. Considering the mean velocity it takes approximately tc = 5 4 iteration steps to cover the distance from the nostrils to the. Therefore, the initialization and averaging time span consist of tc, = 4tc and tc,2 = 2tc iter- ation steps. In the following, first the problem is defined by discussing the various nasal cavity geometries and the mesh resolution of the numerical analysis. The subsequent discussion consecutively juxtaposes the results of the three geometries for the total pressure loss, the vortical flow structures, the wall-shear 7

8 stress, and finally the heating capability. 3.. Problem definition The flow fields of three nasal cavities whose CT-data and surfaces are shown in Fig. 4 and Fig. 5 are investigated. A preliminary characterization of the noses is based on a personal evaluation by the patient and a first validation by rhinologists. The personal evaluation is based on the Rhinosinusitis Disability Index (RSBI) questionnaire covering physical, psychological, emotional, social, and functional aspects of the patient s quality of life [4, 4]. Regarding the results of the questionnaire, the noses are labeled good, medium, andpoor denoted by N g, N m,andn p. Note that no scientific analysis was performed to substantiate this categorization. The geometric analysis of the nasal cavity surfaces and the evaluation by the patient and the rhinologists reveal the special geometric features listed in Tab. 2. Unlike N g and N m the geometry of N p represents a nasal cavity after surgery. In the right cavity the has been removed. The removal of the in the left cavity generates a wide opening to the left paranasal sinus (Fig. 4(c), Fig. 5(e), and Fig. 5(f)). Furthermore, a perforation connects both nasal cavities such that there is a flow interaction between them. The pronounced difference of the surface area Āp of the compared to the surface area of the left and Ā l n and Ār n (Tab. 2) is not observed in the nasal cavities N g and N m. Table 3: Nasal cavity simulation setups. The Reynolds number Re is based on the hydraulic diameter D h,p of the as listed in Tab. 2, the kinematic viscosity of air, and a volume flux of 25 ml/s which corresponds to an average flow rate for adults during inhalation at rest. The slight differences of the cell resolutions and the number of cells is due to the different volumes and extents of the nasal cavities. label Re ρ f num. cells cell res. [mm] N g N m N p z y x zoom area flow direction (c) (d) left cavity p c right cavity right cavity p 3 cross section p 4 p 3 p 2 cross section p c p p p p 2 y z x p 4 left cavity z y x p s Figure 6: Probe locations of points p,...,p 4 and location of a cross section in the. Image shows the zoom area of image, where the point of view is from the rear onto the, and of image (c), where a side view onto the is depicted. The line in image (d) defines the location of the velocity distribution illustrated in Fig. 7. The plane is colored by the velocity magnitude. 8

9 u/u max s [mm] v/v max 5 (c) Figure 7: Streamwise velocity profiles in the at grid resolution G st and G f. The location of the cross section is defined in Figs. 6 and 6(c). Streamwise velocity distribution v = v max at grid resolutions G st and f as a function of arc length s, theline is defined in Fig. 6d. Streamwise velocity distribution at G st. (c) Streamwise velocity distribution at G f. log(psd).... e-5 e-6 e-7 e-8 e-9 e- e- e-5. log(k). Figure 8: Distributions of the power spectral density PSD of the y-velocity component v at p on the different grids G st and G f. It is shown in Fig. 4, Fig. 5(c), and 5(d) that the nasal cavity N m suffers from a deviation forming several nichelike areas. In addition, the lower and the in the right cavity are swollen decreasing the breathing capability. The simulation results for N m will show a strong unsteady generation of small vortices in the channel connecting the left nasal cavity and the. The details of the analysis will be presented further below. The geometry N g suffers only from a somewhat narrow left nasal cavity channel (Fig. 4, Fig. 5, and Fig. 5). G st G f G st G f 2 nasal cavities. The problem setup is complemented by defining the volume flux, or in other words, the Reynolds number, since the geometry and the fluid are fixed, and the mesh resolution. This information is summarized in Tab. 3. To show that the resolution determined by the number of cells given in Tab. 3 suffices to capture the essential flow structures, which describe the differences of the flow fields of the various geometries, a comparison of the solutions for the nasal cavity N m having the largest geometrical variation in the streamwise direction is performed for two grid resolutions. The standard reference grid G st consists of cells at a minimum cell size of δ x st = mm. The other mesh G f possesses a twice as fine resolution such that the minimum cell size is δ x f = mm and the total number of cells is Note that the grids are isotropically refined and that in the LBM the spatial step δ x is used as a reference to obtain dimensionless lengths such that the dimensionless cell length is δx =. Duetothehigherresolutionthenumberof iteration steps for the initialization phase and the number of samples for the averaging process doubles for the finer solution. The various resolutions result in only a small difference of the total integral pressure loss of 2. and.8 for the left and right nasal cavity. Likewise, the temperature difference of.985k in the is small. Furthermore, streamwise velocity distributions in a cross section in the defined in Fig. 6 are juxtaposed in Fig. 7. The velocity contours are colored by the velocity magnitude. The results evidence that both profiles differ only slightly. This is quantitatively corroborated by the comparison of the velocity distribution shown in Fig. 7-(c). The maximum difference of the G st and G f solutions is less than 5%. Note that the gradient of the streamwise velocity normal to the wall which determines the wall-shear stress shows a perfect match (see Fig. 7). Furthermore, the vortical structure of the flow is analyzed by the power spectral density PSD of the lateral y-velocity component v at position p in the defined in Fig. 6. The standard and fine distributions are shown in Fig. 8. The small divergence in the high wave number region shows that the energy transport from large scale to small scale structures differs somewhat on both grids. Nevertheless, the cut-off occurs at approximately k.3, i.e., at almost the same dimensionless wave number, which means that the spectra covered by both grids are comparable. Considering larger vortical structures a good agreement is achieved at p which is located just downstream of the entrance. That is the flow structures are determined by the development of the flow from the nostrils to the. Keeping this in mind, the small differences in the very susceptible PSD distributions are acceptable in the sense that the standard grid possesses a sufficient resolution to capture the essential flow features that will evidence the variations of the flow field due to the geometric differences of the nasal cavities. For this reason, the following analysis of the flow fields of the N g, N m,andn p geometries is based on solutions obtained by the standard mesh resolution given in Tab Total pressure loss The specific total pressure loss between the left and right nasal cavity and the of the geometries N g, N m,and N p is considered along characteristic streamlines indicated in 9

10 β r,g γ r,g α r,g β l,g α l,g γ l,g δ r,g γ r,m α l,m γ l,m α r,m ϵ r,m δ l,m δ r,m β r,m β l,m (c) region of perforation perforation region of missing β l,p β r,p region of missing α r,p α l,p region of Figure 9: Streamlines and labeling of characteristic locations on the streamlines in the nasal cavities N g, N m,andn p for the analysis of the specific total pressure difference p t illustrated in Fig.. Streamlines and characteristic locations forn g. Streamlines and characteristic locations for N m. (c) Streamlines and characteristic locations for N p. Fig. 9. The definition of the streamlines is based on the streamwise development of the core region of the mean velocity distribution in the left and cross sections. These streamlines do not enter a recirculation zone, pass between the center and, and possess approximately the same arc length. Note that the streamlines of the various noses differ from each other due to the massive geometric discrepancies of the nasal cavities. First, an analysis of the flow influencing the local pressure drop is given. Subsequently, the total integral pressure losses δp l and δp r for the

11 left and the right nasal cavities of all geometries are juxtaposed to evaluate the individual respiration efficiency. Note that in the following discussion the pressure is normalized by the dimensionless squared reference velocity ( v avg/ ξ ) 2 of the simulation by using ( ) 2 ( ) 2 ( ) 2 vavg = M cs M = (38) ξ ξ 3 and the reference density ρ =. Using Eqs. 8 and 38 the static pressure p and the dynamic pressure p d,x read p x = ρ p x ( vavg ξ ) 2 = ρ x 3 ( ) 2 = ρx (39) M M 2 3 p d,x = 3 2 ρx v 2 x M 2. (4) They are substituted for the pressure terms p x and p d,x in Eq. 5 to Eq. 9. Analysis for N g. Fig. shows the specific total pressure loss along characteristic streamlines of configuration N g defined in Fig. 9. Considering the left nasal cavity the flow encounters an orifice between the lower and downstream of α l,g. The diameter of the channel remains small until the is reached. The flow passes β l,g between the lower and and the high wall shear in this region causes an almost constantly increasing pressure loss. At position γ l,g the flow reaches the. The left channel expands and the streamline is shifted to the axis of the. Finally, it enters a pipe-like geometry, in which the flow is slightly accelerated and the pressure loss is increased marginally. The flow in the right cavity impinges upon the lower turbinate at α r,g and is deflected upwards, which causes the local pressure loss shown in Fig.. Note that the lines placed in the viscous flow field from the nostrils through the can be considered approximate streamlines. Due to the approximation, however, the fundamental condition of a monotonously increasing pressure loss along a streamline is not valid especially in cross sections where a strong variation of the flow normal to the streamwise direction occurs. In other words, the slight deviation in Fig. at β r,g from this monotone growth is related to the small discrepancy between the illustrated line and the rigorous streamline. Upon passing β r,g, theflowentersthewakeregionoftheturbinates,andis shifted close to the wall such that the higher wall shear causes the increasing pressure loss around γ r,g. Furtherdownstream, the flow enters the and impinges on the backwall of the. Finally, the fluid is accelerated causing the increase of the local pressure loss at location δ r,g. Analysis for N m. The features of the pressure loss in the nasal cavity N m are defined by the deviation. Fig. shows several strong pressure losses followed by local plateaus. In the left cavity, the flow is shifted towards the lower turbinate at α l,m and then deflected upwards. Due to the converging channel, the flow is accelerated resulting in the high local pressure loss. Further downstream, the channel slightly expands generating a local pressure plateau followed by a dissipating flow caused by the deviated at location β l,m. The expansion of the channel downstream of the locally reduces the impact of the wall shear. Subsequently, the flow undergoes an acceleration in the converging channel at γ l,m and vortices are shed generating a pronounced mixing loss. At δ l,m the streamline is located at the center of the in the mixing zone downstream of the bifurcation. The flow is deflected to the right side, where hardly any shed vortices occur, which causes a slight decrease of the pressure loss gradient. Asimilardistributioncanalsobefoundintherightnasal cavity. At α r,m the flow impinges on the swollen lower turbinate and is deflected upwards. The total pressure loss is due to the increased shear that the flow undergoes near the wall. At β r,m the lower and orientate the flow towards the highly deviated. The following pressure loss at γ r,m is also dominated by wall shear caused by a narrow region between the lower and. The total pressure loss reaches a small plateau at δ r,m, where the flow is directed towards the mixing zone downstream of the turbinates. The small decrease of the pressure loss at this location is due to discrepancy between the approximate and the rigorous streamline as it is also present at location β r,g for configuration N g. The following strong interaction between the streams of all turbinate channels around ϵ r,m induces the continuous increase of the pressure loss. Finally, the slope decreases slightly as the streamline enters the right side of the containing hardly any vortical structures. Analysis for N p. In contrast to N g and N m the distribution of the total pressure loss of the configuration N p is plateau-like (Fig. (c)). The pressure loss in the left cavity at location α l,p is slightly influenced by a small local channel bending at the transition zone from the nostrils to the main nasal chamber. The removed center and in the left and right nasal cavity result in wide channels yielding an almost direct transport through the nasal cavities. The reason for the slight pressure loss occurring at location β l,p has been discussed in the analysis of location β r,g in configuration N g above. Considering the right nasal cavity the channel converges slightly at α r,p causing a small increase of the pressure loss. Downstream of α r,p the pressure loss stays almost constant until the is reached. At location β r,p the flow is accelerated and the streamline is shifted to the wall causing high shear and a strongly increasing pressure loss. Comparison of N g, N m,andn p. The total pressure loss distributions along the streamlines of the various configurations are juxtaposed in Fig. (d) and Fig. (e) for the left and right cavity. The categorization of the breathing capability solely by the total pressure loss clearly contradicts the preliminary classification by the rhinologists. Although the pressure loss for the nasal cavity N m is greater than that of N g,which makes sense due to the extremely deviated of N m, the previous categorization of N p to possess the worst respiration ability has to be reconsidered. Regarding Fig. (d) and Fig. (e), which only evidence the variation of the distribution of the mechanical energy, N p offers a good breathing capability. Fig. and Tab. 4 additionally show the total pressure loss along the streamlines in the left and right nasal cavities averaged over the cross section by Eqs. 7 and 8. It goes without saying that the juxtaposition of the mean pressure-loss value corroborates the former local analysis, i.e., the pressure loss is largest for N m and smallest for N p.

12 p t α r,g p t,l (N g,s) p t,r (N g,s).2 β r,g.4 β l,g γ l,g.6 α l,g γ r,g.8 δ r,g p t α r,m p t,l (N m,s) p t,r (N m,s) β r,m.2 γ r,m δ r,m α l,m.4 ε r,m β l,m.6 γ l,m.8 δ l,m s/s t s/s t (c) p t,l (N p,s) p t,r (N p,s) β r,p p t.2.5 α l,p α r,p β l,p s/s t (d) (e) p t,l (N g,s) p t,l (N m,s) p t,l (N p,s) p t,r (N g,s) p t,r (N m,s) p t,r (N p,s).2.2 p t.8 p t s/s t s/s t Figure : Distributions of the specific total pressure loss p t along streamlines in the left ( l ) and right ( r ) nasal cavity. The positions of the streamlines and the characteristic locations are illustrated in Fig. 9. Specific total pressure loss in the nasal cavity N g. Specific total pressure loss in the nasal cavity N m. (c) Specific total pressure loss in the nasal cavity N p. (d) Comparison of the specific total pressure loss in the left cavities. (e) Comparison of the specific total pressure loss in the right cavities Vortical flow structures To analyze in more detail the mixing process in the various geometries the vortical flow structures will be discussed. First, a thorough investigation is presented for the nasal cavity N m, which shows a production of unsteady frequently shed secondary flow structures in the wake region of the turbinates. Then, the less pronounced flow structures in N p and N g are described. Note that in the following discussion the velocity fields, i.e., the streamlines of the various geometries, are compared by considering the dimensionless velocity v m = v / v max, where v max =6.5m/sisthemaximum velocity determined in all the solutions. In contrast, the dimensionless velocity vector v ξ = v / ξ is used to obtain the -criterion c (Eq. 2), the vorticity, the turbulent kinetic energy, and the results of the cross- and autocorrelations of configuration N m.thetemporal development in the analysis of the correlations for iteration step ι is defined by t ι = ι δt δ x M D h,p. (4) 2

13 δp t N g l N g r N m l Figure : Breathing capability of the left ( l ) and right ( r ) nasal cavities N g, N m,andn p. Comparison of the total integral dimensionless pressure difference δp from the nostril to the. Table 4: Comparison of the pressure loss δp t in the left ( l ) and right ( r ) nasal cavity from the nostrils to the, of the temperature increase, and of the wall-shear stress. The averaged wallshear stress τ w,avg and the specific maximum τ w,λ normalized by the overall maximum wall-shear stress τ max w,λ of the individual maxima τ w,λ to obtain the dimensionless averaged normal wall-shear stress τw,λ and the dimensionless specific maximum τ w,λ for the configurations λ {g, m, p}. N m r N p l N p r δt label δp t T T τw,λ τ w,λ τ [ w,λ [K] N ] m 2 N g Ng l.26 Ng r.779 N m Nm l Nm r N p Np l.8597 Np r.859 Analysis for N m. It is clear from the illustration shown in Fig. 2, left and right side view, that in configuration N m the flow is accelerated near the nostrils in the nasal valve. This phenomenon is due to the converging channel in this region and was also observed in [9, ]. Note that due to the steady inflow boundary condition there are no fluctuations in the inflow cross section or in the nostril cross section, the flow is laminar and steady in this region. Upstream of the swollen turbinates the flow is homogeneously distributed in the left and right cavity. In the left cavity the flow is split by the. Partially, the fluid is directed through the small lower channel, which causes a local velocity increase. The fluid exits the turbinate channels in a region dominated by mixing jets, which causes the production of unsteady frequently shed secondary flow structures, which are discussed in the following in more detail. The contour of c (Eq. 2) colored by the turbulent kinetic energy k is shown in Fig. 3, where the region in which the flow of the left and right cavity merge is highlighted. A zone of increased k is observed in the turbinate mixing zone in the left nasal cavity before the flow enters the merging zone upstream of the. This higher turbulent kinetic energy is also visible in Fig. 3, where the k-contours are depicted. The k-distribution of the configuration N m evidences that the left cavity produces a much higher velocity fluctuation level than the right cavity. Nevertheless, vortices are shed from both sides towards the as indicated by the c- distribution in Fig. 3. High-energy vortices from the left cavity spread over almost the whole diameter, while vortices produced in the right cavity possess a much smaller mixing tendency. The level of kinetic energy l k is expressed by the ratio of the local k-value and the squared reference velocity, i.e., l k = k/ ( M/ 3 ) 2. In the turbinate mixing zones at probe locations p 3 and p 4,showninFig.6,alevelof l k (p 3)= and l k (p 4)=9.7 3 is obtained, which also emphasizes the difference of kinetic energy produced in the two channels. To gain more insight into the instantaneous vortex structure, Fig. 3(c) shows c-contours of an instantaneous solution colored by the vorticity v. Small scale vortices are primarily generated in the left nasal cavity. Also, high vorticity peaks are stronger related to the left cavity flow. The converging geometry results in avortexstretchingdistributedoverthewholecrosssectionof the. To determine the eddy transport, the normalized crosscorrelation ˆR xy R xy = max{ ˆR } (42) xy is computed for a single time step at the locations p, p,and p 2 and a time series of the y-component of the velocity v normal to the streamwise direction at the locations p 3 and p 4. The positions of p,...,p 4 are shown in the enlargements in Figs. 6 and 6(c). The zoom area is depicted in Fig. 6. The exact locations are listed in Tab. 5 and are given relative to point p c,alsoshowninfigs.6and6(c). Notethatthe geometry of N m is rotated such that the ground of the nasal cavity is parallel to the x-direction and the is orthogonal to the y-direction. The graphs in Fig. 4 evidence by the higher correlation an increased probability for an eddy generated in the nasal cavities to be transported to the. The peak of the correlation is reached, when the eddy passes the probe. The interval between these peaks corresponds to the time span, in which the eddy already passed the probe and only correlations to different vortical structures are recorded. Comparing the correlations in Figs. 4, 4(c), and 4(e) for positions p 3 with the correlations in Figs. 4, 4(d), and 4(f) for point p 4 shows a clear difference in the distributions. While the correlation for position p 3 is hardly visible, the correlation for p 4 is quite evident. This leads to the conclusion that there is a higher v -correlation between the and the left cavity (p 4) than the right cavity (p 3). Since the y-component of the velocity v is related to vortical structures this means that the major part of the vortices entering the are generated in the left cavity. This result corresponds to the increased vorticity and turbulent kinetic energy illustrated in the left nasal cavity in Fig. 3 and the strong increase of the pressure loss at location γ l,m shown in Fig.. The comparison of the correlations for the positions p, p,andp 2 with either p 3 or p 4 shows the strongest correlations to exist between p and p 4 3

14 velocity main flow is transported between lower and center turbinate homogenous flow distribution.75.5 s.25 channel mixing zone acceleration in lower turbiate cavity mixing zone flow around acceleration in velocity.75 flow is split by lower and converging channel swollen lower and deviation and channel convergence cause acceleration.5.25 deviation left nostril channel mixing zone cavity mixing zone acceleration in swollen lower and cavity mixing zone flow is split by acceleration in small opening below acceleration of flow (c) velocity channel mixing zone missing recirculation zone fluid wall interaction zone.75.5 flow in paranasal sinus inter cavity flow.25 acceleration in slow recirculation zone missing recirculation zone channel mixing zone flow in paranasal sinus acceleration in Figure 2: Streamlines in the nasal cavities N g, N m,andn p colored by the velocity magnitude v r = v/ v max, where v max =6.5 m/s is the maximum dimensional velocity of all configurations. Side views (left and right column) and top views (center column) of the three-dimensional geometries are shown. Streamlines in the nasal cavity N g. Streamlines in the nasal cavity N m. (c) Streamlines in the nasal cavity N p. and between p 2 and p 4.ThisisalsoemphasizedinFig.5, where the normalized rms-values ˆR xy,rms R xy,rms = max{ ˆR xy,rms} (43) are shown. Eddies are primarily shed from the left nasal cavity and are distributed over the complete cross section. The impact of the vortical structures shed from the right nasal cavity can be considered secondary compared to that of the left nasal cavity, i.e., the pressure loss downstream of location ϵ r,m is lower than at location γ l,m. Eddy turnaround times τ e normalized by their maximum are obtained from normalized autocorrelations R xx. Fig.6(f) shows the results for τ e(x),x {p,...,p 4} based on the distribution of the autocorrelations illustrated in Figs. 6-(e). The smallest turnaround time is determined at p in the 4

15 (c) right cavity left cavity k left cavity right cavity k left cavity right cavity vorticity zone of increased k zone of increased k votex stretching voticity production zone small scale vortices Figure 3: Vortical structures in the of the nasal cavity N m. shows contours of the dimensionless -criterion colored by the turbulent kinetic energy k; shows the interval contours of the turbulent kinetic energy in the range [, 3.2 5] ; (c) shows the -criterion for an instantaneous solution colored by the vorticity. Table 5: Exact locations of the sampling points p,...,p 4 relative to the center point p c. The locations are illustrated in Fig. 6 and 6(c). location x[mm] y[mm] z[mm] p c... p p p p p center of the. An increased time is obtained at p 4 which is located in the region of the generation of shed vortices of the left nasal cavity. The vortices in the center of the seem to be stretched due to the favorable pressure gradient such that τ e decreases drastically at p.inp and p 2 larger vortices prevail resulting in a higher turnaround time. The results of the power spectral density PSD distributions of the y-component of the velocity v at p,...,p 4 as afunctionofthedimensionlesswavenumberk are shown in Fig. 7. It is evident that independent of the position the highest energy content is in the large scale vortices. At position p vortices at higher wave number, i.e., frequency, occur due to vortex stretching in this area. The comparison of the distributions at p and p 2 evidences the higher overall energy content at p and the low impact of high frequency, i.e. small, vortices on the PSD at p 2. The spectra at p and p 4 almost coincide showing again the strong correlation between these positions that are located on an almost natural connecting line between the left nasal cavity and the entrance. Analysis for N p. The streamlines colored by the velocity magnitude in Fig. 2(c) evidence the major impact of the missing left in the nasal cavity N p on the overall flow structure. The flow is accelerated near the nostril in the nasal valve and directly guided towards the back of the nasal cavity. The flow performs a clockwise rotation about the streamwise axis before forming a recirculation zone caused by the interaction with the backwall of the cavity. Most of the fluid is deflected towards the, where the left and right cavity merge. Due to the geometric shape, the flow rotates about the streamwise axis. The top and side views show the fluid to enter the paranasal sinus and to form a vortex in this region. The part of the fluid in the nasal cavity is accelerated by the converging channel in the. The perforation causes a direct interaction of the flow fields in the left and the right cavity in the region of the missing, where aslowlytumblingrecirculationzoneisformed(fig.2(c),top view). Analysis for N g. Acomparisonofthestreamlinescoloredby the velocity magnitude in the side views of the configuration N g in Fig. 2 show the flow in the left cavity to be uniformly distributed, whereas the major flux in the right cavity is located in the region of the lower and. The homogeneous distribution of the flow in the entire left cavity is based on the low velocity, which increases at the end of the turbinates. Unlike the streamwise velocity gradient in N p and N m the acceleration of the fluid into the is small Wall-shear stress Fig. 8 shows the dimensionless wall-shear stress distributions τ w = τ w/τ max w,λ of the configurations N g, N m, and N p obtained from Eq. 27 using the dimensional velocities v k,k {, 2, 3} and the overall maximum wall-shear stress τ max w,λ =max{τ w,g, τ w,m, τ w,p} for the individual configurations λ. Note that for better visualization, the paranasal sinuses are removed. The comparison evidences the N g configuration to have a smoother distribution than the configurations N m and N p. That is only some local maxima are observed in regions of converging channels, i.e., in the left nasal cavity close to the nostril and in both cavities upstream of the entrance. The N m configuration is dominated by regions of high wall-shear stress τ w. The overall smaller channel width leads to a higher mean flow at a comparable mass flux resulting in ahigherwallshear. Thesmallercrosssectionclosetothe nostrils also leads to a local maximum. The strong deviation and the swollen result in high wallshear stress in the right nasal cavity. In the left cavity, the fluid impinges upon the resulting in a local maximum. The major increase occurs in the, where 5

16 R xy (p,p 3 ) R xy (p,p 4 ).5.5 R xy R xy t ι t ι (c) (d) R xy (p,p 3 ) R xy (p,p 4 ).5.5 R xy R xy t ι t ι (e) (f) R xy (p 2,p 3 ) R xy (p 2,p 4 ).5.5 R xy R xy t ι t ι Figure 4: Temporal normalized crosscorrelation R xy of the dimensionless y-velocity component v for the positions p, p, p 2, p 3,and p 4 defined in Fig. 6 for configuration N m. Crosscorrelation of p and p 3. Crosscorrelation of p and p 4. (c) Crosscorrelation of p and p 3. (d) Crosscorrelation of p and p 4. (e) Crosscorrelation of p 2 and p 3. (f) Crosscorrelation of p 2 and p 4. RMS R xy (p,p 3 ) R xy (p,p 3 ) R xy (p 2,p 3 ) R xy (p,p 4 ) R xy (p,p 4 ) R xy (p 2,p 4 ) Figure 5: The normalized rms-values R xy,rms of the crosscorrelation R xy of the dimensionless y-velocity component v shown in Fig. 4 for configuration N m. 6 the flow from both sides merges into a mixing zone. The highest shear stress is reached at the back of the, where the fluid is guided downstream to the larynx. Like N g and N m the configuration N p also possesses local maxima near the nostrils. In the left nasal cavity the fluid interacts with the outer backwall causing the increased wallshear stress downstream of the large orifice to the paranasal sinus. This flow behavior occurs since the is removed and the fluid is directed against the nasal cavity wall. The perforation experiences an increased τ w due to the mass flow from the left to the right nasal cavity. In the right nasal cavity the fluid enters a converging channel in the region of the, which is why this region is dominated by increased wall-shear stress. Downstream of the turbinates a ring of high τ w is formed due to the convergingdiverging channel geometry. However, the highest wall-shear stress is again determined further downstream in the. The comparison of the dimensionless specific maximum

17 .8 R xx (p ).8 R xx (p ).6.6 R xx.4.2 R xx t ι t ι (c) (d).8 R xx (p 2 ).8 R xx (p 3 ).6.6 R xx.4.2 R xx t ι t ι (e) (f).8 R xx (p 4 ).6.8 R xx.4.2 τ e t ι p p p 2 p 3 p 4 Figure 6: Autocorrelations and the eddy turnaround time τ e of the dimensionless y-velocity component v at positions p,...,p 4 defined in Fig. 6 for configuration N m. Autocorrelation at p. Autocorrelation at p. (c) Autocorrelation at p 2. (d) Autocorrelation at p 3. (e) Autocorrelation at p 4. (f) Normalized eddy turnaround time defined by the integral time of the autocorrelations at positions p,...,p 4. log(psd).. e-6 e-8 e- e-2 e-4 PSD(p ) PSD(p ) PSD(p 2 ) PSD(p 3 ) PSD(p 4 ) e-5. log(k). Figure 7: Comparison of the power spectral density PSD of the dimensionless y-velocity component v at locations p,...,p 4 defined in Fig. 6 for configuration N m.. wall-shear stress τ w,λ = τ w,λ /τ max w,λ,λ {g, m, p} in Fig. 9 and Tab. 4 shows that locally the skin friction in configuration N p is highest at τ w,p =followedbyτ w,m =.77 and τ w,g =.55 for the configurations N m and N g. Considering the averaged normalized wall-shear stress τw,λ = τ w,avg,λ /τ max w,λ this result is changed in the sense that τw,g <τw,p <τw,m. That is the τ w,λ analysis rather identifies spots of high wall friction which could influence the patient s comfort by local inflammations. In the N p configuration such a region is, e.g., the backwall in the left nasal cavity and the strongly converging region. A more detailed analysis of the skin friction is obtained by considering the wall-shear stress distribution c τ defined in Eq. 3 at a binning interval difference of ϖ τ = For the configuration N g the stress τ w(l) pervertexl shown in Fig. 9 clusters in the small bin intervals, which is caused 7

18 τ w.43 entrance to paranasal sinus τ w entrance to paranasal sinus deviation swollen τ w.43 (c) entrance to paranasal sinus missing.3225 perforation missing Figure 8: Wall-shear stress distribution τ w of the various nasal cavity configurations N g, N m,andn p. The paranasal sinuses were removed for better visualization. Wall-shear stress τ w in the nasal cavity N g. Wall-shear stress τ w in the nasal cavity N m. (c) Wall-shear stress τ w in the nasal cavity N p. by decreased velocity gradients in the paranasal and forehead sinuses covering a large fraction of the total nasal cavity surface (see Fig. 5 and 5). At higher stresses the distribution decreases justifying the low overall wall-shear stress distribution illustrated in Fig. 8. The binning of the configuration N m has a smaller count for low values of τ w,i.e., the surface representing the paranasal and forehead sinuses as shown in Figs. 5(c) and 5(d) is smaller. The spectrum shows asomewhatbroaderrangeofmediumstressesintheinterval [.6,.2]. The pronounced decrease coincides with that for N g,butreachesaslightlyhigherplateauvalue. Thatisthe configuration N m possesses a larger area at higher wall-shear stress. Hardly any low 8

19 .2 τ * w,λ τ w,λ.2.. N g N m Np τ * w,λ τ w,λ log(c τ ) e-5 e-5... N g N m N p log(bin) Figure 9: Comparison of wall-shear stress for the nasal cavities N g, N m,andn p. Averaged normalized (left) and specific maximum (right) wall-shear stress τw,λ and τ w,λ. Wall-shear stress distribution based on the normalized bin count c τ = cnt τ / P as a function of bin..5. N g N m Np.4. δt/t.3.2. log(c q ).. e-5 e N g N m N p bin Figure 2: Heating capability and heat flux distributions of nasal cavities N g, N m,andn p. Heating capability of the left ( l ) and right ( r ) nasal cavities N g, N m,andn p. Heat flux distribution based on the normalized bin count c q = cnt q /A λ as a function of bin. τ w values are evident in the N p configuration. The distribution reaches two local maxima at.9 and.3 before experiencing a decrease followed by an almost constant and a subsequent decay. The fluctuating decay in all three spectra for high wall-shear stress values is caused by the dependence of the number of entries per bin on the size of the interval ϖ τ Heating capability To complement the evaluation of the respiration efficiency of the nasal cavities the temperature increase of the flow, which is a function of the residence time of the fluid in the nasal cavity, is discussed next. That is the heating capability is determined by considering the temperature increase δt from the nostril to the, where T i = T =isthedimensionless inflow temperature. Ideally, the flow through the nasal cavity is heated up to almost body temperature given by the dimensionless temperature T b =.546. Considering the configurations N g, N m,andn p, the nasal cavity N m has the most efficient heating capability at a temperature increase shown in Fig. 2 and given in Tab. 4 by δt (N m)=.539 to almost T b,whichcorrelateswiththefindingsobtainedbynaftali et al. [6]. The heating capability of N g is slightly smaller than that of N m and clearly higher than that of N p, i.e., δt (N m) >δt(n g)=.483 >δt(n p)=.394. The lower temperature increase for N p is due to the reduced surface area (see Tab. 2) and the less intricate geometry of the cavity, which prevents a pronounced mixing and as such reduces the residence time of the fluid in the cavity. This is also emphasized by the analysis of the surface-areaaveraged heat flux q avg, which is normalized by the maximum obtained for the three configurations, i.e., by q g for the good configuration. The small difference q g,m avg = q g avg q m avg = and q g,p avg = q g avg q p avg = of the good and medium configurations, and the good and poor configurations indicates that the surface area difference has a major impact on the heating capability. Fig. 2 shows the dimensionless heat flux distribution q obtained from Eq. 34 for the configurations N g, N m,andn p. Note that q is normalized by its maximum over all configurations q max. The juxtaposition of the distributions shows that the heat flux is best in the sense of smoothly increasing in configuration N g (Fig. 2). Especially in the right nasal cavity, q continuously increases along the streamwise direction. In contrast to the left side q is almost equally distributed between the turbinate channels. The major heat flux occurs in the 9

20 q entrance to paranasal sinus q entrance to paranasal sinus.997 deviation swollen. q.9926 (c) entrance to paranasal sinus missing.997 perforation missing Figure 2: Heat flux distribution q of the various nasal cavity configurations N g, N m,andn p. The paranasal sinuses were removed for better visualization. Heat flux q in the nasal cavity N g. Heat flux q in the nasal cavity N m. (c) Heat flux q in the nasal cavity N p. retral part of the turbinate channels and the. The distribution in configuration N m in Fig. 2 is dominated by local maxima in the retral part of the turbinate channels. A smooth distribution is determined in the channel of the left nasal cavity in the channel mixing zone of the right cavity, and in the region. However, the overall q distribution is decreased compared to configuration N g. The low heat flux distribution of configuration N p in Fig. 2(c) emphasizes the low heating capability of this nasal cavity. To analyze the distribution of the heat flux in more detail, the binning ς q is evaluated at an interval difference of ϖ q = 3 for the various nasal cavities λ. As shown in 2