High Fidelity Modeling of Aerosol Pathogen Propagation in Built Environments with Moving Pedestrians
HHIGH FIDELITY MODELING OF AEROSOL PATHOGEN PROPAGATION INBUILT ENVIRONMENTS WITH MOVING PEDESTRIANS
RAINALD L ¨OHNER AND HARBIR ANTIL
Abstract.
A high fidelity model for the propagation of pathogens via aerosols in the presence ofmoving pedestrians is proposed. The key idea is the tight coupling of computational fluid dynamicsand computational crowd dynamics in order to capture the emission, transport and inhalation ofpathogen loads in space and time.An example simulating pathogen propagation in a narrow corridor with moving pedestrians clearlyshows the considerable effect that pedestrian motion has on airflow, and hence on pathogen prop-agation and potential infectivity. Introduction
Advances in computational fluid and crowd dynamics (CFD, CCD), as well as computer hardwareand software, have enabled fast and reliable simulations in both disciplines. A natural next stepis the coupling of both disciplines. This would be of high importance for evacuation studies wherefire, smoke, visibility and inhalation of toxic materials influence the motion of people, and wherea large crowd can block or influence the flow in turn. The same capability could also be used tosimulate with high fidelity the transmission of pathogens in the presence of moving pedestrians,enabling a much needed extension of current simulation technologies [69] .The present work considers a tight, bi-directional coupling, whereby the flow (and any pathogens init) and the motion of the crowd are computed concurrently and with mutual influences. Enablingtechnologies that made this tight coupling feasible include:a) Development of immersed boundary methods;b) Implementation of fast search techniques for information transfer between codes; andc) Strong scaling to tens of thousands of cores for CFD codes.Before describing the numerical methodologies, a quick overview of pathogen, and in particularvirus transmission is given. This defines the relevant physical phenomena, which in turn definethe ordinary and partial differential equations that describe the flow and the particles. Thereafter,the models used for pedestrian motion are outlined. The sections on numerical methodologiesconclude with a description of the coupling methodology employed. Several examples illustrate theinfluence of pedestrian motion on air motion, and hence aerosol and pathogen transport, and showthe potential of the proposed methodology.2.
Virus Infection
Before describing the proposed model for virus transmission a brief description of virus propa-gation and lifetime is given. Viruses are usually present in the air or some surface, and make their
Key words and phrases.
Viral Infection, Aerosol Transmission, Computational Fluid Dynamics, ComputationalCrowd Dynamics.H. Antil is partially supported by NSF grants DMS-1818772, DMS-1913004, the Air Force Office of ScientificResearch (AFOSR) under Award NO: FA9550-19-1-0036, and Department of Navy, Naval PostGraduate Schoolunder Award NO: N00244-20-1-0005. a r X i v : . [ phy s i c s . s o c - ph ] S e p RAINALD L ¨OHNER AND HARBIR ANTIL way into the body either via inhalation (nose, mouth), ingestion (mouth) or attachment (eyes,hands, clothes). In many cases the victim inadvertently touches an infected surface or viruses aredeposited on its hands, and then the hands or clothes touch either the nose, the eyes or the mouth,thus allowing the virus to enter the body.An open question of great importance is how many viruses it takes to overwhelm the body’s naturaldefense mechanism and trigger an infection. This number, which is sometimes called the viral load or the infectious dose will depend on numerous factors, among them the state of immune defensesof the individual, the timing of viral entry (all at once, piece by piece), and the amount of hair andmucous in the nasal vessels. In principle, a single organism in a favourable environment may repli-cate sufficiently to cause disease [85]. Data from research performed on biological warfare agents[22] suggests that both bacteria and viruses can produce disease with as few as 1-100 organisms (e.g.brucellosis 10-100, Q fever 1-10, tularaemia 10-50, smallpox 10-100, viral haemorrhagic fevers 1-10organisms, tuberculosis 1). Compare these numbers and consider that as many as 3,000 organismscan be produced by talking for 5 minutes or a single cough, with sneezing producing many more[70, 56, 90, 71, 99]. Figure 1, reproduced from [90], shows a typical number and size distribution.3.
Virus Lifetime Outside the Body
Current evidence for Covid-19 points to lifetimes outside the body that can range from 1-2 hoursin air to several days on particular surfaces (so-called fomite transmission mode) [47, 95]. Therehas also been some documentation of lifetime variation depending on humidity.4.
Virus Transmission
Sneezing and Coughing.
In the sequel, we consider sneezing and coughing as the mainconduits of virus transmission. Clearly, breathing and talking will lead to the exhalation of air,and, consequently the exhalation of viruses for infected victims [3, 4]. However, it stands to reasonthat the size and amount of particles released - and hence the amount of viruses in them - is muchhigher and much more concentrated when sneezing or coughing [21, 90, 46, 57, 3, 4].The velocity of air at a person’s mouth during sneezing and coughing has been a source of heated
Figure 1.
Counts of Particles of Various Diameters in Air Expelled by 90 Coughs [70]
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 3 debate, particularly in the media. The experimental evidence points to exit velocities of the orderof 2-14 m/sec [26, 27, 87, 88]. A typical amount and size of particles can be seen in Figure 1.4.2.
Sink Velocities.
Table 1 lists the terminal sink velocities for water droplets in air based onthe diameter [79]. One can see that below diameters of O (0 . mm ) the sink velocity is very low,implying that these particles remain in and move with the air for considerable time (and possiblydistances). Table 1.
Sink Velocities and Reynolds Number For Water Particles in AirDiameter [mm] sink velocity [m/sec] Re1.00E-01 3.01E-01 1.99E+001.00E-02 3.01E-03 1.99E-031.00E-03 3.01E-05 1.99E-061.00E-04 3.01E-07 1.99E-094.3.
Evaporation.
Depending on the relative humidity and the temperature of the ambient air,the smaller particles can evaporate in milliseconds. However, as the mucous and saliva evaporate,they build a gel-like structure that surrounds the virus, allowing it to survive. This implies thatextremely small particles with possible viruses will remain infectious for extended periods of times- up to an hour according to some studies [47, 95].An important question is whether a particle/droplet will first reach the ground or evaporate. Fig-ure 2, taken from [101], shows that below 0.12 mm the particles evaporate before falling 2 m (i.e.reaching the ground). 5. Physical Modeling of Aerosol Propagation
When solving the two-phase equations, the air, as a continuum, is best represented by a setof partial differential equations (the Navier-Stokes equations) that are numerically solved on amesh. Thus, the gas characteristics are calculated at the mesh points within the flowfield. Thedroplets/particles, which are relatively sparse in the flowfield, are modeled using a Lagrangian
Figure 2.
Evaporation Time and Falling Time of Droplets of Varying Diameter
RAINALD L ¨OHNER AND HARBIR ANTIL description, where individual particles (or groups of particles) are monitored and tracked in theflow, allowing for an exchange of mass, momentum and energy between the air and the particles.5.1.
Equations Describing the Motion of the Air.
As seen from the experimental evidence,the velocities of air encountered during coughing and sneezing never exceed a Mach-number of
M a = 0 .
1. Therefore, the air may be assumed as a Newtonian, incompressible liquid, wherebuoyancy effects are modeled via the Boussinesq approximation. The equations describing theconservation of momentum, mass and energy for incompressible, Newtonian flows may be writtenas ρ v ,t + ρ v · ∇ v + ∇ p = ∇ · µ ∇ v + ρ g + βρ g ( T − T ) + s v , (5 . . ∇ · v = 0 , (5 . . ρc p T ,t + ρc p v · ∇ T = ∇ · k ∇ T + s e . (5 . . ρ, v , p, µ, g , β, T, T , c p , k denote the density, velocity vector, pressure, viscosity, gravity vector,coefficient of thermal expansion, temperature, reference temperature, specific heat coefficient andconductivity respectively, and s v , s e momentum and energy source terms (e.g. due to particlesor external forces/heat sources). For turbulent flows both the viscosity and the conductivity areobtained either from additional equations or directly via a large eddy simulation (LES) assumptionthrough monotonicity induced LES (MILES) [7, 24, 25, 38].5.2. Equations Describing the Motion of Particles/Droplets.
In order to describe the inter-action of particles/droplets with the flow, the mass, forces and energy/work exchanged between theflowfield and the particles must be defined. As before, we denote for fluid (air) by ρ, p, T, k, v i , µ and c p the density, pressure, temperature, conductivity, velocity in direction x i , viscosity, and thespecific heat at constant pressure. For the particles , we denote by ρ p , T p , v pi , d, c pp and Q the den-sity, temperature, velocity in direction x i , equivalent diameter, specific heat coefficient and heattransferred per unit volume. In what follows we will refer to droplet and particles collectively asparticles.Making the classical assumptions that the particles may be represented by an equivalent sphereof diameter d , the drag forces D acting on the particles will be due to the difference of fluid andparticle velocity: D = πd · c d · ρ | v − v p | ( v − v p ) . (5 . . drag coefficient c d is obtained empirically from the Reynolds-number Re : Re = ρ | v − v p | dµ (5 . . c d = max (cid:18) . , Re (cid:16) . Re . (cid:17)(cid:19) (5 . . c d = 0 . Re → ∞ . The heat transferred between the particles and the fluid is given by Q = πd · (cid:104) h f · ( T − T p ) + σ ∗ · ( T − T p ) (cid:105) , (5 . . EROSOL PROPAGATION WITH MOVING PEDESTRIANS 5 where h f is the film coefficient and σ ∗ the radiation coefficient. For the class of problems consideredhere, the particle temperature and kinetic energy are such that the radiation coefficient σ ∗ may beignored. The film coefficient h f is obtained from the Nusselt-Number N u : N u = 2 + 0 . P r . Re . , (5 . . P r is the Prandtl-number of the gas
P r = kµ , (5 . . h f = N u · kd . (5 . . ρ p πd · d v p dt = D + ρ p πd g . (5 . . d v p dt = 3 ρ ρ p d · c d | v − v p | ( v − v p ) + g = α v | v − v p | ( v − v p ) + g , (5 . . α v = 3 ρc d / (4 ρ p d ). The particle positions are obtained from: d x p dt = v p . (5 . . ρ p c pp πd · dT p dt = Q , (5 . . dT p dt = 3 k c pp ρ p d · N u · ( T − T p ) = α T ( T − T p ) , (5 . . α T = 3 k/ (2 c pp ρ p d ). Equations (5.2.9, 5.2.10, 5.2.12) may be formulated as a system ofOrdinary Differential Equations (ODEs) of the form: d u p dt = r ( u p , x , u f ) , (5 . . u p , x , u f denote the particle unknowns, the position of the particle and the fluid unknownsat the position of the particle.5.3. Equations Describing the Motion of Diluted Viral Loads.
Viral loads may be obtaineddirectly from the particles in the flowfield. An alternative for small, diluted particles that arefloating in air is the use of a transport equation of the form: c ,t + v · ∇ c = ∇ · d c ∇ c + s c , (5 . . c, d c , s c denote pathogen concentration, the diffusivity and the source terms (due to exhalationor inhalation). RAINALD L ¨OHNER AND HARBIR ANTIL
Numerical Integration of the Motion of the Air.
The last six decades have seen alarge number of schemes that may be used to solve numerically the incompressible Navier-Stokesequations given by Eqns.(5.1.1-5.1.3). In the present case, the following design criteria were imple-mented:- Spatial discretization using unstructured grids (in order to allow for arbitrary geometriesand adaptive refinement);- Spatial approximation of unknowns with simple linear finite elements (in order to havea simple input/output and code structure);- Edge-based data structures (for reduced access to memory and indirect addressing);- Temporal approximation using implicit integration of viscous terms and pressure (the interesting scales are the ones associated with advection);- Temporal approximation using explicit, high-order integration of advective terms ;- Low-storage, iterative solvers for the resulting systems of equations (in order to solvelarge 3-D problems); and- Steady results that are independent from the timestep chosen (in order to have confi-dence in convergence studies).The resulting discretization in time is given by the following projection scheme [61, 62]:- Advective-Diffusive Prediction: v n , p n → v ∗ s (cid:48) = −∇ p n + ρ g + βρ g ( T n − T ) + s v , (5 . . v i = v n + α i γ ∆ t (cid:16) − v i − · ∇ v i − + ∇ · µ ∇ v i − + s (cid:48) (cid:17) ; i = 1 , k − . . (cid:20) t − θ ∇ · µ ∇ (cid:21) (cid:16) v k − v n (cid:17) + v k − · ∇ v k − = ∇ · µ ∇ v k − + s (cid:48) . (5 . . p n → p n +1 ∇ · v n +1 = 0 ; (5 . . v n +1 − v ∗ ∆ t + ∇ ( p n +1 − p n ) = 0 ; (5 . . ∇ ( p n +1 − p n ) = ∇ · v ∗ ∆ t ; (5 . . v ∗ → v n +1 v n +1 = v ∗ − ∆ t ∇ ( p n +1 − p n ) . (5 . . θ denotes the implicitness-factor for the viscous terms ( θ = 1: 1st order, fully implicit, θ = 0 .
5: 2ndorder, Crank-Nicholson). α i are the standard low-storage Runge-Kutta coefficients α i = 1 / ( k +1 − i ).The k − v n by v k − . Theoriginal right-hand side has not been modified, so that at steady-state v n = v k − , preserving therequirement that the steady-state be independent of the timestep ∆ t . The factor γ denotes thelocal ratio of the stability limit for explicit timestepping for the viscous terms versus the timestepchosen. Given that the advective and viscous timestep limits are proportional to: EROSOL PROPAGATION WITH MOVING PEDESTRIANS 7 ∆ t a ≈ h | v | ; ∆ t v ≈ ρh µ , (5 . . γ = ∆ t v ∆ t a ≈ ρ | v | hµ ≈ Re h , (5 . . γ = min (1 , Re h ) . (5 . . O (1), implying that a high-order Runge-Kuttascheme is recovered. Conversely, for regions where Re h = O (0), the scheme reverts back to theusual 1-stage Crank-Nicholson scheme. Besides higher accuracy, an important benefit of explicitmultistage advection schemes is the larger timestep one can employ. The increase in allowabletimestep is roughly proportional to the number of stages used (and has been exploited extensivelyfor compressible flow simulations [45]). Given that for an incompressible solver of the projectiontype given by Eqns.(5.4.1-5.4.7) most of the CPU time is spent solving the pressure-Poisson systemEqn.(5.4.6), the speedup achieved is also roughly proportional to the number of stages used.At steady state, v ∗ = v n = v n +1 and the residuals of the pressure correction vanish, implying thatthe result does not depend on the timestep ∆ t .The spatial discretization of these equations is carried out via linear finite elements. The resultingmatrix system is re-written as an edge-based solver, allowing the use of consistent numerical fluxesto stabilize the advection and divergence operators [62].The energy (temperature) equation (Eqn.(5.1.3)) is integrated in a manner similar to the advective-diffusive prediction (Eqn(5.4.2)), i.e. with an explicit, high order Runge-Kutta scheme for theadvective parts and an implicit, 2nd order Crank-Nicholson scheme for the conductivity.5.5. Numerical Integration of the Motion of Particles/Droplets.
The equations describingthe position, velocity and temperature of a particle (Eqns. 5.2.9, 5.2.10, 5.2.12) may be formulatedas a system of nonlinear Ordinary Differential Equations of the form: d u p dt = r ( u p , x , u f ) . (5 . . u n + ip = u np + α i ∆ t · r ( u n + i − p , x n + i − , u n + i − f ) , i = 1 , k . (5 . . α i = 1 k + 1 − i , i = 1 , k (5 . . k -th order accurate in time. Note that in each step the location ofthe particle with respect to the fluid mesh needs to be updated in order to obtain the propervalues for the fluid unknowns. The default number of stages used is k = 4. This would seemunnecessarily high, given that the flow solver is of second-order accuracy, and that the particlesare integrated separately from the flow solver before the next (flow) timestep, i.e. in a staggeredmanner. However, it was found that the 4-stage particle integration preserves very well the motion RAINALD L ¨OHNER AND HARBIR ANTIL in vortical structures and leads to less ‘wall sliding’ close to the boundaries of the domain [66]. Thestability/ accuracy of the particle integrator should not be a problem as the particle motion willalways be slower than the maximum wave speed of the fluid (fluid velocity).The transfer of forces and heat flux between the fluid and the particles must be accomplished ina conservative way, i.e. whatever is added to the fluid must be subtracted from the particles andvice-versa. The finite element discretization of the fluid equations will lead to a system of ODE’sof the form: M ∆ u = r , (5 . . M , ∆ u and r denote, respectively, the consistent mass matrix, increment of the unknownsvector and right-hand side vector. Given the ‘host element’ of each particle, i.e. the fluid meshelement that contains the particle, the forces and heat transferred to r are added as follows: r iD = (cid:88) el surr i N i ( x p ) D p . (5 . . N i ( x p ) denotes the shape-function values of the host element for the point coordinates x p ,and the sum extends over all elements that surround node i . As the sum of all shape-functionvalues is unity at every point: (cid:88) N i ( x ) = 1 ∀ x , (5 . . f p = ρ p πd (cid:16) v n +1 p − v np (cid:17) ∆ t , (5 . . q p = ρ p c pp πd (cid:16) T n +1 p − T np (cid:17) ∆ t . (5 . . r = 0- DO : For Each Particle:- DO : For Each Runge-Kutta Stage:- Find Host Element of Particle: IELEM , N i ( x )- Obtain Fluid Variables Required- Update Particle: Velocities, Position, Temperature, ...- - ENDDO - Transfer Loads to Element Nodes-
ENDDO
Particle Parcels.
For a large number of very small particles, it becomes impossible to carry everyindividual particle in a simulation. The solution is to:a) Agglomerate the particles into so-called packets of N p particles;b) Integrate the governing equations for one individual particle; andc) Transfer back to the fluid N p times the effect of one particle. EROSOL PROPAGATION WITH MOVING PEDESTRIANS 9
Beyond a reasonable number of particles per element (typically > Other Particle Numerics.
In order to achieve a robust particle integrator, a number of additionalprecautions and algorithms need to be implemented. The most important of these are:- Agglomeration/Subdivision of Particle Parcels: As the fluid mesh may be adaptively refinedand coarsened in time, or the particle traverses elements of different sizes, it may be impor-tant to adapt the parcel concentrations as well. This is necessary to ensure that there issufficient parcel representation in each element and yet, that there are not too many parcelsas to constitute an inefficient use of CPU and memory.- Limiting During Particle Updates: As the particles are integrated independently from theflow solver, it is not difficult to envision situations where for the extreme cases of very lightor very heavy particles physically meaningless or unstable results may be obtained. In orderto prevent this, the changes in particle velocities and temperatures are limited in order notto exceed the differences in velocities and temperature between the particles and the fluid[66].- Particle Contact/Merging: In some situations, particles may collide or merge in a certainregion of space.- Particle Tracking: A common feature of all particle-grid applications is that the particlesdo not move far between timesteps. This makes physical sense: if a particle jumped tengridpoints during one timestep, it would have no chance to exchange information with thepoints along the way, leading to serious errors. Therefore, the assumption that the newhost elements of the particles are in the vicinity of the current ones is a valid one. Forthis reason, the most efficient way to search for the new host elements is via the vectorizedneighbour-to-neighbour algorithm described in [58, 62].5.6.
Immersed Body Techniques.
The information required from CCD codes consists of thepedestrians in the flowfield, i.e. their position, velocity, temperature, as well inhalation and exha-lation. As the CCD codes describe the pedestrians as points, circles or ellipses, a way has to befound to transform this data into 3-D objects. Two possibilities have been pursued here: • a) Transform each pedestrian into a set of (overlapping) spheres that approximate the bodywith maximum fidelity with the minimum amount of spheres; • b) Transform each pedestrian into a set of tetrahedra that approximate the body withmaximum fidelity with the minimum amount of tetrahedra.The reason for choosing spheres or tetrahedra is that one can perform the required interpolation/information transfer much faster than with other methods.In order to ‘impose’ on the flow the presence of a pedestrian the immersed boundary methodologyis used. The key idea is to prescribe at every CFD point covered by a pedestrian the velocity andtemperature of the pedestrian. For the CFD code, this translates into an extra set of boundaryconditions that vary in time and space as the pedestrians move. This is by now a mature technology.Fast search techniques as well as extensions to higher order boundary conditions may be found in[62, 63]. Nevertheless, as the pedestrians potentially change location every timestep, the search forand the imposition of new boundary conditions can add a considerable amount of CPU as comparedto ‘flow-only’ runs. 6. Modeling of Pedestrian Motion
The modeling of pedestrian motion has been the focus of research and development for more thantwo decades. If one is only interested in average quantities (average density, velocity), continuum models [37] are an option. For problems requiring more realism, approaches that model eachindividual are required [91]. Among these, discrete space models (such as cellular automata [5, 6,89, 19, 80, 49, 51, 13, 53]), force-based models (such as the social force model [34, 36, 76, 52, 64]) andagent-based techniques [73, 83, 31, 32, 96, 94, 14] have been explored extensively. Together withinsights from psychology and neuroscience (e.g. [97, 94]) it has become clear that any pedestrianmotion algorithm that attempts to model reality should be able to mirror the following empiricallyknown facts and behaviours:- Newton’s laws of motion apply to humans as well: from one instant to another, we can onlymove within certain bounds of acceleration, velocity and space;- Contact between individuals occurs for high densities; these forces have to be taken intoaccount;- Humans have a mental map and plan on how they desire to move globally (e.g. first gohere, then there, etc.);- Human motion is therefore governed by strategic (long term, long distance), tactical (mediumterm, medium distance) and operational (immediate) decisions;- In even moderately crowded situations ( O (1 p/m )), humans have a visual horizon of O (2 . − . m ), and a perception range of 120 degrees; thus, the influence of other humans beyondthese thresholds is minimal;- Humans have a ‘personal comfort zone’; it is dependent on culture and varies from individualto individual, but it cannot be ignored;- Humans walk comfortably at roughly 2 paces per second (frequency: ν = 2 Hz ); they areable to change the frequency for short periods of time, but will return to 2 Hz wheneverpossible.We remark that many of the important and groundbreaking work cited previously took placewithin the gaming/visualization community, where the emphasis is on ‘looking right’. Here, theaim is to answer civil engineering or safety questions such as maximum capacity, egress timesunder emergency, or comfort. Therefore, comparisons with experiments and actual data are seenas essential [64, 40, 41].6.1. The PEDFLOW Model.
The PEDFLOW model [64] incorporates these requirements asfollows: individuals move according to Newton’s laws of motion; they follow (via will forces) ‘globalmovement targets’; at the local movement level, the motion also considers the presence of otherindividuals or obstacles via avoidance forces (also a type of will force) and, if applicable, contactforces. Newton’s laws: m d v dt = f , d x dt = v , (6 . . m, v , x , f , t denote, respectively, mass, velocity, position, force and time, are integrated intime using a 2nd order explicit timestepping technique. The main modeling effort is centered on f .In the present case the forces are separated into internal (or will) forces [I would like to move hereor there] and external forces [I have been hit by another pedestrian or an obstacle]. For the sake ofcompleteness, we briefly review the main forces used. For more information, as well as verificationand validation studies, see [64, 40, 41, 103, 42, 43, 44]. Will Force.
Given a desired velocity v d and the current velocity v , this force will be of the form f will = g w ( v d − v ) . (6 . . . EROSOL PROPAGATION WITH MOVING PEDESTRIANS 11
The modelling aspect is included in the function g w , which, in the non-linear case, may itself be afunction of v d − v . Suppose g w is constant, and that only the will force is acting. Furthermore,consider a pedestrian at rest. In this case, we have: m d v dt = g w ( v d − v ) , v (0) = 0 , (6 . . . v = v d (cid:16) − e − αt (cid:17) , α = g w m = 1 t r , (6 . . . d v dt ( t = 0) = α v d = v d t r . (6 . . . t r which governs the initialacceleration and ‘time to desired velocity’. Typical values are v d = 1 . m/sec and t r = O (0 . sec ).The ‘relaxation time’ t r is clearly dependent on the fitness of the individual, the current state ofstress, desire to reach a goal, climate, signals, noise, etc. Slim, strong individuals will have low valuesfor t r , whereas fat or weak individuals will have high values for t r . Furthermore, dividing by themass of the individual allows all other forces (obstacle and pedestrian collision avoidance, contact,etc.) to be scaled by the ‘relaxation time’ as well, simplifying the modeling effort considerably.The direction of the desired velocity s = v d | v d | (6 . . . x d ( t d ) that he would like to reach at a certain time t d . If there are no timeconstraints, t d is simply set to a large number. Given the current position x , the direction of thevelocity is given by s = x d ( t d ) − x | x d ( t d ) − x | , (6 . . . x d ( t d ) denotes the desired position (location, goal) of the pedestrian at the desired time ofarrival t d . For members of groups, the goal is always to stay close to the leader. Thus, x g ( t g )becomes the position of the leader. In the case of an evacuation simulation, the direction is givenby the gradient of the perceived time to exit τ e to the closest perceived exit: s = ∇ τ e |∇ τ e | . (6 . . . | v d | depends on the fitness of the individual, and the moti-vation/urgency to reach a certain place at a certain time. Pedestrians typically stroll leisurely at0 . − . m/sec , walk at 0 . − . m/sec , jog at 1 . − . m/sec , and run at 3 . − . m/sec . Pedestrian Avoidance Forces.
The desire to avoid collisions with other individuals is modeled byfirst checking if a collision will occur. If so, forces are applied in the direction normal and tangentialto the intended motion. The forces are of the form: f i = f max / (1 + ρ p ) ; ρ = | x i − x j | /r i , (6 . . . where x i , x j denote the positions of individuals i, j , r i the radius of individual i , and f max = O (4) f max ( will ). Note that the forces weaken with increasing non-dimensional distance ρ . Foryears we have used p = 2, but, obviously, this can (and probably will) be a matter of debate andspeculation (perhaps a future experimental campaign will settle this issue). In the far range, theforces are mainly orthogonal to the direction of intended motion: humans tend to move slightlysideways without decelerating. In the close range, the forces are also in the direction of intendedmotion, in order to model the slowdown required to avoid a collision. Wall Avoidance Forces.
Any pedestrian modeling software requires a way to input geographicalinformation such as walls, entrances, stairs, escalators, etc. In the present case, this is accomplishedvia a triangulation (the so-called background mesh). A distance to walls map (i.e. a function d w ( x )is constructed using fast marching techniques on unstructured grids), and this allows to define awall avoidance force as follows: f = − f max
11 + ( d w r ) · ∇ d w , p = 2 (6 . . . | d w | = 1. The default for the maximum wall avoidance force is f max = O (8) f max ( will ).The desire to be far/close to a wall also depends on cultural background. Contact Forces.
When contact occurs, the forces can increase markedly. Unlike will forces, contactforces are symmetric. Defining ρ ij = | x i − x j | / ( r i + r j ) , (6 . . . ρ ij < f = − [ f max / (1 + ρ pij )] ; p = 2 (6 . . . a ) ρ ij > f = − [2 f max / (1 + ρ pij )] ; p = 2 (6 . . . b )and f max = O (8) f max ( will ). Motion Inhibition.
A key requirement for humans to move is the ability to put one foot in frontof the other. This requires space. Given the comfortable walking frequency of ν = 2 Hz , one isable to limit the comfortable walking velocity by computing the distance to nearest neighbors andseeing which one of these is the most ‘inhibiting’. Psychological Factors.
The present pedestrian motion model also incorporates a number of psycho-logical factors that, among the many tried over the years, have emerged as important for realisticsimulations. Among these, we mention:- Determination/Pushiness: it is an everyday experience that in crowds, some people exhibita more polite behavior than others. This is modeled in PEDFLOW by reducing the collisionavoidance forces of more determined or ‘pushier’ individuals. Defining a determination orpushiness parameter p , the avoidance forces are reduced by (1 − p ).- Comfort zone: in some cultures (northern Europeans are a good example) pedestrians wantto remain at some minimum distance from contacting others. This comfort zone is an inputparameter in PEDFLOW, and is added to the radii of the pedestrians when computingcollisions avoidance and pre-contact forces.- Right/Left Avoidance and Overtaking: in many western countries pedestrians tend to avoidincoming pedestrians by stepping towards their right, and overtake others on the left. How-ever, this is not the norm everywhere, and one has to account for it. EROSOL PROPAGATION WITH MOVING PEDESTRIANS 13
Numerical Integration of the Motion of Pedestrians.
The equations describing the po-sition and velocity of a pedestrian may be formulated as a system of nonlinear Ordinary DifferentialEquations of the form: d u p dt = r ( u p , x , u f ) . (6 . . δ w or exit(s) for any given point of the background grid evaluated via a fast ( O ( N ln( N ))) nearestneighbour/heap list technique ([62, 64]). For cases with visual or smoke impediments, the closestdistance to exit(s) is recomputed every few seconds of simulation time.6.3. Linkage to CFD Codes.
The information required from CFD codes such as FEFLO consistsof the spatial distribution of temperature, smoke, other toxic or movement impairing substances inspace, as well as pathogen distribution. This information is interpolated to the (topologically 2-D)background mesh at every timestep in order to calculate properly the visibility/ reachability of exits,routing possibilities, smoke, toxic substance or pathogen inhalation, and any other flowfield variablerequired by the pedestrians. As the tetrahedral grid used for the CFD code and the triangularbackground grid of the CCD code do not change in time, the interpolation coefficients need to becomputed just once at the beginning of the coupled run. While the transfer of information fromCFD to CCD is voluminous, it is very fast, adding an insignificant amount to the total run-times.7.
Coupling Methodology
The coupling methodology used is shown in Figure 3. The CFD code computes the flowfield,providing such information as temperature, smoke, toxic substance and pathogen concentration,and any other flow quantity that may affect the movement of pedestrians. These variables arethen interpolated to the position where the pedestrians are, and are used with all other pertinentinformation (e.g. will-forces, targets, exits, signs, etc.) to update the position, velocity, inhalation ofsmoke, toxic substances or pathogens, state of exhaustion or intoxication, and any other pertinentquantity that is evaluated for the pedestrians. The position, velocity and temperature of thepedestrians, together with information such as sneezing or exhaling air, is then transferred to theCFD code and used to modify and update the boundary conditions of the flowfield in the regionswhere pedestrians are present.Of the many possible coupling options (see e.g. [8, 82, 9]), we have implemented the simplest one:loose coupling with sequential timestepping ([59, 65]). This is justified, as the timesteps of boththe flow and pedestrian solvers are very small, so that possible coupling errors are negligible.8.
Examples
Corridor With Pedestrians.
This example considers the corridor of 10.0 m x 2.0 m x 2.5 mshown in Figure 4. Both entry and exit sides have two doors each of size 0.8 m x 2.0 m. Forclimatisation, 4 entry vanes and 1 exit vane are placed in the ceiling. The vertical air velocityfor the entry vanes as set to v z = 0 . m/sec , while the horizontal velocity was set as increasing Figure 3.
Coupling CFD and CCD Codesproportional to the distance of the center of the vane to a maximum of v r = 0 . m/sec . Twostreams of pedestrians enter and exit through the doors over time. As stated before, the pedestriandynamics code, which only ‘sees’ the floorplan of the problem at hand, computes position, velocityand orientation of the pedestrians, and then produces a tetrahedral mesh for each pedestrian andthe sends this information to the transfer library. This information is then passed on to the flowsolver, which treats the pedestrians via the immersed body approach in the flowfield. Should therebe smoke, pollutants or pathogens in the flowfield, this information is passed back to the pedestriandynamics code, which interpolates it at the height of pedestrians in order to update inhalation,intoxication and infection information.This simulation was run in 3 phases:Phase 1: Every code is run independently until ‘things settle down’, i.e until the flow reaches a quasi-steady state and the pedestrian streams have formed; for the present case this took 20 secof physical time;Phase 2: The restart files from Phase 1 are taken, and the run continues in fully coupled mode, until‘things settle down’; for the present case this took 20 sec of physical time;Phase 3: The restart files from Phase 2 are taken, and the run continues in fully coupled modeimposing the boundary conditions for a sneezing event. Figure 4.
Simple Channel: Geometry and Boundary Conditions
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 15
In order to see the effects of pedestrians, three simulations were carried out: a) Two pedestriansstreams in counterflow mode; b) Two pedestrians streams in parallel flow mode; and c) No pedes-trians.Figures 5-11, 12-18 and 19-25 show the solutions obtained for these different modes.In the cases shown different temporal scales appear:- The fast, ballistic drop of the larger ( d = 1 mm ) particles, occurring in the range of O (1) sec ;- The slower drop of particles of diameter d = O (0 . mm , occurring in the range of O (10) sec ;and- The transport of the even smaller particles through the air, occurring in the range of O (100) sec .We have attempted to show these phases in the results, and for this reason the results are notdisplayed at equal time intervals. Unless otherwise noted, the particles have been colored accordingto the logarithm of the diameter, with red colors representing the largest and blue the smallestparticles.Note the very large differences in the flowfield with and without pedestrians. The main reason forthis difference is the discrepancy in velocities: humans walk at approximately v = 1 . m/sec , whilethe perception of discomfort due to air motion being at around v − . m/sec , implying that inmost of the volume of any built environment where humans reside, these lower velocities will beencountered. The different velocities between walking pedestrians, and in particular counterflows asthe one shown, lead to large-scale turbulent mixing, enhancing the spread of pathogens emanatingfrom infected victims.Note the very large difference in mixing and viral transmission due to the presence of pedestrians.9. Conclusions and Outlook
A high fidelity model for the propagation of pathogens via aerosols in the presence of movingpedestrians has been implemented. The key idea is the tight coupling of computational fluiddynamics and computational crowd dynamics in order to capture the emission, transport andinhalation of pathogen loads in space and time in the presence of moving pedestrians.The example of a narrow corridor with moving pedestrians clearly shows the considerable effectthat pedestrian motion has on airflow, and hence on pathogen propagation and potential infectivity.At present, the ‘pedestrians’ appear in the flow code as rigid bodies. The incorporation of leg andarm movement while walking would be a possible improvement to the model.
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EROSOL PROPAGATION WITH MOVING PEDESTRIANS 17
Figure 6.
Counterflow Movement: Solution at t = 0 . sec [12] F. Camelli and R. L¨ohner - VLES Study of Flow and Dispersion Patterns in Heterogeneous Urban Areas; AIAA -06-1419 (2006).[13] N. Courty and S. Musse - Simulation of Large Crowds Including Gaseous Phenomena; pp.206212 in
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Figure 7.
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EROSOL PROPAGATION WITH MOVING PEDESTRIANS 19
Figure 8.
Counterflow Movement: Solution at t = 2 . sec [23] J.J. Fruin - Pedestrian Planning and Design ; Metropolitan Association of Urban Designers and EnvironmentalPlanners, New York (1971).[24] C. Fureby and F. Grinstein - Monotonically Integrated Large Eddy Simulation of Free Shear Flows;
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Figure 9.
Counterflow Movement: Solution at t = 5 . sec [29] J.K. Gupta, C-H. Lin and Q. Chen - Risk Assessment of Airborne Infectious Diseases in Aircraft Cabins; IndoorAir
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EROSOL PROPAGATION WITH MOVING PEDESTRIANS 21
Figure 10.
Counterflow Movement: Solution at t = 10 . sec [35] D. Helbing and P. Molnar - Self-Organization Phenomena in Pedestrian Crowds; 569577 in Self-Organization ofComplex Structures: From Individual to Collective Dynamics (F. Schweitzer (Ed.), London: Gordon and Breach(1997).[36] D. Helbing, I.J. Farkas, P. Moln´ar and T. Vicsek - Simulation of Pedestrian Crowds in Normal and EvacuationSituations; pp. 21-58 in
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Figure 11.
Counterflow Movement: Solution at t = 20 . sec [39] M. Ip, J.W. Tang, D.S.C. Hui, A.L.N. Wong, M.T.V. Chan, G.M. Joynt, A.T.P. So, S.D. Hall, P.K.S. Chan andJ.J.Y. Sung - Airflow and Droplet Spreading Around Oxygen Masks: A Simulation Model for Infection ControlResearch; AJIC
35, 10, 684-689 (2007).[40] M. Isenhour and R. L¨ohner - Verification of a Pedestrian Simulation Tool Using the NIST Recommended TestCases;
The Conference in Pedestrian and Evacuation Dynamics 2014 (PED2014), Transportation Research Pro-cedia
2, 237-245 (2014).[41] M. Isenhour and R. L¨ohner - Verification of a Pedestrian Simulation Tool Using the NIST Stairwell Evacua-tion Data;
The Conference in Pedestrian and Evacuation Dynamics 2014 (PED2014), Transportation ResearchProcedia
2, 739-744 (2014).[42] M. Isenhour - Simulating Occupant Response to Emergency Situations;
PhD Thesis , George Mason University,Fairfax, VA (2016).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 23
Figure 12.
Parallel Movement: Solution at t = 0 . sec [43] M. Isenhour and R. L¨ohner - Validation Data from the Evacuation of a Student Center; pp. 472-479 in Proc.Pedestrian and Evacuation Dynamics 2016 (PED 2016) , (W. Song, J. Ma and L. Fu eds.), University of Scienceand Technology Press, Hefei, China, Oct 17-21 (2016).[44] M. Isenhour and R. L¨ohner - Pedestrian Speed on Stairs: A Mathematical Model Based on Empirical Analysisfor Use in Computer Simulations; pp. 529-533 in
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Figure 13.
Parallel Movement: Solution at t = 0 . sec [47] G. Kampf, D. Todt, S. Pfaender, E. Steinmann - Persistence of Coronaviruses on Inanimate Surfacesand Their Inactivation With Biocidal Agents; J. of Hospital Infection
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4, Univ. Duisburg-Essen (2003).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 25
Figure 14.
Parallel Movement: Solution at t = 1 . sec [52] T.I. Lakoba, D.J. Kaup and N.M. Finkelstein - Modifications of the Helbing-Moln´ar-Farkas-Vicsek Social ForceModel for Pedestrian Evolution; Simulation
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PLoS ONE
J.Occup. Environ. Hyg.
9, 443-9. (2012).
Figure 15.
Parallel Movement: Solution at t = 2 . sec [58] R. L¨ohner and J. Ambrosiano - A Vectorized Particle Tracer for Unstructured Grids; J. Comp. Phys.
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AIAA -95-2259 [Invited] (1995). -[60] R. L¨ohner - Multistage Explicit Advective Prediction for Projection-Type Incompressible Flow Solvers;
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Comp. Meth. Appl. Mech. Eng.
Applied CFD Techniques, Second Edition ; J. Wiley & Sons (2008).[63] R. L¨ohner, J.R. Cebral, F.F. Camelli, S. Appanaboyina, J.D. Baum, E.L. Mestreau and O. Soto - AdaptiveEmbedded and Immersed Unstructured Grid Techniques;
Comp. Meth. Appl. Mech. Eng.
Appl. Math. Modelling
34, 2, 366-382 (2010).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 27
Figure 16.
Parallel Movement: Solution at t = 5 . sec [65] R. L¨ohner - Coupling Several CFD and CSD Codes in One Application; pp. 1 - 16 in Special Edition Int. J. ofMultiphysics (2011).[66] R. L¨ohner, F. Camelli, J.D. Baum, F. Togashi and O. Soto - On Mesh-Particle Techniques;
Comp. Part. Mech.
1, 199-209 (2014).[67] R. L¨ohner, M. Baqui, E. Haug and B. Muhamad - Real-Time Micro-Modelling of a Million Pedestrians;
Engi-neering Computations
33, 1, 217-237 (2016).[68] R. L¨ohner and F. Camelli - Tightly Coupled Computational Fluid and Crowd Dynamics; pp. 505-509 in
Proc.Pedestrian and Evacuation Dynamics 2016 (PED 2016) , (W. Song, J. Ma and L. Fu eds.), University of Scienceand Technology Press, Hefei, China, Oct 17-21 (2016).[69] R. L¨ohner, H. Antil, S. Idelsohn and E. O˜nate - Detailed Simulation of Viral Propagation in the Built Environ-ment; arXiv:2006.13792 [physics.soc-ph]
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Figure 17.
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26 (6): 8086 (2006).[74] N. Pelechano, J. Allbeck and N.I. Badler - Virtual Crowds: Methods, Simulation and Control; Morgan &Claypool, San Rafael, CA (2008).[75] W.M. Predtetschenski and A.I. Milinski -
Personenstr¨ome in Geb¨auden - Berechnungsmethoden f¨ur die Projek-tierung ; Verlaggesellschaft Rudolf M¨uller, K¨oln-Braunsfeld (1971).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 29
Figure 18.
Parallel Movement: Solution at t = 20 . sec [76] M.J. Quinn, R.A. Metoyer and K. Hunter-Zaworski - Parallel Implementation of the Social Forces Model; pp.63-74 in Proc. 2nd Int. Conf. in Pedestrian and Evacuation Dynamics (2003).[77] R. Ramamurti and R. L¨ohner - A Parallel Implicit Incompressible Flow Solver Using Unstructured Meshes;
Computers and Fluids
5, 119-132 (1996).[78] R. Ramamurti, W.C. Sandberg and R. L¨ohner - Computation of Unsteady Flow Past Deforming Geometries;
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Boundary Layer Theory ; McGraw-Hill (1979).[80] A. Schadschneider - Cellular Automaton Approach to Pedestrian Dynamics - Theory; pp. 75-86 in
Pedestrianand Evacuation Dynamics (M. Schreckenberg and S.D. Sharma eds.), Springer (2002).[81] M. Schreckenberg and S.D. Sharma (eds.) -
Pedestrian and Evacuation Dynamics , Springer (2002).[82] M. Sch¨afer and S. Turek (eds.) -
Proc. Int. Workshop on Fluid-Structure Interaction: Theory, Numerics andApplications , Herrsching (Munich), Germany, Sept. 29 - Oct 1 (2008).
Figure 19.
No Pedestrians: Solution at t = 0 . sec [83] A. Sud, R. Gayle, E. Andersen, S. Guy, Ming Lin and D. Manocha - Real-time Navigation of Independent AgentsUsing Adaptive Roadmaps; ACM Symposium on Virtual Reality Software and Technology (2007).[84] G.N. Sze To and C.Y. Chao - Review and Comparison Between the Wells-Riley and Dose-Response Ap-proaches to Risk Assessment of Infectious Respiratory Diseases;
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77 213-222 (2011).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 31
Figure 20.
No Pedestrians: Solution at t = 0 . sec [87] J.W. Tang, A.D. Nicolle, J. Pantelic, G.C. Koh, L. Wang, M. Amin, C.A. Klettner, D.K.W. Cheong, C. Sekharand K.W. Tham - Airflow Dynamics of Coughing in Healthy Human Volunteers by Shadowgraph Imaging: AnAid to Aerosol Infection Control; PLoS ONE
7, 4: e34818 (2012). doi:10.1371/journal.pone.0034818[88] J.W. Tang, A.D. Nicolle, C.A. Klettner, J. Pantelic, L. Wang, A. Bin Suhaimi, A.Y.L. Tan, G.W.X. Ong, R.Su, C. Sekhar, D.K.W. Cheong and K.W. Tham - Airflow Dynamics of Human Jets: Sneezing and Breathing -Potential Sources of Infectious Aerosols;
PLoS ONE
8, 4: e59970 (2013). doi:10.1371/journal.pone.0059970[89] K. Teknomo, Y. Takeyama and H. Inamura - Review on Microscopic Pedestrian Simulation Model;
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Morioka, Japan, March (2000).[90] P.F.M. Teunis, N. Brienen, M.E.E. Kretzschmar - High Infectivity and Pathogenicity of Influenza A Virus ViaAerosol and Droplet Transmission;
Epidemics
2, 215222 (2010).[91] D. Thalmann and S.R. Musse -
Crowd Simulation ; Springer-Verlag, London (2007).
Figure 21.
No Pedestrians: Solution at t = 1 . sec [92] R. Tilch, A. Tabbal, M. Zhu, F. Decker and R. L¨ohner - Combination of Body-Fitted and Embedded Grids forExternal Vehicle Aerodynamics; Engineering Computations
25, 1, 28-41 (2008).[93] K. K.-W. To et al. - Temporal Profiles of Viral Load in Posterior Oropharyngeal Saliva Samples and SerumAntibody Responses During Infection by SARS-CoV-2: An Observational Cohort Study;
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Annals of the Association of AmericanGeographers
The New England Journal of Medicine
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 33
Figure 22.
No Pedestrians: Solution at t = 2 . sec [96] G. Vigueras, M. Lozano, J.M. Ordun and F. Grimaldo - A Comparative Study of Partitioning Methods forCrowd Simulations; Applied Soft Computing
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Airborne Contagion and Air Hygiene. An Ecological Study of Droplet Infections ; CambridgeUniversity Press (1955).
Figure 23.
No Pedestrians: Solution at t = 5 . sec [101] X. Xie, Y. Li, A.T.Y. Chwang, P.L. Ho, W.H. Seto - How Far Droplets Can Move in Indoor Environments - Re-visiting the wells Evaporation-Falling Curve; Indoor Air
17, 211-225 (2007). doi:10.1111/j.1600-0668.2006.00469.x[102] T. Zhang, Q. Chen and C.-H. Lin - Optimal Sensor Placement for Airborne Contaminant Detection in anAircraft Cabin;
HVAC&R Research
13, 5, 683-696 (2007).[103] J. Zhang, D. Britto, M. Chraibi, R. L¨ohner, E. Haug and B. Gawenat - Qualitative Validation of PEDFLOWfor Description of Unidirectional Pedestrian Dynamics;
The Conference in Pedestrian and Evacuation Dynamics2014 (PED2014), Transportation Research Procedia
2, 733-738 (2014).[104] Y. Zhang, G. Feng, Z. Kang, Y. Bi and Y. Cai - Numerical Simulation of Coughed Droplets in ConferenceRoom; , October,19-22 Jinan, China (2017),
Procedia Engineering
Building and Environment
44, 437-445 (2009).
EROSOL PROPAGATION WITH MOVING PEDESTRIANS 35
Figure 24.
No Pedestrians: Solution at t = 10 .00
No Pedestrians: Solution at t = 10 .00 sec Rainald L¨ohner, Center for Computational Fluid Dynamics, College of Science, George MasonUniversity,, Fairfax, VA 22030-4444, USA,
E-mail address : [email protected] Harbir Antil, Center for Mathematics and Artificial Intelligence (CMAI), College of Science,,George Mason University, Fairfax, VA 22030-4444, USA
E-mail address : [email protected] Figure 25.
No Pedestrians: Solution at t = 20 .00