detailed simulation of viral propagation and mitigation in
TRANSCRIPT
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CFD Center, George Mason University
Detailed Simulation of Viral
Propagation and Mitigation in the Built
Environment
Rainald LöhnerCFD Center, College of Sciences
George Mason University, Fairfax, VA, USA
www.cfd.gmu.edu/~rlohner
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CFD Center, George Mason University
Acknowledgements• CFD Center, GMU
– Rainald Löhner
– Lingquan Li
– Alejandro Figueroa
– Fernando Camelli
– Chi Yang
– Juan Cebral
– Fernando Mut
• Math, GMU– Harbir Antil
– Carlos Rautenberg
– Thomas Brown
• CHHS, GMU– Amira Albert Roess
• ASI– Joseph Baum
– Orlando Soto
– Fumiya Togashi
– Michael Giltrud
– Charles Charman
• CIMNE, UPC, Barcelona– Eugenio Oñate
– Sergio Idelsohn
• NYT: Mika Gröndahl
• US District Courts of NY/MD
• SLR, Deutsche Bahn
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CFD Center, George Mason University
Outline
• Pathogen Transmission
• Mitigation Options
• Physical Modeling
• Numerical Methods
• Examples
• Reopening After the Crisis
• Conclusions and Outlook
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CFD Center, George Mason University
Pathogen Transmission
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CFD Center, George Mason University
Virus Transmission: Sneezing
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CFD Center, George Mason University
Virus Transmission: Breathing
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CFD Center, George Mason University
Virus Transmission: HVAC/Wakes
0.2-0.3 m/s
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CFD Center, George Mason University
Virus Transmission
• Entering the Body: Nose, Mouth, Eyes
• Exiting the Body: Nose, Mouth
– Sneeze, Cough, Shout, Sing, Speak, …
• ➔ Transmission Modes:
– Person-to-Person : Large/Small Droplets
– Person-Air-Person : Small Droplets
– Person-Surface-Person : Large Droplets
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CFD Center, George Mason University
Droplet Distribution When Coughing
R.G. Loudon and R.M. Roberts - Droplet Expulsion from the Respiratory Tract; Am. Rev.
Respir. Dis. 95, 3, 435–442 (1967).
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CFD Center, George Mason University
Sink Velocities of Droplets in Air
Diameter [mm] Sink Velocity [m/sec] Reynolds-Nr.
1.0e+00 3.01e+01 1.99e+03
1.0e-01 3.01e-01 1.99e+00
1.0e-02 3.01e-03 1.99e-03
1.0e-03 3.01e-05 1.99e-06
1.0e-04 3.01e-07 1.99e-09
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CFD Center, George Mason University
Droplet Evaporation
X. Xie, Y. Li, A.T.Y. Chwang, P.L. Ho, W.H. Seto - How Far Droplets Can Move in Indoor
Environments – Revisiting the Wells Evaporation-Falling Curve; Indoor Air 17, 211-225 (2007).
doi:10.1111/j.1600-0668.2006.00469.x
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CFD Center, George Mason University
Viral Load / Infectious Dose
• Many Factors:
– State of Immune Defenses of Victim
– Timing of Viral Entry (All at Once, Piece by Piece)
– Hair and Mucous in Nasal Vessels, ……
• Data From Biological Warfare Agents [Fra97]
– Brucellosis 10-100
– Q fever 1-10
– Tularaemia 10-50
– Smallpox 10-100
– Viral Haemorrhagic Fevers 1-10
– Tuberculosis 1 (!!)
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CFD Center, George Mason University
Viral Load / Infectious Dose
• Many Factors:
– State of Immune Defenses of Victim
– Timing of Viral Entry (All at Once, Piece by Piece)
– Hair and Mucous in Nasal Vessels, ……
• Data From Biological Warfare Agents [Fra97]
– Brucellosis 10-100
– Q fever 1-10
– Tularaemia 10-50
– Smallpox 10-100
– Viral Haemorrhagic Fevers 1-10
– Tuberculosis 1 (!!)
Cov-19: 100-200
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CFD Center, George Mason University
Covid-19 Lifetime Outside the Body
• Air: 1-2 Hours
• Some Surfaces: 1-2 Days
• Some Variability With Humidity/Temperature
• Some Variability With UV/Sunlight Radiation
– The More Sunlight, The Shorter The Lifetime
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CFD Center, George Mason University
Infectivity of Covid-19
• Different Types: How Effective ?
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CFD Center, George Mason University
Infectivity of Covid-19
• Different Types: How Effective ?
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CFD Center, George Mason University
Covid-19 vs. Influenza/Flu
• Max Yearly US Deaths From Influenza/Flu: 60K
– No Lockdowns, No Preventive Measures
• US Deaths After 12-Months of Covid-19: 400K
– With Lockdowns, With Preventive Measures
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CFD Center, George Mason University
Covid-19 vs. Influenza/Flu
• Max Yearly US Deaths From Influenza/Flu: 60K
– No Lockdowns, No Preventive Measures
• US Deaths After 12-Months of Covid-19: 400K
– With Lockdowns, With Preventive Measures
• ➔ Covid-19 Orders of Magnitude More Deadly
Than Influenza/Flu
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CFD Center, George Mason University
Mitigation Options
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CFD Center, George Mason University
Mitigation of Virus Transmission
• 2 Main Modes:
– Large Droplets → `Spitting’
– Small Droplets → `Smoke’
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CFD Center, George Mason University
Procedure
Measure
Large Droplets
(spitting)
Small Droplets
(cigarette smoke)
Person-Air-
Person
Person-Surface-
Person
2m/6ft Distance
Face Masks
Periodic Hand
Cleaning
Plexiglass
Shields
1-Way Person
Traffic
2x Daily
Cleaning
Nightly UV
Cleaning
Maximize Fresh
Air in HVAC
Hard UV Lamps
in HVAC Ducts
HEPA Filters in
HVAC Ducts
Upper Room UV
Cleaning
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CFD Center, George Mason University
Physics
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CFD Center, George Mason University
Air: Navier-Stokes, Incompressible
Conservation of Mass, Momentum and Energy:
r:Density T: Temperature
v:Velocity cp: Heat Capacitance
p:Pressure k: Thermal Conductivity
m:Viscosity 𝛽: Expansion Coefficient
𝐠: Gravity f,s: External Forces/Heat Sources
∇ ⋅ 𝐯 = 0𝜌𝐯,𝑡 + 𝜌𝐯 ⋅ ∇𝐯 + ∇𝑝 = ∇𝜇∇𝐯 + 𝜌𝐠𝛽(𝑇0 − 𝑇) + 𝐟
𝜌𝑐𝑝𝑇,𝑡 + 𝜌𝑐𝑝𝐯 ⋅ ∇𝑇 = ∇𝑘∇𝑇 + s
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CFD Center, George Mason University
Equations for Diagnostics
• Age of Air
𝑎,𝑡 + 𝐯 ⋅ ∇𝑎 = 1
• Pathogen Concentration
𝑐,𝑡 + 𝐯 ⋅ ∇𝑐 = ∇𝑑𝑐∇𝑐 + sc
• UV Irradiation
𝐼,𝑡 + 𝐯 ⋅ ∇𝐼 = ∇𝑑𝐼∇𝐼 + sI
• …
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CFD Center, George Mason University
Particle Motion and Temperature
• Velocity and Position
• Temperature
p
pp3
;6
vx
Dv
==dt
d
dt
ddp
r
Qdt
dTdcppp =
p3
6
r
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CFD Center, George Mason University
Momentum Transfer
• Drag Force of Each Particle
• Drag Coefficient and Reynolds-Number
( )piip
2
vv2
1
4−−= vvr
di c
dD
( )m
r dcd
p687.0 Re;Re15.01Re
24,1.0max
vv −=
+=
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CFD Center, George Mason University
Heat Transfer
• Heat Flux For Each Particle
• Film Coefficient, Nusselt- and Prandtl-Number
( ) ( ) 442
pp TTTThdQ −+−=
m
kcNu
d
Nukh
p=+=
= Pr;RePr459.02; 55.0333.0
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CFD Center, George Mason University
Numerics
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CFD Center, George Mason University
Basic Elements of Solver (1)• Spatial Discretization: Unstructured Grids
– Arbitrary Geometries
– Adaptive Refinement
• Spatial Discretization: Simple FEM
– One-Type Element Code: Speed and Simplicity
– Use Tetrahedra Even in Boundary Layers
• Temporal Discretization: Explicit Advection
– Physically Interesting Scales ➔ Need Accuracy
– 1-Step Schemes: Moving Grids, Adaptive Refine/Remesh
• Temporal Discretization: Implicit Viscous/Pressure
– Low-Storage Iterative Solvers
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CFD Center, George Mason University
Basic Elements of Solver (2)• Steady Results: Independent of Timestep
– Confidence
– Convergence Study Possible
• Edge-Based Data Structures
– Reduction of Indirect Addressing
– Reduction in Flops
• Extensive Renumbering
– Avoidance of Cache-Misses
– Shared-Memory Parallel Machines
• Preconditioning:
– Diagonal: Isotropic Grid
– Linelets: Stretched (RANS) Grid
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CFD Center, George Mason University
Temporal Discretization: Δ-Scheme (1)
• Advective / Diffusive Prediction:
• Pressure Correction:
v𝑛+1 = v𝑛 + Δv𝑎 + Δv𝑝 = v∗ + Δv𝑝
1
Δ𝑡− θ∇𝜇∇ (v∗ − v𝑛) + v𝑛 ⋅ ∇v𝑛 + ∇𝑝𝑛 = ∇𝜇∇v𝑛
∇ ⋅ v𝑛+1 = 0v𝑛+1 − v∗
Δ𝑡+ ∇(𝑝𝑛+1 − 𝑝𝑛) = 0
⇒ ⇒ ∇2(𝑝𝑛+1 − 𝑝𝑛) =∇ ⋅ v∗
Δ𝑡
v𝑛 → v∗
𝑝𝑛 → 𝑝𝑛+1
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CFD Center, George Mason University
Temporal Discretization: Δ-Scheme (2)
• Velocity Correction:
Remarks:
– Residuals of Pressure Correction Vanish for Steady-
State
➔ Results Not Depend on Projection Scheme
➔ Results Not Depend on Δt
v𝑛+1 = v∗ − Δ𝑡∇(𝑝𝑛+1 − 𝑝𝑛)
v∗ → v𝑛+1
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CFD Center, George Mason University
Particle Motion and Temperature
• Position, Velocity and Temperature: ODEs of Type
• Integrated Explicitly; Typically: 4th Order Runge-
Kutta
),,,( turdt
dufux=
1
1
+−=
iki
),,,( 111 −+−+−++ =− inin
fi
inin turtuu ux
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CFD Center, George Mason University
Particle Tracking
• Need: Flow Variables At Location of Particle
• ➔ Need Host Element for Each Particle
• Initialization: Bins + Near-Neighbour Search
• Incremental: Near-Neighbour Search
– Vectorized and Parallelized for OMP
– Also Running in MPI
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CFD Center, George Mason University
UV Radiation
• Irradiation Function of Distance/Angle
• Shading Possible ➔ Ray-Tracing
– From Element (Gauss-Points) to UV Lamp
𝑠𝐼 ~1
𝑟2
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CFD Center, George Mason University
Raytracing With FEM Grids
• Any Point P:
• ➔
• Given Input Location, Obtain Output
• Output Faces: ξo=0
• Get min(Δti), Δti > 0 ➔ Neighbour Element
𝐱𝑃 = 𝐱𝐴 + 𝐱𝐵𝐴 ξ + 𝐱𝐶𝐴 η + 𝐱𝐷𝐴 ζ = 𝐱𝐴 + 𝐆 𝝃
𝝃 = 𝐆−1𝐱𝑃𝐴
𝝃o = 𝝃i + 𝐆−1Δ𝐱= 𝝃i + 𝐆−1𝐯Δt = 𝝃i + 𝜶 Δt
J. Favre and R. Löhner – Ray Tracing With a Space-Filling Finite Element Mesh;
IJNME 37, 3571-3580 (1994)
i
o
A
B
C
D
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CFD Center, George Mason University
Ceiling UV In Hospital Room
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CFD Center, George Mason University
Coupling of CFD and CCD
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CFD Center, George Mason University
Coupling of CFD and CCD
For Each Gridpoint:
Temperature
Smoke
Toxic Substances
Pathogens
…
3-D to Tria
(CCD Background Grid)
CFD: FEFLO
CCD: PEDFLOW
Ellipse/Point to 3-D Body
For Each Pedestrian:
Position, Velocity
Temperature
Pathogens
…
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CFD Center, George Mason University
CCD → CFD
• Several Options Possible
• Body Fitted
– Transcribe Discrete Surface from CCD → CFD / Merge
– Move/Smooth/Remesh CFD Mesh
• Embedded
– Transcribe Discrete Surface from CCD → CFD
– Obtain Intersections With Edges/New BC
• Immersed
– Transcribe Discrete Volume from CCD → CFD
– Obtain CFD Points Inside CCD Domain/New BC
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CFD Center, George Mason University
CCD → CFD
As Tet-Mesh As Spheres
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CFD Center, George Mason University
Immersed Bodies
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CFD Center, George Mason University
Immersed Body: Options
• Desired: In Body Region: vf = wb
• Kinematic: Impose: vf = wb
• Kinetic : Use Force: f = c0 (vf – wb) [Goldstein]
• Kinetic/Kinematic: [Mohd-Yusof]:
i
n
i
n
iiii
i
ttr
vwMffr
vM −
−=+=
+1
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CFD Center, George Mason University
Immersed Body: Extensions
• Kinematic:
– Extend to Higher Order [Balaras]
– Same as for Embedded
• Kinetic:
– Use Lagrange Multipliers [Glowinsky]
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CFD Center, George Mason University
Search for Points in Bodies: Options
• Option 1:
– Store CFD Points in Bin/Octree
– Loop Over Immersed Body Elements
• Get Bounding Box
• Get CFD Points in Bounding Box
• Detailed In/Out Analysis
• Option 2:
– Store Immersed Body Elements in Bin/Octree
– Loop Over CFD Points
• Get Immersed Body Element in Region
• Detailed In/Out Analysis
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CFD Center, George Mason University
Pedestrian Motion
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CFD Center, George Mason University
Discrete Models
• Any Pedestrian Flow Simulation:
– Global Movement: Strategic, Tactical
– Local Movement: Operational
• Global Movement
– Targets (Regions/Lines/…) ➔Will Force
• Local Movement
– Collision Avoidance
• Social Force/ Contact Models
– Local Geometry Info
• Walls, Paths, Roughness, …
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CFD Center, George Mason University
PEDESTRIAN MOTION
• Newton's Law:
m v,t = f
x,t = v
• m: Mass
• v: Velocity
• x: Position
• f: Sum of All Forces
• Modeling Effort: f
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CFD Center, George Mason University
PEDESTRIAN FORCES• Internal Forces
– Will Force (Get There (in Time))
– Pedestrian Collision Avoidance Forces:
Intermediate Range
– Pedestrian Collision Avoidance Forces : Near
Range
– Wall/Obstacle (Collision) Avoidance Forces
• External Forces
– Contact: Other Pedestrians
– Contact: Walls/Obstacles
• …
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CFD Center, George Mason University
PEDFLOW• Mixture of Agent-Based and Social Force Model
• Forces Via by Minimal Set of Well-Defined
Parameters
– Relaxation Time (Fitness)
– Desired Velocity
– Pushiness (Distraction)
• Strategic and Tactical: Desired Locations/Time
• Operational: Local Collision Avoidance
• Background Grid for Geometry/Spatial Search
• Edge-Based Data Structures for Pedestrians
R. Löhner – On The Modeling of Pedestrian Motion;
Appl. Math. Mod. 34, 2, 366-382 (2010).
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CFD Center, George Mason University
Madrid Metro Station
• 3/11/2004 Attack
• Did Blast Analysis
• Follow-Up
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CFD Center, George Mason University
Madrid Bomb Attack
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CFD Center, George Mason University
Madrid Bomb Attack
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CFD Center, George Mason University
Madrid Bomb Attack
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CFD Center, George Mason University
Evacuation from Medina Mosque
• Over 105 Pedestrians
• Run on Laptop
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CFD Center, George Mason University
Rendering via 3-D Studio Max
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CFD Center, George Mason University
Examples
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CFD Center, George Mason University
Examples (1)
• Physics and Numerics: FEFLO
• Flow Initialization:
– Ambient: Quiescent, 20oC
– Sneeze: Spherical Region, r=5cm, Near Mouth
– V=5f(t) [m/sec] ; T=20+(37-20)f(t) [oC]
• Particle Initialization
– 4 Size Groups; Released Every 0.005 sec for 0.1 sec
– V=5 m/sec ; T=37oC
f(t)
t
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CFD Center, George Mason University
Examples (2)
• 3 Different Timescales, Depending on Size
– O(100) sec: Fast, Ballistic Drop of Large d=1.0 mm
Particles
– O(101) sec: Slower Drop of Intermediate d=0.1 mm
Particles
– O(102) sec: Transport of Small d<0.01 mm Particles
Through Air
• HVAC Systems:
– Good Mixers
– Complex Flowfields
– ➔ Condusive to Pathogen Transmission
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CFD Center, George Mason University
Sneezing in TSA Queues
12.47 Mels
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CFD Center, George Mason University
Sneezing in TSA Queues
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CFD Center, George Mason University
Sneezing in TSA Queues
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CFD Center, George Mason University
Sneezing in Hospital Room
2.25 Mels
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CFD Center, George Mason University
Sneezing in Hospital Room
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CFD Center, George Mason University
Hospital Room With UV Lamp
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CFD Center, George Mason University
Hospital Room With UV Lamp
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CFD Center, George Mason University
Sneezing in a Generic Classroom
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CFD Center, George Mason University
Sneezing in a Generic Classroom
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CFD Center, George Mason University
Sneezing in a Generic Classroom
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CFD Center, George Mason University
Classroom With Ceiling-UV
NBC Universal Today Show, September 30 (2020), see: https://www.today.com/health/ventilation-
covid-19-reduce-spread-proper-airflow-t192366
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CFD Center, George Mason University
Sneezing in Subway
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CFD Center, George Mason University
Sneezing in Subway
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CFD Center, George Mason University
Sneezing in Subway
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CFD Center, George Mason University
M. Gröndahl, Ch. Goldbaum and J. White - What Happens to Viral Particles on the Subway;
New York Times, August 10 (2020);
see also: https://www.nytimes.com/interactive/2020/08/10/nyregion/nyc-subway-coronavirus.html
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CFD Center, George Mason University
Narrow Corridor
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CFD Center, George Mason University
Narrow Corridor
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CFD Center, George Mason University
Narrow Corridor: Viral Load
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CFD Center, George Mason University
Reopening After the Crisis
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CFD Center, George Mason University
Reopening After the Crisis
• Will Have/Need Sensors to Monitor Environment
• Basic Questions:
– How Many ?
– Placed Where ?
• Current Approach:
– Run Many Scenarios Cases
– Place Many Sensors
– Keep (Recursively) the One Detecting the Most Cases
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CFD Center, George Mason University
Hospital Room: Problem Setup
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CFD Center, George Mason University
Hospital Room: Average Velocities
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CFD Center, George Mason University
Case Study: Hospital Room
• Cases 1,2,3:
– Contaminant/Pathogens Through Each of the 3 Different
Entry Vents
– 0-60 sec
• Case 4:
– Virus Production from Patient
– 0-10 sec
• Run for 300 sec (5 min) of Real Time
• Measure Contaminant/Pathogen Concentration on
all Walls
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CFD Center, George Mason University
Hospital Room: Age of Air
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CFD Center, George Mason University
Max Values Measured During 5 Mins
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CFD Center, George Mason University
Wall Data Recorded
• Points With No Cases Measured: 4308
• Points With 1 Case (Out of 4) Measured: 3377
• Points With 2 Cases (Out of 4) Measured: 1010
• Points With 3 Cases (Out of 4) Measured: 0
• Points With 4 Cases (Out of 4) Measured: 0
• Excluded Location:
– Above zmin (Minimum Height Requirement)
– Not on Beds/Furniture/Patient/Attendant
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CFD Center, George Mason University
Result: 2 (Optimal) Sensors
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CFD Center, George Mason University
Conclusions
• Summarized:
– Mechanical Characteristics of Virus Contaminants
– Transmission Via Droplets and Aerosols
• Emphasis on High-Fidelity (Hi-Fi) Physics
– PDEs
– Appropriate Numerical Methods
• Examples from the Built Environment
– TSA Queues, Hospital Rooms, Corridors, Trains, …
• Optimal Placement of Sensors
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CFD Center, George Mason University
Outlook
• Increase Realism
– Boundary Conditions for HVAC Systems [Entry, Mixing, …]
• Streamlining Simulation Toolbox/Workflow
• Field These Tools To Enable Smooth Post-Pandemic
Transition
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CFD Center, George Mason University
Thank You Very Much for
Your Attention
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CFD Center, George Mason University
Additional Material
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CFD Center, George Mason University
Temporal Discretization: Δ-Scheme (1)
• Advective / Diffusive Prediction:
• Pressure Correction:
v𝑛+1 = v𝑛 + Δv𝑎 + Δv𝑝 = v∗ + Δv𝑝
1
Δ𝑡− θ∇𝜇∇ (v∗ − v𝑛) + v𝑛 ⋅ ∇v𝑛 + ∇𝑝𝑛 = ∇𝜇∇v𝑛
∇ ⋅ v𝑛+1 = 0v𝑛+1 − v∗
Δ𝑡+ ∇(𝑝𝑛+1 − 𝑝𝑛) = 0
⇒ ⇒ ∇2(𝑝𝑛+1 − 𝑝𝑛) =∇ ⋅ v∗
Δ𝑡
v𝑛 → v∗
𝑝𝑛 → 𝑝𝑛+1
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CFD Center, George Mason University
Temporal Discretization: Δ-Scheme (2)
• Velocity Correction:
Remarks:
– Residuals of Pressure Correction Vanish for Steady-
State
➔ Results Not Depend on Projection Scheme
➔ Results Not Depend on Δt
v𝑛+1 = v∗ − Δ𝑡∇(𝑝𝑛+1 − 𝑝𝑛)
v∗ → v𝑛+1
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CFD Center, George Mason University
Spatial Discretization: Advection (1)
• Galerkin:
𝐫i = 𝐷𝑖𝑗𝐅ij = 𝐷𝑖𝑗 𝐟𝑖 + 𝐟𝑗
ij
k
ij
k
ij
ij
ij
kk
ijk
ikij
i ddDD
dSFS === ,,f
𝑑𝑘𝑖𝑗=1
2න 𝑁,𝑘
𝑖 𝑁𝑗 − 𝑁,𝑘𝑗𝑁𝑖 𝑑Ω
𝐟𝑖 = (𝑆𝑘𝑖𝑗v𝑖𝑘)𝐯𝑖
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CFD Center, George Mason University
Spatial Discretization: Advection (2)
• Need Consistent Numerical Fluxes
– Integration Along Characteristics
– Taylor-Galerkin/Streamline Diffusion/SUPG/GLS/…
– Edge-Based Upwinding
• Consistent Numerical Flux
• Higher Order Scheme via Limiting (e.g. MUSCL)
( ) ( )k
j
k
i
ij
k
ij
ji
ij
ji S vv2
1v,vFij +=−−+= vvff
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CFD Center, George Mason University
Spatial Discretization: Divergence (1)
• Galerkin:
• Need Consistent Numerical Fluxes (LBB Condition)
– Different Functional Spaces for v,p (e.g. Mini-Element)
– Artificial Viscosity/Stabilization
– Edge-Based Consistent Numerical Flux
ri = 𝐷𝑖𝑗Fij = 𝐷𝑖𝑗 f𝑖 + f𝑗
f𝑖 = 𝑆𝑘𝑖𝑗v𝑖𝑘
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CFD Center, George Mason University
Spatial Discretization: Divergence (2)
• Consistent Numerical Flux
• Higher-Order (4th Order Damping)
( )ij
ijij
ji
ij
jil
tpp
=−−+= ,ffFij
Fij = f𝑖 + f𝑗 − 𝜆𝑖𝑗 𝑝𝑖 − 𝑝𝑗 +𝐥𝑖𝑗
2⋅ ∇𝑝𝑖 + ∇𝑝𝑗
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CFD Center, George Mason University
Walls: Boundary Conditions• Walls: Bouncing, Sticking, Gliding, …
• Embedded Surfaces
Bouncing Sticking Gliding