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Speech can produce jet-like transport relevant to asymptomatic spreading of virus

Manouk Abkarian, Simon Mendez, Nan Xue, Fan Yang, Howard A. Stone

arXiv:2006.10671v3physics.flu-dynphysics.bio-ph

TL;DR

Speech and breathing flows are insufficiently characterized as transport mechanisms for exhaled material, limiting quantitative understanding relevant to virus-transmission guidance. The paper combines estimates, simulations, and experiments to characterize these flows and finds that plosive sounds generate directed puff and jet-like transport across distances reaching about 2 m or farther.

  • Problem

    Few quantitative studies document how speech-generated airflow transports exhaled material, despite asymptomatic and presymptomatic transmission during conversations.

  • Method

    The paper uses order-of-magnitude estimates, numerical simulations, and laboratory experiments to characterize breathing- and speech-induced flows, including phonetic effects.

  • Results

    Plosive sounds such as ‘P’ produce directed vortical puffs and conical jet-like flows, with transport scaling from distances below 0.5 m to about 2 m or farther.

  • Takeaways & Limitations

    Sequential plosives can create long-distance jet-like transport, while airflow speeds at 1–2 m are typically tens of centimeters per second and ambient currents may become dominant.

  • Takeaways & Limitations

    The authors are not trained in public health and caution that their results should not directly determine social-distancing guidelines.

Abstract

from arXiv · show

Many scientific reports document that asymptomatic and presymptomatic individuals contribute to the spread of COVID-19, probably during conversations in social interactions. Droplet emission occurs during speech, yet few studies document the flow to provide the transport mechanism. This lack of understanding prevents informed public health guidance for risk reduction and mitigation strategies, e.g. the "six-foot rule". Here we analyze flows during breathing and speaking, including phonetic features, using order-of-magnitudes estimates, numerical simulations, and laboratory experiments. We document the spatio-temporal structure of the expelled air flow. Phonetic characteristics of plosive sounds like 'P' lead to enhanced directed transport, including jet-like flows that entrain the surrounding air. We highlight three distinct temporal scaling laws for the transport distance of exhaled material including (i) transport over a short distance ($<$ 0.5 m) in a fraction of a second, with large angular variations due to the complexity of speech, (ii) a longer distance, approximately 1 m, where directed transport is driven by individual vortical puffs corresponding to plosive sounds, and (iii) a distance out to about 2 m, or even further, where sequential plosives in a sentence, corresponding effectively to a train of puffs, create conical, jet-like flows. The latter dictates the long-time transport in a conversation. We believe that this work will inform thinking about the role of ventilation, aerosol transport in disease transmission for humans and other animals, and yield a better understanding of linguistic aerodynamics, i.e., aerophonetics.

INTRODUCTION

The paper addresses limited quantitative understanding of breathing and speaking flows as a potential transport mechanism for pathogens during social interactions. It characterizes how speech, especially plosive sounds, structures exhaled-air transport across distances and timescales.

  • Research gap: Quantitative studies of breathing and speaking flows during speech remain limited despite evidence that speech emits droplets.This gap matters for understanding transport during everyday interactions and informing mitigation strategies.
  • Research questions: The study asks how breathing, speaking, laughing, or singing by asymptomatic or presymptomatic individuals affects surrounding air and transports exhaled material.It also examines whether social-interaction position or orientation could reduce potential exposure to exhaled air.
  • Key mechanism: Plosive sounds such as ‘P’ enhance directed transport, producing jet-like flows that entrain surrounding air.Sequential plosives can accumulate into puff-packets and generate conical, jet-like flow structures.
  • Transport regimes: Exhaled material follows three transport regimes: below 0.5 m within a fraction of a second, approximately 1 m through individual vortical puffs, and about 2 m or farther through sequential plosives.The longest-range regime is associated with conical, jet-like flows and dictates long-time transport in conversation.
  • Scope: The authors focus on mouth airflow during inhalation and exhalation, examining how breathing and speech features influence exhaled-material transport.The mouth flow is studied because it is considered more directed toward a potential facing interlocutor.

Orders of Magnitude

The paper estimates mouth-flow Reynolds numbers for breathing and speaking to assess the dominant fluid-mechanical regime. The estimates indicate inertial effects and generally time-dependent, turbulent behavior.

  • Flow scales: Typical mouth airflow speeds are approximately 0.5–2 m/s, with volumetric flow rates of approximately 0.2–0.7 L/s.The order-of-magnitude estimate uses a characteristic mouth radius a = 1 cm.
  • Dimensionless estimate: Reynolds numbers are defined as Re = 2ua/ν, using mouth-flow speed u, characteristic radius a, and air kinematic viscosity ν.The analysis uses ν ≈ 1.5 × 10^-5 m^2/s.
  • Results: Breathing has Re = O(7 × 10^2–3 × 10^3), while speaking has Re = O(1 × 10^3–7 × 10^3).These magnitudes indicate that inertial effects dominate and that the flows are generally time dependent and turbulent.

Breathing and Blowing as Jet-like Flows

Breathing, blowing, and speaking produce directed outflows whose structure and reach depend on flow strength and phonetic content. Plosive-rich speech generates interacting vortical puffs and conical, jet-like transport, while speech with varied sounds produces more variable directions.

  • Breathing and Blowing as Jet-like Flows: Breathing and strong blowing produce qualitatively similar conical, jet-like flows, although blowing reaches higher velocities than ordinary breathing.Breathing velocities are about 0.3–1 m/s, whereas blowing can reach a few meters per second; both show cone angle 2α ≈20°.
  • Asymmetry of Exhalation and Inhalation: Exhalation forms starting jet-like flows that propagate about a meter, while inhalation draws air inward more uniformly from around the mouth.This asymmetry transports large droplets and aerosols away from the speaker during outflow.
  • Speaking, Plosive Sounds and Jet-like Flows: Speech produces jerky, direction-changing flows, with sound-specific puffs that can vary substantially in angle and speed.In one sentence, the ‘coro’ syllables reached almost 8–12 cm/s, while the puff associated with ‘virus’ was directed upward at about 50° and reached 5–7 cm/s.
  • Speaking, Plosive Sounds and Jet-like Flows: A plosive ‘P’ initially drives a jet-like flow scaling as t^1/2, then transitions within 10–100 ms to puff-like t^1/4 transport that can approach a meter.The slower regime is associated with stabilized transport by a vortex ring.
  • Speaking, Plosive Sounds and Jet-like Flows: Sequential plosives in ‘Peter Piper picked a peck’ create interacting vortices and an average conical jet extending about a meter.Average velocities are tens of cm/s, with peak velocities near 1.2–1.5 m/s; accumulated puff-packets transition toward turbulent jet-like flow.
  • Speaking, Plosive Sounds and Jet-like Flows: The study focuses on plosive-containing phrases because they produce directed, approximately conical turbulent jets relevant to aerosol transport during speech.The authors connect this directed transport to the possibility of asymptomatic virus transmission by aerosols.

MODELING

The paper summarizes established mathematical models of steady turbulent jets to interpret its experimental and numerical results. These models describe transport, mixing, and entrainment in jet-like flows.

  • MODELING: The modeling section uses well-known mathematical models to assist interpretation of the paper’s experimental and numerical results.The models are presented as a completeness-oriented summary rather than as the paper’s main new modeling contribution.

Characteristic Features of a Steady Turbulent Jet

A steady turbulent jet is characterized through conservation and growth of momentum and through entrainment of surrounding air. These properties describe how jet speed, volume flux, and dilution change downstream.

  • Characteristic Features of a Steady Turbulent Jet: Steady turbulent-jet analysis tracks volume flux, linear momentum, kinetic energy, and entrainment of surrounding air.Entrainment dilutes the jet and helps characterize mixing with its surroundings.
  • Characteristic Features of a Steady Turbulent Jet: For a conical jet, constant linear-momentum flux gives v^2A = constant, so axial speed decreases as cross-sectional area grows downstream.The model writes v(x)/v0 = (A0/A(x))^1/2 < 1 and uses A(x) ∝(αx)^2 beyond the mouth.
  • Characteristic Features of a Steady Turbulent Jet: Because volume flux satisfies Q = vA, downstream entrainment increases the jet volume flux relative to its exit value.The model gives Q/Q0 = (A(x)/A0)^1/2 > 1.

Starting Jets and Puffs

The paper distinguishes continuous starting jets from finite-momentum puffs and investigates how breathing and speaking generate interacting puff trains. Simulations compare periodic breathing-like and speech signals across different phrases and exhaled volumes.

  • Starting Jets and Puffs: A starting jet has penetration distance L ∝t^1/2, whereas an isolated puff travels as L ∝t^1/4.Starting jets arise from sudden momentum injection; puffs conserve finite linear momentum in inertially dominated flows.
  • Starting Jets and Puffs: Breathing and speaking release interrupted jets and puffs sequentially, producing interacting trains rather than continuous jets or isolated puffs.The simulations investigate how these puff trains grow in space and time.
  • Starting Jets and Puffs: The simulations use 3-D incompressible Navier-Stokes equations driven by periodic flow-rate signals from a 1 cm × 1.5 cm elliptical orifice.Speaking signals contain relatively high-frequency volume-flow changes; studied exhaled volumes range from 0.5–1 L per breath.
  • Starting Jets and Puffs: Four 4-second-cycle cases compare normal breathing, breathing-like exhalation, a phrase with ordinary speech, and a phrase rich in plosive sounds.The phrases are “Sing a song of six pence” and “Peter Piper picked a peck”; simulations use 0.5, 0.75, and 1.0 L per breath.
  • Starting Jets and Puffs: All simulated cases form conical turbulent-like jets that spread transversely and mix exhaled contents with the environment.Continual exhalation and inhalation cycles progressively build the jet, while far-field flow loses an obvious pulsation signature.

Quantifying the Jets

The study quantifies jet angle, length, and cycle-averaged velocity to compare breathing and speaking flows. Higher mean flow rates lengthen jets, while speech modulation broadens them laterally and shortens their axial extent.

  • Quantifying the Jets: Jet angle α encloses 90% of tracer particles, while jet length L places 90% of tracers upstream of x = L.The included angles are stable after initial cycles and are typically 10–14°.
  • Quantifying the Jets: Cycle-averaged axial velocity profiles are tracked downstream to 2.0 m across as many as 14 cycles or 56 s.The plotted v(x) ∝1/x relation serves as a guide for steady turbulent-jet behavior.
  • Quantifying the Jets: The jet-length evolution is evaluated using nondimensional time 2v0t/(aα), mouth equivalent radius a ≈1.22 cm, and average exhalation speed v0.Two power laws are plotted against the numerical evolution of L/a.
  • Quantifying the Jets: Higher mean flow rates produce longer jets, while speech modulation increases lateral growth, cone angle, and reduces jet length.For a fixed exhaled volume per cycle, different exhalation types produce comparable jet lengths.

A Train of Puffs

Multi-cycle averaging shows that both breathing and speaking develop far-field velocity profiles characteristic of turbulent starting jets. A sequence of exhaled puffs therefore becomes a jet-like flow that governs transport downstream.

  • A Train of Puffs: Period-averaged axial velocity falls as v(x) ∝x−1 in the far field for both speech and breathing.The near-mouth flow is laminar, but downstream behavior resembles a steady turbulent starting jet.
  • A Train of Puffs: At breathing and speaking Reynolds numbers, a train of puffs transitions into a turbulent, jet-like flow that dominates exhaled-material transport.Time averaging over each period filters turbulent fluctuations and yields approximate axial-speed profiles.

A Diffusive-like, Directed Cloud of Exhaled Air

The paper models growing exhaled clouds as directed starting jets whose length follows established temporal scaling. The resulting far-field concentration decreases with distance, including substantial dilution near 2 m.

  • A Diffusive-like, Directed Cloud of Exhaled Air: The starting-jet scaling predicts L ∝t1/2 for growing jets at constant angle.The characteristic time is estimated by integrating axial travel time through a jet with angle α.
  • A Diffusive-like, Directed Cloud of Exhaled Air: The theoretical length prediction matches nondimensional numerical trends when v0 is the average mouth-exit speed during exhalation.Peak velocity and flow-rate details do not change the scaling, but they influence the spreading angle.
  • A Diffusive-like, Directed Cloud of Exhaled Air: For one-cycle simulations, the exhaled material behaves as one large puff with L ∝t1/4.This scaling is reported as consistent with the experiments.
  • A Diffusive-like, Directed Cloud of Exhaled Air: At a = 1 cm, α = 10°, and L = 2 m, directed-jet concentration falls by approximately 0.03, with simulations yielding typical dilution levels of 0.04–0.05.The far-field concentration is modeled as c(x)/c0 ∝a/(αx).

EXPERIMENTAL CHARACTERIZATION OF THE SPREADING

Laser-sheet experiments measured speech propagation distance over time and found distinct behavior for plosive speech, breathing, and ambient flow. Plosive-phrase data agreed with a t^1/2 scaling, while ambient currents increasingly affected transport beyond about 2 m.

  • Experimental setup: The experiment measured propagation distance L to a laser sheet as a function of time while repeating plosive and non-plosive sentences.Breathing, background flow, and the phrase “Sing a song of six pence” provided comparison cases.
  • Ambient-flow effects: A weak laboratory ambient flow had speed O(0.05 m/s), compared with initial experimental speeds O(0.5 m/s).It therefore had little effect on initial puff spreading for L < 1 m.
  • Ambient-flow effects: Ambient flow introduced about 20% uncertainty during puff spreading and deceleration at distances such as L > 2 m.The authors note that these results can help estimate the crossover between speech-dominated and ventilation-dominated transport.

DISCUSSION

The discussion identifies three transport regimes for exhaled material during conversations containing plosives, spanning sub-meter delivery, individual vortical puffs, and longer-range accumulated jets. It also emphasizes that ventilation and speech intensity affect interpretation and potential exposure.

  • Three transport regimes: Less than about 50 cm, exhaled material arrives within a fraction of a second through upward, downward, and forward-directed flows.The forward regime, especially for bilabial plosives, follows a t^1/2 starting-jet power law.
  • Three transport regimes: Out to about 1 m, individual vortical puffs from syllables with single plosives drive slower transport following a t^1/4 power law.These puffs represent a distinct intermediate-time transport regime.
  • Three transport regimes: Out to about 2 m or farther, accumulated plosive puffs form a conical, jet-like flow with a t^1/2 power law.Speech behaves as a train of puffs that effectively generates a continuous turbulent jet.
  • Scope and limitations: The study does not account for speaker head or trunk movement, ventilation-driven background motion, or differing vortical strengths among speech sounds.These omissions limit direct interpretation of social-distancing guidance and leave details of jet spreading unresolved.
  • Ventilation and exposure: Typical airflow speeds at 1–2 m are tens of centimeters per second, so ambient currents may dominate at those distances.Estimating potential dose requires both distance from the speaker and time spent in front of the speaker.

Appendix A: Speaker

The appendix describes a single-subject laboratory setup for visualizing breathing and speech flows with seeded fog, laser-sheet imaging, and particle image velocimetry. It also documents the numerical simulation framework used to study turbulent transport.

  • Speaker: Because of pandemic restrictions, one consenting 44-year-old male subject participated in the laboratory experiments.The study received Princeton University IRB approval under protocol #12834.
  • Flow visualization: A laser sheet approximately 2 m long, 1 m high, and 3 mm thick illuminated the seeded flow for high-speed imaging.The sheet was formed from a 532 nm laser expanded with a concave cylindrical lens.
  • Flow visualization: A fog machine generated approximately 1 μm water-based droplets that acted as passive tracers of the local flow.The droplets persisted for tens of minutes with no notable sedimentation during the experiments.
  • Flow visualization: A laser sheet placed perpendicular to the flow measured the axial structure of out-flows at distance L in front of the speaker.The camera was positioned perpendicular to the laser sheet.
  • Velocity measurement: Particle image velocimetry used 16 × 16 pixel interrogation windows, approximately 2 cm × 2 cm, with 50% overlap to estimate local velocities.The image sequences were processed with PIVlab using cross-correlation.
  • Numerical modeling: Large eddy simulations with the YALES2BIO solver modeled turbulent and intermittent speech flows using representative time-periodic mouth flow rates.The simulations solved the incompressible Navier–Stokes equations for an elliptical mouth-like orifice.
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