Numerical Modeling of the Macroscopic Behavior of a Crowd of People under Emergency Conditions Triggered by an Incidental Release of a Heavy Gas: an Integrated Approach
Abstract
In the present study, we focus on modeling and simulations of macroscopic behavior of a crowd of people under emergency conditions. We present a newly developed integrated numerical approach, which combines the time-dependent RANS method in predicting the temporal and spatial evolution of a heavy gas turbulent dispersion with the macroscopic model of a crowd movement. The coupling is imposed through locally determined concentration of the heavy gas which directly impacts the movement of the crowd of people. The potential of the developed approach is demonstrated in a series of cases that include the crowd behavior on an open railway platform and inside a train station, as well as predictions of field studies of turbulent dispersion of a heavy gas under windy conditions. Finally, various scenarios have been analyzed of the coupling between the local concentration of a potentially harmful heavy gas on crowd behavior through different forms of the cost functions. In conclusion, the version of the integrated model is recommended for future optimization studies of crowd dynamics following an incidental release of heavy gas.
Keywords
Crowd modeling Evacuation CFD Heavy gas Turbulent dispersion Integrated approach1 Introduction
Here, we present details of development of an integrated in-house computation code for predictions of a crowd movement under influence of an incidental release of a heavy gas. The first part of the code is based on a macroscopic description of a crowd dynamics (fluid-dynamics similarity) and discretised form of the crowd continuity equation, [6, 12]. This code is then integrated with an existing in-house code based on solving a full set of the time-dependent Navier-Stokes equations (T-RANS) [13, 14] with additional terms due to concentration buoyancy in order to properly mimic the behavior of a heavy gas. The novel contribution is in a fully coupled approach, where we propose how to redefine some of the model parameters (so called cost functions) used in the crowd dynamics model, in terms of the local concentration of a heavy gas.
2 Mathematical Formulation and Governing Equations
We divide the complete set of transport equations into three categories. The first category includes macroscopic description of the behavior of the crowd of people based on the conservation of the crowd mass. The second category lists the time-dependent Reynolds-Averaged Navier-Stokes (T-RANS) for turbulent dispersion of heavy gas. Finally, the third category of equations are redefined cost functions that take into account the local critical concentration of the heavy gas. In the present study we also assume that there is no feedback influence of the local crowd movement on the distribution of heavy gas, i.e. we deal with one-way concentration/crowd dynamics coupling.
2.1 Crowd evacuation model
2.2 A heavy gas turbulent dispersion
The turbulence model coefficients
C_{𝜖1} | C_{𝜖2} | C_{𝜖3} | σ_{k} | σ_{ε} | C_{μ} | C_{c} |
---|---|---|---|---|---|---|
1.44 | 1.92 | 0.8 | 1 | 1.3 | 0.09 | 0.15 |
2.3 Coupling of the crowd evacuation model and a heavy gas dispersion: the redefined cost function
3 Numerical Method
Here we provide the most important details of numerical methods developed and used in the present research for predictions of crowd movement and the turbulent dispersion of heavy gas. In essence, we have two separate algorithms (crowd dynamics and CFD of heavy gas turbulent dispersion), which are coupled through a/the two-dimensional projection of the local distribution of the heavy gas concentration at the particular distance from the ground. These local distributions of the heavy gas concentration are then used as the input parameters to redefine the particular cost functions of the crowd movement.
3.1 Algorithm used for evolution of crowd behavior
- 1.
A numerical mesh of the domain is created and an initial crowd density distribution is set.
- 2.
Using the crowd density distribution, the speed and cost distribution are calculated.
- 3.
The Eikonal equation is solved using the speed and cost distributions, resulting in the potential ϕ.
- 4.
The gradient of ϕ is calculated, and, combined with the speed distribution, the velocity field is obtained.
- 5.
The fluxes in each cell are calculated from the density distribution and the velocity field in the center of control volumes.
- 6.
Using the calculated fluxes, the density distribution at the next time (t + Δt) is calculated.
- 7.
Steps 2 to 6 are repeated until the desired time is reached.
Constants used in the Runge-Kutta discretization, the implicit Radau IA scheme, [16]
b_{1} | b_{2} | c_{1} | c_{2} | a_{11} | a_{12} | a_{21} | a_{22} |
---|---|---|---|---|---|---|---|
\(\displaystyle \frac {1}{4}\) | \(\displaystyle \frac {3}{4}\) | 0 | \(\displaystyle \frac {2}{3}\) | \(\displaystyle \frac {1}{4}\) | \(-\displaystyle \frac {1}{4}\) | \(\displaystyle \frac {1}{4}\) | \(\displaystyle \frac {5}{12}\) |
3.2 Numerical details of the finite-volume code for a heavy gas turbulent dispersion
For the CFD part of the algorithm, we have used in-house finite-volume computer code based on discretization of the mass, momentum, energy and concentration equations within the time-dependent Reynolds-Averaged Navier-Stokes (RANS) approach, for the non-orthogonal geometries with a structured non-uniform numerical mesh based on the hexagonal elements, [13]. This computer code is used as a platform for numerous additional extensions to mimic some essential features of atmospheric and environmental flows. Some of the most important features include: (i) possibility to simulate a complex terrain orography [17], (ii) an efficient mesh generation and visualization tools to take into account the presence of buildings within the complex urban areas [13, 18, 19] (iii) the presence of vegetation and surface roughness effects [20], (iv) ability to use a highly skewed mesh for air flows around hilly or mountain terrains [21], (v) inclusion of complex atmospheric chemistry for proper simulations of the ozone distribution [22], (vi) as well as the radiation and ground/soil conductivity effects as a part of the global energy balance for simulation of the urban heat island phenomenon [23, 24].
Here, we will provide just a short description of the most essential numerical details of the CFD algorithm. For the numerical algorithm, the non-staggered grid was employed for all variables with the Rhie-Chow interpolation to prevent decoupling between velocity and pressure fields. The standard iterative predictor-corrector SIMPLE algorithm was used to couple the instantaneous velocity and pressure to satisfy both continuity and momentum equations. The time-rate of change of transport variables was discretised by the fully-implicit three-consecutive time-step method. The diffusive term of discretised transport equations was approximated with the second-order central-differencing scheme (CDS). The convective term in momentum equation was represented by the second-order quadratic upwind scheme (QUICK), while the second-order linear upwind scheme (LUDS) was applied for the scalar equation.
3.3 Numerical details of the integrated approach
To couple the CFD heavy gas turbulent dispersion model and crowd evacuation model, the pre-specified horizontal cross-sections from a three-dimensional concentration field are extracted and projected onto the underlying two-dimensional mesh by using standard bilinear interpolation, as a part of the ’Remesh’ procedure shown in Fig. 1. Similarly, because of different characteristic time-steps used in CFD and crowd evacuation algorithms, a simple linear interpolation is applied to get values of the concentration at particular time-instants that are required to calculate characteristic redefined cost functions for a fully coupled approach.
4 Results and Discussion
4.1 Validation of the crowd evacuation model
We conclude that the crowd evacuation algorithm applied here showed a good agreement with similar studies in literature and the potentials of its application are demonstrated for real-life cases.
4.2 Validation of heavy gas turbulent dispersion model
Specification of the flow parameters used in numerical simulations of ’Thorney Island Trial Number #26’
Density ratio | Wind at 10 m | Released volume | Surface | Parameter | Re | Ri_{b} |
---|---|---|---|---|---|---|
ρ_{gass}/ρ_{air} | U_{0} | V_{gas} | Roughness | λ | ||
2 | 1.9 m/s | 1970 m^{3} | 0.005 m | 0.07 | 1.7 ⋅ 10^{6} | 34.8 |
Specification of numerical parameters used in numerical simulations of ’Thorney Island Trial Case Number #26’
Mesh: | Min. CV size (x × y × z) in m | Max. CV size (x × y × z) in m | Total number of CVs |
---|---|---|---|
Coarse | 0.5 × 0.5 × 0.5 | 2.5 × 5.0 × 5.0 | 1,653,080 |
Fine | 0.25 × 0.25 × 0.25 | 2.5 × 2.5 × 2.5 | 6,234,920 |
Finally, from the observed behavior of the CFD results for a heavy gas turbulent dispersion, we conclude that the present model (with additional concentration buoyancy terms and GGDH representation of the turbulent concentration fluxes), was able to reasonable predict available field scale concentration measurements at various locations. This is in contrast to some earlier RANS studies reported in the literature (for the identical test case), which failed to show a good agreement with experiments.
4.3 Integrated approach
Test case parameters for perceived cost coupling
Description | α | C_{ref} | c_{2}/c_{1} |
---|---|---|---|
No coupling. | 0 | N/A | 0 |
Perceived cost << regular cost | 1 | 0.5 | 0.04 |
Perceived cost \(\sim \) regular cost | 25 | 0.02 | 1 |
Perceived cost \(\sim \) regular cost | 50 | 0.01 | 2 |
Perceived cost >> regular cost | 250 | 0.002 | 10 |
Test case parameters for movement reduction coupling
Description | f | C_{ref} |
---|---|---|
No coupling | 0 | 0.02 |
Mild slowdown (10%) | 0.1 | 0.02 |
Severe slowdown (50%) | 0.5 | 0.02 |
Standstill at reference concentration | 1.0 | 0.02 |
Standstill at half concentration | 1.0 | 0.01 |
Test case parameters for combined coupling including the critical heavy gas threshold (C_{crit})
Description | C_{crit} | α | f |
---|---|---|---|
No coupling | \(\infty \) | 0 | 0 |
Critical concentration coupling (C) | 0.05 | 0 | 0 |
Critical concentration + perceived cost (CP) | 0.05 | 50 | 0 |
Critical concentration + movement reduction (CM) | 0.05 | 0 | 0.2 |
All couplings (CPM) | 0.05 | 50 | 0.2 |
Parameters for crowd movement simulations (based on the full (CPM) coupling) using different crowd sizes
ρ_{0} [people/m^{2}] | Total crowd [people] | C_{crit} | α | f |
---|---|---|---|---|
0.1 | 300 | 0.05 | 50 | 0.2 |
0.5 | 1,500 | 0.05 | 50 | 0.2 |
2.5 | 7,500 | 0.05 | 50 | 0.2 |
5.0 | 15,000 | 0.05 | 50 | 0.2 |
8.0 | 24,000 | 0.05 | 50 | 0.2 |
5 Conclusions
We have presented a detailed mathematical framework behind the newly developed integral numerical approach that mimics the macroscopic response behavior of a crowd of people when exposed to an incidental release of heavy gas. We have shown that a relatively simple macroscopic evacuation model can produce qualitatively sound results which were in good agreement with similar studies in the literature. The crowd evacuation model we have developed has demonstrated great flexibility and numerical robustness in simulating complex floor configurations which include multiple walls (both straight and slanted) and outlets. We have demonstrated that CFD based on a time dependent two-equation eddy viscosity RANS approach can serve as a simple and reasonable accurate approach in simulations of a heavy gas turbulent dispersion. Here we also stressed the importance of the additional concentration buoyancy terms in both momentum and turbulence parameters equations, as well as the use of the GGDH in modeling of turbulent concentration fluxes. Finally, we presented some originally developed coupling schemes between the evacuation model and CFD, in which we applied various forms of the redefined cost functions and crowd movement reduction, which were based on the local referent (or critical) values of the heavy gas concentration. The presented results revealed the importance of various contributions to the redefined costs functions. The inclusion of the critical heavy gas threshold (which marks a toxic concentration level of the released gas, and consequently, an estimate of potential casualties) additionally improved the predictive capabilities of the fully integrated approach. Integrated simulations of the presented scenario of a hypothetic incidental release of a heavy gas in environmental conditions and its impact on the dynamics of the crowd behavior showed many features which are expected in realistic situations. We conclude that the approach presented here can serve as a solid foundation for further developments of fully coupled crowd dynamics/heavy gas release interactions.
Footnotes
- 1.
Additional term in the momentum equation to account for the presence of a porous structure is \({F_{i}^{D}}=-C_{d} a |U| U_{i}\), where C_{d} is the drag coefficient and a is the leaf area density. Similarly, additional term in k and ε equations are \({S_{k}^{D}}=C_{d} a \left (\beta _{p} |u|^{3} - \beta _{d} |U| k \right )\) and \(S_{\varepsilon }^{D}= C_{d} a \left (C_{\varepsilon 4} \beta _{p} |U|^{3} \varepsilon /k - c_{\varepsilon 5} \beta _{d} |u| \varepsilon \right )\), respectively. The original model coefficients are given in [20].
Notes
Compliance with Ethical Standards
Conflict of interests
The authors declare that they have no conflict of interest.
References
- 1.Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983)MathSciNetCrossRefGoogle Scholar
- 2.Zhao, D.L., Yang, L.Z., Li, J.: Exit dynamics of occupant evacuation in an emergency. Physica A 363, 501–511 (2006)CrossRefGoogle Scholar
- 3.Fredkin, E., Toffoli, T.: Conservative logic. Int. J. Theor. Phys. 21(3/4), 219–253 (1982)MathSciNetCrossRefGoogle Scholar
- 4.Helbing, D., Molnár, P.: Social force model for pedestrian dynamics. Phys. Rev. E. 51, 4282–4286 (1995)CrossRefGoogle Scholar
- 5.Henderson, L.F.: The statistics of crowd fluids. Nature 229, 381–383 (1971)CrossRefGoogle Scholar
- 6.Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res. B 36(6), 507–535 (2002)CrossRefGoogle Scholar
- 7.Hughes, R.L.: The flow of human crowds. Annu. Rev. Fluid Mech. 35, 169–182 (2003)MathSciNetCrossRefGoogle Scholar
- 8.Bonabeau, E.: Agent-based modeling: Methods and techniques for simulating human systems. Proceedings of the National Academy of Sciences of the USA (PNAS) 99 (Suppl. 3), 7280–7287 (2002)CrossRefGoogle Scholar
- 9.Lo, S.M., Huang, H.C., Wang, P., Yuen, K.K.: A game theory based exit selection model for evacuation. Fire Safety J. 41, 364–369 (2006)CrossRefGoogle Scholar
- 10.Altshuler, E., Ramos, O., Nunez, Y., Fernandez, J., Batista-Leyva, A.J., Noda, C.: Symmetry breaking in escaping ants. Am. Nat. 166(6), 643–649 (2005)CrossRefGoogle Scholar
- 11.Zheng, X., Zhong, T., Liu, M.: Modeling crowd evacuation of a building based on seven methodological approaches. Build. Environ. 44, 437–445 (2009)CrossRefGoogle Scholar
- 12.Huang, L., Wong, S.C., Zhang, M., Shu. C-W., Lam, W.K.H.: Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm. Transp. Res. B 43(1), 127–141 (2009)CrossRefGoogle Scholar
- 13.Kenjereš, S., Hanjalić, K.: Tackling complex turbulent flows with transient RANS. Fluid Dyn. Res. 41(1), 1–32 (2009). Art. 012202CrossRefGoogle Scholar
- 14.Kenjereš, S., de Wildt, S., Busking, T.: Capturing transient effects in turbulent flows over complex urban areas with passive pollutants. Int. J. Heat Fluid Flow 51, 120–137 (2015)CrossRefGoogle Scholar
- 15.Zhao, H.: A fast sweeping method for Eikonal equations. Math. Comp. 74, 603–627 (2004)MathSciNetCrossRefGoogle Scholar
- 16.Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn, pp. 224–225. Wiley (2008)Google Scholar
- 17.Kenjereš, S., Hanjalić, K.: Combined effects of terrain orography and thermal stratification on pollutant dispersion in a town valley: A T-RANS simulation. J. Turbul. 3(26), 1–25 (2002)zbMATHGoogle Scholar
- 18.Kenjereš, S., Hanjalić, K., Gunarjo, S.B.: A T-RANS/LES approach to indoor climate simulations. In: Proceedings of ASME Fluids Engineering Division, Montreal, Canada. Paper No FEDSM20002-31400 (2001)Google Scholar
- 19.Liu, D., Kenjereš, S.: Google-Earth based visualizations for environmental flows and pollutant dispersion in urban areas. Int. J. Environ. Res. Public Health 14 (3), 1–16 (2017). Art. 247Google Scholar
- 20.Kenjereš, S., ter Kuile, B.: Modelling and simulations of turbulent flows in urban areas with vegetation. J. Wind Eng. Ind. Aerodyn. 123, 43–55 (2013)CrossRefGoogle Scholar
- 21.Mirkov, N., Rašuo, B., Kenjereš, S.: On the improved finite volume procedure for simulation of turbulent flows over real complex terrains. J. Comput. Phys. 287, 18–45 (2015)MathSciNetCrossRefGoogle Scholar
- 22.Muilwijk, C., Schrijvers, P.J.C., Wuerz, S., Kenjereš, S.: Simulations of photochemical smog formation in complex urban areas. Atmos. Environ. 47, 470–484 (2016)CrossRefGoogle Scholar
- 23.Schrijvers, P.J.C., Jonker, H.J.J., Kenjereš, S., de Roode, S.R.: Breakdown of the night time urban heat island energy budget. Energy Build. 83, 50–64 (2015)CrossRefGoogle Scholar
- 24.Schrijvers, P.J.C., Jonker, H.J., de Roode, S., Kenjereš, S.: The effect of using a high-albedo material on the universal temperature climate index within a street canyon. Urban Climate 17, 284–303 (2016)CrossRefGoogle Scholar
- 25.McQuaid, J., Roebuck, B.: Large scale field trials on dense vapor dispersion. Commission Eu. Comm. Report EUR 10029 (1985)Google Scholar
- 26.McQuaid, J.: Objectives and design of the phase I heavy gas dispersion trials. J. Haz. Mat. 57, 79–103 (1998)CrossRefGoogle Scholar
- 27.Davies, M.E., Singh, S.: The phase II trials: A data set on the effect of obstructions. J. Haz. Mat. 11, 301–323 (1985)CrossRefGoogle Scholar
- 28.Sklavounous, S., Rigas, F.: Validation of turbulence models in heavy gas dispersion over obstacles. J. Haz. Mat. A 108, 9–20 (2004)CrossRefGoogle Scholar
- 29.Tauseef, S.M., Rashtchian, D., Abbasi, S.A.: CFD-based simulation of dense gas dispersion in presence of obstacles. J. Loss Prev. in the Proc. Ind. 24, 371–376 (2011)CrossRefGoogle Scholar
- 30.Hsieh, K.J., Lien, F.S., Yee, E.: Dense gas dispersion modeling of CO2 released from carbon capture and storage infrastructure into a complex environment. Int. J. Greenhouse Gas Control 17, 127–139 (2013)CrossRefGoogle Scholar
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