^{1}

^{*}

^{1}

^{*}

The huge number of pilgrims to the holy Mecca in the Hajj needs high awareness of crowd safety management. The stoning of the Jamarat, which is one of the rituals of the Hajj, undergoes the most dangerous crowd movements where fatal accidents occurred. This work investigates some problems related with the crowd dynamics when stoning the Jamarat pillars and gives some solutions. The main idea of this research is to suppose that the crowd dynamics is assimilated to fluid movement under certain conditions. Numerical simulation using a computational fluid dynamics program is used to solve Navier-Stokes equations governing the mechanics of homogeneous and incompressible fluid in a domain similar to the Jamarat Bridge from the entrance to the middle Jamarah. Some solutions are proposed inspired by the flow solutions to better manage crowd movements in the Jamarat Bridge and eventually in other similar dynamics events like sporting events.

The control and management of large crowds flow is a field of great interest. Interactions between mathematics and applied science in such fields are very strong. A typical example of these crowds is the annual pilgrimage in Makkah city (Hajj).

Pedestrian’s movement from the dangerous areas, under several physical constraints, is a challenging problem. If people fail to escape from a building in time, people may be injured and killed. Failure of obstacles dysfunction or wrong exit gate can be the causes of such accidents (Xiaoping et al. [

Many studies and reviews ( [

There are two main physical approaches for modeling crowd motion. The first approach is supposing crowd motion similar to discrete individual’s motion like granular material behavior. The second approach, applicable only in large crowds, involves treating the crowd as a continuous medium as in fluid mechanics [

Henderson has presented a theory of the flow of people along a channel. This theory stipulated that a crowd fluid may exhibit anomalous gas dynamic behavior caused by a change in the sign of a well-known derivative. This indicated that expansion shocks are possible in crowd fluids like real fluids [

Helbing et al. have analyzed video recordings of the crowd disaster in Mina/Makkah during the Hajj in 1426H on January 12, 2006 and have revealed two subsequent, sudden transitions from laminar to stop-and-go and “turbulent” flows there to “turbulent” crowd motion [

Hughes has resolved equations governing two-dimensional flow of pedestrians derived for flows of both single and multiple pedestrian types. Similarly to some fluids flows, he has demonstrated the existence of two regimes of fluid, a high-density and a low-density regime [

Hughes has presented a good review about the crowd movement where the property of nonlinearity and time-dependence was investigated and the fact that crowd equations are conformably mappable. This property has made many interesting applications [

In reference [

Al-Haboubi has proposed a new design to solve the safety problems about the congesting in Jamarat area. He has suggested to install uni-directional lanes and to restrict stoning to within a circular zone. An experiment was carried out to estimate the throwing time at a lane and used to estimate the waiting time in a simulation mode [

Fluid Models was proposed by Payne [

In a recent paper, Karamouzas et al. have presented a novel statistical-mechanical approach of crowd motion based on resemblance to interaction particle systems. Applied to a large collection of human motion data, this analysis reveals a simple power law interaction that is based not on the physical separation between pedestrians but on their projected time to a potential future collision, and is therefore fundamentally anticipatory in nature [

In this work, we propose a numerical model to study the flow of crowds at the Jamarat Bridge during the Hajj by assuming similarity between the crowd dynamics and a real incompressible fluid flow by means of computational fluid dynamics simulation ANSYS Fluent code.

The objective of this study is to propose some modifications to the Jamarat Bridge to better manage the crowd movement and compare with the present Bridge.

Three modifications to the Jamarat Bridge domain are proposed. The velocity and the pressure for the flow are presented and compared to the present domain.

The paper is organized as follows: In Sections 2 and 3 the description and modelling of crowd dynamics problem is presented. Section 4 is devoted to present the details of the numerical resolution. The numerical parameters and algorithms are also presented in this section. In Section 5, discussions and comparisons about the proposed procedures are given.

In this work, we suppose that there is a similarity between crowd flow and the dynamics of an incompressible and laminar fluid. We investigate to study the crowd behavior in the Jamarat Bridge in the Hajj event where the people number is huge and the similarity is realistic. The governing equations of such a fluid are the classic Navier-Stokes equations. In the Cartesian two dimensional plane, the mass and momentum conservation equations are written as:

where

The partial differential equations governing the flow are discretized and resolved with the finite volume method. The computational domain is taken to be a part of the Jamarat Bridge, between the entry side of the Bridge to the part behind the trailing edge of the middle Jamarah.

In this work the third Jamarah is not included and the whole bridge is truncated to minimize the calculation cost.

To perform the equations resolution, the domain is meshed using a mesh generator software (GAMBIT package). The whole computational domain is divided into 4 blocks (

and all other blocks where the two pillars (Jamarat) are located, are discretized with triangular cells. The whole domain is divided on 365,000 cells.

The Ansys Fluent CFD code is used to resolve the Equations (1)-(3) governing the steady flow. In all investigated cases the inlet velocity is imposed initially to be horizontal and of value equal to 1.0 m/s. Second order central upwind discretization is used for the momentum equation. The time integration is done with an implicit method. The calculations are leaved until the residuals curves decrease down three decades and the pressure and velocity profiles don’t vary.

The velocity is taken initially zero in all the boundary sides of the domain except the inlet and the outlet sides. At the inlet of the bridge, the velocity is constant and the outlet is set up as a pressure outlet. Boundary layers effect is not considered here.

The boundary conditions used in the simulation are recapitalized in

Three practical propositions are performed. The initial domain DOM0 (

The second proposed modification to the initial domain is to divide the whole domain into two subdomains DOM2 (

In the third domain DOM3 proposed (

which provides a rather smooth curve. The leading edge and the trailing edge of shapes surrounding the first and second Jamarah are also modified to be more stretched. The major axis of the ellipse is taken greater.

Inlet | Wall Boundary | Outlet | Boundary layers | Flow |
---|---|---|---|---|

Velocity inlet Horizontal, V = 1.0 m/s | No-slip | Pressure outlet | Not considered here | steady |

In the three new domains proposed, Matlab and Photoshop software are used to design the shapes and exported in the mesh generator.

The crowd flow can be fatal in sharp points especially when irregular and sudden rush occur. The leading edges of the shapes surrounding the Jamarat are high pressure side in the flow dynamics. The trailing edges of the Jamarat are sides where chaotic and turbulent flows can occur. The Three domains proposed in this work try to reduce the flow pressure, in particular in the critical points.

In

The problem of crowd dynamics in the Jamarat Bridge in Mina/Makkah was investigated in this paper with a model based on the likeness between collective crowd and the flow of a fluid. A CFD analysis of an incompressible laminar fluid flowing in

a 2D domain similar to the Jamarat Bridge is performed. Three modifications of the Bridge domain have been proposed and compared to the present domain. Comparisons of velocity and pressure behavior have been showed. The first proposition was to add obstacles to eliminate sharp points in the bridge border before reaching the first Jamarah.

The second proposal is to divide the bridge into two sub-domains with a longitudinal barrier throughout the whole domain. The third idea is to modify the form of the Jamarat contours in aim to eliminate the space between them by a fairly regular line and reduce flow instabilities. The barriers that can be added to the bridge may be permanent or temporary.

With the fluid simulations on three new domains, the three solutions showed good behavior for the velocity and pressure functions. The stopping points and strong gradients of the crowd movements can be avoided and improved.

Authors would like to thank the deanship of scientific research at King Faisal University (KFU), Alhasa, Saudi Arabia for supporting this research (Grant number 165040).

Mnasri, C. and Farhat, A. (2016) Numerical Simulation of the Flow of Crowds at the Jamarat Bridge during the Annual Hajj Event. Open Journal of Fluid Dynamics, 6, 321-331. http://dx.doi.org/10.4236/ojfd.2016.64024