Roger L. Hughes

A continuum theory for the flow of pedestrians

The equations of motion governing the two-dimensional flow of pedestrians are derived for flows of both single and multiple pedestrian types. Two regimes of flow, a high-density (subcritical) and a low-density (supercritical) flow regimes, are possible, rather than two flow regimes for each type of pedestrian. A subcritical flow always fills the space available. However, a supercritical flow may either fill the space available or be self-confining for each type of pedestrian, depending on the boundary location. Although, the equations governing these flows are simultaneous, time-dependent, non-linear, partial differential equations, remarkably they may be made conformally mappable. The solution of these equations becomes trivial in many situations. Free streamline calculations, utilizing this property, reveal both upstream and downstream separation of the flow of pedestrians around an obstacle. Such analysis tells much about the nature of the assumptions used in various models for the flow of pedestrians. The present theory is designed for the development of general techniques to understand the motion of large crowds. However, it is also useful as a predictive tool. The behavior predicted by these equations of motion is compared with aerial observations for the Jamarat Bridge near Mecca, Saudi Arabia. It is shown that, for this important case, pedestrians, that is pilgrims, aim at achieving each immediate goal in minimum time rather than achieving all goals in overall minimum time. Typical of many examples, this case illustrated the strong dependence of path on the psychological state of the pedestrians involved. It is proposed that the flow of pedestrians over the Jamarat Bridge be improved by appropriate barrier placement, that force an effective global view of the goals.

The flow of large crowds of pedestrians

Despite popular belief the motion of a crowd is governed by well-defined rules of behaviour. These rules imply a set of coupled, non-linear, partial differential equations for the density and velocity potential for each type of pedestrian in the crowd. As may be expected, the solution of these equations may, in different regions of space, be supercritical or subcritical with the possibility of a shock wave separating the regions. Less predictable is the remarkable finding that these coupled, non-linear, time dependent equations are conformally mappable and this finding enables solutions to be obtained easily for both supercritical and subcritical flows.

Minimisation of the risk of trampling in a crowd

Over the past decade, there have been many crowd related tragedies. To help avoid such situations a strategy is developed here to improve the safety of pedestrians in densely populated situations. The results of simulations performed on two cases of accidents involving trampling, which occur when pedestrians are moving, illustrate the ability of this modelling strategy for minimising predicted crowding risks in such situations. This study demonstrates that effective crowd control may be achieved either by adjusting the size of the crowd or the complexity of the environment in which pedestrians walk, which effectively influences their speed.

The Hydraulics of Local Separation in a Coastal Current with Application to the Kuroshio Meander

Abstract A hydraulic theory is advanced to explain the large sustained meandering of the Kuroshio Current to the south of Japan. It is shown that in general a promontory induces separation of a coastal current. Once separated the current will meander on the second derivative of the topography (i.e. it is not a shelf wave-like meander) in barotropic waters, and with Rossby wave-like behavior in baroclinic waters. The theory reduces to a hydraulic description of the White and McCreary model of the Kuroshio meander away from the coast but avoids difficulties associated with that theory at the source of the meander at the Kyushu promontory.

Western Boundary Current Separation: Inferences from a Laboratory Experiment

Abstract Observations of a laboratory model of a western boundary current, and its separation and subsequent meandering, are described. The current is established by pumping fluid through a rotating channel that contains a topographic β effect and continental slope topography. The observations are compared with a theoretical model of all three aspects of the current: the structure of the attached current, the process of separation, and the dynamics and path of the meandering jet. This model includes a viscous boundary layer for the attached current, with a thickness of order [ν/(dvI/dy)]1/2, where ν is kinematic viscosity and dvI/dy is the velocity gradient of the inviscid (free slip) flow along the boundary. Comparison between the observations and the model show that the attached boundary current is governed by potential vorticity conservation and the Bernoulli equation, and the pressure decreases along its length. The separation of this current from the sidewall is then caused by the minimum pressure le...

A mathematical determination of von Karman's constant, Κ

One of the few fundamental constants in fluid mechanics is von Karmans constant governing many turbulent flows. Using an energy argument and mathematical symmetry it is argued here that von Karmans constant is κ = 0.414 (with derivable corrections for specific applications)

The behavior of stratified pools in the Wimmera River, Australia

Numerous inland Australian streams contain density-stratified or saline pools, which are usually located on channel bends. Saline pools consist of a layer of saline water underlying a layer of fresh water. Saline pools generally form as a result of saline groundwater seeping into the stream and collecting in scour depressions during periods of low flow. Inflows of saline river water can also collect in scour depressions. Field and laboratory investigations of saline pool mixing by overflowing fresh water reveal that mixing depends on a balance between interfacial shear and buoyancy forces acting on a thin dense layer flowing up the downstream slope of the scour depression, and on the bend sharpness. Convection associated with surface cooling also causes mixing. A model for saline pools formed by groundwater inflows and mixed by fresh overflows is proposed and applied to several saline pools in the Wimmera River.

On detecting anomalous behaviour in runs

Abstract Suppose that values in a finite sequence of data are labelled as either +1 or −1 depending, respectively, on whether they are either above or below the median. Using this sequence of +1 and −1 s a unit-lag autocorrelation coefficient may be determined. The present study establishes the probability distribution for the number of runs of a fixed length of −1 s (or +1 s) occurring in the labelled data. This distribution is calculated both with and without the constraint of the sample autocorrelation. The distribution is compared with observed stream flow data to illustrate its use in detecting both deficits and excesses of low or high flows of a given duration. The use of this distribution, which is not restricted to stream flow data, provides an extremely convenient alternative to the more traditional methods of detecting anomalous behaviour and avoids requiring knowledge of the form of the parent distribution.

Comments on “The Interaction of an Eastward-Flowing Current and an Island: Sub- and Supercritical Flow”

AbstractIn regards to the recent paper by Pedlosky and Spall (Journal of Physical Oceanography, November 2015), this comment maintains that the steady-state solutions of Rossby waves in a uniform eastward current past an island have waves on the upstream side that are not caused by the island because of inappropriate boundary conditions and the assumed form of the solution. The solutions are interesting but are not the solutions to the problem as posed in their paper. Similar upstream waves in a time-dependent numerical model are also inconsistent with linear Rossby wave theory, though the reasons for their presence are uncertain.

Tidal oscillations in shallow bays and sediment movement

The present study seeks to understand the interaction of a directly forced, large amplitude, tidal mode with non-directly forced resonant or nearly resonant tidal modes in a coastal bay or basin despite frictional effects. It is shown that in shallow basins a non-directly forced resonant mode may be indirectly activated with a response that may be time dependent, with the amplitude of this mode increasing and then decreasing before increasing again in a cyclical manner associated with amplitude beats. A laboratory experiment and theoretical arguments are used to support these conclusions and to study the effect when the amplitude of the tidal disturbances is comparable to the depth of the basin. It is noted that the presence of such modes, whether directly or indirectly forced, and whether resonant or non-resonant, may be able to move sediment and thereby change the resonance behaviour of the basin by changing its geometry. The changes in the geometry may then change the resonant frequencies of the basin, possibly causing the decay of the mode that caused the sediment motion. The implications of the mechanism for the modelling of the tidal behaviour in other bays with more complicated geometry are considered.