L. F. Henderson

On the fluid mechanics of human crowd motion

The paper presents a theory of the flow of people along a channel which may be of variable width, or have a partial blockage in it. The results are presented in kinetic form to facilitate comparison with observation. It is shown that a crowd fluid may exhibit anomolous gas dynamic behaviour caused by a change in the sign of a well known derivative. This indicates that expansion shocks are possible in crowd fluids.

The von Neumann paradox for the diffraction of weak shock waves

We present results from our experiments with the irregular reflection of shock waves in argon. We compare the data with the results we obtained numerically; the assumptions for the computational code were that we had unsteady, two-dimensional, compressible, inviscid, flow of a perfect gas. When precautions were taken to reduce the effects of the gas viscosity on the experimental data, we obtained very good agreement between the numerical and the experimental results for the ramp Mach number and the trajectory path triple-point angle, but there were discrepancies with the wave-angle data. The discrepancies were ascribed to the sensitivity of the data to both viscosity and to a singularity. We show that there are actually two weak irregular wave reflections, namely a classic Mach reflection (MR) and a new type, that we call a von Neumann reflection (NR). The structure of the NR is discussed in some detail, and so are the transition criteria for the various wave systems.

On the refraction of shock waves at a slow–fast gas interface

We present the results of numerical computations of the refraction of a plane shock wave at a CO 2 /CH 4 gas interface. The numerical method was an operator split version of a second-order Godunov method, with adaptive grid refinement. We solved the unsteady, two-dimensional, compressible, Euler equations numerically, assuming perfect gas equations of state, and compared our results with the experiments of Abd-El-Fattah & Henderson. Good agreement was usually obtained, especially when the contamination of the CH 4 by the CO 2 was taken into account. Remaining discrepancies were ascribed to the uncertainties in measuring certain wave angles, due to sharp curvature, poor definition, or short length of the waves at large angles of incidence. All the main features of the regular and irregular refractions were resolved numerically for shock strengths that were weak, intermediate, or strong. These include free precursor shock waves in the intermediate and strong cases, evanescent (smeared out) compressions in the weak case, and the appearance of an extra expansion wave in the bound precursor refraction (BPR). The structure of a BPR was elucidated for the first time.

On the refraction of shock waves

This paper discusses the refraction of plane shock waves in media with arbitrary equations of state. Previous work is reviewed briefly, then a rigorous definition of wave impedance is formulated. Earlier definitions are shown to be unsatisfactory. The impedance is combined with the boundary conditions at the media interface to study both head-on and oblique shock incidence. The impedance determines the nature of the reflected and transmitted waves, their intensities, and the fractions of energy and power that are reflected and transmitted. The refractive index is also defined and determines whether or not a wave will be refracted, and also helps determine whether the wave system will be regular or irregular. The fundamental law of refraction is derived and shown to be a consequence of the fact that an arbitrary point on a shock or an expansion wave follows a ray path of minimum time between any two points on the path. This is a generalization of Fermats Principle to media that are deformed and convected by the waves propagating through them.

The refraction of a plane shock wave at a gas interface

The paper deals with the refraction of a plane shock wave by an interface between two different gases. It is shown that the equations of motion can be reduced to a single polynomial equation of degree 12. Detailed numerical results are presented for the air-CH 4 and the air-CO 2 interfaces, which are respectively ‘slowfast’ and ‘fast–slow’ combinations. When the results are compared with experiment good agreement is obtained. The numerical data are multi-valued, but it is found that it is always the weakest solution that agrees with the experimental data. The multiple roots of the equation are often found to be associated with the appearance of degenerate and irregular wave systems and some attempt is made to analyse and discuss these systems.

Shock waves at a slow-fast gas interface

This paper describes the results of our experiments with shock waves refracting at a CO 2 /CH 4 interface. The refraction is slow-fast because the speed of sound in the incident gas (CO 2 ) is less than that in the transmitting gas (CH 4 ). We found three phenomena which apparently have not been reported before and which all have free precursor shocks in their wave systems; schlieren photographs of them are presented. As a result of the present and earlier work, we can assert that there exist at least four different free precursor refractions. Theoretical studies suggest that the slow-fast phenomena can be conveniently classified into three groups characterized by different ranges of values of the inverse strength ζ i of the incident shock i . The classification may be an exhaustive list of the phenomena, at least when the gases are nearly perfect, but we cannot be sure. We present experimental data on all the phenomena in each group, including data on the transition conditions from one wave system to another both within and across the groups.

Further experiments on transition to Mach reflexion

Our 1975 paper reported the results of experiments on shock reflexion in a wind tunnel and a shock tube; further results are presented here. For strong shocks it is shown that transition to Mach reflexion takes place continuously at the shock wave incidence angle ω 0 corresponding to the normal shock point ω 0 = ω N , unless the downstream boundaries form a throat. In this event transition can be promoted anywhere within the range ω 0 [les ] ω N , and it is even possible to suppress regular reflexion altogether! However when ω 0 N the transition is discontinuous and accompanied by hysteresis. Again for strong shocks evidence is presented which suggests that the famous persistence of regular reflexion beyond the ω N point ω 0 > ω N is spurious. For weak shocks the transition condition is not known but it is found that even for regular reflexion a marked discrepancy between theory and experiment develops as the shocks become progressively weaker. Also when weak shocks diffract over single concave corners there is a somewhat surprising discontinuity in the regular reflexion range. It seems that none of these phenomena can be adequately explained by real gas effects such as viscosity and variation of specific heats.

Precursor shock waves at a slow—fast gas interface

This paper presents experimental data obtained for the refraction of a plane shock wave at a carbon dioxide–helium interface. The gases were separated initially by a delicate polymer membrane. Both regular and irregular wave systems were studied, and a feature of the latter system was the appearance of bound and free precursor shocks. Agreement between theory and experiment is good for regular systems, but for irregular ones it is sometimes necessary to take into account the effect of the membrane inertia to obtain good agreement. The basis for the analysis of irregular systems is one-dimensional piston theory and Snells law.

On the Confluence of Three Shock Waves in a Perfect Gas

The paper deals with the behaviour of three shock waves meeting at a point in a perfect gas. It is shown that the equations of motion can be reduced to a single polynomial equation of degree 10. The real roots of this equation are studied to determine their physical significance. In addition, the appearance of degenerate shock systems is shown to be associated with the formation of certain multiple roots of the polynomial equation.

On the von Neumann paradox of weak Mach reflection

Recent experimental and numerical studies of weak Mach reflections are examined. It is shown that the fundamental reason for the von Neumann paradox is that his theory of Mach reflection is based on the assumption that the flow downstream of the reflected wave and the Mach shock near the wave triple point is uniform. The assumption is shown to be valid for strong Mach reflection which agrees with experiment, but invalid for weak Mach reflection which does not agree with experiment. It is also shown that viscous effects are dominant when the incident shock is within about 100 mean free path lengths of the corner, but not otherwise. The analytical theory of the entire subsonic region supports these conclusions.