On the compression wave generated when a high-speed train enters a tunnel with a flared portal

An analysis is made of the compression wave generated when a high-speed tra in enters a tunnel with a flared portal. Nonlinear steepening of the wavefr ont in a very long tunnel is responsible for an intense, environmentally ha rmful, micro-pressure wave, which propagates as a pulse from the distant tu nnel exit when the compression wave arrives, with amplitude proportional to the maximum gradient in the compression wavefront. The compression wave pr ofile can be determined analytically for train Mach numbers M satisfying M- 2 much less than 1, by regarding the local flow near the tunnel mouth durin g train entry as incompressible. In this paper, the influence of tunnel por tal flaring on the initial thickness of the compression wave is examined fi rst in this limit. The shape of the flared portal is "optimal" when the pre ssure gi adient across the front is constant and an overall minimum, so tha t the pressure in the wavefront increases linearly. This linear behaviour i s shown to occur for a flared portal extending a distance l into the tunnel from the entrance plane (x = 0) only when the tunnel cross-sectional area S(x) satisfies S(x)/A = 1/[A/A(E) - (x/l) (1 - A/A(E))], - l < x < 0, where x increases negatively with distance into the tunnel, A is the cross- sectional area in the uniform section of the tunnel (x < - l), and A(E) is the tunnel entrance cross-section. The optimum portal is achieved by adjust ing the value of A/A(E), to make the pressure gradient continuous, and a fo rmula is derived for this ratio for tunnels of semi-circular cross-section. For optimal flaring, the pressure rises linearly as the front of the train traverses the flared section of length l, and the thickness of the compres sion wavefront similar to l/M. A formula is proposed for extrapolating these predictions to train Mach num bers as large as 0.4, which is expected to be typical of future high-speed rail operations. It is validated for the special case of a circular cylindr ical tunnel, for which an exact solution is known for arbitrary subsonic Ma ch numbers, and by comparison with scale model experiments using trains of various nose profiles. (C) 1999 Academic Press.