Abstract
An analysis is made of the compression wave generated when a high-speed train enters a tunnel with a flared portal. Nonlinear steepening of the wavefront in a very long tunnel is responsible for an intense, environmentally harmful, micro-pressure wave , which propagates as a pulse from the distant tunnel exit when the compression wave arrives, with amplitude proportional to the maximum gradient in the compression wavefront. The compression wave profile can be determined analytically for train Mach numbers M satisfying M2⪡1, by regarding the local flow near the tunnel mouth during train entry as incompressible . In this paper, the influence of tunnel portal flaring on the initial thickness of the compression wave is examined first in this limit. The shape of the flared portal is “optimal” when the pressure gradient across the front is constant and an overall minimum, so that the pressure in the wavefront increases linearly . This linear behaviour is 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) satisfiesS (x)A=1[A/AE−(x/ℓ)(1−A/AE)], −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 AEis the tunnel entrance cross-section. The optimum portal is achieved by adjusting the value of A/AEto make the pressure gradient continuous, and a formula 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 compression wavefront ∼ℓ/M .A formula is proposed for extrapolating these predictions to train Mach numbers 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 cylindrical tunnel, for which an exact solution is known for arbitrary subsonic Mach numbers, and by comparison with scale model experiments using trains of various nose profiles.
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