M.S. Howe

THE COMPRESSION WAVE PRODUCED BY A HIGH-SPEED TRAIN ENTERING A TUNNEL

An analytical investigation is made of the compression wave produced when a high–speed train enters a tunnel. The wave propagates ahead of the train within the tunnel at about the speed of sound. In very long tunnels it is transformed by nonlinear steepening into a shock whose amplitude is about one or two per cent of atmospheric pressure. The emergence of the shock from the far end of the tunnel produces an environmental disturbance analogous to the sonic boom generated by an aircraft in supersonic flight. In this paper the train is modeled by a continuous distribution of monopole sources whose strengths are determined by the train nose profile. The initial wavelength greatly exceeds the tunnel diameter at typical train Mach numbers of about 0.2 and the analytical problem of wave generation by interaction of the monopoles with the tunnel can be solved by use of a compact Grees function. The functional form of Grees function depends on the tunnel entrance geometry and on the proximity of other inhomogeneities, such as embankments, buildings and bridge structures. Detailed predictions are given for axisymmetric ‘rain’ entering a long circular cylindrical tunnel. The results are found to be in excellent agreement with experimental data for this configuration available in the literature.

Influence of an unvented tunnel entrance hood on the compression wave generated by a high-speed train

Abstract A theoretical and experimental study is made of the compression wave generated when a train enters a nominally uniform tunnel with a long, unvented entrance hood. The purpose of the hood is to reduce as much as practicable the maximum gradient of the compression wave front. The pressure gradient can increase in a long tunnel as a result of nonlinear wave steepening, and thereby increase the impact on residential dwellings of the acoustic ‘boom’ (or micro-pressure wave) radiated from the far end of the tunnel when the compression wave arrives. Our experiments are conducted at model scale using axisymmetric ‘trains’ projected at speeds up to 350 kph along the axis of a cylindrical tunnel fitted with a cylindrical entrance hood. Theoretical predictions of the compression wave are made using the equation of aerodynamic sound containing a slender body approximation to the effective source representing the moving train, coupled with a small correction that accounts for the ‘vortex’ sources in the free shear layers in the exit flows from the hood and tunnel of the air displaced by the train. The compression wave is generated by the two successive interactions of the train nose with the hood portal and with the junction between the hood and tunnel. The interactions produce a system of compression and expansion waves, each having characteristic wavelengths that are much smaller than the hood length; the waves are temporarily reflected back and forth within the hood prior to transmission into the tunnel, and are resolved analytically by use of an approximate Greens function determined by the hood geometry. Theoretical predictions are found to be in excellent agreement with experiment, including in particular a detailed correspondence between measured and predicted interference patterns produced by the multiple reflections of waves in the hood.

On the infrasound generated when a train enters a tunnel

Abstract An analysis is made of the ‘open air infrasound’ generated when a train enters a tunnel. For a tunnel of nominal radius R and train speed U the acoustic frequency ∼ U / R and wavelength ∼ R / M ≫ R , where M is the train Mach number. Infrasound is inaudible, but the pressure fluctuations generated by a high-speed train ( M >0.2) can cause annoying vibrations and ‘rattles’ in dwellings and other buildings close to a tunnel portal. Detailed calculations are presented in this paper for a ‘hood-like’ portal modelled analytically by the end of a circular cylindrical, thin-walled duct, and for an axisymmetric ‘train’ entering along the axis of the duct. A slender body approximation is used to model the influence of the moving train, and the acoustic problem is solved using the exact Greens function for a semi-infinite cylinder. Predictions are in good agreement with recent track-side measurements reported by Iida et al. (Proceedings of the 50th Japan National Congress on Theoretical and Applied Mechanics) for model scale experiments conducted at Mach numbers M as large as 0.33. However, both measurements and theory indicate that in applications at full scale it may be important to include in estimates of the infrasound the nonacoustic ‘near-field’ pressures produced by the passing train.

The compression wave generated by a high-speed train at a vented tunnel entrance

An analytical investigation is made of the compression wave generated when a high-speed train enters a long tunnel with distributed venting. The compression wave amplitude is determined by train speed and the area ratio of the train and tunnel, but its rise time depends principally on the geometry of the tunnel entrance. Vented tunnel entrance “hoods” are frequently used to increase the rise time, in order to reduce the impact of the micro-pressure pulse radiated from the tunnel exit when the compression wave arrives at the far end of the tunnel. Approximate calculations are performed to determine the initial rise time for a tunnel of rectangular cross section with a continuously variable vented roof near the entrance, for train Mach numbers less than about 0.2 (∼150 mph). The distribution of venting apertures can be optimized to maximize rise time, and a sixfold increase is shown to be possible when the aperture distribution decreases exponentially with distance into the tunnel. The method of this paper is applicable also to more general tunnel entrance geometries, and for higher train Mach numbers.

Flow-induced force fluctuations on a sphere at high Strouhal number

Abstract An empirical model is developed to estimate the broadband unsteady force spectrum induced on a rigid sphere in a nominally steady, uniform flow. The Reynolds number is sub-critical, and the frequency range considered is above the low-mode Strouhal shedding frequency of the sphere (0.5⩽fd/U0⩽100, where f is the frequency, d is the diameter, and U0 is the mean flow speed). The model uses the separation of variables assumption for the cross-power spectral densities of the surface pressure fluctuations. The assumption is shown to be a proper engineering approximation except in the lower part of the considered frequency range. In addition, the flow-induced unsteady lift and drag forces are measured independently of each other using towed spheres in a basin of water. Both estimations, from the empirical model and the data measured in the tow tank, show that the dimensionless power spectral densities of broadband unsteady lift and drag forces are constant for fd/U0

Design of a tunnel-entrance hood with multiple windows and variable cross-section

A theory is proposed for the design of a uniform tunnel-entrance hood whose cross-sectional area A h exceeds the tunnel area A. A train entering the tunnel produces a low-frequency compression wave that can be subject to nonlinear steepening in a long tunnel. An optimized hood of length h increases the initial thickness of the compression wave front from ∼ R/M to h /M, where R« h is the nominal radius of the tunnel and M is the train Mach number. In addition, the pressure rise should be linear across the wave front to obtain an overall minimum value of the subjectively important pressure gradient. This is achieved in a uniform hood by distributing windows along the hood wall to vent away high-pressure air displaced by the train. We consider the problem of determining the distribution and sizes of these windows and the magnitude of the area ratio A h /A to ensure that the hood behaves optimally at low-train Mach numbers (M<0.2), when the hood can he regarded as being acoustically compact. At the projected higher Mach numbers of advanced high-speed trains (∼0.4, say) recent analysis for hoods of uniform cross-section by Howe in 2003 indicates that a hood optimized for low Mach number operations continues to produce an essentially linear pressure rise across a compression wave of thickness ∼ h /M except for a low-amplitude oscillation at the very front of the wave.

Hydrodynamic lift and drag fluctuations of a sphere—empirical model

An empirical model is developed to estimate the unsteady lift and drag induced on a spherical body subjected to a steady, uniform flow. A wind tunnel is used to measure the statistics of the surface pressure fluctuations on a sphere under subcritical Reynolds number conditions. The measurement results are incorporated into the model for predicting the unsteady force induced on the sphere. The model is based on a separable assumption of the cross‐power spectral densities of the surface pressure fluctuations. This assumption is shown to be a proper engineering approximation except in low‐frequency ranges. The model functions are obtained using least mean‐square curve fits of the data. Tow tank experiments are performed, where the flow‐induced unsteady side force and drag are measured independently of each other on spheres configured as acoustic velocity hydrophones. The empirical model and the tow tank measurements are in very good agreement with each other and also with a simplified vortex ring model prese...