A New Simple Equation Governing Distortion of Compression Wave Propagating Through Shinkansen Tunnel with Slab Tracks

Abstract

A high-speed train entering a tunnel generates a compression wave that propagates through the tunnel toward its exit. When the compression wave reaches the tunnel exit, a pressure pulse causing environmental problems (the micro-pressure wave) is radiated from the exit portal. The magnitude of the micro-pressure wave is approximately proportional to the maximum pressure gradient of the compression wave arriving at the tunnel exit. In a long Shinkansen tunnel with concrete slab tracks the compression wavefront steepens due to the nonlinear effect during its propagation because the whole surfaces of the tunnel being smooth. Hence, it is necessary to investigate the distortion of the compression wave in a tunnel and clarify the characteristics of compression wave propagation for estimating and reducing the micro-pressure wave. In this paper, we introduce a new simple equation governing the distortion of the compression wave propagating through a Shinkansen tunnel with concrete slab-tracks and propose a new simple scheme for a numerical calculation. A space evolution type equation with one variable is derived from the three conservation equations (mass, momentum, and energy including the wall friction and heat transfer terms) of the 1D CFD by assuming small disturbances excited by the compression wave. The numerical calculation scheme based on the simple equation remarkably reduces the computing time because its CFL condition is relaxed. The calculation results by the present scheme agree well with those by a conventional scheme based on the 1D CFD and the accuracy of the simple equation is verified.

<b>Prediction of the Micro-pressure Waves Using an Unsteady Acoustic Analysis</b>

“…where ρ : density of the air, U: train speed, R: blockage ratio, M: Mach number, (7) represents the pressure increment in the tunnel based on the linear acoustic theory [4,5], and (8) represents the pressure increment in the tunnel considering the nonlinear effect [12]. From the results shown in Fig.…”

“…In the propagation stage of the compression wave, the 1D theoretical wave equation (1) [7,8] is used.…”

“…where Pr: Prandtl number, Re c : Reynolds number (Re c = c 0 d 0 /ν, ν : the kinematic viscosity of the air), ε s : the coefficient of the steady frictional force, ε us : the coefficient of the unsteady frictional force. Here, the terms related α and β in the right-hand side (1) represent the frictional effect acting on the compression wave from the tunnel wall, and the values of ε s and ε us are referenced [7,8].…”

“…3, type 1 side branches (cross-sectional area: 7.1 m 2 , length: 3 m) and type 2 side branches (cross-sectional area: 7.1 m 2 , length: 5 m) were installed at equal intervals of 1000 m and 500 m, respectively, in the tunnel. The distortion of the compression wave when it passes through the junction of the tunnel and the side branch is calculated by using the acoustic method on the basis of the propagation of the plane wave [6,7,8]. Table 2 Analytical condition for the propagation of the compression wave…”

“…For example, rapid calculation on the basis of linear acoustic analysis is used to research stage (i) [4,5]. For stage (ii), the simultaneous equation of the conservation law for 1D compressible flow is used: conservation of mass, momentum and energy [6] or the propagation analysis on the basis of the acoustic theory [7,8]. For stage (iii), methods which model radiation from the exit portal as monopole sound source [1] or analyses which take into account the geometry around the exit portal [9] are representative.…”