Characteristics analysis of near-field and far-field aerodynamic noise around high-speed railway bridge

Abstract

The aerodynamic noise around the high-speed railway bridge is studied by the train-bridge-flow field numerical model and theory analysis. With the background of the Beijing-Shanghai high-speed railway line in China, based on the broadband noise sources method and acoustic analogy theory, both the intensity characteristics of near-field aerodynamic noise sources of the train and the spatial distribution characteristics of far-field aerodynamic noise around the bridge are analyzed. The results show that there is the largest sound source energy on the nose tip of the high-speed train; the fluctuating pressure between air and train surface contributes the most to aerodynamic noise; along the longitudinal direction of the bridge, the aerodynamic noise energy near the junction of train tail and body is the strongest; along the transverse direction of the bridge, the aerodynamic noise energy decreases gradually with the distance from the centerline of the railway; along the vertical direction, the aerodynamic noise energy is the strongest at a location of 1.2 m above the top surface of the rail; the train speed, train type, and the height variation of the bridge pier can affect the distribution of far-field aerodynamic noise.

Appendices

Appendix 1

If the right side of Eq. (2) is regarded as the sound source term F(x, t), Eq. (2) is a typical acoustic wave equation:

$$ \left(\frac{1}{c_0^2}\frac{\partial^2}{\partial {t}^2}-{\nabla}^2\right)\left[{Hc}_0^2\left(\rho -{\rho}_0\right)\right]=F\left(x,t\right) $$

(9)

which can be solved using Green’s function.

When the high-speed train is running, the ground can be approximated as an interface with good sound reflection, so the sound field induced by the train is a semi-free field, and its Green’s function can be expressed as (Goldstein 1974):

$$ G\left(\mathbf{x},\mathbf{y};t-\tau \right)=\frac{\delta \left(t-\tau -\left|\mathbf{x}-\mathbf{y}\right|/{c}_0\right)}{4\uppi \left|\mathbf{x}-\mathbf{y}\right|}+\frac{\delta \left(t-\tau -\left|\mathbf{x}-\tilde{\mathbf{y}}\right|/{c}_0\right)}{4\uppi \left|\mathbf{x}-\tilde{\mathbf{y}}\right|} $$

(10)

where x = {x₁, x₂, x₃}andy = {y₁, y₂, y₃} are the receiving point vector and the source point vector in the Cartesian coordinate system, respectively; τ and t respectively represent the time emitted at the sound source point should be standardized form y and received at the receiving point x; $\tilde{\mathbf{y}}=\left\{{y}_1,{y}_2,-{y}_3\right\}$ is symmetrical with y = {y₁, y₂, y₃} about the ground.

Herein, the variables ρ^′ = ρ − ρ₀ and p^′ = p − p₀ are introduced. Based on Eq. (10) of Green’s function, Eq. (11) can be derived from Eq. (9):

$$ {\displaystyle \begin{array}{c}H{c}_0^2\left(\rho -{\rho}_0\right)=H{p}^{\prime}\left(x,t\right)={\iiint}_V{\int}_{-\infty}^{\infty }F\left(y,t\right)G\left(x,y;t-\tau \right)\mathrm{d}\tau \mathrm{d}V\\ {}=\frac{\partial^2}{\partial {x}_i\partial {x}_j}{\iiint}_V\left[{T}_{ij}\right]\frac{\mathrm{d}V}{2\uppi R}-\frac{\partial }{\partial {x}_i}{\oint}_S\left[\rho {u}_i\left({u}_j-{v}_j\right)+{p}_{ij}\right]\frac{\mathrm{d}{S}_j(y)}{2\uppi R}\\ {}+\frac{\partial }{\partial t}{\oint}_S\left\{\left[\rho \left({u}_j-{v}_j\right)+{\rho}_0{u}_j\right]\right\}\frac{\mathrm{d}{S}_j(y)}{2\uppi R}\\ {}\approx -\frac{\partial }{\partial {x}_i}{\oint}_S\left[\rho {u}_i\left({u}_j-{v}_j\right)+{p}_{ij}\right]\frac{\mathrm{d}{S}_j(y)}{2\uppi R}\end{array}} $$

(11)

where S and V are the integral surface and integral space, respectively; R is the distance from the sound source point y to the receiving point x; other mathematical symbols are as mentioned above.

By simplifying and solving Eq. (11), the theoretical prediction of Eq. (3) for the far-field aerodynamic noise can be obtained.

Appendix 2

Fig. 23

The discretization and grid of computational domain

Fig. 24

Two different types of train models. a CRH2. b CR400-BF