CHAPTER 5: DEVELOPMENT OF NUMERICAL MODEL FOR TIRE/ROAD NOISE ON POROUS PAVEMENT
5.2 Numerical Representation of the Acoustic Absorption of Porous Pavement
5.2.3 Validation of the Acoustic Representation of Porous Pavement in BEM
191 The noise amplification resulting from horn effect is found to be 10 to 20 dB (Graf et al., 2002; Kropp et al., 2000). Porous pavement is effective in reducing the horn effect through its absorption of acoustic energy during the multi-reflection process when sound wave propagates in the horn-shape geometry.
5.2.3.1 BEM Model of Horn Effect on Porous Pavement
The boundary element method is used to simulate sound propagation in porous pavement and its resulting reduction in horn effect. BEM is formulated such that integral equations are defined on the domain boundaries and integrations will relate boundary solutions to field point solutions. The benefit of BEM arises from the fact that only the boundaries need to be sub-divided, reducing the problem dimension by one. Furthermore, the infinite domain in acoustic radiation and scattering problem can be more efficiently modeled in this approach.
The linear acoustic theory decomposes an acoustic variable into two terms: a constant one corresponding to a steady state and a fluctuating one representing the perturbations of the constant state. The fluctuations are sufficiently small so that a linear pressure-density relationship stands and all products of acoustic perturbation terms can be neglected. In this case, the wave equation is derived as:
2 2
2 2
1 p
p s
c t
(5.27)
where p is the acoustic pressure, c is the sound propagation speed in the fluid, t is the time, s is the acoustic source, and 2 is the Laplacian operator. For a harmonic field, the complex pressure amplitude satisfies the Helmholtz equation:
2 2
p k p0 s
(5.28)
The particle velocity (v) is related to the pressure gradient by the Euler equation:
0 0
( ) 0
grad p jk Z v (5.29)
Three types of boundary conditions are used in the study of the horn effect in porous pavements. The tire walls are assumed to be perfectly-reflecting surfaces,
192
which are acoustically equivalent to zero normal velocity condition vn 0. Surface impedance gets to infinity and no sound energy is absorbed by such boundaries. This condition is represented by:
( )s 0 p x
n
(5.30)
where p(xs) is the pressure at point xs on boundary. The boundary condition on porous pavement surface is defined by its surface normal acoustic impedance, which denotes the acoustic absorption properties on boundaries. It can be derived from Equations (5.5) and (5.29):
0 0
( ) ( ) 0
( )
s
s s
p x Z
jk p x
n Z x
(5.31)
Another boundary is defined at the infinity, where the Sommerfeld condition must be satisfied. It specifies that the acoustic pressure approaches zero and no sound energy is reflected.
0
lim 0
s
s
s s
x s
x p x jk p x
x
(5.32)
The integral form of Helmholtz equation on boundaries relates the boundary pressure and the normal velocity by a matrix equation:
A p B v n (5.33)
where [A] and [B] are coefficient matrices. Boundary conditions [Equations (5.30) to (5.32)] are applied to Equation (5.33) to calculate the pressures and particle velocities at the boundaries. The acoustic pressure at any field point is then solved through the integration of boundary solutions:
f T T n
p x C p D v (5.34)
where C and D are coefficient vectors derived from integration of Helmholtz equation on boundaries.
193 ( , )
( ) f s ( )
i i s s
G x x
C N x d x
n
(5.35)
( ) ( , ) ( )
i air i s f s s
D j N x G x x d x
(5.36)
where Ω is the boundary of the calculation domain, Ni(xs) is the interpolation function for boundary element, and G(xf, xs) is the Green function.
Porous pavement is represented by an absorbent panel in the BEM model, on which the acoustic impedance is defined to model its sound absorption properties. In the open space problem of in-field noise measurement, porous pavement is supposed to be infinite horizontally. However, in numerical simulation, the boundary has to have a limited dimension to make sub-division possible. Convergence studies conducted by Anfosso-Lédée (1997) suggested that the finite extension of a boundary can approximate the solutions if the boundary extends over the source and receiver by a distance longer than that between the source/receiver and the boundary and this distance must be longer than one wavelength. This principle is adopted in this study.
For the space above pavement surface, the air at 25°C is modeled as the sound propagating medium. The mass density of air is set to be 1.225 kg/m3 and the sound velocity in it is 340 m/s. Tire walls are modeled by perfectly-reflecting panels according to the geometry of tire used in the test, and a monopole with white noise is adopted as sound source in the simulations. With such a configuration, the acoustic pressure field around the tire-pavement horn is computed from the BEM model.
5.2.3.2 Model Validation Results
The acoustic representation of porous pavement in the BEM model is next validated against published experimental results. The validation work is based on the differences between sound pressure levels on porous and non-porous pavements.
Comprehensive measurements conducted by Schwanen et al. (2007) on various types of pavements are used to validate the model.
194
The experimental set-up is sketched in Figure 5.2. A tire with 195 mm width, 381 mm rim diameter and 310 mm outer radius is placed on the tested surface without vertical loading. Two source positions are selected on the center line of tire tread, 10 and 15 cm from the center of contact patch, respectively. Two receiver positions are defined with the coordinates shown in Figure 5.2. Receiver A is right in front of the tire-pavement contact patch, 428 mm from its center; and Receiver B is perpendicular to the tire center plane, at a distance of 297.5 mm. Both receivers are placed 100 mm above the pavement surface. Two porous pavements and one dense-graded pavement are involved in the model validation work. The acoustic impedance and absorption coefficient of tested pavements were measured by the extended surface method (ISO, 2002) and the results are shown in Figure 5.3.
The effects of porous pavements on sound absorption are represented by the differences in sound pressure levels on porous and dense-graded surfaces. The results computed from the numerical simulations are compared against values measured in the experiments (see Figure 5.4). Model validation is performed for different source positions, receiver locations and porous surfaces to better demonstrate the feasibility of simulation approach. It is seen that the BEM model is capable to predict the main peak frequency of sound absorption on porous pavement, as well as the magnitude of noise reduction at the peak frequency. However, the range of peak frequency detected by receiver B in simulation is wider than that obtained in the measurement. This may result from the setup configuration of experiment. Moreover, it is observed that the errors at high frequencies are larger than those at low frequencies and the estimation of secondary absorption peaks is below satisfaction. Recognizing that the differences between numerical and experimental results shown in Figure 5.4 cover the errors in acoustic impedance and sound pressure level measurements, the accuracy of BEM formulation is considered sufficient for the representation of porous pavement acoustic absorption.