CHAPTER 5: DEVELOPMENT OF NUMERICAL MODEL FOR TIRE/ROAD NOISE ON POROUS PAVEMENT
5.2 Numerical Representation of the Acoustic Absorption of Porous Pavement
5.2.2 Modeling the Acoustic Absorption of Porous Pavement
Measurement of pavement acoustic characteristics requires specialized device and high investment, which may not always be available. As a result, various models have been developed to relate the acoustic properties of porous mixture to its composition and volumetric characteristics. Acoustic absorption can be modeled by either phenomenological models (Attenborough and Howorth, 1990; Hamet and Bérengier, 1993; Hubelt, 2003) or microstructural models (Champoux and Stinson, 1992; Neithalath et al., 2005; Kim and Lee, 2010). This section shall introduce the representative phenomenological and microstructural approaches. It was found that the phenomenological model is in close agreement with the microstructural model in the case of porous pavements (Bérengier et al ., 1997). Different models rely on different variables to derive acoustic absorption, therefore, the selection of model depends on what variables are available in the measurement.
5.2.2.1 Phenomenological Model
The phenomenological approach considers the porous mixture and the air within it as a whole dissipative compressible fluid. This method introduces a viscous dissipation resulting from the velocity gradients within the material and a thermal dissipation due to the thermal gradients (Bérengier et al ., 1997). Among the various models, the one developed by Hamet and Bérengier (1993) allows the acoustical characteristics of a porous pavement to be determined using only three parameters (i.e.
porosity, tortuosity and airflow resistance). All these parameters can be separately measured by laboratory tests, making this model very attractive to implementation.
187 In Hamet and Bérengier 's model, the dynamic density and bulk modulus are expressed as
2(1 )
g airq j fu f
(5.9)
1
0 1 ( 1) (1 )
Kg P j f f (5.10) where ρg is the complex dynamic density, Kg is the bulk modulus, ρair is the density of air, γ is the specific heat ratio (1.4 for ideal gas), q2 is the media tortuosity, P0 is the ambient atmospheric pressure and f is the frequency. fμ and fθ are functions describing the viscous and thermal dependencies respectively.
(2 2)
s air
f r q (5.11)
(2 )
s air pr
f r N (5.12)
where ϕ is effective porosity, rs is the air flow resistance of the porous pavement, and Npr is Prandtl number (about 0.71 in the air at 25°C). Using Equations (5.13) and (5.14), the acoustic impedance of porous layer (Zp) and its complex wave number (k) can be written as Equations (5.15) and (5.16).
1
F j f f (5.13)
1
F j f f (5.14)
12 1 12
cq F F
Zp air (5.15)
1 2
1 2
0 ( 1)
kk qF F (5.16)
where c is the homogeneous sound speed in the air, and k0 2f c.
Assuming that the porous surface layer is laid on a perfectly reflecting dense- graded base course, whose impedance could be taken as infinite, surface impedance of the whole porous pavement is derived as:
jkl
Z
Z pcoth (5.17)
where l is the thickness of the porous surface layer. With this surface impedance, the acoustic absorption coefficient can be calculated from Equation (5.8). This model has
188
been validated against experimental measurements (Bérengier et al ., 1997) and was successfully applied in the analysis of sound propagation over porous pavements (Wai and Kai, 2004).
5.2.2.2 Microstructural Model
The complex pore structure within a porous layer is simplified as individual pores with simple and well-defined geometries in the microstructural models. The entire porous medium field is then assembled by such representative pores. Viscous effects and thermal motions are usually considered separately for each single pore in a microstructural model and the results are corrected by a so-called shape factor to take into account the pore structure complexity, such as the variations in tortuosity and pore size. The acoustic effect at macroscopic scale is then generalized from the corrected single-pore results. Although it is more computationally expensive than a phenomenological model, the microstructural model provides more physical insights of sound propagation mechanisms in the porous medium. This study adopts the model developed by Neithalath et al. (2005), which relates acoustic absorption of porous pavements to three geometric parameters of pore structure (i.e. porosity, characteristic pore size and porous layer thickness).
Recognizing the fact that air is alternatively compressed and expanded when acoustic waves propagate through a porous mixture, Neithalath's model simplifies the pore network as a series of alternating cylinders with varying diameters (see Figure 5.1). Each unit of the pore structure consists of a pore (with diameter Dp and length Lp) and an aperture (with diameter Da and length La). Porosity value (ϕ) is related to the pore structure dimensions and wall thickness (d) through:
2 2
( )( )2
a a p p
a p p
D L D L L L D d
(5.18)
The characteristic pore size (Dp) is defined using the median of all pore sizes larger than 1 mm, which can be obtained from an image analysis on porous mixtures. The
189 other dimension parameters are related to the pore size and porosity, and can be determined by an iterative process or an optimization algorithm (Losa and Leandri, 2012). The effective porosity is maintained constant in the simplification of a pore structure.
The pore network structure is next modeled using an electro-acoustic analogy consisting of a series of electro resistors and inductors (see Figure 5.1) to simulate the acoustic behavior of porous pavements. The air impedance in a pore (Zp) is modeled by an inductor, the value of which is a function of the pore diameter (Dp) and can be calculated by:
cot( )
p air p
Z j c D c (5.19) The acoustic impedance of an aperture (Za) is modeled by a resister and an inductor, representing the real component (Ra) and the imaginary component (Ma) of acoustic impedance, respectively. The impedance of apertures is thus computed by:
a a a
Z R j M (5.20)
2 2
32 1
32 4
a a
a
a a
L D
R D L
(5.21)
2
8 1 1
9 2 3
a
a air a
a
M L D
L
(5.22)
where η is the air dynamic viscosity, and β is the acoustic Reynolds number deriving from Equation (5.23).
2
a air
D
(5.23)
For the single-cell situation, the acoustic impedance is given by:
2
1 2
p
a p
a
Z Z D Z
D (5.24)
190
For a porous layer with thickness l, which is composed by n cells, the acoustic impedance is determined by applying the electro-acoustic analogy to all the cells and calculated using the following iterative relationship.
2 2
1
1
1 1
p
n a
a p n
Z Z D
D Z Z
(5.25)
where Zn is the acoustic impedance of n-cell case, and Zn-1 is the acoustic impedance of the situation with n-1 cells.
In the above calculations, air density (ρair) should be multiplied by a structure factor to take into account that the air in lateral pores appears to be "heavier" than the air in main pores due to the different mechanisms in acoustic energy dissipation. The structure factor (ks) is defined by Equation (5.26) and is based on the fact that all pores do not contribute equally in sound absorption.
2 2 2 2
2 2 2
( ) ( )
( )
a a p p a p p a
s
a p a p
L D L D L D L D
k L L D D
(5.26)
After the acoustic impedance of porous pavement is determined by Equation (5.25), the absorption coefficient can be derived using Equation (5.8). This model has been validated against experimental measurements (Neithalath et al., 2005) and was applied on the analysis of the acoustic characteristics of porous pavements (Losa and Leandri, 2012).