Numerical investigation of microstructure effect on acoustic properties of underwater anechoic coatings

NUMERICAL INVESTIGATION OF MICROSTRUCTURE EFFECT ON ACOUSTIC PROPERTIES OF UNDERWATER ANECHOIC COATINGS Trinh Van Hai 1* , Nguyen Van Tap 2 1 Le Quy Don Technical University 2 Milita

NUMERICAL INVESTIGATION OF MICROSTRUCTURE EFFECT ON ACOUSTIC PROPERTIES OF UNDERWATER

ANECHOIC COATINGS

Trinh Van Hai 1* , Nguyen Van Tap 2

1 Le Quy Don Technical University

2 Military Institute of Mechanical Engineering

Abstract

This paper presents a simulation-based model for predicting the acoustic properties of underwater anechoic structures with its main layer made of viscoelastic sound absorbing materials including air-filled cavities Some preliminary numerical results are first compared with the published analytical and experimental data to validate the proposed modeling Then, a developed design of anechoic using two arrays of air-filled bubbles is considered It is observed from the investigation results that the new coating shows an interesting sound absorbing performance (i.e., absorbing > 80% incident energy in a large frequency range 25 MHz) compared with the original one having a single air cavity array The developed structure allows broadening and tailoring their acoustic performance (peak frequency and averaging level) by tuning some microstructural parameters of the air-filled cavity (shape and size) and its distribution (location and fraction)

Keywords: Air-filled cavity; acoustic property; anechoic coating; resonant sound absorber

1 Introduction

The high-quality acoustic stealth coatings are a complicated scale and multi-component structure, by considering their microstructure, these anechoic structures can

be categorized into three main types: air-filled cavity, multi-layer composite, pressure-resisting [1] In such layered configurations, there are two types of resonance mechanisms which are due to either the radial motion of the hole wall or the drum-like oscillations of the cover layer [2] Additionally, the developed structure allows us to broaden and tailor its acoustic performance by tuning some geometrical parameters of the air bubble distribution It can be stated that macroscopic acoustic properties are highly dependent on the local microstructural features of each individual layer as well

as the layer configuration [3]

Different approaches have been proposed in the literature to characterize the link between the microstructural parameter of anechoic structures and their macroscopic acoustic performance: analytical [4-7], numerical [8-11], and experimental methods

* Email: hai.tv@lqdtu.edu.vn

[6, 12, 13] Various analytical studies addressing scattering from gratings and resonant sound absorbers exist: transfer matrix method [14], effective medium model [15, 16], and cavity resonances mode [17, 18] However, these analytical models often make simplifications on the displacement field and geometry, thereby imposing limitations on the type of problem to be solved On the contrary, the numerical approach such as the finite element method (FEM) is more flexible in dealing with complex structures which allows analyzing harmonic wave propagation in viscoelastic gratings with periodic or random distributions with single- or multi-layer structures

As a survey introduced in [1], Russian designs enable to lower the noise level of Akula and Sierraz class submarines by 10~20 dB by anechoic coating using modified rubber Britain technology has used polyurethane as anechoic material, lowering conventional submarine noise to 90 dB and meeting the qualifications of silent submarines. Furthermore, in US submarines, the double-layer anechoic structure composed of polyurethane and glass fibers can decrease noise by up to 40 dB and the radiated noise level of nuclear-powered submarines to less than 120 dB, approximately the ocean noise level (for more detailed about design and construction of US and Soviet submarines, see [19])

The remaining part of the paper is organized as follows In Sec 2, we first briefly introduce the theoretical background of acoustic aspects of wave propagation and the definitions of several acoustic properties used for characterizing sound absorbing structures Then, we present a numerical framework at a relevant scale to implement the acoustical model of a multi-layer anechoic coating Sec 3 provides an example validation together an extended investigation about acoustical behavior of air-filled cavity anechoic structure with a single-cavity array in order to demonstrate the verification of the proposed numerical framework In Sec 4, a developed anechoic configuration and an analysis on its acoustical potential performance are proposed Finally, some concluding remarks are given

2 Theoretical formulation and numerical framework

2.1 Governing equations for structure-acoustic analysis

The structure-acoustic problem is sketched schematically in Fig 1 It consists of a fluid domain Ωf, and a solid domain Ωs The boundary between the fluid domain and the structure domain is denoted, Ω For the structure-acoustic system, in three-dimensional space, the structure is described by the differential equation of motion for a continuum body assuming small deformations [20, 21]:

3 3 3

1 j 1

( 1, 2, 3) in ,



where t the time variable, u si (i = 1,2,3) are three components of the displacement us, c ijkl

(i,j,l,k = 1,2,3) are the elastic constants of stiffness tensor C, ρ s represents the density of

the solid, and x j represents the coordinate variable x, y and z respectively

The fluid is governed by the acoustic wave equation [21, 22]:

2

2 2 2

2p c p c q f in f ,





where p is the dynamic pressure related with the dynamic density  by the constitutive f

condition pc20 f q f is the added fluid mass per unit volume and c0 is the speed of

sound Operator  denotes a gradient of a variable as  1 2 3

T

Fig 1 Fluid domain  f embedded in an elastic structure of infinite extent  s by an interface 

At the solid-fluid interfaces, the coupling conditions ensure the continuity in displacement and pressure between the domains as [21, 22]:

s 3

where u f the displacement in fluid linked with the pressure as p =-02u f 2t with

ρ0 is the static density of air, and n is the normal direction of the boundary I 3 is a 33

unitary matrix and S s is the Cauchy stress tensor Noted that the components of stress tensor are estimated from stress vector  s C: s The strain vector  is deduced s

from the strain-displacement relation as  s us in which the differential operator 

is introduced,

T



Based on the above governing equations and boundary conditions, using the FEM schematic, the discretized form of the governing equation for the coupled structure-acoustic problem can be written in matrix form as follows [22, 23]:

T

0 0

0

,





where M, C and K are the global mass, damping, and stiffness matrices, respectively The subscripts ‘s’ and ‘f’ denote the solid and the fluid domains, respectively u s is the

nodal displacement vector in the structural domain and p f is the nodal pressure vector in

the acoustic domain, whereas f s and f f are the nodal structural force and the nodal acoustic pressure vectors, respectively

2.2 Unit cell modeling and parameter calculations

In acoustics, the pressure of sound wave can be calculated as a function of time and location [24] In general, complex form is often used for representing the sound waves [25], and the harmonic wave is depicted at frequency  and wavenumber k as,

where A0 is the amplitude, 0 is the initial phase, and j is the imaginary unit

Fig 2 Diagram of an acoustic material layer excited by an acoustic wave

In order to understand the acoustic behavior of sound absorbing materials, we first need to understand what happens when sound wave travels through these media It can

be known that the sound wave interacts with the material or object surface and may be absorbed, transmitted and reflected (see Fig 2) Therefore, the incident wave energy would be partly separated into three corresponding components

The structure is excited by a harmonic plane wave from the semi-infinite fluid medium The expression of the incident wave is:

where the time dependence exp(-j t) as shown in Eq (6) has been omitted for clarity,

 and  denote the direction of wave incidence (see Fig 3a)

Fig 3 (a) Absorption structure includes a viscoelastic material and a steel backing,

and (b) schematic description of the representative unit cell

According to Bloch’s theorem, if the structure has the periodic distance d x in the x direction and d y in the y direction, any space function  (e.g., pressure, displacement…)

satisfies the following relation [9]:

x d x,y d zy,  x y z, , expjd k x sin cos expjd k y sin sin 

Fig 3b shows the computing unit cell model among the multilayer slab, where S

and S surfaces parallel to the xOy plane limit the finite element mesh from the half infinite backing and water domains for incidence, respectively The external incident wave and the reflected wave in the domain above the S surface and the transmitted wave in the domain above the S surface can be written in the general forms:

in

r

m,n

m,n

p x, y, z p x, y, z p x, y, z

















(9)

where k mx 2m  d xksin cos ,  k ny 2n  d yksin cos , and  k mn2 k2k mx2 k ny2

R mn and T mn are the reflection coefficient and transmission coefficients corresponding to

the (m, n)th mode

By solving Eq (5), the nodal values of the pressure on the incident surface ( p)

and transmission surface ( p) can be obtained Thus, the unknown coefficient R mn and

T mn are can be deduced from two sets of equations establishing based on the number of

known pressure values in the corresponding surfaces It can be noted that each unknown coefficient requires one nodal pressure value Thus, the anechoic performance of an

acoustic sound absorbing medium is defined by the sound reflection (R), transmission (T) and absorption coefficients ( ) as [9, 23, 26]:

3 Verification and comments on single-cavity configuration

3.1 Verification of the developed FEM procedure

Fig 4 Comparison between the proposed numerical results and analytical model and experimental data proposed in [12]: the reflectance (top, left), transmittance (top, right), and absorption (bottom) coefficients The results correspond to a case of L x = 50  m

In this validation step, the obtained numerical results are compared with the available data proposed somewhere in the literature Here, to verify our FEM work, the analytical models and measurement data proposed by V Leroy et al in [12] are used The acoustical model of anechoic tile are structured with a soft elastic layer within an air cavity array and backing by a steel layer A detailed configuration of the absorbing layer is drawn in Fig 3a In this sample, the elastic material layer has a thickness of 230

z

L  m and cylindrical cavities of diameter D 24 µm and height H 12 µm The Young’s modulus, density, and Poisson’s ratio of the steel plate are 2.161011 Pa,

7800 kg/m3, and 0.3, respectively As shown in Fig 4, the comparison between the proposed numerical results and analytical model and experimental gives a good agreement which validates the finite element modeling procedure

3.2 Tuning the fraction of air cavity

Fig 5 Acoustic performance of anechoic tiles with tuning air fraction The results from top to bottom are respectively for the reflectance, transmittance and absorption coefficients

In this section, we present how the distribution density of air cavity affects the acoustical behavior of the single air array anechoic As demonstrated in [12], for the steel block alone, 88% of the energy is reflected and the remaining energy fraction (12%) is transmitted Fig 5 presents the energy proportions that are reflected (top),

transmitted (middle) and absorbed (bottom) for various configurations of L x (50 µm,

75 µm, 100 µm, 125 µm, and 150 µm) As expected, when the block is covered by the

L x = 120 µm meta-screen, the reflectance is drastically reduced (line with markers in

Fig 6 (top)), especially between 1.3 and 2.8 MHz where less than 6% of the incident energy is reflected, with the measured reflectance dropping nearly to zero at 1.3 MHz

and 2.8 MHz In terms of absorption (Fig 6 (bottom)), all anechoic configurations manage to dissipate a significant part of the energy over a broad frequency range

Interestingly, the configuration with L x = 100 µm (markers in Fig 5 (bottom)) provides

a very high absorption over the entire 1.32.8 MHz range, throughout which more than

91% of the incoming energy is dissipated, with a maximum absorbance of nearly 100%

at 1.3 MHz and 2.8 MHz The obtained trends here are in very good agreement with the suggested results illustrated in [12] for cases of L x = 50 µm and L x = 120 µm

4 Results and discussion on double-cavity configuration

4.1 Design configuration

Fig 6 Configuration of viscoelastic absorbing layer with double void array

In this section, acoustical properties of anechoic tile with an absorbing panel of two lattice layers of air cylindrical cavities is investigated (Fig 7) Compared to the original one, the geometric properties of the additional layer of voids are also introduced

by the dh cylindrical cavities with their distances L x and L y in the x and y-direction, respectively, at the location l within the elastic layer

The potential acoustical performance of this developed coating structure is demonstrated in the following part by tuning several structural parameters Here, we focus only on the reflection and absorption property, for simplicity, the added air cavity

is identical with the original one in terms of shape (d = D, h = H) and distributions (L x and L y) The effect of the configuration difference between two air layers on its acoustical properties will be characterized in the forthcoming works

4.2 Cylindrical cavity size effect (D)

Fig 7 shows the performance of anechoic tile based on air cavity having

H = 13 m, L x = L y = 100 m, l = L x /4, and varying its diameter D from 10 m to 25 m

It is observed that by the acoustical behavior of the double air array coating depends strongly on the air-filled cavity diameter The anechoic within small void seems to have good acoustical behavior at low frequencies nearly 1.6 MHz (see the shape peak in Fig 7 with more than 96% of the incoming energy is absorbed while remaining less than 3% reflected energy) Interestingly, compared with the results demonstrated in

Fig 5 with 24-diameter air cavities, the double array configuration (with D = 15 m and

D = 20 m representing respectively by thin dashed and thick solid line) can provide a

good acoustical functional factors ( 94.5%) at high frequency range (4  5 MHz)

Fig 7 Effects of the cylindrical cavity size on acoustical performance of anechoic tiles:

reflection (left) and absorption (right)

4.3 Air cavity density effect (L x )

Fig 8 Effects of the air cavity distribution on acoustical performance of anechoic tiles:

reflection (left) and absorption (right)

In order to show the effect of the air-filled cavity density, this section investigates

anechoic tile based on air cavity having H = 13 m, D = 24 m, l = 100 m, and

varying cavity distance from 50 m to 125 m It can be seen from the results from

Fig 8 that the length L x affects both in the level of reflected and absorbed energy and the frequency occurring the second shape peak In detail, the absorption property

increases monotonically when L x increases for L x < 100 m and then decreases

monotonically when L x continues to increase above the critical value L x = 100 m

Generally, the air cavity with length around 100 m shows good acoustical performance (see the thick solid line)

4.4 Cavity array distance effect (l)

The third parameter of anechoic tile is the distance between two cavity layers

l (50 m, 100 m, 150 m and 250 m) Here, the remaining parameters are kept as:

D = 24 m, H = 13 m, and L x = L y = 100 m From the Fig 9, it is seen that both fluctuations of reflection and absorption property are related fully to the air cavity

distance Within the distance l ranging from 50 m to 250 m, we enable to tailor the

number of shape peaks and the fluctuated amplitude of acoustic performance It should

finally be noticed that the configuration with l = 150 m provides a good absorption

(>80% of the incident energy, see the thick solid line) in the frequency range 25 MHz

Fig 9 Effects of the cavity distance on acoustical performance of anechoic tiles:

reflection (left) and absorption (right)

5 Conclusions

In this paper, the simulation-based approach is presented to investigate the link between microstructure and acoustic properties of an anechoic structure with periodic air cavities To this regard, the multi-layer anechoic coating is reconstructed, and the acoustic-structure model is generated including a cover layer, a viscoelastic absorbing panel and the steel layer as the backing From this, the reflection, absorption, and transmission properties of the anechoic coating backing a steel plate are calculated numerically Very good agreements are observed between the present numerical results

of and an anechoic layer within a single cavity array with the analytical model, numerical results, and experimental data from the literature, which validates the suggested simulation-based procedure From investigated results of the developed configuration, it is seen that all tuning geometrical parameters of the added air cavity affect strongly the acoustical properties The air cavity size and its density have an effect on the level of reflected and absorbed energy, whereas the air cavity distance has only effect on the fluctuating behavior This interesting point shows a good opportunity

to achieve a desired reflection or absorption properties in an entire frequency range by tuning together all three parameters mentioned here and also others fixed (e.g.,

thickness layer L, ratios d/D, h/H)