1. Introduction
When sound waves propagate through vortical flow, the flow–sound interaction affects the sound propagation [
1], resulting in the acoustic scattering phenomena [
2,
3,
4]. Two-dimensional vortex scattering is a classical model in the understanding of flow–sound interaction [
5,
6,
7,
8,
9]. The study of the characteristics of vortex scattering is important for both exploring the coupling mechanism of flow and acoustic [
10] and capturing the features of the sound propagation in nonlinear flow in engineering, such as sound propagation through the wake of the aircraft jets and acoustic imaging in medicine.
Numerical algorithms are becoming more accurate [
11,
12,
13,
14,
15]. Numerical simulation methods are the main approach to deal with the problem of vortex acoustic scattering [
16,
17]. Coloninus [
18] analyzed the propagation of acoustic waves through a Gaussian vortex flow based on a high-order numerical scheme. Karabasov [
19] solved the Euler equation with the extended upwind leapfrog scheme and analyzed the acoustic scattering from steady-state Gaussian vortex and Rankine vortex flow at different Mach numbers. Blanc-Benon [
20] used geometric acoustic theory and parabolic wave equations to simulate the propagation of acoustic waves through the turbulent flow. Iwatsu and Tsuru [
21] used Cin’s integral method and the compact finite-difference scheme to solve the linear Euler equations and calculated the scattering acoustic field from the Rankine vortex and Burgers vortex, respectively. Ke [
2] analyzed the acoustic scattering characteristics of plane waves propagating through the Gaussian vortex and vortex pairs with the WENO scheme, and the results were consistent with the direct numerical simulation obtained by Coloninus [
18]. In the study of underwater vortex scattering, the finite-difference time-domain method (FDTD) was applied [
22,
23,
24]. Zhang [
25] introduced the simulation of vortex scattering based on the FDTD method and discussed the directivity of the scattering sound.
To improve the stability in the numerical simulation of vortex scattering, the finite-difference frequency-domain (FDFD) algorithm with numerical discretization on the staggered grid is introduced to the simulation of vortex scattering.
Meanwhile, the construction of an acoustic absorption boundary is especially important for numerical simulation. By adding several absorption layers around the computational domain, the perfectly matched layers (PML) can decrease the boundary reflection and thus turn the infinite domain into a finite domain. For acoustic propagation in the moving flow, PML can be used as the acoustic absorbing boundary [
26]. The acoustic absorption effect of PML depends on the absorption coefficient, and setting a proper absorption coefficient improves the acoustic absorption [
27]. Moreira [
28] solved the propagation equation of acoustic waves in uniform and inhomogeneous medium using the frequency-domain finite-difference method combined with PML and analyzed the influence of the PML absorption coefficient on the acoustic absorption effect. Jie [
29] adopted PML for the broadband scattering model of the underwater target and set different absorption coefficients according to the frequency of the incident wave. It shows that the acoustic absorption effect works well at each frequency. Park [
30] applied the PML to solve the 2D Helmholtz equation and symmetrically discretized the PML matrix, which can effectively reduce the computational resource. However, the effect of PML on the calculation of vortex scattering and the suitable value of parameters for vortex scattering simulation are unclear, such as the effect of the relative thickness of PML, the size of the computational domain, and the vortex core radius. In order to study the above issues quantitatively, we have introduced an attenuation term in the equation of motion in this paper, which increases from the inner to the outer boundary, instead of the diffusion term multiplied by the attenuation factor according to the aforementioned works. This method can ensure energy absorption, and the attenuation variation inside the PML layers is relatively smooth. This ensures the precision of numerical simulations regarding the scattering of vortex acoustics.
In this paper, a FDFD algorithm combined with the PML is established to evaluate the acoustic scattering caused by a vortex, and the relevant parameters suitable for 2D vortex acoustic scattering are specially discussed and determined. In the end, the validated results agree well with the reference results. In
Section 1, a two-dimensional vortex scattering model is introduced. In
Section 2, the numerical scheme to solve the perturbation equation is given. In
Section 3, the algorithm is introduced, and the grid independence is verified. In
Section 4, the relationship between the PML absorption coefficient, PML layers, and Mach number is explored, and the algorithm is verified.
Section 5 contains the conclusion.
2. Vortex Scattering Model
In this paper, the classical two-dimensional Burgers vortex model is used. The schematic diagram of vortex scattering is shown in
Figure 1:
The tangential velocity of the vortex flow is [
2]:
where
is the radius of the vortex core.
is the distance from the point in the flow to the center of the vortex core.
= 1.256431 is a constant chosen so that the maximum velocity occurs at
.
is the circulation of the vortex. The Mach number
is the ratio of the maximum velocity to the sound speed.
The sound pressure of the incident plane wave is defined as [
31]:
where
is the amplitude of the incident wave,
is the frequency,
is the wavenumber,
is the
-direction coordinate, and
t is the time.
3. Governing Equations
For vortex flow with a low Mach number, the amplitude of acoustic variables is assumed to be smaller than the fluid variables. Therefore, based on the first-order perturbation method, the Navier–Stokes equations can be separated into two parts based on the magnitude. Then, the wave equation of the flow–sound interaction is obtained.
For flow without external force, according to the isentropic adiabatic principle, the governing equations can be simplified [
32]:
where
,
, and
represent the density, velocity and pressure of the flow, respectively.
is the kinematic viscosity, and
c is the sound speed.
Assuming that the magnitude of the acoustic variable is much smaller than the corresponding fluid variable [
33],
p and
u can be defined as follows:
where
and
represent the pressure and velocity of the flow without the acoustic disturbance, respectively.
and
are the perturbed acoustic pressure and velocity generated by the sound wave.
We consider
, so that
and
satisfy the following formulas:
For low Mach number flow, the following assumptions are met: (1)
, (2) the time scale of the flow is much larger than the period of the incident acoustic wave, and (3) the influence of the acoustic disturbance on the flow is negligible. The vortex scattering phenomenon caused by the flow–sound interaction is then mainly controlled by the nonlinear term. By retaining the term and neglecting the high-order amount, the simplified governing equation of acoustic disturbance can be obtained:
where
is the density of the medium.
The above time domain equation can be transformed into a frequency domain equation. It can be assumed that
and
satisfy
,
respectively, and then Equations (8) and (9) can be transformed into:
where
is the circular frequency.
The PML is adopted as the acoustic absorption boundary, as shown in
Figure 2. The computational domain is a square with N-layers PML. In order to effectively absorb the sound, the attenuation factor of the
i-th layer satisfies the following formula:
where
is the maximum absorption coefficient.
Obviously, in the PML, Equation (11) can be modified to:
It can be seen from Equation (13) that the attenuation term adds an attenuation factor to the velocity of the particle inside the PML layers.
5. Discussion of PML Parameters
5.1. Relationship between Layer Numbers and Absorption Coefficient
Firstly, the effect of the absorption coefficient on the result for a given number of PML layers is discussed. By choosing sufficient thickness of PML as 75 layers (5 times the wavelength) and changing
from 200 to 30,000
, the sound scattering directivity results of different cases are compared in
Figure 5.
We define the relative error as follows:
where
is the effective scattered sound pressure at the monitoring point
of the test.
is the theoretical value of effective scattered sound pressure at the monitoring point
.
As can be seen from
Figure 5, obvious fluctuations in the backward scattering region arise with an absorption coefficient too small or too large. When the attenuation is too small (200 kg/(m
3·s), the PML boundary absorbs less acoustic energy, resulting in excessive reflection. When the attenuation is too large, the boundary behaves like a rigid boundary and results in a stronger reflection. Therefore, an appropriate absorption coefficient should be chosen.
In order to analyze the attenuation of sound pressure inside the PML layers, the distribution of effective scattering sound pressure of the PML boundary is shown in
Figure 6. When
= 200
, the scattered sound pressure is not completely attenuated in PML layers, and the distribution along the PML layers shows significant fluctuations. When
= 1000–15,000
, the scattered acoustic pressure in PML layers can be attenuated smoothly, and the larger the absorption coefficient is, the fast it attenuates. When
= 30,000
, the scattered acoustic pressure can be completely attenuated in PML layers, but local fluctuations occur near the first layer of the PML layers, which produces a strong reflection into the internal computational domain.
It can be seen that the appropriate absorption coefficient should satisfy the following conditions: the scattered acoustic pressure inside the PML layers can be completely attenuated, and no strong reflection is produced in the internal computational domain.
Another problem that should be considered is that too many PML layers bring low computing efficiency.
Figure 7 shows the variation in the calculation time with the number of PML layers. As can be seen from the figure, when the number of PML layers increases, the calculation time also increases rapidly.
Based on this study of this section, when a larger number of layers is chosen, a better acoustic absorption can be obtained, but it may have a larger calculation load. Therefore, a reasonable number of layers should be chosen, which will be analyzed shortly.
5.2. Dimensionless Analysis of PML Parameters
This section analyzes the relationship between the absorption coefficient and related parameters based on the dimensionless analysis method. There are eight related physical quantities, namely medium density
, sound speed
, incident wave frequency
, number of nodes per wavelength
, the size of the computational domain
, Mach number
, vortex core radius
, number of PML layers
, and PML absorption coefficient
. We choose
as the basic quantity. the dimensional relationship is:
where
is a function.
is the ratio of the thickness of the PML boundary to the wavelength.
is the ratio of the size of the computational domain to the wavelength.
is the ratio of the radius of the vortex to the wavelength.
The absorption coefficient relationship is (17):
In Equation (17), is the main parameter characterizing the thickness of the PML boundary. From Equation (1), it can be seen that the flow velocity distribution is determined using , therefore , , and can be regarded as the main parameters characterizing the flow velocity.
5.3. Parameters of Flow
In this section, the relationship between the absorption coefficient and the flow field velocity parameters , , and are analyzed. Since a large causes a large amount of calculations, we first analyze , then , and finally with proper . The frequency is set to be 85 Hz, the sound speed is 340 m/s, the number of nodes per wavelength is 15, and .
5.3.1. Size of the Computational Domain
This section discusses the influence of the size of the computational domain on the simulation. The effective scattering attenuation distribution in the PML with different sizes of calculation domains (2.5, 5.0, 10.0) is shown in
Figure 8. Results show that sound pressure is completely attenuated within the same number of layers, and there is no obvious local fluctuation. The size of the computational domain has little effect on the choice of PML parameters. Therefore, the following analysis of PML parameters can use a smaller calculation domain to reduce computational cost.
5.3.2. Vortex Core Radius
This section analyzes the radius of the vortex core. The incident sound frequency is 85 Hz, the sound speed is 340 m/s, the size of the computational domain is 10 m, and the radius of the vortex core is from 0.5 to 1.5 m. Therefore
is from 0.125 to 0.375. The effective scattering sound distribution in PML layers is shown in
Figure 9. It can be seen from the figure that the acoustic pressure scattered from different vortex is completely attenuated within the same number of layers, and there is no obvious local fluctuation. The vortex core radius has little effect on the choice of PML parameters.
5.3.3. Mach Number
This section analyzes the relationship between the Mach number and the absorption coefficient. The frequency is 85 Hz, the sound speed is 340 m/s, the size of the computational domain is 10 m, and the Mach number is 0.10–0.25. The scattering sound distribution in PML layers is shown in
Figure 10. It can be seen from the figure that the acoustic pressure is completely attenuated in the same number of layers at different Mach numbers, and there is no obvious local fluctuation. The Mach number has little effect on the choosing of PML parameters.
It can be seen from the results in this section that the flow velocity parameters
,
, and
have little effect on the parameter choosing of the PML boundary. When the flow velocity increases, the particle velocity increases. If the PML attenuation factor does not change, the scattered acoustic is attenuated at the same attenuation ratio in the PML layers. At the same time, due to the large attenuation factor, the PML absorption coefficient approximately satisfies the following formula:
5.4. Relative Thickness of PML
This section analyzes the effect of PML thickness on acoustic absorption. The number of nodes per wavelength is chosen to be 15, and the number of PML layers
is from 15 to 75. Therefore,
is from 1.0 to 5.0. For different thicknesses, according to the function
g (
), cases of different absorption coefficients are conducted. The comparison of the directivity of different cases is shown in
Figure 11. It can be seen from the figure that when the thickness of PML is smaller than 2.5 times the wavelength, there is no obvious reflection.
In order to ensure effective acoustic absorption while minimizing the amount of calculation resource, a PML thickness of 2.5 times the wavelength is suggested.
5.5. Comparison with Direct Numerical Simulation (DNS)
In this section, the results obtained by our algorithm are compared with DNS results [
18]. The density of the fluid medium is
, the characteristic radius of the vortex core is
, the Mach number is
, the frequency of the incident plane wave is
, the sound speed is
, and the computational domain is a
square. The monitoring points are
away from the center of the vortex core.
From
Section 5.4, we know that when the PML boundary thickness is 2.5 times the wavelength, the reasonable
is 23.5–40. The scattering sound pressure directivity compared with the DNS result is shown in
Figure 12. It can be seen from the figure that the results of this method are consistent with the results obtained through DNS [
18]; thus, the algorithm can be used to predict the vortex scattering field.
6. Conclusions
In this paper, a finite-difference algorithm in the frequency domain combined with the PML absorbing boundary is developed to simulate acoustic scattering from a two-dimensional vortex. The mesh independence is verified, and when the mesh corresponds to over 15 nodes per wavelength, the result has a good convergence. The value of the PML absorption coefficient and the number of PML layers are analyzed. The limitation of this algorithm is that it is only applicable to 2D vortex acoustic scattering. But it also provides a reliable algorithm for the numerical study of the effect of the vortex on sound propagation, such as target detection and acoustic imaging.
By comparing with the literature data, the accuracy of the algorithm is verified, and the following conclusions are drawn:
In the PML region, when the scattered sound pressure can be completely attenuated before reaching the outermost boundary and there is no obvious fluctuation near the boundary between the computational domain and the PML, the scattering sound can be regarded as fully absorbed.
When the thickness of the PML boundary is larger than 2.5 times the wavelength, a qualified sound absorption effect can be obtained by selecting a suitable PML absorption coefficient.
The algorithm established in this paper can effectively calculate the two-dimensional vortex scattering. The value of the absorption coefficient of PML is suggested to be linear with the density of the medium and the frequency of the incident wave. The Mach number shows little effect on the acoustic absorption effect of PML.