Noise reduction structures are important for the vibration and noise reduction design of aerospace engines. The design of noise-reducing structures often needs to be quickly evaluated via numerical simulations. Hence, the simulation results of the corresponding system are very important for guiding the design of noise-reducing structures. High temperature is one of the key environmental factors that need to be considered when evaluating the sound attenuation process via numerical methods. In this study, numerical simulations of acoustic wave propagation on an acoustic liner structure considering air temperature variations are carried out by using compressible Navier–Stokes equations and the ideal gas equation of state. The results showed that the effect of temperature on sound attenuation under grazing flow conditions is complex. Moreover, an increase in temperature will reduce the transmission loss of the acoustic liner in the grazing flow at high air speed.

## I. INTRODUCTION

To reduce noise, acoustic liners are often placed in the ducts of large systems, such as aeroengines.^{1} Acoustic impedance is used to reflect the noise absorption performance of an acoustic liner and needs to be considered in the design of sound liners. Several acoustic impedance models have been developed.^{2–4} These models can be used to obtain the relevant parameters of acoustic liners and greatly simplify the calculation process. The presence of grazing streams significantly alters the acoustic impedance of the acoustic liner.^{5,6}

However, older models do not work well when considering grazing flow. Dickey *et al.*^{7} devised an experimental method by comparing the results of perforated plates with different aperture shapes under grazing flow. He concluded that the empirical formula for one perforated plate does not apply to the other perforated plate when both have different aperture shapes. Armstrong measured acoustic impedance in flow environments based on waveguide theory to aid in designing mathematical models for acoustic linings. Lee and Ih^{8} proposed a new circular aperture perforated plate impedance model applicable under grazing mean flow conditions. Peat *et al.*^{9} devised an analytical method for describing the effect of grazing flow on the acoustic impedance of rectangular apertures and compared it with earlier approximate theories, pointing out the limitations of these theories regarding accuracy and application. The effect of the presence of grazing streams on the various parameters^{10–13} of the sound lining has begun to be investigated in search of ways to improve it. The modeling of acoustic impedance under grazing flow conditions has been gradually refined over the years.^{14–19} By using a combination of previous acoustic impedance models and empirical impedance models that have been progressively refined in recent years, it is possible to characterize the acoustic liner under the action of simple grazing flows, contributing to the design of acoustic liners.^{20–26}

In many of the above-mentioned studies, only the effects of grazing flows were considered. However, with regard to specific application environments, such as engine ducts, the effect of temperature on the acoustic lining in the presence of grazing flow conditions needs to be considered. Ćosić *et al.*^{27} investigated the acoustic response of a Helmholtz damper in the presence of thermal grazing flow and the effect of the temperature difference in the grazing flow and the crossflow in the Helmholtz damper at linear and nonlinear amplitudes. Relevant prediction models have been proposed, but experimental results are still needed. Färm *et al.*^{28} investigated the effect of temperature gradients and flow variations on the surface impedance of structures and proposed relevant numerical algorithms. However, the combined effects of temperature gradients and grazing flows have still not been considered. Esnault *et al.*^{29} analyzed the effect of upper turbulence on the heat transfer between acoustic liner walls. Lafont *et al.*^{30} investigated the relationship between the sound pressure intensity and the acoustic lining temperature in the presence of grazing flows. Lafont later proposed multiphysics field coupling among the acoustic, flow, and thermal fields, measured the test structure,^{31} and verified his conjecture that there would be coupling among the three. Although temperature-related studies of acoustic liners under grazing flow conditions have been initiated and predictions can be made for some structures based on a small amount of experimental data, there is still a need for a numerical approach to reflect the combined effects of temperature and grazing flows on acoustic liners.

In this paper, we propose a numerical computational model based on the concept of the numerical solution of compressible Navier–Stokes equations and the ideal gas equation of state. This approach allows for the consideration of the simultaneous effects of temperature and grazing flows on the acoustic liner. First, through the governing equations of the model, we can couple three physical fields, namely, the temperature, the flow velocity, and the acoustic field, to consider the effects of the temperature and the flow rate on the model comprehensively. Second, the governing equations are solved via numerical methods, and the parameters of the model are obtained. In the numerical calculation, the parallel computation method is adopted to fully utilize the performance of our computer, which significantly improves the calculation efficiency. We expect parallel computing to increase the computational scale and thus get better computational accuracy, but the use of parallel computing introduces errors, mainly due to errors caused by the closure of the internal scheme to the subdomain boundaries.^{32–35} For the problem of subdomain boundary closure, different processing methods will lead to different results. An unsuitable method will lead to a decrease in computational accuracy and deviation from the initial goal;^{36–38} therefore, a suitable subdomain boundary closure method is something we must consider.^{39–41} In Sec. II, where the numerical methods are presented, further information is given on the subdomain boundary closure problem, the computational problem for solid and liquid subdomains, and the time step advancement method. Notably, our numerical model can reflect not only the effect of temperature on the sound lining but also the effect of temperature on grazing flows. Overall, the model enables the coupling of the three physical fields and the observation of the effects of changes in each physical field on the overall model; e.g., when considering the effects of temperature and grazing flow together on the acoustic lining, we can also observe the interactions between the two. The design concepts of this study are instructive for future research on structures such as aero-engines, where we need to consider the complex interactions among temperature, grazing flow, and acoustic liner.

## II. COMPUTATIONAL MODELS AND NUMERICAL ALGORITHM

### A. Geometric model

The numerical computational model proposed in this paper is designed according to Huang *et al.*’s model. The model consists of the acoustic liner where the tests were performed and the flow tube for the passage of the sound and the grazing streams, as shown in Fig. 1. The acoustic liner consists of eight periodic structures, with the individual structures consisting of two parts: the microperforated plate above and its back cavity, and the hypersurface below. The deflector walls are considered acoustically rigid due to the significant difference in impedance between the deflector walls and the air, except for the location of the acoustic lining arrangement. As shown in Fig. 1, we set the airflow and the sound pressure wave entering from the left side, *U*, which represents the velocity of the passed grazing flow, and *p*, which represents the pressure of the incident sound wave. We define the flow direction as follows: the x axis is positive, the y axis is perpendicular to the acoustic lining, and the z axis is perpendicular to the XY plane.

The model data for a single acoustic liner unit are based on Huang *et al.*’s data and are shown in Fig. 2, where the hypersurface below consists of a plurality of Helmholtz resonators. The numerical model in this paper has a computational domain, and by importing the mesh data of the solid part, the geometric model can be imported into the numerical model and subsequently computed. Figure 2(a) shows the approximate structure of the acoustic liner. Figure 2(b) shows the meshing of the solid part of the geometric model.

The effects of temperature and grazing flow rate are investigated by solving the numerical model under different conditions. Case 1 serves as the benchmark. Compared to case 1, only the rate of the grazing flow is changed in cases 2–6, only the temperature level is changed in case 7, and only the temperature gradient is changed in case 8. Compared to case 8, only the rate of grazing flow is changed in case 9. Table I provides a summary of the cases simulated in this study. In this paper, a complex study of the temperature is considered, in which the temperature gradient is set mainly to *T*_{0} and *T*_{1}. *T*_{0} is the temperature of the wall above the flow tube, and *T*_{1} is the temperature of the lowermost part of the acoustic lining.

Cases . | Fluid velocity U (m/s)
. | Upper wall temperature T_{0} (K)
. | Bottom temperature of the liner T_{1} (K)
. |
---|---|---|---|

1 | 0 | 293 | 293 |

2 | 10 | 293 | 293 |

3 | 20 | 293 | 293 |

4 | 40 | 293 | 293 |

5 | 60 | 293 | 293 |

6 | 120 | 293 | 293 |

7 | 0 | 773 | 773 |

8 | 0 | 773 | 293 |

9 | 60 | 773 | 293 |

Cases . | Fluid velocity U (m/s)
. | Upper wall temperature T_{0} (K)
. | Bottom temperature of the liner T_{1} (K)
. |
---|---|---|---|

1 | 0 | 293 | 293 |

2 | 10 | 293 | 293 |

3 | 20 | 293 | 293 |

4 | 40 | 293 | 293 |

5 | 60 | 293 | 293 |

6 | 120 | 293 | 293 |

7 | 0 | 773 | 773 |

8 | 0 | 773 | 293 |

9 | 60 | 773 | 293 |

### B. Governing equations

*ρ*,

*t*,

*x*

_{j}, and

*u*

_{j}are the density, time, spatial coordinates, and velocity components, respectively. Where

*u*

_{j}for

*j*= 1, 2, 3 are the components of the velocity vector $u=u,v,w$, and

*x*

_{j}for

*j*= 1, 2, 3 are the components of the position vector $x=x,y,z$.

*p*is the pressure, which can be calculated by Eq. (4). Where

*R*= 287 J K

^{−1}kg

^{−1}is the ideal gas constant.

*τ*

_{ij}is the stress tensor, which can be calculated by Eq. (5), and

*μ*and

*λ*are the dynamic viscosity coefficient and the Lamé coefficient.

*T*is the temperature.

*E*is the energy, which is expressed as follows:

*c*

_{V}is the constant volume specific heat capacity. Based on the above governing equations, a set of computational model equations applicable to the simulation of flow and acoustic wave propagation under high-temperature conditions can be obtained. For high-temperature and low Mach number gas flows, the differential equation of pressure can be obtained from the gas equation of state, which is written as follows:

*α*and thermal expansion coefficient

*β*under low-pressure conditions can be calculated as follows:

*et al.*’s model.

^{42}In the model, the exit of the tube is the acoustic open boundary, and the acoustic liner and the tube surroundings are set as acoustic hard boundaries except for the contact boundaries, and the acoustic liner is regarded as a rigid body. The bottom surface of the tube is set as no slip, and the remaining surfaces are set as slip.

### C. Numerical algorithm

Equations (14)–(16) are discretized using the finite element difference method. The left plane of the flow tube is set to be the inflow plane, and the right plane is set to be the outflow plane, flowing in the x axis direction. In the latter, the temperature setting is mainly reflected in the temperature setting in the Z-axial direction. Dividing the model along the x axis into 20 parallel computational units. The grid size of each computational cell is set to 0.5 × 0.5 × 0.5 mm^{3}, and the number of grids is 200 × 100 × 200.

In calculating the above control equations, the first-order windward format is taken for the boundary points and the grid points next to the boundary, and the second-order windward format is taken for the rest of the grid points. We set initial values for the corresponding parameters in the model, such as pressure, velocity, and temperature. From these initial values, the values of pressure, velocity, and temperature on the next time step are calculated, respectively, and so on, advancing the time step until the calculation is completed.

In this study, a uniform grid is used for the computation, which ensures continuity between the individual computational cells. Considering the boundary closure problem between each parallel subdomain, it is necessary to add imaginary grids at both ends for numerical computation calculations at the boundary points and the closure problem between the parallel subdomains while performing the computations. For example, when calculating the grid points in the x axis direction, it is necessary to calculate the 1 to Nx grids; however, in the actual calculation, considering the calculation method of the second-order windward format of the boundary points, it is necessary to calculate the 0 to Nx + 1 grids. In the subdomain boundary closure problem, which is mainly an x axis imaginary grid computation problem, the inflow and outflow planes are removed, and the rest of the subdomains are mapped to the right imaginary grid of the left computational subdomain and the left imaginary grid of the right computational subdomain by mapping their own 1 and Nx grid point data, respectively, thus realizing the subdomain boundary closure problem. In recent studies, new subdomain boundary closure schemes, such as the NOHAP scheme,^{43–49} have been found, and new treatments will be attempted in subsequent work to achieve better working results. In this study, 200 000 time steps were set, with individual time steps spaced at 0.02 s. Advancement at the time step relies on Euler’s method for realization, the derivation of which is shown in Eqs. (19) and (20).

To solve the governing equations and boundary conditions, we adopt the finite difference method, the upwind difference scheme, and the central difference scheme to solve the above governing equations.

^{50}with the general integration kernel formulation. The integral kernel formulation in the above method combines the domain interface conditions of the governing equations of different subdomains to obtain unified equations that can describe the whole domain. The meta-equation of the method is written as follows:

*d*denotes the distance to the interface between the two subdomains, and

*ɛ*is a small positive number representing the artificial thickness of the interface. When the absolute value of the distance

*d*of the point waiting to be solved from the distance between the boundary surfaces of the two subdomains is less than the chosen small positive number

*ɛ*, the $\mu 0\epsilon $ is given by the above-mentioned equation. When the absolute value of

*d*exceeds

*ɛ*, it is considered that the governing equations at that point are given by the corresponding subdomain.

Similarly, the pressure and temperature governing equations for the entire computational domain can be derived in this way. Compared to the frequency-domain numerical calculation method, the time-domain numerical calculation method in this paper can better reflect the interaction between the gazing flow, the temperature, and the acoustic lining to demonstrate the phenomenon. In the process of numerical computation, the parallel computation technique is employed. Parallel computation through MPI (Message Passing Interface) and OpenMP coupling can fully utilize the arithmetic power of the computer and significantly improve the computational efficiency and computational scale.

## III. MODEL VALIDATION OF THE NUMERICAL ALGORITHM

*et al.*’s model.

^{32}

^{,}Figure 3 shows a comparison of the numerical calculation method and Huang

*et al.*’s data

^{32}for cases 1 and 5. The transmission loss is a good indicator of the performance of the sound lining and can be calculated from the following expression:

*P*

_{0}indicates the average sound pressure across the inlet cross-section on the left side of the deflector, and

*P*

_{1}indicates the average sound pressure across the cross-section of the right outlet of the deflector.

A comparison of the results shows that the numerical calculation method in this paper can coincide with the results of Huang *et al.* and reflect the trend of the resulting curve well. The numerical algorithm proposed in this paper is rational, and the results are within a reasonable range.

## IV. RESULTS AND DISCUSSION

### A. Effects of the grazing flow velocity

*P*

_{e}is the effective value of the measured sound pressure, and

*P*

_{ref}is the reference sound pressure with a value of 20

*µ*Pa.

Figure 5 shows the variations in the fluid pressure in different places of the flow duct with time under different grazing flow rates. Figure 6 shows the pressure results of the fluid when the velocity of the grazing flow is different and when the temperature is 293 K. The magnitude of the pressure is the most intuitive reflection of the performance of the acoustic liner, and the transmission loss of the liner at this point can be obtained by measuring the acoustic pressure at the inlet and outlet. As the velocity of the grazing flow increases, the pressure will be disturbed and fluctuations will occur, which in turn will affect the performance of the acoustic liner.

Specifically, when the velocity of the grazing stream is high, the changes in the velocity and velocity dispersion of the fluid are more obvious. Figures 7(a) and 7(b) show the velocities of the fluids in the flow tubes at the different grazing flow rates. Here, we define *u* as the fluid velocity component in the x-direction. Figure 8 shows the variations in the fluid velocity divergences in the presence of grazing flows at different air speeds. The fluid velocity divergence represents the sound wave distribution in the flow. An increase in the grazing flow velocity leads to more visible sound wave movement, which affects the performance of the structure. The numerical calculation method in this paper can better reflect the phenomena in the model. Moreover, this model can not only reveal the phenomena in the duct but also reveal the different phenomena in the underlying area because of the different conditions.

### B. Effect of temperature

Figure 9(a) illustrates the change in the sound pressure level in the model for different temperatures. Figure 9(b) shows the effect of different temperatures on the transmission loss of the metal liner in cases 1 and 7. The structure performs better at higher temperatures, but the change tends to level off. Figures 10 and 11 show the changes in the fluid velocity and velocity divergence, respectively, with increasing temperature. When the grazing flow velocity is 0 m/s, an increase in temperature causes the fluid velocity caused by the acoustic wave to attenuate faster, as illustrated in Fig. 11(b). Figure 12 shows the change in the fluid velocity component *u* in the model for different temperatures, verifying the results presented in Figs. 10 and 11.

These comparisons show that at a grazing flow velocity of 0 m/s, an increase in temperature leads to stabilization of the fluid in the flow tube. However, the fluid phenomenon is more pronounced in the acoustic liner, giving it a slight increase in performance.

### C. Effect of the temperature gradient

In this paper, in addition to investigating the effect of temperature on the acoustic performance of the structure, the effect of the temperature gradient is also taken into account. Two temperature differences are provided between the upper wall surface of the flow tube and the bottom of the sound liner, thus creating a temperature gradient perpendicular to the grazing stream. Considering the practical application scenario, the upper wall surface of the flow tube is set at a high temperature. Temperature gradients play different roles when grazing stream velocities are different, as displayed in Figs. 13(a) and 13(b). Figure 13(a) shows results similar to those in Fig. 9(b). The temperature gradient causes the acoustic performance of the structure to increase and causes the trends in transmission loss to level off. However, Fig. 13(b) shows a different result. Figure 13(b) shows that an imposed temperature gradient leads to a reduction in transmission loss at a grazing flow velocity of 60 m/s.

Figures 14(a)–14(d) show the effect of the temperature gradient on the fluid velocity caused by grazing flow velocities of 0 and 60 m/s, respectively. Figures 14(a) and 14(b) show that the temperature gradient leads to a more complex fluid velocity distribution in the flow tube. However, the fluid velocity distribution in the lower portion of the sound liner does not change much. Figures 14(c) and 14(d) show that for a grazing flow velocity of 60 m/s, the temperature gradient stabilizes the velocity distribution of the fluid in the flow tube.

Figure 15 shows that the temperature gradient leads to a more complex velocity dispersion distribution for the fluid in the model. Considering the results displayed in Fig. 14, it is hypothesized that the change in the velocity dispersion components in the other two directions leads to a more complex velocity dispersion distribution in the model.

Figure 16 illustrates the effect of the velocity of the grazing flow on temperature. Figure 16(a) shows that when the velocity of the grazing flow is 0 m/s, the temperature distribution is highly uniform. However, when the velocity of the grazing flow is 60 m/s, there is a very clear fluctuation in the temperature distribution at the lower wall of the flow tube, which causes fluctuations in the temperature distribution throughout the pipeline, as demonstrated in Fig. 16(b). These results indicate that differences in fluid velocities can have an effect on the temperature distribution.

Based on the above-mentioned discussion, temperature changes affect the fluid on the one hand, differences in fluid velocities in turn affect the temperature distribution, and both the fluid and the temperature affect the performance of the sound lining. It can be hypothesized that some coupling relationships exist between the flow field, acoustic field, and thermal field, which influence each other and thus lead to the complex phenomena observed in related research. The results showed that the joint influence of flow and thermal fields is not a simple superposition and that more complex mechanisms remain to be explored.

## V. CONCLUSION

In this paper, we propose a computational method that enables the numerical simulation of acoustic wave propagation processes in grazing flows while considering air temperature variations. Moreover, the reasonableness of the numerical calculation method is verified by comparing the results of this approach with the experimental results. The effects of the grazing flow velocity, the air temperature level, and the temperature gradients on the flow field are investigated.

As the grazing flow velocity increases, the movement of the fluid in the flow tube becomes more complex, affecting the structure’s acoustic performance. When the grazing flow rate is not considered, the fluid’s tendency to move decreases as the structure’s temperature increases, and its acoustic performance improves. When the temperature gradient is considered, complex couplings occur between fluids, thermal fields, and acoustic waves, which influence each other and lead to complex phenomena in this study.

The design solutions of noise reduction structures often need to be quickly evaluated through numerical simulation, so the calculation results of the corresponding simulation system are very important for guiding the design of noise reduction structures. The numerical method proposed in this paper can better reflect the coupling relationship between the flow field, the thermal field, and the acoustic field, and the parameters of each part of the model can be obtained through numerical computation, thus reflecting the various phenomena in the model. With the coupled parallel computation of threads and processes in numerical computation, we can realize a larger computational scale and higher computational accuracy, which can better show the details of the model in the illustration. The numerical methods in this paper allow numerical simulation of the propagation process of acoustic waves in noise-reducing structures considering high-temperature gases, which can aid in the design of acoustic linings and promote further research on related theories.

In this paper, we could develop a theoretical analysis for specific cases,^{51} but given the limitations of our capabilities, we only perform numerical studies. In later studies, we will try to do further work on this aspect.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grant No. 2022YFB3303500).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Hongwei Jiang**: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Xin Zhao**: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*High Accuracy Computing Methods: Fluid Flows and Wave Phenomena*

*Sustained Simulation Performance*

*44th AIAA Thermophysics Conference, June 24-27, 2013, San Diego, CA*