In this paper, the effects of porous media parameters on circular cylinder wake flow and radiation noise are investigated using large eddy simulations and Ffowcs Williams–Hawkings acoustic analogy. We performed three-dimensional numerical simulations for flow around the cylinder coated with a porous layer of different pores per inch in a subcritical flow regime ( $ Re D = 4.7 \xd7 10 4$) to explore the control mechanism of porous media on wake and radiation noise. The results show that the application of porous media significantly alters the separation pattern behind the cylinder and stabilizes the shear layer detached from the cylinder. The existence of porous layers leads to the transformation of chaotic and irregular vortex structures into more orderly vortices. Moreover, this study also reveals that the cylinder coated with high pore density can provide the desired noise reduction. The analysis of vortex sound theory indicates that porous media reduces the interaction area and magnitude of the positive and negative Lamb vector divergences, which is beneficial for drag reduction and noise attenuation. In addition, the comparison of sound pressure contours shows that the application of porous media does not change the radiation mode of noise, but the porous media with high pore density helps to decrease the generation of noise and intensity of the sound source.

## I. INTRODUCTION

Flow around a blunt body is a fundamental phenomenon in various industrial applications such as offshore wind turbine towers, aircraft landing gear, offshore rig risers, submarine pipelines, etc. The periodic shedding vortices generated by the flow around a blunt body typically produce significant pressure variations on the leeward side of the body, which increases flow drag and may lead to unwanted fluid–structure interaction, i.e., flow-induced vibration and radiation noise.^{1,2} Therefore, exploring and developing new strategies for effective control of flow and radiated noise generated by blunt bodies is of great academic significance and engineering value and has attracted much attention of scholars.

The classical categorization of flow control for bluff bodies is divided into active control and passive control according to auxiliary requirements. Active control requires the application of external energy input to change the flow structure, thereby achieving the goal of flow control and noise reduction, such as blowing and sucking,^{3} buoyancy-driven devices,^{4} and vibrating walls.^{5} Active control contains fine off-design performance, but it requires complex mechanical devices or systems. In contrast, passive control methods require no additional devices and have the advantage of structural simplicity. Passive control is usually implemented by geometry optimization and adjustment to control flow structure and radiated noise, such as wavy walls,^{6} serrations,^{7} grooves,^{8} or roughness elements.^{9} Among passive controls, porous media treatment inspired by bionics stands out a potential passive control technology, due to its simple structure and easy implementation. Porous media is a multiphase structural material with a solid framework and many pores inside, and fluid can flow through pores and cause changes in pressure. Due to the unique properties of porous media between solid and fluid, it can generate a pressure reduction mechanism through fluid penetration and discharge across the porous–fluid interface.^{10,11}

In the past decade, porous media has attracted widespread attention as a potential method for flow control. There have been many studies on blunt bodies such as circular cylinders coated with porous media for flow control and noise reduction. Sueki *et al.*^{12} conducted wind tunnel tests on circular cylinders coated with different types of porous materials, and the experimental results showed that the open-cell porous material can effectively stabilize the free shear layer, suppress vortex shedding as well as reduce flow-induced noise. Klausmann and Ruck^{13} experimentally investigate the effect of porous coating on the leeward side of a circular cylinder on drag reduction and showed that drag reduction mainly depends on layer thickness and coating angle, and the vortex street formation occurs further downstream, which leads to a substantial decrease in the pressure fluctuations. Liu *et al.*^{14} conducted a numerical study on cylinders with porous coating, and the results showed that the porous media stabilize the chaotic flow wake, decrease the vortex shedding frequency, and cause a significant reduction in radiated noise. Geyer^{15} experimentally analyzed the optimal porous materials for noise reduction and found that porous materials cannot eliminate completely the tonal noise, and the porous materials with low airflow resistivity exhibit better noise reduction performance. Xu *et al.*^{16} clarify the effectiveness of porous coating on reducing the pressure drag and aerodynamic noise of the cylinder, and the porous materials with higher pores per inch (PPI) are beneficial for reducing pressure drag. However, the optimal value of the PPI for noise reduction was not found from the experimental results.

Other studies focus on how the layout of porous materials affects the flow and its induced noise. Hu *et al.*^{10} numerically studied the flow field and noise characteristics of circular cylinders with full and partial porous coating. Consequently, the full porous coating cylinder with a reasonable porous layer thickness can suppress vortex shedding and reduce noise. The porous coating around the separation point has better effects on flow control and noise reduction. In addition, Zhang *et al.*^{17} conducted numerical simulations to further confirm that the application of porous media with high porosity in the separation zone is an effective scheme for flow control and noise reduction. Recently, Xu *et al.*^{18} numerically studied three schemes with porous treating on wake evolution, and the results illustrate that porous coating increases mean drag and diminishes the fluctuations of lift and drag forces. Du *et al.*^{19} experimentally measured the flow around a circular cylinder with porous coating and found that applying porous materials with 20 PPI on the leeward side of a cylinder can achieve optimal drag reduction. Moreover, the results of dynamic mode decomposition (DMD) analysis revealed that the porous materials can effectively weaken the energy of vortices in different modes.

In summary, the research works mentioned earlier have demonstrated that porous media coating has great potential in flow control and reducing radiated noise of circular cylinder. Therefore, promoting and deepening the study of wall flow over porous media as a passive control method is extremely beneficial. On the one hand, issues regarding the flow characteristics of porous cylinders, such as the changes in near wake structure, the porous–fluid interface seepage phenomenon, etc., have not been fully answered. Meanwhile, some inherent physical mechanisms of the control effects of porous coating are still essentially ambiguous. More investigations and analyses on the interaction of porous coating with the near wake of a cylinder need to be conducted, aiming to improve understanding of the mechanism of flow modification caused by porous cladding. On the other hand, no definitive explanation about the mechanisms involved in the noise mitigation for a porous circular cylinder has been found yet. So far, the investigations on the mechanisms of radiated noise reduction for the porous cylinder are limited to noise spectral analysis, and the sound field around the cylinder should be analyzed, especially the propagation mode of the sound wave.

Therefore, the present study aims to investigate wake flow and radiated noise characteristics of circular cylinders coated with porous media based on large eddy simulation (LES) and Ffowcs Williams–Hawkings (FW–H). The porous layer coated on the surface of the circular cylinder has high porosity with uniform thickness, and fluid fills the pores of the porous media. Four groups of cases including solid cylinder are considered to investigate the wake flow and radiated noise characteristics of cylinders coated with porous media. The underlying mechanism associated with the wake modification and noise reduction of porous cladding is analyzed based on the vortex sound theory. The remainder of this paper is organized as follows. In Sec. II, the computational domain, numerical method, and porous model are introduced. To validate the numerical method, the flow quantities and acoustic results of solid and porous cylinders were compared with data from the reference literature. In Sec. III, the results are discussed in terms of the dynamic parameters, time-averaged flow field, instantaneous flow structures, velocity power spectral density (PSD) as well as typical acoustic characteristics. Finally, the conclusions of this paper are presented in Sec. IV.

## II. NUMERICAL METHOD

### A. Model description

The model of flow around a solid cylinder with porous media cladding that we considered here is three-dimensional, as shown in Fig. 1. The solid cylinder has a diameter *D* of 0.025 m, and the thickness of porous cladding is $ h = 0.25 D$. The Reynolds number $ R e D = D U \u221e / \upsilon = 47 \u2009 000$ is based on the cylinder diameter *D* and free inflow velocity $ U \u221e$ (Mach number $ M a = 0.08$), which is within the range of the subcritical Reynolds number. The computational domain extends 35*D* in the *x*-direction, while the velocity inlet and pressure outlet are located 10*D* and 25*D* upstream and downstream of the central axis of the cylinder, respectively. The top and bottom boundaries both are located at 10*D* away from the cylinder center and are specified as symmetry conditions. Based on the results of previous numerical studies by Zhang *et al.*^{17} and Liu and Azarpeyvand,^{20} the spanwise length of the computation domain is chosen as $ L Z = 6 D$ to guarantee the fully developed flow in the spanwise direction. Moreover, symmetry boundary conditions are applied on the side boundaries, and the no-slip condition is defined at the solid cylinder.

The computational domain was divided into an O-type and rectangular sub-domains as shown in Fig. 2, which is discretized with hexahedral meshes. The refined grids were utilized near the cylinder surface as well as near the wake including the porous cladding region. To guarantee the quality of the mesh and the reliability of the numerical result, the height of the first layer mesh near the cylinder surface is set to be 0.0002*D*, with a stretching factor of 1.08, which satisfies the requirement of the LES for $ y + < 1$. For the mesh in the porous region, as shown in Fig. 2(b), the length of the porous region in the radial direction is discretized with 75 grids. In addition to the grid resolution requirements near the solid cylinder surface, the refined grids were also applied to resolve the boundary layer of the porous–fluid interface, which also corresponds to $ y + < 1$. Moreover, the circumference and spanwise length are discretized with 240 and 80 grids, respectively.

### B. Numerical method

^{21}The dynamics of large-scale structures are explicitly calculated, and a subgrid-scale (SGS) model is used to model the eddies with scales smaller than the filter size. In the LES of an incompressible flow, the filtered continuity and momentum equations are obtained as follows:

*x*,

*y*, and

*z*directions, respectively.

*ρ*is the density of the fluid, $ \tau i j$ represents the subgrid-scale (SGS) stress, which reflects the influence of small-scale vortex motion on the solved governing equation, and

*μ*is the dynamic viscosity of the fluid. The standard Smagorinsky–Lilly model is employed in this study, which models the isotropic part of the SGS term with formulations as follows:

In order to minimize numerical dissipation and avoid numerical oscillation, different spatial and temporal discretization schemes are used for governing equations. The second-order upwind and second-order central difference schemes are used to discretize the convection and diffusion terms in the conservation equations, respectively. Simultaneously, an implicit second-order scheme is applied to temporal discretization. Time step $ \Delta t = 1 \xd7 10 \u2212 5$ s for the present simulation is used to ensure temporal resolution, which corresponds to the non-dimensionless time step of $ \Delta t \u2217 = 0.011$.

### C. Acoustic computation methodology

^{22}to predict aerodynamic noise. The FW–H equation is an essentially inhomogeneous wave equation and can be derived from the continuity and Navier–Stokes equations. The FW–H equation is as follows:

The terms on the right side of Eq. (6) represent three different sound source terms, respectively, i.e., monopole, dipole, and quadrupole sources. In the case of low-speed flow around a bluff body, quadrupole source can be negligible compared with dipole source.^{23} Therefore, the acoustic source terms in the FW–H equation are mainly determined by the pressure fluctuation on the wall of the bluff body. Furthermore, the solution of Eq. (6) will no longer be described in detail.

In this study, the acoustic sampling time has been carried out for 0.25 s, excluding the initial transient. The sampling resolution time is $ 1 \xd7 10 \u2212 5$ s, which is the same as the time resolution of the flow field. The solid cylinder surface is chosen as the FW–H integral surface for solid case, and the porous–fluid interface is defined as the noise integral source surfaces of the FW–H equation for all porous cases. A virtual monitoring point is set at the position of (0, 40*D*, 3*D*) to collect the far-field acoustic signal. Moreover, to measure the directivity characteristics of the noise generated by various cylinders in midspan plane, 36 virtual monitoring points are uniformly arranged at a distance of 40*D* from the center of the circular cylinder.

### D. Model of porous media

^{24,25}based on the volume average method. For the porous media zone, the momentum equation has an additional momentum source term to represent the pressure drop in the porous media. The viscous and inertial resistances of the porous zone are described by the Ergun equation

^{26}as follows:

*μ*is the molecular viscosity,

*ε*is the porosity defined as the percentage of pore volume in a porous material to the total volume in its natural state, and $ D p$ is the mean particle diameter; the Ergun constants

*a*and

*b*are defined as $ \alpha = 150$ and $ b = 1.75$, on the basis of a lot of experimental tests.

*K*and inertial loss coefficient $ C F$ (or Forchheimer coefficient) of the porous media stand for the Darcy and non-Darcy effects, respectively. Also, there are two key parameters that describe the internal flow of the porous media calculated based on the Ergun equation. The expression is as follows:

*K*and inertial loss coefficient $ C F$ have become key parameters for describing the internal flow of porous media. The classical continuity boundary conditions

^{27}are applied to the interface between the mainstream fluid region and the porous zone, that is, the velocity and stress are continuous, which can be written as

### E. Grid independence study and numerical validation

To evaluate the sensitivity of simulation results to grid resolution, grid independence investigations are performed on the solid cylinders at $ R e D = 47 \u2009 000$. Three different numbers of mesh including the coarse, medium, and fine are considered, and the corresponding mean drag coefficient $ C d \xaf$ and Strouhal number *St* ( $ S t = f s D / U \u221e$) are reported in Table I. As is shown in Table I, the relative change of $ C d \xaf$ and *St* between coarse and medium meshes is 11.0% and 6.63%, respectively. However, as the grid resolution further increases, the relative change in these aerodynamic coefficients is all less than 2.1%, which means that the medium mesh resolution is sufficient for the current simulation. Therefore, the discretization of the porous cylinders computational domain also uses a similar grid resolution, and the total number of grid cells is 7.21 × 10^{6}.

Mesh . | $ N grid$ . | $ C d \xaf$ . | Change (%) . | St
. | Change (%) . |
---|---|---|---|---|---|

Coarse | $ 4.78 \xd7 10 6$ | 1.18 | ⋯ | 0.181 | ⋯ |

Medium | $ 6.62 \xd7 10 6$ | 1.31 | 11.0 | 0.193 | 6.63 |

Fine | $ 8.75 \xd7 10 6$ | 1.29 | 1.53 | 0.189 | 2.07 |

Mesh . | $ N grid$ . | $ C d \xaf$ . | Change (%) . | St
. | Change (%) . |
---|---|---|---|---|---|

Coarse | $ 4.78 \xd7 10 6$ | 1.18 | ⋯ | 0.181 | ⋯ |

Medium | $ 6.62 \xd7 10 6$ | 1.31 | 11.0 | 0.193 | 6.63 |

Fine | $ 8.75 \xd7 10 6$ | 1.29 | 1.53 | 0.189 | 2.07 |

To verify the reliability of the numerical method in this paper, the time-averaged flow quantities of the solid cylinders are first compared with the available numerical and experimental data in other literature at $ R e D = 4.7 \xd7 10 4$. Some flow variables such as mean drag coefficient $ C d \xaf$ and Strouhal number *St* are summarized in Table II. The mean drag coefficient of the present simulation is 1.31, which is within the range of listed reference values from 1.24 to 1.35. Meanwhile, the Strouhal number obtained from the present numerical simulation is close to the numerical results of Seo and Moon^{28} and Zhang *et al.*^{17} and is consistent with other experimental results.^{29,30} Therefore, it indicates that the present three-dimensional simulation can effectively capture the periodic characteristics of the vortex shedding generated from flow around the circular cylinder. Figure 3 shows a comparison between the time-averaged pressure coefficient obtained from different cylinders and the reference data, and the time-averaged pressure coefficient is defined as $ C p = ( p \u2212 p \u221e ) / ( 0.5 \rho \u221e U \u221e 2 )$, where $p$ and $ p \u221e$ denote the local static pressure and the freestream static pressure, respectively. It can be seen from Fig. 3(a) that the present simulation of the solid cylinder is in good agreement with previous results.^{19,28,31–33} Furthermore, to verify the accuracy of the application method for porous media, a similar validation work was conducted on a cylinder coated with 0.97 porosity and 10 PPI porous materials at $ R e D = 4.7 \xd7 10 4$, as shown in Fig. 3(b). The present time-averaged pressure coefficient is consistent with the numerical results of Zhang *et al.*,^{17} though there exists a slight deviation with reference data from Ref. 22.

Author . | Method . | $ Re / 10 4$ . | $ C d \xaf$ . | Deviation (%) . | St . | Deviation (%) . |
---|---|---|---|---|---|---|

Seo and Moon^{28} | LES | 4.6 | 1.24 | 5.6 | 0.187 | 3.2 |

Zhang et al.^{17} | LES | 4.7 | 1.26 | 3.9 | 0.193 | 0 |

Szepessy and Bearman^{29} | Exp | 4.3 | 1.35 | 2.9 | 0.190 | 1.5 |

Cantwell and Cloes^{30} | Exp | 4.7 | 1.0–1.35 | ⋯ | 0.18–0.20 | ⋯ |

Present | LES | 4.7 | 1.31 | ⋯ | 0.193 | ⋯ |

Author . | Method . | $ Re / 10 4$ . | $ C d \xaf$ . | Deviation (%) . | St . | Deviation (%) . |
---|---|---|---|---|---|---|

Seo and Moon^{28} | LES | 4.6 | 1.24 | 5.6 | 0.187 | 3.2 |

Zhang et al.^{17} | LES | 4.7 | 1.26 | 3.9 | 0.193 | 0 |

Szepessy and Bearman^{29} | Exp | 4.3 | 1.35 | 2.9 | 0.190 | 1.5 |

Cantwell and Cloes^{30} | Exp | 4.7 | 1.0–1.35 | ⋯ | 0.18–0.20 | ⋯ |

Present | LES | 4.7 | 1.31 | ⋯ | 0.193 | ⋯ |

Finally, to evaluate the reliability of predicting far-field noise, the calculated acoustic spectra of the solid and porous cylinders are compared with reference data at $ R e D = 4.7 \xd7 10 4$. Figure 4 presents the sound pressure level (SPL) spectra obtained for the solid and porous cylinders with a virtual monitoring point positioned at $ 90 \xb0$ from the front stagnation point. The present simulation results are in good agreement with the experimental results of Sueki *et al.*^{12} and the numerical result of Zhang *et al.,*^{17} both for the tonal peak caused by vortex shedding and the broadband component, as shown in Fig. 4(a). However, there was a considerable discrepancy between the present simulation results with the numerical results of Liu *et al.*,^{14} particularly in the broadband noise, which may be attributed to the difference between two-dimensional simulation and three-dimensional simulation of flow around a solid cylinder. Moreover, Fig. 4(b) displays a comparison between the 1/3 octave frequency spectra obtained from the present simulation and the reference data. As can be seen, the SPL profile agrees well with the experimental data of Sueki *et al.*^{12} and the numerical results obtained by Liu *et al.*^{14} in terms of predominant peak frequency and tonal peak. The aforementioned analysis indicates that the numerical method used in this study is deemed reliable for capturing the key characteristics of the flow field and acoustic field.

## III. RESULTS AND DISCUSSION

### A. Aerodynamic comparison

Figure 5 compares the time-dependent variation of lift and drag coefficients between solid and porous cylinders with different PPI. The porous cladding has significantly change the flow characteristics of the cylinder, in terms of both the time-averaged and fluctuations of the coefficients. As shown in Fig. 5(a), the amplitude of the lift coefficient reduces for the porous cylinders, and the time history curves of the lift coefficient for various cylinders have an obvious phase difference. Also indicated in Fig. 5(b) are drag coefficients for the solid and porous cylinders. The porous cladding changes the features of the drag coefficient, including amplitude, frequency as well as irregularity, and the mean drag coefficients decrease with the increase in the PPI. Compared with the solid cylinder, we observe a slight change in the mean drag of the porous cylinder with 8 PPI, and note that the mean drag of 13 PPI and 20 PPI porous cylinder is less than that of the solid cylinder with a drag reduction up to 16.7% and 33.6%, respectively.

Figure 6(a) shows the distribution of the time-averaged pressure coefficients along the cylinder surface, which explains the variation of the drag in Fig. 5(b). In the cases of porous cladding, the fact that the windward pressure increases slightly is accompanied by a decrease in velocity near the cylinder wall. Meanwhile, the base pressure on the leeward side of the porous cylinders has increased significantly, which suggests that the porous cladding can recover leeward pressure and cause a decrease in drag. What is more, the distribution of the time-averaged pressure coefficient shows that a pressure plateau appears behind the separation point of the cylinder with porous cladding, which indicates that the porous cladding has changed the flow characteristics of the near wake. This phenomenon is more pronounced in cylinders with higher PPI porous cladding, and the present results are consistent with the result of Liu *et al.*^{14} The distribution of pressure fluctuations is shown in Fig. 6(b), and the surface pressure fluctuation on the cylinder surface increases by more than three-fourths ( $ 0 \xb0 < \theta < 135 \xb0$) for porous cylinder with the low PPI (PPI = 8). As the PPI increases, the pressure fluctuations on the cylinder surface decreased significantly. This indicates that porous cladding with higher PPI can more effectively attenuate the pressure fluctuations on the cylinder surface. According to the theory of FW–H equations for predicting noise, the reduction of pressure fluctuation on the cylinder surface indicates the potential reduction in far-field radiated noise.

### B. Time-averaged flow field

In the time-averaged streamwise velocity profile along the *y*-direction at $ \theta = 90 \xb0$, as shown in Fig. 7(a), the results show that porous cladding increases the thickness of the boundary layer on the cylinder surface from 0.52*D* to about 0.77*D*, which significantly reduces the reverse pressure gradient and is conducive to stabilize the free shear layer. Meanwhile, there exists a slip velocity at the porous–fluid interface for the cylinder coated with porous media. This slip velocity weakens the Kelvin–Helmholtz instability on the cylinder surface, which stabilizes the free shear layer, suppresses vortex shedding, and improves wake flow.^{17} Therefore, the suppression of vortex shedding generated by cylinders with 13 PPI and 20 PPI porous cladding explains the significant attenuation of lift fluctuation in Fig. 6(b). In addition, the porous cladding with high permeability can lead to an increase in the average velocity of the porous–fluid interface and accelerate the flow of fluid in the porous media, which may trigger more instability mechanisms.^{34} More importantly, when the flow passes through a cylinder, the flow separation occurs in the cylinder surface, which produces a reverse circulation flow in the wake region. The recirculation length $ L r$ as a quantitative parameter describes the degree and range of reverse circulation flow behind the cylinder, which is defined as the distance from the center point of the cylinder to the location, where the time-averaged streamwise velocity in the centerline changes from negative to zero. Figure 7(b) shows the time-averaged streamwise velocity along the horizontal centerline in the wake of the cylinders. Compared with the solid cylinder, the length of the recirculation region is found to be shortened for the cylinder with 8 PPI porous cladding but elongates the recirculation region for cylinders with 13 PPI and 20 PPI porous cladding.

To investigate the effect of the porous cladding on the flow field, Fig. 8 shows the contours of the time-averaged pressure and streamlines on the mid-plane of $ z / D = 0$. Compared with the solid cylinder, the recirculation zone of 8 PPI porous case in the cylinder near the wake slightly decreases, and a pair of small secondary vortex pairs is generated on the inner cylinder surface. However, the porous cladding with 13 PPI and 20 PPI elongates the recirculation regions behind the circular cylinder to some extent, and the negative pressure in the wake was more obviously attenuated. The recirculation zone length of 13 PPI porous case reached 1.6*D*, while that of the solid case is 1.4*D*. The recirculation zone length of 20 PPI porous case reaches the maximum among all the cases. This is mainly because the porous cladding enlarges the free shear layer, and the flow reattempts to end the stable free shear layer further downstream leading to the creation of a larger reverse flow region.^{20} More importantly, as shown in Figs. 8(c) and 8(d), the pressure in the near wake of the porous cylinders increases significantly, leading to the reduction in drag. Meanwhile, two insignificant secondary vortex structures can be observed in the near-wall region in the solid and 8 PPI porous cases, whereas they disappear in 13 PPI porous case. This means that the flow reattachment does not happen for the 13 PPI porous case on the inner wall surface. Moreover, with the PPI increase to 20, the two main counter-rotating vortices on the leeward side appear asymmetrically on both sides of the wake centerline.

Figure 9 shows the contours of the time-averaged streamwise and vertical velocity components for the different cylinders. The presented contours of velocity only show the upper half of the flow field due to the symmetric appearance of the time-averaged flow structure in the near wake. As shown in Fig. 9(a), there is a narrow low-speed region in the wake behind the solid cylinder. For the porous cases, the area of the low-speed region for the streamwise velocity is extended and shifted downstream. A similar phenomenon can be observed in Fig. 9(b), and the low-speed region of the vertical velocity is also shifted downstream.

To further figure out the effects of porous cladding on the wake, Fig. 10 shows the time-averaged streamwise and vertical velocity profiles in the downstream wake. The corresponding streamwise velocity at $ x / D = 1$ for solid case increases monotonically with distance when $ | y / D | < 0.8$, as shown in Fig. 10(a). The streamwise velocities for 8 PPI and 13 PPI cases show a similar distribution to the solid cylinder. In contrast, a U-shape streamwise velocity profile appears at $ x / D = 1$ for the 20 PPI case, which is different from the V-shape profile of other cylinders. When $ x / D = 2$, the porous case of 13 PPI also shows a similar profile, which behaves as a flat top as that of the 20 PPI porous case. Compared with the solid cylinder, it can be observed from Fig. 10(a) that porous cladding reduces the streamwise velocity in the wake but slightly increases the velocity near the free shear layer on both sides. Evidence supporting of this result can be found in Fig. 9(a). In other words, the fluid in the low-speed region downstream of the cylinders is essentially stagnant. The fluid permeates into the porous layer on the windward side of the porous cylinder, ejects from the leeward side with a velocity component in the streamwise direction, and produces an effect similar to that of a micro-jet. The micro-jet acts on the separation point of the cylinder, resulting in the flow exhibiting behavior similar to early separation, which is similar to the previous reported studies.^{12,35} At the further downstream location, the streamwise velocity profile changes from a U-shape to a V-shape, while the disparities near the horizontal centerline gradually decrease. Moreover, the momentum deficit of the 13 PPI and 20 PPI porous cases in $ x / D \u2265 2$ is larger than that of the solid case owing to the global modification of the flow field by porous media.

Figure 10(f)–10(j) displays the time-averaged vertical velocity $ v \xaf / U \u221e$ profiles for the various cylinders in downstream. Each vertical velocity profile at different axial locations is more or less mirror symmetric about the wake centerline, and vertical velocity switch sign at the wake centerline. At the location of $ x / D = 1$, while other porous cases partially exhibit narrow S-type profiles with significantly reduced velocity magnitudes, the 20 PPI case even shows an inverse S-type profile. This is due to the recirculation region that has already happened for the solid cylinder in the near wake, and the porous cladding generates a more stable free shear layer, which hinders the portion of the fluid escape from the recirculation region into the freestream in the vertical direction. Further downstream at $ x / D = 2$, the variations of profiles for the solid case and porous case are considerably different. At $ x / D = 3$, the vertical velocity magnitude of the solid cylinder has decreased significantly, which is in accordance with the smaller wake region and the short separated shear layer. Meanwhile, each of the porous cases exhibits similar vertical velocity profiles with a smaller velocity magnitude than that at $ x / D = 2$. Finally, the solid cylinder and the porous cylinder show similar profiles and vertical velocity magnitudes close to zero at further downstream.

### C. Instantaneous flow structures

The turbulent motion could be regarded as a superposition of vortices of various scales and intensities. In the process of fluid motion, the vortices are constantly breaking up and merging. Therefore, elucidating the size, change, and transport of vortices in the flow field is important for analyzing the energy loss. Figure 11 shows the contours of the instantaneous vorticity on the mid-plane of $ z / D = 0$ for the different cylinders. For the solid cylinder, a quantity of multiple-scale and chaotic vortical structures are generated behind the cylinder. Meanwhile, the wake exhibits significant oscillations in the y-direction. Among these complex vortex structures, the large-scale and strong vortices are the main factors leading to negative pressure region in the wake flow and cylindrical lift fluctuation. Compared with the solid cylinder, the porous cylinder with 8 PPI generates the vortex shedding on the inner cylinder surface and porous–fluid interface. The secondary shedding vortex generated by the inner cylinder surface is significantly stronger than that in the porous–fluid interface. Meanwhile, similar phenomena can be found in 13 PPI case as shown in Fig. 11(c). The difference is that the magnitude of the secondary shedding vortex weakens. Comparatively, as shown in Fig. 11(d), the porous cladding with 20 PPI can effectively eliminate the small-scale vortical structures in the wake despite enlarging the recirculation region, and the secondary shedding vortex behind the cylinder quickly dissipated with the effect of porous cladding. The elongated separation shear layer causes the shedding position of the vortex to move downstream and results in the reduction of force acting on the cylinder.

^{36,37}indicates that for the low Mach number and isentropic flow, the basic physical mechanism of fluid sound generation originates from the stretching and breaking of vortices in the flow field. Therefore, Fig. 12 shows the instantaneous vortical structures colored by the streamwise velocity base on the

*Q*-criterion. In the 3D Cartesian coordinate system, Q-criterion can be written as follows:

*u*,

*v*, and

*w*represent the velocity components in the

*x-*,

*y-*, and

*z-*directions, respectively. For the solid cylinder, as shown in Fig. 12(a), the developing shear layer on the leeward of the cylinder exhibits strong three-dimensional, and the wake region is strongly disturbed. The laminar shear detaches from the cylinder surface and forms three-dimensional spanwise vortices due to the Kelvin–Helmholtz (K–H) instability of the shear layer. The braid vortices are initiated behind the K–H vortex, which indicates that the K–H vortex is a short-lived structure. With the further development of flow downstream, braid vortices quickly become unstable and break into a series of hairpin-like vortexes and more complex structures, where the flow reaches a turbulent state. For the porous cylinder with 8 PPI, the instantaneous vortical structures behind the cylinder are substantially different from those of the solid cylinder. The separated shear layer from the inner cylinder surface rolls up and forms quasi-two-dimensional braid vortices, and every two adjacent braid vortices with opposite signs usually form a streamwise vortex pair. Meanwhile, the porous cladding also eliminates small-scale vortex structures in the wake. For the porous cylinder with 13 PPI, the porous cladding elongates the low-speed recirculation region and stabilizes the shear layer generated by the porous–fluid interface, and the position where laminar flow transforms into turbulent flow structures shifts further downstream. As the pore density increases to 20, the free shear layer of the porous cylinder becomes more stable than other cases. The braid vortices become unstable and convert into small-scale streamwise vortex structures further downstream. Meanwhile, the three-dimensionality of the vortex structure of the porous cylinder is suppressed and more synchronous in the spanwise direction, which confirms that vortex shedding in the porous case is more two-dimensional than that in the solid case.

To better understand the effect of porous cladding in the wake evolution, Fig. 13 shows the distribution of turbulence kinetic energy on the mid-plane of $ z / D = 0$ for the different cylinders. For the solid cylinder, as shown in Fig. 13(a), the distribution of the turbulent kinetic energy presents a quasi-crescent-shaped due to the interaction of the shear layer on both sides of the cylinder. While the turbulence kinetic energy in the wake is suppressed for all porous cases, and this effect is enhanced as the PPI increases. Also, compared with the solid cylinder, the high-intensity turbulence kinetic energy region is pushed toward downstream of the cylinder by porous cladding.

### D. Velocity power spectral density

Figure 14 shows the velocity power spectral density (PSD) profiles for velocity signals at various probes and provides a comparison between the solid cylinder and the porous cylinder with different PPI. In order to estimate the energy content at different frequencies, the signals of time domain velocity are processed using the Welch's power spectral density method with the time segments with $ 2 / 3$ overlapping. All the numerical measurements (forces, pressures, velocity, etc.) are saved in the minimum time step, resulting in a sampling frequency of 20 000 Hz. According to the Shannon–Nyquist sampling theorem, the maximum resolved frequency is 10 000 Hz. The frequency resolution based on sampling time is 4 Hz, which satisfies the requirement of convergence of the spectra and resolves the low-frequency region. As shown in Figs. 14(a), 14(f), and 14(k) (i.e., along $ y / D = 0$), the solid cylinder (shown in black) exists a single tone, referred to as $ f 1$-tone here, which corresponds to twice the vortex shedding frequency. The dominant peak decreases in magnitude as the position moves downstream (increasing $ x / D$). Similarly, the porous cases all possess broadband signal with the dominant peak, and the peak value of PSD decreases with the increase in PPI. Meanwhile, the Strouhal number for the porous cases is smaller than the solid case, and this may be attributed to the fact that the porous cladding elongates the separated shear layer, delaying the shedding of vortices.

Away from the centerline of the wake, $ y / D = 0.5$, it can be seen in Fig. 14(b) that there exists a typical $ f 1$-tone for all cases. Comparing the PSD profiles along $ y / D = 0.5$ in Figs. 14(b), 14(g), and 14(l), both the $ f 1$-tone and broadband energy of solid cylinder decrease in magnitude with the increase in $ x / D$. For the porous cases, the energy change of $ f 1$-tone is similar to that of the solid case. However, there seems to be a slight increase in high-frequency energy, especially in the case of $ PPI = 20$. Interestingly, the spectra of porous cases also contain a “secondary dominant” peak (herein referred to as the $ f 2$-tone) with lower amplitude at a higher frequency, where $ f 2$-tone frequency is twice the $ f 1$-tone frequency. Casalino and Jacob^{38} discussed that the odd harmonic of unsteady drag is twice the even harmonic of unsteady lift (mainly corresponding to the $ f 1$-tone and $ f 2$-tone Strouhal number). More recently, Maryami *et al.*^{39} confirmed these findings in subsequence studies and found that $ f 2$-tone is caused by the drag fluctuation of the solid cylinder. The $ f 2$-tone is also observed in the sound pressure level spectra of Geyer,^{15} and the results indicate that there is no direct relationship between $ f 2$-tone and the porosity of the porous materials. As the PPI increases, the characteristics of $ f 2$-tone become more pronounced, though the magnitude decreases. In addition, the porous cladding with high PPI weakens the effect of reducing broadband energy in the far wake region near the centerline.

Comparing the PSD profiles along $ x / D = 1.5$ from $ y / D = 0$ to 2 [in Figs. 14(a)–14(e)], the $ f 2$-tone is only present at $ y / D = 1.0$ and 1.5 for the solid case, while the porous cases all possess a strong $ f 2$-tone. In addition, the PSD reaches the highest value at $ y / D = 0.5$, which indicates the occurrence of strong shear stress here. Figures 14(f)–14(j) show the PSD profiles of different vertical positions at $ x / D = 3.0$; note that $ f 1$-tone exists in all cases from $ y / D = 0$ to 2. At the position above $ y / D = 0.5$, the $ f 2$-tone decreases in magnitude as the position moves away from the centerline of the wake. Further downstream at $ x / D = 7.0$, as shown in Figs. 14(f)–14(j), the distribution of $ f 1$-tone is similar to that at $ x / D = 3.0$, whereas the $ f 2$-tone has decreased significantly in magnitude. This indicates that porous cladding with high PPI can effectively suppress the energy in the wake, whereas the suppression effect weakens with the increase in distance.

### E. Aeroacoustic analysis

Figure 15(a) shows the acoustic spectral characteristics of the solid and porous cylinders. Compared with the solid cylinder, the porous cladding can achieve radiated noise attenuation, especially for the porous cladding with higher PPI. Remarkably, the porous cladding with 20 PPI achieves the best noise reduction effect in the entire spectra. Meanwhile, the predominant peak frequency and corresponding SPL amplitude decrease with the increase in pore density, which means that vortex shedding is suppressed and reduction of the overall sound pressure level (OASPL). However, for being coated with 8 PPI, the noise in low-frequency bands is larger than that of the solid cylinder. In addition, the noise attention of the cylinder coated with porous media is also reflected in the far-field noise directive pattern, as shown in Fig. 15(b). Comparing the profiles of OASPL, the noise reduction level reaches up to 27.6 dB for the porous cylinder coated with 20 PPI. Furthermore, the porous cladding produces an equal amount of noise attention toward both the upper and lower sides of the simulation domain, which causes the directivity of the lobes of the porous cylinder present in the shape of a dipole.

^{40,41}has been widely utilized to simulate flow-induced noise, but the physical relationship between flow structure and noise cannot be revealed clearly. Therefore, the vortex sound theory extended by Powell

^{36}and Howe

^{42}connects the sound source with the aerodynamic parameters in the flow field, which can understand noise source from an aerodynamic perspective and propose more effective control measures. For low Mach number and isentropic flow, the vortex sound equation can be written as follows:

*B*is the total enthalpy of the fluid; $ c 0$, $ \omega \u2192$, and $ U \u2192$ represent the speed of sound, velocity vector, and flow vortex vector, respectively. The left side of Eq. (14) describes the propagation of sound in fluid, and the right term represents the sound source of unsteady flow. Especially, the cross-product term vorticity vector and velocity vector $ \omega \u2192 \xd7 U \u2192$ are called “Lamb Vector.” The Lamb vector divergence $ div ( \omega \u2192 \xd7 U \u2192 )$ is a fundamental factor in flow-induced sound production, and positive and negative values represent stretching and vorticity-bearing motion, respectively. Meanwhile, The Lamb vector divergence $ div ( \omega \u2192 \xd7 U \u2192 )$ describes the evolution process of high-momentum and low-momentum fluid motions and represents the interaction between strong strain rate region and vorticity region.

^{43}The mechanisms associated with the Lamb vector divergence play an essential role in the evolution of the shear layer separated from the cylinder.

Contours of the instantaneous Lamb vector divergence on the mid-plane of $ z / D = 0$ are displayed in Fig. 16. For the solid cylinder, it can be found from Fig. 16(a) that two-layer structures with opposite signs appear near the cylindrical boundary layer. Meanwhile, as shown in Fig. 16(b)–16(d), all porous cylinders also exhibit a similar phenomenon. Hamman *et al.*^{43} investigated the relationship between the Lamb vector divergence and the drag (or lift) on a stationary bluff body in the flow around a bluff body and found that decreasing the area over which regions of positive and negative Lamb vector divergence can lead to drag reduction. Compared with the solid cylinder, the interaction area behind the 13 PPI case and 20 PPI case decreases, resulting in the drag reduction. The Lamb vector divergence appears as an acoustic term in Powell's vortex sound equation, which indicates a strong correlation between the Lamb vector divergence and the noise radiation. Therefore, the sound field characteristics near the trailing edge of the cylinder are dominated by the evolution of turbulent shear expansion. The magnitude of Lamb vector divergence of three porous cylinder cases is smaller than that of solid cylinder, especially for the porous cladding with 20 PPI, which indicates that acoustic source intensity is weakened by porous media. This result demonstrates that reducing the spatial distribution and magnitude of the Lamb vector divergence in the flow field of a cylinder coated with porous media is an important factor for noise reduction.

Figure 17 presents contours of acoustic pressure for different cylinders at vortex shedding frequency. Within the range of Reynolds and Mach number range considered in current work, the acoustic radiation in all four cases is mainly dominated by the lift dipole. Consistent with the lift fluctuations, the sound waves propagate in the *y*-direction and −*y*-direction with a phase difference of $ 180 \xb0$. For the solid cylinder, vortices shed from the upper and lower sides of the cylinder are responsible for triggering alternate positive and negative disturbance pressure pulses of the cylinder, which is consistent with the research of Inoue and Hatakeyama.^{44} Meanwhile, the wake flow is symmetrical along the horizontal centerline and is not deviate from the upper and lower halves of the cylinder. As a result, the generated sound field presents similar behavior in the upper and lower halves of the simulation domain. Similarly, all porous cases also maintain the similar noise radiation mode as the solid cylinder. The estimated wavelength of solid cylinder is $ \lambda \u2248 D / ( M a \xd7 S t ) \u2248 64.26 D$. For the 8 PPI case, the porous cylinder generates a sound wave with a longer wavelength than the solid cylinder, i.e., $ \lambda \u2248 64.60 D$. With the increase in pore density, the 13 PPI case has an acoustic wavelength of $ \lambda \u2248 69.34 D$, and the 20 PPI case generates sound with a wavelength of approximately $ \lambda \u2248 92.28 D$. Moreover, the magnitudes of acoustic pressure decrease with the increase in pore density.

## IV. CONCLUSION

In this study, we investigate the wake flow and radiated noise characteristics of the circular cylinder coated with porous media at the subcritical Reynolds number using LES and FW–H acoustic analogy. The present numerical simulation is deemed reliable for capturing the key features of the flow field and acoustic field. Specifically, the results suggest that the permeability of the coating plays an essential role in determining the flow and acoustic properties of the system. Moreover, our study reveals the underlying physics behind the interactions between the flow field and porous medium, which is crucial for the development of effective noise reduction strategies.

We found that cylinders coated with porous media help attenuate lift fluctuation and reduce drag. The effect of drag reduction increases with the increase in pores per inch, with a maximum drag reduction of 33.6%. Meanwhile, the porous cladding with higher PPI can increase the slip velocity at the porous–fluid interface, suppress vortex shedding, weaken the intensity of turbulent kinetic energy in the wake, and make its peak region shift downstream. More importantly, the drag reduction of a cylinder coated with porous media is attributed to the increase in base pressure on the leeward side.

Flow characterization indicated that the porous media significantly changes the separation pattern behind the cylinder. Specifically, porous cladding elongates the low-speed recirculation region and moves the center of the vortex further away from the cylinder. Moreover, the complex irregular vortex structures convert into small-scale streamwise vortex structures, and the wake oscillation in porous cases is substantially contracted in the spatial range, which helps to attenuate the turbulent fluctuations in the wake near the cylinder.

The analysis based on vortex sound theory indicates that the porous cladding reduces the interaction area and magnitude of the positive and negative Lamb vectors divergence in the flow field, which is beneficial for drag reduction and noise attenuation. A maximum noise reduction of 27.6 dB is obtained for a porous cladding with 20 PPI. Moreover, the acoustic field analysis shows that porous media does not change the noise radiation mode, but the porous media with high PPI is beneficial to reduce the noise generation of noise and intensity of the noise source. Overall, this research contributes to the broader understanding of flow-acoustic coupling in porous media and provides insight into the design of better sound-absorbing materials for practical use.

## ACKNOWLEDGMENTS

The authors acknowledge financial support from the National Natural Science Foundation of China (Nos. 12172207 and 92052201) and the Open Project of Key Laboratory of Aerodynamic Noise Control of China Aerodynamics Research and Development Center (No. ANCL20190305).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Huanhuan Feng:** Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (equal). **Linfeng Chen:** Data curation (equal); Methodology (equal). **Yuhong Dong:** Methodology (equal); Project administration (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.