Effects of periodic suction-blowing excitation on the aerodynamic sound generated by a laminar flow past a square cylinder using the direct numerical simulation approach

While designing several aerodynamic and hydrodynamic devices, knowledge about the nature of fluid flow around bluff bodies is essential. Usually, the flow gets separated on the rear portion of a bluff body due to a strong adverse pressure gradient. Vortex shedding takes place above a certain critical Reynolds number,¹ which brings unsteady nature to the flow field. Consequently, forces imparted by the fluid on the body display unsteadiness. The unsteady nature of the flow field is often undesirable as it may lead to a structural failure due to vortex-induced vibrations and is also responsible for generating aerodynamic sound. Thus, it is necessary to control flow over bluff bodies.

Researchers have used various active and passive flow control techniques to suppress the vortex shedding behind a bluff body for reducing fluctuating forces acting on the body^2,3 and reducing undesired aerodynamic sound/noise.^4,5 Shear flows can be effectively altered by employing uniform blowing or suction from the wall. In this flow control technique, a secondary flow is injected in and out of a body surface for controlling flow separation and the vortex shedding in the wake region.

Cohen⁶ employed a uniform suction and blowing over a porous circular cylinder and observed that the boundary layer became thinner, and the Strouhal frequency was increased with a rise in the suction velocity. In contrast to an increase in the blowing velocity, the boundary layer got thicker, and the Strouhal frequency was decreased. Delaunay and Kaiktsis⁷ investigated the effects of steady suction or blowing from the base of a circular cylinder on the stability and wake dynamics at low Reynolds number (Re < 90) and observed that slight blowing can stabilize the wake by reducing backflow without affecting the shedding frequency. Delaunay and Kaiktsis⁷ also noted that a relatively high level of suction is required to stabilize the near wake, drastically increasing the drag coefficient and decreasing the Strouhal number.

Mathelin et al.^8,9 experimentally investigated the effects of a blowing excitation on flow past a circular cylinder at various Reynolds numbers (Re = 3900, 5400, 7000, and 10 500). An increase in the wake width has been reported due to the blowing excitation as the apparent cylinder diameter is increased.⁸ Injecting a low-velocity fluid into the growing boundary layer increases boundary layer thickness and lowers the viscous drag of the cylinder.⁸ Blowing excitation causes a decrement in the wake static pressure defect due to the mass injection in the wake.⁹ Fransson et al.¹⁰ experimentally investigated the effects of continuous suction or blowing from the surface of a porous circular cylinder on wake modifications at Re = 8300. It was observed that suction delays flow separation, resulting in a narrower wake width accompanied by a drag reduction in contrast to the blowing excitation, which widened the wake.¹⁰ 

Researchers were able to suppress the vortex shedding and control the wake geometry using suction-blowing excitation. Park et al.¹¹ numerically investigated modifications in the flow over a circular cylinder using a pair of suction/blowing slots with feedback control and completely suppressed the vortex shedding for Re = 60. Guo-ping and Jian-wen¹² numerically investigated the effects of strength and position of a suction or blowing strip kept over a circular cylinder for Re = 100. Their results displayed that the suction should be prescribed on the shoulder of the cylinder, or the blowing should be prescribed on the rear portion of the cylinder for efficiently suppressing the asymmetry of the wake in the transverse direction. Dong et al.¹³ numerically investigated the idea of using a uniform suction on the windward surface and uniform blowing on the leeward surface of a cylinder for suppressing vortex shedding at Re = 500, 1000. It was observed that for a small amount of forcing, vortex shedding was suppressed, attenuating lift fluctuations significantly.¹³ Muralidharan et al.¹⁴ investigated the control of vortex structures past a circular cylinder using two suction actuators at the top and bottom surfaces in addition to blowing actuators at the rear side of the cylinder. Muralidharan et al.¹⁴ reported significant wake suppression at Re = 100 and Re = 3900.

These studies highlight the application of suction and blowing excitation in controlling bluff body flows.

Mahato et al.¹⁵ used the direct numerical simulation (DNS) approach to understand sound generation from a triangular wedge in a uniform flow, corresponding to the Mach number 0.2 and the Reynolds number of Re = 100. The authors have shown the connection between the vortex shedding behind the wedge and the aerodynamic sound generation. It was observed that when the vortex is shed from the top side of the body, a negative and a positive pressure pulse is generated from the top and bottom sides of the cylinder, respectively. The Strouhal frequency associated with the vortex shedding phenomenon and the frequency of sound waves was observed to be the same. The resultant sound field had a dipolar nature dominated by the lift dipole equivalent sources. Margnat¹⁶ used a hybrid prediction approach to evaluate the aerodynamic noise radiated by a rectangular cylinder when kept at incidence to the incoming flow. It was observed that the contribution of the drag dipole is significant and affects the directivity.¹⁶ Karthik et al.¹⁷ reported aeroacoustic results for the flow past finite-length circular cylinders at the Re = 84 770 for various length-to-diameter ratios. The authors used the large eddy simulation approach to solve the incompressible Navier–Stokes equations and evaluated the hydrodynamic fluctuations. This information was further used to predict the acoustic field using the Ffwocs Williams and Hawkings approach. Karthik et al.¹⁷ observed that the cylinder L/D ratio controls vortex shedding and, hence, the flow-induced sound generation.

Mahato et al.⁴ and Ganta et al.⁵ followed the DNS approach to study the reduction in aerodynamic sound using active and passive control means. Mahato et al.⁴ suggested an innovative arrangement of the splitter plates, which reduced the aerodynamic sound by more than 20 dB, for a laminar flow past a square cylinder. Introduction of the splitter plates in the aft portion of a cylinder modified the interaction of the separated shear layers from the cylinder, providing sound reduction. In contrast, Ganta et al.⁵ prescribed rotary oscillations to a circular cylinder to control the aerodynamic sound field for flow past a cylinder. It was observed that the frequency of the radiated sound waves is the same as the forcing frequency of the cylinder subjected to rotary oscillations, and the directivity of the sound field can be controlled by the amplitude of the prescribed rotary oscillations.⁵ 

In the past, many researchers have looked at the suction-blowing excitation technique from the flow control perspective. In the literature, the effects of a periodic suction-blowing excitation on the generation and propagation of aerodynamic sound have not been studied in detail and have been attempted here for a laminar flow past a square cylinder. Sound source-related information has been evaluated from the flow field information and related to the aerodynamically generated sound.

A periodic suction-blowing excitation can be applied on a square cylinder’s top and bottom surfaces in either an in-phase or an out-phase manner. In the case of in-phase excitation, either suction or blowing is prescribed at any instant on a square cylinder’s top and bottom surfaces. In contrast, for out-phase excitation, if suction excitation is specified on the top surface at a particular moment, then blowing is applied on the bottom surface of the cylinder. The present study aims at understanding how the in-phase and the out-phase excitation techniques affect the sound field directivity. Effects of changes in the excitation amplitude, frequency, and phase delay between the vortex shedding phenomenon and the excitation on the radiated sound field have also been studied. The bias created in the sound field directivity for different excitation cases has been explained using Powell’s sound source term (PSST)¹⁸ and by decomposing the sound field into time-varying dominant modes using the proper orthogonal mode decomposition (POD)¹⁹ technique.

This paper is organized in the following way. Section II discusses the mathematical formulation and numerical procedures in detail. Subsequently, in Secs. III and IV, we discussed and analyzed results for the no-excitation case and the cases with periodic suction-blowing excitation, respectively. This article ends with the summary and conclusions in Sec. V.

Here, we provide a detailed description of the flow model, numerical procedure, and the grid used to solve the problem considered under the present investigation.

A schematic of the flow model is shown in Fig. 1. A square cylinder has been immersed in a uniform flow. The center of the cylinder is located at the origin. The top and the bottom surfaces of the cylinder are parallel to the flow direction. Let P(r, θ) be any point in the domain whose position can be identified with the distance r between the center of the cylinder and the point P and the angle θ, as shown in Fig. 1. Flow approaches the cylinder from the left-hand side along the (+ve) x axis direction. Let u and v denote the components of the velocity vector $(V⃗=uî+vĵ)$ along the (+ve) x axis and (+ve) y axis, respectively. Let L be the length of one of the sides of the square cylinder. The width of the periodic suction-blowing excitation strip is 0.2L and has been located in between x = ±0.1L on the top and the bottom surfaces (y = ±0.5L) of the cylinder. The schematic shows a case for the in-phase excitation.

In this work, effects of a periodic suction-blowing excitation on the aerodynamic sound field for a 2D laminar flow past a square cylinder at a Reynolds number of Re = 100 have been studied. The Reynolds number is obtained using the relation $Re=ρ∞U∞Lμ∞$⁠, where the parameters ρ_∞, U_∞, μ_∞, and L indicate the free-stream properties, namely, density, velocity, viscosity, and the length of one of the sides of the square cylinder, respectively. The prescribed Prandtl number $(Pr=μ∞Cpk)$ is 0.7, and the Mach number $(M=U∞c∞)$ is 0.2, where C_p, k, and c_∞ are the specific heat at constant pressure, thermal conductivity, and speed of sound at the free-stream conditions. Simulations have been performed corresponding to the free-stream conditions with density ρ_∞ = 1.22 kg/m³, temperature T_∞ = 288°K, and viscosity coefficient μ_∞ = 1.8 × 10⁻⁵ kg/m s. The ratio of specific heats is kept as γ = 1.4.

Here, we have solved 2D, unsteady, compressible Navier–Stokes equations, expressed in the conservative form, using an in-house developed solver based on accurate finite difference schemes. Detailed information related to the mass, momentum, and energy conservation equations has been provided in Refs. 4, 20, and 21 and not mentioned here to avoid repetition. The flow conservation equations have been solved in the non-dimensional form. The distance between any two points in the domain has been non-dimensionalized with respect to the side length (L) of the square cylinder. Velocity components are non-dimensionalized using the free-stream velocity U_∞. Similarly, density (ρ), temperature (T), and the viscosity coefficient (μ) have been non-dimensionalized with respect to corresponding free-stream values ρ_∞, T_∞, and μ_∞, respectively. Pressure has been non-dimensionalized using the factor $ρ∞U∞2$⁠, while time has been non-dimensionalized using the ratio L/U_∞. All the physical quantities have been mentioned in the respective dimensionless form from here onward.

In the present in-house developed solver, the unsteady terms in the mass, momentum, and energy conservation equations have been discretized using the computationally explicit Runge–Kutta (CERK) time integration scheme,²¹ while the first-order spatial differential terms have been discretized using the sixth-order compact (C6) scheme.²² The C6-CERK scheme was used by Yadav et al.²¹ for performing DNS of sound generation due to the laminar flow past a circular cylinder and an airfoil. The second-order spatial discretization terms have been discretized using a second-order, explicit central difference (CD2) scheme.

A structured, nearly orthogonal, body-fitted grid has been generated around a square cylinder of unit length, forming an O-shaped domain. There are 497 points along the surface of the cylinder, and 1086 points are positioned in the wall-normal direction. The outer boundary of the domain has been located at r = 1200. The first grid point has been kept at a distance of 0.005 from the surface of the cylinder in the wall-normal direction. Grid point spacing close to the surface of the cylinder in the wall-normal direction is sufficiently small to resolve the boundary layer accurately. The computational domain has been divided into the sound and buffer zones. There are 497 × 936 points in the sound zone (0.5 ≤ r ≤ 100). Spacings between the grid points in the wall-normal direction have been increased in a geometric progression. The corresponding grid stretching factor has been kept below 5% in the sound zone to resolve the sound waves accurately. Maximum spacing in the wall-normal direction (Δr_max) in the sound zone is 0.5. Numerical solutions from the sound zone have been used for analyzing hydrodynamic and sound disturbances.

Spacing between the grid points in the buffer zone is rapidly increased to attenuate hydrodynamic and sound disturbances propagating toward the outer domain boundary. Sound waves do not decay as quickly as hydrodynamic disturbances^15,23 and travel a large distance before undergoing a considerable reduction in amplitude. A large buffer domain (100 ≤ r ≤ 1200) is necessary to attenuate unphysical sound waves reflected from the domain outer boundary.¹⁵ The buffer domain acts similar to the sponge region.²⁴ The grid stretching factor in the buffer zone is kept below 12%. A smooth transition in the distribution of grid points has been ensured at the interface of the sound and buffer zones. A close-up view of the generated grid around the top-right corner of the square cylinder is shown in Fig. 2(a), while Fig. 2(b) displays grid point distribution in the larger section of the domain.

Simulations have been performed by initially prescribing free-stream conditions everywhere in the domain. For the case without any suction-blowing excitation, the no-slip boundary condition has been prescribed on the surface of the cylinder, which results in the development of shear layers. Separated shear layers from the top and the bottom surfaces of the cylinder interact to cause vortex shedding.¹ Simulations have been performed over a sufficiently long duration so that the transient effects related to the initial condition do not affect the sound field prediction. The no-excitation case serves as a basis to compare modifications in the flow and sound fields caused by the periodic suction-blowing excitation. The periodic suction-blowing excitation has been applied in the region −0.1 ≤ x ≤ 0.1 on the top and the bottom surfaces by prescribing the velocity components on the cylinder surface as follows:

In Eq. (1), the amplitude “A” indicates the maximum magnitude of time-varying velocity component “v.” The parameter f_f = St₀ × fr indicates the forcing frequency in terms of the forcing frequency ratio (fr) and the Strouhal frequency (St₀) associated with the no-excitation case. The phase lag between the vortex shedding process and the suction-blowing excitation has been indicated by “ϕ,” and the symbol “t” denotes the solution time. Equation (1) displays an in-phase excitation in which either suction or blowing takes place at any instant on the top and bottom surfaces of the square cylinder. In contrast, one can also prescribe an out-phase excitation in which if suction takes place on the top surface, then blowing is prescribed on the bottom surface of the cylinder and vice versa. The out-phase excitation is given in the following equation:

The adiabatic boundary condition has been prescribed on the surface of the cylinder, while characteristic-based boundary conditions²⁵ have been prescribed at the outer boundary of the domain. The buffer domain along with the characteristic-based boundary conditions²⁵ helps to control unphysical reflections from the domain boundary.

We have carried out detailed validation and grid independence studies by solving the 2D laminar flow past a square cylinder without excitation.

For validating the present in-house developed solver, we have considered the no-excitation case related to a laminar, uniform flow past a square cylinder at Re = 100 and M = 0.2. The flow past a square cylinder at Re = 100 is unsteady⁴ and displays periodic vortex shedding, as shown in Fig. 3(a), at the indicated instant. Time variation of the lift (C_l) and the drag (C_d) coefficients is also shown in Figs. 3(b) and 3(c), respectively. Fast Fourier Transforms (FFTs) of the lift and drag coefficient variations with time are shown in Fig. 3(d). Figure 3 shows a periodic variation of lift and drag coefficients with time. The amplitude of C_l fluctuations is far more than C_d fluctuations. The Strouhal frequency St₀ = fL/U_∞ is a non-dimensional number where f is the vortex shedding frequency. The Strouhal frequency for the no-excitation case is St₀ = 0.1407, as observed from the lift coefficient’s dominant peak in the FFT. In contrast, the dominant peak in the drag coefficient variation is at 2St₀, as the vortices are periodically shed from the top and bottom sides of the cylinder. Table I compares the time-averaged drag coefficient (C_D), the Strouhal number (St₀), and the maximum amplitude of the lift coefficient fluctuation $(Cl′)$ for a laminar flow past a square cylinder at Re = 100, obtained in the present study with the results available in the literature. As observed from Table I, the present results are in good agreement with the results reported in the literature. Next, we have performed a grid independence study to understand the effects of the change in grid size on flow properties.

We have reported the grid independence study results for a laminar flow past a square cylinder at Re = 100. It has been shown that the chosen grid adequately captures the hydrodynamic features of the flow field and also accurately resolves the disturbance pressure waves associated with the sound field.

The grid independence study has been performed using four different grids, as highlighted in Table II. Grid-A is a coarse grid with 397 × 240 grid points, while Grid-D is a very fine grid with 993 × 2170 grid points. The minimum grid spacings in the azimuthal and the wall-normal directions have been reported in the second and third rows. The time steps prescribed while performing simulations on different grids have been mentioned in the subsequent row. The values of the root mean square (rms) of the time-varying lift coefficient (Cl_rms), the time-averaged drag coefficient (C_D), and the Strouhal number (St₀) have been compared. Table II displays that there is a visible difference only in the rms of time-varying lift coefficient (Cl_rms) for Grid-A and Grid-B. However, differences in the time-averaged drag coefficient and the Strouhal number obtained for different grids are insignificant. We have chosen Grid-C for performing computations with and without suction-blowing excitation. As discussed next, Grid-C resolves the hydrodynamic scales appropriately and accurately captures the amplitude of disturbance pressure waves in the sound zone.

As mentioned before, the no-excitation case serves as a reference case to understand modifications in the flow and sound fields due to periodic suction-blowing excitation. We have used the solution of the no-excitation case at t = 293.966 as an initial solution while starting periodic suction-blowing excitation. The vorticity contours in Fig. 3(a) have been shown for the same instant. Figure 3(b) indicates that C_l = 0 at t = 293.966 and the lift coefficient further becomes negative as time progresses.

Figure 4(a) displays the contours for the instantaneous pressure p(x, y, t) distribution around the square cylinder at t = 293.966, while Figs. 4(b) and 4(c) display the contours for time-averaged pressure [P_avg(x, y)] and the instantaneous disturbance pressure [p′(x, y, t)] field, respectively. We have considered around 71 snapshots of the flow field per vortex shedding cycle, and the time averaging operation has been performed over 14 vortex shedding cycles to obtain information about the mean pressure [P_avg(x, y)] field. Information about the instantaneous disturbance pressure field [p′(x, y, t)] has been derived from the difference between the instantaneous pressure field [p(x, y, t)] and the time-averaged pressure field [P_avg(x, y)]. We have used 26 equispaced levels between ±0.001 while displaying disturbance pressure field contours. Close to the body, the disturbance pressure field contains information about the hydrodynamic and sound disturbances. In contrast, the disturbance pressure field displays information about the sound field far away from the body.¹⁵ The generated sound field can be correlated to the fluctuating forces acting on the square cylinder using Curle’s analogy.²⁹ The sound field for such a low Reynolds number flow displays disturbance pressure waves generated due to the lift and the drag dipole equivalent sources.¹⁵ As the amplitude of fluctuations associated with the lift coefficient is far more compared to that due to the drag coefficient, one observes the formation and propagation of dominant disturbance pressure fluctuations p′(x, y, t), along the normal direction to the flow propagation¹⁵ and Fig. 4(c) confirms this observation.

Figure 5(a) displays the decay of maximum amplitude of disturbance pressure wave (|Δp′|) along different radial locations (r) and θ = 90° for different grids mentioned in Table II. As shown, obtained decay rates of maximum amplitude of the disturbance pressure wave, away from the square cylinder, match the theoretical decay rate $(|Δp′|theo∼1/r)$¹⁵ for Grid-B, Grid-C, and Grid-D. Grid-A, being the coarse grid, is not able to capture the disturbance pressure field details correctly.

Figure 5(a) shows that the hydrodynamic field and the sound field details have been adequately resolved by Grid-C, used in present simulations. Figures 5(b) and 5(c) display time variation of disturbance pressure [p′(x, y, t)] at r = 70, θ = ±90°, and θ = 180°, respectively. One observes that the variation of p′(x, y, t) vs time at θ = ±90° is anti-symmetric. For a laminar flow past a body, if a vortex is shed from either the top or the bottom side of a body, a negative pressure pulse is originated from that side.¹⁵ In contrast, a positive pressure pulse is originated from the opposite side.¹⁵ These tiny pressure pulses travel in the domain with the speed of sound. Due to periodic vortex shedding, one observes an anti-symmetric nature in a disturbance pressure variation at a far-field location r = 70 and θ = ±90°. The FFT of the time-varying disturbance pressure variation in Figs. 5(b) and 5(c) is shown in Fig. 5(d). It displays that the dominant frequency of sound waves at r = 70 and θ = ±90° is the Strouhal frequency at which the lift coefficient varies. In contrast, at r = 70 and θ = 180°, the sound waves vary with twice the Strouhal frequency, similar to the drag coefficient. The maximum amplitude of p′(x, y, t) with time is significantly more at θ = ±90° compared to θ = 180°, as the amplitude of the lift coefficient fluctuations is far more compared to the drag coefficient fluctuations.¹⁵ The sound variations at r = 70 and θ = ±90° and θ = 180° are dominated by the Strouhal frequency and twice the Strouhal frequency, respectively. The contribution of the individual dominant frequencies to the disturbance pressure field has been separated using the proper orthogonal decomposition (POD) technique to understand the relative importance of various sound sources.

Researchers and engineers have analyzed complex fluid flows using proper orthogonal decomposition techniques to extract coherent flow field structures.^19,30,–32 As the turbulent and transitional fluid flows involve a broad range of spatiotemporal scales, the POD analysis helps to extract important information related to the dominant coherent structures present in the flow field. In the present study, the POD technique is applied to decompose the disturbance pressure fields to understand the nature of radiated sound fields. The POD technique has been implemented using the method of snapshots.^19,31,32 Here, the disturbance pressure fields are expressed as follows:

where ϕ_k(x, y) and a_k(t) represent the kth spatial POD mode and its corresponding time-varying POD mode amplitude, respectively. Here, N denotes the number of snapshots considered for the POD analysis. Equation (3) represents the instantaneous disturbance pressure field as a linear combination of the product of a_k(t) and ϕ_k(x, y). The POD modes ϕ_k(x, y) are eigenfunctions of a covariance matrix (R_ij), which is expressed as

where S denotes the domain for POD analysis and the values of indices are given as i, j = 1, 2, 3, …, N. The eigenvalues (λ_k) of the covariance matrix R_ij denote the contribution by the corresponding POD mode ϕ_k(x, y) to the disturbance pressure field. There exist large numbers of POD modes for transitional and turbulent flows due to the presence of a wide range of spatiotemporal scales in the flow field.^31,32 As the present simulations are in the laminar flow regime, few dominant POD modes are sufficient to represent the disturbance pressure field accurately.

The POD analysis of the disturbance pressure fields has been carried out for the no-excitation case. Simulation data over 14 vortex shedding cycles with around 71 snapshots in each cycle have been considered for the POD analysis. Figure 6 shows the first four dominant POD modes along with their time-varying POD amplitudes and frequency information. The first four dominant POD modes contribute more than 99% (λ₁ + λ₂ + λ₃ + λ₄ > 0.99) to the disturbance pressure fields. The disturbance pressure pulses in the first two dominant POD modes (ϕ₁ and ϕ₂) propagate almost in the vertical direction, as shown in Figs. 6(a) and 6(b). Their respective time-varying POD amplitudes displayed in Fig. 6(e) vary with vortex shedding frequency, as displayed in Fig. 6(f). These first two POD modes represent almost 89% (λ₁ + λ₂ ≈ 0.89) of the instantaneous disturbance pressure field. The next two dominant POD modes (ϕ₃ and ϕ₄) display disturbance pressure pulses propagating along and opposite to the flow direction, as shown in Figs. 6(c) and 6(d). Corresponding time-varying POD amplitudes shown in Fig. 6(g) vary with a frequency of twice the vortex shedding cycle, as shown in Fig. 6(h). The contribution of these two POD modes to the disturbance pressure fields is around 10% (λ₃ + λ₄ ≈ 0.10). Based on these observations, the first two POD modes represent the sound fields contributed due to fluctuating lift forces acting on the cylinder surface, resembling a lift dipolar nature. In contrast, the next two POD modes are contributed by the fluctuating drag forces acting on the cylinder surface, representing a drag dipolar behavior.

As per Powell’s theory of vortex sound,¹⁸ the aerodynamic sound is generated in an unsteady fluid flow due to the vorticity and motion of vortices. Powell¹⁸ has shown that the changes in the circulation or area of a vortex ring create a dipole sound field, and vorticity in a slightly compressible fluid is responsible for inducing the flow and the sound field. Following this important work, many researchers^33,34 implemented Powell’s vortex sound theory to predict far-field noise.

Suction-blowing excitation is supposed to alter the formation and interaction of the shear layers developed on the surface of the cylinder. Thus, the modifications in the vorticity field can be related to the corresponding changes in the sound field. Here, Powell’s analogy has been used to study the nature of sound sources for the cases with and without excitation. The non-homogeneous wave equation with Powell’s sound source terms³⁵ mimicking sound propagation is given as

The term on the right-hand side of Eq. (5) represents the Powell’s sound source term $(PSST=−∇⋅f⃗P)$⁠. The sound source function $(f⃗P)$ in Eq. (5) is given as

In Eq. (5), ρ′ and c_∞ indicate the density fluctuations and phase speed of a sound wave. The vector terms $ω⃗$ and $V⃗$ indicate vorticity and velocity, respectively, where vorticity is given by $ω⃗=∇×V⃗$⁠. The term $(−ω⃗×V⃗)$ in Eq. (6) denotes Coriolis acceleration and displays sound generation due to the variation of Coriolis acceleration in the source region.³⁵ The other term in Eq. (6) indicates sound created by the fluctuating kinetic energy.

The information about Powell’s sound source term (PSST) has been evaluated using the unsteady flow field obtained from the present DNS computations. Liow et al.³⁴ have suggested that the sound sources (PSST), as evaluated from the instantaneous hydrodynamic field, should not be directly used to predict the far-field sound as the hydrodynamic field contains both the time-invariant and the fluctuating component. As the time-invariant part does not contribute to sound generation, it was advised to be removed while solving Eq. (5) numerically.^33,34 The time-averaged mean component of PSST is obtained by performing time averaging operation over multiple vortex shedding cycles. The fluctuating part of PSST has been obtained by subtracting the time-averaged mean component of PSST from the instantaneous sound source.

Figure 7 displays information related to Powell’s sound sources PSST. The mean component of PSST is displayed in Fig. 7(a), which shows symmetrical structures on both the top and bottom sides of the cylinder. Figures 7(b) and 7(c) show the fluctuating components of PSST obtained at two different instants at which the time-varying lift coefficient exhibits maximum and minimum values, respectively. One observes that the sound sources are localized in the shear layers formed over a cylinder surface and the shed vortices in the wake region. As the sound sources are displayed at the instants of maximum and minimum values of the time-varying lift coefficient, the sound source structures present on the top and bottom surfaces of the cylinder shown in Fig. 7(b) are exactly opposite to the structures displayed in Fig. 7(c). Periodic suction-blowing excitation can introduce a bias in the sound field directivity. Information about the fluctuating sound sources has been further evaluated at two locations, close to the top (x = 0, y = 0.55) and the bottom (x = 0, y = −0.55) surfaces of the cylinder. These locations are specifically chosen as they are located symmetrically about the y = 0 line. Any bias in the sound source distribution and corresponding time variation can be correlated to the observed bias in the sound field directivity. Figure 7(d) shows the time variations of the fluctuating components of the PSST at the indicated locations. In the absence of suction-blowing excitation, the maximum fluctuating amplitude of PSST at the chosen locations is $∼1.25$⁠, and there is no bias in the sound field directivity about y = 0 line, as observed from Fig. 4(c). The corresponding FFTs of the fluctuating components of PSST in Fig. 7(d) are shown in Fig. 7(e), which confirms the Strouhal frequency is the dominant frequency associated with the sound source variation. It is also observed that the amplitude corresponding to the drag dipole frequency is very small compared to the amplitude of the lift dipole frequency, signifying the dominance of lift dipole equivalent sources.

Next, we discuss results related to the modifications in the flow and sound fields due to the application of the periodic suction-blowing excitation on the surface of the square cylinder. As given in Eq. (1), the periodic suction-blowing strip has been applied on the top and bottom surfaces of a square cylinder between x = ±0.1. The amplitude of the wall-normal component of the excitation velocity varies with time. The amplitude (A), the non-dimensional frequency ratio (fr), and the phase delay (ϕ) between the vortex shedding and the applied excitation characterize the applied excitation. As mentioned before, we have used the solution of the no-excitation case at t = 293.966 as an initial condition while starting suction-blowing excitation cases. A summary of all the cases reported in this work with periodic suction-blowing excitation is given in Table III. The cases have been performed for Re = 100 and M = 0.2.

To understand the difference between the in-phase and the out-phase excitation on the modification of the flow and sound fields, we have considered a case with amplitude of excitation A = 0.05, non-dimensional frequency ratio fr = 1, and phase delay ϕ = 0°. The forcing frequency ratio fr = 1 suggests that the forcing frequency (f_f) of excitation is the same as the vortex shedding frequency (St_o) of the no-excitation case, as f_f = fr × St_o.

In the case of in-phase excitation, if the blowing excitation is applied when the vortex is about to shed from the top side of the cylinder, then the suction is used when the vortex is about to shed from the bottom side of the cylinder. This is expected to create a kind of bias in the evolved sound field. In contrast, for the out-phase excitation, either suction or blowing excitation is applied when the vortex is about to shed from either side of the cylinder. Thus, it is expected that the flow and the sound field should not display any bias in the case of the out-phase excitation.

Table IV compares the aerodynamic parameters obtained for the in-phase and out-phase excitation cases. The considered aerodynamic parameters are the time-averaged drag coefficient, the maximum fluctuating lift and drag coefficients, and the Strouhal frequency. One observes that the in-phase and out-phase excitation techniques do not significantly alter the mean drag coefficient (C_D) and the Strouhal number (St) compared to the no-excitation case, as the amplitude of suction-blowing excitation is small. However, from Table IV, we can observe that there is a change in the amplitude of fluctuations of the lift and drag coefficient. In comparison with the no-excitation case, the lift coefficient fluctuations for the out-phase excitation increase, while these fluctuations are almost similar for the in-phase case. The increase in lift fluctuation for the out-phase case increases the sound directivity in the normal direction, as observed in Fig. 8(d). The drag fluctuations for the in-phase excitation case increase significantly, while for out-phase excitation, these fluctuations remain almost similar. This is visible in the sound directivity in Fig. 8(c) where the sound directivity in the flow direction increases in comparison with the no-excitation case. Sound fields containing tiny pressure fluctuations are considerably affected due to the bias introduced by the in-phase excitation technique, as discussed next.

Figure 8 displays the disturbance pressure contours for the in-phase excitation in (a) and the out-phase excitation in (b) at an instant corresponding to the maximum lift coefficient. One observes that sound waves predominantly travel in the top-left direction for the in-phase excitation. For the out-phase excitation, the sound field on the top and the bottom sides of a cylinder displays anti-symmetric nature about the y = 0 line. The directivity of the sound field has been shown at a radial location r = 70 for the in-phase and out-phase excitations in Figs. 8(c) and 8(d), respectively. For the in-phase excitation, the maximum amplitude of sound waves is noted around θ = 121°. In contrast, in the case of out-phase excitation, the maximum amplitude of sound waves is noted around θ = 102°, similar to the no-excitation case.

Figure 9 shows the time variation of the lift coefficient and velocity amplitude of suction-blowing excitation on the upper and lower surfaces of the cylinder for the in-phase and the out-phase cases in (a) and (c), respectively. For the in-phase excitation case, the lift coefficient is maximum at t = 296.9. Suction is applied on both the upper and the lower surfaces (y = ±0.5) of the cylinder at that instant. In contrast, at t = 300.5, the lift coefficient is minimum, and blowing is applied on the top and bottom surfaces of the cylinder. The time-varying velocity amplitudes of the in-phase excitation show the in-phase behavior of suction-blowing excitation. In the case of out-phase excitation, Fig. 9(c) displays the maximum and minimum lift coefficients at t = 270.1 and t = 273.7, respectively. At these instances, the amplitude of suction-blowing excitation is minimal, close to zero.

Figures 9(b) and 9(d) show the schematic of sound sources for the in-phase excitation at t = 296.9 and out-phase excitation case at t = 270.1, respectively. The lift dipole equivalent sources associated with the vortex shedding significantly contribute to the sound field.¹⁵ Periodic suction-blowing excitation results in a monopole equivalent sound source as the fluid mass has been sucked in and injected out through the cylinder surface. Applied excitation also modifies the effective cross-sectional area of the body,^8,9 which results in the additional drag dipole. The blowing operation increases the effective area of the cross section of a body, enhances the boundary layer displacement thickness, and increases drag. At the same time, suction reduces the displacement thickness and the effective area of the cross section, reducing the drag.^10,13 Thus, the resultant sound field is due to the combined effects of the lift dipole due to vortex shedding and monopole excitation along with the drag dipole introduced by the periodic suction-blowing excitation. In Figs. 9(b) and 9(d), symbols (+) and (−) denote the positive and negative pressure pulses generated due to vortex shedding (lift dipole), respectively. The positive and negative disturbance pressure pulses generated by the monopole equivalent of sound sources due to the periodic suction-blowing excitation are denoted by the symbols + and −, respectively. In addition, the characters ⊕ and ⊖ indicate the positive and negative disturbance pressure pulses generated due to the drag dipole.

At t = 296.9 for the in-phase excitation case, as the vortex is about to shed from the upper surface, a negative pressure pulse “(−)” is generated on the upper side and a positive pulse “(+)” is generated from the lower side,¹⁵ as shown in Fig. 9(b). At this instant, a negative pressure pulse will be generated around the cylinder due to suction excitation, as indicated by the “−” symbol. As suction reduces drag, a negative disturbance pressure pulse “⊖” will be created in the upstream region, while a positive disturbance pressure pulse “⊕” will be created in the downstream part due to the negative drag dipole equivalent source. One observes that the resultant strength of the sound sources will be increased on the top and the left surfaces of the cylinder, while the strength of sound sources on the bottom and the right surfaces will be suppressed. Due to such sound sources distribution, resultant sound waves predominantly propagate toward the top-left direction for the in-phase excitation case, marked by an arrow. For the out-phase excitation case, the direction of sound waves propagation is primarily decided by the lift dipole created by the vortex shedding process as the excitation velocity amplitude is close to zero at t = 270.1 and t = 273.7. Thus, it is expected that the propagation of sound waves for the out-phase excitation case will be predominantly normal to the flow direction.

The observations from Fig. 9 are also verified by the instantaneous disturbance pressure waves in the far-field, as shown in Fig. 10. The Mach number for the present simulations is 0.2, and it would take about 14 non-dimensional time for a sound pulse to cover r = 70 distance from the surface of the cylinder. The disturbance pressure pulses triggered by sound sources for the in-phase excitation at t = 296.9 and 300.5 will reach at r = 70 at t = 310.9 and 314.5 and are shown in Figs. 10(a) and 10(b), respectively. As seen from Fig. 9, (a) and (b) display dominant −ve and +ve disturbance pressure pulses in the top-left region of the domain.

For the out-phase excitation, the disturbance pressure pulses created at t = 270.1 and t = 273.7, by the sound sources marked in Fig. 9, will reach at r = 70 at t = 284.1 and 287.7, respectively. Corresponding variations of disturbance pressure amplitude of these waves at r = 70 are shown in Figs. 10(c) and 10(d), respectively. As concluded from Fig. 9, one observes dominant −ve and +ve disturbance pressure pulses at t = 284.1 in the top and bottom regions, respectively.

Figure 11 shows the instantaneous sound source amplitudes of PSST obtained for the in-phase and for the out-phase excitation of the suction-blowing arrangement. Their respective time-varying amplitudes obtained at indicated locations close to the cylinder surface are also shown. Figures 11(a) and 11(b) represent the sound sources obtained for an in-phase excitation at the indicated instants corresponding to the minimum and maximum values of the lift coefficient, respectively [as displayed in Fig. 9(a)]. In contrast, those of the sound sources obtained at maximum and minimum values of the lift coefficient [as shown in Fig. 9(c)] for an out-phase excitation are displayed in Figs. 11(d) and 11(e), respectively. For an out-phase excitation, the sound source structures near the cylinder surface at the instance of maximum lift coefficient (at t = 270.1) show an opposite behavior compared to sound source structures considered at the instance of minimum lift coefficient (at t = 273.7). Therefore, there is no bias in the distribution of sound sources on the top and bottom sides of the cylinder in one shedding cycle for an out-phase excitation. Hence, the net radiated sound field due to the out-phase excitation displays symmetric nature on the top and bottom sides of the cylinder. However, for an in-phase excitation, the sound source structures observed at the instants correspond to the maximum and minimum peak instants of the lift coefficient (at t = 296.9 and t = 300.5) that does not exhibit such opposite symmetric kind behavior. Therefore, the net-generated sound field due to in-phase excitation exhibits a bias in the sound field on the top and bottom sides of the cylinder. Next, the bias in the distribution of sound sources has been quantitatively shown by considering the fluctuating amplitudes of PSST at two different locations on either side of the cylinder. Figures 11(c) and 11(f) represent the time-varying amplitudes of PSST-related sound sources for both in-phase and out-phase excitations, respectively. These amplitudes are evaluated at the indicated symmetric positions close to the top and bottom surfaces of the cylinder (x = 0, y = 0.55) and (x = 0, y = −0.55). The fluctuating amplitudes display a 180° out of phase behavior for an out-phase excitation but the same peak values at the top and bottom sides, as shown in Fig. 11(f). Consequently, no bias in the generated sound field is observed for out-phase excitation. In contrast, sound sources have different peak values on the top and bottom sides of the cylinder for an in-phase excitation, as shown in Fig. 11(c). Hence, a bias is triggered in the generated sound field.

Next, the POD analysis of the disturbance pressure fields has been carried out for the suction-blowing cases with either in-phase excitation or out-phase excitation. The forcing frequency of the suction-blowing excitation has been prescribed as the natural shedding frequency of a no-excitation case. The shedding frequency of the excitation case is the same as the forcing frequency of excitation, as the system is synchronous with the forcing frequency of excitation.

Figures 12(a)–12(d) show the first four dominant POD modes of the disturbance pressure fields for the in-phase excitation case, and their respective time-varying POD amplitudes along with frequency content are shown in Figs. 12(e)–12(f). Although the first four dominant POD modes contribute more than 99% (λ₁ + λ₂ + λ₃ + λ₄ > 0.99) to the disturbance pressure field, the contribution of the first two POD modes is ∼90% (λ₁ + λ₂ ≈ 0.90) to the disturbance pressure field. The frequency of time-varying amplitudes of the first two POD modes (a₁ and a₂) is the same as the forcing frequency of suction-blowing excitation. In addition, a bias in the distribution of pressure pulses is also observed for the first two modes, which show pressure pulses traveling more toward the upper left portion of the cylinder. Hence, the net radiated sound fields for the in-phase excitation must exhibit a similar bias in the directivity patterns as the first two POD modes contribute 90% to the disturbance pressure field. The first two POD modes are contributed by the fluctuating lift force and the monopole source due to suction-blowing excitation. On the other hand, the next two POD modes (ϕ₃ and ϕ₄) together contribute less than 10% to the disturbance pressure field and their POD amplitudes vary (a₃ and a₄) with twice the shedding frequency of suction-blowing excitation. Furthermore, the disturbance pressure pulses in these two POD modes (ϕ₃ and ϕ₄) travel along and opposite to the flow direction, which resembles a drag dipole contributed by the fluctuating drag forces acting on the cylinder. In addition, the fluctuating drag forces vary with twice the shedding frequency. Therefore, these two POD modes (ϕ₃ and ϕ₄) are contributed by the fluctuating drag forces acting on the cylinder.

For an out-phase excitation, Figs. 13(a)–13(d) display the first four dominant POD modes of the disturbance pressure fields. Their corresponding time-varying POD amplitudes, along with frequency information, are shown in Figs. 13(e)–13(h). Although the first four dominant POD modes contribute around 99% (λ₁ + λ₂ + λ₃ + λ₄ ≈ 0.99) to the overall disturbance pressure fields, a significant energy contribution of almost 89% (λ₁ + λ₂ ≈ 0.89) to the disturbance pressure field is due to the first two POD modes, which resemble a lift dipole nature. No bias in the distribution and magnitude of pressure pulses is observed for the first two modes. Furthermore, the time-varying amplitudes of the first two POD modes (a₁ and a₂) vary with the shedding frequency of the cylinder. On the other hand, the disturbance pressure pulses in the next two POD modes (ϕ₃ and ϕ₄) propagate toward upstream and downstream directions of the flow, and their corresponding POD amplitudes (a₃ and a₄) vary with twice the shedding frequency of the cylinder subjected to out-phase excitation. Therefore, the net radiated sound fields obtained for the out-phase excitation case do not exhibit any bias in the directivity patterns. Thus, the in-phase excitation introduces a significant bias in the sound field directivity. From here onward, we have considered cases with the in-phase excitation only.

To understand and quantify the effects of excitation amplitude on the radiated sound field, excitation cases with four amplitudes A = 0.005, 0.01, 0.05, and 0.1 have been simulated. For these four excitation cases, the forcing frequency ratio is kept as fr = 1, and the phase delay has been prescribed as ϕ = 0°.

Figure 14 displays the effects of the change in excitation amplitude on the variation of the instantaneous lift and drag coefficients in (a) and (b), respectively. For comparison, the time-varying lift and the drag coefficients for the no-excitation case have also been shown in the respective frames. As shown in Fig. 14, the maximum amplitude of the lift coefficient for individual cases does not show any appreciable change, as the time-varying amplitude of the applied suction-blowing excitation is small. However, as we increase the amplitude of excitation, the fluctuating amplitude of the drag coefficient increases. The blowing operation results in the increased effective area of the cross section of the body due to the enhancement of the displacement thickness of the boundary layer, while the suction operation reduces the effective area of the cross section of the body.^8,9 As a result, the periodic suction-blowing excitation increases the maximum amplitude of time-varying fluctuations in the drag coefficient as compared to the no-excitation case. The frequency of the time-varying lift coefficient for different excitation amplitude cases remains the same as that of the no-excitation case. FFT of the time-varying drag coefficient is displayed in Fig. 14(c). The dominant frequency associated with the drag coefficient variation changes from twice the Strouhal frequency (2St₀) for the no-excitation case and low excitation amplitude (A = 0.005, 0.01) cases to the Strouhal frequency (St₀) for the higher excitation amplitude cases (A = 0.05, 0.1). One of the important observations from the FFT of C_d for the excitation cases is excitation that introduces an additional fluctuating component in the drag coefficient variation at the Strouhal frequency. This nature will introduce a drag dipole equivalent sound source due to periodic suction-blowing excitation, in addition to the drag dipole equivalent source due to periodic vortex shedding.

Figure 15 shows the time variation of fluctuating lift coefficients along with the sound source (PSST) distribution for the indicated excitation amplitude cases. Here, the distribution of sound sources is shown at the instant of maximum and minimum lift coefficients. Figures 15(a), 15(d), and 15(g) display fluctuating lift coefficients for A = 0.005, A = 0.05, and A = 0.1, respectively. Here, the filled red circular symbols display the instant for which maximum and minimum lift coefficients are noted. Corresponding sound source distribution close to the cylinder is shown in Figs. 15(b), 15(c), 15(e), 15(f), 15(h), and 15(i). Consider a case with a low excitation amplitude of A = 0.005. Here, the distribution of sound source structures in the wake region at the minimum and maximum lift coefficient instants exhibits almost opposite nature, as shown in Figs. 15(b) and 15(c). For a given shedding cycle, no appreciable bias is found in the sound source distribution near the top and bottom portions of the cylinder. Hence, the net radiated sound fields for this case must not exhibit any bias on the top and bottom sides of the cylinder. However, for higher excitation amplitude cases with A = 0.05 and A = 0.1, there is a bias in the distribution of sound sources close to the cylinder surface when compared at the minimum and maximum lift coefficients instants. Note that the bias is significantly higher for the case with A = 0.1 than other cases.

The bias in the sound field has been further explained by evaluating fluctuating sound sources at locations (x = 0, y = 0.55) and (x = 0, y = −0.55) following Eq. (6). The time variation of the fluctuating sound sources is shown in Figs. 16(a) and 16(c) for comparatively lower (A = 0.005) and higher (A = 0.1) excitation amplitude cases, respectively. Corresponding FFTs are shown in Figs. 16(b) and 16(d), respectively. With the increase in the suction-blowing excitation amplitude, the difference in the strength of fluctuating sound sources on the cylinder’s upper and lower surfaces increases. As expected, the amplitude of sound source fluctuation increased with excitation amplitude. The lower excitation amplitude (A = 0.005) case behaves similar to the no-excitation case with dominant sound sources at the Strouhal frequency. Thus, the resultant sound field has the Fig. 8 kind of directivity pattern, as shown in Fig. 17. The higher excitation amplitude (A = 0.1) case displays multi-periodic nature associated with the fluctuating sound sources, with the dominant lift and the drag dipole frequencies resulting in an eccentric-circular kind of directivity pattern for the A = 0.1 case, as shown in Fig. 17. The sound sources are dominant in the top-half portion compared to the bottom-half part of the cylinder for the A = 0.1 case; the resultant sound field intensity is more in the top-half region.

Figure 17 displays the directivity of the sound field for the different excitation amplitude cases using the root mean square values of the disturbance pressure fields evaluated at r = 70. As discussed before, one observes asymmetry in the radiated sound field due to the nature of in-phase excitation. The amplitude of sound waves is more in the top half of the domain than in the bottom-half portion. For the no-excitation case, the amplitude of the fluctuating lift coefficient is almost ten times higher than the drag coefficient, which resulted in the Fig. 8 kind of directivity pattern. However, as we increase the suction-blowing excitation amplitude, the increased fluctuating amplitude of the drag coefficient and the enhanced drag dipole contribution from the suction-blowing excitation increase the net drag dipole contribution to the sound field and bring progressive asymmetry in the radiated sound field. For the case of A = 0.1, the maximum amplitude of the drag coefficient fluctuation is comparable to that of the lift coefficient fluctuation. The contribution from the comparable lift and drag dipole equivalent sources along with the monopole equivalent of sound source associated with the periodic suction-blowing excitation results in an eccentric and stretched circular kind of directivity pattern for the A = 0.1 case, as shown in Fig. 17.

Next, we have studied flow and sound field changes due to variations in the suction-blowing excitation frequency. Here, we have considered ten different cases with the forcing frequency ratio (fr) as 0.25, 0.5, 0.75, 0.9, 1.0, 1.1, 1.25, 1.5, 1.75, and 2. We are interested in discovering how the excitation at a frequency other than the Strouhal frequency modifies the flow and sound field. Here, the excitation amplitude has been prescribed as 0.1, while the phase delay has been kept as ϕ = 0°.

Figure 18 displays the time-varying lift and drag coefficients for the different forcing frequency cases. The Strouhal frequency dominates the time variation of the lift coefficient as the excitation forcing amplitude is relatively small. Although the lift coefficient variation is not much affected by the suction-blowing forcing, time variation of the drag coefficient does display the effects of excitation, as the amplitude of fluctuating drag coefficient is considerably small compared to the lift coefficient. The topmost frame displays the lowest forcing frequency (fr = 0.25) excitation. In this case, one observes the superposition of low and high-frequency oscillations. The observed low-frequency drag coefficient oscillations for this case with a period around ΔT = 28.42 are due to the periodic suction-blowing excitation. In contrast, high-frequency oscillations occur at the frequency 2St_o due to vortex shedding in the wake. The multi-periodic nature of the drag coefficient variation is evident at all the forcing frequencies except fr = 2, where the forcing frequency and the frequency of drag coefficient variation for the no-excitation case are the same.

Figure 19 displays the variation of the mean drag coefficient and variation of the rms of amplitudes of fluctuating lift and drag coefficients in (a) and (b), respectively, for the indicated forcing frequency cases. One observes that the mean drag is not much affected over a range of forcing frequencies due to the smaller forcing amplitude. However, the drag coefficients’ fluctuations increase considerably compared to the no-excitation case (fr = 0). As the forcing amplitude for all the cases shown in Fig. 19 is the same (A = 0.1), the rms values of the fluctuating drag coefficients do not display appreciable variation for different excitation cases and are almost one order higher than the no-excitation case.

Figure 20 displays contours for the disturbance pressure (p′) field at an instant when the lift coefficient becomes maximum for the indicated forcing frequency cases. Here, we have shown ten equispaced contour levels in between ±0.001. As shown in Fig. 20, the lower frequency cases (fr = 0.25, 0.5) display disturbance pressure waves with very large wavelengths. As forcing frequency increases to fr = 2.0, wavelengths of associated sound waves decrease, and one observes more number of −ve and +ve oscillations as marked by the dashed and solid lines, respectively, in the same region of the domain. As discussed before, the sound field for the no-excitation case is contributed by the fluctuating lift and drag dipole equivalent sources. However, periodic suction-blowing excitation decreases and increases the effective body dimension, enhancing the drag fluctuations. Excitation creates an additional drag dipole due to the increment and decrement of the boundary layer displacement thickness, as explained before. The suction-blowing excitation also results in an additional monopole equivalent of the sound source. Thus, the resultant sound field is contributed by the action of the lift dipole and the drag dipole caused by the vortex shedding and the monopole equivalent of sound sources along with the drag dipole due to suction-blowing excitation. Figure 20 displays that the sound propagation is more intense toward the forward direction.

The variation of the directivity of disturbance pressure field at r = 70 is shown in Fig. 21 for the indicated forcing frequency cases by plotting rms of p′ fluctuations vs θ. The directivity for the case of fr = 1 displays the bias in the sound field toward the top-left direction due to the reasons explained for the case of in-phase excitation in Figs. 9 and 10. The combined effects of sound generated by the lift dipole equivalent sources associated with vortex shedding, the monopole, and the drag dipole equivalent sources associated with the periodic suction-blowing excitation, result in the directivity bias for the fr = 1 case. However, for the excitation cases other than fr = 1, the contribution of various sound sources evens out any bias in the sound field directivity as the vortex shedding frequency and the excitation frequency are different. Thus, the bias in the sound field directivity can be removed by the exciting sound field at a frequency other than the vortex shedding frequency.

Figure 22 displays disturbance pressure p′(r = 70, θ = 90°) variation with time in the left-hand side frames and corresponding FFTs in the right-hand side frames for the indicated forcing frequency cases. Although the time-varying lift coefficient for different forcing cases is not much altered by the forcing, as shown in Fig. 18, the disturbance pressure (p′) variation with time is significantly affected by the forcing, even though the forcing amplitude is small (A = 0.1). The p′ variation with time shown in Fig. 22 helps to organize the obtained sound field into following five categories.

1. Pre-synchronous non-beating cases (fr = 0.25, 0.5): Here, the p′ variation with time displays variation with both the forcing frequency and the Strouhal frequency. As the forcing frequency is quite small compared to the Strouhal frequency, the slow varying nature in p′ variation is essentially brought by the forcing frequency.

2. Pre-synchronous beating cases (fr = 0.75, 0.9): Here, the p′ variation with time displays variation with both the forcing frequency and the Strouhal frequency. In addition, the interaction between the forcing frequency and the Strouhal frequency results in the formation of sound beats. This is essentially due to the small difference between the forcing frequency and the Strouhal frequency. The smaller the difference between the forcing frequency and the Strouhal frequency, the greater the modulation period.

3. Synchronous case (fr = 1.0): In this case, the forcing frequency and the Strouhal frequency are the same. Thus, the p′ variation with time displays a synchronous phenomenon.

4. Post-synchronous beating cases (fr = 1.1, 1.25): Here, the p′ variation with time again displays the beating phenomenon as the forcing frequency and the Strouhal frequency are close to each other. The nature of the beats formed in the post-synchronous beating cases is different from the pre-synchronous beating cases as the forcing frequency is larger for the post-synchronous beating cases.

5. Post-synchronous non-beating cases (fr = 1.5, 1.75, 2.0): Here, the p′ variation displays the forcing and the Strouhal frequencies. However, the p′ variation with time is decoupled as the forcing and the Strouhal frequencies are significantly different.

Next, we have investigated modifications in the flow and sound field due to applying a periodic suction-blowing excitation with a phase delay. It is expected that the phase delay between the vortex shedding cycle and the suction-blowing excitation would change the nature of the resultant sound field, which gets contribution from the lift dipole equivalent sources due to vortex shedding, the monopole, and drag dipole equivalent sound sources associated with the periodic suction-blowing excitation.

We have simulated seven phase delay cases with ϕ = 0°, 30°, 60°, 90°, 120°, 150°, and 180°. The amplitude for these cases has been fixed as A = 0.1, and the frequency ratio has been kept as fr = 1. Panels (a) and (b) of Fig. 23 display the time variation of the lift and drag coefficients, respectively, for indicated values of phase delay. One observes that the amplitudes of the fluctuating lift and drag coefficients do not vary much for different excitation cases as the amplitude of excitation is the same for all the cases. However, the amplitude of fluctuating drag coefficients obtained for these phase delay cases is different from that of the no-excitation case. The Strouhal frequency for all the excitation cases with the phase delay is the same as that of the no-excitation case due to the lock-on phenomenon. The corresponding FFTs of the drag coefficients also reveal that the periodic suction-blowing excitation results in dominant variation at the Strouhal frequency, highlighting that excitation introduced an additional drag dipole equivalent sound source.

Next, sound source (PSST) distribution is evaluated using the right-hand side term of Eq. (5) for two different phase delay cases with ϕ = 0° and ϕ = 180° and is displayed in Fig. 24. The left side frames represent a phase delay case of ϕ = 0°, while the phase delay case of ϕ = 180° has been displayed in the right-hand side frames. Figures 24(a) and 24(d) display the time variation of fluctuating lift coefficients for the cases ϕ = 0° and ϕ = 180°, respectively. Here, the filled red colored circular symbols represent the instants of maximum and minimum values of lift coefficients. For the ϕ = 0° case, Figs. 24(b) and 24(c) represent the sound source distribution obtained at the instants of maximum and minimum lift coefficients. The sound source structures at these two opposite instants are more dominant on the top surface of the cylinder. As a result, the sound field distribution should be more intense toward the top region of the cylinder, as displayed later in sound field directivity patterns. On the other hand, the distribution of sound sources for the ϕ = 180° case [as shown in Figs. 24(e) and 24(f)] is more intense toward the bottom surface of the cylinder. Hence, for the ϕ = 180° case, the sound field intensity should be more dominant toward the bottom region of the cylinder, as shown later in sound field directivity patterns.

The bias in the sound source distribution patterns has further been analyzed by evaluating fluctuating sound sources at different locations (x = 0, y = 0.55) and (x = 0, y = −0.55). These two locations (x, y) = (0, 0.55) and (x, y) = (0, −0.55) are symmetrically present near the top and bottom surfaces of the cylinder, respectively. Figures 25(a) and 25(c) display the fluctuating sound sources obtained for the cases with ϕ = 0° and ϕ = 180°, respectively. In both cases, the maximum amplitudes of the sound sources at these two locations are different. This further confirms a bias in the distribution of sound sources on the top and bottom surfaces of the cylinder in any given shedding cycle. Furthermore, the corresponding frequency details are displayed in Figs. 25(b) and 25(d). The frequency spectrum of both phase-delay cases is governed by shedding frequency and its super harmonics.

Figure 26 displays the variation of disturbance pressure (p′) contours for the indicated phase delay cases that are considered at an instant when the lift coefficient for the individual case is maximum. We have shown 15 contour levels between ±0.01. The bias in the generated sound fields is visible for the phase delay cases compared to the no-excitation case. The bias in the sound field is also visible in the directivity plots shown in Fig. 27. The reasons behind the bias in the sound field directivity introduced by the excitation with the phase delay are similar to the explanation provided for the in-phase excitation case in Figs. 9 and 10 and not mentioned here to avoid repetition.

So far, we have discussed modifications in the sound field directivity due to the periodic suction-blowing excitation. Effects of the change in excitation amplitude, frequency of excitation, and phase of excitation on the sound field directivity have been discussed. Effects of changes in the periodic suction-blowing excitation conditions on the sound field directivity have been quantified by measuring sound pressure levels (SPL) at r = 70 and θ = ±90°, as shown in Fig. 28. The first and second column displays SPL for θ = 90° and θ = −90°, respectively. Figures 28(a) and 28(b) display the effects of the change in amplitude. Figures 28(c) and 28(d) display the effects of variation of the forcing frequency. Figures 28(e) and 28(f) display the effects of the change in-phase angle ϕ on the SPL. For the different excitation cases, the SPL at r = 70 and θ = ±90° is evaluated as

where p^rms and $prefrms$ are the rms values of the disturbance pressure at a particular location for a case with and without excitation, respectively. Here, the rms value of the disturbance pressure p^rms for the no-excitation case serves as a reference level while evaluating SPL. Figure 28 shows that among the various excitation cases, A = 0.05, fr = 1, and ϕ = 0° case displays a reduction in SPL by more than 12 dB for r = 70 and θ = −90° location. SPL in most of the other cases displays an increase in noise levels due to additional sound sources introduced by the periodic suction-blowing excitation.

In this work, DNS of the 2D, unsteady, laminar compressible flow was performed using accurate space-time schemes to understand the modifications in the radiated sound field due to the application of the periodic suction-blowing excitation for the flow past a square cylinder at Re = 100 and M = 0.2. The maximum amplitude of the disturbance pressure field displays the theoretical decay rate $(p′∼1/r)$ for the no-excitation case. This highlights that the chosen grid and the used discretization schemes resolve the hydrodynamic quantities accurately and also resolve the disturbance pressure fields accurately. The present work provided a detailed discussion on the modifications in the radiated aerodynamic sound field due to the in-phase and the out-phase excitation techniques. This article also discussed the effects of variation of the excitation amplitude, frequency, and the phase difference between the excitation and the vortex shedding phenomenon on the sound field directivity. Highlights of the no-excitation and periodic suction-blowing excitation in terms of important sound sources and their effects on modifying sound field directivity are summarized in Fig. 29. Some of the important conclusions from the present work are listed as follows.

1. In the no-excitation case, the aerodynamic sound field is dominated by the lift dipole equivalent sources compared to the drag dipole equivalent sources, providing a Fig. 8 kind of directivity pattern.

2. The periodic suction-blowing excitation introduces additional monopole equivalent sources and the drag dipole equivalent sources. The interaction between these additional sound sources and the lift dipole equivalent sources due to vortex shedding provides a resultant sound field.

3. The directivity of the resultant sound field strongly depends on the kind of periodic suction-blowing excitation (in-phase vs out-phase) prescribed on the cylinder surfaces. The directivity can also be altered by changing the amplitude, frequency, and phase delay associated with the prescribed excitation. It is possible to significantly alter the sound field directivity without modifying the hydrodynamic properties.

4. The phase delay between the vortex shedding phenomenon and the applied suction-blowing excitation dictates the sound field directivity. To reduce the aerodynamic sound field either in the lower or upper half portion of the domain, phase delay can be introduced either by the in-phase excitation technique or by prescribing phase delay between the suction-blowing excitation and the vortex shedding.

5. Vortex shedding happens periodically from the top and bottom sides of the cylinder. The in-phase excitation technique prescribes either suction or blowing excitation on both the surfaces of a cylinder at any instant. This introduces a bias in the generated sound field when excitation’s prescribed forcing frequency ratio is fr = 1. For other frequency ratios, the vortex shedding frequency and the forcing frequency are different, and directivity patterns are symmetric about the y = 0 line.

6. For a small amplitude of excitation (A = 0.005, 0.01), the sound field directivity displays a Fig. 8 kind of directivity pattern. It is still dominated by the lift dipole equivalent sources due to vortex shedding as for the no-excitation case. As the forcing amplitude increases (A = 0.05, 0.1), the additional monopole and drag dipole equivalent sound sources become dominant, providing an eccentric, stretched circular kind of directivity pattern.

7. Powell’s vortex sound theory¹⁸ helps to identify any bias in the sound source distribution around the cylinder, which can be further related to the observed bias in the sound field directivity.

8. Beats of sound have been observed when the forcing frequency and the vortex shedding frequency are sufficiently close. The aerodynamic sound field has been categorized into five categories depending on the excitation frequency and the vortex shedding frequency.

9. Among the various excitation cases, A = 0.05, fr = 1, and ϕ = 0° case displays a reduction in SPL by more than 12 dB for r = 70 and θ = −90° location. In most of the other cases, SPL displays an increase in noise levels due to additional sound sources introduced by the periodic suction-blowing excitation. By choosing appropriate excitation parameters, the aerodynamic sound can be altered.

The present work can be further extended in the future to find out the sound field characteristics at a much higher Reynolds number, corresponding to engineering applications. The flow field at a higher Reynolds number is three-dimensional, and the resultant sound field displays a band of frequencies rather than a single dominant frequency. Effects of a periodic suction-blowing excitation on the hydrodynamic and sound field can be studied in detail for such high Reynolds number applications.

The authors have no conflicts to disclose.

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