Aerodynamic noise from long circular and non-circular cylinders using large eddy simulations

Abstract

Flow-induced aerodynamic noise from four cylindrical shapes of infinite length at a low subcritical flow regime is studied using Large Eddy Simulation (LES) and acoustic analogy. Numerical simulations are performed for short-span (length to diameter ratio of 3) cylinders, and a sound correction method based on equivalent/spatial coherence length has been applied to estimate noise from long-span cylinders. An attempt is made to compare spatial coherence lengths of four cross-sections at the same Reynolds number (Re). The sound correction method that is well established for circular cylinders proved effective for non-circular cross-sections also. Owing to the limitation in computational capacity, a well-resolved LES is still unachievable for higher Re flows and long-span cylinders without adopting a sound correction methodology. A grid resolution based on the characteristic length and velocity scale was adopted in simulation and proved effective for computing aerodynamic and aeroacoustic characteristics. An ‘effective frequency band’ of sound pressure level-frequency curve is proposed that predicts over 99.5% of the overall sound pressure level, and features of this band for four cross-sections are presented.

Keywords

Aerodynamic noise aeroacoustics spatial coherence length large eddy simulation acoustic analogy

Introduction

Aerodynamic noise radiated from bluff bodies is one of the common noise problems in engineering, examples being landing gear of aircraft, chimneys, marine risers, submarine masts, pipelines, etc. Understanding the generation, transport mechanism, and estimation of noise is essential in devising better methods for controlling noise. For a rigid cylinder immersed in a fluid, aerodynamic noise is produced by two different phenomena. The first one is due to the aerodynamic load on the body surface, which generates pressure fluctuations, and the other one is due to turbulence. Sound from surface pressure fluctuations is mainly tonal, and sound from turbulence has a broadband frequency spectrum. Studies on aeolian tones started as early as 1878, when Strouhal did experiments to measure the frequency of the tones and defined the Strouhal number (St = ƒD/U), relating frequency of the tone (ƒ), the diameter of the cylinder (D), and velocity of mean flow (U).

Etkin et al.¹ reported that the dominant sound (noise) correlates with the lift force’s fundamental frequency. The broadband noise correlates to quadrupole noise from the turbulent wake. Lighthill² represented the sound produced in a turbulent isentropic flow as the difference between fluid flow and a quiescent medium at rest and proposed an analogy to estimate it. Ffocws Williams and Hawkings³ (FWH) modified the Lighthill analogy for conditions when a moving aerodynamic surface is introduced in the fluid flow.

Revell⁴ experimentally investigated and established quantitative relationships between far-field noise and drag coefficient for a circular cylinder at diameter based Reynolds number (Re) range of 4.5×10⁴ to 4.5×10⁵. One of the earlier numerical studies in flow-induced noise was by Hardin and Lamkin.⁵ They performed two-dimensional (2D) simulation to compute sound generated by uniform flow over a circular cylinder at a Re = 200 using Curle’s acoustic analogy.⁶ Results obtained were not accurate, but it paved the way for using computational aeroacoustics (CAA) as a useful tool for computing sound. Kato et al.⁷ proposed the concept of equivalent coherent length (L_c) to calculate sound pressure level (SPL) radiated from the whole span of the body using surface pressure fluctuations from a small portion of the body. He proposed that the spanwise surface pressure fluctuations have the same phase angle within L_c and are entirely independent beyond L_c. The study predicted far-field sound from a circular cylinder at Re = 10,000 using large eddy simulation (LES) in conjunction with Lighthill acoustic analogy. Three dimensional (3D) numerical simulations were performed for a circular cylinder of aspect ratio 2 (cylinder length to diameter ratio, L/D) to predict noise from a cylinder of aspect ratio 50, and the results favorably agreed with experiments. Another 2D simulation was performed by Cox et al.⁸ on circular cylinder using Reynolds Averaged Navier Stokes (RANS) and Lighthill acoustic analogy at Re = 90,000. They concluded that 2D simulations fail to capture key features of the turbulent flow, thus requiring correlation lengths from experiments to predict the far-field noise.

Hedges et al.⁹ computed flow around a landing gear using detached eddy simulation (DES) and unsteady RANS equation with Spalart-Allmaras one-equation model. The time averaged flow features computed by two methods were similar, DES capturing detailed information on wake flow and turbulent intensity. As wider range of unsteady scales of motion were captured by DES, the method promised better noise prediction.

Greschner et al.¹⁰ used detached eddy simulations (DES) and FWH acoustic analogy to predict sound generated from a rod-airfoil configuration at subcritical flow regime. The flow features and spectral content of the flow was predicted well with DES, but observed that selection of RANS turbulence model in the near wall region significantly influences the capturing of fluctuating velocities. By introducing explicit algebraic stress model (EASM)¹¹ in DES model, predictions improved and were comparable to LES. In DES, the switch from RANS (near wall) to LES is based on local grid spacing and hence selection of grid resolution is very important. Also, the transition from RANS to LES requires special attention as the inherent numerical dissipation in RANS equation can affect the generation of turbulent characteristics in the LES region.¹²

Boudet et al.¹³ carried out numerical simulations using LES and RANS on rod-airfoil configuration at Re 48,000. FWH acoustic analogy was used to estimate far field sound. RANS equations failed to capture broadband components of turbulence as they are modeled by averaged quantities. LES yielded favorable agreement with experiments and this study established the usage of LES and FWH acoustic analogy for far field sound estimation of complex configurations. Wang and Moin¹⁴ studied the influence of span length on the noise from the beveled trailing edge of a flat strut and concluded that the spanwise domain of the order of coherence length of the source field was the minimum needed to predict far-field noise accurately. They concluded that LES coupled with FWH acoustic analogy is a promising tool for estimating far field sound characteristics. Kim¹⁵ carried out large eddy simulations for flow around smooth cylinders at various subcritical Reynolds numbers to understand the flow features. He proposed a grid resolution based on characteristic eddy scales and obtained agreeable results for the flow features. The present work explores the possibility of employing the proposed grid resolution for far field pressure fluctuation estimation also.

Seo and Moon¹⁶ introduced corrections to sound pressure level values for all frequencies, rather than for a single frequency (aeolian tone) of Kato et al.⁷ They obtained favorable results for flow in the low subcritical regime. Broadband noise prediction was made for Re = 46,000 for a circular cylinder of L/D =30 using a simulation length of L/D = 3. This method accounted for the dependency of spanwise coherence lengths for different frequencies, and favorable results were obtained. LES with Smagorinsky subgrid stress model was used for the numerical analysis, and the acoustic field computation was done with linearized perturbed compressible equations (LPCE). Orselli et al.¹⁷ did LES for flow around a circular cylinder at Re = 90,000 with L/D = 2.5 and used the corrections proposed by both Kato et al.⁷ and Seo and Moon¹⁶ to obtain the far-field sound for a cylinder with L/D = 25.3, for which experimental data were available. The comparisons with experiments were favorable with both methods. Agrawal and Sharma¹⁸ used LES and Curle’s analogy to estimate far-field noise on a rod aerofoil configuration at Re = 48,000 and the results matched closely with experiments.

Fujita¹⁹ experimentally investigated aerodynamic noise from 2D cylinders of four cross-sections, circular, square, ‘half square-half circular’ (flow from the square side), and ‘half circular-half square’ (flow from the circular side) to understand the effect of geometry in the generation of far-field sound. The diameter D and side length S are equal. Experiments were done with a cylinder of diameter (D) 20 mm for a span length of 200 mm at Re = 18,000, keeping the microphone at an angle of 90°to the cylinder axis, 50D away. He observed that the ‘half circular-half square’ shape produces the least sound among the four shapes under the same flow conditions. Fujita’s work is taken as the reference experiment in the present study. Recently, Geyer and Sarradj²⁰ did experiments to understand the influence of porous materials on flow-induced noise of cylinders for a wide range of Re from 16,000 to 100,000. This study remains a comprehensive study on flow-induced noise in the subcritical flow regime.

Research on flow-induced noise largely depends on available experimental facilities and analytical methods. With the advent of powerful computational facilities, computational fluid dynamics (CFD) is widely being used in studying fluid flow problems. Using the correction method proposed by Kato et al.,⁷ far-field sound from long-span bodies could be estimated using numerical simulations on shorter lengths. This correction method was developed keeping circular cylinder in view, and was proven effective for circular cylinders as well as airfoil, rod-airfoil configurations. The present study, for the first time, explores the possibility of using this correction method for any non-circular cylinder cross sections by comparing the numerical estimates with available experimental results.

From previous studies^{7,13,14,16,17,18} it is clear that LES is a promising tool for estimating far-field pressure fluctuations accurately. LES coupled with FWH acoustic analogy was adopted in this work to estimate sound from long-span non-circular cylinders using simulations of shorter segments to keep computational effort manageable. Extensive numerical studies exist that bring out the grid resolution required for capturing flow characteristics around cylinder cross sections in low Re subcritical flow regime. The present study explores the possibility of adopting such grid resolution parameters for the estimation of far field acoustic pressure for both circular and non-circular cylinders.

The objective of this study is to capture the physics of sound generation from flow past a variety of cylinder cross sections using a reasonably resolved LES simulation. The flow induced noise from cylinders in low Re subcritical flow regime is mainly tonal in nature.^1,21 Present study attempts to confirm the tonal nature of sound in this flow regime by defining an effective frequency region for each cylinder cross section, which contributes 99.5% of the total sound generated.

Problem description

In the present study, the LES method was used for cross-sections of four cylinders shown in Figure 1 at a single value of Re (= 19,800). These are Circular (diameter D), Square (side D), Half Square-Half Circular (HSHC, side S = diameter D), and (d) Half Circular-Half Square (HCHS, S = D). Flow is from left to right in all cases. For all cylinders, D = S = 30 mm, and the numerically modeled cylinders have L_s/D = 3, where L_s is the length of the cylinder. Correction method based on spatial coherence length was employed to obtain the far-field sound corrections for longer span lengths (L_e), where L_e > L_s, i.e., for larger L_e/D. Numerical simulations on the circular cylinder were used for validating the flow features, grid resolution, and far-field sound correction method. Numerical simulations on the non-circular shapes were performed to understand the effectiveness of the correction method on geometries other than circular cross-sections. Experiments conducted by Fujita¹⁹ and Geyer and Sarradj²⁰ were taken as benchmarks for the validation and comparison of numerical simulation results.

Figure 1.

Cylinder cross-sectional shapes (flow is from left to right) (a) Circular (b) Square (c) Half Square - Half Circular (HSHC) and (d) Half Circular - Half Square (HCHS). S = D in (c) and (d).

A schematic of the computational domain and flow direction is shown in Figure 2. Uniform free-stream flow in the x-direction with a velocity of U = 9.56 m/s normal to the cylinder axis (z-axis) was used so that Re = UD/ν or US/ν = 19,800, where ν is the kinematic viscosity of air (= 1.45×10⁻⁵ m²/s). Sound monitoring points (microphone locations in experiments) were at 10° intervals in the x-y plane at a distance (H) of either 498 mm or 1000 mm from the cylinder axis (z-axis) to obtain the far-field sound and its directivity pattern (see Figure 2(b)).

Figure 2.

Computational domain for circular cylinder.

Computational domain, grid details, and boundary conditions

A computational domain of size 29D × 17D × 3D with D = 30 mm was chosen for LES simulations and is shown in Figure 2 for circular cylinder alone. The cylinder axis (z-axis) was placed 8.5D from the inlet, and the downstream length of the domain extended up to 20.5D from the cylinder axis. The top and bottom extents of the domain were fixed at 8.5D from the cylinder axis. Based on the literature^17,22,23 and the available computational capacity, a span length of L_s = 3D (L_s is the length of the cylinder) was chosen in simulations.

A grid resolution based on turbulent eddy characteristic scales was adopted as suggested by Kim¹⁵ and Orselli et al.¹⁷ The length (l) and velocity (u^′) scales of turbulent eddies were considered as l = 0.05D and u^′= 0.2U. Viscous wall unit normal to cylinder wall (y⁺) was maintained below 1 (i.e., y⁺ < 1) on the cylinder surface to resolve the viscous sublayer accurately. The wall unit is given by y⁺ = yu^*/ν, where u^* = √(τ_w/ρ) is the friction velocity obtained from time-averaged wall shear stress (τ_w), and ρ is the density of air (= 1.225 kg/m³). For this, the first grid point normal to the cylinder surface (y) was 9×10⁻⁵ m away and successive grid points followed a growth ratio of 1.05 up to y = 2.8D within which there were 100 grid points. This region (domain Ω₁) was generated as O-grid with approximately 2.2 million cells. The distance of the first grid point and the growth ratio was found by trial and error. In the spanwise direction (z-axis), a constant wall resolution (Δz⁺) of 82 was maintained, which corresponds to a cell length (i.e., distance between successive grid points in the z-direction) of about 1.5 mm, where Δz⁺ = Δzu*/ν. Circular cylinder circumference was discretized with 360 grid points, and the spanwise length of 3D was discretized with 60 grid points. Computational domain Ω₂ is discretized such that there are 30 and 35 (= N in Figure 2(c)) uniformly spaced grid points in the upstream, top, and bottom sections of this sub-domain. The downstream region of the cylinder is discretized with 100 grid points following a growth ratio of 1.02 such that a gradual gradation in grid is ensured (see Figure 2(d)). In Figure 2(d), the location A is on the cylinder surface, location B is at the boundary between Ω₁ and Ω₂, and location C is in the downstream end of the domain. Based on this, domain Ω₂ has approximately 1.2 million cells. The whole structured grid, therefore, has about 3.4 million cells (2.2 million in Ω₁ and 1.2 million in Ω₂) for the circular cylinder. Similar grid resolution was maintained for other cylinders also. Figure 3 shows the grids close to the cylinders.

Figure 3.

Computational grid close to the cylinder.

The nondimensional time is defined as t^* = tU/D so that t = 3.13807×10⁻³t^*. A time step of Δt = 4×10⁻⁵ s (Δt^* = 0.01274) was used, which resolves the vortex shedding cycle and the smallest eddy timescale with 420 time steps and 20 time steps, respectively. This value of Δt was chosen based on resolving the time scales alone and gives CFL (Courant-Friedrichs-Lewy) number less than one near the cylinder surface. Since semi-implicit iterative algorithm (SIMPLE) was used, numerical stability was not of concern in the choice of ‘CFL no. < 1’ criterion. Simulations were run for five flow through time (FLT), and data for 200D/U time units^15,24 was taken for processing the results, which corresponds to about 38 vortex shedding cycles.

A uniform flow velocity U with zero turbulence level was employed as the inlet boundary condition, and the outflow boundary condition was used at the outlet of the domain. Spanwise boundaries were given periodic boundary conditions, top and bottom boundaries have zero shear slip boundary condition.

The LES computations were performed on a 32 CPU cluster using Intel dual-core Xeon processors. 32 CPUs took nearly 36 s to resolve the flow for each time step, which translates to roughly 7 days of run time to complete 200D/U units (= 0.6276 s) of time.

Numerical method

In a turbulent flow, there exists a spectrum of eddies whose size ranges from the dimension of the object to the lengths of the smallest possible eddies. In LES, large scale eddies are resolved directly, and the effect of the smallest scale eddies (subgrid stress) are modeled. A general-purpose finite volume code Ansys FLUENT was used for solving the flow features as well as far-field acoustic pressure fluctuations.

The Ansys FLUENT theory guide²⁵ gives the details of filtered Navier-Stokes equations and various subgrid-scale models that are used in LES, of which constant subgrid-scale turbulent viscosity (μ_t) model with Smagorinsky constant (C_s) of 0.1 is used in the present calculations. Ffowcs-Williams and Hawkins (FWH) acoustic analogy is used in the present study and the details are described in section 15.2 of Ansys FLUENT theory manual.²⁵ In acoustic calculations, the surface f = 0, the source (emission) surface, is the stationary cylinder surface. The LES and FW-H equations are, therefore, not reproduced here.

Node-based Green-Gauss theorem was used to obtain solution gradients at cell centers, as suggested by Kim.¹¹ Semi-Implicit, iterative algorithm (SIMPLE) was used to solve for pressure and velocity in volume cells based on Rajani et al.²³ Upwind schemes of higher-order accuracy are more dissipative than central schemes of second-order accuracy and hence bounded central differencing scheme (BCD) was employed in simulations.²² Usage of the second-order accurate numerical scheme is common in engineering applications of LES, and an implicit second-order transient scheme was used for temporal discretization.²⁴ Acoustic model in Ansys FLUENT employs Ffowcs-Williams and Hawkins equations (FWH) to obtain the far-field pressure fluctuations. The speed of sound in air is assumed 340 m/s, and a reference acoustic pressure of 2×10⁻⁵ Pa is used in expressing sound pressure levels in dB. The solution of the FWH equation has surface and volume integrals, which have monopole, dipole and quadrupole acoustic sources. Only surface integrals were considered in the study, which implies that the quadrupole acoustic sources are neglected.

Numerical simulations were performed for a smaller section of the cylinder (of length L_s), and a correction method based on equivalent/spatial coherence length (L_c) was used to estimate radiated noise from the longer (of length L_e) cylinder used in experiments. Information on the phase and intensity of surface pressure fluctuations is important to obtain the sound pressure spectrum from the whole span of the cylinder. The intensity of pressure fluctuations near the surface along the span length is nearly the same for all locations, whereas the phase angles are expected to vary, which contributes to the sound source. The relation between surface pressure fluctuations and phase angle could be explained using magnitude squared coherence estimate C_xy between signals x and y, which is a function of frequency f (in Hz). When surface pressure fluctuates with a constant phase difference (highly correlated), C_xy takes a value of 1, and when surface pressures fluctuate independently (uncorrelated), C_xy takes a value of 0. It is given by

C_{x y} (f) = \frac{{| P_{x y} (f) |}^{2}}{P_{x x} (f) P_{y y} (f)}

(1)

where P_xx (ƒ) and P_yy (ƒ) are the auto-spectral densities of two signals x and y separated by a distance Δz along the span length, and P_xy (ƒ) is the cross-spectral density between them. Surface pressure fluctuations were captured along the span length of the cylinder, and C_xy of these pressure fluctuations were estimated. The coherence values obtained along the span length are expected to follow a Gaussian distribution^16,17 of the form

\exp (- Δ z^{2} / L_{g}^{2})

, where L_g is the Gaussian length that fits the graph best.

The far-field pressure fluctuation from LES data was converted to power spectral density S_pp (f), using Matlab function ‘pwelch’ and its unit is Pa²/Hz. In 200D/U units of time (≈0.6276 s sample length), approximately 15,000 data points (N) were taken and divided into two segments. S_pp (ƒ) of each segment was averaged with an overlap of 50%.^26,27 The sound pressure level (SPL) spectrum can be obtained as

S P L (f) = 10 \log_{10} \frac{S_{p p} (f)}{P_{r e f}}

(2)

where P_ref is the reference pressure for air (2×10⁻⁵ Pa or 20 μPa). In equation (2), S_pp (f) is multiplied with Δf to obtain power in Pa² and used to obtain SPL at each frequency.

Kato et al.⁷ proposed that surface pressure fluctuations are highly correlated to a specific spanwise length (Δz), and beyond which, it gradually becomes uncorrelated. For Δz = 0, C_xy = 1, and for Δz→∞, C_xy = 0. The particular span length (i.e., Δz) where C_xy = 0.5 is the equivalent (or spatial coherence) length L_c. If the length of the cylinder in an experiment is L_e and the length of the cylinder in simulation is L_s (L_s < L_e for computational economy), then the sound radiated (SPL) from the experimental span length (L_e) may be obtained using

\begin{array}{l} S P L (f) = S P L_{S} (f) + S P L_{C}; \\ S P L_{C} = {\begin{matrix} 10 \log_{10} \frac{L_{e}}{L_{s}} (f o r L_{c} \leq L_{s}) \\ 20 \log_{10} \frac{L_{c}}{L_{s}} + 10 \log_{10} \frac{L_{e}}{L_{s}} (f o r L_{s} \leq L_{c} \leq L_{e}) \\ 20 \log_{10} \frac{L_{e}}{L_{s}} (f o r L_{e} \leq L_{c}) \end{matrix} \end{array}

(3)

where SPL_S is obtained from numerical simulation with span length L_s and SPL_C is a constant correction to obtain SPL(f). The lengths L_s and L_e are shown in Figure 4.

Figure 4.

Schematic of surface pressure receivers.

The spatial coherence length L_c in equation (3) can be less than L_s, between L_s and L_e, or greater than L_e, depending on the condition that C_xy = 0.5 at L_c.

From the SPL spectrum, the overall sound pressure level (OASPL) can be estimated either by (i) integrating S_pp (f) with respect to the frequency, or (ii) by logarithmically summing up all noise amplitudes of the spectrum.^17,28 The OASPL of the sound spectrum can be calculated using either of the formula:

\begin{array}{l} O A S P L = 10 \log_{10} \sum_{i = 1}^{n} 10^{\frac{S P L_{i}}{10}} \\ o r \\ O A S P L = 10 \log_{10} \int^{​} \frac{S_{p p} (f)}{p_{r e f}^{2}} d f \end{array}

(4)

where SPL_i is the sound pressure level at discrete frequencies which are ΔSt apart, n in number, that define SPL(f) in equation (3).

Results and discussion

Circular cylinder

Aerodynamic characteristics

Numerical simulations were performed for flow past the circular cylinder (Figure 1(a)), and the time-averaged results were compared with experiments^29,30 and previous numerical simulations³¹ at similar Reynolds number. The time histories of the coefficient of drag (C_d) and coefficient of lift (C_l) from the present LES are shown in Figure 5, and their power spectral densities (ϕ) are shown in Figure 6.

Figure 5.

C_d and C_l for circular cylinder (t = 3.13807 × 10-3 t*).

Figure 6.

Power spectral densities of C_d and C_l for circular cylinder.

The statistical parameters of the fluid flow, namely, Strouhal number (St), the rms value of lift coefficient

({C^{'}}_{l})

, time-averaged drag coefficient

({\bar{C}}_{d})

and separation angle (θ_s), are compared with experiments in Table 1. Aerodynamic parameters for circular cylinder show good agreement with experiments whereas square cylinder results show deviation from experiment values. Experiments on square cylinder³² itself reported large variation of

{C^{'}}_{l}

in the range of 0.82–1.33. LES study on square cylinder³³ reported

{C^{'}}_{l}

in the range of 1.15–1.79 and DNS study on square cylinder by Trias et al.³⁴ reported

{C^{'}}_{l}

as 1.71. The

{C^{'}}_{l}

value obtained in the present study is 1.69, which is very close to previous numerical studies.

Table 1.

Comparison of aerodynamic characteristics (with reference studies) of circular and square cylinders.

Shape	Case	St	${C^{'}}_{l}$	$θ_{s}$ (Deg.)	${\bar{C}}_{d}$
Circle	Re = 20000 (Norberg³⁰)	0.194	0.47	78	1.0–1.4
	Re = 140000 (Cantwell and Coles²⁹)	0.19–0.215	—	—	1.0–1.4
	Re = 13000 (Abrahamsen et al.³¹)	0.203	0.545	87.6	1.31
	Re = 19800 LES (present)	0.19	0.6	85	1.24
Square	Re 13000 to 69000 (Liu et al.³²)	0.12–0.135	0.82–1.33	—	2.04–2.19
Square	Re = 19800 LES (present)	0.13	1.69	—	2.36

Note. LES = Large Eddy Simulation.

The time-averaged coefficient of pressure (C_p) around the circular cylinder is compared with previous studies in Figure 7. In this figure, Re = 13,100 in Abrahamsen et al.,³¹ Re = 140,000 in Cantwell and Coles,²⁹ Re = 20,000 in Norberg³⁰ and Re = 19,800 in the present LES. Mean pressure distribution is captured very well, but the minimum value of C_p is found somewhat lower than the experimental value.

Figure 7.

Time averaged Cp distribution around circular cylinder (see Figure 2).

The peak frequency of drag fluctuations was reported at double the peak frequency of lift fluctuations in experiments, and this is borne out in the present calculations also from the data in Table 2 for the circular cylinder.

Table 2.

Aerodynamic characteristics of 4 shapes using large eddy simulation.

Shape	St	peak freq. of Cl (f in Hz)	peak freq. of C_d (f in Hz)	${C^{'}}_{l}$	${\bar{C}}_{d}$
Circular	0.19	61	122	0.60	1.24
Square	0.13	41	53	1.69	2.36
HCHS	0.21	67	137	0.10	0.95
HSHC	0.13	41	80	1.54	2.85

Note. NCHS = Half Square-Half Circular.

In addition to the statistical flow parameters, capturing the flow field in the wake region of the cylinder is also essential to assess the quality of a numerical simulation. Velocity profiles in different planes in the wake region are compared with various Re flows, namely, Re = 3900 in Parnaudeau et al.,³⁵ Re = 13,100 in Abrahamsen et al.,³¹ Re = 140,000 in Cantwell and Coles²⁹ and Re = 140,000 in Breuer.³⁶ Streamwise velocity (u) profile in the (x, y) plane at y/D = 0 and profile at x/D = 1.06 in the (x, y) plane is shown in Figure 8 and Figure 9 respectively. The velocity profile in Figure 8 is explained in terms of recirculation length (L_r), which is defined as the distance between the cylinder surface (x = 0.5D, y = 0 and z = L/2) and the x location where the sign of the time-averaged streamwise velocity component (u) changes sign. From Figure 8, L_r ≈ 1.5D for Re = 3900,³⁵ L_r ≈ 0.75D for Re = 13,100³¹ and L_r ≈ 0.41D for Re = 140,000.²⁹ In general, L_r decreases with Re. In the present work, for Re = 19,800, L_r from Figure 8 gives L_r ≈ 0.76D, which is almost the same as that for Re = 13,100, a value which is close to 19,800.

Figure 8.

Time averaged stream wise velocity profile (u) in the wake region : y/D = 0 in x-y plane at z = L/2.

Figure 9.

Time averaged stream wise velocity profile (u) in the wake region : x/D = 1.06 in x-y plane at z = L/2.

Figure 9 shows the streamwise velocity component along the vertical section located 1.06D behind the cylinder. For lower Re (= 3900³⁷), L_r is higher (1.5D), and the velocity minima is higher at a given x location, whereas for higher Re (= 140,000²⁹), L_r and the velocity minima is lower at the same x location. In the present work, the value of Re (= 19,800) lies between 3900 and 140,000, and as a result, the velocity minima also lie in between. The Re = 19,800 curve is close to Re = 13,100 curve, the two Re values being close. The agreement of L_r (Figure 8) and velocity minima (Figure 9) with published results is good. Overall, the aerodynamic and near wake region results from the present LES agree well with experiments and numerical simulations reported in the literature.

Acoustic characteristics

Flow-induced sound at location H = 498 mm (see Figure 2) away at an angle of 90° to the cylinder axis is compared with experimental data²⁰ of a cylinder with L = 10D and Re = 19,800. In the present LES, receivers were kept at an interval of 10° in the x-y plane to obtain the overall sound pressure level (OASPL) as well as the directivity pattern. Far-field pressure fluctuation (p) at H = 498 mm and θ = 90° to cylinder axis is shown in Figure 10.

Figure 10.

Acoustic pressure time series at H = 498 mm, θ = 90°, U/D = 318.67.

In the present study, numerical simulations were performed for a cylinder with L/D = 3, and the correction method based on equivalent/spatial coherence length (L_c) was used to estimate radiated noise from the longer (experimental) cylinder (L/D = 10). Surface pressure fluctuations were monitored near the wall, and C_xy was estimated for various frequencies. Since periodic boundary condition is used at the span ends, pressure fluctuations were monitored up to half of the cylinder length only. C_xy of the pressure fluctuations was obtained using equation (1), and the data were fitted with a Gaussian function, as mentioned in Numerical Method. The distribution of C_xy along the span for various frequencies is shown in Figure 11 and Figure 12.

Figure 11.

In C_xy, as shown in the Y axis title of Figure 11. for circular cylinder at various frequencies. Symbols indicate large eddy simulation results and bold lines indicate data fit with Gaussian function.

Figure 12.

In C_xy at fundamental frequency for circular cylinder. Symbol indicate large eddy simulation results and boldline indicate data fit with Gaussian function.

It is observed that surface pressure fluctuations are highly correlated (i.e. C_xy > 0.5 at z/D < 1.5) at frequency of vortex shedding (St = 0.19) and higher frequencies. Far-field sound from flow around cylinders largely depends on the vortex shedding frequencies, and hence L_c corresponding to vortex shedding frequency only was considered for sound correction calculations mentioned in equation (3). For circular cylinder, at the fundamental frequency of St ∼ 0.19, L_c was obtained as 2.4D (where C_xy = 0.5), which is less than the simulated length (L_s) of 3D. Using equation (3), the correction factor for the non-simulated length (L_e = 10D) was obtained as 5.23 dB. Corrected SPL spectrum, given by SPL(f) is compared with the experiment²⁰ in Figure 13, showing good comparison.

Figure 13.

Sound pressure level spectrum at H = 498 mm for circular cylinder.

Pressure data equivalent to 38 periods of vortex shedding cycle (= 0.6276 s, with Δt = 4×10⁻⁵ s) were divided into 2 segments with an overlap of 50%, and the final SPL spectrum was obtained by averaging the power spectral density.^26,27 Table 3 shows the comparison of St and OASPL at H = 498 mm location. The SPL spectrum is captured well, and the experimental OASPL (dB) is close to the computed value. Favorable results from the circular cylinder simulations confirm the effectiveness of numerical methods and grid resolution adopted.

Table 3.

Comparison of overall sound pressure level at H = 498 mm for circular cylinder.

Method	St	Overall sound pressure level (dB)
Experiment (Geyer and Sarradj²⁰)	0.19	67.11
Large eddy simulation (present)	0.19	69.21

Overall sound pressure level directivity at various locations in the x-y plane at H = 498 mm is shown in Figure 14. The cylinder is positioned at the centre of the plot, and the flow direction is from left to right (0°–180°). Maximum OASPL was observed at 90° and 270°. LES simulations are able to capture the dipole characteristics of sound well. Quadruple noise sources have negligible effect on the overall sound pressure level and hence these noise sources (volume sources) were not considered in the calculation of OASPL.²¹

Figure 14.

Overall sound pressure level directivity for circular cylinder at H = 498 mm.

Non-circular cylinders

Aerodynamic characteristics

Experimental investigation of the characteristics of aerodynamic sound radiated from cylinders of four cross-sections shown in Figure 1 was reported by Fujita¹⁹ at Re = 18,000 for cylinders with D = S = 20 mm and L/D = 10. In the present study, numerical simulations were performed for cylinders with S = D = 30 mm and L/D or L/S = 3 (see Fig. 1(b), (c) and (d)) at Re = 19,800, which is quite close to the experimental Re. The aerodynamic characteristics of all cylinders are presented in Table 2, and the time histories of their drag and lift coefficients are shown in Figure 15–17. The time-averaged drag coefficient is maximum for HSHC, followed by square, circle, and HCHS, in that order. The rms lift coefficient is maximum for square, followed by HSHC, circle, and HCHS, in that order.

Figure 15.

C_d and C_l for square cylinder.

Figure 16.

C_d and C_l for half circular-half square cylinder.

Figure 17.

C_d and C_l for half circular-half square cylinder cylinder.

The statistical parameters of fluid flow, namely, Strouhal number (St), rms lift coefficient (C_l^′), time-averaged drag coefficient $({\bar{C}}_{d})$ for square cylinder are compared with experiments in Table 1. The LES simulations predicted somewhat higher values of ${\bar{C}}_{d}$ and C_l^′ for the square cylinder. The PSD of the lift coefficient for all four shapes are shown in Figure 18

Figure 18.

Power spectral density of lift coefficient

and those of the drag coefficient in Figure 19

Figure 19.

Power spectral density of drag coefficient.

. The ratio of the peak frequency of the drag coefficient to that of the lift coefficient is about 2 for circular (which is well known), HSHC and HCHS shapes, but is about 1.3 for the square shape (see Table 2).

Acoustic characteristics

Far-field pressure fluctuations were monitored at H = 1 m away, 90° to the cylinder axis. The spanwise surface pressure fluctuations were monitored near the cylinder surface along the span length, and C_xy was estimated. C_xy values along the span length were fitted with a Gaussian function as discussed in Numerical Method and these distributions for all four cylinders at their respective vortex shedding frequencies are shown in Figure 20.

Figure 20.

L_c = z/D at C_xy = 0.5 for 4 cross sections. Symbol indicates large eddy simulation results and bold line indicate data fit with Gaussian function.

For circular, HCHS, and HSHC cylinders, the surface pressure fluctuations were observed non-coherent within the simulated span length. For the square cylinder, surface pressure fluctuations were observed coherent for span lengths larger than the experiment length, which accounts for a larger correction factor. Estimated spatial coherence length and the corresponding sound correction factor for various cylinder cross-sections are reported in Table 4, following the same procedure as outlined in Circular Cylinder.

Table 4.

Correction to sound pressure level. (L_e = 10D, L_s = 3D).

Shape	L _c	Type	Sound pressure level_C (dB)
Circular	2.4D	L_c ≤ L_s	5.23
Square	18.4D	L_e ≤ L_c	10.46
HCHS	2.8D	L_c ≤ L_s	5.23
HSHC	2.2D	L_c ≤ L_s	5.23

Correction methodology explained in Results and Discussion (see Table 4 for correction that has to be added) was applied to all cross-sections, and the SPL spectra obtained from LES and compared with experiment¹⁵ are shown in Figures 21–24. Numerical simulations captured the acoustic characteristics quite well, the maximum sound being reported from the square cylinder and minimum sound from the HCHS cylinder. Estimated OASPL (see equation (4)) for all four cylinders are compared with experimental values and are shown in Table 5. As can be seen, the agreement is good, thus establishing the effectiveness of using the correction methodology for cylinders of non-circular shapes.

Figure 21.

Sound pressure level spectrum at H = 1 m for circular cylinder.

Figure 22.

Sound pressure level spectrum at H = 1 m for square cylinder.

Figure 23.

Sound pressure level spectrum at H = 1 m for half square-half circular cylinder.

Figure 24.

Sound pressure level spectrum at H = 1 m for half circular-half square cylinder.

Table 5.

Overall sound pressure level summary (from St = 0 to St = 1). (Microphone at H = 1 m, θ = 90°, see Figure 2).

Shape	L _c	OASPL (dB)		OASPL_L (dB)	OASPL_R (dB)	St_max^**
Shape	L _c	LES	Expt^*	OASPL_L (dB)	OASPL_R (dB)	LES	Expt^*
Circular	2.4D	59.37	59.26	55.92 (45.16%)	56.76 (54.84%)	0.19	0.21
Square	18.4D	72.62	74.30	68.19 (36.04%)	70.68 (63.96%)	0.13	0.14
HCHS	2.8D	45.51	43.68	42.39 (48.66%)	42.62 (51.34%)	0.22	0.22
HSHC	2.2D	66.29	70.97	59.96 (23.27%)	65.14 (76.73%)	0.12	0.12

Note. OASPL = Overall sound pressure level; HCHS = Half Square-Half Circular.

*: Fujita (2010); ^**ƒ (Hz) = 318.6 St.

It can be observed from Figures 21–24 that the SPL at peak frequencies matches closely with experiments and deviations are observed in the broadband region of SPL spectrum. Noise from the broadband region correlates to quadrupole sound sources. The quadrupole sound sources were neglected in the study as their contribution is considered negligible.^1,21

An attempt is made to quantify the tonal nature of sound in flow around cylinders by defining an effective frequency region for each cylinder cross section. Consider OASPL given by equation (4), which consists of summation of n discrete values of SPL(f) which are Δf (or ΔSt) apart. In other words, SPL_i = SPL(f_i) = SPL(f_i−1+Δf). Also SPL(f₁) = SPL(0) = 0. Let the peak of SPL(f) be at f_N which corresponds to St_max, i.e. St_max = f_ND/U. Let the SPL(f) curve be partitioned in two, one from i = 1 to N−1 (left curve) and the other from i = N to n (right curve). Then one can define OASPL_L and OASPL_R such that

\begin{array}{l} O A S P L = O A S P L_{L} + O A S P L_{R} \\ O A S P L_{L} = 10 \log_{10} \sum_{i = 1}^{N - 1} 10^{S P L_{i} / 10}; O A S P L_{R} = 10 \log_{10} \sum_{i = N}^{n} 10^{S P L_{i} / 10} \end{array}

(5)

From the SPL curves in Figures 21–24, it is clear that the contribution of frequencies far separated (either towards left or towards right) from the peak frequency (f_N or St_max) to OASPL_L or OASPL_R is negligibly small because of the dB scale. Thus, one can define OASPL_ΔL and OASPL_ΔR as

O A S P L_{Δ L} = 10 \log_{10} \sum_{i = n_{L}}^{N - 1} 10^{S P L_{i} / 10}; O A S P L_{Δ R} = 10 \log_{10} \sum_{i = N}^{n_{R}} 10^{S P L_{i} / 10}

(6)

where the frequencies

f_{n_{L}} and f_{n_{R}}

may be so chosen that OASPL_ΔL and OASPL_ΔR are very close (say 99% or more) to OASPL_L and OASPL_R, respectively. This will give a ‘effective’ left and right frequency bands

f_{L} = f_{N - 1} - f_{n_{L}} and f_{R} = f_{n_{R}} - f_{N}

, respectively so that the full ‘effective’ frequency band is

f_{e f f} = f_{n_{R}} - f_{n_{L}}

. The Strouhal numbers corresponding to

f_{n_{L}} and f_{n_{R}}

are St_L and St_R respectively, and the effective Strouhal number band corresponding to f_eff is St_eff. In other words, if one calculates OASPL using

O A S P L = 10 \log_{10} \sum_{i = n_{L}}^{n_{R}} 10^{S P L_{i} / 10}

(7)

it will approximate the value given by equations (4) to a desired level of accuracy (say 99.5%).

Contributions to total OASPL by OASPL_L and OASPL_R for various cylinders are shown in Table 5. For cylinders with flat frontal shapes (square and HSHC), OASPL_R contributes significantly more than OASPL_L, whereas for cylinders with circular frontal shapes (circular and HCHS) the contributions are nearly equal. The values of St_L and St_R and also St_eff, based on 99.5% criterion (i.e. OASPL_ΔL and OASPL_ΔR are 99.5% of OASPL_L and OASPL_ΔR respectively), are reported in Table 6 for all four cylinders. For HCHS and HSHC cylinders, 99.5% of the contribution to OASPL is from a larger St_eff, whereas the same for circular and square cylinders are from a smaller St_eff.

Table 6.

Aeolian tonal noise region.

Shape	St_L	St_max	St_R	Overall sound pressure level (dB)(% of actual)	(St_eff)St_R−St_L
Circular	0.177	0.19	0.223	59.15 (99.6)	0.046
Square	0.086	0.13	0.147	72.41 (99.7)	0.061
HCHS	0.142	0.22	0.248	45.31 (99.5)	0.106
HSHC	0.041	0.12	0.146	66.06 (99.6)	0.105

Note. NCHS = Half Square-Half Circular.

The distribution of OASPL_ΔL and OASPL_ΔR for circular cylinder is shown in Figure 25 and Figure 26 respectively. Using the 99.5% criterion, the values of St_L and St_R (and hence St_eff) are obtained from this figure (highlighted) and reported in Table 6. The variation for other 3 cylinders is similar and are not shown. Figures 27–30 shows the effective tonal frequency region (St_eff) for 4 cylinders.

Figure 25.

Overall sound pressure level_ΔL distribution for circular cylinder.

Figure 26.

Overall sound pressure level_ΔR distribution for circular cylinder.

Figure 27.

Aeolian tonal power band region at H = 1 m for circular cylinder.

Figure 28.

Aeolian tonal power band region at H = 1 m for square cylinder.

Figure 29.

Aeolian tonal power band region at H = 1 m for half square-half circular cylinder cylinder.

Figure 30.

Aeolian tonal power band region at H = 1 m for half circular-half square cylinder cylinder.

Total sound generated from each cylinder is generated from an effective frequency region around the fundamental frequency. The effective frequency region (St_eff) for each cylinder is shown in Table 6. OASPL corresponding to this frequency constitutes 99.5% of total OASPL and hence confirms the insignificance of volumetric sound sources in this flow regime.

Overall sound pressure level directivity for the four cylindrical shapes are shown in Figures 31–34. All cylindrical cross-sections exhibit dipole like characteristics.

Figure 31.

Overall sound pressure level (dB) directivity plot for circular cylinder at H = 1 m.

Figure 32.

Overall sound pressure level (dB) directivity plot for square cylinder at H = 1 m.

Figure 33.

Overall sound pressure level (dB) directivity plot for half square-half circular cylinder cylinder at H = 1 m.

Figure 34.

Overall sound pressure level (dB) directivity plot for half circular-half square cylinder cylinder at H = 1 m.

Conclusion

Aerodynamic noise radiated from circular and three non-circular cylinder cross-sections are investigated using LES and compared with available experimental data. A sound correction method that is established for a circular cylinder could be effectively used for predicting sound from non-circular cylinders also. A reasonably resolved LES solution which predicts flow characteristics well was found effective in capturing acoustic characteristics also. Among the four cross-sections, the maximum and minimum sound was generated from the square and HCHS cylinders, respectively. Spatial coherence length and root mean square value of lift coefficient is found maximum for square cylinder, which also generates maximum sound. For cylinders with square frontal shapes, the contribution of sound from the higher frequencies is found more, and for those with circular frontal shapes, the contribution of sound from lower and higher frequencies is nearly the same. The directivity plots of the cylinders of various cross-sections are presented, and they exhibit dipole like characteristics.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Joemon Jacob

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