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Aerodynamic sound from a square cylinder with a downstream wedge
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Aerodynamic sound from a square cylinder with a downstream wedge Siti Ruhliah Lizarose Samion a , Mohamed Sukri Mat Ali a , Aminudin Abu a , Con J. Doolan b , Ric Zong-Yang Porteous c
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Malaysia–Japan International Institute of Technology, Universiti Teknologi Malaysia, 54100 Kuala Lumpur, Malaysia b School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, 2052, Australia c School of Mechanical Engineering, The University of Adelaide, South Australia, 5005, Australia
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Article history: Received 21 September 2015 Received in revised form 8 March 2016 Accepted 10 March 2016 Available online xxxx
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Keywords: Bluff body Passive flow control Passive sound control High Reynolds number
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The effect of placing a wedge in the wake of a square cylinder (side length D) at a Reynolds number of 22,000 is numerically investigated. In particular, the effect of the wedge on aerodynamic noise is observed along with its effect on the flow field. Wedge base height (h) and its gap distance (G) downstream of the cylinder are systematically varied. Flow simulations are carried out using an unsteady RANS model employing the k–ω SST turbulence model, whereas the calculation of aerodynamic noise radiated from the flow is solved using Curle’s equation. A special correction technique is applied to consider spanwise effects on noise production and validation is provided using new aeroacoustic data for a square cylinder in cross-flow. It is found that the flow behavior can be divided into two main regimes (regime I and regime II), with a linking transition regime. For regime I, the generated sound is lower than that of the isolated square cylinder case. The thinnest wedge produced the best sound reduction (11.79 dB) when the wedge is placed at G = 2D. For regime II, the calculated sound level is higher than the case of an isolated square cylinder. This is because the sound emitted from both bodies have about the same magnitude and are in phase. For this case, the maximum increase of sound pressure is 6.24 dB, when the medium wedge is at G = 2.5D. © 2016 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Bluff bodies create flow separation over a significant part of their surfaces in their wakes. For a fixed flow condition, the aerodynamic forces are mainly influenced by the geometry of the bluff bodies [11–13]. This is due to the different flow structures, even though the vortex generation process is topologically similar. Therefore, different types of geometry of bluff bodies can create different kinds of sound [14]. However, published flow noise data are mostly focused on circular cylinders [15–17] and there are very few reported investigations of aerodynamic sound from square cylinders [25]. The study of aerodynamic sound from a square cylinder is of interest as the flow separation is fixed and gives different flow behavior than that of the circular cylinder. Bearman [18] and Lyn et al. [19] are among the few experimental studies concerning the wake and surface pressures on a rectangular cylinder. Other investigations of flow over rectangular cylinder have focused on the effect of the aspect ratio on the vortex shedding pattern. Nakaguchi et al. [20], Norberg [21] and Ohya
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E-mail address: [email protected] (S.R.L. Samion). http://dx.doi.org/10.1016/j.ast.2016.03.007 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.
[22] are among the experimental studies, while Okajima [23], Shimada and Ishihara [24] are numerical studies; each investigated the flow over a rectangular cylinder with different aspect ratios. However, these investigations are limited to the flow field and did not examine aerodynamic sound. Furthermore, almost all practical applications occur at high Reynolds number, but very few investigations of aerodynamic sound are performed at high Reynolds number. In bluff body flow problems, a Reynolds number of 22,000 is commonly taken as representative of a high Reynolds number. For example, Sohankar [25] studies the flow over a square cylinder with moderate to high Reynolds numbers ranging from 1 × 103 to 5 × 106 and classifies Re > 20,000 as a high Reynolds number. For a fixed geometry, a change in Reynolds number can affect the vortex shedding frequency and wake. Concerning aerodynamic sound for a square cylinder in cross flow, there are very limited validation data for the case of high Reynolds numbers. Fujita et al. [26] have investigated the sound generated by flow over a square cylinder when the angle of attack is altered at a fixed Re = 1.3 × 104 . Nakato et al. [27] is the sole previous investigation of the aerodynamic sound generated by flow around a rectangular cylinder at a range of high Reynolds
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Nomenclature
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D f h Lr Re St u x
diameter of square cylinder [m] frequency [Hz] height of wedge [m] length of reattachment [m] Reynolds number Strouhal number velocity [m/s] streamwise coordinate
y
vertical coordinate
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λ
sound pressure wavelength [m]
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i
streamwise direction index
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j
vertical direction index
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0
initial value according to ambient pressure
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rms
root-mean-squared value
mean
time-averaged value
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number, i.e. Re = 6.4 × 10 to 2.3 × 10 . Margnat [28] numerically investigated the flow and noise created by rectangular cylinders at incidence but at low Reynolds number. Rokugou et al. [29] investigated the relationship between force fluctuations of a square cylinder at high Reynolds number with the aerodynamic sound generation. There are also previous studies that consider the spanwise length effect on the production of aerodynamic noise. Seo and Moon [30] develop a correction that allows the prediction of noise from long-span bodies using the numerical simulation of only a small span-wise portions of the body. Casalino and Jacob [31] use an ad-hoc statistical model to account for random phase differences along the span of cylinders with turbulent wakes. Also, Doolan [32] predicts spanwise-corrected noise using a statistical treatment on the sound source itself. Passive flow control is an attractive method to control aerodynamic sound [33]. The use of a downstream object, placed in the wake of the upstream body, is one technique that can affect the aerodynamic sound level. A critical study by Leclercq and Doolan [34] on the vortex–wake interaction of two-tandem rectangular blocks finds that the unsteady force amplitude and phase differences can either reinforce or cancel noise in the far-field. Another type of geometry (for the downstream, secondary body) that has been studied is the thin flat plate, either detached or attached (as a splitter plate) to the base of the upstream body. You et al. [35] studied the effect of splitter plate length on the sound radiation for a circular cylinder at low Reynolds number (Re = 100). Ali et al. [36,37] then adopt the splitter plate for the case of square cylinder at Re = 150. Shear layer reattachment was found to significantly alter sound generation. The sound cancellation mechanism was also investigated by Ali et al. [38] at low Reynolds number (Re = 150). They found that there were two distinct regimes of sound generation. In regime I, a sound reduction of 2.3 dB is achieved when the gap distance of plate is G = 0. While in regime II, as the plate length is altered to 0.26D and placed at location where plate lift is out-of-phase with the cylinder lift (5.6D), a significant reduction of 6.3 dB was obtained. However, due to differences in the acoustical waveforms generated by the cylinder and plate, the maximum amount sound reduction is limited. These differences are due to the non-linear fluid dynamics experienced by the plate inserted in the cylinder wake. Hence, there is an opportunity to investigate new downstream body geometries that may improve the sound cancellation effect observed in previous seminal research reviewed above. There are some evidences that different downstream geometries may be effective for passive noise control. A study by Uffinger et al. [39] correlated the flow and sound fields from square cylinders with wedge or elliptical bodies placed in their wake. A square cylinder with a wedge in its wake shows a 4.5 dB lower sound pressure level when compared with a downstream elliptical body. No significant parameters such as the gap distance between bodies were investigated in this study, leaving these particular aspects unexplored. Cheng et al. [40] also studied the effect of down3
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stream body shape on the wake of a square cylinder. For the case of a wedge, the separated shear layers were able to approach each other and interact more rapidly than without the wedge. However, no investigation on the sound generation was presented. Prasath et al. [41] presented a complete explanation of the flow physics of a single wedge with different aspect ratios (AR = Length/Height) and orientations with respect to the flow. Due to the direct impact of eddies on the surface of base-facing wedge, the lift coefficients of the wedge with its base facing the flow was much higher compared with a wedge with its apex facing the flow. Also, as AR increases, a delay in the onset of vortex shedding was observed, especially for the case when the base faced the flow. If seen from the perspective of vortex shedding frequency, Iungo et al. [42] found that the vortex shedding frequency increases upon decreasing AR of the wedge. From the review, AR and gap distance may alter the flow structures in the wake, which may also affect noise generation. Therefore, it is desirable to extend previous work on passive sound control using a downstream wedge as a passive control device, so that it can contribute to more complete understanding and eventual practical solution to industrial noise problems. The current study has two objectives. First, to investigate the applicability of 2D Unsteady Reynolds Averaged Navier–Stokes (URANS) to study noise from square cylinder in high Reynolds number (i.e. Re = 22,000). Second, to investigate the use of a downstream wedge as a passive noise control device in more detail than has been previously reported upon. This paper presents a procedure to predict noise from two-dimensional bluff bodies using a URANS flow solver and an acoustic analogy based on Curle’s theory. The methodology is validated against new experimental data obtained for the case of a single square cylinder in cross flow. Then the numerical methodology is used to investigate the effect of a downstream wedge on noise production.
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2. Numerical procedure
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The aerodynamic noise calculation starts with simulating a low Mach number (Ma = 0.09) turbulent flow using an incompressible URANS method. Using this flow simulation, the sound source (in this case the time gradient of the lift force) is obtained. Then, after the source is obtained, the sound propagation is then calculated using Curle’s equation. Later, a spanwise correction is used to further improve the accuracy of the noise prediction by considering the spanwise effects of the flow field.
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2.1. Flow simulations
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The governing equations for flow are given by
∂ ui =0 ∂ xi ∂ ∂ ui 1 ∂p ∂ 2ui =− +ν − u i u j − u i u j ∂t ρ ∂ xi ∂x j∂x j ∂xj
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where subscripts i , j = 1, 2. Here x1 and x2 denote streamwise and cross-stream directions, respectively, and u 1 and u 2 are the corresponding time-averaged velocity components; u i depicts the fluctuating part of the velocity; ρ is the density of fluid and p represents the fluid pressure; u i u j is the Reynolds stress term. The k–ω SST model (Wilcox [43]) is used to provide closure to the system of equations formed by Reynolds averaging. The open source computational fluid dynamics (CFD) code OpenFOAM is used for simulating the fluid. The temporal term is discretized using second order backward scheme and the convection term is discretized using second order QUICK scheme.
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2.2. Acoustic simulations
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The sound propagation is governed by the equations from the Lighthill acoustic analogy [9]
∂2Tij ∂2 2 2 (ρ − ρ0 ) = − c ∇ (x, t ) 0 2 ∂ xi ∂ x j ∂t
T i j = ρ u i u j − τi j + δi j (( p − p 0 ) − c 02 (ρ − ρ0 ))
F i (tr ) ≈
[ P i j ]τ =tr n j dS( y )
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S
the Curle’s solution for a fixed rigid compact body is
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ρ (x, t ) − ρ0 ] = −
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xi x j
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c 02 [
4π r 2 rc 02
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Equation (3) is an inhomogenous wave equation derived from rearrangement of Navier–Stokes equations [9]. T i j is Lighthill’s stress tensor, where the first term at the right hand side of Eq. (4) (ρ u i u j ) is the Reynolds stress tensor; τi j denotes viscous stress, δi j is Kronecker delta, ρ and p are the instantaneous density and pressure, respectively. Subscript ‘0’ represents reference value of the parameter. The ambient pressure is the reference value in the current study. In the current study, considering a compact body present in the flow (i.e. the square cylinder), the free-field Green’s function is used to solve Lighthill’s equations. This method is introduced by Curle and explained in Curle’s theory [8]. As the dimension of the body is very small compared to the wavelength (ratio of wavelength to body dimension is 88.2), the sound source are assumed compact. In the case of a compact, fixed, and rigid body, emission time variation along the body can be neglected. Hence, r ≈ |x|. Therefore, taking P i j as pressure, y as the point on the rigid surface, and instantaneous force F i of fluid on the body (i.e. lift and drag) as
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xj ∂
4π r 2 c 03 ∂ t
T i j (y, t − r /c 0 ) dV( y ) V
F j (t − r /c 0 )
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Hence, the radiated sound is obtained by a quadrupolar volume integration and a dipolar surface integration. Inoue and Hatakeyama [44] showed that the magnitude of sound pressure can be roughly 2 2 estimated as p q /ρ U ∞ ∝ AMa7/2 /r 1/2 and pd /ρ U ∞ ∝ Ma5/2 /r 1/2 , where subscripts ‘q’ and ‘d’ are for quadrupole and dipole, respectively, and A is a constant. Hence, as the quadrupole sound is small compared to the dipole sound, it is neglected in this study. This leaves the following as solution to calculate the radiated sound.
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∂ p (x, t ) = − 4π ∂ x i
Fi r
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xi
4π c 0 r 2
∂ Fi ∂t
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Good agreement is found between the calculations using Curle’s solution with the DNS results as has been proved by Ali et al. [36].
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Only the small deviation between the lines in the upstream direction is due to the Doppler effect and is not taken into account in the current study as the Mach number for the current study is small (Ma = 0.09).
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3. Experimental method
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For the experimental validation, a square cross-sectioned cylinder with a diameter of 10 mm was tested in an anechoic wind tunnel at the University of Adelaide. The tunnel has internal dimensions of 1.4 m × 1.4 m × 1.6 m. The facility contains a rectangular contraction outlet with a height of 75 mm and a width of 275 mm. The wind tunnel walls are lined with foam wedges which provide an anechoic environment above 300 Hz. The schematic diagram of the experimental setup is provided in Fig. 1. The cylinder had a span of 400 mm (so that it spanned the entire width of the contraction) and was mounted 30 mm from the exit plane of the nozzle. The mounting plate was the same that was used by Moreau et al. [45]. The cylinder was mounted in such a manner so that the ends were located in a zero velocity environment. For the experiment, the free-stream velocity of the jet was set at 32 m/s. A single B&K 4190 microphone was used to measure the flowinduced noise. The microphone’s frequency range is ±1 dB between 50–20000 Hz as stated in the transducer documentation. The microphone was located 0.5 m from the cylinder axis at a 90 degree angle to the flow direction directly above the cylinder. The acoustic pressure was sampled at 21 6 Hz for 10 seconds using a DAQ with an automatic anti-aliasing filter. The microphone records were not adjusted for interference from the shear layer.
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4. Validation test case
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A single square cylinder of diameter D in cross flow is used as a validation test case, of which the numerical prediction of the emitted sound is compared with the experimental measurement. The computational domain size is 31.5D × 21D and the coordinate origin is at the center of the square cylinder (see Fig. 2). The upstream, top, and bottom boundaries are 10D away from cylinder. The boundary conditions are as shown in Fig. 2. The inlet boundaries are constant velocity boundary conditions set to the free-stream value (Dirichlet BC), with a zero pressure gradient (Neumann BC) conditions also applied. The outlet boundary has a zero velocity gradient and a fixed pressure value. A uniform structured mesh is made surrounding the cylinder until 1D from the surface of square cylinder, where the mesh is highly refined. This type of mesh is adapted from the grid convergence study of Ali et al. [46] for the case of square cylinder. Fig. 3 shows the mesh adopted in current study. A grid refinement study assessing three levels of mesh density was performed and the results are compared with similar previous studies. Table 1 shows the description of the cells with comparison to previous information from the literature. Here, y w / D refers to the unit mesh size in the buffer zone near the surface (the smallest cell size). The values of y + are also taken into account to investigate about the best grid for the simulation. y + denotes nondimensionalized distance of grid points from the cylinder wall and is defined as y + = u τ y /ν , where u τ is the friction velocity, and y is the distance from the body wall [43]. y + and y + max are values of min the smallest and the biggest y + value respectively, while the average value of y + is shown in the last column. Murakami et al. [47] and Arslan et al. [48] performed LES simulations of a rectangular cylinder at similar high Reynolds number flow and are compared in Table 1. The fine mesh has similar levels of refinement to the previously published LES studies.
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Fig. 1. Schematic diagram of the experimental setup.
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Table 1 Computational meshed used in this study, compared with those in the literature.
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Case
No. of cells
y w /D
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Coarse Medium Fine Murakami et al. [47] Arslan et al. [48]
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on the calculation using Eq. (7), the root mean square value of sound pressure, P rms for numerical simulation is 8.85 × 10−5 Pa, i.e. 12.9 dB when measured at R = 50D, θ = 90◦ above the cylinder. Sound pressure level (SPL) is calculated using Eq. (8) in which the P ref is taken as 2 × 10−5 [Pa].
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Fig. 2. Schematic diagram of computational domain and the boundary conditions applied.
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Numerical data used for calculating global, averaged results are taken when the solution has reached a statistically stable state. This was done by taking data from the non-dimensional time tU / D = 150 for at least 10 vortex shedding cycles. Table 2 shows comparison of global results obtained using the fine grid with other previous studies. Almost identical results are obtained. Thus the fine mesh density (353 × 306) is used for the results presented here. The sound calculated is compared with the experimental results obtained in the anechoic wind tunnel. The results show that the sound radiated due to the drag is small and the dominant sound source is due to the lift fluctuations, a similar finding by Ali et al. [36]. Therefore, only the sound source due to the unsteady lift force is considered in this study. The location to measure the sound is chosen to be at R = 50D, θ = 90◦ from the center of the square cylinder, the same as that of experiment. Based
SPL = 20 log(
P rms P ref
)
(8)
√ SPLc = 10 log( L c / L s ) + 10 log( π N )
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The simulation acoustic result is validated by comparing it with that of experiment’s. Following the spanwise correction technique as has been suggested by Seo and Moon [30] and Doolan [32], the acoustic power spectral density (PSD) of the current results is used for comparison. To calculate the value to be corrected to the sound pressure level (SPL) of the simulation, the following Eq. (9) is used.
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(9)
Here, L c represents the spanwise coherence length, L s denotes the simulated span length as defined as L s = L / N, in which considering a long-span body with length L divided into N subsections. Since the vortex shedding in flow over circular cylinder at Re = 22,000 is correlated over a distance of about 6.5D, the value of L c in current study is taken as 5D, following the procedure used by Doolan [32] and Casalino and Jacob [31]; here, N = 10 giving L s = 2.75D. The numerical and experimental acoustic power spectral densities are compared in Fig. 4. The simulated PSD is able to
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Fig. 3. Mesh of the computational domain.
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Table 2 Comparison of global results of the current study with the literature.
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No
Author
Method
Re
C D ,mean
C L ,rms
St
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Present (Fine) Tian et al. [49] Bosch and Rodi [55] Lyn [19] Bearman [18] Shimada and Ishihara [24] Murakami and Mochida [47] Arslan et al. [48]
k–ω SST k–ω SST k–ε LDV Exp. – k–ε 3D LES 3D LES
22,000 21,400 22,000 21,400 22,000 22,000 22,000 22,000
2.10 2.06 2.108 2. 1 2.1 2.05 2.09 2.25
1.43 1.492 1.012 – 1.2 1.43 1.60 1.43
0.126 0.138 0.146 0.13 0.13 0.141 0.132 0.127
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Fig. 5. Problem geometry for the case of a downstream wedge. The parameters that are varied are height, h and gap distance, G.
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reasonably reproduce the Aeolian tone frequency and magnitude. The broadband levels at higher frequency are not well predicted and this may be due to the URANS methodology not able to represent smaller eddies in the near wake as well as a breakdown in the assumption of acoustic compactness. As this study is concerned with the flow and noise at the Aeolian tone only, the methodology is considered sufficiently accurate to proceed with a fundamental investigation of passive flow and noise control using a downstream wedge.
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Fig. 4. Comparison of computed and experimental acoustic power spectral densities. Experimental results (black solid line) are compared with the corrected numerical calculation results (blue dotted line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5. Investigation of the effect of a downstream wedge
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Fig. 6. Root-mean-square sound pressure for various gap distances for all three cases compared with the isolated square cylinder.
h = 1D (thick wedge), h = 0.5D (medium wedge) and h = 0.25D (thin wedge) known here as Case 1, Case 2 and Case 3, respectively. The gap distance is varied 0 ≤ G ≤ 7D so that the sensitivity of the gap distance with the flow and noise generation can be investigated.
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5.1. Test case description
5.2. Summary of flow regimes and sound generation
The effect of gap distances in the downstream (G) and wedge height (h) are the important factors relating to the sound generation being investigated in this study. Fig. 5 shows the problem geometry. Three types of wedge height (h) were used in this study;
Fig. 6 shows the variations of root-mean-square sound pressure 2 (p rms /ρ U ∞ ) at different gap distances. It is found there are two types of regimes (regime I and regime II) present in the change of sound with gap distance. There is also a small transition regime
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cylinder) returns to the vicinity of that of single square cylinder.
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Fig. 7. Sound source, ∂∂Ft 2 for each test case.
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5.3. Lift
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between regimes I and II. The classification of regime is based on the flow behavior properties and flow patterns. Regime I shows a significant sound reduction compared with the isolated square cylinder. The highest reduction is achieved when the wedge is h = 0.25D at G = 2D i.e. 11.79 dB. On the other hand, in regime II, the magnitude of sound at first exceeds, but returning to the vicinity of sound magnitude of that of single square cylinder case when approaching G = 6D. There is also sound reduction obtained in regime II cases but at the gap distance G ≥ 6D. The two regimes are due to the effect of the wedge which has altered the sound generation mechanism [36]. The sound source (i.e. the time gradient of the lift, ∂∂Ft2 ) for both the square cylinder and the wedge are shown in Fig. 7. It is observed that there is an extreme difference of root-mean-square of lift gradient between that in regime I and regime II, for both the square cylinder and the wedge. The downstream flow structures are governed by the gap distance between the square cylinder and the wedge. Time-averaged streamlines shown in Fig. 8 illustrate the different regimes observed. In regime I, the saddle point is found behind the wedge. This happens when the separated shear layer from the square cylinder reattaches behind the wedge and a recirculation region is formed in the wake of the wedge. In regime I, the square cylinder and wedge experience reduced magnitude of lift acting upon them. Hence, the sound source in regime I shows reduced value leading to the reduced magnitude of rootmean-square sound pressure. On the contrary in regime II, the recirculation region forms within the gap, and shows the same pattern with the case of single square cylinder. Therefore, the sound source (gradient of lift force experienced by the square
As flow past over a bluff body, shear layers from two sides are shed alternately downstream the body. When one side sheds a shear layer, it will experience a low pressure and the opposite side will experience a high pressure. Hence, the fluctuating forces are experienced by the bluff bodies (square cylinder and wedge) are the result of the alternation of periodic pressure fluctuations from both sides of the body. This unsteady force happens at the frequency of the vortex shedding. In regime I, the lift acting on the square cylinder and wedge are sensitive to the gap distance as shown in Fig. 9. Especially in medium and thin wedge case, there is a gradual reduction in lift of square cylinder with gap distance, but the magnitude of lift acting on the wedge increases. The same pattern was found by Ali et al. [51]. This can be explained by the wake–wedge interaction. As the wedge is placed nearer to the core of growing vortex, the velocity fluctuations around the wedge intensify and consequently the surface pressure fluctuations on wedge increase. Thus, the lift acting on the wedge increases due to this wake– wedge interaction. In regime II, the lift acting on square cylinder is weakened before it slowly returns to its original value while the lift acting on the wedge at first is approximately the same magnitude as that of square cylinder but gradually decreases as the gap distance increases. As the wedge is placed further from the core of the growing vortex of the upstream body, the magnitude of the velocity fluctuations reduce. Thus, at small gaps in regime II, both the lift on the square (cylinder) and wedge decrease. However, as the wedge is placed further downstream, the presence of the wedge has a diminished effect on the lift. Therefore, the value returns to the value of C L ,rms of the isolated square. The gradual changes of the lift on the wedge relate to the roll-up of the shear layer from the square cylinder. This has more chance to dissipate when the wedge is placed further downstream. Thus the wedge is in the vicinity of a weaker vortex thus experiencing a lower unsteady aerodynamic force. Furthermore, a thinner wedge experiences lower lift than on the thicker wedge in regime II. This relates with the increasing of vortex diffusion and the loss of flow symmetry as the wedge becomes thinner. A similar pattern found in Iungo et al. [52] where the lift acting on the body is reduced when the aspect ratio (length/height) of the wedge is higher.
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Fig. 8. Streamline (time-averaged velocity) for case 3 at representative gap distances of different regimes. (a) Single square cylinder, (b) regime I, G = 0, (c) transition regime, G = 2.26D, (d) regime II, G = 2.76D.
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as the wedge is placed far from the square cylinder, the wake– wedge interaction becomes weak. Thin wedge cases show the lowest drag value followed by medium wedge case when compared to regime II. Thus it can be concluded that the thinner the wedge gets, the lower the drag acting on the wedge.
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Fig. 9. Root-mean-square lift coefficient for various gap distances for all test cases.
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5.4. Drag The two regimes can also be defined by mean drag coefficient. Generally, the magnitude of mean drag acting on the square cylinder and wedge decrease in regime I. When the wedge is placed in regime I, the wake width and the vortex formation length increase, resulting in lower drag. In regime II, the magnitude of mean drag acting on square cylinder increases and returns to the value of an isolated square cylinder. The drag acting on the wedge also increases and stays constant at a fixed value. When the wedge is placed in regime II, the vortex formation length returns to the same length with that of the isolated square cylinder, i.e. the development of the vortex in the wake of square cylinder is no longer affected by the presence of the wedge when it is placed further downstream in regime II. Thus the mean drag acting on the square cylinder returns the same as the drag acting on isolated square as can be seen in Fig. 10. Also, the reattachment of the shear layers that occurs within the gap has constrained the recirculation region. This leads the base pressure magnitude experienced by the wedge to be constant, so the drag acting on the wedge is constant. However, the fixed magnitude of drag is lower for the thinner wedge. In Ganga Prasath et al. [53] who studied the effect of aspect ratio of triangular prism, it is found that the drag coefficient decreases with the increase of aspect ratio (the triangular prism becomes longer) due to the separated flow zone shrinking. This relates to current study; when the wedge is thinner (aspect ratio increases), the drag acting on the wedge itself is lower. Generally in regime II, lift and drag acting on the square cylinder gradually return to their value the same as that of the isolated square case. Different behavior is found for the wedge. Lift of the wedge in regime II gradually decreases, while drag stay constant at fixed value when the wedge is in regime II. This is because
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5.6. Sound radiation
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Figs. 11(a), 12(a) and 13(a) show instantaneous vorticity contours at a time when the lift is maximum for G = 0 for cases 1, 2 and 3 respectively. The vortex shedding pattern from square cylinder with wedge in regime I is the same as the vortex shedding of a low Re case as has been found in Inoue et al. [44,50], Ali et al. [51] and Doolan [54]. Figs. 11(b), 12(b) and 13(b) represent instantaneous vorticity contours in the transition regime for thick, medium and thin wedge cases respectively. Transition regime flow depicts both the characteristics of regime I and regime II. While, Fig. 11(c), 12(c) and 13(c) show instantaneous vorticity contours for regime II for cases 1 to 3 respectively. Broadly, regime I is characterised by vortex formation occurring behind the wedge, regime II is characterised by vortex formation within the gap between the cylinder and wedge, while the transition region is more complicated in that there exists an unsteady flow region between the cylinder and wedge, flow reattachment on the wedge that affects vortex shedding in the wake. The Strouhal number (St = f D /U , where f is vortex shedding frequency) is presented in Fig. 14. The vortex shedding frequency is obtained from power spectrum density of lift fluctuations. Almost all cases have values of St lower than that of isolated square. This is expected as the formation of the vortex has been disturbed by the presence of the wedge.
Fig. 10. Mean drag coefficient for various gap distances for all test cases.
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5.5. Vortex shedding
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Figs. 15 to 17 show contours of sound pressure, (that is calculated using Eq. (7)) for of regime I, the transition regime, and regime II for thick, medium and thin wedge cases respectively. Sound radiates in the direction of 90◦ from the free stream, in a dipolar pattern, which is as expected as the source of noise is unsteady lift. The estimated wavelength of an isolated square cylinder is λ ≈ D /(Ma × St ) ≈ 88.2D (St = 0.0113). In regime I, thick wedge case generates a sound wave with a shorter wavelength than the isolated square cylinder, i.e. λ ≈ 75.3D (St = 0.0133) for G = 0. In the transition regime, the sound wavelength is λ ≈ 83.3D (St = 0.012) and regime II the sound wavelength is λ ≈ 88.1D (St = 0.0114) approaching that value of isolated square cylinder case. For medium thin wedge case, the placement of the wedge in regime I and the transition regime leads to an increase in the acoustic wavelength. The medium wedge in regime I gives out sound of wavelength of λ ≈ 107.1D (St = 0.0093), while in transition regime (G = 1.5D) the sound wavelength is λ ≈ 102.4D (St = 0.0098). For thin wedge case, regime I has an acoustic wavelength of λ ≈ 95.2D (St = 0.0105) and the transition regime, generates sound with a wavelength of approximately λ ≈ 107.1D (St = 0.0093). Both medium and thin wedges, for regime II, are found to radiate the wavelength that approaches the value of the isolated square cylinder. They radiate sound with a wavelength λ ≈ 83.3D (St = 0.012) for both medium and thin wedge cases. Fig. 18 shows the phase difference between the sound emitted from the square cylinder and the sound emitted from the wedge for various gap distances for all three cases. The phase lag becomes linear once the gap distance is sufficient to establish vortex shedding between the cylinder and the wedge (regime II). This linearity can be explained by the nearly constant convection velocity of the shed vortices from the upstream cylinder in regime II. In contrast, the phase difference is non-linear in regime I and the transition
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Fig. 11. Instantaneous vorticity contours for thick wedge case (case 1) taken at time when lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 12. Instantaneous vorticity contours for medium wedge case (case 2) taken at a time when the lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 13. Instantaneous vorticity contours for thin wedge case (case 3) taken at a time when the lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 14. Strouhal number for various gap distances, for all test cases.
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regime when constant vortex shedding has not been established within the gap. The pattern is similar to the study of Ali et al. [38] who investigated effect of placing a detached plate in the wake of a square cylinder. Linearity relation for thick wedge case starts at
This paper has presented a method for calculating the flow and noise from bluff bodies at high Reynolds number using twodimensional URANS flow simulations and Curle’s acoustic analogy. The method was validated using new experimental data for the test case of a single square cylinder in a cross flow with Re = 22,000 and Ma = 0.09. The validated methodology was then used to investigate the effect of a downstream, detached wedge as a form of passive flow and noise control. It was found that the flow behavior can be divided into two regimes based on the analysis done on the flow properties and flow patterns. There is also the presence of a transition regime in which the flow behaves as if it is about to change from regime I to regime II. Noise generated in regime I showed a reduction in level compared with the single square cylinder. However, noise generated in regime II case showed an increase in the sound level. The reduction on noise level is due to the reduction
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2 Fig. 16. Case 2: Contours of sound pressure (p /ρ U ∞ ). Blue-lines represent positive signs. Red-dashed-lines represent negative signs. Contour levels are from 1.0 × 10−9 to 8.0 × 10−9 with constant increment of 1.0 × 10−10 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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and wedge causing an intense vortex–wedge interaction, causing high levels of noise radiation. For the current test cases, it was found that the wedge cannot completely cancel the sound created by a square cylinder in high Reynolds number flow; however, modification of the shape and position of the downstream wedge may allow a significant reduction of sound, if the phase and strength of the noise source of the wedge can be controlled in such a manner so that it can significantly cancel the noise generated by the upstream cylinder.
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Fig. 18. Sound phase lag for various gap distance in different cases.
Conflict of interest statement
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in aerodynamic forces on the cylinder and wedge in regime I. In regime II, vortex shedding occurs in the gap between the cylinder
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Acknowledgements
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The authors gratefully acknowledge the use of the services and facilities of the HPC-UTM at the Universiti Teknologi Malaysia, FRGS PY/2015/05383 and TWAS 13-272 research grants and MJIIT fellowship award to the first author. Also, the support from the University of Adelaide for the experimental results is gratefully acknowledged.
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Aerodynamic sound from a square cylinder with a downstream wedge Siti Ruhliah Lizarose Samion a , Mohamed Sukri Mat Ali a , Aminudin Abu a , Con J. Doolan b , Ric Zong-Yang Porteous c
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Malaysia–Japan International Institute of Technology, Universiti Teknologi Malaysia, 54100 Kuala Lumpur, Malaysia b School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, 2052, Australia c School of Mechanical Engineering, The University of Adelaide, South Australia, 5005, Australia
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Article history: Received 21 September 2015 Received in revised form 8 March 2016 Accepted 10 March 2016 Available online xxxx
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Keywords: Bluff body Passive flow control Passive sound control High Reynolds number
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The effect of placing a wedge in the wake of a square cylinder (side length D) at a Reynolds number of 22,000 is numerically investigated. In particular, the effect of the wedge on aerodynamic noise is observed along with its effect on the flow field. Wedge base height (h) and its gap distance (G) downstream of the cylinder are systematically varied. Flow simulations are carried out using an unsteady RANS model employing the k–ω SST turbulence model, whereas the calculation of aerodynamic noise radiated from the flow is solved using Curle’s equation. A special correction technique is applied to consider spanwise effects on noise production and validation is provided using new aeroacoustic data for a square cylinder in cross-flow. It is found that the flow behavior can be divided into two main regimes (regime I and regime II), with a linking transition regime. For regime I, the generated sound is lower than that of the isolated square cylinder case. The thinnest wedge produced the best sound reduction (11.79 dB) when the wedge is placed at G = 2D. For regime II, the calculated sound level is higher than the case of an isolated square cylinder. This is because the sound emitted from both bodies have about the same magnitude and are in phase. For this case, the maximum increase of sound pressure is 6.24 dB, when the medium wedge is at G = 2.5D. © 2016 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Bluff bodies create flow separation over a significant part of their surfaces in their wakes. For a fixed flow condition, the aerodynamic forces are mainly influenced by the geometry of the bluff bodies [11–13]. This is due to the different flow structures, even though the vortex generation process is topologically similar. Therefore, different types of geometry of bluff bodies can create different kinds of sound [14]. However, published flow noise data are mostly focused on circular cylinders [15–17] and there are very few reported investigations of aerodynamic sound from square cylinders [25]. The study of aerodynamic sound from a square cylinder is of interest as the flow separation is fixed and gives different flow behavior than that of the circular cylinder. Bearman [18] and Lyn et al. [19] are among the few experimental studies concerning the wake and surface pressures on a rectangular cylinder. Other investigations of flow over rectangular cylinder have focused on the effect of the aspect ratio on the vortex shedding pattern. Nakaguchi et al. [20], Norberg [21] and Ohya
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E-mail address: [email protected] (S.R.L. Samion). http://dx.doi.org/10.1016/j.ast.2016.03.007 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.
[22] are among the experimental studies, while Okajima [23], Shimada and Ishihara [24] are numerical studies; each investigated the flow over a rectangular cylinder with different aspect ratios. However, these investigations are limited to the flow field and did not examine aerodynamic sound. Furthermore, almost all practical applications occur at high Reynolds number, but very few investigations of aerodynamic sound are performed at high Reynolds number. In bluff body flow problems, a Reynolds number of 22,000 is commonly taken as representative of a high Reynolds number. For example, Sohankar [25] studies the flow over a square cylinder with moderate to high Reynolds numbers ranging from 1 × 103 to 5 × 106 and classifies Re > 20,000 as a high Reynolds number. For a fixed geometry, a change in Reynolds number can affect the vortex shedding frequency and wake. Concerning aerodynamic sound for a square cylinder in cross flow, there are very limited validation data for the case of high Reynolds numbers. Fujita et al. [26] have investigated the sound generated by flow over a square cylinder when the angle of attack is altered at a fixed Re = 1.3 × 104 . Nakato et al. [27] is the sole previous investigation of the aerodynamic sound generated by flow around a rectangular cylinder at a range of high Reynolds
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Nomenclature
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D f h Lr Re St u x
diameter of square cylinder [m] frequency [Hz] height of wedge [m] length of reattachment [m] Reynolds number Strouhal number velocity [m/s] streamwise coordinate
y
vertical coordinate
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λ
sound pressure wavelength [m]
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i
streamwise direction index
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vertical direction index
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initial value according to ambient pressure
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rms
root-mean-squared value
mean
time-averaged value
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number, i.e. Re = 6.4 × 10 to 2.3 × 10 . Margnat [28] numerically investigated the flow and noise created by rectangular cylinders at incidence but at low Reynolds number. Rokugou et al. [29] investigated the relationship between force fluctuations of a square cylinder at high Reynolds number with the aerodynamic sound generation. There are also previous studies that consider the spanwise length effect on the production of aerodynamic noise. Seo and Moon [30] develop a correction that allows the prediction of noise from long-span bodies using the numerical simulation of only a small span-wise portions of the body. Casalino and Jacob [31] use an ad-hoc statistical model to account for random phase differences along the span of cylinders with turbulent wakes. Also, Doolan [32] predicts spanwise-corrected noise using a statistical treatment on the sound source itself. Passive flow control is an attractive method to control aerodynamic sound [33]. The use of a downstream object, placed in the wake of the upstream body, is one technique that can affect the aerodynamic sound level. A critical study by Leclercq and Doolan [34] on the vortex–wake interaction of two-tandem rectangular blocks finds that the unsteady force amplitude and phase differences can either reinforce or cancel noise in the far-field. Another type of geometry (for the downstream, secondary body) that has been studied is the thin flat plate, either detached or attached (as a splitter plate) to the base of the upstream body. You et al. [35] studied the effect of splitter plate length on the sound radiation for a circular cylinder at low Reynolds number (Re = 100). Ali et al. [36,37] then adopt the splitter plate for the case of square cylinder at Re = 150. Shear layer reattachment was found to significantly alter sound generation. The sound cancellation mechanism was also investigated by Ali et al. [38] at low Reynolds number (Re = 150). They found that there were two distinct regimes of sound generation. In regime I, a sound reduction of 2.3 dB is achieved when the gap distance of plate is G = 0. While in regime II, as the plate length is altered to 0.26D and placed at location where plate lift is out-of-phase with the cylinder lift (5.6D), a significant reduction of 6.3 dB was obtained. However, due to differences in the acoustical waveforms generated by the cylinder and plate, the maximum amount sound reduction is limited. These differences are due to the non-linear fluid dynamics experienced by the plate inserted in the cylinder wake. Hence, there is an opportunity to investigate new downstream body geometries that may improve the sound cancellation effect observed in previous seminal research reviewed above. There are some evidences that different downstream geometries may be effective for passive noise control. A study by Uffinger et al. [39] correlated the flow and sound fields from square cylinders with wedge or elliptical bodies placed in their wake. A square cylinder with a wedge in its wake shows a 4.5 dB lower sound pressure level when compared with a downstream elliptical body. No significant parameters such as the gap distance between bodies were investigated in this study, leaving these particular aspects unexplored. Cheng et al. [40] also studied the effect of down3
4
stream body shape on the wake of a square cylinder. For the case of a wedge, the separated shear layers were able to approach each other and interact more rapidly than without the wedge. However, no investigation on the sound generation was presented. Prasath et al. [41] presented a complete explanation of the flow physics of a single wedge with different aspect ratios (AR = Length/Height) and orientations with respect to the flow. Due to the direct impact of eddies on the surface of base-facing wedge, the lift coefficients of the wedge with its base facing the flow was much higher compared with a wedge with its apex facing the flow. Also, as AR increases, a delay in the onset of vortex shedding was observed, especially for the case when the base faced the flow. If seen from the perspective of vortex shedding frequency, Iungo et al. [42] found that the vortex shedding frequency increases upon decreasing AR of the wedge. From the review, AR and gap distance may alter the flow structures in the wake, which may also affect noise generation. Therefore, it is desirable to extend previous work on passive sound control using a downstream wedge as a passive control device, so that it can contribute to more complete understanding and eventual practical solution to industrial noise problems. The current study has two objectives. First, to investigate the applicability of 2D Unsteady Reynolds Averaged Navier–Stokes (URANS) to study noise from square cylinder in high Reynolds number (i.e. Re = 22,000). Second, to investigate the use of a downstream wedge as a passive noise control device in more detail than has been previously reported upon. This paper presents a procedure to predict noise from two-dimensional bluff bodies using a URANS flow solver and an acoustic analogy based on Curle’s theory. The methodology is validated against new experimental data obtained for the case of a single square cylinder in cross flow. Then the numerical methodology is used to investigate the effect of a downstream wedge on noise production.
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2. Numerical procedure
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The aerodynamic noise calculation starts with simulating a low Mach number (Ma = 0.09) turbulent flow using an incompressible URANS method. Using this flow simulation, the sound source (in this case the time gradient of the lift force) is obtained. Then, after the source is obtained, the sound propagation is then calculated using Curle’s equation. Later, a spanwise correction is used to further improve the accuracy of the noise prediction by considering the spanwise effects of the flow field.
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2.1. Flow simulations
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The governing equations for flow are given by
∂ ui =0 ∂ xi ∂ ∂ ui 1 ∂p ∂ 2ui =− +ν − u i u j − u i u j ∂t ρ ∂ xi ∂x j∂x j ∂xj
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(1)
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where subscripts i , j = 1, 2. Here x1 and x2 denote streamwise and cross-stream directions, respectively, and u 1 and u 2 are the corresponding time-averaged velocity components; u i depicts the fluctuating part of the velocity; ρ is the density of fluid and p represents the fluid pressure; u i u j is the Reynolds stress term. The k–ω SST model (Wilcox [43]) is used to provide closure to the system of equations formed by Reynolds averaging. The open source computational fluid dynamics (CFD) code OpenFOAM is used for simulating the fluid. The temporal term is discretized using second order backward scheme and the convection term is discretized using second order QUICK scheme.
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2.2. Acoustic simulations
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The sound propagation is governed by the equations from the Lighthill acoustic analogy [9]
∂2Tij ∂2 2 2 (ρ − ρ0 ) = − c ∇ (x, t ) 0 2 ∂ xi ∂ x j ∂t
T i j = ρ u i u j − τi j + δi j (( p − p 0 ) − c 02 (ρ − ρ0 ))
F i (tr ) ≈
[ P i j ]τ =tr n j dS( y )
(5)
S
the Curle’s solution for a fixed rigid compact body is
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ρ (x, t ) − ρ0 ] = −
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xi x j
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c 02 [
4π r 2 rc 02
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(4)
Equation (3) is an inhomogenous wave equation derived from rearrangement of Navier–Stokes equations [9]. T i j is Lighthill’s stress tensor, where the first term at the right hand side of Eq. (4) (ρ u i u j ) is the Reynolds stress tensor; τi j denotes viscous stress, δi j is Kronecker delta, ρ and p are the instantaneous density and pressure, respectively. Subscript ‘0’ represents reference value of the parameter. The ambient pressure is the reference value in the current study. In the current study, considering a compact body present in the flow (i.e. the square cylinder), the free-field Green’s function is used to solve Lighthill’s equations. This method is introduced by Curle and explained in Curle’s theory [8]. As the dimension of the body is very small compared to the wavelength (ratio of wavelength to body dimension is 88.2), the sound source are assumed compact. In the case of a compact, fixed, and rigid body, emission time variation along the body can be neglected. Hence, r ≈ |x|. Therefore, taking P i j as pressure, y as the point on the rigid surface, and instantaneous force F i of fluid on the body (i.e. lift and drag) as
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(3)
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xj ∂
4π r 2 c 03 ∂ t
T i j (y, t − r /c 0 ) dV( y ) V
F j (t − r /c 0 )
(6)
Hence, the radiated sound is obtained by a quadrupolar volume integration and a dipolar surface integration. Inoue and Hatakeyama [44] showed that the magnitude of sound pressure can be roughly 2 2 estimated as p q /ρ U ∞ ∝ AMa7/2 /r 1/2 and pd /ρ U ∞ ∝ Ma5/2 /r 1/2 , where subscripts ‘q’ and ‘d’ are for quadrupole and dipole, respectively, and A is a constant. Hence, as the quadrupole sound is small compared to the dipole sound, it is neglected in this study. This leaves the following as solution to calculate the radiated sound.
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∂ p (x, t ) = − 4π ∂ x i
Fi r
=
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xi
4π c 0 r 2
∂ Fi ∂t
(7)
Good agreement is found between the calculations using Curle’s solution with the DNS results as has been proved by Ali et al. [36].
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Only the small deviation between the lines in the upstream direction is due to the Doppler effect and is not taken into account in the current study as the Mach number for the current study is small (Ma = 0.09).
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3. Experimental method
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For the experimental validation, a square cross-sectioned cylinder with a diameter of 10 mm was tested in an anechoic wind tunnel at the University of Adelaide. The tunnel has internal dimensions of 1.4 m × 1.4 m × 1.6 m. The facility contains a rectangular contraction outlet with a height of 75 mm and a width of 275 mm. The wind tunnel walls are lined with foam wedges which provide an anechoic environment above 300 Hz. The schematic diagram of the experimental setup is provided in Fig. 1. The cylinder had a span of 400 mm (so that it spanned the entire width of the contraction) and was mounted 30 mm from the exit plane of the nozzle. The mounting plate was the same that was used by Moreau et al. [45]. The cylinder was mounted in such a manner so that the ends were located in a zero velocity environment. For the experiment, the free-stream velocity of the jet was set at 32 m/s. A single B&K 4190 microphone was used to measure the flowinduced noise. The microphone’s frequency range is ±1 dB between 50–20000 Hz as stated in the transducer documentation. The microphone was located 0.5 m from the cylinder axis at a 90 degree angle to the flow direction directly above the cylinder. The acoustic pressure was sampled at 21 6 Hz for 10 seconds using a DAQ with an automatic anti-aliasing filter. The microphone records were not adjusted for interference from the shear layer.
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4. Validation test case
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A single square cylinder of diameter D in cross flow is used as a validation test case, of which the numerical prediction of the emitted sound is compared with the experimental measurement. The computational domain size is 31.5D × 21D and the coordinate origin is at the center of the square cylinder (see Fig. 2). The upstream, top, and bottom boundaries are 10D away from cylinder. The boundary conditions are as shown in Fig. 2. The inlet boundaries are constant velocity boundary conditions set to the free-stream value (Dirichlet BC), with a zero pressure gradient (Neumann BC) conditions also applied. The outlet boundary has a zero velocity gradient and a fixed pressure value. A uniform structured mesh is made surrounding the cylinder until 1D from the surface of square cylinder, where the mesh is highly refined. This type of mesh is adapted from the grid convergence study of Ali et al. [46] for the case of square cylinder. Fig. 3 shows the mesh adopted in current study. A grid refinement study assessing three levels of mesh density was performed and the results are compared with similar previous studies. Table 1 shows the description of the cells with comparison to previous information from the literature. Here, y w / D refers to the unit mesh size in the buffer zone near the surface (the smallest cell size). The values of y + are also taken into account to investigate about the best grid for the simulation. y + denotes nondimensionalized distance of grid points from the cylinder wall and is defined as y + = u τ y /ν , where u τ is the friction velocity, and y is the distance from the body wall [43]. y + and y + max are values of min the smallest and the biggest y + value respectively, while the average value of y + is shown in the last column. Murakami et al. [47] and Arslan et al. [48] performed LES simulations of a rectangular cylinder at similar high Reynolds number flow and are compared in Table 1. The fine mesh has similar levels of refinement to the previously published LES studies.
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Fig. 1. Schematic diagram of the experimental setup.
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Table 1 Computational meshed used in this study, compared with those in the literature.
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Case
No. of cells
y w /D
y+ min
Coarse Medium Fine Murakami et al. [47] Arslan et al. [48]
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0.0500 0.0285 0.0200 0.022 0.025
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on the calculation using Eq. (7), the root mean square value of sound pressure, P rms for numerical simulation is 8.85 × 10−5 Pa, i.e. 12.9 dB when measured at R = 50D, θ = 90◦ above the cylinder. Sound pressure level (SPL) is calculated using Eq. (8) in which the P ref is taken as 2 × 10−5 [Pa].
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Fig. 2. Schematic diagram of computational domain and the boundary conditions applied.
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Numerical data used for calculating global, averaged results are taken when the solution has reached a statistically stable state. This was done by taking data from the non-dimensional time tU / D = 150 for at least 10 vortex shedding cycles. Table 2 shows comparison of global results obtained using the fine grid with other previous studies. Almost identical results are obtained. Thus the fine mesh density (353 × 306) is used for the results presented here. The sound calculated is compared with the experimental results obtained in the anechoic wind tunnel. The results show that the sound radiated due to the drag is small and the dominant sound source is due to the lift fluctuations, a similar finding by Ali et al. [36]. Therefore, only the sound source due to the unsteady lift force is considered in this study. The location to measure the sound is chosen to be at R = 50D, θ = 90◦ from the center of the square cylinder, the same as that of experiment. Based
SPL = 20 log(
P rms P ref
)
(8)
√ SPLc = 10 log( L c / L s ) + 10 log( π N )
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The simulation acoustic result is validated by comparing it with that of experiment’s. Following the spanwise correction technique as has been suggested by Seo and Moon [30] and Doolan [32], the acoustic power spectral density (PSD) of the current results is used for comparison. To calculate the value to be corrected to the sound pressure level (SPL) of the simulation, the following Eq. (9) is used.
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(9)
Here, L c represents the spanwise coherence length, L s denotes the simulated span length as defined as L s = L / N, in which considering a long-span body with length L divided into N subsections. Since the vortex shedding in flow over circular cylinder at Re = 22,000 is correlated over a distance of about 6.5D, the value of L c in current study is taken as 5D, following the procedure used by Doolan [32] and Casalino and Jacob [31]; here, N = 10 giving L s = 2.75D. The numerical and experimental acoustic power spectral densities are compared in Fig. 4. The simulated PSD is able to
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Table 2 Comparison of global results of the current study with the literature.
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No
Author
Method
Re
C D ,mean
C L ,rms
St
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Present (Fine) Tian et al. [49] Bosch and Rodi [55] Lyn [19] Bearman [18] Shimada and Ishihara [24] Murakami and Mochida [47] Arslan et al. [48]
k–ω SST k–ω SST k–ε LDV Exp. – k–ε 3D LES 3D LES
22,000 21,400 22,000 21,400 22,000 22,000 22,000 22,000
2.10 2.06 2.108 2. 1 2.1 2.05 2.09 2.25
1.43 1.492 1.012 – 1.2 1.43 1.60 1.43
0.126 0.138 0.146 0.13 0.13 0.141 0.132 0.127
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reasonably reproduce the Aeolian tone frequency and magnitude. The broadband levels at higher frequency are not well predicted and this may be due to the URANS methodology not able to represent smaller eddies in the near wake as well as a breakdown in the assumption of acoustic compactness. As this study is concerned with the flow and noise at the Aeolian tone only, the methodology is considered sufficiently accurate to proceed with a fundamental investigation of passive flow and noise control using a downstream wedge.
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Fig. 4. Comparison of computed and experimental acoustic power spectral densities. Experimental results (black solid line) are compared with the corrected numerical calculation results (blue dotted line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5. Investigation of the effect of a downstream wedge
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Fig. 6. Root-mean-square sound pressure for various gap distances for all three cases compared with the isolated square cylinder.
h = 1D (thick wedge), h = 0.5D (medium wedge) and h = 0.25D (thin wedge) known here as Case 1, Case 2 and Case 3, respectively. The gap distance is varied 0 ≤ G ≤ 7D so that the sensitivity of the gap distance with the flow and noise generation can be investigated.
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5.1. Test case description
5.2. Summary of flow regimes and sound generation
The effect of gap distances in the downstream (G) and wedge height (h) are the important factors relating to the sound generation being investigated in this study. Fig. 5 shows the problem geometry. Three types of wedge height (h) were used in this study;
Fig. 6 shows the variations of root-mean-square sound pressure 2 (p rms /ρ U ∞ ) at different gap distances. It is found there are two types of regimes (regime I and regime II) present in the change of sound with gap distance. There is also a small transition regime
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cylinder) returns to the vicinity of that of single square cylinder.
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Fig. 7. Sound source, ∂∂Ft 2 for each test case.
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between regimes I and II. The classification of regime is based on the flow behavior properties and flow patterns. Regime I shows a significant sound reduction compared with the isolated square cylinder. The highest reduction is achieved when the wedge is h = 0.25D at G = 2D i.e. 11.79 dB. On the other hand, in regime II, the magnitude of sound at first exceeds, but returning to the vicinity of sound magnitude of that of single square cylinder case when approaching G = 6D. There is also sound reduction obtained in regime II cases but at the gap distance G ≥ 6D. The two regimes are due to the effect of the wedge which has altered the sound generation mechanism [36]. The sound source (i.e. the time gradient of the lift, ∂∂Ft2 ) for both the square cylinder and the wedge are shown in Fig. 7. It is observed that there is an extreme difference of root-mean-square of lift gradient between that in regime I and regime II, for both the square cylinder and the wedge. The downstream flow structures are governed by the gap distance between the square cylinder and the wedge. Time-averaged streamlines shown in Fig. 8 illustrate the different regimes observed. In regime I, the saddle point is found behind the wedge. This happens when the separated shear layer from the square cylinder reattaches behind the wedge and a recirculation region is formed in the wake of the wedge. In regime I, the square cylinder and wedge experience reduced magnitude of lift acting upon them. Hence, the sound source in regime I shows reduced value leading to the reduced magnitude of rootmean-square sound pressure. On the contrary in regime II, the recirculation region forms within the gap, and shows the same pattern with the case of single square cylinder. Therefore, the sound source (gradient of lift force experienced by the square
As flow past over a bluff body, shear layers from two sides are shed alternately downstream the body. When one side sheds a shear layer, it will experience a low pressure and the opposite side will experience a high pressure. Hence, the fluctuating forces are experienced by the bluff bodies (square cylinder and wedge) are the result of the alternation of periodic pressure fluctuations from both sides of the body. This unsteady force happens at the frequency of the vortex shedding. In regime I, the lift acting on the square cylinder and wedge are sensitive to the gap distance as shown in Fig. 9. Especially in medium and thin wedge case, there is a gradual reduction in lift of square cylinder with gap distance, but the magnitude of lift acting on the wedge increases. The same pattern was found by Ali et al. [51]. This can be explained by the wake–wedge interaction. As the wedge is placed nearer to the core of growing vortex, the velocity fluctuations around the wedge intensify and consequently the surface pressure fluctuations on wedge increase. Thus, the lift acting on the wedge increases due to this wake– wedge interaction. In regime II, the lift acting on square cylinder is weakened before it slowly returns to its original value while the lift acting on the wedge at first is approximately the same magnitude as that of square cylinder but gradually decreases as the gap distance increases. As the wedge is placed further from the core of the growing vortex of the upstream body, the magnitude of the velocity fluctuations reduce. Thus, at small gaps in regime II, both the lift on the square (cylinder) and wedge decrease. However, as the wedge is placed further downstream, the presence of the wedge has a diminished effect on the lift. Therefore, the value returns to the value of C L ,rms of the isolated square. The gradual changes of the lift on the wedge relate to the roll-up of the shear layer from the square cylinder. This has more chance to dissipate when the wedge is placed further downstream. Thus the wedge is in the vicinity of a weaker vortex thus experiencing a lower unsteady aerodynamic force. Furthermore, a thinner wedge experiences lower lift than on the thicker wedge in regime II. This relates with the increasing of vortex diffusion and the loss of flow symmetry as the wedge becomes thinner. A similar pattern found in Iungo et al. [52] where the lift acting on the body is reduced when the aspect ratio (length/height) of the wedge is higher.
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Fig. 8. Streamline (time-averaged velocity) for case 3 at representative gap distances of different regimes. (a) Single square cylinder, (b) regime I, G = 0, (c) transition regime, G = 2.26D, (d) regime II, G = 2.76D.
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as the wedge is placed far from the square cylinder, the wake– wedge interaction becomes weak. Thin wedge cases show the lowest drag value followed by medium wedge case when compared to regime II. Thus it can be concluded that the thinner the wedge gets, the lower the drag acting on the wedge.
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Fig. 9. Root-mean-square lift coefficient for various gap distances for all test cases.
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5.4. Drag The two regimes can also be defined by mean drag coefficient. Generally, the magnitude of mean drag acting on the square cylinder and wedge decrease in regime I. When the wedge is placed in regime I, the wake width and the vortex formation length increase, resulting in lower drag. In regime II, the magnitude of mean drag acting on square cylinder increases and returns to the value of an isolated square cylinder. The drag acting on the wedge also increases and stays constant at a fixed value. When the wedge is placed in regime II, the vortex formation length returns to the same length with that of the isolated square cylinder, i.e. the development of the vortex in the wake of square cylinder is no longer affected by the presence of the wedge when it is placed further downstream in regime II. Thus the mean drag acting on the square cylinder returns the same as the drag acting on isolated square as can be seen in Fig. 10. Also, the reattachment of the shear layers that occurs within the gap has constrained the recirculation region. This leads the base pressure magnitude experienced by the wedge to be constant, so the drag acting on the wedge is constant. However, the fixed magnitude of drag is lower for the thinner wedge. In Ganga Prasath et al. [53] who studied the effect of aspect ratio of triangular prism, it is found that the drag coefficient decreases with the increase of aspect ratio (the triangular prism becomes longer) due to the separated flow zone shrinking. This relates to current study; when the wedge is thinner (aspect ratio increases), the drag acting on the wedge itself is lower. Generally in regime II, lift and drag acting on the square cylinder gradually return to their value the same as that of the isolated square case. Different behavior is found for the wedge. Lift of the wedge in regime II gradually decreases, while drag stay constant at fixed value when the wedge is in regime II. This is because
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5.6. Sound radiation
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Figs. 11(a), 12(a) and 13(a) show instantaneous vorticity contours at a time when the lift is maximum for G = 0 for cases 1, 2 and 3 respectively. The vortex shedding pattern from square cylinder with wedge in regime I is the same as the vortex shedding of a low Re case as has been found in Inoue et al. [44,50], Ali et al. [51] and Doolan [54]. Figs. 11(b), 12(b) and 13(b) represent instantaneous vorticity contours in the transition regime for thick, medium and thin wedge cases respectively. Transition regime flow depicts both the characteristics of regime I and regime II. While, Fig. 11(c), 12(c) and 13(c) show instantaneous vorticity contours for regime II for cases 1 to 3 respectively. Broadly, regime I is characterised by vortex formation occurring behind the wedge, regime II is characterised by vortex formation within the gap between the cylinder and wedge, while the transition region is more complicated in that there exists an unsteady flow region between the cylinder and wedge, flow reattachment on the wedge that affects vortex shedding in the wake. The Strouhal number (St = f D /U , where f is vortex shedding frequency) is presented in Fig. 14. The vortex shedding frequency is obtained from power spectrum density of lift fluctuations. Almost all cases have values of St lower than that of isolated square. This is expected as the formation of the vortex has been disturbed by the presence of the wedge.
Fig. 10. Mean drag coefficient for various gap distances for all test cases.
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5.5. Vortex shedding
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Figs. 15 to 17 show contours of sound pressure, (that is calculated using Eq. (7)) for of regime I, the transition regime, and regime II for thick, medium and thin wedge cases respectively. Sound radiates in the direction of 90◦ from the free stream, in a dipolar pattern, which is as expected as the source of noise is unsteady lift. The estimated wavelength of an isolated square cylinder is λ ≈ D /(Ma × St ) ≈ 88.2D (St = 0.0113). In regime I, thick wedge case generates a sound wave with a shorter wavelength than the isolated square cylinder, i.e. λ ≈ 75.3D (St = 0.0133) for G = 0. In the transition regime, the sound wavelength is λ ≈ 83.3D (St = 0.012) and regime II the sound wavelength is λ ≈ 88.1D (St = 0.0114) approaching that value of isolated square cylinder case. For medium thin wedge case, the placement of the wedge in regime I and the transition regime leads to an increase in the acoustic wavelength. The medium wedge in regime I gives out sound of wavelength of λ ≈ 107.1D (St = 0.0093), while in transition regime (G = 1.5D) the sound wavelength is λ ≈ 102.4D (St = 0.0098). For thin wedge case, regime I has an acoustic wavelength of λ ≈ 95.2D (St = 0.0105) and the transition regime, generates sound with a wavelength of approximately λ ≈ 107.1D (St = 0.0093). Both medium and thin wedges, for regime II, are found to radiate the wavelength that approaches the value of the isolated square cylinder. They radiate sound with a wavelength λ ≈ 83.3D (St = 0.012) for both medium and thin wedge cases. Fig. 18 shows the phase difference between the sound emitted from the square cylinder and the sound emitted from the wedge for various gap distances for all three cases. The phase lag becomes linear once the gap distance is sufficient to establish vortex shedding between the cylinder and the wedge (regime II). This linearity can be explained by the nearly constant convection velocity of the shed vortices from the upstream cylinder in regime II. In contrast, the phase difference is non-linear in regime I and the transition
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Fig. 11. Instantaneous vorticity contours for thick wedge case (case 1) taken at time when lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 12. Instantaneous vorticity contours for medium wedge case (case 2) taken at a time when the lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 13. Instantaneous vorticity contours for thin wedge case (case 3) taken at a time when the lift is at maximum. Red-lines represent clockwise direction with contour level ranging −10 ≤ D /U ∞ ≤ −0.3. Green lines represent anticlockwise direction with contour level ranging 0.3 ≤ D /U ∞ ≤ 10. : spanwise vorticity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 14. Strouhal number for various gap distances, for all test cases.
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G = 1D, while medium wedge case starts at G = 3.5D and thin wedge case starts at G = 2.5D.
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regime when constant vortex shedding has not been established within the gap. The pattern is similar to the study of Ali et al. [38] who investigated effect of placing a detached plate in the wake of a square cylinder. Linearity relation for thick wedge case starts at
This paper has presented a method for calculating the flow and noise from bluff bodies at high Reynolds number using twodimensional URANS flow simulations and Curle’s acoustic analogy. The method was validated using new experimental data for the test case of a single square cylinder in a cross flow with Re = 22,000 and Ma = 0.09. The validated methodology was then used to investigate the effect of a downstream, detached wedge as a form of passive flow and noise control. It was found that the flow behavior can be divided into two regimes based on the analysis done on the flow properties and flow patterns. There is also the presence of a transition regime in which the flow behaves as if it is about to change from regime I to regime II. Noise generated in regime I showed a reduction in level compared with the single square cylinder. However, noise generated in regime II case showed an increase in the sound level. The reduction on noise level is due to the reduction
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2 Fig. 16. Case 2: Contours of sound pressure (p /ρ U ∞ ). Blue-lines represent positive signs. Red-dashed-lines represent negative signs. Contour levels are from 1.0 × 10−9 to 8.0 × 10−9 with constant increment of 1.0 × 10−10 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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and wedge causing an intense vortex–wedge interaction, causing high levels of noise radiation. For the current test cases, it was found that the wedge cannot completely cancel the sound created by a square cylinder in high Reynolds number flow; however, modification of the shape and position of the downstream wedge may allow a significant reduction of sound, if the phase and strength of the noise source of the wedge can be controlled in such a manner so that it can significantly cancel the noise generated by the upstream cylinder.
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Fig. 18. Sound phase lag for various gap distance in different cases.
Conflict of interest statement
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in aerodynamic forces on the cylinder and wedge in regime I. In regime II, vortex shedding occurs in the gap between the cylinder
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Acknowledgements
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The authors gratefully acknowledge the use of the services and facilities of the HPC-UTM at the Universiti Teknologi Malaysia, FRGS PY/2015/05383 and TWAS 13-272 research grants and MJIIT fellowship award to the first author. Also, the support from the University of Adelaide for the experimental results is gratefully acknowledged.
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Uncited references
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[1] [2] [3] [4] [5] [6] [7] [10]
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