Spanwise Coherence Measurements of Porous Coated Cylinders in Uniform Flow

Zilun Xiang 1 , Elias J. G. Arcondoulis 2 , Reza Maryami 3 , Yu Liu 4

Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology 1088 Xueyuan Blvd, Nanshan, Shenzhen 518055, P.R. China

ABSTRACT

A porous coated cylinder (PCC) significantly reduces vortex shedding noise of a smooth cylinder placed in uniform flow. A key understanding of how the porous media relates to vortex shedding reduction is linked to the spanwise coherence length of the PCC at the near-wall region. To date, few studies estimate the PCC spanwise correlation length to be five-to-seven outer diameters, yet a more comprehensive study is needed. This paper presents an experimental investigation conducted in an anechoic wind tunnel using a set of polyurethane PCCs tested with varying porosity, thickness and at several Reynolds numbers. Spanwise coherence and correlation lengths are estimated using two hot-wire anemometry probes in the near-wall flow field, where one probe remains in place and the other is shifted along the span. The results presented here give some new visions of the spatial structure of vortex shedding caused by porous materials, noise generating mechanisms and a fitting formula of spanwise correlation length of PCCs.

1. INTRODUCTION

The flow around a stationary cylinder in uniform flow reveals key elementary turbulence and wake structures. When a cylinder is placed in a uniform flow, a periodic vortex shedding phenomenon occurs behind the cylinder called the Karman vortex street, that can lead to the generation of the predictable yet potentially troublesome Aeolian tone. In recent years, it has been found that adding a layer of porous material on the surface of the cylinder can significantly reduce this noise. Porous media can change the properties of the flow field around the blu ff body, such as surface pressure distribution, near velocity field, vortex shedding frequency and wake structure. Therefore, it is of great engineering significance to study the flow field around the cylinder and this passive flow control method.

1 11910615@mail.sustech.edu.cn

2 elias@sustech.edu.cn

3 r.maryami@sustech.edu.cn

4 liuy@sustech.edu.cn

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

Porous Coated Cylinders (PCCs) have been implemented successfully as a form of passive flow control, resulting in vortex shedding tone reduction and some broadband noise reduction in many cases [1–13]. Most studies used a randomised open-cell porous structure, such as metal foam and polyurethane, that can be quantified by the number of Pores Per Inch (PPI) it possesses. Typically for PCCs, the best tonal noise suppression is observed with porous materials that have 10 PPI to 30 PPI [1,2,8,10,14–16]. Despite the growing number of experimental studies of PCCs, there exist many shortcomings in the understanding of PCC spanwise flow behaviour. Understanding the spanwise correlation length of a PCC helps determine the suitable dimensions of a numerical simulation flow field, to ensure the development of a true 3-D flow field. Studies suggest that the spanwise correlation length of a PCC is at least 6 D (that are limited by the experimental test section dimensions [17] and computational domain size [6]). In comparison, the spanwise correlation length of a smooth bare cylinder may vary between 2 . 5 D and 3 . 5 D [18] in a similar Reynolds number range. As a result, there is little understanding of the e ff ect of PPI, porosity and porous layer thickness (relative to the bare cylinder diameter) on the spanwise development of velocity fluctuations at the near-wall region, yet it appears to be understood that the application of porous media to a bare cylinder significantly increases spanwise coherence. Conversely, randomized porous materials applied to airfoil trailing edges significantly decrease the spanwise coherence [19, 20]. Therefore, to adequately estimate the true 3-D flow field along a PCC span is a challenging topic and a small contribution to this field of research is presented in this study. Wind tunnel experiments were conducted to investigate the spanwise coherence and correlation length of PCCs using two di ff erent layer thickness and PPI values of 10, 20 and 30. For the smaller diameter PCC we are able to analyse a span length of 11 D allowing us to potentially determine the PCC spanwise correlation length explicitly, or estimate it via a Gaussian distribution curve fit. A smooth bare cylinder is also investigated for validation and reference purposes. Two hot-wire probes and a single far-field microphone are recorded simultaneously. One probe location is fixed near one end of the cylinder, the other is gradually shifted further away along the span to measure the spanwise relationship of velocity fluctuations near the cylinder outer diameter. Acoustic spectra are presented here yet it is future work to investigate the complex relationship of two flow measurement locations and far-field acoustic pressure. We provide an overview of previous PCC studies and material properties, and then explain the PCC design, followed by the experimental methodology. Results of coherence between flow measurements are presented for each cylinder and flow velocity over a range of frequencies. From these results, some preliminary estimates of spanwise correlation length and a discussion are provided.

2. CYLINDER DESIGN

The polyurethane porous materials used in this study (and previous studies [10,21]) are presented in Fig. 1. We quote the same material properties here: the 10 PPI has φ = 95%, 20 PPI φ = 90% and 30 PPI φ = 80%, where porosity is calculated via φ = 100 × (1 − V ) (%) and V is the ratio of the PCC volume to the equivalent solid volume. Note that these polyurethane material properties are very similar to those used by others [2,14,15,22]. The inner solid cylinder (referred to as Bare) has a diameter d = 20 mm. The outer diameter of the cylinder, D , is 40 mm and 60 mm using a porous layer thickness of t = 10 mm and 20 mm, respectively. The outer diameter is calculated via D = d + 2 t . It should be noted that the e ff ective outer diameter of a PCC is still in question, as to whether the Reynolds number that most accurately represents the PCC flow field should use D or d , and whether it is a function of PPI and / or porosity [21]. Nonetheless, we define the Reynolds number in this study to be associated with D , Re = uD /ν , where ν = 1 . 51 × 10 − 5 (m 2 / s) is the dynamic viscosity of air. The span of the cylinder was the same as the height of the test section, being L = 550 mm. The

Figure 1: Polyurethane porous materials used to coat the bare cylinder. Three PPIs are shown: PPI = 10, 20 and 30.

cylinder was designed to be able to easily connect to the end-plates from one side to another and workable for flush-mounted transducers (note that the results obtained by flush-mounted transducers will be investigated in a future study). This geometrical set-up provides a test range along spanwise to 11 D (where D = 40 mm) and 7.33 D (where D = 60 mm), allowing ± 1 D clearance from each end of the end-plate (to account for boundary layer e ff ects). Note that the Bare cylinder has a diameter of d = 20 mm and therefore over 14 D can be tested; yet, it is well known that the spanwise correlation of a bare cylinder rapidly diminishes beyond 3 D [18].

3. METHODOLOGY

Experiments were conducted in an open-jet anechoic wind tunnel at the Southern University of Science and Technology (SUSTech) in Shenzhen, China [23] over freestream flow speeds u = 10 m / s, 20 m / s, 30 m / s and 40 m / s. The wind tunnel has a test section area of 600 mm × 550 mm. The closed-loop tunnel generates a free stream with a turbulence intensity of less than 0.15% and the anechoic cut-o ff frequency is approximately 100 Hz. Figure 2(a) presents a schematic diagram of the experimental set-up. Two rectangular end-plates supported the cylinders that were placed x = 600 mm from the nozzle exit to avoid any interaction between the contraction and cylinder. The axis of cylinder is defined as the z -axis. The direction of the freestream velocity is the x -direction and its normal axis in the horizontal plane is defined as the y -direction. Constant-temperature hot-wire measurements were conducted to record velocity fluctuations near the cylinder outer diameter along the cylinder span. Data were collected using a 24 Bit National Instruments Data Acquisition (NI DAQ) card. Measurements were obtained at the sampling frequency of 2 15 = 32,708 Hz for 16 seconds. The pressure-time data were converted into the frequency domain by using a Fast Fourier Transformation with 50% overlap and Hanning-windowed blocks of 51 . 2 × 10 3

samples using Welch’s method [24], yielding a frequency resolution of 1 Hz. The Power Spectral Density (PSD) of velocity fluctuations, Φ uu (dB / Hz), were calculated without a reference velocity (of chief importance is the spectral content and the relationship of this spectral content between two probe locations, rather than absolute values of Φ uu ). Two Dantec single wire probes recorded data simultaneously at the same angle θ r = 60 ◦ and distance from the surface, δ = 0 . 75 D , as depicted in Fig. 2(b). One probe was fixed at z 0 (stationary probe), and the other recorded data at location z j . Once the moving probe arrived at z j , data were recorded for both probes (over 16 s), then a traverse located the moving probe further from the

Figure 2: Experimental set-up schematic diagram (a) placement of the probes relative to the end-plates and (b) angular placement of the probe and microphone relative to the outer PCC layer.

stationary probe by an additional ∆ z = 10 mm, and the process was repeated until 11 D was reached (for D = 40 mm) or 7 . 3 D was reached (for D = 60 mm). Acoustic signals were recorded simultaneously with the hot-wire anemometry measurements. A 1 / 4-inch Brüel & Kjær (B&K) free-field microphone was placed 1.5 m from the cylinder at 90 ◦ as shown in Fig. 2(a). The microphone was calibrated using a B&K TYPE 4228 piston-phone calibrator (1 kHz, 1 Pa). Using the same DAQ system, the data were recorded and processed identically to the hot-wire measurements to generate an acoustic PSD, Φ pp (dB / Hz), using a reference pressure of 20 µ Pa.

4. RESULTS

4.1. Acoustic Spectra Acoustic PSDs, Φ pp (dB / Hz), are presented for each cylinder and flow speed in Fig. 3. The data are plotted against Strouhal number, S t = f D / u , to help compare the acoustic spectral characteristics of cylinders with di ff erent diameters. By casual observation the PCCs clearly reduce the vortex shedding tone, as expected and presented by many others for porous materials with PPI 10 to 30 [1,2,8,10,14– 16]. In addition, the Bare (20 mm) Strouhal number of 0.18 to 0.20 is consistent with others in this Reynolds number regime [18]. Most importantly, these acoustic PSDs reveal that every PCC shows significant tonal noise reduction relative to the Bare (20 mm) cylinder, thereby providing validation that the PCC design is adequate for spanwise correlation analysis and relevant to other PCC studies. For each PCC there exists two tones: one lower and higher than the Bare (20 mm) Strouhal number. These tonal noise characteristics have been observed prior in separate experimental facilities [10, 21]. With increasing velocity, the Strouhal number shift of each of these tones appears to remain constant. In terms of noise reduction, the D = 40 mm PCCs possess superior performance.

(a) Endplate rN Uu ' . ; 1 Moving Z f _— probe (at z;) L in Az = 10 mm , vy increments ——_>> Z Stationary a probe (at Zo) Endplate (b) Solid (bare) cylinder Porous coating Microphone (v= 1.5 m)

Figure 3: Acoustic PSDs, Φ pp (dB / Hz), for all cylinders at each flow velocity, u = 10 m / s, 20 m / s, 30 m / s and 40 m / s.

4.2. Fluctuations in Velocity Spectra Figure 4 presents the velocity fluctuation PSDs, Φ uu (dB / Hz), recorded at the cylinder midspan. Similar to the acoustic PSDs in Fig. 3 each PCC reveals significant decreases in velocity fluctuation spectral content. Interestingly, two PCCs (PPI 10 (40 mm) and PPI 20 (40 mm)) possess the same Strouhal number as the Bare (20 mm) cylinder ( S t ≈ 0 . 2) at u = 10 m / s, yet the other PCCs reveal a Strouhal number shift to S t ≈ 0 . 18. This shift-pattern is observed for all flow speeds. This shift of the vortex shedding tone is documented by others, for both randomized porous materials (such as polyurethane) [10, 15, 17, 21] and even for structured PCCs of similar PPI and porosity [10]. Even more interestingly, the PPI 10 (40 mm) decreases in Strouhal number with increasing flow velocity while the tone diminishes in magnitude. It is future work to further understand the behaviour of this specific PCC, as there may be a potential optimal configuration within this porosity, layer thickness and Reynolds number regime. Despite the Strouhal number shift, each PCC shows similar tonal reduction relative to the Bare (20 mm) cylinder between 12 dB to 15 dB at u = 10 m / s. At higher flow speeds (and therefore

Figure 4: Velocity fluctuation PSDs, Φ uu (dB / Hz), for all cylinders at each flow velocity, u = 10 m / s, 20 m / s, 30 m / s and 40 m / s.

Reynolds numbers) each PCC again shows similar reduction yet the PPI 10 (40 mm) cylinder shows superior reduction. At u = 20 m / s, the PPI 10 (40 mm) has an additional 3 dB reduction, relative to the other PCCs (that have a 15 dB reduction from the Bare (20 mm) cylinder). The superior performance of the PPI 10 (40 mm) cylinder is further pronounced with increasing flow speed, where at u = 30 m / s and 40 m / s, a further 7 dB and 15 dB reduction is recorded relative to the other PCCs, respectively. Other studies agree that porous media with PPI 10 yield best passive flow and noise control capability as compared to higher PPI materials [1,2,10,21].

4.3. Spanwise Coherence The magnitude-squared coherence function, γ 2 0 , j , between a reference signal (denoted by subscript 0) and another signal (denoted by subscript j ) as a function of frequency, f , can be estimated as

G 0 j   2

γ 2 0 , j =

G 00 G j j (1)

where G 0 j is the cross PSD of the two signals, and G 00 and G j j are the auto spectral densities of each signals corresponding to 0 and j , respectively. Each term is calculated using Welch’s method [24], using the same criteria as Φ uu and Φ pp and with a resolution of 1 Hz. Equation (1) was used for each cylinder and calculated for every ∆ z / D step to generate contour maps of coherence, as a function of

frequency and spanwise length. These results are presented in Figs. 5 and 6 for PCCs with D = 40 mm and 60 mm, respectively, for each flow speed conducted in this study.

Figure 5: Near-wall velocity fluctuation coherence along the span, γ 2 0 , z / D , for the Bare (20 mm) cylinder and D = 40 mm PCCs at each flow velocity, u = 10 m / s, 20 m / s, 30 m / s and 40 m / s.

In Fig. 5 every cylinder possesses strong coherence along the span at S t ≈ 0 . 18. The Bare (20 mm) cylinder decays rapidly at S t = 0 . 18 within ∆ z / D = 4. This decay rate is increased with increasing velocity (and hence Reynolds number). The PCCs, however, show strong coherence relationship along the majority of the span at S t ≈ 0 . 18. For PPI 10 (40 mm) at u = 10 m / s the spanwise coherence resembles the Bare (20 mm) coherence: short and also relatively broad in frequency. However, with increasing flow speed, this coherence quickly increases along the span and then starts to diminish at u = 40 m / s. The other PCCs show strengthening of spanwise coherence from u = 10 m / s to 20 m / s yet retain their spanwise coherence pattern with further increasing flow speed. The PPI 30 PCCs show continuous increasing spanwise coherence magnitudes with increasing flow speed. From these data, it may be deduced that the spanwise coherence is indeed impacted by flow speed (Reynolds number flow regimes) and seeking a single spanwise coherence length for PCCs in general will be an oversimplification of the spanwise flow behaviour that is clearly dependent on PPI, t / d and Reynolds number.

Figure 6: Near-wall velocity fluctuation coherence along the span, γ 2 0 , z / D , for the D = 60 mm PCCs at each flow velocity, u = 10 m / s, 20 m / s, 30 m / s and 40 m / s.

The D = 60 mm PCC spanwise coherence results are presented in Fig. 6. Clearly these PCCs have a much stronger spanwise coherence relationship than the equivalent D = 40 mm PCCs. For all PPI values and flow speeds, each PCC at S t = 0 . 18 has very strong coherence magnitudes along the entire span. This further complicates the pursuit of understanding spanwise coherence relationships and shows that a Reynolds number-based dependence is insu ffi cient to determine the spanwise correlation length of the PCC, it must also be a function of t / d .

4.4. Spanwise Correlation Length The spanwise correlation length of each cylinder, Λ , can be investigated by integrating the spanwise coherence function along the span via

Z + ∞

q

γ 2 0 , z / D d z (2)

0

where γ 2 0 , z / D is estimated over the entire frequency domain. In this study, we define the vortex shedding frequency, of any cylinder, to be f 0 (Hz). We take the γ 2 0 , z / D values (as shown in Figs. 5 and 6) only at f 0 such that we have a series of points recorded along ∆ z / D corresponding to γ 2 0 , z / D | f 0 . These coherence values are plotted in Fig. 7. The spanwise correlation of a bare cylinder typically possesses an exponential decay along the span [6, 18, 25]. We observe this for each cylinder in Fig. 7. Clearly, and consistently with Figs. 5 and 6, the decay of the PCCs is considerably decreased relative to the Bare (20 mm) cylinder. The decay of the PCC curves appears to be proportional to its PPI for D = 40 mm. For D = 60 mm, both PCCs have much slower decay rates are nearly indistinguishable from each other. This presented spanwise correlation ldata is technically associated with the velocity fluctuations associated with the f 0 -frequency. Considering the spectral content of the Bare and PCCs coherence contours in Figs. 5 and 6, it can be seen that the majority of the energy is contained within a narrow band about f 0 . It is therefore assumed that using an f 0 -based calculation provides a reasonable estimation for the correlation length. It is future work to integrate the entire frequency domain to develop a more accurate estimate of the spanwise correlation length and validate this assumption. From trial-and-error, it was found that both the bare cylinder and PCC γ 2 0 , z / D | f 0 distributions are best fit using a Gaussian decay distribution, as compared to an exponential distribution used by others [6].

Figure 7: Spanwise coherence at the vortex shedding frequency, γ 2 0 , z / D | f 0 , calculated at u = 40 m / s for each cylinder in this study.

In the absence of spanwise data over significantly longer distances ( L > 20 D ) it is di ffi cult to know what curve best fits these data. The Gaussian curves, g ( z / D ), used to fit γ 2 0 , z / D | f 0 are defined as

 2 (3)

g ( z ) = e −  z C √

2

where g ( z ) | max = 1 at z = 0 and C is a fitted coe ffi cient. Example Gaussian fitted curves of the Bare (20 mm) and the PPI 10 (60 mm) cylinders are presented in Fig. 8 alongside the recorded data. The curves fit the data well in both cases presented, which represent extremes of each other: slow and rapid decay Gaussian curves. For each cylinder, its Gaussian curve, g ( z ), was numerically integrated over the z -domain in the same manner as Eq. (2) to estimate the spanwise correlation length at the vortex shedding frequency, Λ ( f 0 ), via

Λ ( f 0 ) = Z z max

z 0 g ( z ) d z (4)

where the integration bound z 0 is defined in Fig. 2 and z max = 50 D to capture the vast majority of Gaussian tail in the integration. Using Eq. (4) the correlation length of the Bare (20 mm) cylinder was calculated as Λ ( f 0 ) = 3 . 2 D . This is consistent with the large collated dataset presented by Norberg [18] where at Re = 5 . 3 × 10 4 , Λ ≈ 3 D to 3 . 5 D . This provides some validation of the approach presented here, which involves using an estimate of the spanwise correlation length at the f 0 -frequency (i.e., Λ ≈ Λ ( f 0 )). It is important to note that these estimates of Λ ( f 0 ) must be treated with caution, as the curve fitting of a Gaussian function without including its tail decay may lead to significant errors. From our preliminary analysis, we estimate that Λ ( f 0 ) = 9 D for PPI 10 (60 mm). Estimations of Λ ( f 0 ) for other flow speeds, u = 10 m / s, 20 m / s and 30 m / s and other PPI values will be future work.

Figure 8: Gaussian fitted curves applied to the Bare (20 mm) and PPI 10 (60 mm) cylinders using data presented in Fig. 7.

To date, the only other comparable results were conducted by Geyer [17] at Re = 7 . 7 × 10 4 using two-probe hot-wire anemometry and by Liu and Azarpeyvand [6] at Re = 1 . 66 × 10 5 via delayed detached eddy simulations. The numerical results of Liu and Azarpeyvand show that the spanwise coherence of a windward PCC (in a tandem pair of PCCs) possesses spanwise coherence values in excess of 0.98 over 6 D . Geyer showed using Eq. (1) that PCCs with lower porosity ( φ ≈ 99 %) possess longer spanwise coherence lengths than PCCs with φ ≈ 75% to 85%. In the case of the higher porosity PCCs, Geyer’s spanwise coherence was in excess of 7 D , which is in agreement with the results presented here.

5. CONCLUSIONS

This paper presents an investigation of the spanwise behaviour of PCCs over a range of flow speeds, porous layer thickness and material properties (PPI). Each PCC displayed significant tonal noise reduction relative to a bare cylinder of the same inner diameter, which was to be expected. Each PCC presented two tones, both of which were much lower in magnitude than the bare cylinder tone. The spectra of the velocity fluctuations revealed that the shedding tone of some PCCs shift in Strouhal number and that the PPI 10 (40 mm) cylinder shows unique shedding tone abatement. By investigating the velocity fluctuations at the near-wall of PCCs along the span using a dual-probe configuration, deeper insight into the spanwise flow behaviour and the resulting acoustic spectra was obtained. It was shown that along the span, the coherence of the velocity fluctuations is dependent on PPI, porous layer thickness and Reynolds number. PCCs with a larger diameter typically had a greater coherence extent along the span. Calculation of the spanwise correlation length, at the vortex shedding frequency, showed good agreement with the bare cylinder published data. Values in excess (or equal to) seven-diameters agree with the limited available published data. From this preliminary study, it is revealed that the PCC spanwise correlation length can vary from seven to over ten diameters, depending on PPI, porous layer thickness and Reynolds number. It is future work to investigate the spanwise correlation length of PCCs using surface pressure microphones at the inner (bare) cylinder diameter. These data will be recorded simultaneously with

the near-wall velocity fluctuations using a hot-wire anemometry probe (in the same procedure as this study) in conjunction with a far-field microphone. The transfer function between the coherence of the inner and outer diameter of a PCC in terms of pressure fluctuations could be estimated. This transfer function would provide important insight into the mechanism by which porous media suppress vortex shedding tones and the spanwise development of near-wall flow features, as a function of PPI, porous layer thickness and freestream flow velocity.

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