Proceedings of the Institute of Acoustics

 

 

Fluid-structure interactions and aeroacoustic coupling of airfoil with flexible membrane(s)

 

Arif Irsalan¹, The Hong Kong Polytechnic University, Hong Kong, P. R. China

Randolph C. K. Leung², The Hong Kong Polytechnic University, Hong Kong, P. R. China

Garret C. Y. Lam³, The Hong Kong Polytechnic University, Hong Kong, P. R. China

Muhammad Rehan Naseer⁴, The Hong Kong Polytechnic University, Hong Kong, P. R. China

 

ABSTRACT

 

In this paper, fluid-structure interactions of a NACA 0012 airfoil mounted with short membrane(s) and its coupling effect on airfoil aeroacoustics are presented. A time-domain direct aeroacoustic simulation coupled with structural dynamics is carried out at a low Reynolds number of 50,000 to explore the aeroacoustic-structural interactions. Two different airfoil configurations based on single and dual membranes are analyzed. The membrane displacements and their impact on the flow field are characterized in wavenumber-frequency domain to analyze the structural dynamics due to flow unsteadiness within the laminar boundary layer and the resulting acoustic waves emanating from the airfoil trailing edge. A strong correlation between the membrane displacement and downstream propagating flow is observed for both configurations whereas the correlation is considerably weakened between the membrane displacement and upstream acoustic waves which ultimately results in the airfoil self-noise reduction without affecting the airfoil aerodynamics. The extent of noise reduction for dual membranes airfoil configuration is observed to be considerably higher than the single membrane airfoil configuration which corresponds to a much lower correlation among the upstream propagating acoustic waves and membrane displacement resulting in the redistribution of upstream flow energy into different frequencies.

 

1. INTRODUCTION

 

Self-noise generation of an airfoil operating at low/moderate freestream Reynolds number (Re) is one of the most undesirable aspects associated with its operations. The mechanism of airfoil self-noise generation has been studied over the years by several researchers since the early stages of airfoil aeroacoustics and has been explored to date. Brooks et al. [1] gave a classification of the different flow physical mechanisms that can lead to noise radiation from airfoil. They categorized different airfoil self-noise mechanisms at subsonic flow conditions. The first category is turbulent boundary layer (TBL) trailing edge noise which is associated with the flow turbulence passing over the trailing edge at high Re. Secondly, at low Re, the laminar boundary layer (LBL) develops over the airfoil and the noise generation is associated with the vortex shedding. Another form of airfoil self-noise is due to vortex shedding occurring at the aft region of blunt trailing edge. Also, at high angles of attack (AoAs), the separated flow may cause stall which results in low-frequency noise radiation. Lastly, noise can be radiated due to the formation of tip vortices occurring at the tips of wings or blades.

 

The airfoil tonal noise generation phenomenon involves the interaction among different physical mechanisms involving hydrodynamics, acoustics, and even structural dynamics in some cases. Hence, the study of airfoil tonal noise require in-depth understanding of all the physical processes involved and their inter-dynamics. One of the most interesting aspects on the airfoil tonal noise generation is to identify whether the events occurring on the pressure side of the airfoil or the suction side are responsible for eventual tonal noise generation. Earlier studies mostly focused on airfoil pressure side flow separation and the resulting tonal noise generation; however, some of the later studies even noticed the role of the suction surface of the airfoil in tonal noise generation at some flow conditions. Paterson et al. [2] observed no change in tonal noise behavior when the airfoil suction surface was tripped, whereas the tonal noise was almost eliminated on tripping the pressure side of the airfoil. Hence, the tonal noise generation was mostly associated with the events occurring on the pressure surface of airfoil. However, Jones et al. [3] observed that the events on the suction surface are responsible for tonal noise generation. In their experimental study, Inasawa et al. [4] observed that the suction surface of airfoil is responsible for feedback loop at low Re up to 2.1×10⁵. The involvement of both suction and pressure side of airfoil in tonal noise generation indicates their dependence based on a number of flow conditions such as Re, freestream Mach number (M), airfoil profile, and incidence of flow. A comprehensive study on tonal noise regime for a NACA 0012 airfoil at low to moderate Re (3 × 10⁴- 2.3 × 10⁵) and effective AoA from 0^◦to 6.3^◦ was carried out by Probsting et al. [5]. Their study presents a very valuable overview of the tonal noise regime and their dependence on either suction side or pressure side or interaction of both sides of airfoil based on Re and AoA. Chong et al. [6] and Plogmann et al. [7] observed multiple tones and ladder-like structure in their experimental studies. [7] showed that the boundary layer becomes turbulent when the pressure side boundary layer was tripped and subsequently separation bubble and tonal noise was suppressed. Desquesnes et al. [8] in their numerical investigation observed the involvement of both the pressure and suction surfaces in tonal noise generation. A recent study by Sanjose et al. [10] shows a significant connection between tonal noise and T-S waves.

 

In recent years, flow control methods based on the fluid-structure interactions have been attempted and employed on the airfoil; however, most of them are aimed to enhance the aerodynamic characteristics such as airfoil stall [11–13, 15]. Hence, the rationale behind the design of flexible panel(s)/membrane in these studies is entirely incompatible/unsuitable for airfoil noise control. The utilization of flexible structures for airfoil noise control has received very little attention in the literature. Recently, Arif et al. [16] proposed a passive method for airfoil tonal noise reduction by utilizing a short elastic surface on the airfoil. Based on their comprehensive numerical analysis, an elastic surface located within the laminar separation bubble is found to be highly effective in airfoil tonal noise reduction. In this paper, we aim to further develop the understanding of the fluid-structure interactions occurring between membrane(s) mounted on the airfoil suction surface and their effect on the airfoil tonal noise reduction. Two different airfoil configurations based on single and dual membranes are analyzed and their aeroacoustic characteristics are compared with a baseline rigid airfoil.

 

2. NUMERICAL METHODOLOGY

 

2.1. Direct Aeroacoustic Simulation

 

In this study, we utilize Direct aeroacoustic simulation (DAS) due to its high accuracy in capturing flow dynamics and acoustics features. DAS solves the unsteady compressible Navier-Stokes (N-S) equations and equation of state simultaneously. Its utility in wide range of aeroacoustics research including jet flows and cavity [17, 18] makes it a suitable choice for current scope of work. The unsteady N-S equations are solved by conservation element and solution element (CE/SE) method which is a multidimensional method for solving conservation laws with high resolution [19]. CE/SE methods has been widely applied and validated to analyze complex physical problems including jet noise, airfoil self noise, shock interaction and wave propagation [16, 17, 20].

 

Taking the fluid properties of freestream with velocity _∞ and airfoil chord as reference, the two-dimensional N-S equations in dimensionless strong conservative form can be written as:

 

 

where U = [ ρ     ρu     ρv     ρE ]^T, F = [ ρu, ρu² + p − 𝜏_xx / Re_c, ρuv − 𝜏_xy / Re_c, ( ρE + p)u − ( 𝜏_xxu + 𝜏_xyv − q_x)/Re_c ]^T, G = [ ρv, ρuv − τ_xy / Re_c, ρv ² + p − τ_yy / Re_c, (ρE + p)v − (𝜏_xyu + 𝜏_yyv − q_x ) / Re_c ]^T are the flow flux conservation variables, ρ is the density of fluid, u and v are the velocities in x and y direction respectively, t is the time, normal and shear stresses 𝜏_xx = (2/3) µ (2∂u / ∂x − ∂v / ∂y), 𝜏_xy = µ (2∂u / ∂y − ∂v / ∂x), 𝜏_yy = (2/3)µ (2∂v / ∂y − ∂u / ∂x), µ is the viscosity, total energy E = p / ρ(γ − 1) + (u² + v²) / 2, pressure p = ρT / γM ², heat flux q_{x =} [ µ / (γ − 1)PrM² ] (∂T / ∂x), q_y = [ µ / (γ − 1)PrM² ] (∂T / ∂y), the specific heat ratio γ = 1.4, the reference Mach number M =  /  where ˆu₀ is the reference velocity,   is the speed of sound, the specific gas constant for air = 287.058J/(kg · K), the reference Reynolds number Re =  , and Prandtl number Pr =  = 0.71. All the dimensional quantities are indicated with caret.

 

2.2. Structural Solver and Coupling

 

The present work involves the complex fluid-structure interactions between the membrane and unsteady flow passing over the airfoil. The dynamics of membrane continuously evolves due to fluid loading and as a result, the surrounding fluid is also affected. The nonlinear dynamics of the membrane is solved by 1-D plate equation [21] written as:

 

 

where w is the membrane displacement,  is the bending stiffness,  is the Young′s Modulus, ν is the Poisson′s ratio,  is the external tensile stress in tangential direction,  is the internal tensile stress in the tangential direction, L isthe length of the membrane, h is the thickness,  is the structural damping coefficient,  is the stiffness of the foundation support and  is the net pressure exerted on the membrane. A very thin membrane is considered for the present study, therefore, C and K can be neglected [22]. The membrane dynamic equation (Eq. 2) is solved by the standard finite difference method and the nonlinear coupling between flow fluctuation and structural dynamics is resolved with a monolithic scheme [22].

 

Table 1: Mesh parameters

 

 

2.3. Numerical Setup

 

For the present study, NACA0012 is selected as a baseline airfoil due to its practical application in a number of devices operating at low Re and vast amount of aeroacoustic data available in literature studies [5, 8]. A flow condition of M = 0.4 at an angle of attack AoA = 5^◦and a low Re = 5 × 10⁴ is considered as the flow tends to be laminar in this regime [3, 5]. The schematics of the computational domain for the present study is shown in Figure 1(a) where all the dimensions are made dimensionless with airfoil chord (i.e. ). A buffer zone of width 1.5 surrounding the physical domain is set to eliminate any possible erroneous numerical reflection. All domain boundaries adopt non-reflecting boundary condition except the inlet boundary. On the airfoil, no-slip boundary condition using near wall approach is applied [17]. A similar grid generated in Arif et al. [16] is adopted in the present study (Figure 1(b)) which is subjected to a comprehensive grid independence study and validated with literature [23]. The amount of mesh elements along the airfoil surface N_suction, airfoil wake N_wake and amount of layers within an offset of 0.05 from airfoil surface N_{o f f set}  are listed in Table 1. A time step size of 1 × 10⁻⁵ is selected to maintain Courant-Friedrichs-Lewy condition (CFL) ≤ 1.

 

 

Figure 1: (a) Schematic sketch of computational domain and (b) mesh around airfoil.

 

For the present study, two different airfoil configurations based on single and dual membranes are analyzed and their aeroacoustic characteristics are compared with a baseline rigid airfoil. The baseline airfoil is designated as "Base" in this study, whereas the single membrane and dual membranes airfoil configurations are designated as "SM" and "DM" respectively. The length of membrane for both SM and DM configurations are set as L_EP = 0.05 and their structural properties are chosen based on localized fluid loading characteristics over the baseline airfoil. The normalized h, T and ρ for the membrane are set as 0.009, 0.725 and 6367.35 respectively such that the membrane(s) vibrate in structural resonance with their third natural modes under the excitation of airfoil boundary instability. For SM, the membrane is mounted at 40% of airfoil chord on the suction surface based on its favorable noise reduction characteristics [24]. For DM, the membranes are mounted apart with a distance of one convective disturbance wavelength at 40% and 55% of airfoil chord respectively. A schematic sketch of both the configurations is shown in Figure 2.

 

 

Figure 2: Schematic of designed airfoil configurations. (a) SM; (b) DM. 1 , first membrane; 2 , second membrane.

 

 

Figure 3: Time history of transverse velocity fluctuations v′. (a) x = 0.7, (b) x = 0.8, (c) x = 0.9, and (d) x = 0.99. , Base; , SM; , DM.

 

3. RESULTS AND DISCUSSION

 

3.1. Aeroacoustic Analysis

 

We initially analyze the airfoil aeroacoustic characteristics for Base, SM and DM configurations for comparative analysis. The effect of membrane dynamics on flow instabilities is analyzed by evaluating the time histories of transverse velocity fluctuations at different chord locations downstream of boundary layer reattachment point over the suction surface of airfoil as shown in Figure 3. For brevity, only a time episode of t = 4 comprising of ∼ 13 periodic cycles is shown. The overall pattern of velocity fluctuations is periodic with respect to time for all cases where the localized flow-induced vibration of membranes in DM provide significant reduction of flow instability growth at all chord locations as compared to Base and SM which indicate that the dual membranes absorb the flow fluctuation energy more effectively and leaves less flow distortion for the scattering at airfoil trailing edge.

 

The aerodynamic coefficients for Base, SM, and DM configurations are also evaluated and their temporal statistics are compared in Table 2. The time-averaged C_L,mean for SM and DM are found to be slightly higher than that of Base configuration due to reduction in the length of laminar separation bubble and effective mean camber provided by the membrane [25]. The overall aerodynamic efficiency of the airfoil is essentially remains unaffected due to weak membrane deformations for SM and DM.

 

The acoustic characteristics of Base, SM, and DM configurations are evaluated and compared by analyzing their SPL spectra above the airfoil trailing edge at a location of (x, y) = (1, 2.5) at each

 

Table 2: Aerodynamic coefficients for Base, SM and DM configurations. Values in brackets show the relative changes from Base configuration.

 

 

 

Figure 4: Comparison of SPL spectra at (x, y) = (1, 2.5). (b) Comparison of ∆SPL reduction at all azimuth locations. , Base; , SM; , DM.

 

AoA and shown in Figure 4(a). The spectral content of all the configurations shows similar behavior which confirms that the weak membrane vibrations does not alter the flow characteristics of the airfoil and does not affect the airfoil aerodynamics. A sharp peak is observed at the fundamental frequency of flow instabilities f = 3.37 along with its higher harmonics at f = 6.6 and 10 with much lower magnitudes. A maximum noise reduction of 10 dB is observed for the DM configuration at f = 3.37 whereas a maximum noise reduction of 3 dB is observed for SM configuration. The noise reduction at higher harmonics (f = 6.6 and 10) for SM and DM configurations is much lower than that at the fundamental frequency.

 

Figure 4(b) shows the reduction in the overall sound pressure level (∆SPL = −20 × log₁₀(p′_rms,SM/DM / p′_rms,Base)) for SM an'd DM configurations evaluated at all azimuth locations AoAs. The acoustic solution within the sector θ = ±30^◦is ignored due to contamination of aerodynamic pressure fluctuations downstream of airfoil trailing edge. The overall noise reduction for DM configuration is observed to be much higher than SM at all locations. The maximum noise reduction for both configurations is observed at θ = 115^◦. An average ∆SPL reduction of ∼ 7.7 dB is observed all around the airfoil for DM as compared to ∼ 2.6 dB for SM. The acoustic analysis clearly indicate that the overall effect of dual membranes configuration in tonal noise reduction is more than double the noise reduction achieved by the single membrane configuration.

 

3.2. Wavenumber-Frequency Analysis

 

To analyze the fluid-structure interactions between unsteady airfoil flow and the membrane(s) in SM and DM configurations, the wavenumber-frequency analysis is carried out. For the analysis, the fluid pressure fluctuations p′ right above the membrane and its displacement w′ signals are

 

 

Figure 5: Wavenumber-frequency spectra for SM configuration. (a) w′; and (b) p′. Yellow dashed lines, U_∞; white dashed lines, 0.65U_∞; purple dashed lines, acoustic velocity.

 

transformed from space-time domain Y(x, t) to wavenumber-frequency domain Y(k, f) by performing two-dimensional Fourier transform (2D-FFT). The 2D-FFT is evaluated by:

 

 

For 2-D FFT, the p′ and w′ data over the membrane is arranged in a space-time matrix. The temporal data is extracted with a length of 1 × 10⁵samples. The Fourier transform is initially applied to the time signal at each location with a Hamming window without any segmentation or data overlapping. Subsequently, for each frequency, the complex-Fourier transform is applied in space domain. Figures 5 and 6 show the spectra of k− f for the case of SM and DM respectively. For brevity, only positive axis for frequency is shown which is representative for all physical wave propagation phenomenon for the present study. The wave propagating with positive k indicates downstream travelling waves whereas the wave propagating with negative k represents upstream travelling waves. The spectrum of w′ in Figure 5(a) shows a symmetric pattern of upstream and downstream waves travelling through the membrane induced by the fluid loading and vice versa. The travelling waves are observed to be convecting with a phase speed of 0.65U_∞ which is indicated by a line of constant slope (dashed white line). The symmetric wave propagation at different frequencies indicates the presence of standing wave pattern for the membrane. The high energy content can be identified at f = 1.1, 2.2 and 3.3 indicating membrane dominant vibration modes. The amplitude of third mode f = 3.3 is observed to be much higher than lower modes which even coincides with the fundamental frequency of flow instabilities. A similar trend of wave propagating phenomenon is observed for both the membranes of DM configuration as shown in Figures 6(a) and (b). However, a variation in magnitude of w′ is observed where the magnitude of first membrane in DM is higher than that of SM. Furthermore, the magnitude of w′ for second membrane is higher than first membrane in DM which corresponds to the high strength of boundary layer instabilities at this particular location.

 

The spectrum of p′ for SM in Figure 5(b) shows asymmetry between downstream and upstream travelling waves. The range of wavenumber for downstream travelling waves spans between 0.65 − 0.7U_∞ which corresponds to approximate hydrodynamic convection speed. The high energy content is observed at f = 3.3 which is coincident with the third structural mode of membrane. Low magnitude energy content is also observed at the first harmonics f = 6.6 for both downstream and upstream waves. The upstream travelling waves propagate with the acoustic phase speed (indicated

 

 

Figure 6: Wavenumber-frequency spectra for DM configuration. (a) w′ and (c) p′ for first membrane; (b) w′ and (d) p′ for second membrane. Yellow dashed lines, U_∞; white dashed lines, 0.65U_∞; purple dashed lines, acoustic velocity.

 

by purple dotted line) with similar frequency content which shows a strong correlation between hydrodynamic fluctuations and acoustic waves.

 

The spectrum of p′ for both the membranes of DM case in Figures 6(c) and (d) show a similar trend of wave propagation where the downstream waves travels with hydrodynamic convection velocity while the upstream waves propagates with the acoustic phase speed. However, the p′ spectra for DM also indicate the presence of very low magnitude energy content at f = 1.1, 4.4 and 5.6 respectively. Although the magnitude of p′ at these frequencies is much lower than the f = 3.3, they still indicate that the membrane dynamics in the DM configuration have introduced additional frequency content in the flow.

 

The correlation analysis between the w′ and p′ data over the membrane is carried out to further uncover the underlying fluid-structure coupling phenomenon and noise reduction mechanism for SM and DM configurations. The cross-correlation _pw between the two signals is calculated by:

 

 

where m is the lag coefficient and N is the sample size which is taken as 1,000,00 for the present study for fine resolution. The cross-correlation is normalized by:

 

 

where _pp(0) and _ww(0) are the autocorrelations of p′ and w′ at zero lag respectively. Figure 7(a) shows the spectrum of normalized cross-correlation R_pw for SM between the downstream pressure p′ waves and the point of maximum displacement of membrane w′_max (x = 0.407) to identify the

 

 

Figure 7: Spectra of cross-correlation for SM configuration between time signal of w′ at location of maximum displacement and (a) downstream propagating p′ waves; (c) upstream propagating p′ waves. Cross-correlation plot of w′ and (b) downstream propagating p′ at w′_max (x = 0.407) ; (d) upstream propagating p′ at w′_max (x = 0.407).

 

existing relation between these two variables. The vertical axis represent the time shift 𝜏. For brevity, the vertical axis is limited to a range of -2 to 2. A strong correlation can be observed between the downstream propagating p′ waves and w′_max except at the rear end of membrane. Furthermore, a slight inclination for any constant correlation value indicates that the correlation between p′ and w′_max signals does not occur at a fixed membrane location rather it follows downstream wave propagation. Figure 7(b) shows the R_pw plot between the downstream p′ waves and w′_max. A high value of R_pw is observed between the two signals with regular peak to peak intervals. A negative value of R_pw at 𝜏 = 0 shows the opposite behavior between p′ and w′ as the positive pressure tends to cause negative membrane displacement and vice versa.

 

The spectrum of R_pw between the upstream p′ waves and w′_max is shown in Figure 7(c). The upstream p′ waves and w′ are weakly correlated from membrane leading edge up to 50% of the membrane length. Subsequently, the correlation between the two signals completely diminishes. The spectrum reveals that the membrane displacement is weakly related with the upstream propagating acoustic waves which are generated by the flow scattering at the trailing edge of the airfoil. This weak correlation is attributed to the difference in the phase speeds of upstream acoustic waves and membrane displacement as observed in Figure 5. The observed phenomenon may lead to weakening of the strength of upstream propagating acoustic waves and results in reduction of tonal noise. Figure 7(d) shows the R_pw between the downstream p′ waves and w′_max. A significant reduction in the magnitude of R_pw is observed between the two signals. Again a negative value of cross-correlation at 𝜏 = 0 is observed depicting opposite behavior of p′ and w′.

 

Figures 8(a) and (b) show the spectra of normalized cross-correlation R_pw between the downstream p′ waves and the point of w′_max for the first and second membrane of DM configuration respectively. A strong correlation can be observed between the downstream propagating p′ waves and w′ for the second membrane. However, the correlation is slightly reduced at the rear half of first membrane. The spectrum of R_pw between the upstream pressure waves and the point of maximum displacement of first and second membrane of DM configuration is shown in Figures 8(c) and (d). For the first membrane, the upstream p′ waves and w′ are weakly correlated from membrane leading edge up

 

 

Figure 8: Spectra of cross-correlation for DM configuration between time signal of w′ at location of maximum displacement and (a) downstream propagating p′ waves for first membrane; (c) upstream propagating p′ waves for first membrane. Spectrum of cross-correlation for DM configuration between time signal of w′ at location of maximum displacement and (b) downstream propagating p′ waves for second membrane; (d) upstream propagating p′ waves for second membrane.

 

to 30% of the membrane length. Subsequently, the correlation between the two signals completely diminishes. On the other hand, for the second membrane, the correlation between upstream p′ waves and w′ completely diminishes after 10% of membrane length from the leading edge. The correlation slightly increases again at the rear end of second membrane. The observed phenomenon clearly indicates that the second membrane has much stronger effect on the upstream p′ waves as compared to first membrane. Furthermore, the reduction in correlation for DM is observed to be much more than SM configuration.

 

4. CONCLUSIONS

 

Fluid-structure interactions of an airfoil mounted with short membrane(s) and its coupling effect on airfoil aerodynamics and acoustics have been studied in detail at low Re of 5 × 10⁴ and low AoA of 5^◦ using direct aeroacoustic simulations. Two different airfoil configurations based on single membrane ′SM′ and dual membranes ′DM′ are analyzed and compared with baseline airfoil ′Base′. The coupled fluid-structural interactions are investigated by transforming the pressure fluctuations and membrane displacement data from space-time domain into wavenumber-frequency domain by double Fourier transform. The detailed fluid-structure interaction investigation in this study reveals that the membrane dynamics distorts the correlation between the upstream propagating acoustic waves and its displacement which results in a reduction in coherence among the pressure and velocity signals at a distant location. Hence, a significant reduction in airfoil tonal noise of 8 dB for DM and 3 dB for SM is achieved. Furthermore, the airfoil aerodynamics analyses for Base, SM and DM configurations clearly indicate that the overall mean flow is not affected due to membrane dynamics and the airfoil maintains its aerodynamic efficiency without any significant distortion.

 

ACKNOWLEDGEMENTS

 

The authors gratefully acknowledge the support from a research donation from the Philip K. H. Wong Foundation under grant number 5-ZH1X. The first and fourth authors gratefully acknowledge the support with research studentship tenable at Department of Mechanical Engineering, The Hong Kong Polytechnic University (PolyU). The second and third authors are grateful to the support from the Research Grants Council of the Government of Hong Kong Special Administrative Region under grant number A-PolyU 503/15 and the ANR/UGC International FlowMatAc number ANR-15-CE22- 0016-01.

 

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¹ irsalan.arif@connect.polyu.hk

² mmrleung@polyu.edu.hk

³ garret.lam.hk@connect.polyu.hk

⁴ rehan.naseer@connect.polyu.hk