Airfoil Tonal Noise Prediction Using Urans

One of the first comprehensive studies on airfoil tonal noise was an experimental research that was conducted in the 1970s [3]. The ‘ladder-type’ variation of the peak tone frequency f_s was obtained, relating it to the freestream flow speed u_∞, airfoil chord c and air kinematic viscosity vfs=0.011u∞1.5(cv)−0.5$$v{f_s} = 0.011u_\infty ^{1.5}{(cv)^{ - 0.5}}$$. The self-excited aeroacoustic feedback was soon proposed [4], which assumes that the tonal noise is due to the first instability excited by a point downstream. This idea was further investigated, so that the aeroacoustic signal was assumed to originate at the airfoil trailing edge as a result of Tollmien–Schlichting (T–S) waves diffraction. T–S waves are in turn triggered by the acoustic signal, which closes the feedback loop [5]. The validity of aeroacoustic feedback theory was discussed in the study of McAlpine et al. [6], and within that study it was concluded that the feedback is not necessary for the tonal noise to occur. It was found that the tonal noise is largely dependent on the existence of the separation bubble. In Lowson et al. [7], the authors established that the tones appear only up to a limited angle of incidence and in a certain Reynolds number range. Some more recent attempts can be found in Pröbsting et al. [8], which generally confirms the importance of the separation bubble in relation to the tonal noise and the tones’ existence conditions.

In the light of these findings, LBL-VS is potentially important in the context of low Reynolds number propellers, which operate in low turbulence conditions and at moderate angles of attack [9]. Additionally, an appropriate computational model must be selected, which allows us to identify the existence of tonal noise on arbitrary airfoil, with the associated cost not ruling out the shape optimisation.

The computational analysis of airfoil noise, especially if broadband noise is considered, requires a high-fidelity computational fluid dynamics (CFD) model, capable of resolving scales of turbulence, such as the large eddy simulation (LES) or hybrid RANS-LES [10]. The tonal nature of LBL-VS noise requires appropriate modelling of laminar-TBL transition and formation of hydrodynamic instabilities, which propagate over the airfoil surface up to the trailing edge. Especially, late transition occurring at low turbulence conditions is important. The complexity of LBL-VS, further increased by the aeroacoustic feedback from the airfoil trailing edge, responsible for amplification of T–S waves, requires direct computational aeroacoustics (CAA) methods performed with direct numerical simulations (DNS) [11,12]. Whereas a direct application of CAA would resolve the acoustic field, the use of the relatively inexpensive unsteady Reynolds-averaged Navier–Stokes (RANS) model [13], in conjunction with the use of transition SST for turbulence closure, would suffice for modelling, with good accuracy, the existence of tonal noise as well as the levels of this noise. For the selected computational case of symmetric NACA airfoil, the method was shown whereby promising results, in agreement with the experiment described in the study of De Gennaro et al. [13], could be obtained. However, sufficient investigations have not been conducted regarding the performance associated with, as well as the limitations occurring in, a generic airfoil problem.

In the present work, the validity of an incompressible unsteady Reynolds-averaged Navier–Stokes (URANS)-based modelling approach for airfoil LBL-VS tonal noise was assessed. The generic, asymmetric S834 airfoil was investigated, and the results obtained were compared to the experiment [14]. As the exact reproduction of an experimental setup, i.e. modelling the finite-span airfoil, was not feasible, the two-dimensional analysis of an airfoil, i.e. assuming an airfoil of infinite span, was conducted. The focus was thus on the qualitative aspects of tonal noise rather than the quantitative comparison of noise levels. Specifically, the existence and frequency of tones were of primary interest.

The present article is structured as follows: in Section 2, the details of aerodynamic and acoustic models are described. In Section 3, aerodynamic and acoustic results are presented and discussed. In Section 4, the summary and conclusions are stated.

Navier–Stokes (N–S) equations are used to describe the viscous fluid motion, which in the present instance is represented by the following continuity equation: 1∂ρ∂t+∇⋅(ρu→)=0,$${{\partial \rho } \over {\partial t}} + \nabla \cdot (\rho \vec u) = 0,$$ and the following momentum conservation equation: 2∂∂t(ρu→)+∇⋅(ρu→u→)=−∇p+∇⋅(τ¯¯),$${\partial \over {\partial t}}(\rho \vec u) + \nabla \cdot (\rho \vec u\vec u) = - \nabla p + \nabla \cdot (\overline {\bar \tau } ),$$ where p is the static pressure, ρ is the density, t is the time and τ¯¯$$\overline {\bar \tau } $$ is the stress tensor, defined as: 3τ¯¯=μ[ (∇u→+∇uT→)−23∇⋅u→I ],$$\overline {\bar \tau } = \mu \left[ {\left( {\nabla \vec u + \nabla \overrightarrow {{u^T}} } \right) - {2 \over 3}\nabla \cdot \vec uI} \right],$$ where μ is the molecular viscosity and I is the unit tensor.

So far as practical engineering applications are concerned, owing to the magnitude of the computational cost involved, derivation of direct numerical solutions to the N–S equations is not performed. The most popular simplification of N–S equations in industrial practice involves averaging; and among approaches involving averaging, the most popular, Reynolds averaging, gives RANS equations. RANS equations are of the form: 4∂∂t(ρui)+∂∂xj(ρuiuj)=−∂p∂xi+∂∂xj[ μ(∂ui∂xj+∂uj∂xi−23δij∂uk∂xk) ]+∂∂xj(−ρui′uj′)$${\partial \over {\partial t}}\left( {\rho {u_i}} \right) + {\partial \over {\partial {x_j}}}\left( {\rho {u_i}{u_j}} \right) = - {{\partial p} \over {\partial {x_i}}} + {\partial \over {\partial {x_j}}}\left[ {\mu \left( {{{\partial {u_i}} \over {\partial {x_j}}} + {{\partial {u_j}} \over {\partial {x_i}}} - {2 \over 3}{\delta _{ij}}{{\partial {u_k}} \over {\partial {x_k}}}} \right)} \right] + {\partial \over {\partial {x_j}}}\left( { - \rho u_i^\prime u_j^\prime } \right)$$ where u represents the mean velocity and u’ is the fluctuating term resulting from Reynolds averaging. Additional modelling of the fluctuating Reynolds stress term −ρui′uj′¯$$\overline { - \rho u_i^\prime u_j^\prime } $$ is required to close RANS equations. Eddy-viscosity models employ the Boussinesq hypothesis, which relates Reynolds stresses to the mean velocity gradients using the turbulent viscosity μ_T: 5−ρui′uj′¯=μT(∂ui∂xj+∂uj∂xi)−23(ρk+μT∂uk∂xk)δij.$$\overline { - \rho u_i^\prime u_j^\prime } = {\mu _T}\left( {{{\partial {u_i}} \over {\partial {x_j}}} + {{\partial {u_j}} \over {\partial {x_i}}}} \right) - {2 \over 3}\left( {\rho k + {\mu _T}{{\partial {u_k}} \over {\partial {x_k}}}} \right){\delta _{ij}}{\rm{.\;}}$$

The selection of eddy-viscosity model is case-specific. In this work, the requirement to accurately model the laminar bubble and the associated transition to the TBL led to the selection of a transition SST model [15], a common choice in the literature [16]. Transition SST is a four-equations, correlation-based turbulence model, which solves transport equations for turbulence kinetic energy k, specific dissipation rate ω, intermittency γ and momentum thickness Reynolds number Re_θ, which define the transition onset criteria. The model shows very good agreement with experiments and outperforms many other eddy-viscosity models in airfoil analysis requiring laminar-to-turbulent transition prediction [17].

In the present work, airfoil analysis at low Mach and moderate Reynolds number (Re) at the angle of attack (α) of 4° is considered. The case summary is presented in Table 1. In addition to the baseline case (u_∞ = 32 m/s), two additional analyses (u_∞ = 22.4 m/s and u_∞ = 47.9 m/s) are performed to confirm or refute the baseline conclusions.

2.1.1.

Computational domain and boundary conditions

The analyses were performed with a 2D model, which exploits the two-dimensional nature of LBL-VS and significantly reduces the computational cost. The domain extends over 20 chords upstream and almost 40 chords downstream from the airfoil. The problem was defined using common boundary conditions for incompressible flow, namely velocity-inlet with required freestream velocity and pressureoutlet in the downstream of domain (Fig. 2).

Figure 2.

Due to the turbulence dissipation in the computational domain, its value is computed from the results of CFD model at the location directly above the leading edge of an airfoil. The turbulence at the inlet of the computational domain is controlled using the turbulence intensity and turbulent viscosity ratio.

Turbulence intensity is defined as: TI=u′U,$${\rm{TI}} = {{u'} \over U},$$ where u’ is the root mean square of the turbulent velocity fluctuations: u′=13(ux′2¯+uy′2¯+uz′2¯)=23k,$$u' = \sqrt {{1 \over 3}\left( {\overline {u_x^{\prime 2}} + \overline {u_y^{\prime 2}} + \overline {u_z^{\prime 2}} } \right)} = \sqrt {{2 \over 3}k} ,$$ and U is the mean (Reynolds-averaged) velocity: U=Ux2+Uy2+Uz2.$$U = \sqrt {U_x^2 + U_y^2 + U_z^2} .$$

2.1.2.

Mesh

The selected turbulence model (transition SST requires a mesh that allows resolution of the viscous sublayer, and specifically in such a way that y⁺ does not exceed unity. The mesh is particularly important in the region of laminar-turbulent transition (laminar bubble), where the hydrodynamic instabilities are formed. Moreover, the region downstream of the laminar bubble should be well-modelled, such that the VS directly influencing pressure oscillations on the airfoil surface is properly resolved. A structured C-mesh was used to achieve orthogonal mesh in the region surrounding the airfoil (Fig. 3). The mesh consists of 305,000 quadrilateral elements. The near wall mesh is presented in Figure 4, whereas the mesh near the trailing edge is presented in Figure 5. The airfoil trailing edge is blunt, with three elements across its thickness.

Figure 3.

Figure 4.

Figure 5.

The hybrid approach to modelling aeroacoustics is based on the separation of flow dynamics and acoustics. Firstly, the aerodynamic model is calculated. Then the noise sources are computed and propagated to the observer location using a selected method. The possible approaches include Curle analogy [19], Ffowcs Willimas–Hawkings analogy [20], Lighthill analogy [21,22], and others. The latter approach, schematically presented in Figure 6, is used in this work.

Figure 6.

Lighthills analogy is formulated by rearranging N–S equations with acoustic density perturbation ρ’ = ρ – ρ₀: 6∂2ρ′∂t2−c0∂2ρ′∂xi∂xi=∂2Tij∂xi∂xj,$${{{\partial ^2}\rho '} \over {\partial {t^2}}} - {c_0}{{{\partial ^2}\rho '} \over {\partial {x_i}\partial {x_i}}} = {{{\partial ^2}{T_{ij}}} \over {\partial {x_i}\partial {x_j}}},$$ where the left side of equation describes a propagation operator for all regions of the domain and the right side describes an equivalent acoustic source. It is described for source region only and is based on the Lighthill tensor T_i approximation for low Mach, incompressible flow, lending itself to expression as: 7Tij≈ρuiuj,$${T_{ij}} \approx \rho {u_i}{u_j},$$ i.e. sources are calculated from velocity fluctuations. While the solution in the source region is typically discarded due to the limited accuracy, the calculated acoustic far-field is accurate. The Lighthill analogy can only be used under low freestream velocity conditions. It is based on the assumption that no acoustic sources are present in the far-field, which restricts the source region location. Important in the context of the present work is the separation of aerodynamics and acoustics. The model is unable to account for a potential aeroacoustic feedback unless the aerodynamic model captures it.

In the present work, Lighthill analogy, with volume sources in the region surrounding the airfoil, was used. The analysis was performed using a commercial software, Actran 2021 [23]. To begin with, aeroacoustic sources were computed from the unsteady CFD results and transformed to the frequency domain. Thereafter, the finite element method-based solver was used to calculate the acoustic pressure propagation in the acoustic domain, with the corresponding mesh shown in Figure 7 (the yellow zone depicts the aeroacoustic source region). The mesh sizing was selected to allow modelling of frequencies between 300 Hz and 5,000 Hz, assuming at least six quadratic finite element (FE) cells per wavelength. The 1/3 octave band sound pressure levels were calculated at microphones surrounding the trailing edge at a distance of 0.63 m (as in the experiment).

Figure 7.

To recreate the experimental environment, turbulence intensity not exceeding 1% had to be ensured at the airfoil [14]. However, the wind tunnel test section turbulence properties were not described in sufficient detail to allow more exact reproduction of the freestream conditions. For this reason, the influence of freestream turbulence, specifically turbulence intensity, on the computational model was investigated. In Figure 8, the example of turbulence dissipation is presented with turbulence intensity levels around the airfoil.

Figure 8.

Several analyses with turbulence intensity at the leading edge of an airfoil ranging from 0.18% to 0.99% were conducted. Figure 9 presents the instantaneous pressure coefficient on the rear part of the airfoil for turbulence intensities of 0.18% and 0.55%. The hydrodynamic instabilities are formed at the position of laminar bubble, approximately at 60% chord and 87% chord for the suction and pressure sides, respectively. Hence, on the suction side the pressure oscillations reach their maximum amplitude and begin to decay. On the pressure side, the oscillations are only observed for the lower turbulence case.

Figure 9.

A comparison of pressure contours for cases with different turbulence intensities is presented in Figure 10. Lower turbulence results in pressure oscillations at both pressure and suction sides, whereas for higher turbulence only suction side oscillations are present. It can be noted that the strength of vortices generated at suction side is higher for the lower turbulence (as assessed by density of the isopressure lines). Additionally, the vortices in the wake of an airfoil are only observed in the lower turbulence case.

Figure 10.

The boundary layer in the vicinity of hydrodynamic instabilities formation is presented in Figure 11 for the low turbulence case (TI = 0.18%). The three snapshots were taken in equal time steps of 0.4 t_f, and time was scaled to match the period of suction side oscillations, i.e. t_f = 1/f. The centres of vortices were joined with lines to allow easy tracking of the vortices’ movement. Initially, the vortices are very slow, as a result of which they are subject to acceleration, as can be observed from comparing the gradients of the joining lines.

Figure 11.

Similarly, in Figure 12, the vortices pattern is presented for the low turbulence case (TI = 0.18%), where the vortices are generated at both pressure and suction sides. The location and movement of these are visualised in the rear part of the airfoil using q-criterion. It can be observed that the vortex at the suction side, initially downstream of the pressure side vortex, is moving faster, and eventually passes the latter. Consequently, different frequencies of oscillations at both sides are expected to be observed during quantitative analysis.

Figure 12.

In Figure 13, drag monitor frequency domain representation is shown. Two peaks are observed: the major at f = 1,300 Hz corresponds to the frequency of oscillations at the pressure side and the secondary at f = 1,660 Hz corresponds to the frequency of oscillations at the suction side. Similar plots for remaining turbulence intensities show only single peak, corresponding to the suction side VS.

Figure 13.

A summary of pressure oscillations’ presence and strength is shown in Figure 14. The variation of peak amplitudes in spectral analysis of pressure signals, collected from four different points on the airfoil surface, is plotted against turbulence intensity. The general trend of decreasing amplitude with increasing turbulence intensity is observed. Vortices are only present on the pressure side for the lowest turbulence intensity case, while on the suction side they are observed up to the intensity of 0.65%. It should be noted that the amplitude of pressure oscillations on the suction side halves between the 87% chord and 96% chord locations, while on the pressure side the amplitudes are approximately doubled. In Table 2, a summary of peak tone frequencies and amplitudes for pressure signals collected at 87% chord and drag counts (whole airfoil) is presented. An important result is shown for the case with a turbulence intensity of 0.18%, where the lower pressure amplitude at 1,300 Hz results in the peak frequency for drag signal. This result agrees with the generally observed dominant role of pressure side instabilities in the generation of tonal noise. For higher turbulence levels, where the pressure side oscillations are very small, their frequency is the same as that of the suction side oscillations, which suggests that these oscillations are generated by flow pattern of the suction side.

Figure 14.

The final assessment of the URANS modelling is performed by comparing the frequency and amplitude of drag oscillations for S834 airfoil at different freestream speeds, and thus different Reynolds numbers; a summary of the comparison results is presented in Table 3. It is observed that the frequency of oscillations for different Reynolds numbers is well-matched with the peak tone frequency measured in the experiment. These results were obtained at the turbulence intensities of 0.44% and 0.69% for freestream velocity values of 22.4 m/s and 47.9 m/s, respectively. Hence, the intensities were in the regime, where the expected LBL-VS frequency matches the experiment tonal noise peak, as shown for the baseline case. The trend of increasing sound power level (PWL) with increasing amplitude of drag oscillations is also maintained; however, due to various turbulence levels between cases, further conclusions cannot be drawn.

The acoustic pressure field is presented in Figure 15 for the frequency of 1,600 Hz. The following conclusions can be drawn:

The noise source is located at the trailing edge of an airfoil.

Figure 15.

The quantitative comparison of noise levels obtained in the present work and in the experiment presented in the study of Oerlemans [14] is provided in Figure 16. The frequency of oscillations from the aerodynamic model was transferred to the noise tonal peak observed in the noise spectrum. The tone frequency from CAA matches the peak tone frequency measured in the experiment. The level of noise is, however, significantly higher than in the experiment. The difference can be attributed to the differences in the computational model setup, which assumes an ideal, infinite span airfoil, while the experiment was conducted on the finite wing. Noise levels’ differences at the remaining frequencies result from turbulence averaging in RANS. To model higher frequencies scale resolving CFD is needed, which was not in the scope of the present work.

Figure 16.