Data-driven determination of low-frequency dipole noise mechanisms in stalled airfoils

To extract the far-field acoustics, it is first necessary to determine the instantaneous pressure. Planar pressure fields were reconstructed from the instantaneous velocity fields using a Poisson solver approach (De Kat and Van Oudheusden 2012; Laskari et al. 2016) for which the divergence of Navier–Stokes momentum equations is invoked:where \({\varvec{u}}\) is the instantaneous velocity vector, p is the pressure and \(\rho\) is the (constant) density. Neumann boundary conditions were applied on the inlet, outlet, and suction (upper) boundaries of the domain and Dirichlet boundary conditions were employed using Bernoulli’s equation in the free stream on the pressure (lower) domain boundary. The application of Dirichlet boundary conditions on both the suction and pressure boundaries was also tested, but was found to give slightly worse agreement in comparison to numerical simulations that will be discussed shortly. This is likely due to separation phenomena partially invalidating the assumption of irrotational flow near the suction boundary. The unsteady velocity term was directly computed from the time-resolved Eulerian fields (Jakobsen et al. 1997) using finite differences. Good agreement using central differences up to sixth order was found; therefore, second-order gradients were opted for in favor of processing speed. Spatial derivatives were also computed using second-order central differences.

To provide a comparison for the pressure reconstructions a two-dimensional Reynolds-averaged Navier–Stokes (RANS) simulation was performed using OpenFOAM software. The simulation utilized a k-\(\omega\) shear stress transport (SST) closure model. A standard C-type domain and grid geometry was chosen with a horizontal and vertical extent of 20 chords lengths. The simulation properties (e.g. kinematic viscosity, inlet velocity, chord length) were chosen to match identically those of the experiment. The comparison between the surface pressure coefficient \(C_p=P/\frac{1}{2}\rho U_{\infty }^2\) (with P the mean pressure) in the PIV and RANS is shown in Fig. 2. The agreement in the attached case (Fig. 2a) is qualitatively comparable, but with quantitative discrepancies on the suction side particularly at the leading edge (LE) where the PIV underestimated the suction peak. At higher angles of attack (Fig. 2b,c) the PIV was found to closely compare in the magnitude of the surface pressure on the suction side, but again struggled to capture the suction peak at the leading edge. The difficulty of the PIV in capturing pressure near the leading edge is likely due to the thin boundary layers that require exceedingly high spatial resolution.

Naturally, it is important to investigate whether and how inaccuracies in the mean pressure on the suction side impacts the time-varying behavior critical for estimating the far-field noise. This is assessed via the standard deviation of the surface pressure coefficient on the suction side plotted in Fig. 2d. The magnitude of \(C_{p,std}\) compares well with results previously reported in the literature via measurements (Sicot et al. 2006; Lee et al. 2013) and simulations (Golubev et al. 2016). The erratic behaviour of \(C_{p,std}\) for the attached case (\(\alpha =4^{\circ }\)) near the LE is likely measurement bias error. As mentioned previously, the effective resolution at the leading edge relative to the developing boundary layer is low. This issue is less problematic for the stalled cases (the focus of the present work) which do not exhibit such discontinuities in \(C_{p,std}\) and have a greater effective resolution since the boundary layer separates in these cases. Taken together, the results of Fig. 2 lend confidence for the application of the Poisson solver to capture the fluctuating pressure fields necessary for the noise extrapolation.

The far-field acoustics surrounding the airfoil physically belongs to a domain much too large to measure directly via experiment. Therefore, Curle’s acoustic analogy is adopted (Curle 1955) utilizing the pressure fluctuations obtained from the PIV to estimate the far-field noise. Following the work of Larsson et al. (2004), the original solution presented by Curle is modified and the far-field dipole noise emanating from the surface of the airfoil are isolated in the surface integral (see also Koschatzky et al. 2011a):where \(p_a\) is the acoustic pressure and \(p_{a,0}\) the steady far-field pressure, \(l_i\) is the listener unit vector pointing from the source to the listener position, \(n_j\) is the surface normal unit vector, \(\dot{p}=\partial p/\partial t\) is the unsteady pressure, \(a_0\) is the speed of sound, \(r=|y_i-x_i\vert\) is the distance to the listener position \(x_i\) from the source \(y_i\), and the surface S is the airfoil surface. The integrand is evaluated at the retarded time \(t-r/a_0\). Quadrupole sources are ignored in the estimation of the acoustic fluctuations, as the Mach number is sufficiently low in the present case (Curle 1955). A similar result was confirmed in the work of Larsson et al. (2004) for an open cavity flow. For stalled airfoils, previous work suggests the dipole sources are likewise dominant (Moreau et al. 2009; Laratro et al. 2014).

To carry out the integration, surface normal vectors were defined at each PIV vector location along the mask of the airfoils for all cases considered. A Savitsky–Golay filter was employed for smoothly varying surface normal estimates shown in Fig. 3. This was found to be necessary particularly near the LE with significant curvature. The unsteady pressure \(\dot{p}\) along the surface of the airfoil was obtained directly through finite second-order temporal differences of the pressure. To reduce noise, a sliding average was used to smooth the unsteady pressure estimates with a width of 25 ms, corresponding to a non-dimensional frequency \(f^{\star }=12\). This does not impact the acoustic frequencies of interest in the present study that are restricted to \(f^{\star }\le 3\).

Listener positions are selected in a semi-circular array of positions at a specified radius from the airfoil mid-chord position for \(0^{\circ }\le \theta \le 180^{\circ }\) spanning the suction side of the airfoil at \(N=30\) locations. Here, \(\theta =0^{\circ }\) is defined with respect to the LE at each angle of attack (not with respect to the direction of free stream velocity). The listener distance \(r=a_0/f_{\textrm{min}}\) is selected to automatically satisfy the far-field approximation for the frequencies of interest, where \(f_{\textrm{min}}=0.19\) Hz is the lowest resolved frequency. We remark that the far-field approximation in not consequential for the relative frequency content of \(p_a({\varvec{x}},t)\) but does impact the magnitude linearly with increasing r (Eq. 2). The choice of r is therefore inconsequential for the correlation-based analysis that is the focus of the present study. For reference, the resulting difference between a typical wind-tunnel choice of \(r=10 c\) and the chosen r of this study is 74.4 dB.

To elucidate flow structures responsible for the dipole noise generation, the spectral Linear Stochastic Estimation (sLSE) is leveraged between the instantaneous velocity fields and the far-field acoustic pressure (Tinney et al. 2006). Specifically, the sLSE is performed between the acoustic fluctuations at each listener position and the coefficients of the modes of the velocity fields via the proper orthogonal decomposition (POD; Sirovich 1987). In the following we present a brief outline on the application of this technique for the present analysis. The reader is referred to, e.g. Taylor and Glauser (2004) and Podvin et al. (2018) for more details on combining POD with LSE. The POD is performed on the fluctuating velocity vectors \(u_i'\) aswhere k is the mode number, \(K_{tot}\) is the total number of modes, \(a_k(t)\) is the k-th instantaneous POD coefficient, and \(\phi _k\) the k-th orthogonal spatial mode. For the present study \(K_{tot}=1075\) modes are used to generate the POD basis and up to \(K=30\) modes are retained for the sLSE analysis, as higher modes were not found to impact the results for all cases.

In the present context, the sLSE leverages the temporal cross-correlation between the mode coefficients and the acoustic fluctuations to obtain the portion of the velocity fields most correlated to the noise. Following the work of Tinney et al. (2006), two matrices are defined aswhere the angled brackets \(\langle \cdot \rangle\) denote ensemble averaging and N is the total number of listener positions. These two matrices are used to obtain the spectral LSE coefficients aswhere \(W_{jk}(f)=\{W_{jk}(t)\}\) and \(V_{ji}(f)=\{V_{ji}(t)\}\) are the Fourier transforms of (4) and the dependence on f implies the Fourier domain. In practice the sLSE coefficients are obtained for each run individually and ensemble averaged for each case. The estimated POD coefficients are then calculated in each run aswhere \(P_{a,ji}(f)\) is the Fourier transform of the matrix of acoustic fluctuations of size \([N\times n_t]\) where \(n_t\) is the number of times steps in the run, \(\{\cdot \}^*\) denotes the inverse Fourier transform, and \(\tilde{A}_{ki}\) is the matrix of estimated POD coefficients of size \([K\times n_t]\). The velocity fields correlated with the acoustic fluctuations are then reconstructed viawhere \(\tilde{a}_k(t)\) is the k-th mode coefficient and time instant of the matrix \(\tilde{A}_{ki}\). The main advantage of performing the LSE in the Fourier domain is the automatic incorporation of time delays in the estimation of the LSE coefficients (Tinney et al. 2006).

The results of the acoustic extrapolation via Curle’s analogy are reported in Fig. 4. The overall sound power level (OASPL) in Fig. 4a shows the expected increase in noise at the onset of stall. The variation with listener angle indicates that the noise directivity is largely driven by the geometry of the airfoil, i.e. the alignment of \(l_i\) and \(n_j\), with maximum noise level at \(\theta =90^{\circ }\) (normal to the airfoil chord) in all cases. Only the case at \(\alpha =15^{\circ }\) exhibits a notable asymmetry with approximately 2 dB more sound power for \(\theta >120^{\circ }\) (in proximity to the TE) with respect to \(\theta <60^{\circ }\) (in proximity to the LE). No significant differences in the shape of the distributions were found after considering band-limited directivity.

The spectra reported in Fig. 4b are obtained using Welch’s method with a Hamming window in 3 segments of equal length with 50% overlap and averaged over all orientations and runs. Both stalled cases exhibit significant broad-band noise indicative of turbulence. A range of bluff-body shedding frequencies at Strouhal numbers \(St\equiv f^{\star }\sin \alpha\) between 0.1 and 0.2 is indicated in the figure to delineate between bluff-body shedding related frequencies and shear-layer flapping at higher frequencies (Derakhshandeh and Alam 2019). Here we focus on the two cases in stall as this is the focus of this manuscript (in addition, the attached case contains likely unphysical variations at the LE, see Fig. 2d). Both stalled cases exhibit local peaks in the bluff-body shedding range and the shear-layer flapping range (indicated by the arrows in the figure), though these peaks do not necessarily coincide between cases. These peaks suggest that both shear layer flapping and bluff body shedding mechanisms play a role in noise generation in both transient and deep stall (Lacagnina et al. 2019).

Having quantified the directivity and spectral content, we turn our attention to the relationship between structures in the velocity field and the generation of noise using the framework presented in Sect. 3.3. To investigate spatial distribution of structures correlated to the noise, the planar turbulent kinetic energy for the fluctuating velocity fields \(q({\varvec{x}})=\frac{1}{2}\langle u_i'u_i'\rangle\) and for the acoustic-correlated fluctuating velocity fields \(\tilde{q}({\varvec{x}})=\frac{1}{2}\langle \tilde{u}_i'\tilde{u}_i'\rangle\) is plotted in Fig. 5. Here, the contours are shown relative to the maximum value of each quantity. It is therefore important to note the relative magnitude \(\tilde{q}_{\textrm{max}}/q_{\textrm{max}}=0.16,0.30\) for the \(\alpha =13^{\circ },15^{\circ }\) cases, respectively. In other words, at its maximum roughly 16% and 30% of the energy of the fluctuating velocity fields was found to correlate with the far-field noise for each case.

The contours of \(\tilde{q}({\varvec{x}})\) concentrate in two areas. First, in the shear layer but shifted downstream, i.e. closer to the TE, with respect to the contours of \(q({\varvec{x}})\). This contour is elongated along the shear layer, with a clear maximum shortly downstream of the TE location in both cases. For the case in deep stall, another maximum is seen in the shear layer further upstream. Second, contours appear in close proximity to the TE itself. This is likely related to the bluff body shedding alternating at the TE and in the shear layer. The location of the contours of \(\tilde{q}({\varvec{x}})\) in proximity to the TE reflects the tendency of the TE to scatter noise efficiently, consistent with the results of Lacagnina et al. (2019) and Mayer et al. (2020).

It is of further interest to isolate the structures in the velocity field that are responsible for the most intense acoustic fluctuations. To this end, we present a conditional analysis of the acoustic-correlated POD coefficients. The acoustic fluctuations are segmented based on a threshold chosen to be \(\pm 2\) standard deviations. (The ensuing results were found not to be qualitatively sensitive to this threshold down to \(\pm 1\) standard deviation.) This is chosen to isolate the top 5% most intense fluctuations by magnitude. In Fig. 6a, a time segment of 2.5 s of data is shown for the case at \(\alpha =13^{\circ }\). This particular segment reveals three instants of positive extreme events and two negative extreme events, however the overall number of peaks for both was found to be approximately equal across all runs for each case.

The conditional analysis was carried out by identifying the local maximum and minimum within each time segment of intense acoustic fluctuations. At the extrema, the time lag \(\tau\) is defined to be zero and the average of the first five POD coefficients is tabulated at all maxima and minima, respectively. (We note that beyond five modes the conditional averages were found to be negligibly small, and therefore restrict to just the first five.) The time lag is shifted at all possible temporal bins across \(\pm 0.5\tau /T_L\) in order to capture the evolution of the velocity fields leading up to and immediately following intense acoustic fluctuations. From Fig. 6b, c, the conditional averages for the \(\alpha =13^{\circ }\) case, it is evident that most of the variation in the conditional POD coefficients is in the range \(\tau /T_L=\pm 0.2\).

The fluctuating velocity fields conditioned on local maxima and minima for the case at \(\alpha =13^{\circ }\) are shown in Fig. 7 at time lags indicated in Figs. 6b, c. It can be seen that the structures leading to local maxima and local minima in the acoustic fluctuations are almost mirrored. For the local maxima, a downwash of fast-moving fluid can be seen originating in the shear layer above the trailing edge. At \(\tau =0\) (panel e), a simultaneous low-speed structure at the incipient shear layer emerges. Together, these structures manifest a diverging pattern of slow and fast-moving fluid at the incipient shear layer and above the trailing edge. At the trailing edge itself, the fluid is seen to be low-speed.

The opposite action is seen to occur for the local minima. An upwash of slow-moving fluid can be seen originating in the shear layer above the trailing edge. At \(\tau =0\) (panel f), a simultaneous high-speed structure at the incipient shear layer emerges. These structures manifest a converging pattern of fast and slow-moving fluid at the incipient shear layer and above the trailing edge. The behaviour at the TE is likewise opposite to the local maxima. This can be seen in the contours of out-of-plane vorticity \(\tilde{\omega }_3=\frac{\partial \tilde{v'}}{\partial x}-\frac{\partial \tilde{u'}}{\partial y}\) as well, with the sign of the vorticity flipping between the maxima and the minima.

The conditional fields for the case at \(\alpha =15^{\circ }\) are shown in Fig. 8. Despite this case being well beyond transient stall and firmly in a state of deep stall originating at the LE, the qualitative structure of the fluctuating velocity fields correlated to extreme noise events is remarkably similar to the \(\alpha =13^{\circ }\) case. The magnitude of the out-of-plane vorticity is seen to comparatively increase particularly at \(\tau =0\). In addition, likely due to the larger region of separation, the downwash and upwash structures appear to occupy a comparatively larger area.

A small difference of note between the two cases can be seen at the largest presented time lag of \(\tau /T_L=0.2\) (panels i and j of Figs. 7 and 8). For the case in transitional stall at \(\alpha =13^{\circ }\), the acoustic minima (panel i) no longer appears to mirror the maxima (panel j), unlike for the case in deep stall at \(\alpha =15^{\circ }\). Taking into account the changes relative to the previous time lag (\(\tau /T_L=0.1\)) this suggests an asymmetry exists in transitional stall such that the structures associated to the acoustic minima are not as long-lived as the maxima.