Trailing-edge noise from the scattering of spanwise-coherent structures

A large-eddy simulation of turbulent, compressible flow around a NACA 0012 airfoil at zero angle of attack and Mach number 0.115 is used to study mechanisms of trailing-edge noise. The boundary layers at both sides of the airfoil have a forced transition near the airfoil leading edge, and are turbulent near the trailing-edge. Flow-acoustic correlations and spectral (frequency-domain) proper orthogonal decomposition (SPOD) are used to evaluate turbulent structures that are relevant for the radiated sound. Homogeneity in the spanwise direction allows application of a Fourier decomposition in span prior to both correlations and SPOD. It is known that acoustic theory, based on an analysis of the tailored Green's function modeling trailing-edge scattering, shows that only spanwise wave numbers $k_z$ satisfying $k_z<k$, where $k$ is the acoustic wave number, lead to radiated sound; two-dimensional disturbances ($k_z=0$) always satisfy this criterion, and thus spanwise-coherent structures are expected to be important for trailing-edge noise. Analysis of turbulence statistics of the boundary layer close to the trailing edge shows that the well-known, dominant streaky structures have $k_z>k$ and thus should not contribute to the radiated sound. To investigate this further using simulation data, flow-acoustic correlations are obtained using either the standard two-point analysis or considering two-dimensional disturbances in velocity and pressure fields, and results show significant correlation coefficients (of about 0.5) once two-dimensional disturbances near the trailing edge are isolated. A further increase of correlation peaks (up to 0.7) is obtained once the antisymmetric parts of the fields is considered, reflecting the classical antisymmetric nature of trailing-edge scattering. SPOD is then used for frequencies around the peak radiated sound to examine the structure of two-dimensional disturbances in the trailing-edge region and their contribution to radiated sound. Leading SPOD modes show coherent hydrodynamic waves propagating from the region of boundary-layer tripping toward the trailing edge, characterizing a noncompact source akin to wave packets seen in turbulent jets. These leading SPOD modes have significant contribution to the radiated sound, as two modes lead to 50% of the acoustic intensity for the lower studied frequencies. The present results point to the scattering of spanwise-coherent boundary-layer structures as the dominant mechanism of trailing-edge noise in this flow.

Figure 1

Mesh details: (a) full view of Cartesian grid composed by 896 × 511 × 64 nodes; (b) body-fitted O-grid block composed by $1536\times 125\times 128$ nodes; (c) detailed rounded trailing edge. Panels (a) and (b) show only 1 in every 4 points of the grid.

Figure 2

Sound pressure level at observer location $x=c$, $y=7.9c$ and midspan.

Figure 3

(a) The base flow and (b) mean square streamwise velocity at three different stations of the profile: $x/c=0.7$, $x/c=0.8$, and $x/c=0.9$.

Figure 4

Instantaneous streamwise velocity fluctuation field in the wall-parallel plane. $\left(\mathrm{a}\right)y^{+}\approx 15\mathrm{and}\left(\mathrm{b}\right)y^{+}\approx 100.$

Figure 5

Spanwise averaged velocity fluctuations. $\left(\mathrm{a}\right)y^{+}\approx 15\mathrm{and}\left(\mathrm{b}\right)y^{+}\approx 100.$

Figure 6

Spanwise premultiplied power spectral density of the streamwise velocity $u$, $k_z{E}_{uu}$, at three different stations of the profile. $\left(\mathrm{a}\right)x/c=0.7,\left(\mathrm{b}\right)x/c=0.8,\mathrm{and}\left(\mathrm{c}\right)x/c=0.9.$

Figure 7

Spanwise power spectral density of the streamwise velocity $u$ at three different stations of the profile. $\left(\mathrm{a}\right)x/c=0.7,\left(\mathrm{b}\right)x/c=0.8,\mathrm{and}\left(\mathrm{c}\right)x/c=0.9.$

Figure 8

Two-dimensional premultiplied energy spectrum, $k_z\omega E_{uu}({\lambda }_z,{\lambda }_t)$, of the streamwise velocity for wall-normal positions $y^{+}\approx 15$ at (a) $x/c=0.7$, (b) $x/c=0.8$, and (c) $x/c=0.9$, and for $y^{+}\approx 100$ at (d) $x/c=0.7$, (e) $x/c=0.8$, and (f) $x/c=0.9$. The red dashed line represents the ${\lambda }_a^{+}=c_0/u_{\tau }{\lambda }_t^{+}$, which delimits the propagative (P) and evanescent (E) regions.

Figure 9

Power spectral densities of pressure at $x/c=1$ and $y/c=1$.

Figure 10

Correlation coefficient of surface pressure at $x/c=0.97$, and far-field pressure at $x/c=1.0$ and $y/c=1.0$. The correlation $p_{ff}\times p_{nf}$ (far-field pressure, $p_{ff}$, and near-field pressure, $p_{nf}$), indicated by $\left(\mathrm{——}\right)$, is calculated with the far-field pressure fluctuation at $x/c=1.0$ and $y/c=1.0$ and correlated with wall pressure fluctuation at $x/c=0.97$; correlation of $p_{ff}\times \mathrm{\Delta }{p}_{nf}$, indicated by $\left(\mathrm{——}\right)$ is computed between far-field pressure fluctuation and wall pressure difference between upper an lower surface at the same location.

Figure 11

Peak correlation between grid 1 pressure and far-field pressure. Both quantities are spanwise averaged.

Figure 12

Peak correlation coefficient between near-field pressure in the trailing-edge region and acoustic pressure. (a) Correlation between the quantities without spanwise-average and (b) Correlation between spanwise-averaged quantities.

Figure 13

Peak correlation between grid 1 pressure difference of the upper and lower surface of the airfoil and far-field pressure; spanwise-averaged quantities.

Figure 14

Peak negative correlation between $u$-component and acoustic field pressure. Peak negative correlation is plotted in a reversed scale to allow direct comparison with the color scales of Fig. 12 and other similar plots. (a) Correlation between the quantities without spanwise-average and (b) Correlation between spanwise-averaged quantities.

Figure 15

Minimum correlation between grid 1 $u$-component, with difference between the upper and lower surfaces of the airfoil, and acoustic field pressure. Fluctuations were spanwise averaged. See comments in the caption of Fig. 14.

Figure 16

Comparison between the RMS of grid 1 $u$-component without and with spanwise-averaging. (a) RMS of grid 1 u-component without spanwise-average and (b) RMS of grid 1 u-component with spanwise-average.

Figure 17

(a) SPOD eigenvalues; (b) cumulative contribution of SPOD modes to total TKE.

Figure 18

SPOD mode 1. (a) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=4.95$, (b) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=4.95$, (c) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=7.42$, (d) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=7.42$, (e) , (f) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=12.37$, (g) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=17.32$, (h) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=17.32$, (i) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=22.26$, and (j) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=22.26$.

Figure 19

SPOD mode 2. (a) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=4.95$, (b) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=4.95$, (c) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=7.42$, (d) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=7.42$, (e) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=12.37$, (f) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=12.37$, (g) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=17.32$, (h) , (i) ${\mathrm{\Phi }}_u\mathrm{at}\mathrm{He}=22.26$, and (j) ${\mathrm{\Phi }}_v\mathrm{at}\mathrm{He}=22.26$.

Figure 20

First and second SPOD mode for pressure corresponding to the two lower frequencies: (a), (b) $\mathrm{He}=4.95$ and (c), (d) $\mathrm{He}=7.42$.

Figure 21

First and second SPOD modes for pressure corresponding to the two higher frequencies: (a), (b) $\mathrm{He}=17.32$ and (c), (d) $\mathrm{He}=22.26$.

Figure 22

(a) Region of the acoustic field $\mathrm{\Omega }$ (dashed line) used to integrate the energy representation of pressure. The continuous line represents the NACA 0012 profile. (b) Contribution of first $k$ SPOD modes to radiated pressure on the $\mathrm{\Omega }$.

Figure 23

Convergence of the SPOD modes, following Eq. (A1). $\left(\mathrm{a}\right)\mathrm{He}=4.95,\left(\mathrm{b}\right)\mathrm{He}=7.42,\left(\mathrm{c}\right)\mathrm{He}=12.37,\left(\mathrm{d}\right)\mathrm{He}=17.32,\mathrm{and}\left(\mathrm{e}\right)\mathrm{He}=22.26.$