Modeling the nonlinear aeroacoustic response of a harmonically forced side branch aperture under turbulent grazing flow

Hydrodynamic modes in the turbulent mixing layer over a cavity can constructively interact with the acoustic modes of that cavity and lead to aeroacoustic instabilities. The resulting limit cycles can cause undesired structural vibrations or noise pollution in many industrial applications. To further the predictive understanding of this phenomenon, we propose two physics-based models which describe the nonlinear aeroacoustic response of a side branch aperture under harmonic forcing with variable acoustic pressure forcing amplitude $p_a$. One model is based on Howe's classic formulation that describes the shear layer as a thin vortex sheet, and the other is based on an assumed vertical velocity profile in the side branch aperture. These models are validated against experimental data. Particle image velocimetry (PIV) was performed to quantify the turbulent and coherent fluctuations of the shear layer under increasing $p_a$. The specific acoustic impedance $Z$ of the aperture was acquired over a range of frequencies for different bulk flow velocities $U$ and acoustic pressure forcing amplitudes $p_a$. In this work, we show that once the handful of parameters in the two models for $Z$ have been calibrated using experimental data at a given condition, it is possible to make robust analytical predictions of this impedance over a broad range of the frequency, bulk flow velocity, and forcing amplitude. In particular, the models allow prediction of a necessary condition for instability, implied by negative values of the acoustic resistance $\mathrm{Re}\left(Z\right)$, which corresponds to a reflection coefficient $R$ of the aperture with magnitude larger than 1. Furthermore, we demonstrate that the models are able to describe the nonlinear saturation of the aeroacoustic response caused by alteration of the mean flow at large forcing amplitudes, which was recently reported in literature. This effect stabilizes the coupling between the side branch opening and the acoustic field in the cavity, and its quantitative description may be of value for control of aeroacoustic instabilities. We visualize and compare the models' representations of the hydrodynamic response in the side branch aperture and of the saturation effect under increasing $p_a$.

Figure 1

PIV data obtained from the center plane of the side branch aperture visualizing the spatiotemporal evolution of the hydrodynamic disturbance in the shear layer for a mean bulk flow velocity $U=74.1\mathrm{m}/\mathrm{s}$ under acoustic forcing with frequency $f=1500\mathrm{Hz}$ and acoustic pressure forcing amplitudes $p_a$ of $10\mathrm{Pa}$ (top row), $50\mathrm{Pa}$ (middle row), and $300\mathrm{Pa}$ (bottom row), respectively. Shown is the phase-averaged streamwise velocity $\langle v_x(\mathit{x},t)\rangle ={\overline{v}}_x\left(\mathit{x}\right)+{\tilde{v}}_x(\mathit{x},t)$ at four equally spaced time instants over a full acoustic forcing cycle. The phase $\omega t$ of the acoustic forcing is displayed above the frames in the top row.

Figure 2

PIV data obtained from the center plane of the side branch aperture visualizing the spatiotemporal evolution of the hydrodynamic disturbance in the shear layer for a mean bulk flow velocity $U=74.1\mathrm{m}/\mathrm{s}$ under acoustic forcing with frequency $f=1500\mathrm{Hz}$ and acoustic pressure forcing amplitudes $p_a$ of $10\mathrm{Pa}$ (top row), $50\mathrm{Pa}$ (middle row), and $300\mathrm{Pa}$ (bottom row), respectively. Shown is the vector field of the coherent velocity fluctuations $\tilde{\mathit{v}}(\mathit{x},t)$, superimposed on the coherent vorticity fluctuations ${\tilde{\omega }}_z(\mathit{x},t)$ at four equally spaced time instants over a full acoustic forcing cycle. The phase $\omega t$ of the acoustic forcing is displayed above the frames in the top row.

Figure 3

Sketch of the experimental configuration. The input parameters of model 1 are shown: the mean bulk flow velocity $U$, the side branch width $W$, the channel height $H$, and the acoustic pressure forcing $p_{a}cos\omega t$ that is applied above the side branch opening. In the cutout, the real part of the vortex sheet displacement ${\zeta }_R(\xi ,t)=\mathrm{Re}(\zeta \left(\xi \right)e^{-i\omega t})$, the mean streamwise velocities just above and below the vortex sheet, $U_{+}$ and $U_{-}$, respectively, and the scaled streamwise coordinate $\xi =2x/W$ are shown.

Figure 4

Sketch of model 2. The coordinate system used for model 2 is shown, as well the convective speed of the perturbations $u_c$ and a typical distribution of the coherent vertical velocity field in the aperture $v_{y,c}(x,t)$ at a given time instant $t$. The contributions of the different regions in the aperture to the coherent volume flux $Q$ defined in Eq. (2) are also indicated. Note that the origin is not the same for Figs. 3 and 4.

Figure 5

Comparison of the specific acoustic impedance $Z$ of the side branch opening, computed from model 1 (solid green line) and model 2 (solid blue line) with the experimentally measured values of the same from Ref. [20] (black dots). Shown are the real (first and third rows) and imaginary parts (second and fourth rows) of $Z$ for different bulk flow velocities $U$ for the acoustic pressure forcing amplitude $p_a=20\mathrm{Pa}$. The model parameters were calibrated to the data for the case $U=74.1\mathrm{m}/\mathrm{s}$. Note that the frequency range considered for the cases $U=56.5$ and $U=60.0\mathrm{m}/\mathrm{s}$ is different than for the other cases. The curves for which the calibration of the empirical parameters in the models was performed are indicated by an increased line thickness compared to the other cases.

Figure 6

Comparison of the specific acoustic impedance $Z$ of the side branch opening, computed from model 1 (solid green line) and model 2 (solid blue line) with the experimentally measured values of the same from Ref. [20] (black dots). Shown are the real (first and third rows) and imaginary parts (second and fourth rows) of $Z$ for different acoustic pressure forcing amplitudes $p_a$ at the bulk flow velocity $U=74.1\mathrm{m}/\mathrm{s}$. The model parameters were calibrated to the data for the cases $p_a=20$ and $p_a=800\mathrm{Pa}$, as detailed in Sec. 2. The curves for which the calibration of the empirical parameters in the models was performed are indicated by an increased line thickness compared to the other cases.

Figure 7

Hydrodynamic response in the side branch opening: Shown are the vertical velocity induced by the vortex sheet displacement, $\mathrm{Re}(-i\omega \zeta (x,t))$ (dashed curves), predicted by model 1, and the velocity field $v_{y,c}(x,t)$ (continuous curves), given by model 2, in the aperture at the forcing frequency $f=1050\mathrm{Hz}$ and bulk flow velocity $U=74.1\mathrm{m}/\mathrm{s}$ for increasing forcing pressure amplitudes $p_a=20\mathrm{Pa}$ (blue curves), $p_a=200\mathrm{Pa}$ (red curves), and $p_a=400\mathrm{Pa}$ (green curves). The phase $\omega t$ of the acoustic forcing is displayed above each frame.

Figure 8

(a) Magnitude and (b) phase of the reflection coefficient $R=(Z-1)/(Z+1)$ at the forcing frequency $f=1050\mathrm{Hz}$ and grazing flow speed $U=74.1\mathrm{m}/\mathrm{s}$ for increasing forcing pressure amplitudes $p_a=20\mathrm{Pa}$ (blue dots), $p_a=200\mathrm{Pa}$ (red dots), and $p_a=400\mathrm{Pa}$ (green dots). The dashed and continuous lines correspond to models 1 and 2, respectively. Also shown are the experimentally measured values of $\left|R\right|$ and $\angle \left(R\right)$ at the same conditions, indicated by the black dots and the dash-dotted curves. The conditions for which the calibration of the empirical parameters in the models was performed are indicated by an increased marker size compared to the other cases.

Figure 9

Visualization of the saturation effect over a range of frequencies and acoustic pressure forcing amplitudes $p_a$ for increasing bulk flow velocities $U$. Shown are the contour plots of $\left|R\right|$ for different values of $U$. The black arrow indicates the direction of increasing $U$. The condition for which the calibration of the parameters in the models was performed is indicated by a black frame around the respective insets.

Figure 10

Sensitivity $S$, defined in Eq. (C2), for models 1 and 2 at $U=74.1\mathrm{m}/\mathrm{s}$ and $p_a=20\mathrm{Pa}$ with respect to the empirical parameters shown in the $x$ axis. The parameters are ordered by increasing $S$. The bar plot shows the relative increase of $D_{\mathrm{RMS}}$ between the model and the experimentally acquired values of $Z$ after calibration when a given empirical parameter is excluded from the respective model.