Analysis of Resonance in Jet Screech with Large-Eddy Simulations

Screech is an aero-acoustic resonance phenomenon in imperfectly expanded supersonic jets first discovered by Powell (1953). Screech produces high-intensity tones at distinct frequencies, making it a notoriously riling component of supersonic jet noise. In addition, jet screech can induce high-dynamic loading which, if it is close to the structural resonance frequencies, could cause premature fatigue failure of components around the jet engines such as those reported on VC10, F-15, B1-B, and F-35 (Raman, 1997; Raman et al., 2012; Majumdar, 2014).

Previous research on supersonic jet screech has been comprehensively reviewed by Raman (1999) and Edgington-Mitchell (2019), which highlight the experimental observations from 1950s to date and the theories based on these empirical findings. The classical description of screech contains four components: the downstream traveling instability waves originated from the jet initial shear layer, the interactions between instability waves and shock cells in the jet plume, the upstream traveling component that re-excites the initial shear layer, and lastly the receptivity process of the shear layer near nozzle exit. The oppositely moving coherent structures give rise to the presence of spatially modulating standing wave patterns, first documented by Panda (1999). More recent work by Edgington-Mitchell et al. (2021) further investigates the distinct groups of waves in jet screech. Two groups of waves are found to be traveling downstream: the Kelvin-Helmholtz (K-H) instability wave and a duct-like mode. The upstream-traveling component includes external acoustic waves and a guided jet mode, first discovered by Tam & Hu (1989).

The role of upstream-traveling waves in screech closure has been a topic of debate until recently. The original depiction of screech by Powell (1953) considers upstream-traveling acoustic waves outside the jet plume to be the closure mechanism. On the other hand, various experimental and numerical evidence suggests the guided jet mode closes the screech feedback path (Bogey & Gojon, 2017; Edgington-Mitchell et al., 2018; Gojon et al., 2019). This new insight leads to improved model for screech frequency prediction. Mancinelli et al. (2021) uses a locally parallel linear stability analysis about the fully expanded base state to predict A1 and A2 mode frequencies in under-expanded axisymmetric jets, but the model requires a posteriori calibration of reflection coefficients for the phase resonance condition. Nogueira et al. (2022a, b) construct a linear stability model by including spatial periodicity in the base state due to shock cells, and finds an absolute instability mechanism for screech generation. This semi-local model is able to accurately predict the staging of screech modes across a range of NPR values.

Despite the success of these recent models, it is still difficult to predict the screech amplitude due to two major problems. The first problem is the sound generation mechanism. Given the presence of the waves and shock cells, the emission of the highly directive screech tones could be attributed to localized shock leakage (Edgington-Mitchell et al., 2020; Suzuki & Lele, 2003; Manning & Lele, 2000) or distributed sources by the Mach-wave radiation mechanism (Tam et al., 1986). The second problem is how the gain of screech feedback is determined. In order to answer this, the spatially non-local characteristics of screech feedback need to be considered, including the receptivity close to the nozzle exit, the saturation region where K-H waves and the guided jet mode grow to maximum magnitudes, and the dissipation region where large-scale waves turn into smaller-scale turbulence at the end of the jet potential core. The purpose of the current work is to shed light on the second problem.

High-fidelity large-eddy simulations (LES) data for a 4:1 rectangular screeching jet with intense anti-symmetric flapping are used to analyze the global behaviors of the screech feedback. Through previous works, the numerical results have been validated against experiments conducted at Florida State University. Excellent agreement between the LES data and the experimental data is found (Wu et al., 2022). Using the LES data and spectral proper orthogonal decomposition (SPOD) (Lumley, 1970; Towne et al., 2018; Schmidt & Colonius, 2020), the dominant coherent structures at the screech fundamental frequency are examined. The magnitude and phase of the external acoustic wave, the guided jet mode and the K-H wave are analyzed. This paper is organized as follows: Section 2 summarizes the numerical and data analysis methods, Section 3 characterizes the dominant SPOD modes at the screech fundamental frequency, Section 4 examines the role of upstream-traveling guided jet mode versus the external acoustic wave for screech closure, Section 5 quantifies the wave energy and describes the global behavior of these instability waves, Section 6 complements the analysis with results from three different NPR conditions, Section 7 relates the SPOD modes with the acoustic radiation of the screech tone in the far field, and lastly Section 8 concludes with the main findings of the study.

With the procedure outlined in Section 2.2, the leading-order SPOD mode for the case at $M_{j}=1.69$ is computed using $700$ acoustic time units of data. The acoustic time unit is defined by the time it takes for sound in the ambient to travel a distance of $h$, the minor dimension of the nozzle exit. Figure 4 (a) shows the energy spectra of the SPOD modes for transverse velocity fluctuations $v^{\prime}$. A high-energy peak is seen at the screech fundamental frequency, $St=0.23$, with clear rank separation between the first mode and the others. Figure 4(b) shows the modulus of the first SPOD mode function at $St=0.23$ with respect to the wavenumber in the streamwise direction $k_{x}$ and the transverse coordinate $y$. The contour plot indicates the presence of waves with opposite signs of wavenumbers. Here, the sign of the wavenumber can be used as a proxy for the direction of wave propagation. The interference of these oppositely-traveling waves give rise to spatially modulating standing waves in the jet plume, which have been observed and studied in detail in previous screech experiments (Panda, 1999).

By only using the positive or negative $k_{x}$ for spatial reconstruction, figure 5 separates the downstream-traveling and upstream-traveling components from the SPOD mode. The downstream-traveling waves are the K-H instability waves commonly seen in subsonic and supersonic jets (Schmidt et al., 2018; Nekkanti & Schmidt, 2020). The upstream-traveling waves contain both external acoustic waves and the guided jet mode (Tam & Hu, 1989). To facilitate the remainder of the discussion, we use $c_{-}$ to denote the upstream-traveling acoustic wave, $k_{-}$ to denote the upstream-propagating guided mode and $k_{+}$ to denote the downstream-propagating K-H wave.

Figure 4(b) shows both the upstream- and the downstream-traveling waves are composed of multiple Fourier modes, with one dominant group close to the acoustic wavenumber and additional subdominant groups at higher wavenumbers. The peak wavenumbers for $k_{-}$ and $k_{+}$ are summarized in Table 2, alongside with the peak wavenumber of the shock cells $k_{s}$ from the mean flow data. Figure 4(b) indicate the peak wavenumbers for $k_{-}$ and $k_{+}$ have a phase velocity slightly less than the ambient speed of sound. This feature agrees with other experimental and theoretical works (Tam & Ahuja, 1990; Edgington-Mitchell et al., 2021). Values of $k_{-}$ and $k_{+}-|k_{s}|$ indicate the guided jet mode, the K-H wave and the shock cell structures could exchange energy via triadic interactions. Strong interactions between the K-H wave and the shock cells could excite the guided jet mode, and the growth of the guided jet mode could further reinforce the K-H and shock interactions. This has also been suggested by recent experimental and theoretical works (Edgington-Mitchell et al., 2018, 2021, 2022). The SPOD modes for other variables, including fluctuations of density, fluctuations of velocity in $x$ and $z$ direction as well as pressure, have also been computed. The results are presented in Appendix A.

Having obtained the dominant coherent structures at the screech frequency using SPOD, one can further examine the upstream-traveling waves to investigate the screech closure mechanism, the pathway that redirects energy from downstream back to the nozzle exit. To do this, the spatial cross correlation method outlined in Section 2.3 is used to analyze the leading-order SPOD mode for pressure fluctuations $p^{\prime}$. Using Eq. (14), the time delay $\tau$ and relative magnitude variation $\alpha$ for $c_{-}$ and $k_{-}$ with respect to the nozzle exit are computed. To differentiate between acoustic and hydrodynamic fluctuations, the spatial cross-correlation for $c_{-}$ is computed along $y/h=4$, and that for $k_{-}$ is done along $y/h=0.2$. These locations are marked by the dashed blue lines in Figure 6(a). Figure 6(a) also marks the acoustic wavenumber with the red solid lines. The acoustic waves outside the jet plume center around the acoustic wave number. The $k_{-}$ wave for $p^{\prime}$ appears as a pair of symmetric lobes with respect to the jet centerline, and the peak wavenumber lies to the left of the acoustic wavenumber. Figure 6(b) shows the modulus of the SPOD shape function at a slightly higher frequency than the screech frequency. At this non-resonant frequency, there are still upstream-traveling acoustic waves outside the jet plume, but there is no $k-$ wave inside the jet plume, judging by the lack of symmetry with respect to the jet centerline. In addition, the $k+$ wave can be found at both frequencies, so energy still propagates in the downstream direction. Therefore, the $k-$ wave seems to be the differentiating factor between the screech and the non-screech conditions.

Results from the cross-correlation further confirms the growth of the $k-$ wave is the key for screech to take place. Figure 7(a) shows the relative magnitude $\alpha$ for the wave signal tracked along $y/h=5$. Because of the relatively large distance away from the jet boundary, this signal would mostly be associated with the acoustic wave $c-$. Figure 7(b) cross correlates the wave signal along $y/h=0.2$, indicating the growth of the $k-$ wave inside the jet plume. The spatial growth pattern between $c_{-}$ and $k_{-}$ differs in two ways. Firstly, the maximum of $\alpha_{k_{-}}$ is larger than $\alpha_{c_{-}}$ by one order of magnitude at the screech fundamental frequency. This shows the spatial growth of $k_{-}$ wave relative to the nozzle exit is much more significant than that of $c_{-}$. Secondly, $\alpha_{k_{-}}$ is highly sensitive to frequency; the spatial growth is the largest at the screech frequency, $St_{1}$. In contrast, the main peak of $\alpha_{c_{-}}$ does not vary significantly between $St_{1}$ and a few of its neighboring frequencies. At the two higher non-resonant frequencies, there are secondary peaks in the ${c_{-}}$ wave between $x/h=20$ and $x/h=25$ that reach similar or slightly higher values of $\alpha$ than the screech acoustic wave, and these may be the upstream component of the super-directive broadband acoustic radiation, due to the K-H wave near the end of the jet potential core, as discussed by Nekkanti & Schmidt (2021). Overall, unlike $k_{-}$, the growth of $c_{-}$ is non-sensitive to the range of frequencies close to the neighborhood of the screech frequency. These differences are direct evidence showing the dominance of the $k_{-}$ wave over the $c-$ wave in closing the screech feedback loop.

Section 3 shows energy transfer is possible via triadic interactions among the shock cells $k_{s}$, the $k_{+}$ and the $k_{-}$ waves. Section 4 then shows the $k_{-}$ wave is the deciding factor for propagating energy in the upstream direction. These arguments can be further supported by quantifying the wave energy carried by these oppositely-traveling waves using Eq. (16).

Figure 8 shows $E_{k_{-}}$, $E_{k_{+}}$, and $E_{k_{+}}/E_{k_{-}}$ with respect to $x/h$ using the SPOD data for $v^{\prime}$. These results can be best interpreted together with figures 2 and 5. $E_{k_{-}}$ and $E_{k_{+}}$ are reflective of the nonlinear growth and decay of the $k_{-}$ and $k_{+}$ wave over the streamwise direction. They are also functions dependent on frequency. At the screech fundamental frequency, the maxima of $E_{k_{-}}$ and $E_{k_{+}}$ are found to be at $x/h=14.4$ and $x/h=16.9$, respectively. Interestingly, the value of $E_{k_{+}}/E_{k_{-}}$, with focus on the screech tone $St_{1}$, indicates four distinct regions in the jet plume: $x/h$:$[0,5]$, $[5,15]$, $[15,25]$, $[25,30]$. The first region ($\mathrm{I}$) is between $x/h=0$ and $x/h=5$, where $E_{k_{+}}/E_{k_{-}}$ undergoes growth from a low initial value. This likely corresponds to the receptivity process where energy of the $k_{-}$ excites the jet initial shear layers to form embryonic $k_{+}$ wave. The second region ($\mathrm{II}$) is at $5\leq x/h\leq 15$, where $E_{k_{+}}/E_{k_{-}}$ reaches a plateau. In this region, both $E_{k_{-}}$ and $E_{k_{+}}$ grow with respect to the downstream direction at a similar rate. The first two regions also coincide with the jet supersonic core where effects of the shock cells are visible. The third region ($\mathrm{III}$) is located between $x/h=15$ and $x/h=25$, where $E_{k_{+}}/E_{k_{-}}$ increases rapidly due to the fast fall-off of $E_{k_{-}}$. At $x/h=25$, $E_{k_{+}}/E_{k_{-}}$ has a maximum value. The maximum of $E_{k_{+}}/E_{k_{-}}$ indicates the endpoint where the $k_{-}$ wave reaches a significant magnitude with respect to the $k_{+}$ wave. Recall that the $k-$ wave propagates in the $-x$ direction, so the profile between $x/h=15$ and $x/h=25$ indicates that $E_{k_{-}}$ grows, saturates and decays in the $-x$ direction. Therefore, the third region corresponds to the excitation of the guided jet mode. Considering the mean velocity contours in figure 2, the third region extends to the end of the jet supersonic core. Additionally, both the second and third regions align with locations where acoustic waves are emitted as seen in the SPOD structures in figure 5. This is further verified in Section 7. Lastly, the fourth region ($\mathrm{IV}$) is located downstream of $x/h=25$, where $E_{k_{+}}/E_{k_{-}}$ drops rapidly as the $k_{+}$ wave breaks down at the end of the jet potential core.

The four distinct regions are also observed across other flow variables, as shown by figure 9 using the SPOD data for $\rho^{\prime}$, $u^{\prime}$, $v^{\prime}$ and $p^{\prime}$. In summary, the values of $E_{k_{-}}$, $E_{k_{+}}$, and $E_{k+}/E_{k-}$ highlight the spatially non-local and non-periodic characteristics of various processes involved in screech resonance. $E_{k+}/E_{k-}$ can be used as a metric to identify distinct regions in the jet plume that correspond to different processes of screech. For the K-H wave: region $\mathrm{I}$ corresponds to the receptivity/excitation of the K-H wave ($k_{+}$) in the initial shear layer, region $\mathrm{II}$ corresponds to the growth of the $k_{+}$ wave and interaction with shock-cells, and regions $\mathrm{III}$ and $\mathrm{IV}$ are related to the saturation and decay of the k+ wave. For the guided jet mode $k_{-}$, region $\mathrm{III}$ corresponds to the excitation and growth in the $-x$ direction, and region $\mathrm{II}$ is related to the saturation and shock associated modulation of the $k_{-}$ wave.

This section repeats the same analysis carried out in Section 5 for two more NPR conditions. For all three screech cases considered, similar spatial variation regarding $E_{k_{-}}$, $E_{k_{+}}$, and $E_{k+}/E_{k-}$ are observed in Figure 10. Figure 10(c) shows as NPR increases, region I shortens while region II and III lengthens. At higher NPR values, the increased velocity difference between the jet plume and the ambient makes the initial shear layer more unstable and more receptive to Kelvin-Helmholtz instability. The stronger shock cells and longer supersonic core also enable stronger excitation of the $k-$ wave. The maxima of $E_{k+}/E_{k_{-}}$ are marked by the solid circles in Figure 10(c). Their streamwise locations are also indicated in the other subfigures. Figure 10 (d) indicates the endpoint can be taken to be the end of the jet supersonic core which moves further downstream with increasing NPR. Table 3 summarizes the screech frequency and the screech amplitude measured at $100h$ away from the nozzle exit from two jet polar angles. As NPR increases, the screech amplitude increases, which is consistent with the observed increase in length for regions II and III. As verified later in Section 7, these two regions are indeed associated with the distributed acoustic source for the screech tone.

At the screech frequency, the near-field hydrodynamic fluctuations are dominated by spatial-temporal coherent structures, represented by the leading order SPOD mode, as shown in previous sections. In this section, we present direct evidence linking the acoustics of screech with the SPOD structures using Lighthill’s equation (Lighthill, 1952). By an exact rearrangement from mass and momentum balance, Lighthill shows the pressure fluctuations, $p^{\prime}=p-p_{0}$ in the far field from any flow with arbitrary motion are analogous to the acoustic response in an ideal medium at rest due to quadrupole acoustic sources.

and

The term $T_{ij}$ is Lighthill’s stress tensor, which includes all the actual flow effects on noise generation in the acoustic analogy, where $\rho u_{i}u_{j}$ is the momentum component, $\tau_{ij}$ is the viscous stress, and $p^{\prime}-c_{0}^{2}\rho^{\prime}\delta_{ij}$ is due to entropy changes. In the following analysis, we show that the SPOD modes can be directly used to approximate the $T_{ij}$ associated with screech tone.

In the frequency domain, the solution for $p^{\prime}$ far away from the acoustic source region at a location $\mathbf{y}$ can be written as

where $\omega$ is the angular frequency, $V$ is the control volume enclosing the acoustic source, $\hat{T}_{ij}(\mathbf{x},\omega)$ is the Fourier component of $T_{ij}$, and $G_{0}$ is the free-space Green’s function. Applying integration by part and the divergence theorem, and taking the surface integrals to be zero, we can shift the spatial derivatives from $\hat{T}_{ij}$ to the Green’s function.

In non-heated high-Reynolds-number jets, effects of viscous stress and entropic inhomogeneity are small compared to momentum, so

Using Reynolds decomposition, $\rho u_{i}u_{j}$ can be expressed as

where $\bar{()}$ denotes the mean quantity in time and $()^{t}$ denotes the unsteady component. For the purpose of calculating the far-field acoustic pressure, we ignore the steady term in time and only keep the first-order unsteady terms.

Then the time-Fourier component of $T_{ij}$ simply becomes

At the screech frequency, the near-field hydrodynamic fluctuations are dominated by spatial-temporal coherent structures as shown in previous sections. This suggests the generation of screech noise could be well approximated by the waves from the leading-order SPOD mode. To verify this hypothesis, we substitute the SPOD data in place for $\hat{\rho}^{t}$, $\hat{u}_{i}^{t}$ and $\hat{u}_{j}^{t}$ to calculate $\hat{T}_{ij}$. Figure 11 shows the amplitude and phase for the complex valued $\hat{T}_{ij}$ at the screech frequency for the case at $\mathrm{NPR}=4.86$. The $\hat{T}_{11}$ and $\hat{T}_{12}$ are the most dominant terms because of the large streamwise velocity fluctuations, and the spatial distribution spans between $x/h=5$ and $x/h=25$, which aligns with regions $\mathrm{II}$ and $\mathrm{III}$ identified by $E_{k+}/E_{k_{-}}$ in Section 5.

The term involving the free-space Green’s function in the integral solution for $\hat{p}^{\prime}$ is

where $\mathbf{r}=\mathbf{y}-\mathbf{x}$ is the displacement vector from the source to the observer location, and $k_{c}=\omega/c_{0}$ is the acoustic wavenumber. For $|\mathbf{r}|/L_{s}>>1$, where $L_{s}$ is the length scale associated with the distributed acoustic source, the term involving the Green’s function can be approximated as

where $\mathbf{r_{0}}=\mathbf{y}-\mathbf{x_{c}}$ is the displacement from the center of the acoustic source to the observer, and $\mathbf{\xi}=\mathbf{x_{c}}-\mathbf{x}$. For the case analyzed, $x_{c}$ is chosen to be $15h$.

The far-field acoustic pressure at $270h$ away from the nozzle exit in the minor-axis plane is calculated using Eq. 20. $T_{ij}$ is assumed to have a uniform distribution in the major-axis direction between $z/h=-5$ and $z/h=5$ and zero elsewhere. This is a crude simplification of the acoustic source, which is not homogeneous in the $z$ direction due to the rectangular nozzle geometry. Nevertheless, the result of the screech tone amplitude compares favorably with the LES-FWH calculation, as shown in Figure 12. In particular, values marked by the black circles are calculated with Eq. 20, Eq. 24 and Eq. 25. Since $T_{11}$ and $T_{12}$ are much larger compared to the other components of $T_{ij}$, values shown by the red crosses are computed only using these two as the source terms. Those indicated by the purple line are obtained with further simplification on $\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}G_{0}$ using Eq. 26. The various forms of simplification do not yield significant variation in the calculated screech tone amplitude. With varying jet polar angles in the minor-axis plane, the acoustic analogy solutions also capture the screech tone directivity very well.

The good agreement between the acoustic analogy and the FW-H results have a few important implications. First, this verifies the coherent wave structures from SPOD are good representation of the noise generation mechanism that produces far-field screech tones. The SPOD data include the $k_{-}$ and $k_{+}$ waves, which propagate energy in the upstream and downstream directions inside the jet plume and establish the feedback resonance. Simultaneously, areas where the growth and interaction of these waves with the shock cells take place (regions $\mathrm{II}\&\mathrm{III}$) also correspond to the region of a spatially distributed acoustic source for screech tones. The outcome of the interaction of the $k_{+}$ wave with the mean flow, including the mean shock cell structures, includes the $k-$ and the acoustic waves. The acoustic waves are also subject to refraction in propagation across the mean jet shear layer. These processes are reflected in the rapid phase variation of $\hat{T}_{ij}$ across the jet in Figure 11. The far-field Green’s function 26 contains the geometric angular factors as well as phase variation associated with retarded time variation over the source region. When this is combined with the spatially distributed source in Equation 20 the far-field acoustic radiation embodies the constructive and destructive interference of the effective acoustic sources, resulting in the screech tone directivity pattern. In future, these results can be used to improve screech amplitude prediction models, by constructing the acoustic sources with a functional form suggested by the SPOD mode and the mean jet state.

Using high-fidelity large-eddy simulation data, this work studies the screech generation mechanism in a 4:1 rectangular jet under three under-expanded NPR conditions. The analysis combines spectral proper orthogonal decomposition (SPOD), Fourier decomposition in the streamwise direction and spatial cross correlation using the LES data. For the case at NPR = 4.86, a detailed assessment of the dominant coherent structures and their relevance in screech generation is conducted. The effects of NPR variation are evaluated by repeating the same analysis for the other two NPR conditions. The first SPOD mode at the screech fundamental frequency is composed of acoustic waves, the upstream-traveling guided jet mode and the downstream-traveling Kelvin-Helmholtz wave. Estimates of the wave energy indicate global nonlinear characteristics in the streamwise direction.

The main findings of the analysis agree with recent theoretical understanding of screech instability and shed light on the spatial separation of individual processes for screech generation. Direct evidence of the guided jet mode being the screech closure mechanism, not the external acoustic wave, is observed by considering the growth of waves across screech frequency and its neighboring frequencies. Quantification of the wave energies ($E_{k_{+}}$, $E_{k_{-}}$, and their ratio $E_{k_{+}}/E_{k_{-}}$) identifies regions where distinct processes in screech generation take place, including initial shear layer receptivity, sound generation, guided jet mode excitation and coherence decay due to turbulence. The initial shear layer receptivity region is marked by a rapid spatial growth of $E_{k_{+}}/E_{k_{-}}$ near the nozzle. From there, $E_{k_{+}}/E_{k_{-}}$ plateaus then sharply increases in value, corresponding to the origin, growth and saturation of the guided jet mode in the $-x$ direction. This region also aligns with the location where Lighthll’s stress tensor $T_{ij}$, approximated by the SPOD mode and the mean flow (including the shock cells), is large in amplitude. The acoustic analogy formulation with $T_{ij}$ and the free-space Green’s function provides a good estimate of the far-field screech tone noise and matched well with the FW-H calulations using the original LES data. This further verifies the coherent structures highlighted by the leading-order SPOD mode capture both the dominant hydrodynamic and acoustic processes involved in screech generation. The current findings can be extended construct amplitude prediction model for jet screech in future.

We would like to thank Cascade Technologies for their LES solver and Dr. Guillaume A. Brès for his substantial support and guidance on our work. We thank Professor Rajan Kumar from Florida State University for sharing with us the experimental data.

This work is supported by the ONR grant N00014-18-1-2391 with Dr. Steve Martens as the project manager and the XSEDE (TG-CTS190021) program.

The authors report no conflict of interest.