Global analysis of nonlinear second-mode development in a Mach-6 boundary layer from high-speed schlieren data

Abstract

The second-mode instability on a 7° half-angle sharp cone at Mach 6 is analyzed using high-speed calibrated schlieren imagery at a frame rate near the expected fundamental frequency. Experiments were conducted in the NASA Langley 20-Inch Mach 6 facility at unit Reynolds numbers between $6.56 \; \times \; 10^{6}$ and $9.71 \;\times \; 10^{6} \text {m}^{-1}$. Time-resolved pixel intensity signals throughout the boundary layer are reconstructed using spatially available data in the schlieren images to recover an effective sampling rate of over 10 MHz; these are then converted to quantitative density gradients using a thin-lens-based calibration technique. A global analysis is performed on the schlieren data to investigate the nonlinear growth of the second-mode fundamental and harmonic content. Pointwise measures of the autobicoherence are used to identify specific triadic interactions and the locations of their highest levels of quadratic phase coupling. Significant resonance interactions between the second-mode fundamental and harmonic instabilities were found along with interactions between these and the mean flow. Bispectral mode decomposition is employed to educe the flow structures associated with these interactions. A similar analysis is performed for the power spectrum, with power spectral densities computed for each pixel’s time-series and spectral proper orthogonal decomposition employed to derive the modal structure and energy of the flow at specific frequencies. Comparisons between the bispectral quantities and second-mode power show that nonlinear interactions, particularly resonance interactions, are closely correlated with spatiotemporal modulation of disturbances during the nonlinear stage of transition.

Appendix 1

1.1 Effect of propagation speed on reconstructed signal

The application of the reconstruction technique in the present global analysis approach requires the assumption that all content has the same propagation speed. This is adequate for analysis of the second-mode fundamental and harmonics (Unnikrishnan and Gaitonde 2020) but will not be true in general, especially if other disturbance types (e.g., first-mode waves) are present in the boundary layer. In this section, we briefly analyze the effect of changing the propagation speed used in the reconstruction technique on the pixelwise power-spectral-density and autobicoherence.

As mentioned in Sect. 2.4, the reconstruction technique computes the linearly weighted average of the backward and forward spatial signals around the pixel of interest from images taken at times $t_1$ and $t_1 + \varDelta t$, where the width of spatial signals used in each image is equal to $U_p\varDelta t$. The linearly weighted average of the spatial signals are then concatenated sequentially over all images, creating a temporal signal. An error in the propagation speed of the disturbance changes the width $U_p\varDelta t$, resulting in a phase-shift between the backward and forward spatial signals used in the linearly weighted averaging. We will refer to this phase-shift between the spatial signals as $\phi$.

Suppose first that equal averaging (i.e., constant weights of 1/2) was used on the backward and forward spatial signals of images at $t_1$ and $t_1 + \varDelta t$ and an error in the propagation speed of a disturbance wave exists. The computed average wave will have a reduced amplitude (with complete destructive interference at $\phi =180^{\circ }$) and a phase equal to the average phase of the two parent waves (representing a phase error from the true physical wave of $\phi /2$). Additionally, at the point of concatenation with the subsequent averaged wave, taken from times $t_1 + \varDelta t$ and $t_1 + 2\varDelta t$, there will be a discrete phase-shift in the reconstructed signal equal to $\phi$, which occurs because the spatial signal at time $t_1 + \varDelta t$ goes from leading the signal at time $t_1$ to lagging the signal at $t_1 + 2\varDelta t$. This error creates a “ringing" in the spectrum of the reconstructed signal, with artifactual spikes arising at $nf_s+f$ and $nf_s-f$, where n is an integer, $f_s$ the original sampling rate (i.e., the camera frame rate), and f the disturbance frequency. This is the same mechanism that produces the spikes shown in Fig. 5, which originate from stationary irregularities in the images (e.g., window blemishes) and result in artifacts at $nf_s$ because f is 0. In addition, spatial changes in the mean flow (e.g., the boundary-layer height) over the propagation distance used in the reconstruction will produce narrow-banded artifacts at $nf_s$. A sufficiently high frame rate ensures that this effect is small, as changes in the mean flow state will become negligible. In these experiments, the boundary-layer growth over the schlieren field of view was estimated to lie between 2.25 and 2.75 pixels for all conditions. With an average propagation distance of approximately 38 pixels, and twice the distance used in each reconstruction procedure (incorporating forward and backward signals), this results in a minimal boundary-layer growth of 0.108 to 0.144 pixels over the spatial length used in reconstruction. Using instead a linearly weighted averaging of the backward and forward spatial signals (as is done in the present implementation) assures the averaged wave is exact at either end. This removes the discrete phase-shift at concatenation and the phase error, $\phi$, is gradually covered over the distance $U_p\varDelta t$. The artifacts remain but at a significantly decreased power.

If such errors in propagation speed exist, the reconstruction technique will therefore still correctly measure the disturbance frequency, but with amplitude and phase error. At $\phi =180^{\circ }$, the weighted averaging prevents complete destructive interference (as would be the case with equal averaging), but the artifact power is at a maximum. Past 180$^{\circ }$ phase error, the artifact power decreases but the disturbance begins to be aliased to $f + f_s$, with complete aliasing (and complete artifact removal) occurring when an entire wavelength has been either added or removed from the spatial distance $U_p\varDelta t$. This occurs when the error in propagation speed is equal to $\pm \; f_s/\kappa$, where $\kappa$ is the disturbance longitudinal wavenumber. For this reason, higher-wavenumber disturbances will experience greater amplitude and phase error from errors in their propagation speeds.

In Fig. 20, PSD and autobicoherence results (plotted with the same color scale) are presented using three different propagation speeds. The signal used is taken from condition Re97 at $s = 330$ mm and a vertical height corresponding to the maximum second-mode disturbance (approximately 0.91 mm above the cone surface). Condition Re97 was chosen for this analysis due to the higher amplitudes of second-mode harmonics. The propagation speeds used in the reconstruction were $0.9U_{p_0}$, $U_{p_0}$, and $1.1U_{p_0}$, where $U_{p_0}$ is the calculated propagation speed shown in Table 2. A 10% error encompasses the majority of variability in the propagation speed of the disturbances of current interest. Adopting $1.1U_{p_0}$ more accurately reconstructs disturbances propagating at the edge velocity, such as entropic and vortical disturbances. A propagation speed of $0.9U_{p_0}$ encompasses first-mode waves (on similar geometries at Mach 6, the first-mode has been measured to have a longitudinal phase velocity of 89% of the edge velocity (Maslov et al. 2006)) and, to some degree, free-stream disturbances (which have been shown to possess propagation speeds in the range of 63–81% of the free-stream velocity in this facility (Chou et al. 2018)). Additionally, nonlinear interactions between the second-mode fundamental or harmonics with disturbances of other families, which propagate at different speeds, will produce content that propagates at an intermediate speed, $(f_1 + f_2)/(\kappa _1 + \kappa _2)$ (Ball 1963). Disturbances of other families that are likely to be interacting in this case (e.g., first-mode waves and low-frequency disturbances) will be of lower frequency and wavenumber than the second mode. Therefore, $(f_1 + f_2)/(\kappa _1 + \kappa _2)$ will be more heavily weighted toward the second-mode propagation speed and likely to fall into the range of 10% deviation from $U_{p_0}$.

In either case of error ($\pm \;10\%$), there are minimal differences compared to the original results. The PSD (Fig. 20a) in the range of the fundamental second mode remains nearly unchanged in amplitude and frequency. However, the error in the PSD gets progressively worse at higher second-mode harmonics. This is expected, as the same error in propagation speed will produce larger phase errors during the signal reconstruction for higher-wavenumber disturbances. For this same reason, errors in the propagation speed also have the largest effect on higher-harmonic-producing nonlinear interactions observed in the autobicoherence. The quantitative error remains small for interactions involving the second-mode fundamental and the lower harmonics. For example, the $f_0 + f_0 \rightarrow 2f_0$ interaction has peak $b^2$ values of 0.84, 0.81, and 0.81 in Figs. 20b through 20d, respectively, and the $2f_0 + f_0 \rightarrow 3f_0$ interaction has peak $b^2$ values of 0.60, 0.57, and 0.53. The quantitative error increases for triadic interactions involving higher harmonics; however, the lobes of all notable interactions remain qualitatively consistent.

Finally, one may be concerned that, because the second-mode fundamental consists of an approximately 150 kHz band centered near the camera frame rate (286 kHz), the harmonics may instead themselves only be artifacts occurring at frequencies $nf_s+f$ and $nf_s-f$, where f is a fundamental frequency. Performing a quick spatial Fourier transform of the rows of pixels in the raw images discredits this, as the resulting spectra (not shown) clearly show harmonic content at the expected wavenumbers.

Fig. 20

a PSDs using prorogation speeds of $U_{p_0}$ (blue), $0.9U_{p_0}$ (red), and $1.1U_{p_0}$ (yellow). b–d Autobicoherence using propagation speeds of $U_{p_0}$ (b), $0.9U_{p_0}$ (c), and $1.1U_{p_0}$ (d)