Abstract
The Rossiter modes of an open cavity were studied using bi-global linear analysis, local instability analysis and nonlinear numerical simulations. Rossiter modes are normally seen only for short cavities; hence, in the study, the length over depth ratio was two. We focus on the critical region; hence, the Reynolds numbers based on cavity depth were close to 1000. We investigated the effect of the ratio boundary layer thickness to cavity depth, a parameter often overlooked in the literature. Increasing this ratio is destabilizing and increases the number of unstable Rossiter modes. Local instability analysis revealed that the hierarchy of unstable modes was governed by the mixing in the cavity opening. The effect of Mach number was also studied for thin and thick boundary layers. Compressibility had a very destabilizing effect at low Mach numbers. Analysis of the Rossiter mode eigenfunctions indicated that the acoustic feedback scaled to \(\mathrm{Ma}^3\) and explained the strong destabilizing effect of compressibility at low Mach numbers. At moderate Mach numbers, the instability either saturated with Mach number or had an irregular dependence on it. This was associated with resonances between Rossiter modes and acoustic cavity modes. The analysis explained why this irregular dependence occurred only for higher-order Rossiter modes. In this parameter region, three-dimensional modes are either stable or marginally unstable. Two-dimensional simulations were performed to evaluate how much of the nonlinear regime could be captured by the linear stability results. The instability was triggered by the \(10^{-13}\) flow solver noise floor. The simulations initially agreed with linear theory and later became nonlinearly saturated. The simulations showed that, as the flow becomes more unstable, an increasingly more complex final stage is reached. Yet, the spectra present distinct tones that are not far from linear predictions, with the thin boundary layer cases being closer to empirical predictions. The final stage, in general, was dominated by first Rossiter mode, even though the second one was the most unstable linearly. It seems this may be associated with nonlinear boundary layer thickening, which favors lower frequency in the mixing layer, or vortex pairing of the second Rossiter mode. The spectra in the final stages are well described by the mode R1 and a cascade of nonlinearly generated harmonics, with little reminiscence of the linear instability.
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Acknowledgements
The authors would like to thank the São Paulo Research Foundation (FAPESP/Brazil), for Grants 2018/04584-0 and 2017/23622-8; the National Council for Scientific and Technological Development (CNPq/Brazil) for Grants 134722/2016-7, 307956/2019-9 and 304859/2016-8; the US Air Force Office of Scientific Research (AFOSR) for Grant FA9550-18-1-0112, managed by Dr. Geoff Andersen from SOARD; the University of Liverpool for the access to the Barkla cluster, provided by Prof. Vassilios Theofilis; and the Center for Mathematical Sciences Applied to Industry (CeMEAI) funded by São Paulo Research Foundation (FAPESP/Brazil), Grant 2013/07375-0, for access to the Euler cluster, provided by Prof. José Alberto Cuminato.
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Appendices
A Code validation and grid independence tests
1.1 A.1 Flow solver
The test case used as a reference for the DNS validation is described by Colonius et al. [8]. Results of the validation are presented here, but further details can be found in Mathias and Medeiros [25]. The cavity’s aspect ratio is \(L/D=4\) and \(L/\theta =102\), and Reynolds and Mach numbers are, respectively, \(\mathrm{Re}_\theta =60\) (\(\mathrm{Re}_D=1530\)) and \(\mathrm{Ma}=0.6\).
Two meshes were used to verify that the results were grid independent (Table 2). Both meshes covered the same domain, from \(x_i=-2\) to \(x_f=15\) and from \(y_i=-1\) to \(y_f=4\). The cavity ends are at \(x_1=5.34\) and \(x_2=9.34\). Both meshes are stretched so that the most refined region is in the mixing layer at the cavity opening. The validation grids also shared the same buffer zone parameters, which added 20 nodes at each open domain boundary.
Figure 14 illustrates the flow at an arbitrary time after a periodic state is established. It shows a single vortex inside the cavity and vortices being shed from the cavity. Figure 15 shows the wall-normal velocity as a function of time at the point shown in Fig. 14, three quarters across the cavity opening. Data extracted from the reference paper are also plotted. The phases were manually adjusted for better comparison. The results from our computations were grid independent and agreed with the reference results.
1.2 A.2 Global stability analysis
The instability results have to be independent of the computational grid, domain and other computational parameters. The analysis is presented for the thick boundary layer case at \(\mathrm{Ma}=0.1\), as an example, but similar conclusions were obtained for other cases. Corresponding tests were carried out for the base flow, but this is much less demanding and not presented.
For the mesh independence analysis, four meshes were used, as shown in Table 3. All these meshes are for the same domain, which spans from \(x_i=-2\) to \(x_f=10\) and from \(y_i=-1\) to \(y_f=4\). The cavity is placed from \(x_1=2.96\) to \(x_2=4.96\). The buffer zone used employed the same parameters of the DNS validation tests.
For meshes 1 to 4, the time steps used were \(8 \times 10^{-4}\), \(6 \times 10^{-4}\), \(5 \times 10^{-4}\) and \(4 \times 10^{-4}\), respectively, and the number of steps, 500, 667, 800 and 1000, so that for all cases the total physical time simulated was identical. The stretching parameters were kept constant; therefore, Mesh 4 is twice as refined as Mesh 1 in both directions.
Figure 16 shows that the 12 leading eigenvalues obtained for all meshes in the complex plane are very close. The values of the 15 leading eigenvalues are shown in Table 4 and, for the most refined meshes, agree within three decimal places. There is also no visible difference between the eigenmodes obtained for each mesh as well.
A set of results for Rossiter mode stability was provided by Sun et al. [40], where eigenvalues for both modes 1 and 2 were given for Mach numbers between 0.3 and 1.4, \(\mathrm{Re}_D=1500\), \(D/\theta =26.4\), and \(L/D=2\). Figure 17 compares those results to ours at the same parameters. The agreement is good, particularly the trends of all unstable modes with \(\mathrm{Ma}\). It can also be added that perfect agreement can be difficult in global instability analysis of open flows because often not enough information is given of the infinity domain boundary conditions used in every study and different conditions at these boundaries can significantly affect the instability results [29]. In our case, the inlet is upstream of the leading edge, and the cavity position is chosen so that the \(D/\theta \) equals the selected value for a Blasius boundary layer. In Sun et al. [40], the domain inlet is downstream of the flat plate leading edge and is given by a Blasius boundary layer profile. This difference in setup, for example, may lead to significant difference in the boundary layer thicknesses over the cavity.
Further validation results for both the flow solver and the global stability routine can be found in Mathias and Medeiros [25].
B Verification of absolute stability
The existence of relatively large reversed flow in the cavity raises the possibility of a local absolute instability. If a localized spot of absolute instability exists, modes other than the Rossiter ones could become globally unstable. We investigated this possibility by performing an absolute local instability analysis. The analysis did not include compressibility because this has a small and stabilizing effect at subsonic conditions. The profile with highest reversed flow (23.8%) was used in the analysis and corresponded to the \(D/\theta =100\) case. Changing the Mach number caused only a negligible change to the profile; hence, \(\mathrm{Ma}=0.5\) was chosen. Figure 18a depicts this velocity profile.
For the local instability analysis, we solved the Orr–Sommerfeld equation and followed Juniper et al. [19]. Figure 18b shows \(\omega \) (the complex frequency) as a function of \(\alpha \) (the complex wavenumber). The black lines (and the colors) are isocontours of the imaginary part, and the white lines are of the real part. The absolute instability is determined by the condition at the saddle point. The magenta lines mark the location where either \(\partial \omega _i / \partial \alpha _r\) or \(\partial \omega _i / \partial \alpha _i\) vanish. They cross each other at the saddle point, where \(\partial \omega / \partial \alpha =0\), i.e., the group velocity is null. In this case, it happens at \(\alpha = 10.5 - 4.8i\), where \(\omega = 4.9 - 2.8i\). The negative imaginary part of \(\omega \) indicates that this flow is locally absolutely stable.
The fact that this flow is not locally absolutely unstable was already expected because several studies on this parametric region [27, 36, 44], despite having not carried out this analysis, have also not reported modes other than the Rossiter one for \(L/D=2\).
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Mathias, M.S., Medeiros, M.A.F. The effect of incoming boundary layer thickness and Mach number on linear and nonlinear Rossiter modes in open cavity flows. Theor. Comput. Fluid Dyn. 35, 495–513 (2021). https://doi.org/10.1007/s00162-021-00570-2
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DOI: https://doi.org/10.1007/s00162-021-00570-2