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Resolvent analysis of swirling turbulent jets

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This study explores coherent structures in a swirling turbulent jet. Stationary axisymmetric solutions of the Reynolds–Averaged Navier–Stokes equations at \(Re=200,000\) were obtained using an open source computational fluid dynamics code and the Spalart–Allmaras eddy viscosity model. Then, resolvent analysis with the same eddy viscosity field provided coherent structures of the turbulent fluctuations on the base flow. As in many earlier studies, a large gain separation is identified between the optimal and sub-optimal resolvent modes, permitting a focus on the most amplified response mode and its corresponding optimal forcing. At zero swirl, the results indicate that the jet’s coherent response is dominated by axisymmetric (\(m=0\)) structures, which are driven by the usual Kelvin–Helmholtz shear amplification mechanism. However, as swirl is increased, different coherent structures begin to dominate the response. For example, double and triple spiral (\(|m|=2\) and \(|m|=3\)) modes are identified as the dominant structures when the axial and azimuthal velocity maxima of the base flow are comparable. In this case, distinct co- and counter-rotating \(|m|=2\) modes experience vastly different degrees of amplification. The physics of this selection process involve several amplification mechanisms contributing simultaneously in different regions of the mode. This is analysed in more detail by comparing the alignment between the wavevector of the dominant response mode and the principal shear direction of the base flow. Additional discussion also considers the development of structures along the exterior of the jet nozzle.

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Funding

This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the CleanSky2 Joint Undertaking Grant Agreement No. 785303, and under the Marie Skłodowska-Curie Grant Agreement No. 899987. The first author has been supported by the Direction Générale de l’Armement. Results reflect only the authors’ view and none of the above are responsible for any use that may be made of the information it contains.

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  1. Laboratoire d’Hydrodynamique, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Quentin Chevalier, Christopher M. Douglas & Lutz Lesshafft

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  1. Quentin Chevalier
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Correspondence to Quentin Chevalier.

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Appendices

Validation case

In this section results from the code developed internally for the purposes of this study will be compared to known publications. The emphasis here is therefore proving the validity of design choices outlined in Sects. 2.1, 2.2.1 and 2.3.2, not introducing new phenomena. To increase confidence in our model, reproductions of figure 3(a) and 7(a) of Meliga et al. [47], as well as figure 6.6 from Garnaud [26] were made. They will not be discussed here for the sake of brevity, but may be found in Chevalier [15].

The case considered instead is closer to the one described in Sect. 2.1, namely the non-swirling jet based on Pickering et al. [56]. To better compare with the reference, the nozzle was cut out and calculations performed on \(x\in [0;49]\). Figure 20 is to be compared with Fig. 4b of that work. This graph differs notably from its reference - gains are lower than expected, the peak of the \(m=0\) curve happens at a higher frequency, and the other curves have a marked increase before the expected monotonic decrease.

However, there are several points where our model substantially differs from the reference:

  • Base flows come from different methods—one is the result of a LES calculation, the other a RANS process,

  • Eddy viscosity models also differ, one being a length scale model and the other being the SA model—even though qualitatively similar, they vary by about a factor three.

Fig. 20
figure 20

Squared gains as a function of Strouhal number for a variety of azimuthal wavenumbers with no nozzle

Full size image

A qualitative comparison of response and forcing modes was also done on Fig. 21, which is to be compared to figures 9(b) and (c) of Pickering et al. [56]. These structures are qualitatively very close to the reference, though somewhat shorter. This is probably due to a difference in the eddy viscosity used, which leads to different localised dam**. It can be observed that the eddy viscosity used in this study steadily increases with x whereas that of Pickering et al. [56] peaks and eventually decays.

Fig. 21
figure 21

Resolvent modes for \((m,St)\in \{0,1,3\}\times \{0.05,0.2,0.6\}\). Scaling is adjusted to facilitate comparaison to Pickering et al.[56]

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Mesh refinement

As part of the validation process of the code used in this work, convergence with respect to mesh refinement was verified. In the usual manner, a very refined mesh was taken as reference, and it was checked that results do converge quickly to the fine mesh as the number of elements increases, see Fig. 22. Mesh refinement was also verified for fluctuations. Once more, this point will be omitted here and made available in Chevalier [16],

Fig. 22
figure 22

Velocity magnitude for a variety of slices at different number of elements

Full size image

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Chevalier, Q., Douglas, C.M. & Lesshafft, L. Resolvent analysis of swirling turbulent jets. Theor. Comput. Fluid Dyn. (2024). https://doi.org/10.1007/s00162-024-00704-2

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