Resolvent-based modeling of coherent wave packets in a turbulent jet

Abstract

Coherent turbulent wavepacket structures in a jet at Reynolds number 460 000 and Mach number 0.4 are extracted from experimental measurements, and are modelled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing/response wavepackets ('resolvent modes') are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instability and the non-modal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and sub-optimal forcing/response pairs. A theoretical development in the framework of spectral covariance dynamics then explicates the link between linear harmonic forcing/response structures and the cross-spectral density (CSD) of stochastic turbulent fluctuations. A lowrank model of the CSD at given Strouhal number is formulated from a truncated set of linear resolvent modes. Corresponding experimental CSD matrices are constructed from extensive two-point velocity measurements. Their eigenmodes (spectral proper orthogonal decomposition or SPOD modes) represent coherent wavepacket structures, and these are compared to their counterparts obtained from the linear model. Close agreement is demonstrated in the range of 'preferred mode' Strouhal numbers, around a value of 0.4, between the leading coherent wavepacket structures as educed from the experiment and from the linear resolventbased model.

Resolvent-based analysis of streaks in turbulent jets

“…In some sense, the results shown in figure 1(a,b) are similar to what was found by [26] concerning the behaviour of optimals and suboptimals as a function of frequency. As in the previous work, the two curves can be very different from each other and still point to the same mechanism (in their case, the Kelvin-Helmholtz instability).…”

“…It is known from [20] that SPOD and resolvent modes should match if the the forcing term is uncorrelated in space, which can be a strong assumption for a turbulent flow. However, if the optimal forcing has a gain much larger than suboptimal ones, flow fluctuations are dominated by the optimal response [20,25]; thus, resolvent analysis has been useful in the determination of the underlying mechanisms in wall-bounded [15,17] and free-shear flows [20][21][22]26], encouraging the application of the method for this flow to better understand the role of the lift-up effect in a turbulent jet. The SPOD-mode energies for the first 4 modes is shown in figure 1(a), showing a dominance of the first mode in the low azimuthal wavenumber region, with the difference between optimal and suboptimal decreasing as we shift to higher values of m. This dominance is usually related to a preeminence of a physical mechanism (such as the Kelvin-Helmhotlz instability or the lift-up effect); in [11], the same dominance was identified for St = 0, with gains peaking at m = 3.…”

Large-scale streaky structures in turbulent jets

“…Solving the eigenvalue problem defined by equation 3.3 using standard methods, we can obtain the energy of each SPOD mode σ n (m, ω) and the respective eigenfunction ξ (n) (x, r, m, ω). Figure 5(a) shows the leading SPOD-mode energies for several Strouhal numbers St = ωD/(2πU jet ) and azimuthal modes m. The high-energy region for m = 0 was shown to be related to a coherent wavepacket structure, as analysed by Semeraro et al (2016); Lesshafft et al (2018), and is not the focus of the present work. Instead, we consider the m > 1 region of the spectrum, where the reported streaky structures occur; in this region, the highest energies are found for St = 0 (except for m = 1 and 2), meaning that the structures are nearly steady in time, being related to a slow dynamic of the flow; here, SPOD modes for St = 0 should be interpreted as the St → 0 limit, with structures whose time scale is larger than the segment size used in Welch's method.…”

A Parabolised Stability Equation based Broadband Shock-Associated Noise Model

“…The detection of wavepackets has been supported by numerous near-field pressure measurement studies [27][28][29][30]. Their results are further supported in the comparisons made between velocity fields and stability calculations by Cavalieri et al [31] and recent resolvent analysis studies by Schmidt et al [25], Lesshafft et al [32]. Sasaki et al [33] have also found that the evolution of the lower-order azimuthal modes can be predicted using linear wavepackets up to a Strouhal value of St = 4.…”