Wave-Packet Models for Jet Dynamics and Sound Radiation

Abstract

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Large-scale streaky structures in turbulent jets

“…With the defined formulation, the output of the resolvent analysis is the optimal forcing that would maximise streamwise velocity in regions where the turbulence intensity is higher (r/D < 1). The most amplified response corresponds to turbulent structures most likely to be excited by non-linear dynamics (Cavalieri et al 2019). Moreover, if non-linear terms in the Navier-Stokes system are modelled as white noise, the resulting SPOD modes should be equal to response modes from resolvent analysis Towne et al (2018).…”

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

“…A formal justification for a direct comparison between optimal linear response structures and SPOD modes has been suggested by Beneddine et al [28], and, with an increasing level of detail, in two conference papers [19,29], and by Towne et al [20], who also examine the link between resolvent modes and dynamic mode decomposition (DMD). A recent review article [30] provides a didactical introduction to resolvent-based modelling of SPOD modes, including numerical codes for a simple model problem, and a discussion of its relevance for the study of jet noise. Schmidt et al [21] present a detailed comparison between resolvent analysis results and SPOD modes, extracted from LES data, for high-Reynolds-number turbulent jets at Mach numbers 0.4, 0.9 and 1.5.…”

Resolvent-based analysis of streaks in turbulent jets

“…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.…”