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Register a new userResolvent-based modelling of coherent wavepackets in a turbulent jet
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Lutz Lesshafft
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Coherent turbulent wave-packet structures in a jet at Reynolds number 460000 and Mach number 0.4 are extracted from experimental measurements and are modeled 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 wave packets (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 nonmodal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and suboptimal 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 low-rank 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 or SPOD modes) represent coherent wave-packet 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 wave-packet structures as educed from the experiment and from the linear resolvent-based model.
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The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at friction Reynolds number Re$_tau$ = 1007. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated to the turbulent mean flow, is included in the resolvent operator. The agreement is further improved when the flat forcing power spectrum (white noise) is replaced with a power spectrum matching the measures. Such a good agreement is not observed when the eddy-viscosity terms are not included in the resolvent operator. In this case, the estimation based on the resolvent is unable to select the right peak frequency and wall-normal location of buffer-layer motions. Similar results are found when comparing truncated expansions of measured streamwise velocity power spectral densities based on a spectral proper orthogonal decomposition to those obtained with optimal resolvent modes.
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