Nonlinear dynamics of large-scale coherent structures in turbulent free shear layers

Abstract

Fully developed turbulent free shear layers exhibit a high degree of order, characterized by large-scale coherent structures in the form of spanwise vortex rollers. Extensive experimental investigations show that such organised motions bear remarkable resemblance to instability waves, and their main characteristics, including the length scales, propagation speeds and transverse structures, are reasonably well predicted by linear stability analysis of the mean flow. In this paper, we present a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation is based on the triple decomposition of the instantaneous flow into a mean field, coherent fluctuations and small-scale turbulence but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organised motion. The system is closed by employing gradient type of models for the time-and phase-averaged Reynolds stresses of fine-scale turbulence. In the high-Reynolds-number limit, the nonlinear non-equilibrium critical-layer theory for laminar-flow instabilities is adapted to turbulent shear layers by accounting for (a) the enhanced non-parallelism associated with fast spreading of the mean flow, and (b) the influence of small-scale turbulence on coherent structures. The combination of these factors with nonlinearity leads to an interesting evolution system, consisting of the coupled amplitude and vorticity equations, in which non-parallelism contributes the so-called translating critical-layer effect. Numerical solutions of the evolution system capture vortex roll-up, which is the hallmark of turbulent mixing layer, and the predicted amplitude development mimics the qualitative feature of oscillatory saturation that has been observed in a number of experiments. A fair degree of quantitative agreement is obtained with one set of experimental data.

Nonlinear evolution and acoustic radiation of coherent structures in subsonic turbulent free shear layers

“…As a first step towards a mathematical formulation for the nonlinear evolution of CS in a mixing layer, the instantaneous field (u u u, T, p, ρ, μ) is decomposed into the mean flow (Ū U U ,T ,P ,ρ,μ), the quasi-periodic coherent motion (ũ u u,T ,p,ρ,μ) and the remaining fluctuations (u u u , T , p , ρ , μ ) , which are predominately small scale (so that energy cascade takes place among components in this part) Wu & Zhuang 2016),…”

“…Furthermore, the non-parallelism is to appear at leading order, if we set R T ∼ Re 2/3 (cf. Wu & Zhuang 2016), which can be expressed as…”

“…CS in these flows were studied by a great number of experimentalists, and the major findings have been reviewed, e.g. by Hussain (1983) and Wygnanski & Petersen (1987), and a fairly detailed summary was given by Wu & Zhuang (2016). In particular, Gaster et al (1985) found that the spatial and temporal characteristics (wavelength, propagation speed and transverse distribution) of CS in a mixing layer were fairly well predicted by the linear instability analysis for the mean flow, suggesting that CS may be treated as wavepackets of instability modes.…”

“…These local modes are more amiable to theoretical and computational treatments, and equally importantly they possess the key attribute observed of CS: being directly related to external disturbances (such as acoustic noise or turbulence in the oncoming flow), and the link between them could be understood in terms of receptivity mechanisms near the nozzle or the trailing edge of the splitter plate. The present theoretical development will be based on the previous work on nonlinear instability of a compressible laminar shear layer (Sparks & Wu 2008), and on CS in an incompressible turbulent mixing layer (Wu & Zhuang 2016). The ensuing analysis amounts to extending either the former to the CS in a turbulent shear flow, or the latter to the compressible regime.…”

The Simulation of Vortex Structures Induced by Different Local Vibrations at the Wall in a Flat-Plate Laminar Boundary Layer

“…Wu presented a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation was based on a triple decomposition of the instantaneous flow into a mean field, coherent fluctuations, and small-scale turbulence, but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion [4]. Mcmullan implemented large eddy simulations of the plane mixing layer for the purpose of reducing the stream-wise vortex structure that may exist in these flows.…”