On coherent structure in wall turbulence

Abstract

A new theory of coherent structure in wall turbulence is presented. The theory is the first to predict packets of hairpin vortices and other structure in turbulence, and their dynamics, based on an analysis of the Navier-Stokes equations, under an assumption of a turbulent mean profile. The assumption of the turbulent mean acts as a restriction on the class of possible structures. It is shown that the coherent structure is a manifestation of essentially low-dimensional flow dynamics, arising from a critical layer mechanism. Using the decomposition presented in McKeon & Sharma (J. Fluid Mech, 658, 2010 ), complex coherent structure is recreated from minimal superpositions of response modes predicted by the analysis, which take the form of radially-varying travelling waves. The leading modes effectively constitute a low-dimensional description of the turbulent flow which is optimal in the sense of describing the resonant effects around the critical layer and which minimally predicts all types of structure. The approach is general for the full range of scales. By way of example, simple combinations of these modes are offered that predict hairpins and modulated hairpin packets. The example combinations are chosen to represent observed structure, consistent with the nonlinear triadic interaction for wavenumbers that is required for self-interaction of structures. The combination of the three leading response modes at streamwise wavenumbers 6, 1, 7 and spanwise wavenumbers ±6, ± 6, ± 12 respectively, with phase velocity 2/3, is understood to represent a turbulence "kernel", which, it is proposed, constitutes a self-exciting process analogous to the near-wall cycle. Together, these interactions explain how the mode combinations may self-organise and self-sustain to produce experimentally observed structure. The phase interaction also leads to insight into skewness and correlation results known in the literature. It is also shown that the very large scale motions act to organise hairpin-like structures such that they co-locate with areas of low streamwise momentum, by a mechanism of locally altering the shear profile. These energetic streamwise structures arise naturally from the resolvent analysis, rather than by a summation of hairpin packets. In addition, these packets are modulated through a "beat" effect. The relationship between Taylor's hypothesis and coherence is discussed and both are shown to be the consequence of the localisation of the response modes around the critical layer. A pleasing link is made to the classical laminar inviscid theory, whereby the essential mechanism underlying the hairpin vortex is captured by two obliquely interacting Kelvin-Stuart (cat's eye) vortices. Evidence for the theory is presented based on comparison to observations of structure in turbulent flow reported in the experimental and numerical simulation literature and to exact solutions reported in the transitional literature.arXiv:1301.7580v2 [physics.flu-dyn]

Experimental manipulation of wall turbulence: A systems approach

“…This phenomenon is described at length in Sharma and McKeon, 25 but we demonstrate the complexity of vortical structure that can arise under the superposition of three modes in Figure 12. The three modes K 1 = (6, 6, 2/3), K 2 = (1, 6, 2/3) (VLSM), and K 3 = (7, 12, 2/3) are triadically FIG.…”

“…More information on the organization of the flow, essentially the amplitudes and relative phases of the velocity response modes, appears to be available from consideration of the triadic interactions such as those presented by our turbulence kernel; this is beyond the scope of the current manuscript, but is explored in Sharma and McKeon. 25 Another benefit of using the systems paradigm is the potential for carrying over closed-loop control thinking and technology to explore the possibilities for turbulence control in the presence of an almost ideal model. We explore strategies for control in Secs.…”

“…Most notably, the positive shear associated with the mean profile can be shown 25 to increase (suppress) the swirl value associated with prograde (retrograde) hairpin vortex heads. This is a thresholding problem: rotational motion is still captured for both senses of vortex rotation, but the usual thresholding techniques applied to velocity fields lead to a systematic under-accounting of retrograde vortices.…”

“…A brief summary and outlook concludes the paper. The material discussed here represents an integration and expansion of the work presented in assembled papers from this group, especially McKeon and Sharma, 1 Jacobi and McKeon, 24 and Sharma and McKeon, 25 where full details of the underpinning analytical and experimental studies can be found. An abbreviated version was presented as an invited lecture during the 2011 APS-DFD meeting.…”

“…The symmetries associated with the pipe geometry imply that the resolvent is normal under the integrals over the wall-parallel coordinates and time (see Sharma and Mckeon 25 for further discussion on this point). In these homogeneous directions, then, the forcing and response modes are equal, orthogonal and are the Fourier modes, which then represent the appropriate choice for basis functions.…”