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Register a new userThe genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow
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The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can co-exist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate the origin of the first localized solution in a modulational symmetry-breaking Hopf bifurcation from a known global travelling wave that has 2-fold rotational symmetry about the pipe axis. Similar behaviour is found for a global wave of 3-fold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that all global solutions should be expected to lead to more realistic localised counterparts through such bifurcations, which provides a constructive route for their generation.
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The spectral model of Perry, Henbest & Chong (1986) predicts that the integral length-scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scales variation to be more realistic while keeping with the Townsend-Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high Reynolds number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high Reynolds number data by Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The finding of Dallas, Vassilicos & Hewitt (2009) that it is the eddy turnover time and not the mean flow gradient which scales with distance to the wall and skin friction velocity in the intermediate layer implies, when combined with Townsends (1976) production-dissipation balance, that the mean flow gradient has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The same approach also predicts that the mean flow gradient has a logarithmic decay at distances to the wall larger than the position of the outer peak, a qualitative prediction which the aforementioned data also support.
Direct numerical simulations of turbulent pipe flow subjected to streamwise-varying wall rotation are performed. This control method is observed to be able to significantly reduce the friction drag and even laminarize the flow under certain control parameters, which are dictated by velocity amplitude and wavelength, for friction Reynolds number Re{tau} =180. Net energy saving is achievable and the variation of wavelength is found to be more efficient than velocity amplitude in reducing the drag. A series of turbulence statistics are discussed in order to elucidate the impact of steady spatially oscillatory forcing, including budgets of transport equation, turbulence intensity, two-point correlation and one-dimensional spectra. An overall assessment of global energy balance identifies a trend toward laminar regime. The control-induced boundary layer, whose thickness is closely related to control wavelength, is shown to induce a streamwise wavy streak pattern, with its orientation governed by the shear force stemming from gradients of mean velocity. Such strong spatial non-homogeneity is found to significantly reduce the streamwise scale of flow structure. The analysis of conditional-averaged fields reveals an emergence of strong transverse advection, which is observed to cause asymmetrical modification of near-wall quasi-streamwise vortex pair, accompanied by the transverse tilt or diffusion of low-speed streaks and the suppression of its surrounding sweep events, leading to the disruption of near-wall quasi-organized flow structure and hence in turn contributing to the decline of turbulent shear stress.
We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming PDE systems. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.
Turbulence is the major cause of friction losses in transport processes and it is responsible for a drastic drag increase in flows over bounding surfaces. While much effort is invested into developing ways to control and reduce turbulence intensities, so far no methods exist to altogether eliminate turbulence if velocities are sufficiently large. We demonstrate for pipe flow that appropriate distortions to the velocity profile lead to a complete collapse of turbulence and subsequently friction losses are reduced by as much as 95%. Counterintuitively, the return to laminar motion is accomplished by initially increasing turbulence intensities or by transiently amplifying wall shear. The usual measures of turbulence levels, such as the Reynolds number (Re) or shear stresses, do not account for the subsequent relaminarization. Instead an amplification mechanism measuring the interaction between eddies and the mean shear is found to set a threshold below which turbulence is suppressed beyond recovery.
The impact of wall roughness on fully developed laminar pipe flow is investigated numerically. The roughness is comprised of square bars of varying size and pitch. Results show that the inverse relation between the friction factor and the Reynolds number in smooth pipes still persists in rough pipes, regardless of the rib height and pitch. At a given Reynolds number, the friction factor varies quadratically with roughness height and linearly with roughness pitch. We propose a single correlation for the friction factor that successfully collapses the data.
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