The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low speed fluid away from the wall, has been studied experimentally, theoretically, and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. We produce five new three-dimensional solutions of turbulent plane Couette flow, one of which is periodic while four others are relative periodic. Each of these five solutions demonstrates the break-up and re-formation of near-wall coherent structures. Four of our solutions are periodic but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. We argue that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the break-up and re-formation of coherent structures. We also compute a new periodic solution of plane Couette flow that could be related to transition to turbulence.The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. We address this computationally demanding requirement with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas -namely the Newton-Krylov iteration and the locally constrained optimal hook step. Each of our six solutions is accompanied by an error estimate.In the concluding discussion, we discuss dynamical principles that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier-Stokes equation.

Plane Couette flow transitions to turbulence for Re ≈ 325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are quite distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. Computing such connections is essential for understanding the global dynamics of spatially localized structures that occur in transitional plane Couette flow. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

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We report the computation of a family of travelling wave solutions of pipe flow up to ReZ75 000. As in all lower branch solutions, streaks and rolls feature prominently in these solutions. For large Re, these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as Re/N at rates given by Re K0.41 and Re K0.87 ; surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the Re/N limit could be universal to lower branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.

For the familiar Fibonacci sequence (defined by f 1 = f 2 = 1, and fn = f n−1 + f n−2 for n > 2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 = 1.61803398. .. . But for a simple modification with both additions and subtractions-the random Fibonacci sequences defined by t 1 = t 2 = 1, and for n > 2, tn = ±t n−1 ± t n−2 , where each ± sign is independent and either + or − with probability 1/2-it is not even obvious if |tn| should increase with n. Our main result is that n |tn| → 1.13198824. .. as n → ∞ with probability 1. Finding the number 1.13198824. .. involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.