Fernando Mellibovsky

Streamwise-localized solutions at the onset of turbulence in pipe flow

Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulent transients arise from one such solution branch.

Bone tissue properties measurement by reference point indentation in glucocorticoid-induced osteoporosis

Glucocorticoids, widely used in inflammatory disorders, rapidly increase bone fragility and, therefore, fracture risk. However, common bone densitometry measurements are not sensitive enough to detect these changes. Moreover, densitometry only partially recognizes treatment‐induced fracture reductions in osteoporosis. Here, we tested whether the reference point indentation technique could detect bone tissue property changes early after glucocorticoid treatment initiation. After initial laboratory and bone density measurements, patients were allocated into groups receiving calcium + vitamin D (Ca+D) supplements or anti‐osteoporotic drugs (risedronate, denosumab, teriparatide). Reference point indentation was performed on the cortical bone layer of the tibia by a handheld device measuring bone material strength index (BMSi). Bone mineral density was measured by dual‐energy X‐ray absorptiometry (DXA). Although Ca+D‐treated patients exhibited substantial and significant deterioration, risedronate‐treated patients exhibited no significant change, and both denosumab‐ and teriparatide‐treated participants exhibited significantly improved BMSi 7 weeks after initial treatment compared with baseline; these trends remained stable for 20 weeks. In contrast, no densitometry changes were observed during this study period. In conclusion, our study is the first to our knowledge to demonstrate that reference point indentation is sensitive enough to reflect changes in cortical bone indentation after treatment with osteoporosis therapies in patients newly exposed to glucocorticoids.

Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the twofold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov–Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme, we unfold a double-zero (Takens–Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (κ). This scenario is extended, by the inclusion of higher-order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. We provide evidence that these modulated waves undergo a fold-of-cycles and compute some solutions on the unstable branch. These waves are shown to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, in accordance with the bifurcation scenario proposed. The travelling-wave upper-branch solutions are stable within the subspace of twofold periodic flows, and their subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.

From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

We study numerically a succession of transitions in pipe Poiseuille flow that leads from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade possess a shift-reflect symmetry and are both axially and azimuthally periodic with wave numbers {\kappa} = 1.63 and n = 2, respectively. As the Reynolds number is increased, successive transitions result in a wide range of time dependent solutions that includes spiralling, modulated-travelling, modulated-spiralling, doubly-modulated-spiralling and mildly chaotic waves. We show that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift-reflect-symmetric modulated-travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The states studied here belong to a subspace of discrete symmetry which makes many of the bifurcation and path-following investigations presented technically feasible. However, we expect that most of the phenomenology carries over to the full state-space, thus suggesting a mechanism for the formation and break-up of invariant states that can sustain turbulent dynamics.

Pipe flow transition threshold following localized impulsive perturbations

A numerical study of the destabilizing effects of localized impulsive perturbations in pressure-driven Hagen-Poiseuille or pipe flow is presented. The numerics intend to ellucidate the intrinsic mechanisms of subcritical transition to turbulence in pipe flow by reproducing very recent experimental explorations carried out by Hof, Juel, and Mullin [Phys. Rev. Lett. 91, 244502 (2003)], concluding that the minimum amplitude of a perturbation required to cause transition scales as the inverse of the Reynolds number, i.e., O(Re−1). The numerical model simulates the experimental disturbance generator based on impulsive injection of fluid through six slits azimuthally equispaced on a perimeter around the pipe. This is accomplished by introducing a local time-dependent impulsive volume force term in the Navier-Stokes equations for the perturbation velocity field, fulfilling incompressibility constraints. A comprehensive exploration of the critical amplitudes that trigger transition as a function of the injection ...

Critical threshold in pipe flow transition

This study provides a numerical characterization of the basin of attraction of the laminar Hagen–Poiseuille flow by measuring the minimal amplitude of a perturbation required to trigger transition. For pressure-driven pipe flow, the analysis presented here covers autonomous and impulsive scenarios where either the flow is perturbed with an initial disturbance with a well-defined norm or perturbed by means of local impulsive forcing that mimics injections through the pipe wall. In both the cases, the exploration is carried out for a wide range of Reynolds numbers by means of a computational method that numerically resolves the transitional dynamics. For , the present work provides critical amplitudes that decay as Re−3/2 and Re−1 for the autonomous and impulsive scenarios, respectively. For Re=2875, accurate threshold amplitudes are found for constant mass-flux pipe by means of a shooting method that provides critical trajectories that never relaminarize or trigger transition. These transient states are used as initial guesses in a damped Newton–Krylov method formulated to find periodic travelling wave solutions that either travel downstream or exhibit a helicoidal advection.

The role of streamwise perturbations in pipe flow transition

The phenomenon of subcritical transition in Hagen-Poiseuille or pipe flow is explored for a wide range of Reynolds numbers within the interval Re∊[2.5×103,1.26×104] by means of a computational method that numerically resolves the transitional dynamics with nearly 3.5×104 degrees of freedom on a medium aspect-ratio domain of length 32π∕5. The aim of this exploration is to provide a theoretical characterization of the basin of attraction of the basic regime by measuring the minimal amplitude of an initial global perturbation leading to transition. The analysis is based on a particular theoretical scenario that considers streamwise-independent finite amplitude initial vortical perturbations that trigger global transition via optimal inflectional instabilities of streamwise-dependent modes with selected axial wave numbers. Disturbances consisting of 1, 2, and 3 pairs of vortices are investigated. Special attention is given to relaminarization phenomena that is frequently observed for low Reynolds numbers. Lon...

Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow

Turbulent-laminar patterns are ubiquitous near transition in wall-bounded shear flows. Despite recent progress in describing their dynamics in analogy to non-equilibrium phase transitions, there is no theory explaining their emergence. Dynamical-system approaches suggest that invariant solutions to the Navier--Stokes equations, such as traveling waves and relative periodic orbits in pipe flow, act as building blocks of the disordered dynamics. While recent studies have shown how transient chaos arises from such solutions, the ensuing dynamics lacks the strong fluctuations in size, shape and speed of the turbulent spots observed in experiments. We here show that chaotic spots with distinct dynamical and kinematic properties merge in phase space and give rise to the enhanced spatio-temporal patterns observed in pipe flow. This paves the way for a dynamical-system foundation to the phenomenology of turbulent-laminar patterns in wall-bounded extended shear flows.

Global finite amplitude perturbations in medium aspect ratio pipe flow

Results of a numerical study on the finite amplitude global perturbations inducing transition to turbulence in pipe flow are reported. The aim of this analysis is to characterise the basin of attraction of the basic Hagen-Poiseuille flow (which is believed to be lineary stable for all Reynolds numbers Re) by means of the minimal amplitude of an initial global perturbation triggering transition. Subcritical transition in pipe flow is extremely sensitive to the shape of the initial perturbation. The analysis is focused on the streak breakdown transition scenario, by which the basic flow, perturbed with streamwise-independent disturbances of azimuthal wave number n = 1, develops transient streaks that are susceptible of being destabilised by much smaller streamwise-dependent perturbations. The numerical simulations cover a wide range of Reynolds numbers Re [2500, 104] and the transition dynamics are spectrally resolved by the numerical method. The threshold amplitude of perturbations seems to decrease with Re−3/2 within the studied range.

Shear instabilities in Taylor-Couette flow

Subcritical instabilities in small gap Taylor-Couette (TCF) problem are studied numerically when both cylinders rotate in opposite directions. The computations are carried out for a radius ratio \(\eta = r_{{\rm i}}/r_{{\rm o}} = 0.883.\) A first exploration is focused on the study of spiral flows originated from subcritical Hopf bifurcations of the basic circular Couette solution. The second exploration addresses the transition from laminar flow to the usually termed as spiral turbulence regime characterized by alternating laminar and turbulent spiral bands which coexist even in regions of the parameter space where the circular Couette flow is linearly stable.