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 2-fold 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 (k). 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. These waves are expected to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, thereby leaving stable upper-branch solutions whose subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec

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.Comment: 38 pages, 35 figures, 1 tabl

How does flow in a pipe become turbulent?

The transition to turbulence in pipe flow does not follow the scenario familiar from Rayleigh-Benard or Taylor-Couette flow since the laminar profile is stable against infinitesimal perturbations for all Reynolds numbers. Moreover, even when the flow speed is high enough and the perturbation sufficiently strong such that turbulent flow is established, it can return to the laminar state without any indication of the imminent decay. In this parameter range, the lifetimes of perturbations show a sensitive dependence on initial conditions and an exponential distribution. The turbulence seems to be supported by three-dimensional travelling waves which appear transiently in the flow field. The boundary between laminar and turbulent dynamics is formed by the stable manifold of an invariant chaotic state. We will also discuss the relation between observations in short, periodically continued domains, and the dynamics in fully extended puffs.Comment: for the proceedings of statphys 2

Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018

Large-Scale Evidence for the Effect of the COLIA1 Sp1 Polymorphism on Osteoporosis Outcomes: The GENOMOS Study.

Background Osteoporosis and fracture risk are considered to be under genetic control. Extensive work is being performed to identify the exact genetic variants that determine this risk. Previous work has suggested that a G/T polymorphism affecting an Sp1 binding site in the COLIA1 gene is a genetic marker for low bone mineral density (BMD) and osteoporotic fracture, but there have been no very-large-scale studies of COLIA1 alleles in relation to these phenotypes. Methods and Findings Here we evaluated the role of COLIA1 Sp1 alleles as a predictor of BMD and fracture in a multicenter study involving 20,786 individuals from several European countries. At the femoral neck, the average (95% confidence interval [CI]) BMD values were 25 mg/cm2 (CI, 16 to 34 mg/cm2) lower in TT homozygotes than the other genotype groups ( p < 0.001), and a similar difference was observed at the lumbar spine; 21 mg/cm2 (CI, 1 to 42 mg/cm2), ( p = 0.039). These associations were unaltered after adjustment for potential confounding factors. There was no association with fracture overall (odds ratio [OR] = 1.01 [CI, 0.95 to 1.08]) in either unadjusted or adjusted analyses, but there was a non-significant trend for association with vertebral fracture and a nominally significant association with incident vertebral fractures in females (OR = 1.33 [CI, 1.00 to 1.77]) that was independent of BMD, and unaltered in adjusted analyses. Conclusions Allowing for the inevitable heterogeneity between participating teams, this study—which to our knowledge is the largest ever performed in the field of osteoporosis genetics for a single gene—demonstrates that the COLIA1 Sp1 polymorphism is associated with reduced BMD and could predispose to incident vertebral fractures in women, independent of BMD. The associations we observed were modest however, demonstrating the importance of conducting studies that are adequately powered to detect and quantify the effects of common genetic variants on complex diseases

Extended Thromboprophylaxis with Betrixaban in Acutely Ill Medical Patients

Background Patients with acute medical illnesses are at prolonged risk for venous thrombosis. However, the appropriate duration of thromboprophylaxis remains unknown. Methods Patients who were hospitalized for acute medical illnesses were randomly assigned to receive subcutaneous enoxaparin (at a dose of 40 mg once daily) for 10±4 days plus oral betrixaban placebo for 35 to 42 days or subcutaneous enoxaparin placebo for 10±4 days plus oral betrixaban (at a dose of 80 mg once daily) for 35 to 42 days. We performed sequential analyses in three prespecified, progressively inclusive cohorts: patients with an elevated d-dimer level (cohort 1), patients with an elevated d-dimer level or an age of at least 75 years (cohort 2), and all the enrolled patients (overall population cohort). The statistical analysis plan specified that if the between-group difference in any analysis in this sequence was not significant, the other analyses would be considered exploratory. The primary efficacy outcome was a composite of asymptomatic proximal deep-vein thrombosis and symptomatic venous thromboembolism. The principal safety outcome was major bleeding. Results A total of 7513 patients underwent randomization. In cohort 1, the primary efficacy outcome occurred in 6.9% of patients receiving betrixaban and 8.5% receiving enoxaparin (relative risk in the betrixaban group, 0.81; 95% confidence interval [CI], 0.65 to 1.00; P=0.054). The rates were 5.6% and 7.1%, respectively (relative risk, 0.80; 95% CI, 0.66 to 0.98; P=0.03) in cohort 2 and 5.3% and 7.0% (relative risk, 0.76; 95% CI, 0.63 to 0.92; P=0.006) in the overall population. (The last two analyses were considered to be exploratory owing to the result in cohort 1.) In the overall population, major bleeding occurred in 0.7% of the betrixaban group and 0.6% of the enoxaparin group (relative risk, 1.19; 95% CI, 0.67 to 2.12; P=0.55). Conclusions Among acutely ill medical patients with an elevated d-dimer level, there was no significant difference between extended-duration betrixaban and a standard regimen of enoxaparin in the prespecified primary efficacy outcome. However, prespecified exploratory analyses provided evidence suggesting a benefit for betrixaban in the two larger cohorts. (Funded by Portola Pharmaceuticals; APEX ClinicalTrials.gov number, NCT01583218. opens in new tab.

Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries

Novel spectral methods are formulated in terms of divergence-free vector fields in order to compute finite amplitude time-dependent solutions of incompressible viscous flows in cylindrical and/or annular geometries. The numerical discretization of the method leads to a simple dynamical system of amplitudes from which the stability properties of the solution can be analyzed easily. In addition, the formulation allows easy implementation of continuation algorithms to track solutions that have bifurcated from a known state, or the search for disconnected solution branches by means of homotopy transformations of the Navier–Stokes equations. The method is succesfully applied to the study of generic double Hopf bifurcations in pressure-driven helicoidal flows and to the search of unstable travelling wave solutions in pipe flow

Instability mechanisms and transition scenarios of spiral turbulence in Taylor-Couette flow

Alternating laminar and turbulent helical bands appearing in shear flows between counterrotating cylinders are accurately computed and the near-wall instability phenomena responsible for their generation identified. The computations show that this intermittent regime can only exist within large domains and that its spiral coherence is not dictated by endwall boundary conditions. A supercritical transition route, consisting of a progressive helical alignment of localized turbulent spots, is carefully studied. Subcritical routes disconnected from secondary laminar flows have also been identified