We report an experimental evidence of a global bifurcation on a highly turbulent von Karman flow. The mean flow presents multiple solutions: the canonical symmetric solution becomes marginally unstable towards a flow which breaks the basic symmetry of the driving apparatus even at very large Reynolds number. The global bifurcation between these states is highly subcritical and the system thus keeps a memory of its history. The transition recalls low-dimension dynamical systems transitions and exhibits a very peculiar statistics. We discuss the role of turbulence in two ways: the multiplicity of hydrodynamical solutions and the effect of fluctuations on the nature of transitions.
We report the experimental study of the bifurcations of a large-scale circulation that is formed over a turbulent flow generated by a spatially periodic forcing. After shortly describing how the flow becomes turbulent through a sequence of symmetry breaking bifurcations, we focus our study on the transitions that occur within the turbulent regime. They are related to changes in the shape of the probability density function (PDF) of the amplitude of the large scale flow. We discuss the nature of these bifurcations and how to model the shape of the PDF.
Characterisation of a multistable turbulent wake: application of an improved regime identification with analytical model training
Centre INRIA de l’université de Rennes, IRMAR UMR 6625（雷恩大学INRIA中心，IRMAR UMR 6625）；Univ. Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille, UMR 9014 - LMFL - Laboratoire de Mécanique des Fluides de Lille - Kampé de Fériet（里尔大学，法国国家科学研究中心，法国航空与航天研究院，巴黎国立高等工艺技术学院，中央理工学院里尔分校）
Localised turbulent structures such as puffs or oblique stripes are building blocks of the intermittency regimes in subcritical wall-bounded shear flows. These turbulent structures are investigated in incompressible pressure-driven annular pipe flow using direct numerical simulations in long domains. For low enough radius ratio $η$, these coherent structures have a dynamics comparable to that of puffs in cylindrical pipe flow. For $η$ larger than 0.5, they take the shape of helical stripes inclined with respect to the axial direction. The transition from puffs to stripes is analysed statistica…
We study the bifurcations of the large scale jets in the turbulent regime of a forced shear flow using direct numerical simulations of the Navier-Stokes equations. The bifurcations are seen in the probability density function (PDF) of the largest scale mode with the control parameter being the Reynolds number based on the friction coefficient denoted as $Rh$. As one increases $Rh$ in the turbulent regime, the PDF of the large scale mode first bifurcates from a Gaussian to a bimodal behaviour, signifying the emergence of reversals of the large scale flow where the flow fluctuates between two di…
This note studies the mechanism of turbulent energy cascade through an opportune bifurcations analysis of the Navier--Stokes equations, and furnishes explanations on the more significant characteristics of the turbulence. A statistical bifurcations property of the Navier--Stokes equations in fully developed turbulence is proposed, and a spatial representation of the bifurcations is presented, which is based on a proper definition of the fixed points of the velocity field. The analysis first shows that the local deformation can be much more rapid than the fluid state variables, then explains th…
Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and…
We evaluate the effect of two experimental parameters on the slow dynamics of a Von-Kármán swirling flow driven by two propellers in a closed cylinder. The first parameter is the inertia mo- mentum of the propellers, and the second parameter is the aspect ratio, i.e. the distance between the propellers $H$ divided by the diameter $D$. We use a cell with a fixed diameter $D$ but where the distance between the propellers can be turned continuously and where the inertia from the pro- pellers can also be changed using different gears. No change on the dynamics is observed when the momentum of iner…
This work analyses the homogeneous isotropic turbulence by means of the equivalence between Euler and Lagrange representations of motion, adopting the bifurcation rates associated with Navier--Stokes and kinematic equations, and an appropriate hypothesis of fully developed chaos. The equivalence of these motion descriptions allows to show that kinetic and thermal energy cascade arise both from the convective term of Liouville equation. Accordingly, these phenomena, of nondiffusive nature, correspond to a transport in physical space linked to the trajectories divergence. Both the bifurcation ra…
Abrupt transitions to the state of thermoacoustic instability (TAI) in gas turbine combustors are a significant challenge plaguing the development of next-generation low-emission aircraft and power generation engines. In this paper, we present the observation of abrupt transition in three disparate turbulent thermoacoustic systems: an annular combustor, a swirl-stabilized combustor, and a preheated bluff-body stabilized combustor. Using a low-order stochastic thermoacoustic model, we show that the reported abrupt transitions occur when an initially stable, supercritical limit cycle becomes uns…
We demonstrate that turbulent zonal jets, analogous to Jovian ones, which are quasi-stationary, are actually metastable. After extremely long times, they randomly switch to new configurations with a different number of jets. The genericity of this phenomenon suggests that most quasi-stationary turbulent planetary atmospheres might have many climates and attractors for fixed values of the external forcing parameters. A key message is that this situation will usually not be detected by simply running the numerical models, because of the extremely long mean transition time to change from one clim…
A method is developed to estimate the properties of a global hydrodynamic instability in turbulent flows from measurement data of the limit-cycle oscillations. For this purpose, the flow dynamics are separated in deterministic contributions representing the global mode and a stochastic contribution representing the intrinsic turbulent forcing. Stochastic models are developed that account for the interaction between the two and that allow determining the dynamic properties of the flow from stationary data. The deterministic contributions are modelled by an amplitude equation, which describes th…
The ubiquity of turbulent flows in nature and technology makes it of utmost importance to fundamentally understand turbulence. Kolmogorov's 1941 paradigm suggests that for strongly turbulent flows with many degrees of freedom and its large fluctuations, there would only be \emph{one} turbulent state as the large fluctuations would explore the entire higher-dimensional phase space. Here we report the first conclusive evidence of multiple turbulent states for large Reynolds number $\text{Re}=\mathcal{O}(10^6)$ (Taylor number $\text{Ta}=\mathcal{O}(10^{12})$) Taylor-Couette flow in the regime of …
Over a century of research into the origin of turbulence in wallbounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. At slightly higher speeds the situation changes distinctly and the entire flow is turbulent. Neither the origin of the different states encountered during transition, nor their front dynamics, let alone the transformation to full turbulence could be explained to date. Combining experiments, theory and computer simulations here we uncover the bifurcation scenario organising the …
Although the roll/streak structure is ubiquitous in pre-transitional wall-bounded shear flow, this structure is linearly stable if the idealization of laminar flow is made. Lacking an instability, the large transient growth of the roll/streak structure has been invoked to explain its appearance as resulting from chance occurrence in the free-stream turbulence (FST) of perturbations configured to optimally excite it. However, there is an alternative interpretation which is that FST interacts with the roll/streak structure to destabilize it. Statistical state dynamics (SSD) provides analysis met…
When subcritical shear flows transition to turbulence, laminar and turbulent flow often coexists in space, giving rise to turbulent-laminar patterns. Most prominent are regular stripe patterns with large-scale periodicity and oblique orientation. Oblique stripes are a robust phenomenon, observed in experiments and flow simulations, yet their origin remains unclear. We demonstrate the existence of an invariant equilibrium solution of the fully nonlinear 3D Navier-Stokes equations that resembles the oblique pattern of turbulent-laminar stripes in plane Couette flow. We uncover the origin of the …
The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, show…
We introduce a novel way to extract information from turbulent datasets by applying an ARMA statistical analysis. Such analysis goes well beyond the analysis of the mean flow and of the fluctuations and links the behavior of the recorded time series to a discrete version of a stochastic differential equation which is able to describe the correlation structure in the dataset. We introduce a new intermittency parameter $Υ$ that measures the difference between the resulting analysis and the Obukhov model of turbulence, the simplest stochastic model reproducing both Richardson law and the Kolmogor…
The stability of annular electromagnetic pumps (EMPs) is investigated through 3D direct numerical simulations. When induction effects dominate dissipative processes, linear induction EMPs become unstable and lead to non-axisymmetric states relatively different from what was predicted by previous theoretical models. We show that the 3D destabilization of the flow is deeply connected to the axisymmetric state, and always appears as a secondary bifurcation from a locally stalled flow, even if the applied magnetic field is axisymmetric. Finally, we model a configuration aiming to optimize the effi…
Computer Simulations of Pulsatile Human Blood Flow Through 3D-Models of the Human Aortic Arch, Vessels of Simple Geometry and a Bifurcated Artery: Investigation of Blood Viscosity and Turbulent Effects
In order to obtain insights into dynamics of developed plane Couette turbulence, this paper considers bifurcation structure of unstable periodic orbits (UPOs) in the large-eddy-simulation (LES) system with the Smagorinsky-type eddy viscosity model. Treating the Smagorinsky constant as a bifurcation parameter, we detect the bifurcation points connecting two known UPOs which were separately discovered in the Navier--Stokes system [G. Kawahara and S. Kida, J. Fluid Mech., 449, 291--300 (2001)]. At the moderately high Reynolds number, the LES UPO of the present study possesses the spanwise vortice…
Large-scale motions in wall-bounded turbulent flows are frequently interpreted as resulting from an aggregation process of smaller-scale structures. Here, we explore the alternative possibility that such large-scale motions are themselves self-sustained and do not draw their energy from smaller-scale turbulent motions activated in buffer layers. To this end, it is first shown that large-scale motions in turbulent Couette flow at Re=2150 self-sustain even when active processes at smaller scales are artificially quenched by increasing the Smagorinsky constant Cs in large eddy simulations. These …
It is shown that the Truncated Euler Equations, i.e. a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined 2D Navier-Stokes flow with bottom friction and a spatially periodic forcing. In particular, the random reversals of the large scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation velocity that are described by the related microcanonical distribution which dis…
We perform a bifurcation analysis of the steady states of Rayleigh--Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need for any prior knowledge of the solutions. One of the disconnected branches we find contains an S-shaped curve with hysteresis, which is the origin of a fl…
In wall-bounded flows, the laminar regime remain linearly stable up to large values of the Reynolds number while competing with nonlinear turbulent solutions issued from finite amplitude perturbations. The transition to turbulence of plane channel flow (plane Poiseuille flow) is more specifically considered via numerical simulations. Previous conflicting observations are reconciled by noting that the two-dimensional directed percolation scenario expected for the decay of turbulence may be interrupted by a symmetry-breaking bifurcation favoring localized turbulent bands. At the other end of the…
Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear traveling wave solutions to the Navier--Stokes equations in the channel flow geometry -- exact coherent states (ECS) -- that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number Re=666. At the bifurcation, the solution has a highly symmetric quasistreamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, h…
The dynamics, structure and stability of zonal jets in planetary flows are still poorly understood, especially in terms of coupling with the small-scale turbulent flow. Here, we use an experimental approach to address the questions of zonal jets formation and long-term evolution. A strong and uniform topographic $β$-effect is obtained inside a water-filled rotating tank thanks to the paraboloidal fluid free upper surface combined with a specifically designed bottom plate. A small-scale turbulent forcing is performed by circulating water through the base of the tank. Time-resolving PIV measurem…
In wall-bounded parallel flows, sustained turbulence can occur even while laminar flow is still stable. Channel flow is one of such flows and displays spatio-temporal fluctuating patterns of localized turbulence along its way from/to featureless turbulence. By direct numerical simulation, we study the observed inconsistency between turbulence decay according to a two-dimensional directed-percolation (2D-DP) scenario and the presence of sustained oblique localized turbulent bands (LTBs) below the DP critical point. Above Reynolds number Reg \sim 700 sustained LTBs are observed; most LTBs have t…
Multistability is a ubiquitous feature in systems of geophysical relevance and provides key challenges for our ability to predict a system's response to perturbations. Near critical transitions small causes can lead to large effects and - for all practical purposes - irreversible changes in the properties of the system. The Earth climate is multistable: present astronomical and astrophysical conditions support two stable regimes, the warm climate we live in, and a snowball climate, characterized by global glaciation. We first provide an overview of methods and ideas relevant for studying the c…
