A dynamical system description of the transition process in shear flows with no linear instability starts with a knowledge of exact coherent solutions, among them travelling waves (TWs) and relative periodic orbits (RPOs). We describe a numerical method to find such solutions in pipe flow and apply it in the vicinity of a Hopf bifurcation from a TW which looks to be especially relevant for transition. The dominant structural feature of the RPO solution is the presence of weakly modulated streaks. This RPO, like the TW from which it bifurcates, sits on the laminar-turbulent boundary separating initial conditions which lead to turbulence from those which immediately relaminarise.
We present a unique pipe flow rig capable where simultaneous particle tracking and flow velocity measurements in a dilute, neutrally buoyant particulate pipe flow in regimes of transition to turbulence. The flow consists of solid glass spheres for the disperse phase, and a density-matching fluid for the carrier phase. The measurements are conducted using a bespoke, combined 2D PIV/ PTV technique. The technique takes advantage of a phase discrimination approach that involves separating the disperse and carrier phases based on their respective image characteristics. Our results show that the rig…
Several experimental and numerical studies have shown that turbulent motions in circular pipe flow near transitional Reynolds numbers may not persist forever, but may decay. We study the properties of these decaying states within direct numerical simulations for Reynolds numbers up to 2200 and in pipes with lengths equal to 5, 9 and 15 times the diameter. We show that the choice of the ensemble of initial conditions affects the short time parts of lifetime distributions, but does not change the characteristic decay rate for long times. Comparing lifetimes for pipes of different length we notic…
As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, non-laminar flow which remains poorly known. We show experimentally that the fluid friction of flash flow is diagnostic of turbulence. We also show that the statistics of flash flow are in keeping with Kolmogorov's phenomenological theory of turbulence (so that, e.g., the energy spectra of both flash flow and turbulent …
Pipe flow often traverses a regime where laminar and turbulent flow co-exist. Prandtl and Tietjens explained this intermittency as a feedback between the fluctuations of the internal flow resistance and the constant pressure drop driving the flow. However, because the focus has moved towards studying intermittency without flow fluctuations near the universal critical Reynolds number, their explanation has largely disappeared. Here we refine the mechanism, which has never been put to a quantitative test, to develop a model that agrees with experiments at higher Reynolds numbers, enabling us to …
Numerical and experimental studies of transitional pipe flow have shown the prevalence of coherent flow structures that are dominated by downstream vortices. They attract special attention because they contribute predominantly to the increase of the Reynolds stresses in turbulent flow. In the present study we introduce a convenient detector for these coherent states, calculate the fraction of time the structures appear in the flow, and present a Markov model for the transition between the structures. The fraction of states that show vortical structures exceeds 24% for a Reynolds number of abou…
The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface which separates initial conditions which uneventfully relaminarise an…
Transitional pipe flow is modeled as a one-dimensional excitable and bistable medium. Models are presented in two variables, turbulence intensity and mean shear, that evolve according to established properties of transitional turbulence. A continuous model captures the essence of the puff-slug transition as a change from excitability to bistability. A discrete model, that additionally incorporates turbulence locally as a chaotic repeller, reproduces almost all large-scale features of transitional pipe flow. In particular it captures metastable localized puffs, puff splitting, slugs, a continuo…
The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them g…
The flow instability and further transition to turbulence in a toroidal pipe (torus) with curvature (tube-to-coiling diameter) 0.049 is investigated experimentally. The flow inside the toroidal pipe is driven by a steel sphere fitted to the inner pipe diameter. The sphere is moved with constant azimuthal velocity from outside the torus by a moving magnet. The experiment is designed to investigate curved pipe flow by optical measurement techniques. Using stereoscopic particle image velocimetry, laser Doppler velocimetry and pressure drop measurements, the flow is measured for Reynolds numbers r…
This study numerically investigates the bifurcation aspect of the wide-gap spherical Couette flow (SCF), with an emphasis on the competition among polygonal coherence with different wave numbers observed over transitional Reynolds numbers. Focusing on a representative case, the half-radius ratio, we confirm that the axisymmetric state becomes unstable over the first transitional Reynolds number at which the 4-fold spiral state bifurcates, using the continuation method based on the Newton-Raphson algorithm. The Galerkin-spectral method was employed to numerically solve the governing equations. …
One-dimensional models are presented for transitional shear flows. The models have two variables corresponding to turbulence intensity and mean shear. These variables evolve according to simple equations based on known properties of transitional turbulence. The first model considered is for pipe flow. A previous study modeled turbulence using a chaotic tent map. In the present work turbulence is modeled instead as multiplicative noise. This model captures the character of transitional pipe flow and contains metastable puffs, puff splitting, and slugs. These ideas are extended to a limited mode…
Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider $\textit{et al.}$ 2010a). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re < 175$) to transitional ones ($Re \approx 325$), these spatially localized states traverse various regimes …
Turbulent stripe, which would occur in turbulent channel flows at transitional Reynolds numbers, was studied experimentally by flow visualization using reflective flake particles. In a range of bulk mean Reynolds number Re = 1700-2000, the turbulent stripe was observed to be inclined at angles of 20-30 degree against the streamwise direction, and its streamwise wave length was about 60 times of the channel half width ($δ$). The longitudinal streaks with the spanwise spacing of 1.6$δ$ were found in the quasi-laminar regions. The critical Reynolds number was Re = 1300. Time traces of the streamw…
In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers R. We show that in periodic domains that contain a few bands, for given values of R and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics i…
Cavitating flow over a circular cylinder is investigated over a range of cavitation numbers for both laminar and turbulent regimes. We observe non-cavitating, cyclic and transitional cavitation regimes with reduction in freestream $σ$. The cavitation inside the Karman vortices in the cyclic regime, is significantly altered by the onset of "condensation front" propagation in the transitional regime. At the transition, an order of magnitude jump in shedding Strouhal number is observed as the dominant frequency shifts from periodic vortex shedding in the cyclic regime, to irregular-regular vortex…
Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear f…
The transition to turbulence exhibits remarkable spatio-temporal behavior that continues to defy detailed understanding. Near the onset to turbulence in pipes, transient turbulent regions decay either directly or, at higher Reynolds numbers through splitting, with characteristic time-scales that exhibit a super-exponential dependence on Reynolds number. Here we report numerical simulations of transitional pipe flow, showing that a zonal flow emerges at large scales, activated by anisotropic turbulent fluctuations; in turn, the zonal flow suppresses the small-scale turbulence leading to stochas…
Flows through medical devices as well as in anatomical vessels despite being at moderate Reynolds number may exhibit transitional or even turbulent character. In order to validate numerical methods and codes used for biomedical flow computations, the U.S. food and drug administration (FDA) established an experimental benchmark, which was a pipe with gradual contraction and sudden expansion representing a nozzle. The experimental results for various Reynolds numbers ranging from 500 to 6500 were publicly released. Previous and recent computational investigations of flow in the FDA nozzle found …
Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front veloc…
In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al., Phys. Rev. Let. 110, 224502 (2013) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low speeds. Combining slicing methods for continuous symmetry reduction with Poincaré sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it contains a heteroclinic connection to the upper branch …
Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow in its way to/from turbulence as the Reynolds number R is varied. Many previous theoretical and numerical works on the problem have considered small aspect-ratio systems subjected to periodic boundary conditions, while experiments correspond to the opposite limit of large aspect-ratio. Here, by means of fully resolved direct numerical simulations of the Navier-Stokes equations at decreasing R, we scrutinize the transition from temporal to spatiotemporal behav…
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 mo…
Chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in…
Spatio-temporally chaotic dynamics of transitional plane Couette flow may give rise to regular turbulent-laminar stripe patterns with a large-scale pattern wavelength and an oblique orientation relative to the laminar flow direction. A recent dynamical systems analysis of the oblique stripe pattern demonstrated that the Navier-Stokes equations have unstable equilibrium solutions that capture the three-dimensional spatial structure of the oblique stripe patterns. Here we identify unstable periodic orbits that not only show oblique large-scale amplitude modulation but also have a characteristic …
The recent theoretical discovery of finite-amplitude travelling waves in pipe flow has re-ignited interest in the transitional phenomena that Osborne Reynolds studied 125 years ago. Despite all being unstable, these waves are providing fresh insight into the flow dynamics. Here we describe two new classes of highly-symmetric travelling waves (possessing rotational, shift-&-reflect and mirror symmetries) and report a new family of mirror-symmetric waves which is the first found in pipe flow not to have shift-&-reflect symmetry. The highly-symmetric waves appear at lower Reynolds numbers than th…
We study turbulent natural convection in enclosures with conjugate heat transfer. The simplest way to increase the heat transfer in this flow is through rough surfaces. In numerical simulations often the constant temperature is assigned at the walls in contact with the fluid, which is unrealistic in laboratory experiments. The DNS (Direct Numerical Simulation), to be of help to experimentalists, should consider the heat conduction in the solid walls together with the turbulent flow between the hot and the cold walls. Here the cold wall, $0.5h$ thick (where $h$ is the channel half-height) is sm…
In this essay, we recall the specificities of the transition to turbulence in wall-bounded flows and present recent achievements in the understanding of this problem. The transition is abrupt with laminar-turbulent coexistence over a finite range of Reynolds numbers, the transitional range. The archetypical cases of Poiseuille pipe flow and plane Couette flow are first reviewed at the phenomenological level, together with a few other flow configurations. Theoretical approaches are then examined with particular emphasis on the existence of special nontrivial solutions to the Navier-Stokes equat…
We investigate the propagation of MHD fast waves into a cylindrical coronal loop through an inhomogeneous stationary flow region. The background flow is assumed to have a small, spatially periodic structure in addition to a constant speed. We focus on the absorption of the wave energy in Alfvén resonance, comparing with the constant flow case. A new flow (absorption) regime is induced by the periodic flow structure which enhances the absorption for the antiparallel flow and inverse absorption (overreflection) for the parallel flow with respect to the axial wave vector, depending on the transit…
Laminar-turbulent intermittency is intrinsic to the transitional regime of a wide range of fluid flows including pipe, channel, boundary layer and Couette flow. In the latter turbulent spots can grow and form continuous stripes, yet in the stripe-normal direction they remain interspersed by laminar fluid. We carry out direct numerical simulations in a long narrow domain and observe that individual turbulent stripes are transient. In agreement with recent observations in pipe flow we find that turbulence becomes sustained at a distinct critical point once the spatial proliferation outweighs the…
We derive necessary conditions that traveling wave solutions of the Navier-Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution irrespective of the Reynolds number ($Re$). Other conditions are asymptotic in the limit $Re\to\infty$. The exact conditions are likely to be useful tools in the study of transitional structures. For the pipe flow geometry, we give computations up to $Re=100000$ showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at hi…
The complex flow features resulting from the laminar-turbulent transition (LTT) in a sudden expansion pipe flow, with expansion ratio of 1:2 subjected to an inlet vortex perturbation is investigated by means of direct numerical simulations (DNS). It is shown that the threshold for LTT described by a power law scaling with -3 exponent that links the perturbation intensity to the subcritical transitional Reynolds number. Additionally, a new type of instability is found within a narrow range of flow parameters. This instability originates from the region of intense shear rate which is a result of…
The transitional boundary layer flow over a flat plate is investigated. The boundary layer flow is known to develop unstable Tollmien-Schlichting waves above a critical value of the Reynolds number. However, it is also known that this transition can be observed for sub-critical Reynolds numbers. In that case, the basin of attraction of the laminar state coexists with the sustained turbulence. In this article, the trajectory on the separatrix between these two states is simulated. The state on the separatrix is independent from the initial condition and is dynamically connected to both the lami…
Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied 1-dim…
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 connectio…
Laminar as well as turbulent oscillatory pipe flows occur in many fields of biomedical science and engineering. Pulmonary air flow and vascular blood flow are usually laminar, because shear forces acting on the physiological system ought to be small. However, frictional losses and shear stresses vary considerably with transition to turbulence. This plays an important role in cases of e.g. artificial respiration or stenosis. On the other hand, in piston engines and reciprocating thermal/chemical process devices, turbulent or transitional oscillatory flows affect mixing properties, and also mass…
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.
Numerical simulation has been conducted for the chaotic flow in a 3D closed-loop pulsating heat pipe (PHP). Heat flux and constant temperature boundary conditions were applied for evaporator and condenser sections, respectively. Water and ethanol were used as working fluids. Volume of Fluid (VOF) method has been employed for two-phase flow simulation. Spectral analysis of temperature time series was carried out using Power Spectrum Density (PSD) method. Existence of dominant peak in PSD diagram indicated periodic or quasi-periodic behavior in temperature oscillations at particular frequencies.…
Drag reduction and subcritical turbulence in controlled pipe flows
Politecnico di Milano（米兰理工大学）；Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille（里尔大学、法国国家科学研究中心、法国航空发展局、巴黎国立高等工艺学院、里尔中央理工学院）；Massachusetts Institute of Technology（麻省理工学院）
The nature of dynamo action in shear flows prone to magnetohydrodynamic instabilities is investigated using the magnetorotational dynamo in Keplerian shear flow as a prototype problem. Using direct numerical simulations and Newton's method, we compute an exact time-periodic magnetorotational dynamo solution to the three-dimensional dissipative incompressible magnetohydrodynamic equations with rotation and shear. We discuss the physical mechanism behind the cycle and show that it results from a combination of linear and nonlinear interactions between a large-scale axisymmetric toroidal magnetic…
