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 the originally-discovered non-mirror-symmetric waves found by Faisst & Eckhardt 2003 and Wedin & Kerswell 2004 and have much higher wall shear stresses. The first M-class comprises of the various discrete-rotationally-symmetric analogues of the mirror-symmetric wave found in Pringle & Kerswell (2007) and have a distinctive double layer structure of fast and slow streaks across the pipe radius. The second N-class has the more familiar separation of fast streaks to the exterior and slow streaks to the interior and looks the precursor to the class of non-mirror-symmetric waves already known.
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…
New families of three-dimensional nonlinear travelling waves are discovered in pipe flow. In contrast to known waves (Faisst & Eckhardt Phys. Rev. Lett. 91, 224502 (2003), Wedin & Kerswell, J. Fluid Mech. 508, 333 (2004)), they possess no rotational symmetry and exist at much lower Reynolds numbers. Particularly striking is an `asymmetric mode' which has one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. This family originates in a pitchfork bifurcation from a mirror-symmetric travelling wave which can be traced down to a Reynolds number of 773.…
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…
A family of three-dimensional travelling waves for flow through a pipe of circular cross section is identified. The travelling waves are dominated by pairs of downstream vortices and streaks. They originate in saddle-node bifurcations at Reynolds numbers as low as 1250. All states are immediately unstable. Their dynamical significance is that they provide a skeleton for the formation of a chaotic saddle that can explain the intermittent transition to turbulence and the sensitive dependence on initial conditions in this shear 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 R…
The laminar-turbulent boundary S is the set separating initial conditions which relaminarise uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow look to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognising approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on S are organised around only a few travelling waves and their heteroclinic connections. If the…
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 κ = 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 …
Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter $C$, our DNS show that shear-driven turbulence may either be completely laminarised or transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, …
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 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…
Fully 3-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a `2+epsilon' dimensional model of pipe flow which is a minimal 3-dimensionalisation of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained, yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic loca…
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 …
We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio $ε=(r_o-r_i)/r_i$ is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere's rotation rate is increased to $\sim10\%$ beyond the first bifurcation from the basic state flow. For $ε=0.04$ the initial bifurcations are the same as in previous numerical work at $ε=0.154$, and result in steady one- and two-vortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the $ε\to0$ limit.…
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）
We study propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm) in long-wave approximation.The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when balance of nonlinearity, dispersion and dissipation in such a medium holds the model equations are reduced to a version of the Korteweg-deVries-Burgers equation with variable coefficients. Exact travelling-wave solution of this equation is obtained by the modif…
We present a study on the nonlinear dynamics of small long-wave disturbances to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. At high Reynolds numbers, the associated Navier-Stokes equations can be reduced to a set of coupled Korteweg-de Vries-type (KdV) equations that support inviscid and smooth travelling waves numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices concentrated near the pipe boundaries (wall vortexons) or that wrap around the pipe axis (centre vortexons), in agreement with the analytical so…
Slug flows are extremely interesting multiphase regime phenomena which frequently occur in flexible marine risers used by the petroleum industry in offshore environments and have both a liquid and gaseous phase. This paper describes two simple models of the slug flow regime by means of an equivalent monophase flow with a non-constant density. The slug regime is modelled as a monophase density-varying flow with a sinusoidal density, travelling along the pipe itself towards the top end node of the riser. Starting from the bottom end, it is characterized by adiabatic processes and energy loss alo…
In this paper, we study the nonlinear dynamics of an axisymmetric disturbance to the laminar state in non-rotating Poiseuille pipe flows. In particular, we show that the associated Navier-Stokes equations can be reduced to a set of coupled Camassa-Holm type equations. These support inviscid and smooth localized travelling waves, which are numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices that concentrate near the pipe boundaries (wall vortexons) or wrap around the pipe axis (centre vortexons) in agreement with the analytical so…
We present experimental results obtained from pipe flows generated by fractal shaped orifices or openings. We compare different fractal orifices and their efficiencies to re-generate axi-symmetric flows and to return to the standard flow generated by a perforated plate or a circular orifice plate. We consider two families of fractal openings: mono-orifice and complex-orifice and emphasize the differences between the two fractal families. For the Reynolds number we use, we found that there is an optimum iteration for the fractal level above which no improvement for practical application such as…
We discuss propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm). The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when long-wave approximation holds the model equations are reduced to a version of the perturbed Korteweg-deVries equation. Exact travelling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simples…
The flow past a bullet-shaped blunt body moving in a pipe is investigated through global linear stability analysis (LSA) and direct numerical simulation (DNS). A cartography of the bifurcation curves is provided thanks to LSA, covering the range of parameters corresponding to Reynolds number $Re = [50-110]$, confinement ratio $a/A = [ 0.01 - 0.92]$ and length-to-diameter ratio $L/d = [ 2-10]$. Results show that the first bifurcation is always a steady bifurcation associated to a non-oscillating eigenmode with azimuthal wavenumber $m=\pm 1$ leading to a steady state with planar symmetry. For we…
Plane Poiseuille flow shows turbulence at a Reynolds number that is lower than the critical one for the onset of Tollmien-Schlichting waves. The transition to turbulence follows the same route as the by-pass transition in boundary layers, i.e. finite amplitude perturbations are required and the flow is dominated by downstream vortices and streaks in the transitional regime. In order to relate the phenomenology in plane Poiseuille flow to our previous studies of plane Couette flow (Kreilos & Eckhardt, 2012), we study a symmetric subspace of plane Poiseuille flow in which the bifurcation cascade…
We investigate the formation of acoustic horizons for an inviscid fluid moving in a pipe in the case of stationary and axi-symmetric flow. We show that, differently from what is generally believed, the acoustic horizon forms in correspondence of either a local minimum or maximum of the flux tube cross-section. Similarly, the external potential is required to have either a maximum or a minimum at the horizon, so that the external force has to vanish there. Choosing a power-law equation of state for the fluid, $P\propto ρ^{n}$, we solve the equations of the fluid dynamics and show that the two p…
