scholarly journals Revealing the state space of turbulent pipe flow by symmetry reduction

2013 ◽  
Vol 721 ◽  
pp. 514-540 ◽  
Author(s):  
A. P. Willis ◽  
P. Cvitanović ◽  
M. Avila
Keyword(s):  
State Space ◽  
Periodic Orbits ◽  
Pipe Flow ◽  
Travelling Wave ◽  
Symmetry Reduction ◽  
Turbulent Pipe Flow ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Unstable Manifolds ◽  
Reduced State

AbstractSymmetry reduction by the method of slices is applied to pipe flow in order to obtain a quotient of the streamwise 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 infinite-dimensional symmetry-reduced state space onto suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. Visualizations of the flow within the slice and its linearization at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic saddle, which capture turbulent dynamics at transitional Reynolds numbers.

Travelling wave states in pipe flow

2016 ◽  
Vol 791 ◽  
pp. 284-328 ◽  
Author(s):  
Ozge Ozcakir ◽  
Saleh Tanveer ◽  
Philip Hall ◽  
Edward A. Overman
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Travelling Wave ◽  
Large Reynolds Number ◽  
Travelling Wave Solutions ◽  
Asymptotic Structure ◽  
Pipe Flows ◽  
Wave Solutions ◽  
Wave States ◽  
Viscous Core

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

scholarly journals The critical layer in pipe flow at high Reynolds number

2008 ◽  
Vol 367 (1888) ◽  
pp. 561-576 ◽  
Author(s):  
D Viswanath
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
High Reynolds Number ◽  
Shear Flows ◽  
Travelling Wave ◽  
Critical Layer ◽  
Travelling Wave Solutions ◽  
Lower Branch ◽  
Wave Solutions ◽  
High Reynolds

We report the computation of a family of travelling wave solutions of pipe flow up to Re =75 000. As in all lower branch solutions, streaks and rolls feature prominently in these solutions. For large Re , these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as Re →∞ at rates given by Re −0.41 and Re −0.87 ; surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the Re →∞ limit could be universal to lower branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.

scholarly journals Relative periodic orbits form the backbone of turbulent pipe flow

2017 ◽  
Vol 833 ◽  
pp. 274-301 ◽  
Author(s):  
N. B. Budanur ◽  
K. Y. Short ◽  
M. Farazmand ◽  
A. P. Willis ◽  
P. Cvitanović
Keyword(s):  
Periodic Orbits ◽  
Pipe Flow ◽  
Chaotic Dynamics ◽  
Stokes Equations ◽  
Turbulent Pipe Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Unstable Manifolds ◽  
Low Dimensional

The 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 is also 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 lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

scholarly journals Experimental and theoretical progress in pipe flow transition

2008 ◽  
Vol 366 (1876) ◽  
pp. 2671-2684 ◽  
Author(s):  
A.P Willis ◽  
J Peixinho ◽  
R.R Kerswell ◽  
T Mullin
Keyword(s):  
Pipe Flow ◽  
Quantitative Agreement ◽  
Travelling Wave ◽  
Turbulent Pipe Flow ◽  
Transition To Turbulence ◽  
Reynolds Number Flow ◽  
Threshold Amplitude ◽  
Number Flow ◽  
The Stability ◽  
Laminar State

There have been many investigations of the stability of Hagen–Poiseuille flow in the 125 years since Osborne Reynolds' famous experiments on the transition to turbulence in a pipe, and yet the pipe problem remains the focus of attention of much research. Here, we discuss recent results from experimental and numerical investigations obtained in this new century. Progress has been made on three fundamental issues: the threshold amplitude of disturbances required to trigger a transition to turbulence from the laminar state; the threshold Reynolds number flow below which a disturbance decays from turbulence to the laminar state, with quantitative agreement between experimental and numerical results; and understanding the relevance of recently discovered families of unstable travelling wave solutions to transitional and turbulent pipe flow.

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