A swirling spiral wave solution in pipe flow

2013 ◽  
Vol 737 ◽  
Author(s):  
K. Deguchi ◽  
A. G. Walton
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Wave Solution ◽  
Asymptotic Theory ◽  
Spiral Wave ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Nonlinear Solution

AbstractA numerically exact full Navier–Stokes counterpart of the asymptotic nonlinear solution in Hagen–Poiseuille flow proposed by Smith & Bodonyi (Proc. R. Soc. A, vol. 384, 1982, pp. 463–489) is discovered. The solution takes the form of a spiral travelling wave, with a novel feature being a strong induced component of swirl. Our solution shows excellent quantitative agreement with the asymptotic theory at Reynolds numbers of the order of $1{0}^{8} $.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

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.

scholarly journals On the transient nature of localized pipe flow turbulence

2010 ◽  
Vol 646 ◽  
pp. 127-136 ◽  
Author(s):  
MARC AVILA ◽  
ASHLEY P. WILLIS ◽  
BJÖRN HOF
Keyword(s):  
Pipe Flow ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Critical Threshold ◽  
Lifetime Data ◽  
Low Reynolds Numbers ◽  
Transient Nature ◽  
Detailed Statistical Analysis ◽  
Flow Turbulence ◽  
Shed Light

The onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarizes, raising the question of whether it is transient in nature or becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the lifetime data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarization process is no longer memoryless.

scholarly journals Linear stability of boundary layers under solitary waves

2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen
Keyword(s):  
Linear Stability ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Linear Instability ◽  
Navier Stokes ◽  
Acceleration Region ◽  
Critical Reynolds Numbers ◽  
Tollmien Schlichting ◽  
The Stability

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

scholarly journals Subcritical transition in channel flows

2002 ◽  
Vol 451 ◽  
pp. 35-97 ◽  
Author(s):  
S. JONATHAN CHAPMAN
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Domain Of Attraction ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Laminar State

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

Eigenvalue Sensitivity to Base Flow Variations in Hagen-Poiseuille Flow

2002 ◽  
Author(s):  
Isabella M. Gavarini ◽  
Alessandro Bottaro ◽  
Frans T. M. Nieuwstadt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Transient Growth ◽  
Cylindrical Pipe ◽  
Low Reynolds Numbers ◽  
Parabolic Profile ◽  
Eigenvalue Sensitivity ◽  
The Ideal

Transition in a cylindrical pipe flow still eludes thorough understanding. Most recent advances are based on the concept of transient growth of disturbances, but even this scenario is not fully confirmed by DNS and/or experiments. Based on the fact that even the most carefully conducted experiment is biased by uncertainties, we explore the spatial growth of disturbances developing on top of an almost ideal, axially invariant Poiseuille flow. The optimal deviation of the base flow from the ideal parabolic profile is computed by a variational tecnique, and unstable modes, driven by an inviscid mechanism, are found to exist for very small values of the norm of the deviation, at low Reynolds numbers.

Time-dependent helical waves in rotating pipe flow

1990 ◽  
Vol 221 ◽  
pp. 289-310 ◽  
Author(s):  
Michael J. Landman
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Helical Symmetry ◽  
Navier Stokes Equations ◽  
Turbulent Transition

The Navier-Stokes equations for flow in a rotating circular pipe are solved numerically, subject to imposing helical symmetry on the velocity field v = v(r, θ + αz,t). The helical symmetry is exploited by writing the equations of motion in helical variables, reducing the problem to two dimensions. A limited study of the pipe flow is made in the parameter space of the wavenumber α, and the axial and azimuthal Reynolds numbers. The steadily rotating waves previously studied by Toplosky & Akylas (1988), which arise from the linear instability of the basic steady flow, are found to undergo a series of bifurcations, through periodic to aperiodic time dependence. The relevance of these results to the mechanism of laminar-turbulent transition in a stationary pipe is discussed.

Critical threshold in pipe flow transition

2008 ◽  
Vol 367 (1888) ◽  
pp. 545-560 ◽  
Author(s):  
Fernando Mellibovsky ◽  
Alvaro Meseguer
Keyword(s):  
Pipe Flow ◽  
Basin Of Attraction ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Computational Method ◽  
Critical Threshold ◽  
Krylov Method ◽  
Numerical Characterization ◽  
Wide Range ◽  
Critical Amplitudes

This study provides a numerical characterization of the basin of attraction of the laminar Hagen–Poiseuille flow by measuring the minimal amplitude of a perturbation required to trigger transition. For pressure-driven pipe flow, the analysis presented here covers autonomous and impulsive scenarios where either the flow is perturbed with an initial disturbance with a well-defined norm or perturbed by means of local impulsive forcing that mimics injections through the pipe wall. In both the cases, the exploration is carried out for a wide range of Reynolds numbers by means of a computational method that numerically resolves the transitional dynamics. For , the present work provides critical amplitudes that decay as Re −3/2 and Re −1 for the autonomous and impulsive scenarios, respectively. For Re =2875, accurate threshold amplitudes are found for constant mass-flux pipe by means of a shooting method that provides critical trajectories that never relaminarize or trigger transition. These transient states are used as initial guesses in a damped Newton–Krylov method formulated to find periodic travelling wave solutions that either travel downstream or exhibit a helicoidal advection.

Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states

2009 ◽  
Vol 619 ◽  
pp. 213-233 ◽  
Author(s):  
ASHLEY P. WILLIS ◽  
RICH R. KERSWELL
Keyword(s):  
Pipe Flow ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Edge State ◽  
Travelling Wave ◽  
Edge States ◽  
Huge Number ◽  
Laminar And Turbulent Flow ◽  
Flow Through

Fully three-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+ε-dimensional’ model of pipe flow, which is a minimal three-dimensionalization 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 localized ‘puff-like’ structures which can decay abruptly after long times, as well as global ‘slug’ turbulence. Relaminarization statistics of puffs reproduce the memoryless feature of pipe flow and indicate the existence of a Reynolds number about which lifetimes diverge rapidly, provided that the pipe is sufficiently long. Exponential divergence of the lifetime is prevalent in shorter periodic domains. In a short pipe, exact travelling-wave solutions are found near flow trajectories on the boundary between laminar and turbulent flow. In a long pipe, the attracting state on the laminar–turbulent boundary is a localized structure which resembles a smoothened puff. This ‘edge’ state remains localized even for Reynolds numbers at which the turbulent state is global.

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