scholarly journals Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos

2012 ◽  
Vol 702 ◽  
pp. 415-443 ◽  
Author(s):  
Chris C. T. Pringle ◽  
Ashley P. Willis ◽  
Rich R. Kerswell
Keyword(s):  
Variational Problem ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Initial Perturbation ◽  
Transition To Turbulence ◽  
Critical Threshold ◽  
General Strategy ◽  
Transient Growth ◽  
Initial Amplitude ◽  
Energy Growth

AbstractWe propose a general strategy for determining the minimal finite amplitude disturbance that triggers transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude which respect the boundary conditions, incompressibility and the Navier–Stokes equations, to maximize a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approach using the ratio of the final to initial perturbation kinetic energies (energy growth) as the functional and the energy norm to measure amplitudes in the context of pipe flow. Our results indicate that the variational problem yields a smooth converged solution provided that the initial amplitude is below the threshold for transition. This optimal is the nonlinear analogue of the well-studied (linear) transient growth optimal. At the critical threshold, the optimization seeks out a disturbance that is on the ‘edge’ of turbulence during the period. Above this threshold, when disturbances trigger turbulence by the end of the period, convergence is then practically impossible. The first disturbance found to trigger turbulence as the amplitude is increased identifies the ‘minimal seed’ for the given geometry and forcing (Reynolds number). We conjecture that it may be possible to select a functional such that the converged optimal below threshold smoothly converges to the minimal seed at threshold. Our choice of the energy growth functional is shown to come close to this for the pipe flow geometry investigated here.

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 Transient growth in linearly stable Taylor–Couette flows

2014 ◽  
Vol 742 ◽  
pp. 254-290 ◽  
Author(s):  
Simon Maretzke ◽  
Björn Hof ◽  
Marc Avila
Keyword(s):  
Linear Stability ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Energy Growth ◽  
Transient Energy ◽  
Taylor Couette ◽  
Semi Empirical ◽  
Essential Prerequisite ◽  
Cylinder Rotation

AbstractNon-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor–Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers$\mathit{Re}$, the optimal transient energy growth always follows a$\mathit{Re}^{2/3}$scaling, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows, the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations, we show that linear stability and transient growth are independent of the cylinder rotation ratio and we derive a universal$\mathit{Re}^{2/3}$scaling of optimal energy growth using Wentzel–Kramers–Brillouin theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.

Oscillating Pipe Flow: High-Resolution Simulation of Nonlinear Mechanisms

2006 ◽  
Author(s):  
Arash Ghasemi
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Stationary Flow ◽  
Current Approach ◽  
Laminar Flows ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Frequency Change ◽  
Starting Point ◽  
Corner Points

A new perspective suitable for understanding the details of nonlinear pumping (formation of traveling shocks) inside a pressurized cavity is constructed in this paper. Full compressible axisymmetric three-dimensional Navier-Stokes equations are used as the starting point to cover all complexities of the problem that exceedingly increase for particular ranges of Mach, Reynolds and Prandtl numbers. Then a very high-order numerical method is introduced to preserve the user-defined order of accuracy for practical simulations. For removal of spurious waves, higher-order compact filters are derived. All equations are marched in time using the classical Runge-Kutta algorithm which is appropriate for problems involving fine-scale temporal fluctuations. As the most important part of simulation, Navier-Stokes Characteristic Boundary Conditions are used for accurate calculation of wave reflection specially at singular points, i.e., corner points and points across the axis of symmetry. A simultaneous characteristic-decomposition is devised in this paper which completely resolves stability problems arising from problem-dependent treatment of corner points. Numerical experiments are performed for high-Reynolds laminar flows inside the shock region to determine the effect of frequency change on both shock formation (stationary flow) and transient solution. The current approach which favorably compares to the previous experimental data, may be used as a robust tool for understanding the less-understood problem of shock/Stokes-Layer interaction and its consequences on transition to turbulence in Oscillating Pipe Flow.

The Effect of Non-Normality of Navier-Stokes Operator Mechanism on Dynamics of an Axisymmetric Swirling Flow

2013 ◽  
Vol 681 ◽  
pp. 72-78
Author(s):  
Cheng Chen ◽  
Ya Yong Shi
Keyword(s):  
Swirling Flow ◽  
Initial Perturbation ◽  
Fluid Motion ◽  
Nonlinear Behavior ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Transient Growth ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Oseen Vortex

The effect of non-normality of the Navier-Stokes operator on the dynamics of an axisymmetric swirling flow, namely, the Oseen vortex, has been investigated. The eigenvalue analysis and transient growth analysis have been employed in order to obtain the least stable eigenmode and the global optimal perturbation, which are both considered as the initial perturbation. Three stages of dynamic process have been put into evidence for the evolution of the optimal perturbation. The early (linear) stage is characterized by the amplification of radial perturbation, consistent with the prediction of transient growth theory. Having come into the nonlinear stage, the perturbation energy growth is suppressed by the interaction between the vortex ring and the Oseen vortex core. Finally, the phenomena of secondary energy growth are also observed. Compared with the results obtained by applying the least stable eigenmode as the initial disturbance, the nonlinear behavior of the optimal perturbation features radial fluid motion and the rapid production of small eddies, which are both thought to be beneficial to fluid entrainment or mixing. The effect of perturbation amplitude on the nonlinear evolution of flows is also studied.

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

2008 ◽  
Vol 367 (1888) ◽  
pp. 509-527 ◽  
Author(s):  
Jacob Cohen ◽  
Jimmy Philip ◽  
Guy Ben-Dov
Keyword(s):  
Growth Mechanism ◽  
Pipe Flow ◽  
Scaling Laws ◽  
Scaling Law ◽  
Base Flow ◽  
Initial Disturbance ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Hairpin Vortices ◽  
Linear And Nonlinear

The failure of normal-mode linear stability analysis to predict a transition Reynolds number ( Re tr ) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re . The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re , are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al ., indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Re tr of approximately 2000 associated with the bifurcation between two deviation solutions.

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 Pipe flow with large particles and their impact on the transition to turbulence

2020 ◽  
Vol 5 (11) ◽  
Author(s):  
Martin Leskovec ◽  
Fredrik Lundell ◽  
Fredrik Innings
Keyword(s):  
Pipe Flow ◽  
Transition To Turbulence ◽  
Large Particles

Self-limiting and regenerative dynamics of perturbation growth on a vortex column

2013 ◽  
Vol 718 ◽  
pp. 39-88 ◽  
Author(s):  
Fazle Hussain ◽  
Eric Stout
Keyword(s):  
Reynolds Stress ◽  
Stokes Equations ◽  
Initial Energy ◽  
Navier Stokes ◽  
Axisymmetric Mode ◽  
Energy Growth ◽  
Decay Behaviour ◽  
Peak Energy ◽  
Trailing Vortices ◽  
Instability Energy

AbstractWe study the mechanisms of centrifugal instability and its eventual self-limitation, as well as regenerative instability on a vortex column with a circulation overshoot (potentially unstable) via direct numerical simulations of the incompressible Navier–Stokes equations. The perturbation vorticity (${\boldsymbol{\omega} }^{\prime } $) dynamics are analysed in cylindrical ($r, \theta , z$) coordinates in the computationally accessible vortex Reynolds number, $\mathit{Re}({\equiv }\mathrm{circulation/viscosity} )$, range of 500–12 500, mostly for the axisymmetric mode (azimuthal wavenumber $m= 0$). Mean strain generates azimuthally oriented vorticity filaments (i.e. filaments with azimuthal vorticity, ${ \omega }_{\theta }^{\prime } $), producing positive Reynolds stress necessary for energy growth. This ${ \omega }_{\theta }^{\prime } $ in turn tilts negative mean axial vorticity, $- {\Omega }_{z} $ (associated with the overshoot), to amplify the filament, thus causing instability. (The initial energy growth rate (${\sigma }_{r} $), peak energy (${G}_{\mathit{max}} $) and time of peak energy (${T}_{p} $) are found to vary algebraically with $\mathit{Re}$.) Limitation of vorticity growth, also energy production, occurs as the filament moves the overshoot outward, hence lessening and shifting $\vert {- }{\Omega }_{z} \vert $, while also transporting the core $+ {\Omega }_{z} $, to the location of the filament. We discover that a basic change in overshoot decay behaviour from viscous to inviscid occurs at $Re\sim 5000$. We also find that the overshoot decay time has an asymptotic limit of 45 turnover times with increasing $\mathit{Re}$. After the limitation, the filament generates negative Reynolds stress, concomitant energy decay and hence self-limitation of growth; these inviscid effects are enhanced further by viscosity. In addition, the filament transports angular momentum radially inward, which can produce a new circulation overshoot and renewed instability. Energy decays at the $\mathit{Re}$ studied, but, at higher $\mathit{Re}$, regenerative growth of energy is likely due to the renewed mean shearing. New generation of overshoot and Reynolds stress is examined using a helical ($m= 1$) perturbation. Regenerative energy growth, possibly resulting in even vortex breakup, can be triggered by this new overshoot at practical $\mathit{Re}$ (${\sim }1{0}^{6} $ for trailing vortices), which are currently beyond the computational capability.

scholarly journals The non-modal onset of the tearing instability

2018 ◽  
Vol 84 (5) ◽  
Author(s):  
D. MacTaggart
Keyword(s):  
Initial Conditions ◽  
Linear Equations ◽  
Initial Perturbation ◽  
Transient Growth ◽  
Tearing Mode ◽  
Eigenvalue Spectrum ◽  
Tearing Instability ◽  
Set Up ◽  
Solution Behaviour ◽  
Physical Quantities

We investigate the onset of the classical magnetohydrodynamic (MHD) tearing instability (TI) and focus on non-modal (transient) growth rather than the tearing mode. With the help of pseudospectral theory, the operators of the linear equations are shown to be highly non-normal, resulting in the possibility of significant transient growth at the onset of the TI. This possibility increases as the Lundquist number$S$increases. In particular, we find evidence, numerically, that the maximum possible transient growth, measured in the$L_{2}$-norm, for the classical set-up of current sheets unstable to the TI, scales as$O(S^{1/4})$on time scales of$O(S^{1/4})$for$S\gg 1$. This behaviour is much faster than the time scale$O(S^{1/2})$when the solution behaviour is dominated by the tearing mode. The size of transient growth obtained is dependent on the form of the initial perturbation. Optimal initial conditions for the maximum possible transient growth are determined, which take the form of wave packets and can be thought of as noise concentrated at the current sheet. We also examine how the structure of the eigenvalue spectrum relates to physical quantities.

Sign in / Sign up

Export Citation Format

Share Document