Transient growth and minimal defects: Two possible initial paths of transition to turbulence in plane shear flows

2004 ◽  
Vol 16 (10) ◽  
pp. 3515-3529 ◽  
Author(s):  
Damien Biau ◽  
Alessandro Bottaro
Keyword(s):  
Shear Flows ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Plane Shear

scholarly journals Non-normal origin of modal instabilities in rotating plane shear flows

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190550
Author(s):  
Sharath Jose ◽  
Rama Govindarajan
Keyword(s):  
Reynolds Number ◽  
Shear Flows ◽  
Critical Reynolds Number ◽  
Flow System ◽  
Neutral Line ◽  
Plane Couette Flow ◽  
Plane Shear ◽  
Perturbation Amplitude ◽  
Fundamental Reason ◽  
The Stability

Small variations introduced in shear flows are known to affect stability dramatically. Rotation of the flow system is one example, where the critical Reynolds number for exponential instabilities falls steeply with a small increase in rotation rate. We ask whether there is a fundamental reason for this sensitivity to rotation. We answer in the affirmative, showing that it is the non-normality of the stability operator in the absence of rotation which triggers this sensitivity. We treat the flow in the presence of rotation as a perturbation on the non-rotating case, and show that the rotating case is a special element of the pseudospectrum of the non-rotating case. Thus, while the non-rotating flow is always modally stable to streamwise-independent perturbations, rotating flows with the smallest rotation are unstable at zero streamwise wavenumber, with the spanwise wavenumbers close to that of disturbances with the highest transient growth in the non-rotating case. The instability critical rotation number scales inversely as the square of the Reynolds number, which we demonstrate is the same as the scaling obeyed by the minimum perturbation amplitude in non-rotating shear flow needed for the pseudospectrum to cross the neutral line. Plane Poiseuille flow and plane Couette flow are shown to behave similarly in this context.

scholarly journals Nature and dynamics of overreflection of Alfvén waves in MHD shear flows

2014 ◽  
Vol 80 (5) ◽  
pp. 667-685
Author(s):  
D. Gogichaishvili ◽  
G. Chagelishvili ◽  
R. Chanishvili ◽  
J. Lominadze
Keyword(s):  
Shear Flows ◽  
Alfven Waves ◽  
Wave Number ◽  
Alfvén Waves ◽  
Transient Growth ◽  
Mean Velocity ◽  
Linear Dynamics ◽  
Linear Shear ◽  
Basic Physics ◽  
Cascade Processes

Our goal is to gain new insights into the physics of wave overreflection phenomenon in magnetohydrodynamic (MHD) nonuniform/shear flows changing the existing trend/approach of the phenomenon study. The performed analysis allows to separate from each other different physical processes, grasp their interplay and, by this way, construct the basic physics of the overreflection in incompressible MHD flows with linear shear of mean velocity, U0=(Sy,0,0), that contain two different types of Alfvén waves. These waves are reduced to pseudo- and shear-Alfvén waves when wavenumber along Z-axis equals zero (i.e. when kz=0). Therefore, for simplicity, we labeled these waves as: P-Alfvén and S-Alfvén waves (P-AWs and S-AWs). We show that: (1) the linear coupling of counter-propagating waves determines the overreflection, (2) counter-propagating P-AWs are coupled with each other, while counter-propagating S-AWs are not coupled with each other, but are asymmetrically coupled with P-AWs; S-AWs do not participate in the linear dynamics of P-AWs, (3) the transient growth of S-AWs is somewhat smaller compared with that of P-AWs, (4) the linear transient processes are highly anisotropic in wave number space, (5) the waves with small streamwise wavenumbers exhibit stronger transient growth and become more balanced, (6) maximal transient growth (and overreflection) of the wave energy occurs in the two-dimensional case – at zero spanwise wavenumber.To the end, we analyze nonlinear consequences of the described anisotropic linear dynamics – they should lead to an anisotropy of nonlinear cascade processes significantly changing their essence, pointing to a need of revisiting the existing concepts of cascade processes in MHD shear flows.

scholarly journals Hydrodynamic turbulence in unbounded shear flows

2000 ◽  
Vol 18 (2) ◽  
pp. 183-187
Author(s):  
J.G. LOMINADZE
Keyword(s):  
Shear Flows ◽  
Numerical Test ◽  
Threshold Value ◽  
Nonlinear Process ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Wave Number ◽  
Small Scale ◽  
Test Analysis ◽  
Vortex Disturbances

A new conception of subcritical transition to turbulence in unbounded smooth shear flows is discussed. According to this scenario, the transition to turbulence is caused by the interplay between the four basic phenomena: (a) linear “drift” of spatial Fourier harmonics (SFH) of disturbances in wave-number space (k-space); (b) transient growth of SFH; (c) viscous dissipation; (d) nonlinear process that closes a feedback loop of transition by angular redistribution of SFH in k-space; The key features of the concept are: transition to turbulence only by the finite amplitude vortex disturbances; anisotropy of the process in k-space; onset on chaos due to the dynamic (not stochastic) process. The evolution of 2D small-scale vortex disturbances in the parallel flows with uniform shear of velocity is analyzed in the framework of the weak turbulence approach. This numerical test analysis is carried out to prove the most problematic statement of the conception—existence of positive feedback caused by the nonlinear process (d). Numerical calculations also show the existence of a threshold: if amplitude of the initial disturbance exceeds the threshold value, the self maintenance of disturbances becomes realistic. The latter, in turn, is the characteristic feature of the flow transition to the turbulent state and its self maintenance.

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.

scholarly journals Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow

2013 ◽  
Vol 731 ◽  
pp. 1-45 ◽  
Author(s):  
A. Riols ◽  
F. Rincon ◽  
C. Cossu ◽  
G. Lesur ◽  
P.-Y. Longaretti ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Shear Flows ◽  
Three Dimensional ◽  
Transition To Turbulence ◽  
Global Bifurcations ◽  
Magnetic Field Generation ◽  
Dynamo Action ◽  
Systems Research ◽  
Hydrodynamic Shear

AbstractMagnetorotational dynamo action in Keplerian shear flow is a three-dimensional nonlinear magnetohydrodynamic process, the study of which is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics with transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles born out of saddle-node bifurcations. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to injection of both kinetic and magnetic energy for the problem of transition to turbulence and dynamo action in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to understand better the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows.

Sign in / Sign up

Export Citation Format

Share Document