scholarly journals Sustaining turbulence in spectrally stable shear flows – interplay of linear transient growth and nonlinear transverse cascade

2019 ◽  
Vol 82 ◽  
pp. 423-434
Author(s):  
D. Gogichaishvili ◽  
G. Mamatsashvili ◽  
W. Horton ◽  
G. Chagelishvili
Keyword(s):  
Shear Flows ◽  
Nonlinear Process ◽  
Transient Growth ◽  
Fourier Space ◽  
Nonlinear Perturbations ◽  
Nonlinear Processes ◽  
2D And 3D ◽  
Fourier Harmonics ◽  
Nonlinear State

We analyze the sustaining mechanism of nonlinear perturbations/turbulence in spectrally stable smooth shear flows. The essence of the sustenance is a subtle interplay of linear transient growth of Fourier harmonics and nonlinear processes. In spectrally stable shear flows, the transient growth of perturbations is strongly anisotropic in spectral (k-)space. This, in turn, leads to anisotropy of nonlinear processes ink-space and, as a result, the main (new) nonlinear process appears to be not a direct/inverse, but rather a transverse/angular redistribution of harmonics in Fourier space referred to as the nonlinear transverse cascade. It is demonstrated that nonlinear state is sustained owing to the interplay of the linear nonmodal growth and the transverse cascade. The possibility of such course of events has been described ink-space byG. Chagelishvili, J.-P. Zahn, A. Tevzadze and J. Lominadze, A&A, 402, 401 (2003)that reliably exemplifies the well-known bypass scenario of subcritical turbulence in spectrally stable shear flows. We present selected results of the simulations performed in different (HD and MHD; 2D and 3D; plane and Keplerian) shear flows to demonstrate the transverse cascade in action.

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.

Discrete-time sliding mode control for a class of nonlinear process

2019 ◽  
Vol 37 (2) ◽  
pp. 513-534
Author(s):  
Luning Ma ◽  
Dongya Zhao ◽  
Shuzhan Zhang ◽  
Jiehua Feng ◽  
Lei Cao
Keyword(s):  
Sliding Mode Control ◽  
Discrete Time ◽  
Sliding Mode ◽  
External Disturbance ◽  
Nonlinear Process ◽  
Nonlinear Processes ◽  
Mode Control ◽  
Efficient Control ◽  
Equivalent Control ◽  
Nonlinear Process Control

Abstract The efficient control of nonlinear processes is generally considered to be challenging. The development of digital computers promotes the study of nonlinear process control technology. Due to the discrete sampling of digital computer, it is necessary to develop the corresponding control algorithms for nonlinear processes. In this paper, a new equivalent control-based discrete-time sliding mode control is proposed for a class of nonlinear process with uncertainty and external disturbance. An adaptive law and a disturbance observer are designed to estimate the uncertainty and the disturbance, respectively. By combining with them, the new discrete-time sliding mode control is developed with good performance. The corresponding theoretical analysis is well verified by using Lyapunov function. Finally, the proposed approach is demonstrated by case studies in light of MATLAB.

scholarly journals Transient Dynamics of Nonsymmetric Perturbations versus Symmetric Instability in Baroclinic Zonal Shear Flows

2010 ◽  
Vol 67 (9) ◽  
pp. 2972-2989 ◽  
Author(s):  
G. R. Mamatsashvili ◽  
V. S. Avsarkisov ◽  
G. D. Chagelishvili ◽  
R. G. Chanishvili ◽  
M. V. Kalashnik
Keyword(s):  
Shear Flows ◽  
Stable Stratification ◽  
Linear Operators ◽  
Vertical Shear ◽  
Transient Dynamics ◽  
Transient Growth ◽  
Coriolis Parameter ◽  
Linear Dynamics ◽  
Wide Range ◽  
Symmetric Instability

Abstract The linear dynamics of symmetric and nonsymmetric perturbations in unbounded zonal inviscid flows with a constant vertical shear of velocity, when a fluid is incompressible and density is stably stratified along the vertical and meridional directions, is investigated. A small–Richardson number Ri ≲ 1 and large–Rossby number Ro ≳ 1 regime is considered, which satisfies the condition for symmetric instability. Specific features of this dynamics are closely related to the nonnormality of linear operators in shear flows and are well interpreted in the framework of the nonmodal approach by tracing the linear dynamics of spatial Fourier harmonics (Kelvin modes) of perturbations in time. The roles of stable stratification, the Coriolis parameter, and vertical shear in the dynamics of perturbations are analyzed. Classification of perturbations into two types or modes—vortex (i.e., quasigeostrophic balanced motions) and inertia–gravity wave—is made according to the value of potential vorticity. The emerging picture of the (linear) transient dynamics for these two modes at Ri ≲ 1 and Ro ≳ 1 indicates that vortex mode perturbations are able to gain basic flow energy and undergo exponential transient amplification and in this process generate inertia–gravity waves. Transient growth of the vortex mode and, consequently, the effectiveness of the wave generation both increase with decreasing Ri and increasing Ro. This linear coupling of perturbation modes is, in general, specific to shear flows but is not fully appreciated yet. A parallel analysis of the transient dynamics of nonsymmetric perturbations versus symmetric instability is also presented. It is shown that the nonnormality-induced transient growth of nonsymmetric perturbations can prevail over the symmetric instability for a wide range of Ri and Ro. The current analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations rather than by symmetric ones, as was accepted in earlier papers. It is noteworthy that the transient growth of perturbations is asymmetric in the wavenumber space—the constant phase plane of maximally amplified perturbations is inclined in a direction northeast to the zonal one and the inclination angle is different for different Ri and Ro.

scholarly journals Mountain Waves in Two-Layer Sheared Flows: Critical-Level Effects, Wave Reflection, and Drag Enhancement

2008 ◽  
Vol 65 (6) ◽  
pp. 1912-1926 ◽  
Author(s):  
Miguel A. C. Teixeira ◽  
Pedro M. A. Miranda ◽  
JoséL. Argaín
Keyword(s):  
Linear Theory ◽  
Shear Flows ◽  
Wave Reflection ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Drag ◽  
Critical Levels ◽  
Nonlinear Processes ◽  
Mountain Waves ◽  
The Impact

Abstract Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

Mechanical picture of the linear transient growth of vortical perturbations in incompressible smooth shear flows

2016 ◽  
Vol 1 (4) ◽  
Author(s):  
George Chagelishvili ◽  
Jan-Niklas Hau ◽  
George Khujadze ◽  
Martin Oberlack
Keyword(s):  
Shear Flows ◽  
Transient Growth
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