An optimal perturbation bound

2019 ◽  
Vol 42 (11) ◽  
pp. 3791-3798
Author(s):  
Youming Liu ◽  
Chunguang Ren
Keyword(s):  
Perturbation Bound ◽  
Optimal Perturbation

scholarly journals The perturbation bound for the T-Drazin inverse of tensor and its application

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1565-1587
Author(s):  
Ying-Nan Cui ◽  
Hai-Feng Ma
Keyword(s):  
Drazin Inverse ◽  
Tensor Equation ◽  
Perturbation Bound ◽  
Optimal Perturbation

In this paper, let A and B be n x n x p complex tensors and B = A + E. Denote the T-Drazin inverse of A by AD. We give a perturbation bound for ||BD-AD||=||AD|| under condition (W). Considering the solution of singular tensor equation A* x = b, (b ? R(AD)) at the same time. The optimal perturbation of T-Drazin inverse of tensors and the solution of a system of tensor equations have been given.

scholarly journals Resonance in Optimal Perturbation Evolution. Part II: Effects of a Nonzero Mean PV Gradient

2007 ◽  
Vol 64 (3) ◽  
pp. 695-710 ◽  
Author(s):  
H. de Vries ◽  
J. D. Opsteegh
Keyword(s):  
Potential Vorticity ◽  
Propagation Speed ◽  
Building Blocks ◽  
Joint Effect ◽  
Optimal Perturbation ◽  
Orr Mechanism ◽  
Nonzero Mean ◽  
Low Levels ◽  
Pv Gradient

Abstract Optimal perturbations are constructed for a two-layer β-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without β. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution. The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the f plane only the PVB at the surface and at the tropopause can be amplified. In the presence of β, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer f-plane model, the inclusion of β slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface. If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

2008 ◽  
Vol 614 ◽  
pp. 315-327 ◽  
Author(s):  
UWE EHRENSTEIN ◽  
FRANÇOIS GALLAIRE
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Perturbation Analysis ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Layer Flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Transient growth in the near wake region of the flow past a finite span wing

2019 ◽  
Vol 866 ◽  
pp. 399-430 ◽  
Author(s):  
Navrose ◽  
V. Brion ◽  
L. Jacquin
Keyword(s):  
Periodic Structures ◽  
Time Integration ◽  
Tip Vortex ◽  
Wake Region ◽  
Wind Tunnel Experiments ◽  
Near Wake ◽  
Optimal Perturbation ◽  
Tip Vortices ◽  
Naca0012 Airfoil ◽  
Flow Past

We investigate optimal perturbation in the flow past a finite aspect ratio ($AR$) wing. The optimization is carried out in the regime where the fully developed flow is steady. Parametric study over time horizon ($T$), Reynolds number ($Re$), $AR$, angle of attack and geometry of the wing cross-section (flat plate and NACA0012 airfoil) shows that the general shape of linear optimal perturbation remains the same over the explored parameter space. Optimal perturbation is located near the surface of the wing in the form of chord-wise periodic structures whose strength decreases from the root towards the tip. Direct time integration of the disturbance equations, with and without nonlinear terms, is carried out with linear optimal perturbation as initial condition. In both cases, the optimal perturbation evolves as a downstream travelling wavepacket whose speed is nearly the same as that of the free stream. The energy of the wavepacket increases in the near wake region, and is found to remain nearly constant beyond the vortex roll-up distance in nonlinear simulations. The nonlinear wavepacket results in displacement of the tip vortex. In this situation, the motion of the tip vortex resembles that observed during vortex meandering/wandering in wind tunnel experiments. Results from computation carried out at higher $Re$ suggest that, even beyond the steady flow regime, a perturbation wavepacket originating near the wing might cause meandering of tip vortices.

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