Shell Model for Buoyancy-Driven Turbulent Flows

Interior and H-∞ feedback stabilization for sabra Shell model of turbulence

10.22541/au.161074101.15800831/v1 ◽

2021 ◽

Author(s):

Tania Biswas ◽

Sheetal Dharmatti

Keyword(s):

Control Problem ◽

Shell Model ◽

Turbulent Flows ◽

Robust Stabilization ◽

Feedback Stabilization ◽

Algebraic Riccati Equation ◽

Finite Dimensional ◽

Shell Models ◽

Linearized System ◽

Unknown Disturbance

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations which exhibit at most resemblance with turbulent flows. One of the mathematically well studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model namely feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus only finitely many modes of the shell model would suffice to stabilize the system. Later we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic riccati equation. Using the same riccati operator we establish asymptotic stability of the nonlinear system.

Download Full-text

PREDICTABILITY AND THE BUTTERFLY EFFECT IN TURBULENT FLOWS: A SHELL MODEL STUDY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493001239 ◽

1993 ◽

Vol 03 (06) ◽

pp. 1581-1585

Author(s):

A. CRISANTI ◽

A. VULPIANI ◽

M. H. JENSEN ◽

G. PALADIN

Keyword(s):

Shell Model ◽

Turbulent Flows ◽

Stokes Equations ◽

Navier Stokes ◽

Length Scales ◽

Navier Stokes Equations ◽

Gaussian Shape ◽

Model Study ◽

Large Length ◽

The Mean

We discuss the dynamical growth of a disturbance in turbulent flows based on numerical calculations of an approximative model of the Navier–Stokes equations, a so-called shell model. A disturbance at small length scales is observed to propagate (and increase) towards large length scales by an inverse cascade of duration T, the predictability time. At increasing Reynolds number Re, the mean predictability time Tt is found to decrease proportionally to Re−0.47. Moreover, the probability distribution of (T − Tt)/Tt changes its shape as Re increases: at relatively small values of Re it has an almost Gaussian shape, while at large Re it gets an exponential tail, indicating the possibility of large excursions for Tt.

Download Full-text