Highly optimized transitions to turbulence

We study the Navier-Stokes equations in three dimensional plane Couette flow geometry subject to stream-wise constant initial conditions and perturbations. The resulting two dimensional/three component (2D/3C) model has no bifurcations and is globally (non-linearly) stable for all Reynolds numbers R, yet has a total transient energy amplification that scales like R/sup 3/. These transients also have the particular dynamic flow structures known to play a central role in wall bounded shear flow transition and turbulence. This suggests a highly optimized tolerance (HOT) model of shear flow turbulence, where streamlining eliminates generic bifurcation cascade transitions that occur in bluff body flows, resulting in a flow which is stable to arbitrary changes in Reynolds number but highly fragile in amplifying arbitrarily small perturbations. This result indicates that transition and turbulence in special streamlined geometries is not a problem of linear or nonlinear instability, but rather a problem of robustness

A Streamwise Constant Model of Turbulence in Plane Couette Flow

Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behavior of fully turbulent plane Couette flow using a streamwise constant projection of the Navier Stokes equations. This results in a two-dimensional, three velocity component ($2D/3C$) model. We first use a steady state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the $2D/3C$ model under small amplitude Gaussian forcing of the cross-stream components are compared to DNS data. The results indicate that a streamwise constant projection of the Navier Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A system theoretic approach is used to demonstrate the presence of large input-output amplification through the forced $2D/3C$ model. It is this amplification coupled with the appropriate nonlinearity that enables the $2D/3C$ model to generate turbulent behaviour under the small amplitude forcing employed in this study.Comment: Journal of Fluid Mechanics 2010, in pres

Highly optimized transitions to turbulence

We study the Navier-Stokes equations in three dimensional plane Couette flow geometry subject to stream-wise constant initial conditions and perturbations. The resulting two dimensional/three component (2D/3C) model has no bifurcations and is globally (non-linearly) stable for all Reynolds numbers R, yet has a total transient energy amplification that scales like R/sup 3/. These transients also have the particular dynamic flow structures known to play a central role in wall bounded shear flow transition and turbulence. This suggests a highly optimized tolerance (HOT) model of shear flow turbulence, where streamlining eliminates generic bifurcation cascade transitions that occur in bluff body flows, resulting in a flow which is stable to arbitrary changes in Reynolds number but highly fragile in amplifying arbitrarily small perturbations. This result indicates that transition and turbulence in special streamlined geometries is not a problem of linear or nonlinear instability, but rather a problem of robustness

DESCRIPTOR APPROACH FOR ELIMINATING SPURIOUS EIGENVALUES IN HYDRODYNAMIC EQUATIONS

Abstract. We describe a general framework for avoiding spurious eigenvalues -unphysical unstable eigenvalues that often occur in hydrodynamic stability problems. In two example problems, we show that when system stability is analyzed numerically using descriptor notation, spurious eigenvalues are eliminated. Descriptor notation is a generalized eigenvalue formulation for differential-algebraic equations that explicitly retains algebraic constraints. We propose that spurious eigenvalues are likely to occur when algebraic constraints are used to analytically reduce the number of independent variables in a differential-algebraic system of equations before the system is approximated numerically. In contrast, the simple and easily generalizable descriptor framework simultaneously solves the differential equations and algebraic constraints and is well-suited to stability analysis in these systems. Key words. spurious eigenvalue, descriptor, differential algebraic, spectral method, incompressible flow, hydrodynamic stability, generalized eigenvalue, collocation 1. Introduction. Spurious eigenvalues are unphysical, numerically-computed eigenvalues with large positive real parts that often occur in hydrodynamic stability problems. We propose that these unphysical eigenvalues occur when the incompressible Navier Stokes equations are analytically reduced -i.e., the algebraic constraints are used to reduce the number of independent variables before the system is approximated using spectral methods. An alternative approach to analyzing differential-algebraic equations is the descriptor framework, posed as a generalized eigenvalue problem, which explicitly retains the algebraic constraints during the numerical computation of eigenvalues. We reformulate two common hydrodynamic stability problems using descriptor notation and show that this method of computation avoids the spurious eigenvalues generated by other methods. The descriptor formulation is a simple, robust framework for eliminating spurious eigenvalues that occur in hydrodynamic stability analysis. Additionally, this formulation reduces the order of the numerically approximated differential operators and accommodates complex boundary conditions(BCs), such as a fluid interacting with a flexible wall. Resolving the spectrum of hydrodynamic operators is critical for time integration, linear stability Researchers have developed special methods to avoid or filter these modes and uncover the true spectrum of the model problem. Perhaps the first description of these unphysical values is given by Gottlieb and Orsza

Nonlinear effect on quantum control for two-level systems

The traditional quantum control theory focuses on linear quantum system. Here we show the effect of nonlinearity on quantum control of a two-level system, we find that the nonlinearity can change the controllability of quantum system. Furthermore, we demonstrate that the Lyapunov control can be used to overcome this uncontrollability induced by the nonlinear effect.Comment: 4 pages, 5 figure