Linear Optimal Perturbation Research Articles

Laminar-turbulent separatrix in a boundary layer flow

The transitional boundary layer flow over a flat plate is investigated. The boundary layer flow is known to develop unstable Tollmien-Schlichting waves above a critical value of the Reynolds number. However, it is also known that this transition can be observed for sub-critical Reynolds numbers. In that case, the basin of attraction of the laminar state coexists with the sustained turbulence. In this article, the trajectory on the separatrix between these two states is simulated. The state on the separatrix is independent from the initial condition and is dynamically connected to both the laminar flow and the turbulence. Such an edge state provides information regarding the basic features of the transitional flow. The solution takes the form of a low speed streak, flanked by two quasi-streamwise sinuous vortices. The shape of the streaks is close to that simulated with the linear optimal perturbation method. This solution is compared to existing results concerning streak breakdown. The simulations are realized in a temporal framework for a local boundary layer, with periodic boundary conditions in the streamwise direction. A dedicated model, based on a scale separation, is presented. The mean flow is a solution of the Prandtl boundary layer equations while the superposed small-scale fluctuations are a solution of the periodic Navier-Stokes equations. The model is validated with turbulent flow simulations and satisfactorily reproduces the physical characteristics of a boundary layer flow, especially in the outer region, where external fluid is entrained toward the boundary layer.

Further analysis of singular vector and ENSO predictability in the Lamont model—Part I: singular vector and the control factors

In this study, singular vector analysis was performed for the period from 1856 to 2003 using the latest Zebiak-Cane model version LDEO5. The singular vector, representing the optimal growth pattern of initial pertur- bations/errors, was obtained by perturbing the constructed tangent linear model of the Zebiak-Cane model. Variations in the singular vector and singular value, as a function of initial time, season, ENSO states, and optimal period, were investigated. Emphasis was placed on exploring relative roles of linear and nonlinear processes in the optimal per- turbation growth of ENSO, and deriving statistically robust conclusions using long-term singular vector analysis. It was found that the first singular vector is dominated by a west-east dipole spanning most of the equatorial Pacific, with one center located in the east and the other in the central Pacific. Singular vectors are less sensitive to initial conditions, i.e., independence of seasons and decades; while singular values exhibit a strong sensitivity to initial conditions. The dynamical diagnosis shows that the total linear and nonlinear heating terms play opposite roles in controlling the optimal perturbation growth, and that the linear optimal perturbation is more than twice as large as the nonlinear one. The total linear heating causes a warming effect and controls two positive perturbation growth regions: one in the central Pacific and the other in the eastern Pacific; whereas the total linearized nonlinear advection brings a cooling effect controlling the negative perturbation growth in the central Pacific.

A Method to Find Perturbations That Trigger Blocking Onset: Conditional Nonlinear Optimal Perturbations

Abstract In this paper the following question is addressed: assuming that some information is available about initial perturbations (e.g., that they belong to an ensemble, which consists of the perturbations whose magnitudes are less than a given value), how can one determine which perturbations belong to this ensemble and trigger the blocking onset? The applicability of linear singular vectors (LSVs) and conditional nonlinear optimal perturbations (CNOPs) is investigated by a T21L3 quasigeostrophic (QG) model and its tangent linear and adjoint versions. Particular attention is focused on the roles of nonlinear processes and the importance of choosing a proper objective function. LSVs are the fastest-growing perturbations when the evolutions of the initial perturbations are well described by the tangent linear version of the nonlinear model. CNOPs are a natural generalization of LSVs into the nonlinear category, that is, the initial perturbation whose nonlinear evolution attains the maximum of the objective function at a prescribed forecast time under some initial constraint conditions. The results of this research show that in some cases for the given initial ensemble perturbations CNOPs trigger a transition to a blocking regime (whereas LSVs may not generate such a transition), which shows that nonlinear advection processes are fundamental for studying the weather regime transitions from zonal flow to blocking in the medium range. By choosing two objective functions and investigating the resulting CNOPs, it is found that CNOPs obtained from the objective function of the blocking-index form (type-1 CNOPs) may trigger a transition to a blocking regime under some circumstances, whereas CNOPs related to the streamfunction squared norm (type-2 CNOPs) fail to yield such a transition. This demonstrates the importance of selecting a proper objective function when aiming at finding the perturbations yielding such a transition. The mechanism of blocking onset triggered by perturbations is also explored. It is shown that the approach of type-1 CNOP remains a viable tool to capture the spatial structure of initial perturbations that trigger a blocking onset. The planetary-scale projection of the nonlinear interaction of such initial perturbations contributes to the amplification of the blocking downstream and then triggers a blocking onset.

An optimal path to transition in a duct

This paper is concerned with the transition of the laminar flow in a duct of square cross section. As in the similar case of pipe flow, the motion is linearly stable for all Reynolds numbers, rendering this flow a suitable candidate for a study of the 'bypass' path to turbulence. It has already been shown that the classical linear optimal perturbation problem, yielding optimal disturbances in the form of longitudinal vortices, fails to provide an 'optimal' path to turbulence, i.e. optimal perturbations do not elicit a significant nonlinear response from the flow. Previous simulations have also indicated that a pair of travelling waves generates immediately, by nonlinear quadratic interactions, an unstable mean flow distortion, responsible for rapid breakdown. By the use of functions quantifying the sensitivity of the motion to deviations in the base flow, the optimal travelling wave associated with its specific defect is found by a variational approach. This optimal solution is then integrated in time and shown to display a qualitative similarity to the so-called 'minimal defect', for the same parameters. Finally, numerical simulations of an 'edge state' are conducted, to identify an unstable solution that mediates laminar-turbulent transition and relate it to results of the optimization procedure.

Transition to turbulence in duct flow

The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the ‘bypass’ path to turbulence. Initially the so-called ‘linear optimal perturbation problem’ is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous sub-optimal disturbances are focused upon; nonlinear quadratic interactions are immediately caused by such initial perturbations and an unstable streamwise-homogeneous large-amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the boundary between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large-scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.

Subcritical instability of liquid metal channel flow in the presence of a spanwise magnetic field

AbstractThe linear and nonlinear evolution of perturbations is investigated in a magnetohydrodynamic channel flow with electrically insulating walls. The applied magnetic field is parallel to the walls and orthogonal to the stream. Linear optimal perturbations and their maximum amplifications over finite time intervals are computed using a scheme based on the direct and adjoint governing equations. It is shown that dominant optimal perturbations are no more the classical streamwise modes and how the flow is two‐dimenzionalized for high enough Hartmann numbers. For fixed Reynolds and Hartmann numbers, direct numerical simulations are applied to investigate how the transition to turbulence is affected by the magnetic field. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)