Anders Lundbladh

A mechanism for bypass transition from localized disturbances in wall-bounded shear flows

The linear, nonlinear and breakdown stages in the transition of localized disturbances in plane Poiseuille flow is studied by direct numerical simulations and analysis of the linearized Navier–Stokes equations. Three-dimensionality plays a key role and allows for algebraic growth of the normal vorticity through the linear lift-up mechanism. This growth primarily generates elongated structures in the streamwise direction since it is largest at low streamwise wavenumbers. For finite-amplitude disturbances such structures will be generated essentially independent of the details of the initial disturbance, since the preferred nonlinear interactions transfer energy to low streamwise wavenumbers. The nonlinear interactions also give a decrease in the spanwise scales. For the stronger initial disturbances the streamwise vorticity associated with the slightly inclined streaks was found to roll up into distinct streamwise vortices in the vicinity of which breakdown occurred. The breakdown starts with a local rapid growth of the normal velocity bringing low-speed fluid out from the wall. This phenomenon is similar to the low-velocity spikes previously observed in transition experiments. Soon thereafter a small turbulent spot is formed. This scenario represents a bypass of the regular Tollmien–Schlichting, secondary instability process. The simulations have been carried out with a sufficient spatial resolution to ensure an accurate description of all stages of the breakdown and spot formation processes. The generality of the observed processes is substantiated by use of different types of initial disturbances and by Blasius boundary-layer simulations. The present results point in the direction of universality of the observed transition mechanisms for localized disturbances in wall-bounded shear flows.

Direct simulation of turbulent spots in plane Couette flow

The development of turbulent spots in plane Couette flow was studied by means of direct numerical simulation. The Reynolds number was varied between 300 and 1500 (based on half the velocity difference between the two surfaces and half the gap width) in order to determine the lowest possible Reynolds number for which localised turbulent regions can persist, i.e. a critical Reynolds number, and to determine basic characteristics of the spot in plane Couette flow. It was found that spots can be sustained for Reynolds number above approximately 375 and that the shape is elliptical with a streamwise elongation that is more accentuated for high Reynolds numbers. At large times though there appears to be a slow approach towards a circular spot shape. Various other features of this spot suggest that it may be classified as an interesting intermediate case between the Pouseuille and boundary-layer spots. In the absence of experiments for this case the present results represent a true prediction of the physical situation.

Very large structures in plane turbulent Couette flow

A direct numerical simulation was carried out of plane turbulent Couette flow at a Reynolds number of 750, based on half the velocity difference between the walls and half the channel width. Particular attention was paid to choosing a computational box that is large enough to accommodate even the largest scales of the turbulence. In the central region of the channel very large elongated structures were observed, in accordance with earlier findings. The study is focused on the properties of these structures, but is also aimed at obtaining accurate turbulence statistics. Terms in the energy budget were evaluated and discussed. Also, the limiting values of various quantities were determined and their relevance in high Reynolds number flows discussed. The large structures were shown to be very sensitive to an imposed system rotation. They could be essentially eliminated with a stabilizing system rotation (around the spanwise axis) small enough for only minor damping of the rest of the scales. Despite the fact that the large structures dominate the appearance of the flow field their energy content was shown to be relatively small, on the order of 10% of the total turbulent kinetic energy.

Bounds for threshold amplitudes in subcritical shear flows

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number ( R ). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R −21/4 . Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R −1 . The discrepancy is discussed in the light of a model problem.

Spatial simulations of oblique transition in a boundary layer

Simulations of oblique transition in the spatial domain are presented, covering the complete transition process into the turbulent regime. It is conjectured that the three stages identified here and elsewhere are universal for oblique transition in all shear flows: first a nonlinear generation of a streamwise vortex by the oblique waves, second a transient growth of streaks from the vortex by the lift‐up effect, and third a breakdown of the streaks due to secondary instability.

Threshold Amplitudes for Transition in Channel Flows

The threshold amplitude for subcritical transition in Poiseuille and Couette flows is investigated. For Poiseuille flow we present examples where the threshold scales like ≈ R-7/4 for 1500 ≤ R ≤ 5000, where R is the Reynolds number. For Couette flow we present an example where the threshold scales like ≈ R-5/4 for 500 ≤ R ≤ 2000.

Spatial Evolution of Disturbances in Plane Poiseuille Flow

The spatial evolution of disturbances in plane Poiseuille flow is considered. For disturbances governed by the linearized equations potential for significant transient growth of the amplitude has been found. The maximum amplification occurs for disturbances with zero or low frequencies. Spatial numerical simulations of the complete transition scenario involving a pair of oblique waves have also been conducted. A modal decomposition indicates that non-linear excitation of the transient growth is responsible for the rapid emergence of low-frequency structures. Physically, this results in streaky flow structures, as seen from the results of a numerical amplitude expansion. Thus, this spatial transition scenario has been found to be similar to the corresponding temporal one (Schmid & Henningson, 1992).

On the Evolution of Localized Disturbances in Laminar Shear Flows

Disturbances present in any flow will either decay or grow depending on the stability of the background flow field. If a disturbance grows the original flow field cannot be sustained and transition to a new, normally turbulent flow, occurs. An effective method of analyzing infinitesimal disturbances in wall-bounded shear flows is that of decomposition into Fourier modes in the homogeneous directions and eigenmodes in the direction perpendicular to a wall. Classically, the least stable mode in various types of flows has been analyzed in detail while the behaviour of disturbances made up many modes has received less attention. Experimentally determined growth rates for the least stable two-dimensional mode in boundary-layer and channel flows show good agreement with results calculated from the linearized equations of motion. As the l and Orszag & Patera, 1983). The complete breakdown process has also been simulated numerically by Spalart & Yang (1987) and Gilbert (1988).

Spatial Simulations of Bypass Transition in Boundary Layers

Investigations of the mechanisms underlying bypass transition are presented, and it is shown that a linear growth mechanism is required for energy growth. In shear flows such a mechanism is identified in the rapid transient growth of streaks produced by the three-dimensional lift-up effect. In streak breakdown, oblique transition and receptivity of three-dimensional free-stream disturbances it is shown that this transient growth is utilized as a main mechanism for disturbance growth.

Direct Simulation of the Development of Turbulent Spots in Plane Couette Flow

Turbulent spots in plane Couette flow were studied by means of direct numerical simulation in order to determine their basic characteristics. For instance, the lowest Reynolds number for which localized turbulent regions can persist, i.e. a critical Reynolds number, was found to be slightly lower than 375 (based on half the velocity difference between the two surfaces and half the gap width).