Session BN: Boundary Layer Instabilities I

Eigenmodes of the cavity-driven boundary layer flow

The two-dimensional eigenmodes of a cavity-driven separated boundary layer are computed, and are used for analysis of the stability properties of this flow. The local convective instability of initial conditions along the shear layer of the recirculating cavity is seen to be represented in the basis of the eigenmodes by a temporal transient energy growth. The structure of the global eigenmodes and corresponding growth rate can be explained with a simple vortex convection problem, with growth along the shear layer due to inflectional instability, and disturbance regeneration at the upstream lip of the cavity due to a pressure perturbation. We finally analyze the exponential spatial growth of the temporally damped eigenmodes in the downstream direction, and discuss that this is a common features of eigenmodes generated in a strongly nonparallel pocket of an otherwise parallel flow.   

Triad Resonance Mechanism for Turbulent Transition Over a Rotating Disk

This work is part of an experimental study of mechanisms for transition to turbulence in the boundary layer on a rotating disk. The focus here is on a triad resonance between Type I traveling and a stationary cross-flow modes. This made use of a method for introducing controlled initial disturbances consisting of a distributed array of ink dots placed on the disk surface to enhance a narrow band of wave numbers. Hot-wire sensors primarily sensitive to the azimuthal velocity component, were positioned at different spatial ($r,\theta$) locations on the disk to document the growth of disturbances. Spatial correlation measurements were used with two simultaneous sensors to obtain wavenumber vectors. Measured velocity time series were decomposed into those traveling with the disk rotation speed (stationary modes) and those traveling with respect to the disk frame of reference. Cross-bicoherence was used to identify three-frequency phase locking between pairs of traveling modes and a stationary mode. Determination of the wave-numbers associated with these frequencies confirmed that these involved a triad resonance. This was found to occur for a large range of wave numbers, high and low, but the most energetic involved low azimuthal mode number ($n=3$ to 5) stationary modes. At transition to turbulence, these modes had the largest amplitude and dominated the integral development of the boundary layer. Evidence of this mechanism can be found in past flow visualization by others.   

Higher Heat Transfer Efficiency in Laminar Structured Boundary Layers than in Turbulent Boundary Layers, Numerical and Experimental Approach

Unexpected behavior has been observed in heat transfer in a concave boundary layer at a low free-stream velocity, a configuration surprisingly ignored in the literature. The boundary layer is laminar with strong embedded G\"ortler vortices. Precise measurements of the wall heat transfer in this situation demonstrate that heat transfer enhancement was extremely large even compared to the turbulent flat-plate case. Analysis demonstrates that the boundary-layer transition is dominated by the centrifugal instability at a free-stream velocity below a critical value. Further results reinforce the discussion by analysis of the influence of forced upstream perturbations coupled with a set of flow visualizations. Numerical computation of the linear and nonlinear development of steady G\"ortler vortices and their most amplified primary instability have been developed and these results are compared with the experimental measurements.   

Absolute mode of instability and transition toward turbulence in a rotating boundary layer

The three dimensional boundary layer over a rotating disk is accordingly subject to an absolute mode of instability, whose role in transition process is still not well understood. Depending upon the methods of investigation and the associated assumptions, the scenario for the last stage of transition appears either convectively or absolutely dominated (see Davies {\&} Carpenter [J. FLUID MECH. $486$~287 2003] and Pier [J. FLUID MECH. $487$~315 2003]). Here, high accuracy spectral D.N.S. has been applied to two co-rotating disks with a forced inflow above the expected critical Reynolds for the convective/absolute transition. Computations bring new information on the transition process, as the sustained existence of a global non-linear mode is shown, and give insight into how non-linear effects work. Progressive broadening of the spectrum is followed by sudden evolution toward very small scales as time goes on. Moreover, the presence of an absolute mode appears clearly to be a prerequisite to turbulence, and a rich description of its spectrum content is needed to achieve transition.   

Optimal energy growth in swept Hiemenz flow

It has been shown in Gu\'egan, Schmid \& Huerre 2006 that the kinetic energy of optimal G\"ortler-H\"ammerlin (GH) perturbations in swept Hiemenz flow can be transiently amplified by two orders of magnitude at Reynolds numbers ranging from 400 to 1000 and spanwise wavenumbers from 0.1 to 0.5. In this configuration an array of counter-rotating chordwise vortices is compressed by the spanwise shear, as in the well-known Orr mechanism. We show that stronger transient growth can be achieved when the GH assumption is relaxed. In this case the optimal initial perturbation consists in vorticity sheets stacked in the chordwise direction, at a small angle from the symmetry plane of the base flow. Although the spatial structure of the GH perturbations is lost, wall-normal-spanwise plane cuts show that the amplification mechanism is mostly unchanged. The GH assumption thus provides a reasonably good estimate for transient energy amplification levels in swept Hiemenz flow. Extension of this analysis to the spatial growth problem is under way and preliminary results will be shown.   

Boundary layer flow on a long thin rotating cylinder.

The development and stability of the boundary layer flow over a long thin cylinder aligned with the main flow and which rotates around its axis is considered. Numerical results show that the introduction of rotation has an important effect on the behavior of the basic flow. When the swirl increases, the shear stress at the wall also increases due to the changes in the pressure distribution along the cylinder surface. A nonparallel linear stability analysis of the basic flow is also performed using Parabolized Stability Equations (PSE). Even at moderately low rotation, we have found the existence of unstable centrifugal modes, in addition to the shear ones found in previous stability analysis of the boundary layer flow on a cylinder with no rotation. These centrifugal instabilities develop at Reynolds numbers much lower than those required for the growing of the shear instabilities. Our analysis shows that non parallel effects play a key role in the development of these instabilities, being the mode with azimuthal wave number $n=1$ the most unstable one.   

The effects of a uniform axial magnetic field on the global stability of the rotating-disk boundary-layer

Following on from the earlier discovery by Lingwood (1995) that the rotating-disk boundary-layer is absolutely unstable, Jasmine $\&$ Gajjar (2005) have shown that the application of a uniform axial magnetic field can raise the critical Reynolds number for the onset of absolute instability. As with Lingwood's analysis, a \lq parallel-flow' type of approximation is needed in order to derive this locally-based stability result. The approximation amounts to a \lq freezing out' of the underlying radial variation of the mean flow. Numerical simulations have been conducted to investigate the behaviour of linearized disturbances in the genuine rotating disk boundary layer, where the radial dependence of the mean flow is fully accounted for. This extends the work of Davies $\&$ Carpenter (2003), who studied the more usual rotating-disk problem, in the absence of any magnetic field. The simulation results suggest that globally unstable behaviour can be promoted when a uniform axial magnetic field is applied. Impulsively excited disturbances were found to display an increasingly rapid growth at the radial position of the impulse, albeit without any selection of a dominant frequency, as would be more usual for an unstable global mode. This is very similar to the behaviour to that was observed in a recent investigation by Davies $\&$ Thomas (2005) of the effects of mass transfer, where suction was also found to promote global instability.   

Development of a Three-Dimensional Code for a Direct Numerical Simulation of Compressible Flow Instabilities

The engineering research and design requirements of today pose great challenges in computer simulation to engineers and scientists who are called on to analise phenomena in continuum mechanics. In the current work, the instability of the compressible boundary layer at relatively low Reynolds number was investigated. In the aerospace context, important applications involve compressible flows at relatively low Reynolds number. Among them, the flow on gas turbine blades and the flow on high lift devices such as slats and flaps at high angle of attack are particulary important. In aerodynamic applications in low Reynolds number, often a substancial portion of the flow is in the a transition regime, or in the initial stages of a turbulent flow. Recent work in boundary layers has shown in the nonlinear regime, modulated waves behave diferently from regular wavetrains. This is particularly important in view that the waves that arise in natural transition are also strongly modulated. It was also shown that wavepackets and wavetrains emanating from a point source provide a good model for instability natural transition. Inspired on the worked devoted to modulated waves in boundary layers, the current work investigates the evolution of wavepackets and wavetrains from a point source in a boundary layer.