Steep nonlinear global modes in spatially developing media

Pier, Benoı̂t; Huerre, Patrick; Chomaz, Jean‐Marc; Couairon, Arnaud

doi:10.1063/1.869784

Cited by 66 publications

(100 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hannemann & Oertel 1989;Zielinska & Westfreid 1995;Pier & Huerre 1996) suggests that the form of the linear global mode is qualitatively preserved, with nonlinear effects leading to saturation and in some cases causing the global frequency to rise slightly. The exception to this is the steep nonlinear global mode discovered by Pier et al (1998) which may be more relevant in the present case.…”

Section: Referred To As I)mentioning

confidence: 67%

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

2003

View full text Add to dashboard Cite

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.

show abstract

Section: Referred To As I)mentioning

confidence: 67%

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

2003

View full text Add to dashboard Cite

show abstract

“…The corresponding extended spatial structure displays smoothly varying amplitude and wavenumber everywhere. By contrast, steep global modes, as described by Pier et al (1998), obey a marginal stability criterion: the steep global frequency coincides with the real absolute frequency at the transition station between linear convective and absolute instability. More specifically, the steep global mode is triggered at the upstream boundary X ca of the absolutely unstable domain and is tuned at the associated real absolute frequency…”

Section: Introductionmentioning

confidence: 99%

Nonlinear self-sustained structures and fronts in spatially developing wake flows

2001

Self Cite

View full text Add to dashboard Cite

A family of slowly spatially developing wakes with variable pressure gradient is numerically demonstrated to sustain a synchronized finite-amplitude vortex street tuned at a well-defined frequency. This oscillating state is shown to be described by a steep global mode exhibiting a sharp Dee-Langer-type front at the streamwise station of marginal absolute instability. The front acts as a wavemaker which sends out nonlinear travelling waves in the downstream direction, the global frequency being imposed by the real absolute frequency prevailing at the front station. The nonlinear travelling waves are determined to be governed by the local nonlinear dispersion relation resulting from a temporal evolution problem on a local wake profile considered as parallel. Although the vortex street is fully nonlinear, its frequency is dictated by a purely linear marginal absolute instability criterion applied to the local linear dispersion relation.

show abstract

“…The frequency (and nature) of such global modes can be explored by analysing the complex, nonlinear GinzburgLandau equation as a model of the flow (e.g. Pier et al 1998;Pier & Huerre 2001). Alternatively, it is possible to undertake (usually numerically) a linear stability analysis of full pre-computed solutions of spatially developing flows, to identify their stability and the frequency of any global mode which might result (e.g.…”

Section: Introductionmentioning

confidence: 99%

The stability of laminar symmetric separated wakes

Castro

2005

J. Fluid Mech.

View full text Add to dashboard Cite

Time-dependent computations of the two-dimensional incompressible uniformvelocity laminar flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the well-known anti-symmetric (von Kármán-type) vortex shedding is suppressed by the imposition of a symmetry plane on the downstream plate centreline. With a further symmetry plane at the channel's upper boundary, the only two governing parameters in the problem are the channel half-width, H , and the Reynolds number, Re (based on the body half-width and the upstream velocity, U ). The former is restricted to the range 3 6 H 6 30 and the interest lies in determining the nature of the initial instability which occurs in the separated wake as Re is gradually increased. It is found that for sufficiently large H and at a critical Re, a long-time-scale global (supercritical) instability is initiated, which in its saturated (limit) state takes the form of 'lumps' of vorticity being periodically shed from the tail end of the separated bubble. Stability calculations of corresponding mean flow profiles (typical of those found in the separated wake) are undertaken by examining the impulse response of particular profiles via appropriate solution of the Orr-Sommerfeld equation. The results of this analysis extend those available from related published work and are consistent with the behaviour found from the numerical computations. Taken together, all the results suggest that this type of global instability may be generic to many kinds of separated wakes and, indeed, may provide the fundamental explanation for the very low-frequency oscillations often noticed in fully turbulent wake bubbles.

show abstract

Steep nonlinear global modes in spatially developing media

Cited by 66 publications

References 16 publications

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

Nonlinear self-sustained structures and fronts in spatially developing wake flows

The stability of laminar symmetric separated wakes

Contact Info

Product

Resources

About