Interaction of two-dimensional separated flows with a free surface at low Froude numbers

Triantafyllou, George S.; Dimas, Athanassios A.

doi:10.1063/1.857507

Cited by 58 publications

(40 citation statements)

References 11 publications

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“…The shaded regions are the unstable regions. There are two distinct regions of unstable modes for a given Froude number for this entire family of velocity profiles, previously referred to as branches I and II by Triantyfallou & Dimas (1989) for the case of n = 2. The low-wavenumber branch I widens as the Froude number decreases.…”

Section: Instabilities Of Horizontal Shear Flows With a Free Surfacementioning

confidence: 99%

“…Surface shear flows with sech type profiles A well-studied case is the linear instability of a shear flow with a continuous velocity profile of the form U = U o sech 2 (bz). The shear flow with this profile was numerically studied by Triantyfallou & Dimas (1989) and Dimas & Triantyfallou (1994), and was found to fit experimental data for the shear flow in the wake of a hydrofoil. For the linear instability analysis, the neutral modes form the stability boundaries.…”

Section: Instabilities Of Horizontal Shear Flows With a Free Surfacementioning

confidence: 99%

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Short surface waves on surface shear

Zhang

2005

J. Fluid Mech.

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The problem of short wind waves propagating on surface wind drift is considered here. Under the assumption of small monochromatic surface waves on a steady and horizontally uniform surface shear of an inviscid fluid, the governing equation becomes the well-known Rayleigh instability equation. Perturbation solutions exist for the surfacewave problem; however, the conditions for these approximations are violated in the case of short wind waves on wind drift shear. As an alternative approach, the piecewise-linear approximation (PLA) is explored. A proof is given for the rate of convergence of the piecewise-linear approximation for solving the Rayleigh equation without limitations on boundary conditions. The artificial modes of the piecewiselinear flow system are also discussed. The method is numerically efficient and highly accurate. Applying this method, the linear instability of various boundary-layer flows is examined. Short waves propagating with surface shear-flows are stable, while it is possible for waves that are travelling against a shear current to become unstable when the surface speed of the shear is greater than the wavespeed in stagnant fluid. PLA is also applied to examine the applicability of other perturbation approaches to the problem of propagation of short waves on wind drift shear. It is found that the existing approximations cannot fit the whole range of short wind waves. To bridge the gap, new approximations are derived from an implicit form of the exact dispersion relation based upon the variational principle.

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Section: Instabilities Of Horizontal Shear Flows With a Free Surfacementioning

confidence: 99%

Section: Instabilities Of Horizontal Shear Flows With a Free Surfacementioning

confidence: 99%

Short surface waves on surface shear

Zhang

2005

J. Fluid Mech.

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“…Examples are given by Huerre & Monkewitz (1990). These include countercurrent mixing layers in circular jets (Strykowski & Niccum 1991), and the wakes behind circular cylinders (Mathis, Provansal & Boyer 1984;Koch 1985;Triantafyllou, Triantafyllou & Chryssostomidis 1986;Monkewitz 1988;Strykowski & Sreenivasan 1990), blunt bodies (Hannemann & Oertel 1989;Oertel 1990) and a floating cylinder (Triantafyllou & Dimas 1989). But reverse flow certainly does not guarantee absolute instability, nor is it always necessary.…”

Section: Referred To As I)mentioning

confidence: 99%

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

2003

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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.

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“…(13) was taken by Longuet-Higgins (11) in considering a shear layer lying on a deep undisturbed fluid. In this study, the velocity profile is approximated by a two-step piecewise-linear profile such that (17) where ∆ 1 and ∆ 2 are the depth of upper and lower nodes of velocity profile. Eqs.…”

Section: Linear Analysis For Simplified Velocity Profilesmentioning

confidence: 99%

Linear Stability Analysis on Free-Surface Liquid Jet with Different Simplification of Velocity Profile

Itoh

Inoue

Kumamaru

et al. 2007

JFST

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Free surface wave arisen from the shear layer instability on a liquid jet is investigated experimentally and theoretically. The stability analysis is conducted for initially laminar free-surface shear layer with different levels of simplification of velocity profile (i.e., numerical shear-layer profile separated from nozzle exit and two kinds of simplified velocity profiles obtained through linear and piecewise-linear fitting of numerical profile). It is shown that the linear perturbation equation has temporally-neutral, spatially-unstable solutions by substituting all kinds of local velocity profiles. On the other hands, the periodic two-dimensional waves on liquid jet are measured by using a laser beam refraction technique. The frequency of the most-unstable solutions shows fair agreement with the measured dominant frequency of free surface waves. The linear stability analysis can be simplified drastically by introducing linear or piecewise linear approximation of the velocity profile. Such simplification does not result in large discrepancy from the detailed-model, or experimental data, except in the initial region of the jet where arbitrariness exists in simplification of velocity profiles.

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Interaction of two-dimensional separated flows with a free surface at low Froude numbers

Cited by 58 publications

References 11 publications

Short surface waves on surface shear

Short surface waves on surface shear

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

Linear Stability Analysis on Free-Surface Liquid Jet with Different Simplification of Velocity Profile

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