Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions

Gargano, Francesco; Sammartino, M.; Sciacca, Vincenzo; Cassel, Kevin W.

doi:10.1017/jfm.2014.153

Cited by 32 publications

(46 citation statements)

References 58 publications

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“…The response produces only a small disturbance from λ over the quasi-steady temporal scale (i) of §3 until a critical value λ = λ c is encountered at which stage (ii) weakly nonlinear amplification can occur ( §4), leading to strongly nonlinear evolution over the faster time scale (iii) of (2.4a-2.4d), (2.6a-2.6b). This is followed by finite-time blowup as in Smith (1988), Peridier, Smith & Walker (1991 which provokes the even faster evolution (iv) described by Davies, Bowles & Smith (2003) (see also Cassel & Conlisk (2014), Gargano, Sammartino, Sciacca & Cassel (2014)) with further restructuring and deep transition towards turbulence taking place. The time scales (i)-(iv) etc for response a are slow, fast, faster, faster, etc.…”

Section: Further Commentsmentioning

confidence: 82%

Enhanced effects from tiny flexible in-wall blips and shear flow

Pruessner

Smith

2015

J. Fluid Mech.

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Fluid motion at high Reynolds number over a flexible in-wall blip (a compliant bump or dip in an otherwise fixed wall) is considered theoretically for a very short blip buried low inside a boundary layer. Only the near-wall shear of the oncoming flow affects the local motion past the tiny blip. Slowly evolving features are examined first to allow for variations in the incident flow. Linear and nonlinear solutions show that at certain parameter values (eigenvalues) intensifications occur in which the interactive effect on flow and blip shape is larger by an order of magnitude than at most parameter values. Similar findings apply to the boundary layer with several tiny blips present or to channel flows with blips of almost any length. These intensifications lead on to fully nonlinear unsteady motion as a second stage, after some delay, thus combining with finite-time breakups to form a distinct path into transition of the flow.

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Section: Further Commentsmentioning

confidence: 82%

Enhanced effects from tiny flexible in-wall blips and shear flow

Pruessner

Smith

2015

J. Fluid Mech.

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“…In particular, although 'real' singularities presumably do not occur in the Navier-Stokes solutions, singularities can be tracked in the complex plane. Extending the earlier work of Cowley [52], Rocca et al [53] and Gargano et al [54][55][56] have tracked complex singularities in non-interactive boundary-layer and NavierStokes solutions involving unsteady separation. In the most recent of these investigations, the evolving wall shear stress distribution is analysed for complex singularities.…”

Section: (A) Comparison With Asymptotic Stagesmentioning

confidence: 86%

Unsteady separation in vortex-induced boundary layers

Cassel

Conlisk

2014

Phil. Trans. R. Soc. A.

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This paper provides a brief review of the analytical and numerical developments related to unsteady boundary-layer separation, in particular as it relates to vortex-induced flows, leading up to our present understanding of this important feature in high-Reynolds-number, surface-bounded flows in the presence of an adverse pressure gradient. In large part, vortex-induced separation has been the catalyst for pulling together the theory, numerics and applications of unsteady separation. Particular attention is given to the role that Prof. Frank T. Smith, FRS, has played in these developments over the course of the past 35 years. The following points will be emphasized: (i) unsteady separation plays a pivotal role in a wide variety of high-Reynolds-number flows, (ii) asymptotic methods have been instrumental in elucidating the physics of both steady and unsteady separation, (iii) Frank T. Smith has served as a catalyst in the application of asymptotic methods to high-Reynolds-number flows, and (iv) there is still much work to do in articulating a complete theoretical understanding of unsteady boundary-layer separation.

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“…It is well known how Prandtl solutions develop a singularity (see [17] and references therein), and one could therefore expect that the boundary layer solution that we have constructed to show similar blow-up phenomena, and to do this also if initialized with analytic data. In the classical high Reynolds number Navier-Stokes theory the boundary layer singularities (or, to be more precise, the appearance of complex singularities that are precursor of the blow-up [18]) signal the interaction stage-when the pressure profile at the boundary is modified by the vorticity generated in the boundary layer; this interaction ultimately leads to separation of the boundary layer and to the break-up of asymptotic structures like (1.1). It would be interesting to explore if the phenomenology typical of the high Reynolds number Navier-Stokes flows is also shown by the primitive equations solutions.…”

Section: Discussionmentioning

confidence: 99%

“…[7,[15][16][17][18]28,41]), we use the following version of the Abstract CauchyKowalevski Theorem (ACK) (cf. [1,35,47] and references therein).…”

Section: E a Fixed Point Theoremmentioning

confidence: 99%

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

Kukavica

Lombardo

Sammartino

2016

Arch Rational Mech Anal

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The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be O( √ ν). The main assumption is spatial analyticity of the initial datum.

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Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions

Cited by 32 publications

References 58 publications

Enhanced effects from tiny flexible in-wall blips and shear flow

Enhanced effects from tiny flexible in-wall blips and shear flow

Unsteady separation in vortex-induced boundary layers

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

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