Onset of absolutely unstable behaviour in the Stokes layer: a Floquet approach to the Briggs method

2021 ◽  
Vol 928 ◽  
Author(s):  
Alexander Pretty ◽  
Christopher Davies ◽  
Christian Thomas
Keyword(s):  
Growth Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Steady Flows ◽  
Navier Stokes Equations ◽  
Electron Stream ◽  
Stokes Layer ◽  
Symmetry Constraints ◽  
Spatio Temporal ◽  
Spatial Locations

For steady flows, the Briggs (Electron-Stream Interaction with Plasmas. MIT Press, 1964) method is a well-established approach for classifying disturbances as either convectively or absolutely unstable. Here, the framework of the Briggs method is adapted to temporally periodic flows, with Floquet theory utilised to account for the time periodicity of the Stokes layer. As a consequence of the antiperiodicity of the flow, symmetry constraints are established that are used to describe the pointwise evolution of the disturbance, with the behaviour governed by harmonic and subharmonics modes. On coupling the symmetry constraints with a cusp-map analysis, multiple harmonic and subharmonic cusps are found for each Reynolds number of the flow. Therefore, linear disturbances experience subharmonic growth about fixed spatial locations. Moreover, the growth rate associated with the pointwise development of the disturbance matches the growth rate of the disturbance maximum. Thus, the onset of the Floquet instability (Blennerhassett & Bassom, J. Fluid Mech., vol. 464, 2002, pp. 393–410) coincides with the onset of absolutely unstable behaviour. Stability characteristics are consistent with the spatio-temporal disturbance development of the family-tree structure that has hitherto only been observed numerically via simulations of the linearised Navier–Stokes equations (Thomas et al., J. Fluid Mech., vol. 752, 2014, pp. 543–571; Ramage et al., Phys. Rev. Fluids, vol. 5, 2020, 103901).

scholarly journals Numerical stability analysis of a vortex ring with swirl

2019 ◽  
Vol 878 ◽  
pp. 5-36 ◽  
Author(s):  
Yuji Hattori ◽  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Numerical Simulation ◽  
Growth Rate ◽  
Direct Numerical Simulation ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Base Flow ◽  
Critical Layer ◽  
Navier Stokes ◽  
Instability Mode ◽  
Navier Stokes Equations

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

scholarly journals Critical slope for laminar transcritical shallow-water flows

2015 ◽  
Vol 783 ◽  
Author(s):  
O. Thual ◽  
L. Lacaze ◽  
M. Mouzouri ◽  
B. Boutkhamouine
Keyword(s):  
Stokes Equations ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Critical Surface ◽  
Steady Flows ◽  
Surface Slope ◽  
Critical Height ◽  
Navier Stokes Equations ◽  
Shallow Water Flows ◽  
Critical Slope

Backwater curves denote the depth profiles of steady flows in a shallow open channel. The classification of these curves for turbulent regimes is commonly used in hydraulics. When the bottom slope $I$ is increased, they can describe the transition from fluvial to torrential regimes. In the case of an infinitely wide channel, we show that laminar flows have the same critical height $h_{c}$ as that in the turbulent case. This feature is due to the existence of surface slope singularities associated to plug-like velocity profiles with vanishing boundary-layer thickness. We also provide the expression of the critical surface slope as a function of the bottom curvature at the critical location. These results validate a similarity model to approximate the asymptotic Navier–Stokes equations for small slopes $I$ with Reynolds number $Re$ such that $Re\,I$ is of order 1.

Analysis of the dripping–jetting transition in compound capillary jets

2010 ◽  
Vol 649 ◽  
pp. 523-536 ◽  
Author(s):  
M. A. HERRADA ◽  
J. M. MONTANERO ◽  
C. FERRERA ◽  
A. M. GAÑÁN-CALVO
Keyword(s):  
Stability Analysis ◽  
Convective Instability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Outer Interface ◽  
Spatio Temporal ◽  
Capillary Jets ◽  
The Stability

We examine the behaviour of a compound capillary jet from the spatio-temporal linear stability analysis of the Navier–Stokes equations. We map the jetting–dripping transition in the parameter space by calculating the Weber numbers for which the convective/absolute instability transition occurs. If the remaining dimensionless parameters are set, there are two critical Weber numbers that verify Brigg's pinch criterion. The region of absolute (convective) instability corresponds to Weber numbers smaller (larger) than the highest value of those two Weber numbers. The stability map is affected significantly by the presence of the outer interface, especially for compound jets with highly viscous cores, in which the outer interface may play an important role even though it is located very far from the core. Full numerical simulations of the Navier–Stokes equations confirm the predictions of the stability analysis.

scholarly journals Steady streaming in a channel with permeable walls

2013 ◽  
Vol 25 (1) ◽  
pp. 65-82
Author(s):  
KONSTANTIN ILIN
Keyword(s):  
Asymptotic Expansion ◽  
Viscous Fluid ◽  
Steady Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Steady Streaming ◽  
Stokes Layer ◽  
Permeable Walls ◽  
Time Injection

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) injection/suction of a viscous fluid at the walls. We obtain an asymptotic expansion of the solution of the Navier–Stokes equations in the limit when the amplitude of normal displacements of fluid particles near the walls is much smaller than both the width of the channel and the thickness of the Stokes layer. It is shown that the steady part of the flow in this problem is much stronger than the steady flow produced by vibrations of impermeable boundaries. Another interesting feature of this problem is that the direction of the steady flow is opposite to what one would expect if the flow was produced by vibrations of impermeable walls.

Prediction of Melt Flow Effects on Dendrite Growth

2016 ◽  
Vol 850 ◽  
pp. 334-340 ◽  
Author(s):  
Yun Chen ◽  
Xin Bo Qi ◽  
Dian Zhong Li ◽  
Xiu Hong Kang
Keyword(s):  
Growth Rate ◽  
Flow Velocity ◽  
Stokes Equations ◽  
Phase Field Model ◽  
Dendrite Growth ◽  
Solute Diffusion ◽  
Navier Stokes ◽  
Melt Flow ◽  
Navier Stokes Equations ◽  
Flow Effects

The effects of melt flow on dendrite growth during solidification are studied by the quantitative phase field model coupling the Navier-Stokes equations. Through analyzing the relationship between flow velocity and dendrite growth rate in simulations, a flow Péclet number involving with characteristic flow velocity, characteristic length of the zone affected by flow and thermal (solute) diffusion coefficient, is suggested for dendrite growth under convections. The growth rate increment due to flow follows a power-law relationship with the Péclet number. As the Péclet number is much higher than one, the influence of convection on dendrite growth is apparent, whereas as it is below one, the flow effects can be neglected.

scholarly journals Intermittency in solutions of the three-dimensional Navier–Stokes equations

2003 ◽  
Vol 478 ◽  
pp. 227-235 ◽  
Author(s):  
J. D. GIBBON ◽  
Charles R. DOERING
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Average Width ◽  
Binary Character ◽  
Spatio Temporal ◽  
High Reynolds

Dissipation-range intermittency was first observed by Batchelor & Townsend (1949) in high Reynolds number turbulent flows. It typically manifests itself in spatio-temporal binary behaviour which is characterized by long, quiescent periods in the signal which are interrupted by short, active ‘events’ during which there are large excursions away from the average. It is shown that Leray's weak solutions of the three-dimensional incompressible Navier–Stokes equations can have this binary character in time. An estimate is given for the widths of the short, active time intervals, which decreases with the Reynolds number. In these ‘bad’ intervals singularities are still possible. However, the average width of a ‘good’ interval, where no singularities are possible, increases with the Reynolds number relative to the average width of a bad interval.

Onset of Küppers–Lortz-like dynamics in finite rotating thermal convection

2010 ◽  
Vol 644 ◽  
pp. 337-357 ◽  
Author(s):  
A. RUBIO ◽  
J. M. LOPEZ ◽  
F. MARQUES
Keyword(s):  
Hopf Bifurcation ◽  
Centrifugal Force ◽  
Thermal Convection ◽  
Stokes Equations ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Onset Of Convection ◽  
Navier Stokes Equations ◽  
Temporal Complexity ◽  
Spatio Temporal

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

Laminar channel flow driven by accelerating walls

1991 ◽  
Vol 2 (4) ◽  
pp. 359-385 ◽  
Author(s):  
P. Watson ◽  
W. H. H. Banks ◽  
M. B. Zaturska ◽  
P. G. Drazin
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Steady Flows ◽  
Navier Stokes Equations ◽  
Partial Differential System ◽  
Laminar Channel Flow ◽  
Two Dimensional Flow ◽  
Pitchfork Bifurcations ◽  
Route To Chaos ◽  
Physical Result

The two-dimensional flow of a viscous incompressible fluid in a channel with accelerating walls is analysed by use of Hiemenz's similarity solution. Steady flows and their instabilities are calculated, and unsteady flows are computed by solving the initial-value problem for the governing partial-differential system. Thereby, these exact solutions of the Navier–Stokes equations are found to exhibit turning points, pitchfork bifurcations, Hopf bifurcations and Takens–Bogdanov bifurcations along the route to chaos. The substantial physical result is that the chaos previously found for flows with symmetrically accelerating walls is easily destroyed by a little asymmetry.

Instabilities of Numerically Computed Thermocapillary Flows in Liquid Bridges at Medium Prandtl Numbers

2001 ◽  
Author(s):  
Charles Hirsch ◽  
Cristian Dinescu
Keyword(s):  
Marangoni Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Liquid Bridges ◽  
Navier Stokes Equations ◽  
Thermocapillary Flows ◽  
Temporal Structures ◽  
Spatio Temporal ◽  
Prandtl Numbers ◽  
Finite Volume Approach

Abstract The instabilities of the thermocapillary flows in cylindrical liquid bridges are investigated numerically by employing a finite volume approach to solve the complete Navier-Stokes equations accompanied by appropriate boundary conditions on the free surface, which model the surface tension effects. The influences of Marangoni number, aspect ratio A and gravity on the computed spatio-temporal structures of the thermocapillary flow are emphasized. The structures associated with high Marangoni number regimes are analyzed by using specific tools of the nonlinear dynamics: temperature signal, power spectrum, first return maps and Poincare maps.

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