Methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions

1991 ◽  
Vol 434 (1892) ◽  
pp. 719-733 ◽  
Keyword(s):  
Nonlinear Stability ◽  
Stokes Equations ◽  
Multiple Scales ◽  
Navier Stokes ◽  
Nonlinear Stability Analysis ◽  
Centre Manifold ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Seventh Order ◽  
The Relationship

The equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions. The relationship between the Landau equations of the seventh order, which were reduced from the Navier-Stokes equations by applying these two methods, is clarified. How we should define the disturbance amplitude to obtain the complete equivalence between them is discussed.

scholarly journals Dynamics of Single Droplet Splashing on Liquid Film by Coupling FVM with VOF

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 841
Author(s):  
Yuzhen Jin ◽  
Huang Zhou ◽  
Linhang Zhu ◽  
Zeqing Li
Keyword(s):  
Liquid Film ◽  
Weber Number ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Single Droplet ◽  
Navier Stokes Equations ◽  
Volume Method ◽  
The Relationship

A three-dimensional numerical study of a single droplet splashing vertically on a liquid film is presented. The numerical method is based on the finite volume method (FVM) of Navier–Stokes equations coupled with the volume of fluid (VOF) method, and the adaptive local mesh refinement technology is adopted. It enables the liquid–gas interface to be tracked more accurately, and to be less computationally expensive. The relationship between the diameter of the free rim, the height of the crown with different numbers of collision Weber, and the thickness of the liquid film is explored. The results indicate that the crown height increases as the Weber number increases, and the diameter of the crown rim is inversely proportional to the collision Weber number. It can also be concluded that the dimensionless height of the crown decreases with the increase in the thickness of the dimensionless liquid film, which has little effect on the diameter of the crown rim during its growth.

Numerical Simulations of Two-Dimensional Incompressible Flows Over Cavities and Their Control

2007 ◽  
Author(s):  
T. Yoshida ◽  
T. Watanabe
Keyword(s):  
Incompressible Flows ◽  
Stokes Equations ◽  
Control Method ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Mode Iii ◽  
Oscillating Flows ◽  
Navier Stokes Equations ◽  
Oscillation Modes ◽  
The Relationship

We investigate numerically self-sustained oscillating flows over open cavities. The incompressible Navier-Stokes equations are solved using finite difference method for two-dimensional cavities with an upstream laminar boundary layer. A series of simulations are performed for a variety of cavity length to depth ratio. The results show mode switchings among nonoscillations, mode II and mode III oscillations. Variation of Strouhal number is in good agreement with available experimental data. The results of flow fields in the cavity reveal the relationship between the cavity shear layer oscillation modes and recirculating vortices in the cavity. We also demonstrate that oscillations are suppressed by our control method using moving bottom wall.

scholarly journals Weakly nonlinear modelling of a forced turbulent axisymmetric wake

2017 ◽  
Vol 814 ◽  
pp. 570-591 ◽  
Author(s):  
Georgios Rigas ◽  
Aimee S. Morgans ◽  
Jonathan F. Morrison
Keyword(s):  
Turbulent Flows ◽  
Bluff Body ◽  
Stokes Equations ◽  
Nonlinear Models ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Zero Net Mass Flux ◽  
High Reynolds

A theory is presented where the weakly nonlinear analysis of laminar globally unstable flows in the presence of external forcing is extended to the turbulent regime. The analysis is demonstrated and validated using experimental results of an axisymmetric bluff-body wake at high Reynolds numbers, $Re_{D}\sim 1.88\times 10^{5}$, where forcing is applied using a zero-net-mass-flux actuator located at the base of the blunt body. In this study we focus on the response of antisymmetric coherent structures with azimuthal wavenumbers $m=\pm 1$ at a frequency $St_{D}=0.2$, responsible for global vortex shedding. We found experimentally that axisymmetric forcing ($m=0$) couples nonlinearly with the global shedding mode when the flow is forced at twice the shedding frequency, resulting in parametric subharmonic resonance through a triadic interaction between forcing and shedding. We derive simple weakly nonlinear models from the phase-averaged Navier–Stokes equations and show that they capture accurately the observed behaviour for this type of forcing. The unknown model coefficients are obtained experimentally by producing harmonic transients. This approach should be applicable in a variety of turbulent flows to describe the response of global modes to forcing.

On perturbations of Jeffery-Hamel flow

1988 ◽  
Vol 186 ◽  
pp. 559-581 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Stokes Equations ◽  
Temporal Stability ◽  
Pitchfork Bifurcation ◽  
Critical Value ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Nonlinear Perturbations ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Rigid Plane

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

scholarly journals Theory of weakly nonlinear self-sustained detonations

2015 ◽  
Vol 784 ◽  
pp. 163-198 ◽  
Author(s):  
Luiz M. Faria ◽  
Aslan R. Kasimov ◽  
Rodolfo R. Rosales
Keyword(s):  
Stokes Equations ◽  
Period Doubling ◽  
Spatial Dimension ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Dynamical Characteristics ◽  
Reactive Euler Equations ◽  
Doubling Sequence

We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

Nonlinear instability in plane Poiseuille flow: a quantitative comparison between the methods of amplitude expansions and the method of multiple scales

1981 ◽  
Vol 375 (1761) ◽  
pp. 155-167 ◽  
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Model Equations ◽  
Hydrodynamical Stability

This paper shows that two different expansion procedures for hydrodynamical stability problems are equivalent. The method of multiple scales of Stewartson & Stuart (1971) is extended to calculate the stream function up to order ε 2 . Watson’s (1960) rigorous amplitude expansion of the solution of the Navier-Stokes equations is also used to calculate the stream function up to the same order of magnitude, and a complete equi­valence between the two results is found. An analysis of the Eckhaus model equations has been made and the results are equivalent.

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