The Dynamical Systems Approach to the Navier-Stokes Equations of Compressible Fluids

2000 ◽  
pp. 35-66 ◽  
Author(s):  
Eduard Feireisl
Keyword(s):  
Dynamical Systems ◽  
Systems Approach ◽  
Stokes Equations ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-29 ◽  
Author(s):  
James C. Robinson
Keyword(s):  
Dynamical Systems ◽  
Systems Approach ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Open Problems ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Finite Dimensional

The purpose of this review is to give a broad outline of the dynamical systems approach to the two-dimensional Navier-Stokes equations. This example has led to much of the theory of infinite-dimensional dynamical systems, which is now well developed. A second aim of this review is to highlight a selection of interesting open problems, both in the analysis of the two-dimensional Navier-Stokes equations and in the wider field of infinite-dimensional dynamical systems.

Development of mixing and isotropy in inviscid homogeneous turbulence

1982 ◽  
Vol 120 ◽  
pp. 155-183 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Dynamical Systems ◽  
Lower Order ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Fourier Space ◽  
Kolmogorov Entropy ◽  
Navier Stokes Equations ◽  
Constant Energy

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

The engine behind (wall) turbulence: perspectives on scale interactions

2017 ◽  
Vol 817 ◽  
Author(s):  
B. J. McKeon
Keyword(s):  
Coherent States ◽  
Systems Approach ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Scale Interactions ◽  
High Reynolds ◽  
Data Informed

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

scholarly journals Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics

2000 ◽  
Vol 214 (11) ◽  
pp. 1401-1407
Author(s):  
S R Kendall ◽  
H V Rao
Keyword(s):  
Multiple Solutions ◽  
Computational Models ◽  
Stokes Equations ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Algebraic Equations ◽  
Navier Stokes Equations ◽  
Flow Modelling ◽  
Spurious Solutions ◽  
Linear Algebraic Equations

Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Julia García-Luengo ◽  
Pedro Marín-Rubio ◽  
José Real
Keyword(s):  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Existence Of Solution ◽  
Navier Stokes ◽  
Pullback Attractors ◽  
Navier Stokes Equations ◽  
Delay Function ◽  
Measurability Condition ◽  
Delayed Time

AbstractIn this paper we obtain some results on the existence of solution, and of pullback attractors, for a 2D Navier-Stokes model with finite delay studied in [4] and [6]. Actually, we prove a result of existence and uniqueness of solution under less restrictive assumptions than in [4]. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms (for instance, just under a measurability condition on the delay function leading the delayed time). After that, we deal with dynamical systems in suitable phase spaces within two metrics, the L

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