scholarly journals Unstable periodic orbits in the Lorenz attractor

2011 ◽  
Vol 369 (1944) ◽  
pp. 2345-2353 ◽  
Author(s):  
Bruce M. Boghosian ◽  
Aaron Brown ◽  
Jonas Lätt ◽  
Hui Tang ◽  
Luis M. Fazendeiro ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Stokes Equations ◽  
Traditional Approach ◽  
Navier Stokes ◽  
Lorenz Attractor ◽  
Time Averaging ◽  
Unstable Periodic Orbits ◽  
Lorenz Equations ◽  
Navier Stokes Equations ◽  
Generic Orbit

We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computation of unstable periodic orbits of the driven Navier–Stokes equations, which can be simulated using the lattice Boltzmann method.

scholarly journals New variational principles for locating periodic orbits of differential equations

2011 ◽  
Vol 369 (1944) ◽  
pp. 2211-2218 ◽  
Author(s):  
Bruce M. Boghosian ◽  
Luis M. Fazendeiro ◽  
Jonas Lätt ◽  
Hui Tang ◽  
Peter V. Coveney
Keyword(s):  
Periodic Orbits ◽  
Variational Principles ◽  
Stokes Equations ◽  
Iterative Algorithms ◽  
Navier Stokes ◽  
Unstable Periodic Orbits ◽  
Navier Stokes Equations ◽  
New Methods ◽  
Discrete Representations

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier–Stokes equations, simulated using the lattice Boltzmann equation.

Use of Experimental Data for an Efficient Description of Turbulent Flows

1990 ◽  
Vol 43 (5S) ◽  
pp. S240-S245 ◽  
Author(s):  
N. Aubry
Keyword(s):  
Experimental Data ◽  
Turbulent Flows ◽  
Turbulence Modeling ◽  
Stokes Equations ◽  
Original System ◽  
Navier Stokes ◽  
Lorenz Attractor ◽  
Wall Region ◽  
Lorenz Equations ◽  
Chaotic Solution

The proper orthogonal decomposition (POD), also called Karhunen-Loe`ve expansion, which extracts ‘coherent structures’ from experimental data, is a very efficient tool for analyzing and modeling turbulent flows. It has been shown that it converges faster than any other expansion in terms of kinetic energy (Lumley 1970). First, the POD is applied to the chaotic solution of the Lorenz equations. The dynamics of the Lorenz attractor is reconstructed by only the first three POD modes. In the second part of this paper, we show how the POD can be used in turbulence modeling. The particular case studied is the wall region of a turbulent boundary layer. In this flow, the velocity field is expanded into POD modes in the normal direction and Fourier modes in the streamwise and spanwise directions. Dynamical systems are obtained by Galerkin projections of the Navier Stokes equations onto the different modes. Aubry et al. (1988) applied the technique to derive and study a ten dimensional representation which reproduced qualitatively the bursting event experimentally observed. It is shown that streamwise modes, absent in Aubry et al.’s model, participate to the bursting events. This agrees remarkably well with experimental observations. In both examples, the dynamics of the original system is very well recovered from the contribution of only a few modes.

Modelling of By-Pass Transition With Conditioned Navier-Stokes Equations and a K-E Model Adapted for Intermittency

1994 ◽  
Author(s):  
J. Steelant ◽  
E. Dick
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Suction Side ◽  
Navier Stokes ◽  
Intermittent Flow ◽  
Time Averaging ◽  
Bypass Transition ◽  
Navier Stokes Equations ◽  
Direct Excitation ◽  
Energy Equations

Turbomachinery flows are characterized by a very high intensity turbulent mean part. As a consequence, laminar flow in boundary layer regions undergoes transition through direct excitation of turbulence. This is the so-called bypass transition. Regions form that are intermittently laminar and turbulent. In particular in accelerating flows, as on the suction side of a turbine blade, this intermittent flow can extend over a very large part of the boundary layer. Classical turbulence modelling based on global time averaging is not valid in intermittent flows. To take correctly account of the intermittency, conditioned averages are necessary. These are averages taken during the fraction of time the flow is turbulent or laminar respectively. Starting from the Navier-Stokes equations, conditioned continuity, momentum and energy equations are derived for the laminar and turbulent parts of an intermittent flow. The turbulence is described by the classical k-ε model. The supplementary parameter introduced by the conditioned averaging is the intermittency factor. In the calculations, this factor is prescribed in an algebraic way.

scholarly journals On the Role of the Deterministic and Circumferential Stresses in Throughflow Calculations

2008 ◽  
Author(s):  
J.-F. Simon ◽  
O. Le´onard
Keyword(s):  
Flow Field ◽  
Flow Model ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Averaging ◽  
Analysis Tool ◽  
Reynolds Stresses ◽  
Present Contribution ◽  
Navier Stokes Equations ◽  
Throughflow Model

This paper presents a throughflow analysis tool developed in the context of the average-passage flow model elaborated by Adamczyk. The Adamczyk’s flow model describes the 3-D time-averaged flow field within a blade row passage. The set of equations that governs this flow field is obtained by performing a Reynolds averaging, a time averaging and a passage-to-passage averaging on the Navier-Stokes equations. The throughflow level of approximation is obtained by performing an additional circumferential averaging on the 3-D average-passage flow. The resulting set of equations is similar to the 2-D axisymmetric Navier-Stokes equations but additional terms resulting from the averages show up: blade forces, blade blockage factor, Reynolds stresses, deterministic stresses, passage-to-passage stresses and circumferential stresses. This set of equations represents the ultimate throughflow model provided that all stresses and blade forces can be modeled. The relative importance of these additional terms is studied in the present contribution. The stresses and the blade forces are determined from 3-D steady and unsteady databases (a low speed compressor stage and a transonic turbine stage) and incorporated in a throughflow model based on the axisymmetric Navier-Stokes equations. A good agreement between the throughflow solution and the averaged 3-D results is obtained. These results are also compared to those obtained with a more “classical” throughflow approach based on a Navier-Stokes formulation for the endwall losses, correlations for profile losses and a simple radial mixing model assuming turbulent diffusion.

Approximation of attractors for nongradient systems

1995 ◽  
Vol 125 (4) ◽  
pp. 739-758
Author(s):  
I. N. Kostin
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Numerical Approximation ◽  
Stokes Equations ◽  
Hausdorff Metric ◽  
Navier Stokes ◽  
Lorenz Equations ◽  
Navier Stokes Equations ◽  
Suggested Procedure ◽  
Nongradient Systems

The problem of approximation of attractors for semidynamical systems (SDSs) in a metric space is studied. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e. let another (approximate) SDS, which is close in some sense to the exact one, be given. The problem is to construct a set , which is close to M in the Hausdorff metric.The suggested procedure for constructing is finite, which makes it possible to use it in computations. The results obtained are suitable for numerical approximation of attractors for a rather large class of semidynamical systems, including ones generated by the Lorenz equations and the Navier–Stokes equations.

On a New Approach to Slip Flow Using Rayleigh’s Problem

1967 ◽  
Vol 34 (4) ◽  
pp. 837-839 ◽  
Author(s):  
E. P. Russo ◽  
O. A. Arnas
Keyword(s):  
Boundary Condition ◽  
Drag Coefficient ◽  
Flat Plate ◽  
Slip Velocity ◽  
Stokes Equations ◽  
Traditional Approach ◽  
Slip Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
New Approach

A new approach, involving a perturbation of the Navier-Stokes equations, is used to analyze the phenomenon of slip flow over a flat plate. An expression for the coefficient of drag is derived and compared to the drag coefficient obtained by the traditional approach of solving the Navier-Stokes equations with a slip-velocity boundary condition.

A DNS Study of the Dispersion and Deposition of Nano- and Micro-Particles in a Turbulent Channel Flow

2020 ◽  
Author(s):  
Amir A. Mofakham ◽  
Goodarz Ahmadi ◽  
John B. McLaughlin
Keyword(s):  
Deposition Rate ◽  
Lift Force ◽  
Particle Deposition ◽  
Gas Flow ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Navier Stokes ◽  
Time Averaging ◽  
Navier Stokes Equations ◽  
Micro Particles

Abstract Nano- and micro-particles dispersion and deposition in a dilute gas-solid turbulent flow in a channel were studied. A pseudo-spectral DNS code was used to solve the Navier-Stokes equations, and to generate the instantaneous turbulent velocity fluctuation field for the gas flow. Under the one-way coupling assumption, the gas flow carries the particles, but the influence of particles on the flow can be neglected. To provide an understanding of the transport behavior of particles of different sizes, 200,000 monodisperse point particles with Stokes numbers of 0.1, 1, 5, 25, and 125 were introduced with a random distribution in the channel. The corresponding Lagrangian particle equation of motion, including the Stokes drag, the gravity, and the lift forces, were solved, and the trajectories of the particles for the duration of 10,000 wall units were evaluated for dilute suspension. The trap boundary condition on the lower and upper walls of the channel was assumed, and the deposition rates of particles with different sizes were evaluated and recorded as a function of time. Ensemble and time averaging of the simulation results were performed, and the corresponding concentration profiles and the deposition velocities of particles were evaluated for various conditions. A series of simulations were performed, and the effects of wall roughness, lift force, and the gravity direction on the deposition rate were carefully examined. It was found that the surface roughness and the direction of gravity in conjunction with the lift force significantly affect the fine particle deposition rate and could improve the agreement of the DNS simulation with the available experimental data.

scholarly journals Aerodynamic Study of the Wind Flow in the Area of the R2 Expressway in Slovakia

2021 ◽  
Vol 29 (2) ◽  
pp. 55-61
Author(s):  
Olga Hubová ◽  
Marek Macák ◽  
Alžbeta Grmanová
Keyword(s):  
Cfd Simulation ◽  
Stokes Equations ◽  
Turbulence Models ◽  
Navier Stokes ◽  
Time Averaging ◽  
Ansys Fluent ◽  
Navier Stokes Equations ◽  
Wind Speeds ◽  
Centre Of Gravity ◽  
Design Protection

Abstract Our calculation of wind effects was based on the specific wind situation of the planned R2 expressway. Given the topography and the prevailing wind directions, it was necessary to analyse the speeds for winds that could cause vehicles with trailers to be pushed off the roadway, as has been observed in recent years. Using a CFD simulation in the ANSYS FLUENT program, we analysed the entire section of the planned R2 expressway in order to evaluate the wind speeds at the level of the centre of gravity of truck trailers. Statistical turbulence models based on a time-averaging method, i.e., the RANS-Reynolds Averaged Navier-Stokes equations, of turbulent flow quantities and the time-averaging procedure of balance equations are suitable for solving the engineering tasks. In numerical simulations, the Realizable k - ε model was used in which the calculation of the turbulent dynamic viscosity in the equation for Boussinesque’s hypothesis was solved using two transport equations. Plotting and comparing the wind speeds for significant wind directions allowed us to design protection in the dangerous areas using protective walls.

scholarly journals Reply to Basal buoyancy and fast-moving glaciers: in defense of analytic force balance by C. J. van der Veen (2016)

2017 ◽  
Author(s):  
Terence J. Hughes
Keyword(s):  
Stokes Equations ◽  
Traditional Approach ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Geometric Approach ◽  
Force Balance ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Ice Sheet Modeling ◽  
Energy Balances

Abstract. Two approaches to ice-sheet modeling are available. Analytical modeling is the traditional approach. It solves the force (momentum), mass, and energy balances to obtain three-dimensional solutions over time, beginning with the Navier-Stokes equations for the force balance. Geometrical modeling employs simple geometry to solve the force and mass balance in one dimension along ice flow. It is useful primarily to provide the first-order physical basis of ice-sheet modeling for students with little background in mathematics (Hughes, 2012). The geometric approach uses changes in ice-bed coupling along flow to calculate changes in ice elevation and thickness, using floating fraction φ along a flowline or flowband, where φ = 0 for sheet flow, 0 

scholarly journals On the Role of the Deterministic and Circumferential Stresses in Throughflow Calculations

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
J.-F. Simon ◽  
J. P. Thomas ◽  
O. Léonard
Keyword(s):  
Flow Field ◽  
Flow Model ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Averaging ◽  
Analysis Tool ◽  
Reynolds Stresses ◽  
Present Contribution ◽  
Navier Stokes Equations ◽  
Throughflow Model

This paper presents a throughflow analysis tool developed in the context of the average-passage flow model elaborated by Adamczyk. The Adamczyk’s flow model describes the 3D time-averaged flow field within a blade row passage. The set of equations that governs this flow field is obtained by performing a Reynolds averaging, a time averaging, and a passage-to-passage averaging on the Navier–Stokes equations. The throughflow level of approximation is obtained by performing an additional circumferential averaging on the 3D average-passage flow. The resulting set of equations is similar to the 2D axisymmetric Navier–Stokes equations, but additional terms resulting from the averages show up: blade forces, blade blockage factor, Reynolds stresses, deterministic stresses, passage-to-passage stresses, and circumferential stresses. This set of equations represents the ultimate throughflow model provided that all stresses and blade forces can be modeled. The relative importance of these additional terms is studied in the present contribution. The stresses and the blade forces are determined from 3D steady and unsteady databases (a low-speed compressor stage and a transonic turbine stage) and incorporated in a throughflow model based on the axisymmetric Navier–Stokes equations. A good agreement between the throughflow solution and the averaged 3D results is obtained. These results are also compared to those obtained with a more “classical” throughflow approach based on a Navier–Stokes formulation for the endwall losses, correlations for profile losses, and a simple radial mixing model assuming turbulent diffusion.

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