scholarly journals Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations

2019 ◽  
Vol 149 (2) ◽  
pp. 429-446
Author(s):  
Alexey Cheskidov ◽  
Mimi Dai
Keyword(s):  
Fluid Flow ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Critical Value ◽  
Time Dependent ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
All Solutions ◽  
Time Average

AbstractKolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional (3D) fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier–Stokes equations, is a mathematical analogue of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier–Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.

scholarly journals A Code for Simulating Heat Transfer in Turbulent Channel Flow

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 756
Author(s):  
Federico Lluesma-Rodríguez ◽  
Francisco Álcantara-Ávila ◽  
María Jezabel Pérez-Quiles ◽  
Sergio Hoyas
Keyword(s):  
Heat Transfer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Turbulent Channel Flow ◽  
Passive Scalar ◽  
Direct Numerical Simulations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Thermal Flow ◽  
Navier Stokes Equations

One numerical method was designed to solve the time-dependent, three-dimensional, incompressible Navier–Stokes equations in turbulent thermal channel flows. Its originality lies in the use of several well-known methods to discretize the problem and its parallel nature. Vorticy-Laplacian of velocity formulation has been used, so pressure has been removed from the system. Heat is modeled as a passive scalar. Any other quantity modeled as passive scalar can be very easily studied, including several of them at the same time. These methods have been successfully used for extensive direct numerical simulations of passive thermal flow for several boundary conditions.

Time-dependent solutions of the Navier–Stokes equations for spatially-uniform velocity gradients

1994 ◽  
Vol 124 (1) ◽  
pp. 127-136 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Fluid Flow ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Incompressible Fluid Flow ◽  
Navier Stokes Equations ◽  
Temporal Behaviour ◽  
Uniform Velocity ◽  
Velocity Gradients

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.

scholarly journals Estimating intermittency in three-dimensional Navier–Stokes turbulence

2009 ◽  
Vol 625 ◽  
pp. 125-133 ◽  
Author(s):  
J. D. GIBBON
Keyword(s):  
Reynolds Number ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Space Time ◽  
Navier Stokes ◽  
Local Velocity ◽  
Vortex Sheets ◽  
Navier Stokes Equations ◽  
Local Number

The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time $\mathbb{S}$± where $\mathbb{S}$− is comprised of a union of disjoint space–time ‘anomalies’. If $\mathbb{S}$− is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom ± needed to resolve the regions in $\mathbb{S}$± satisfies $\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).

Research of Tubular Pumping Systems on Dual-Directional Operation

2009 ◽  
Author(s):  
Long Li ◽  
Wang Ze ◽  
Xuelin Yang ◽  
Dan Li
Keyword(s):  
Fluid Flow ◽  
Vortex Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Experiment Data ◽  
Navier Stokes Equations ◽  
Diffuse Flow ◽  
Pumping Station ◽  
Pumping System

The tubular pumping system on dual-directional operation is used extensively for drainage and feedwater pumping stations of the cities and towns. The performance of the dual-directional operation of pumping systems is different with that of simple-way operation. The article described the three-dimensional fluid flow and the predicted performance of the numerical investigation inside a tubular pumping station on dual-directional operation, based on the Reynolds time-averaged Navier-Stokes equations and the realizable k-ε turbulent flow model, applied the law-of-wall and sliding mesh technique, and comparing with the experiment data. The main phenomena existing in pressure contours, velocity contours, velocity vectors and flow lines is showed. The disturbance of fluid flow from the pump outlet to pumping station channel is researched. The axial-whirling flow, circulation-vortex flow is discovered inside discharge diffuser of tubular pumping station on feed-directional operation. The axial-whirling flow is strengthened as a result of diffuse flow. The circulation-vortex flow of the impeller outlet is enhanced in the radius and reduced in the middle of discharge diffuser without guide vanes. There is more loss of head in discharge diffuser of the channel, comparing with that of the suction reducer. It was a close predicted performance of numerical simulation with that of the experiment in the best efficiency point. There was a more difference between the predicted performances with that of the experiment data on the feedwater-directional operation, comparing that of the drainagewater -directional operation.

Stabilization of Solution to Navier-Stokes Equations by Feedback Control Defined on the Boundary

2002 ◽  
Author(s):  
Andrei V. Fursikov
Keyword(s):  
Fluid Flow ◽  
Feedback Control ◽  
Stokes Equations ◽  
Time Moment ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Velocity Vector Field ◽  
Navier Stokes Equations ◽  
Real Process ◽  
Mathematical Construction

Let vˆ be a velocity vector field of steady-state fluid flow in a bounded container. We do not suppose that vˆ is stable. For each fluid flow which is close to vˆ at time moment t = 0 we propose a mathematical construction of feedback control from the boundary of the container which stabilize to vˆ this flow, i.e. which forces this flow to tend to vˆ with prescribed exponential rate. We introduce a notion of “real process” which is an abstract analog of fluid flow or (in other version) of numerical solution of Navier-Stokes equations. Real process differs from exact solution of three-dimensional Navier-Stokes equaitons on some small fluctuatons. Alhtough construction of feedback control is based on precise solving of Navier-Stokes equations, feedback control obtained by this method can react on unpredictable fluctuations mentioned above damping them. Such construction can be useful for numerical calculation because there fluctuations appear always.

Stretching effects on the three-dimensional stability of vortices with axial flow

2002 ◽  
Vol 454 ◽  
pp. 419-442 ◽  
Author(s):  
IVAN DELBENDE ◽  
MAURICE ROSSI ◽  
STÉPHANE LE DIZÈS
Keyword(s):  
Dimensional Stability ◽  
Strain Field ◽  
Stokes Equations ◽  
Linear Equations ◽  
Three Dimensional ◽  
Axial Flow ◽  
Time Dependent ◽  
Navier Stokes ◽  
Simultaneous Action ◽  
Navier Stokes Equations

The effect of stretching on the three-dimensional stability of a viscous unsteady vortex is addressed. The basic flow, which satisfies the Navier–Stokes equations, is a vortex with axial flow subjected to a time-dependent strain field oriented along its axis. The linear equations for the three-dimensional perturbations of the stretched vortex are first reduced by using successive changes of variables to equations which are almost identical to those of the unstretched vortex but with time-dependent parameters. These equations are then numerically solved in the particular case of the Batchelor vortex with a strain field which first compresses then stretches the vortex. Through this simulation, it is qualitatively demonstrated how the simultaneous action of stretching and azimuthal vorticity may destabilize a vortex. It is also argued that it provides a possible mechanism for the vortex bursts observed in turbulence experiments.

Numerical Simulations of Three-Dimensional Flows in a Cubic Cavity With an Oscillating Lid

1993 ◽  
Vol 115 (4) ◽  
pp. 680-686 ◽  
Author(s):  
Reima Iwatsu ◽  
Jae Min Hyun ◽  
Kunio Kuwahara
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Time Dependent ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Sinusoidal Oscillations ◽  
Dependent Flow

Numerical studies are made of three-dimensional flow of a viscous fluid in a cubical container. The flow is driven by the top sliding wall, which executes sinusoidal oscillations. Numerical solutions are acquired by solving the time-dependent, three-dimensional incompressible Navier-Stokes equations by employing very fine meshes. Results are presented for wide ranges of two principal physical parameters, i.e., the Reynolds number, Re ≤ 2000 and the frequency parameter of the lid oscillation, ω′ ≤ 10.0. Comprehensive details of the flow structure are analyzed. Attention is focused on the three-dimensionality of the flow field. Extensive numerical flow visualizations have been performed. These yield sequential plots of the main flows as well as the secondary flow patterns. It is found that the previous two-dimensional computational results are adequate in describing the main flow characteristics in the bulk of interior when ω′ is reasonably high. For the cases of high-Re flows, however, the three-dimensional motions exhibit additional complexities especially when ω′ is low. It is asserted that, thanks to the recent development of the supercomputers, calculation of three-dimensional, time-dependent flow problems appears to be feasible at least over limited ranges of Re.

Numerical Computation of Fluid Flow and Heat Transfer in Microchannels

2005 ◽  
Author(s):  
Ningli Liu ◽  
Rene Chevray ◽  
Gerald A. Domoto ◽  
Elias Panides
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Energy Equation ◽  
Stokes Equations ◽  
Numerical Approach ◽  
Time Dependent ◽  
Navier Stokes ◽  
Temperature Profiles ◽  
Navier Stokes Equations ◽  
Flow And Heat Transfer

A finite difference numerical approach for solving slightly compressible, time-dependent, viscous laminar flow is presented in this study. Simplified system of Navier-Stokes equations and energy equation are employed in the study in order to perform more efficient numerical calculations. Fluid flow and heat transfer phenomena in two dimensional microchannels are illustrated numerically in this paper. This numerical approach provides a complete numerical simulation of the development of the fluid flow and the temperature profiles through multi-dimensional microchannels.

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