The number of degrees of freedom of three-dimensional Navier–Stokes turbulence

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method.Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory.Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence.Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself.

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On a Leray–α model of turbulence

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2004.1373 ◽

2005 ◽

Vol 461 (2055) ◽

pp. 629-649 ◽

Cited By ~ 175

Author(s):

Alexey Cheskidov ◽

Darryl D. Holm ◽

Eric Olson ◽

Edriss S. Titi

Keyword(s):

Power Law ◽

Upper Bound ◽

Empirical Data ◽

Degrees Of Freedom ◽

Three Dimensional ◽

Navier Stokes ◽

Large Eddy Simulation Model ◽

Eddy Simulation ◽

Wide Range ◽

Three Dimensional Models

In this paper we introduce and study a new model for three–dimensional turbulence, the Leray– α model. This model is inspired by the Lagrangian averaged Navier–Stokes– α model of turbulence (also known Navier–Stokes– α model or the viscous Camassa–Holm equations). As in the case of the Lagrangian averaged Navier–Stokes– α model, the Leray– α model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray– α model of the order of ( L / l d ) 12/7 , where L is the size of the domain and l d is the dissipation length–scale. This upper bound is much smaller than what one would expect for three–dimensional models, i.e. ( L / l d ) 3 . This remarkable result suggests that the Leray– α model has a great potential to become a good sub–grid–scale large–eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual k −5/3 Kolmogorov power law the inertial range has a steeper power–law spectrum for wavenumbers larger than 1/ α . Finally, we propose a Prandtl–like boundary–layer model, induced by the Leray– α model, and show a very good agreement of this model with empirical data for turbulent boundary layers.

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Estimating intermittency in three-dimensional Navier–Stokes turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112009006089 ◽

2009 ◽

Vol 625 ◽

pp. 125-133 ◽

Cited By ~ 3

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).

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Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.349 ◽

2013 ◽

Vol 730 ◽

pp. 309-327 ◽

Cited By ~ 50

Author(s):

L. Biferale ◽

S. Musacchio ◽

F. Toschi

Keyword(s):

Degrees Of Freedom ◽

Isotropic Turbulence ◽

Three Dimensional ◽

Thin Layers ◽

Energy Cascade ◽

Navier Stokes ◽

Inverse Energy Cascade ◽

Negative Helicity ◽

Homogeneous And Isotropic Turbulence ◽

Transfer Properties

AbstractWe investigate the transfer properties of energy and helicity fluctuations in fully developed homogeneous and isotropic turbulence by changing the nature of the nonlinear Navier–Stokes terms. We perform a surgery of all possible interactions, by keeping only those triads that have sign-definite helicity content. In order to do this, we apply an exact decomposition of the velocity field in a helical Fourier basis, as first proposed by Constantin & Majda (Commun. Math. Phys, vol. 115, 1988, p. 435) and exploited in great detail by Waleffe (Phys. Fluids A, vol. 4, 1992, p. 350), and we evolve the Navier–Stokes dynamics keeping only those velocity components carrying a well-defined (positive or negative) helicity. The resulting dynamics preserves translational and rotational symmetries but not mirror invariance. We give clear evidence that this three-dimensional homogeneous and isotropic chiral turbulence is characterized by a stationary inverse energy cascade with a spectrum ${E}_{back} (k)\sim {k}^{- 5/ 3} $ and by a direct helicity cascade with a spectrum ${E}_{forw} (k)\sim {k}^{- 7/ 3} $. Our results are important to highlight the dynamics and statistics of those subsets of all possible Navier–Stokes interactions responsible for reversal events in the energy-flux properties, and demonstrate that the presence of an inverse energy cascade is not necessarily connected to a two-dimensionalization of the flow. We further comment on the possible relevance of such findings to flows of geophysical interest under rotations and in thin layers. Finally we propose other innovative numerical experiments that can be achieved by using a similar decimation of degrees of freedom.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.33 ◽

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.

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Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

JOURNAL OF MECHANICAL ENGINEERING AND SCIENCES ◽

10.15282/jmes.14.4.2020.05.0579 ◽

2020 ◽

Vol 14 (4) ◽

pp. 7369-7378

Author(s):

Ky-Quang Pham ◽

Xuan-Truong Le ◽

Cong-Truong Dinh

Keyword(s):

Stokes Equations ◽

Three Dimensional ◽

Axial Compressor ◽

Aerodynamic Performance ◽

Numerical Results ◽

Single Stage ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Operating Stability ◽

Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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Experimental and Numerical Analysis of a Motorcycle Air Intake System Aerodynamics and Performance

International Journal of Automotive and Mechanical Engineering ◽

10.15282/ijame.17.1.2020.10.0565 ◽

2020 ◽

Vol 17 (1) ◽

Author(s):

M. A. Abd Halim ◽

N. A. R. Nik Mohd ◽

M. N. Mohd Nasir ◽

M. N. Dahalan

Keyword(s):

Combustion Process ◽

Three Dimensional ◽

Engine Performance ◽

Internal Flow ◽

Rapid Expansion ◽

Navier Stokes ◽

Turbulent Fluctuation ◽

Ansys Fluent ◽

Induction System ◽

Acceleration And Deceleration

Induction system or also known as the breathing system is a sub-component of the internal combustion system that supplies clean air for the combustion process. A good design of the induction system would be able to supply the air with adequate pressure, temperature and density for the combustion process to optimizing the engine performance. The induction system has an internal flow problem with a geometry that has rapid expansion or diverging and converging sections that may lead to sudden acceleration and deceleration of flow, flow separation and cause excessive turbulent fluctuation in the system. The aerodynamic performance of these induction systems influences the pressure drop effect and thus the engine performance. Therefore, in this work, the aerodynamics of motorcycle induction systems is to be investigated for a range of Cubic Feet per Minute (CFM). A three-dimensional simulation of the flow inside a generic 4-stroke motorcycle airbox were done using Reynolds-Averaged Navier Stokes (RANS) Computational Fluid Dynamics (CFD) solver in ANSYS Fluent version 11. The simulation results are validated by an experimental study performed using a flow bench. The study shows that the difference of the validation is 1.54% in average at the total pressure outlet. A potential improvement to the system have been observed and can be done to suit motorsports applications.

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