A Markovian random coupling model for turbulence

1974 ◽  
Vol 65 (1) ◽  
pp. 145-152 ◽  
Author(s):  
U. Frisch ◽  
M. Lesieur ◽  
A. Brissaud
Keyword(s):  
Stochastic Model ◽  
Master Equation ◽  
Turbulent Flows ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Planck Equation ◽  
Navier Stokes ◽  
Coupling Coefficients ◽  
Navier Stokes Equations ◽  
Random Coupling

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

scholarly journals Refinement of a Previous Hypothesis of the Lyapunov Analysis of Isotropic Turbulence

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Nicola de Divitiis
Keyword(s):  
Structure Function ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Longitudinal Velocity ◽  
Navier Stokes ◽  
Lyapunov Analysis ◽  
Velocity Difference ◽  
Navier Stokes Equations ◽  
Previous Hypothesis ◽  
Kolmogorov Theory

The purpose of this paper is to improve a hypothesis of the previous work of N. de Divitiis (2011) dealing with the finite-scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity differenceΔuris derived through a statistical analysis of the Fourier transformed Navier-Stokes equations and by means of considerations regarding the scales of the velocity fluctuations, which arise from the Kolmogorov theory. Due to these latter considerations, this Lyapunov analysis seems to need some of the results of the Kolmogorov theory. This work proposes a more rigorous demonstration which leads to the same structure function, without using the Kolmogorov scale. This proof assumes that pair and triple longitudinal correlations are sufficient to determine the statistics ofΔurand adopts a reasonable canonical decomposition of the velocity difference in terms of proper stochastic variables which are adequate to describe the mechanism of kinetic energy cascade.

On Direct Numerical Simulation of Turbulent Flows

2011 ◽  
Vol 64 (2) ◽  
Author(s):  
Giancarlo Alfonsi
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
High Performance Computing ◽  
Turbulent Flows ◽  
High Performance ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Navier Stokes Equations ◽  
Performance Computing

The direct numerical simulation of turbulence (DNS) has become a method of outmost importance for the investigation of turbulence physics, and its relevance is constantly growing due to the increasing popularity of high-performance-computing techniques. In the present work, the DNS approach is discussed mainly with regard to turbulent shear flows of incompressible fluids with constant properties. A body of literature is reviewed, dealing with the numerical integration of the Navier-Stokes equations, results obtained from the simulations, and appropriate use of the numerical databases for a better understanding of turbulence physics. Overall, it appears that high-performance computing is the only way to advance in turbulence research through the front of the direct numerical simulation.

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

scholarly journals The emergence of localized vortex–wave interaction states in plane Couette flow

2013 ◽  
Vol 721 ◽  
pp. 58-85 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
High Reynolds Number ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Critical Layer ◽  
Navier Stokes ◽  
Turbulent Spots ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractThe recently understood relationship between high-Reynolds-number vortex–wave interaction theory and computationally generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex–wave interaction (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The asymptotic description given reduces the Navier–Stokes equations to the so-called boundary-region equations, for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from $O(h)$ to $O(Rh)$, where $R$ is the Reynolds number and $2h$ is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave–vortex states. The solutions we calculate have an asymptotic error proportional to ${R}^{- 2} $ when compared to the full Navier–Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel, so that the roll part of the flow is driven throughout the flow rather than as in Hall & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize and take on similarities with computationally generated or experimentally observed turbulent spots. In effect, the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier–Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

Approximation of attractors, large eddy simulations and multiscale methods

1991 ◽  
Vol 434 (1890) ◽  
pp. 23-39 ◽  
Keyword(s):  
Turbulent Flows ◽  
Mathematical Theory ◽  
Stokes Equations ◽  
Multiscale Methods ◽  
Large Eddy Simulations ◽  
Navier Stokes ◽  
Theory Approach ◽  
Navier Stokes Equations ◽  
Approximate Inertial Manifolds ◽  
Large Eddy

Recent advances in the mathematical theory of the Navier-Stokes equations have produced new insight in the mathematical theory of turbulence. In particular, the study of the attractor for the Navier-Stokes equations produced the first connection between two approaches to turbulence that seemed far apart, namely the conventional approach of Kolmogorov and the dynamical systems theory approach. Similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows. Our aim in this article is to survey and describe some of the new results concerning the functional properties of the Navier-Stokes equations and to discuss their relevance to turbulence.

Probability Density Functions of Vorticities in Turbulent Channels with Effects of Blowing and Suction

2016 ◽  
Vol 17 (2) ◽  
pp. 127-135
Author(s):  
Can Liu ◽  
Xi Chen
Keyword(s):  
New York ◽  
Probability Density ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Generalized Hyperbolic Distribution ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Turbulent Channel Flows ◽  
Probability Density Function Pdf

AbstractThis paper presents direct numerical simulation (DNS) result of the Navier–Stokes equations for turbulent channel flows with blowing and suction effects. The friction Reynolds number is ${\rm{R}}{{\rm{e}}_\tau} = 394$ and a range of blowing and suction conditions is covered with different perturbation strengths, i. e. $A = 0.05, $ 0.1, 0.2. While the mean velocity profile has been severely altered, the probability density function (PDF) for (spanwise) vorticity – depending on wall distance $({y^ +})$ and blowing/suction strength (A) – satisfies the generalized hyperbolic distribution (GHD) of Birnir [The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci. (2013a), doi: 10.1007/s00332-012-9164–z; The Kolmogorov-Obukhov theory of turbulence, Springer, New York, 2013b] in the bulk of the flow. The latter leads to accurate descriptions of all PDFs (at ${y^ +} = 40, $ 200, 390 and $A = 0.05, $ 0.2, for instance) with only four parameters. The result indicates that GHD is a general tool to quantify PDF for turbulent flows under various wall surface conditions.

scholarly journals Effects of baffle length on turbulent flows generated in stirred vessels

2011 ◽  
Vol 1 (4) ◽  
Author(s):  
Meriem Ammar ◽  
Zied Driss ◽  
Wajdi Chtourou ◽  
Mohamed Abid
Keyword(s):  
Turbulent Flows ◽  
Stokes Equations ◽  
Control Volume ◽  
Numerical Results ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rushton Turbine ◽  
Stirred Vessels ◽  
Hydrodynamic Behaviour ◽  
Effective System

AbstractThe aim of this paper is to study the effect of baffles length on the turbulent flows in stirred tanks. The hydrodynamic behaviour induced by a Rushton turbine (RT6) is numerically predicted by solving the Navier-Stokes equations in conjunction with the Renormalization Group (RNG) of the k-ɛ turbulence model. These equations are solved by a control volume discretization method. The numerical results from the application of the computational fluid dynamics (CFD) code Fluent with the multi-reference frame (MRF) model are presented in the vertical and horizontal planes in the impeller stream region. Our studies were carried out on three different systems. The most effective system was selected based on its calculated power consumption figure. All numerical results showed good agreement with experimental data.

scholarly journals On the analysis of a geometrically selective turbulence model

2020 ◽  
Vol 9 (1) ◽  
pp. 1402-1419 ◽  
Author(s):  
Nejmeddine Chorfi ◽  
Mohamed Abdelwahed ◽  
Luigi C. Berselli
Keyword(s):  
Turbulent Flows ◽  
Turbulence Model ◽  
Large Scale ◽  
Stokes Equations ◽  
A Priori ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Smagorinsky Model ◽  
Navier Stokes Equations ◽  
Velocity Pressure

Abstract In this paper we propose some new non-uniformly-elliptic/damping regularizations of the Navier-Stokes equations, with particular emphasis on the behavior of the vorticity. We consider regularized systems which are inspired by the Baldwin-Lomax and by the selective Smagorinsky model based on vorticity angles, and which can be interpreted as Large Scale methods for turbulent flows. We consider damping terms which are active at the level of the vorticity. We prove the main a priori estimates and compactness results which are needed to show existence of weak and/or strong solutions, both in velocity/pressure and velocity/vorticity formulation for various systems. We start with variants of the known ones, going later on to analyze the new proposed models.

Sign in / Sign up

Export Citation Format

Share Document