MESOSCOPIC MODELLING AND STOCHASTIC SIMULATIONS OF TURBULENT FLOWS (original) (raw)

Stochastic Approach to Turbulence: A Comprehensive Review

Cornell University - arXiv, 2022

Since Kolmogorov's theory, turbulence has been studied using various methods, many of which could be now be understood in a probabilistic framework. Herein, a comprehensive review of the advances made on stochastic theory of turbulence since Kolmogorov has been provided. It has been suggested that stochastic theory would be able to provide a natural foundation for a unified theory of turbulence as a wide range of problems related to turbulence could be treated using stochastic methods. At first, the mathematical theory of stochastic hydrodynamics has been studied and the results such as well-Posedness and ergodicity have been discussed for stochastic Burgers and Navier-Stokes equations. The theory of turbulence was then assessed where a special attention was paid to stochastic methods. Concepts including Burgers turbulence, scalar advection, incompressible turbulence, field theoretic methods, and stochastic variational principle have been studied to provide detailed discussion on the stochastic theory of turbulence. Moreover, an introduction to some stochastic numerical methods has also been provided. None of the covered methods have a counterpart in deterministic theories due to their derivation or inherent limitations of deterministic theories.

A Markovian random coupling model for turbulence

Journal of Fluid Mechanics, 1974

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 t o be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.

Numerical simulation of the stochastic Burgers’ equation using MLMC and CBC algorithm

IOP Conference Series: Materials Science and Engineering

Burgers turbulence is a model for developing tools to study the Navier-Stokes turbulence, many hydrodynamical problemsin the theory related to random Lagrangian systems. Many questions that are generally asked in Navier-Stokes turbulence can be answered using Burgers turbulence. The aim of the present paper is to apply Multi-level Monte Carlo (MLMC) on stochastic Burgers’ equation by using component-by-component algorithm (CBC). CBC algorithm is developed by the concept of circulant matrix that reduces the cost as a Quasi Monte Carlo technique from O(s n) to O(s n log n) where s is the dimension of integral with equi-distributed points. In this paper, Burgers’ equation is discretized using the finite-volume technique, the MLMC with different random samples is applied and the stability is tested. The results show that MLMC is suitable only for some cases of stochastic differential equations (SDEs) when using pseudo random generator, which is Monte Carlo with high cost than using CBC.

Stochastic dynamical model of intermittency in fully developed turbulence

Physical Review E, 2010

A novel model of intermittency is presented in which the dynamics of the rates of energy transfer between successive steps in the energy cascade is described by a hierarchy of stochastic differential equations. The probability distribution of velocity increments is calculated explicitly and expressed in terms of generalized hypergeometric functions of the type n F 0 , which exhibit power-law tails. The model predictions are found to be in good agreement with experiments on a low temperature gaseous helium jet. It is argued that distributions based on the functions n F 0 might be relevant also for other physical systems with multiscale dynamics.

Stochastic Dynamical Modeling of Turbulent Flows

Annual Review of Control, Robotics, and Autonomous Systems

Advanced measurement techniques and high-performance computing have made large data sets available for a range of turbulent flows in engineering applications. Drawing on this abundance of data, dynamical models that reproduce structural and statistical features of turbulent flows enable effective model-based flow control strategies. This review describes a framework for completing second-order statistics of turbulent flows using models based on the Navier–Stokes equations linearized around the turbulent mean velocity. Dynamical couplings between states of the linearized model dictate structural constraints on the statistics of flow fluctuations. Colored-in-time stochastic forcing that drives the linearized model is then sought to account for and reconcile dynamics with available data (that is, partially known statistics). The number of dynamical degrees of freedom that are directly affected by stochastic excitation is minimized as a measure of model parsimony. The spectral content o...

Numerical Simulation of Randomly Forced Turbulent Flows

Journal of Computational Physics, 1998

The present article aims to provide an appropriate numerical method for the simulation of randomly forced turbulent systems. The spatial discretization is based on the classical Fourier spectral method. The time integration is performed by a secondorder Runge-Kutta scheme. The consistency of an extension of this scheme to treat additive noise stochastic differential equations is proved. The random number generator is based on lagged Fibonacci series. Results are presented for two randomly forced problems: the Burgers and the incompressible Navier-Stokes equations with a white-noise in time forcing term characterized by a power-law correlation function in spectral space. A variety of statistics are computed for both problems, including the structure functions. The third-order structure functions are compared with their exact expressions in the inertial subrange. The influence of the dissipation mechanism (viscous or hyperviscous) on the inertial subrange is discussed. In particular, probability density functions of velocity increments are computed for the Navier-Stokes simulation. Finally, for both Burgers and Navier-Stokes problems, our results support the view that random sweeping is the dominant effect of the large-scale motion on the small-scales.

Stochastic trajectory models for turbulent diffusion: Monte Carlo process versus Markov chains

Atmospheric Environment. Part A. General Topics, 1992

Turbulent diffusion of passive scalars and particles is often simulated with either a Monte Carlo process or a Markov chain. Knowledge of the velocity correlation generated by either of these stochastic trajectory models is essential to their application. The velocity correlation for Monte Carlo process and Markov chain was studied analytically and numerically. A genera/relationship was developed between the Lagranglan velocity correlation and the probability density function for the time steps in a Monte Carlo process. The velocity correlation was found to be independent of the fluid velocity probability density function, hut to be related to the time-step probability density function. For a Monte Carlo process with a constant time step, the velocity correlation is a triangle function; and the integral time scale is equal to onehalf of the time-step length. When the time step was chosen randomly with an exponential pdf distribution, the resulting velocity correlation was an exponential function. Other t/me-step probability density functions, such as a uniform distribution and a ha/f-Gaussian distribution, were also tested. A Markov chain, which presumes one-step memory, has a piecewise linear velocity correlation function with a finite time step. For a Markov chain with a short time step, only an exponential velocity correlation function can be realized. Thus, a Monte Carlo process with random time steps is more versatile than a Markov chain. Direct numerical calculation of the velocity correlation verified the analytical results. A new model which combines the ideas of the Monte Carlo process and the Markov chain was developed. By examining the long-t/me mean square dispersion, we found an exact solution for the Lagranglan integral time scale of the new model in terms of the intercorrelation parameter and the mean and the variance of the time steps. Using this new model, we can generate .any velocity correlation, including one with a negative tail. Two approximate solutions that give upper and lower bounds for the Lagrangian velocity correlation are proposed.

Stochastic energy-cascade model for (1+1)-dimensional fully developed turbulence

Physics Letters A, 2004

Geometrical random multiplicative cascade processes are often used to model positivevalued multifractal fields such as the energy dissipation in fully developed turbulence. We propose a dynamical generalization describing the energy dissipation in terms of a continuous and homogeneous stochastic field in one space and one time dimension. In the model, correlations originate in the overlap of the respective spacetime histories of field amplitudes. The theoretical two-and three-point correlation functions are found to be in good agreement with their equal-time counterparts extracted from wind tunnel turbulent shear flow data.

STOCHASTIC MODELING AND SIMULATION OF TURBULENT REACTING FLOWS

2000

The filtered density function (FDF) is implemented for large eddy simulation of three-dimensional, planar jets under both non-reacting and chemically reacting conditions. In this methodology, the effects of the unresolved scalar fluctuations are taken into account by considering the joint probability density function of the scalar quantities in a stochastic manner. It is shown that under reacting flow conditions, the absence of closure for the SGS scalar correlations yields results which are significantly different from those obtained by the FDF. We visualize this data with a combination of volume and isosurface rendering, and introduce methods for reducing the memory and time costs that have historically precluded recording the results of the simulation at all calculated time steps.

A stochastic subgrid model with application to turbulent flow and scalar mixing

Physics of Fluids, 2007

A new computationally cheap stochastic Smagorinsky model which allows for backscatter of subgrid scale energy is proposed. The new model is applied in the large eddy simulation of decaying isotropic turbulence, rotating homogeneous shear flow and turbulent channel flow at Re = 360. The results of the simulations are compared to direct numerical simulation data. The inclusion of stochastic backscatter has no significant influence on the development of the kinetic energy in homogeneous flows, but it improves the prediction of the fluctuation magnitudes as well as the anisotropy of the fluctuations in turbulent channel flow compared to the standard Smagorinsky model with wall damping of C S. Moreover, the stochastic model improves the description of the energy transfer by reducing its length scale and increasing its variance. Some improvements were also found in isotropic turbulence where the stochastic contribution improved the shape of the enstrophy spectrum at the smallest resolved scales and reduced the time scale of the smallest resolved scales in better agreement with earlier observations.