We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

13. 벼잎선충의 약제처리 방법별 방제 효과

An Introduction to Magnetohydrodynamics

Introduction To Stochastic Calculus With Applications

A number of systematic procedures for the identification of vortices/coherent structures have been developed as a way to address their possible kinematical and dynamical roles in structural formulations of turbulence. It has been broadly acknowledged, however, that vortex detection algorithms, usually based on linear-algebraic properties of the velocity gradient tensor, can be plagued with severe shortcomings and may become, in practical terms, dependent on the choice of subjective threshold parameters in their implementations. In two-dimensions, a large class of standard vortex identification prescriptions turn out to be equivalent to the “swirling strength criterion” (λci-criterion), which is critically revisited in this work. We classify the instances where the accuracy of the λci-criterion is affected by nonlinear superposition effects and propose an alternative vortex detection scheme based on the local curvature properties of the vorticity graph (x,y,ω)—the “vorticity curvature criterion” (λω-criter...

/pdf/vortex-identification-from-local-properties-of-the-vorticity-2g7s4568gf.pdf

Vortex identification from local properties of the vorticity field

We develop a stochastic model for the velocity gradient dynamics along a Lagrangian trajectory in isotropic and homogeneous turbulent flows. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the recent fluid deformation (RFD) approximation that was shown to give a realistic picture of small-scale statistics of turbulence only up to moderate Reynolds numbers. As a second step, we constrain the dynamics, in the spirit of Girimaji & Pope (Phys. Fluids A, vol. 2, 1990, p. 242), in order to impose a peculiar statistical structure to the dissipation seen by the Lagrangian particle. This probabilistic closure uses as a building block a random field that fulfils the statistical description of the intermittency, i.e. multifractal, phenomenon. To do so, we define and generalize to a statistically stationary framework a proposition made by Schmitt (Eur. Phys. J. B, vol. 34, 2003, p. 85). These considerations lead us to propose a nonlinear and non-Markovian closed dynamics for the elements of the velocity gradient tensor. We numerically integrate this dynamics and observe that a stationary regime is indeed reached, in which (i) the gradient variance is proportional to the Reynolds number, (ii) gradients are typically correlated over the (small) Kolmogorov time scale and gradient norms over the (large) integral time scale, (iii) the joint probability distribution function of the two non-vanishing invariants and reproduces the characteristic teardrop shape, (iv) vorticity becomes preferentially aligned with the intermediate eigendirection of the deformation tensor and (v) gradients are strongly non-Gaussian and intermittent, a behaviour that we quantify by appropriate high-order moments. Additionally, we examine the problem of rotation rate statistics of (axisymmetric) anisotropic particles as observed in direct numerical simulations. Although our realistic picture of velocity gradient fluctuations leads to better results when compared to the former RFD approximation, it is still unable to provide an accurate description for the rotation rate variance of oblate spheroids.

/pdf/a-multifractal-model-for-the-velocity-gradient-dynamics-in-4s4vwh69n3.pdf

A multifractal model for the velocity gradient dynamics in turbulent flows

We develop a stochastic model for the velocity gradients dynamics along a Lagrangian trajectory Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the Recent Fluid Deformation (RFD) approximation that was shown to give a realistic picture of small scales statistics of turbulence only up to moderate Reynolds numbers As a second step, we constrain the dynamics, in the spirit of Girimaji & Pope (1990), in order to impose a peculiar statistical structure to the dissipation seen by the Lagrangian particle This probabilistic closure uses as a building block a random field that fulfils the statistical description of the intermittency, ie multifractal, phenomenon To do so, we define and generalize to a statistically stationary framework a proposition made by Schmitt (2003) These considerations lead us to propose a non-linear and non-Markovian closed dynamics for the elements of the velocity gradient tensor We numerically integrate this dynamics and observe that a stationary regime is indeed reached, in which (i) the gradients variance is proportional to the Reynolds number, (ii) gradients are typically correlated over the (small) Kolmogorov time scale and gradients norms over the (large) integral time scale (iii) the joint probability distribution function of the two non vanishing invariants $Q$ and $R$ reproduces the characteristic teardrop shape, (iv) vorticity gets preferentially aligned with the intermediate eigendirection of the deformation tensor and (v) gradients are strongly non-Gaussian and intermittent

/pdf/a-multifractal-model-for-the-velocity-gradients-dynamics-in-5i8re1p3t0.pdf

A multifractal model for the velocity gradients dynamics in turbulent flows

The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

An elementary derivation of the harmonic oscillator propagator

It has been broadly acknowledged that vortex detection algorithms, usually based on linear-algebraic properties of the velocity gradient tensor, can be plagued with severe shortcomings and may become, in practical terms, dependent on the choice of subjective threshold parameters in their implementations. In two-dimensions, a large class of standard vortex identification prescriptions turn out to be equivalent to the "swirling strength criterion" ($\lambda_{ci}$-criterion), which is critically revisited in this work. We classify the instances where the accuracy of the $\lambda_{ci}$-criterion is affected by nonlinear superposition effects and propose an alternative vortex detection scheme based on the local curvature properties of the vorticity graph $(x,y,\omega)$ -- the "vorticity curvature criterion" ($\lambda_\omega$-criterion) -- which improves over the results obtained with the $\lambda_{ci}$-criterion in controlled Monte-Carlo tests. We show that the $\lambda_\omega$-criterion is able to cope with strong shear effects, if a subtraction of the mean velocity field background is performed, in the spirit of the Reynolds decomposition procedure. A realistic comparative study for vortex identification is then carried out for a direct numerical simulation (DNS) of a turbulent channel flow, including a three-dimensional extension of the $\lambda_\omega$-criterion. In contrast to the $\lambda_{ci}$-criterion, the $\lambda_\omega$-criterion indicates in a consistent way the existence of small scale isotropic turbulent fluctuations in the logarithmic layer, in consonance with long-standing assumptions commonly taken in turbulent boundary layer phenomenology.

Luca Moriconi

Papers

Vortex identification from local properties of the vorticity field

A multifractal model for the velocity gradient dynamics in turbulent flows

A multifractal model for the velocity gradients dynamics in turbulent flows

An elementary derivation of the harmonic oscillator propagator

Vortex Identification from Local Properties of the Vorticity Field