Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

PDF methods for turbulent reactive flows

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

▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

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DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

▪ Abstract Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

Scale-Invariance and Turbulence Models for Large-Eddy Simulation

A methodology termed the “filtered density function” (FDF) is developed and implemented for large eddy simulation (LES) of chemically reacting turbulent flows. In this methodology, the effects of the unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities. A transport equation is derived for the FDF in which the effect of chemical reactions appears in a closed form. The influences of scalar mixing and convection within the subgrid are modeled. The FDF transport equation is solved numerically via a Lagrangian Monte Carlo scheme in which the solutions of the equivalent stochastic differential equations (SDEs) are obtained. These solutions preserve the Ito-Gikhman nature of the SDEs. The consistency of the FDF approach, the convergence of its Monte Carlo solution and the performance of the closures employed in the FDF transport equation are assessed by comparisons with results obtained by direct numerical simulation ...

Filtered density function for large eddy simulation of turbulent reacting flows

A methodology termed the ‘filtered mass density function’ (FMDF) is developed and implemented for large-eddy simulation (LES) of variable-density chemically reacting turbulent flows at low Mach numbers. This methodology is based on the extension of the ‘filtered density function’ (FDF) scheme recently proposed by Colucci et al. (1998) for LES of constant-density reacting flows. The FMDF represents the joint probability density function of the subgrid-scale (SGS) scalar quantities and is obtained by solution of its modelled transport equation. In this equation, the effect of chemical reactions appears in a closed form and the influences of SGS mixing and convection are modelled. The stochastic differential equations (SDEs) which yield statistically equivalent results to those of the FMDF transport equation are derived and are solved via a Lagrangian Monte Carlo scheme. The consistency, convergence, and accuracy of the FMDF and the Monte Carlo solution of its equivalent SDEs are assessed. In non-reacting flows, it is shown that the filtered results via the FMDF agree well with those obtained by the ‘conventional’ LES in which the finite difference solution of the transport equations of these filtered quantities is obtained. The advantage of the FMDF is demonstrated in LES of reacting shear flows with non-premixed reactants. The FMDF results are appraised by comparisons with data generated by direct numerical simulation (DNS) and with experimental measurements. In the absence of a closure for the SGS scalar correlations, the results based on the conventional LES are significantly different from those obtained by DNS. The FMDF results show a closer agreement with DNS. These results also agree favourably with laboratory data of exothermic reacting turbulent shear flows, and portray several of the features observed experimentally.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112099006643

Filtered mass density function for large-eddy simulation of turbulent reacting flows

Model-free simulations of turbulent reactive flows

A methodology termed the “velocity filtered density function” (VFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function of all of the components of the velocity vector. An exact transport equation is derived for the VFDF in which the effects of the SGS convection appear in closed form. The unclosed terms in this transport equation are modeled. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Ito–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by an Eulerian LES procedure in which the corresponding transport equation...

Velocity filtered density function for large eddy simulation of turbulent flows

: This research was concentrated primarily on developments and applications of the filtered density function (FDF) for subgrid scale (SGS) modeling of turbulent reacting flows. During the past three years, this work addressed: (1) development of the joint velocity-scalar filtered mass density function (VSFMDF), (2) development of the joint frequency-velocity-scalar filtered mass density function (FVS-FMDF), and (3) implementation of the scalar filtered mass density function (SFMDF) and VSFMDF for large eddy simulation of complex turbulent flames. This is a final report of our activities sponsored by AFOSR under Grant FA9550-06-1-0015.

Peyman Givi

Papers

Filtered density function for large eddy simulation of turbulent reacting flows

Filtered mass density function for large-eddy simulation of turbulent reacting flows

Model-free simulations of turbulent reactive flows

Velocity filtered density function for large eddy simulation of turbulent flows

Filtered Density Function for Subgrid Scale Modeling of Turbulent Combustion