## Statistical Mechanics of Turbulent FlowsThe simulation of technological and environmental flows is very important for many industrial developments. A major challenge related to their modeling is to involve the characteristic turbulence that appears in most of these flows. The traditional way to tackle this question is to use deterministic equations where the effects of turbulence are directly parametrized, i. e. , assumed as functions of the variables considered. However, this approach often becomes problematic, in particular if reacting flows have to be simulated. In many cases, it turns out that appropriate approximations for the closure of deterministic equations are simply unavailable. The alternative to the traditional way of modeling turbulence is to construct stochastic models which explain the random nature of turbulence. The application of such models is very attractive: one can overcome the closure problems that are inherent to deterministic methods on the basis of relatively simple and physically consistent models. Thus, from a general point of view, the use of stochastic methods for turbulence simulations seems to be the optimal way to solve most of the problems related to industrial flow simulations. However, it turns out that this is not as simple as it looks at first glance. The first question concerns the numerical solution of stochastic equations for flows of environmental and technological interest. To calculate industrial flows, 3 one often has to consider a number of grid cells that is of the order of 100 . |

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### Contents

Introduction | 1 |

Stochastic variables | 9 |

Theta and delta functions | 29 |

The dynamics of relevant variables | 49 |

The equations of fluid and thermodynamics | 57 |

Second and higherorder RANS equations | 87 |

Consistent turbulence models | 141 |

Stochastic models for smallscale turbulence 153 | 152 |

The unification of turbulence models | 181 |

References | 189 |

201 | |

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¾½ acceleration according adopting Appendix applied approximation assumption asymptotic basic equations calculation chapter closure consideration correlations corresponding defined delta function derived deviatoric Direct Numerical Simulation dissipation DNS data dynamics of relevant eddies expression flow considered fluid dynamic flux Fokker-Planck equation fourth-order Gaussian gradients implies incompressible flow integration kinetic energy length scale Mach number mass fractions mean velocities molecular obtained order of approximation parameters particle PDF F PDF methods Pope problem provides RANS model reacting flows regard relation relevant variables represents rewrite Reynolds number Reynolds stresses right-hand side sample space scalar FDF SGS stress tensor shear flow simulations SML PDF solution stochastic equations stochastic force stochastic model stochastic processes stochastic variables temperature transport equation turbulence models turbulent flows turbulent kinetic energy values vanish vector velocities and scalars velocity fluctuations velocity model velocity-scalar