## 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

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acceleration according adopting appearance Appendix applied approach approximation assumed assumption average basic calculation chapter characteristic closure coefficients combined compared compressibility consideration considered consistent constant construction contributions correlations corresponding defined definition density dependence derived described determined diffusion discussion dissipation dynamics eddies effects explanations expression filter flows fluctuations fluid Fokker-Planck equation force function given implies initial instance integration introduced involve latter leads limit mass means measurements methods mixing molecular moments neglect obtained parameters particle performance pointed Pope positive predictions presented problem processes properties provides quantities question RANS refers regard relation relevant represents requires reveals Reynolds number right-hand side scalar scale seen shear shown shows simulations solution specific standardized statistics stochastic differential equations stochastic model stress tensor structure temperature transport equation turbulent values vanish variables variance variations velocity

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