Turbulent closure models, second moment

In an attempt to circumvent some of these problems, considerable effort has been expended to develop so-called second moment turbulent closure models in which the governing equations are closed by including terms parameterizing various turbulent correlations (see, for example, Lewellen et al., 1974 Wyngaard and Cote, 1974 Lumley and Khajeh-Nouri, 1974 Mellor and Yamada, 1974 Yamada and Mellor, 1975 Zeman and Lumley, 1976, 1979 Zeman and Tennekes, 1977 Freeman, 1977 Yamada, 1977 Manton, 1979 Binkowski, 1979). While second-order closure models are conceptually very appealing, their use in atmospheric... 

Yamada, T. and Bunker S., 1988. Development of a nested grid, second moment turbulence-closure model and an application to the 1982 ASCOT Brush Creek data simulation, J. Atmos. Sci., Tl, pp. 562-578. 

Pope, S.B. 1994. On the relation between stochastic Lagrangian models of turbulence and second-moment closures. J. Physics Fluids 6(2) 973-85. 

The next category of turbulence closures, i.e., impl3ung to be more accurate than the very simple algebraic models, is a hierarchy of turbulent models based on the transport equation for the fluctuating momentum field. These are the first-order closure models, i.e., those that require parameterizations for the second moments and the second-order closure models, i.e., those that... 

The calculation of the six components of the Reynolds stress tensor, that is, six second-order moments of the micro-PDF, f v,yf), is reduced to the calculation of k and the modeling of the turbulent viscosity pf As seen from (12.5.1-2), is a function of a limited number of second-order moments of the micro-PDF. Turbulent viscosity based closure models for the Reynolds-stresses can be used at relatively low computational effort. In the two-equation model approach, the turbulent viscosity is expressed in terms of the turbulent kinetic energy, k, and the turbulence dissipation rate, s, according to ... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

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