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The first efforts in "turbulence" modeling directed towards a practical closure date back to Chou [Chou, 1940] in the early 1940's and to Rotta [Rotta, 1951] in the early 1950's. These early attempts at modeling typically involved a transport equation for the turbulent kinetic energy, and met with limited success. It seemed that something was missing: a time-scale or a length-scale. One of the most successful "recipes" for producing the missing item was the introduction of the turbulent kinetic energy dissipation rate equation ("e"), by Harlow and Nakayama in the late 1960's [Harlow et al, 1967][Harlow et al, 1968]. Rather than attempt to derive the e-transport equation, Harlow and Nakayama produced it from the turbulent kinetic energy equation and dimensional considerations alone, which lead to the still-popular, and useful, k-e and Rij-e [Daly et al, 1970] family of closures.

During this time period there were also attempts to develop theoretical models of turbulence, such as the Quasi-Normal models initiated by Millionshtchikov [Milllionshtchikov, 1993] in the 1940's and the Direct Interaction Approximation of Kraichnan [Kraichnan, 1953] in the 1950's. These models differ from the "practical" closures in some profound ways. The "practical" closures to this day are based on the joint probabilities of the fluctuating fluid quantities at a single point in space and time. The more fundamental theories typically consider the joint probabilities at two points, and in some cases (e.g., DIA) at two times.

Although the single point models provided a tractable set of equations, they sacrificed significant physical fidelity. But, the more fundamental closures are nearly intractable, unless severe restrictions are made (i.e., isotropy, or homogeneity), and then cast in terms of Fourier series. What do the more fundamental theories offer? The two-point models do not require a restrictive coupling of length-scales and time-scales, a type of statistical self-similarity necessary to characterize the multiscale problem by, say, two-scales, k and e. While it will probably be some time before computers are capable of solving the fundamental theories for practical problems, they provide considerable guidance in the assumptions inherent in the derivation of single-point equations, and also serve as a useful bridge between direct numerical simulations and single-point closures.

Current Research in T-3
Much of the work on turbulence in T-3 involves exploiting the more fundamental two-point turbulence modeling approach to derive "enhanced" engineering models. Thus, the effort may be considered as categorized into three subareas.

Derivation and validation of two-point models
A recent development in T-3 by Besnard, Harlow, Rauenzahn and Zemach [14] has been a tractable spectral (two-point) closure by borrowing ideas from both single-point and two-point models. This model does not require an e equation and thus does not invoke many of the self-similarity constraints implicit in the one-point closures. The model has served as the basis of a great deal of the spectral modeling work and practical simulations of turbulent mixing in multiple material problems. Extensions have been made to variable-density turbulence [Clark et al, 1995] and inhomogeneous variable-density turbulence (e.g., Rayleigh-Taylor mixing) [Steinkamp et al, 1995].
More recently, attempts are being made in T-3, most notably by Turner, to construct an Eddy-Damped Quasi-Normal Markovian (EDQNM) model for inhomogeneous flows which does not invoke any assumptions of local homogeneity or local isotropy, and thus fundamentally differs from the work of the French researchers at L'Ecole Centrale de Lyon. The effort follows closely our direct numerical simulations using pseudo-spectral algorithms of inhomogeneous turbulence. Figure 3.6-1 shows results from the EDQNM model. Our research is directed toward understanding the degree to which the EDQNM class of models can represent the strongly intermittent zones at the edges of the turbulent zones, as well as understanding how well "gradient-diffusion" models of the one-point variety can represent the spreading of the turbulence. The work illustrates the relatively strong departures of the statistics from a near-Gaussian distribution and the importance of the action of the triple-velocity correlations on the distribution of the turbulence, and thus highlights the need, and challenge, of higher order turbulence closures.

Direct numerical simulation (DNS) and two-point closures are also being used to investigate the nature of the correlations of the fluctuating pressure-velocity and fluctuating pressure-strain. It is found that the "local" representation in differential form (rather than integral over the field) of these terms as used in most single-point closures may lead to errors at least as large as those due to truncating the hierarchy of moment equations in the "classical" closure problem. This is also true of spectral models which reduce the vector-k-space spectral equations to a scalar k-space. By understanding the nature and extent of this "nonlocal" phenomenon, we hope to derive useable approximations that capture this feature of the pressure correlations.

Fig. 3.6-1. Results of the EDQNM model for inhomogeneous turbulence. Shown is the turbulent kinetic energy (red-high, purple-low) for the self-propagating or "diffusing" turbulence from an initial localized turbulence in the center of a channel.

Examining the two-point closures for emergent scalings and self-similarities. One of the more interesting features observed in the spectral model of Besnard et al. [14] is the emergence of self-similar spectra for turbulence undergoing homogeneous mean shears and strains and during free decay of homogeneous anisotropic turbulence at very high Reynolds numbers [Clark, 1992] [Clark et al, 1995]. The emergence of self-similar spectra has also been observed for the case of a spectral model applied to Rayleigh-Taylor mixing by Steinkamp et al. [91]. The emergence of these self-similar spectral forms indicates that the turbulence model results (and, one hopes, the turbulence itself) can be described by far fewer degrees of freedom than required in the two-point description. The degree to which the spectral models describe actual turbulence is judged by comparisons with direct numerical simulations at low Reynolds numbers and, to whatever degree possible, by comparison to actual experiments.

One-Point "Engineering" Models
The emergence of the self-similar forms suggests that the spectral models may provide a useful tool to judge the applicability of the single-point closures for a given class of flows. If the emergent self-similar form is reasonably simple, or "simplifiable", they can be inserted into the two-point model equations, and a one-point model can be derived by the construction of appropriate integral moments. Besnard et al. have shown that if one chooses to represent the spectra as a particular self-similar form, one can then directly derive a k-e model by constructing appropriate moments of the spectra. If the spectra from the model produce a different self-similar form in different circumstances, then a new set of moment equations may produce an improved k-e model. An example of this has been demonstrated by Clark [Clark, 1992] and Clark and Zemach [Clark et al, 1992].

Current Applications
The turbulence transport models developed in T-3 are being applied to examples at all flow speeds from far subsonic (incompressible) to supersonic, and with various combinations of interpenetrating fluids or clouds of droplets or grains. Some of these examples are: the fuel-air interaction in an internal combustion engine, turbulent flame behavior, unstable deformation of inertial confinement fusion capsules, nozzle flows with aerodynamic applications, two-phase flow of catalytic particles and petroleum in an industrial cracker, research problems for extended model development (free shears and mixing layers), and fluidized beds.

Los Alamos Turbulence Projects
All of the filled circles are linkable projects.
All of the empty circles are navigational guides.

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This is from "The Legacy and Future of CFD at Los Alamos" (LAUR#LA-UR-1426)(365Kb pdf file)






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