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Turbulence
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.
Questions? Contact us!
This is from "The Legacy and Future of CFD at Los Alamos"
(LAUR#LA-UR-1426)(365Kb pdf file)
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