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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 timescale or a lengthscale. 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 etransport equation, Harlow and Nakayama produced it from
the turbulent kinetic energy equation and dimensional considerations alone,
which lead to the stillpopular, and useful, ke and Rije [Daly
et al, 1970] family of closures.
During this time period there were also attempts to develop theoretical
models of turbulence, such as the QuasiNormal 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 twopoint models do not require
a restrictive coupling of lengthscales and timescales, a type of statistical
selfsimilarity necessary to characterize the multiscale problem by, say,
twoscales, 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 singlepoint equations, and also serve as a useful bridge
between direct numerical simulations and singlepoint closures.
Current Research in T3
Much of the work on turbulence in T3 involves exploiting
the more fundamental twopoint turbulence modeling approach to derive
"enhanced" engineering models. Thus, the effort may be considered
as categorized into three subareas.
Derivation and validation of twopoint models
A recent development in T3 by Besnard, Harlow, Rauenzahn
and Zemach [14] has been a tractable spectral (twopoint) closure by borrowing
ideas from both singlepoint and twopoint models. This model does not
require an e equation and thus does not invoke many of the selfsimilarity
constraints implicit in the onepoint 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 variabledensity turbulence [Clark et al,
1995] and inhomogeneous variabledensity turbulence (e.g., RayleighTaylor
mixing) [Steinkamp et al, 1995].
More recently, attempts are being made in T3, most notably by Turner,
to construct an EddyDamped QuasiNormal 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 pseudospectral algorithms of inhomogeneous turbulence.
Figure 3.61 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 "gradientdiffusion"
models of the onepoint variety can represent the spreading of the turbulence.
The work illustrates the relatively strong departures of the statistics
from a nearGaussian distribution and the importance of the action of
the triplevelocity correlations on the distribution of the turbulence,
and thus highlights the need, and challenge, of higher order turbulence
closures.
Direct numerical simulation (DNS) and twopoint closures are also being
used to investigate the nature of the correlations of the fluctuating
pressurevelocity and fluctuating pressurestrain. It is found that the
"local" representation in differential form (rather than integral
over the field) of these terms as used in most singlepoint 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 vectorkspace spectral
equations to a scalar kspace. 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. 3.61. Results of the EDQNM model for inhomogeneous turbulence. Shown
is the turbulent kinetic energy (redhigh, purplelow) for the selfpropagating
or "diffusing" turbulence from an initial localized turbulence
in the center of a channel.
Examining the twopoint closures for emergent scalings and selfsimilarities.
One of the more interesting features observed in the spectral model of
Besnard et al. [14] is the emergence of selfsimilar 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 selfsimilar
spectra has also been observed for the case of a spectral model applied
to RayleighTaylor mixing by Steinkamp et al. [91]. The emergence of these
selfsimilar 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 twopoint 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.
OnePoint "Engineering" Models
The emergence of the selfsimilar forms suggests that the
spectral models may provide a useful tool to judge the applicability of
the singlepoint closures for a given class of flows. If the emergent
selfsimilar form is reasonably simple, or "simplifiable", they
can be inserted into the twopoint model equations, and a onepoint 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
selfsimilar form, one can then directly derive a ke model by constructing
appropriate moments of the spectra. If the spectra from the model produce
a different selfsimilar form in different circumstances, then a new set
of moment equations may produce an improved ke 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 T3 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 fuelair
interaction in an internal combustion engine, turbulent flame behavior,
unstable deformation of inertial confinement fusion capsules, nozzle flows
with aerodynamic applications, twophase 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#LAUR1426)(365Kb pdf file)


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