UCRL-PROP-220113

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**Turbulent Mixing and Hydrodynamics**

February 2006

**Abstract**

The design of efficient, high-gain capsules for inertial confinement
fusion (ICF)^1^, the modeling of supernova implosions and explosions,^2^
and the modeling of shock-induced mixing of multi-phase reactive
energetic materials requires a detailed understanding of the
consequences of material interpenetration, hydrodynamic instabilities
and mixing at molecular (or atomic) scales arising from initial
perturbations at material interfaces, *i.e.*, the Rayleigh-Taylor,
Richtmyer-Meshkov and Kelvin-Helmholtz instabilities (buoyancy-, shock-
and shear-induced instabilities, respectively). From a computational
point of view, this requires the development of models for hydrodynamic
instability growth from initial perturbations through the weakly- and
strongly-nonlinear phases, and finally, to the late-time turbulent
regime. In particular, modeling these processes completely and
accurately is critical for demonstrating the feasibility and potential
success of contemporary ICF capsule designs. In applications to
energetic materials, turbulent mixing of multi-phase mixtures is a key
process in anaerobic and aerobic combustion that can support shock
formation and propagation. A predictive computational capability for the
effects of turbulent mass, momentum, energy and species transport, as
well as material mixing, on the thermonuclear fusion process in ICF
entails the development of turbulent transport and mixing or
subgrid-scale models based on statistically-averaged or filtered
evolution equations, respectively. The former models are typically
referred to as Reynolds-averaged Navier-Stokes (RANS) (and related)
models^3^ and the latter are referred to as large-eddy simulation (LES)
models.^4^ The strong nonlinearity of the equations describing the
hydrodynamics, thermodynamics, material properties and other multi-scale
phenomena, together with the formal ensemble averaging or filtering
procedure, introduce correlations of strongly-fluctuating fields and
other *a priori* unclosed quantities that must be explicitly modeled to
close the set of equations describing the implosion dynamics and burning
of an ICF capsule or the combustion of reactive materials.

**Overview**

The Advanced Simulation and Computing (ASC) Program is pursuing a
multi-disciplinary research effort to develop and quantitatively
validate turbulent transport and mixing models using numerical
simulations, theoretical modeling and experimental data. Specifically,
it is of interest to develop state-of-the-art theoretical and numerical
approaches for modeling the hydrodynamics and turbulent mixing generated
by hydrodynamic instabilities induced by initial interfacial
perturbations. A related objective is to establish the infrastructure
needed to extract model quantities and to visualize data from
large-scale direct numerical simulations (DNS)^5^ and from large-eddy or
monotone-integrated (or implicit) large-eddy simulations (MILES or
ILES).^6^ Statistical information could also be extracted and applied to
probability density function (PDF) or filtered density function (FDF)
approaches to turbulence modeling. Numerical simulations in support of
model development and *a priori* or *a posteriori* validation consider
the spatio-temporal evolution of turbulent mixing^7^ induced by
hydrodynamic interfacial instabilities (and possibly other types of
compressible single- or multi-phase flows) on parallel computing
platforms. Such data can be used to address the proper modeling of
initial conditions in averaged descriptions of turbulent mixing, and to
examine in detail the transition from the linear to the nonlinear and to
the turbulent growth regimes. Hence, a goal is to *address the physical
and mathematical formulation of turbulence models*, with an emphasis on
the principal unclosed quantities that must be modeled using gradient
diffusion and other potentially more sophisticated and
physically-complete closures (*e.g.*, second-moment closures^8^). For
multi-phase flows, this requires including the effects of
compressibility of all phases. Higher order closure models provide
variances that are directly related to mixing properties, and which are
not available from first-order closure models. Another goal is to
*analyze the intrinsic properties of numerical algorithms used to solve
systems of equations containing either embedded turbulent transport and
mixing models or explicit/implicit subgrid-scale models*. It is highly
desirable to relate the work above to experimental data, when available.

Presently, eddy viscosity modeling methods, LES/MILES/ILES methods,
PDF/FDF and kinetic (particle) models are the methods of choice to model
single- and multi-phase turbulent flows. However, the complex nonlinear
physics of turbulence, as well as the proper mathematical and numerical
(discrete) representation of that physics, continue to be open research
areas. The ASC program is interested in research in these areas that can
be applied to the computational modeling of turbulence.

## Turbulent Mixing

The ASC Program requires models describing hydrodynamics and turbulence
over an extensive range of spatio-temporal scales, including the
development of hydrodynamic interfacial instabilities (*i.e.*, the
Rayleigh-Taylor, Richtmyer-Meshkov and Kelvin-Helmholtz instabilities)
from small-amplitude linear growth to weakly- and strongly-nonlinear
growth, and finally to the late-time fully-developed turbulent regime.
This includes the development of DNS capability and of subgrid-scale
models for LES of turbulent mixing. The models should provide detailed
information regarding concentration profiles, and the distribution of,
and correlations among, the sizes and shapes of regions of unmixed or
poorly mixed fluid. It is also necessary to develop improved
constitutive closure models for the Reynolds stresses and other unclosed
quantities for use in RANS or PDF/FDF models, possibly using data from
DNS, LES and MILES/ILES. Research into alternatives to conventional
subgrid-scale models, such as models based on filters and sub-filters,
is also beneficial. Conceptual issues such as explicit filtering in
complex geometries and on unstructured grids for inhomogeneous
turbulence should also be addressed.

The simulations should attain very large Reynolds numbers characteristic
of fully-developed turbulence and mixing. DNS resolves all of the
spatial and temporal scales in the flow, whereas LES resolves only the
largest scales while modeling the averaged effects of the unresolved
(smaller) scales on the resolved scales. DNS is challenging, as
turbulent mixing layers generated by interfacial instabilities can grow
from very small-scale perturbations and a quiescent state, implying a
very wide range of spatial scales. Note that conventional DNS of
shock-driven flows is not possible, due to the inability to fully
resolve shocks. LES is also challenging, as there is a need to model
both the subgrid-scale terms in the large-eddy equations (the explicitly
or implicitly filtered conservation equations) and the unresolved
initial conditions. MILES and ILES rely instead on implicit
subgrid-scale models inherent in the numerical discretization of the
non-dissipative fluid dynamics equations, and do not require explicit
subgrid-scale models. However, such simulations do not exhibit pointwise
convergence and may be ill-posed in multiple dimensions; the small
scales are determined by the regularizing numerical dissipation, so that
the mixing properties depend on the spatial resolution and on other
properties of the algorithm. Additional research is needed in assessing
the applicability of different MILES and ILES methods for simulating
turbulent mixing.^9^

Phenomenological or rigorous mathematical models are needed for the
dependence of turbulent mixing properties on such conditions as the
initial material density ratio at a sharp interface, the direction and
history of the acceleration, the compressibility of the fluid, and the
initial values of symmetry-breaking perturbations (such as initial
density and velocity fluctuations, and surface roughness). These models
are needed in forms suitable for numerical discretization, and it is
particularly desirable that they be implemented and evaluated as RANS or
subgrid-scale models within much more coarsely-zoned simulations than
required to perform DNS. The models must be consistently coupled to
their respective mean flow equations driving the flow.

High-fidelity turbulence modeling is of interest to predictively model
unsteady, highly-nonequilibrium, inhomogeneous, anisotropic,
variable-density and fully-compressible multi-phase turbulence. The
transition of unstable flows to turbulence and the dependence of the
mixing layer properties on the initial conditions are of interest. The
development of models and codes that can incorporate the closed
equations governing the Reynolds stresses and other second-order
quantities, and which predict the dynamics and statistical properties of
evolving hydrodynamic instabilities are sought.

Variable-density and compressibility effects^10^ are important in all of
the applications mentioned above. In particular, it is important to
understand the effects of shocks of various strengths on turbulence and
on turbulent mixing layers, including the effects on coherent structures
and on the structures resulting from the evolution of the instabilities
through baroclinic effects within the mixing layer. Models must
incorporate buoyancy, shock interactions and backscatter effects (energy
transfer from small to larger scales due to nonlinear interactions).

Turbulent combustion,^11^ other exothermic chemical reactions and
thermonuclear reactions must be incorporated and understood in the
presence of all of the aforementioned phenomena. The modeling of
combustion and related processes is particularly challenging, as these
phenomena occur at molecular and atomic scales, which are unresolved on
any possible spatio-temporal computational grid. Consequently,
probabilistic descriptions^12^ have traditionally been used to model the
average effects in reacting flows. The understanding of combustion and
related phenomena is most useful if it spans a large range of ratios of
chemical energy release to the rates of other energetic processes, such
as the conversion of instability energy into turbulence and the
conversion of turbulent kinetic energy into internal energy. Note that
MILES and ILES methods are problematic for simulating combustion, as
their inherent dissipation is likely to be insufficient to adequately
model the subgrid-scale dynamics.^13^

Whenever possible, numerical models of all of these processes should be
tested and validated against high-resolution DNS or experimental data.

## Hydrodynamics

The ASC Program requires the solution of a broad range of hydrodynamics
problems, ranging from low-speed subsonic flows to shocked supersonic
flows. The physical and mechanical phenomena of interest include
turbulence and turbulent mixing of single- phase and multi-phase
materials with full compressibility and material strength,
high-displacement and high-strain rate flows, and highly distorted
hydrodynamic flows with shocks. Three-dimensional multi-scale methods
capable of modeling such phenomena over a broad range of length- and
timescales and relevant dimensionless parameters are needed. Some
problems of interest require the ability to track shock fronts. As the
flow domains of interest can be geometrically complex, unstructured grid
algorithms compatible with high-aspect ratio zoning are highly
desirable.

A variety of methods have previously been used to model flows of
interest, including finite-difference, finite-volume, finite-element and
particle techniques. These methods can be purely Eulerian, purely
Lagrangian or hybrid. Examples of hybrid methods are adaptive mesh
refinement (AMR), which can model shock propagation on an Eulerian grid,
and arbitrary Lagrangian-Eulerian (ALE) methods. While single-fluid
flows are of interest, simulations of multi-fluid/multi-phase flows are
particularly relevant.

Needed algorithm research includes: (1) determining the relative
strengths, weaknesses, and domains of applicability of a wide variety of
hydrodynamics algorithms (such an effort might develop test problems,
some with analytic or semi-analytic solutions that can be used to
evaluate algorithms); (2) evaluation of the behavior of these algorithms
in codes modeling hydrodynamic flows and turbulence, transport,
interphase interactions, material strength and other effects, and; (3)
development of new computational fluid dynamics algorithms to improve
upon deficient algorithms. For example, numerical analysis in support of
turbulence and subgrid-scale model development and assessment could
quantitatively analyze simulation data and apply analytical techniques
(*e.g.*, modified equation or truncation error analysis) to estimate the
effects of the numerical errors and other components of an algorithm on
the properties of the solutions, *i.e.*, intrinsic numerical
dissipation. Such research is essential, as the numerical algorithms
used in either RANS or LES models cannot be developed independently of
the models, *e.g.,* it is important to understand the competition
between the inherent numerical dissipation in the algorithm and the
dissipation contributed by the turbulence model. Issues such as
estimating the errors associated with a subgrid-scale model near the
Nyquist scale imposed by the grid resolution and the relationship
between the grid and filter scales should be addressed for the flows of
interest.

It is essential to compare simulation data with a comprehensive set of
hydrodynamic experiments which complement the algorithm development and
evaluation. Comparisons to experimental data characterizing mixing
morphologies and statistical properties of turbulent mixing would be
particularly valuable.

**Points of Contact**

Oleg Schilling, *Lawrence Livermore National Laboratory*,
schilling1@llnl.gov

Michael J. Steinkamp, *Los Alamos National Laboratory*, steinmj@lanl.gov

Melvin R. Baer, *Sandia National Laboratory*, <mrbaer@sandia.gov>

**Acknowledgement**

This work was, in part, performed by the Lawrence Livermore National
Laboratory, Los Alamos National Laboratory, and Sandia National
Laboratories under auspices of the U.S. Department of Energy.

**References**

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