74 research outputs found
Exploring the beta distribution in variable-density turbulent mixing
In assumed probability density function (pdf) methods of turbulent
combustion, the shape of the scalar pdf is assumed a priori and the pdf is
parametrized by its moments for which model equations are solved. In
non-premixed flows the beta distribution has been a convenient choice to
represent the mixture fraction in binary mixtures or a progress variable in
combustion. Here the beta-pdf approach is extended to variable-density mixing:
mixing between materials that have very large density differences and thus the
scalar fields are active. As a consequence, new mixing phenomena arise due to
1) cubic non-linearities in the Navier-Stokes equation, 2) additional
non-linearities in the molecular diffusion terms and 3) the appearance of the
specific volume as a dynamical variable. The assumed beta-pdf approach is
extended to transported pdf methods by giving the associated stochastic
differential equation (SDE). The beta distribution is shown to be a realizable,
consistent and sufficiently general representation of the marginal pdf of the
fluid density, an active scalar, in non-premixed variable-density turbulent
mixing. The moment equations derived from mass conservation are compared to the
moment equations derived from the governing SDE. This yields a series of
relations between the non-stationary coefficients of the SDE and the mixing
physics. Our treatment of this problem is general: the mixing is mathematically
represented by the divergence of the velocity field which can only be specified
once the problem is defined. In this paper we seek to describe a theoretical
framework to subsequent applications. We report and document several rigorous
mathematical results, necessary for forthcoming work that deals with the
applications of the current results to model specification, computation and
validation of binary mixing of inert fluids.Comment: Added two paragraphs to Introduction + minor changes, Accepted in
Journal of Turbulence, July 19, 201
Extending the Langevin model to variable-density pressure-gradient-driven turbulence
We extend the generalized Langevin model, originally developed for the
Lagrangian fluid particle velocity in constant-density shear-driven turbulence,
to variable-density (VD) pressure-gradient-driven flows. VD effects due to
non-uniform mass concentrations (e.g. mixing of different species) are
considered. In the extended model large density fluctuations leading to large
differential fluid accelerations are accounted for. This is an essential
ingredient to represent the strong coupling between the density and velocity
fields in VD hydrodynamics driven by active scalar mixing. The small scale
anisotropy, a fundamentally "non-Kolmogorovian" feature of
pressure-gradient-driven flows, is captured by a tensorial stochastic diffusion
term. The extension is so constructed that it reduces to the original Langevin
model in the limit of constant density. We show that coupling a Lagrangian
mass-density particle model to the proposed extended velocity equation results
in a statistical representation of VD turbulence that has important benefits.
Namely, the effects of the mass flux and the specific volume, both essential in
the prediction of VD flows, are retained in closed form and require no explicit
closure assumptions. The paper seeks to describe a theoretical framework
necessary for subsequent applications. We derive the rigorous mathematical
consequences of assuming a particular functional form of the stochastic
momentum equation coupled to the stochastic density field in VD flows. A
previous article discussed VD mixing and developed a stochastic Lagrangian
model equation for the mass-density. Second in the series, this article
develops the momentum equation for VD hydrodynamics. A third, forthcoming paper
will combine these ideas on mixing and hydrodynamics into a comprehensive
framework: it will specify a model for the coupled problem and validate it by
computing a Rayleigh-Taylor flow.Comment: Accepted in Journal of Turbulence, Jan 7, 201
A non-hybrid method for the PDF equations of turbulent flows on unstructured grids
In probability density function (PDF) methods of turbulent flows, the joint
PDF of several flow variables is computed by numerically integrating a system
of stochastic differential equations for Lagrangian particles. A set of
parallel algorithms is proposed to provide an efficient solution of the PDF
transport equation, modeling the joint PDF of turbulent velocity, frequency and
concentration of a passive scalar in geometrically complex configurations. An
unstructured Eulerian grid is employed to extract Eulerian statistics, to solve
for quantities represented at fixed locations of the domain (e.g. the mean
pressure) and to track particles. All three aspects regarding the grid make use
of the finite element method (FEM) employing the simplest linear FEM shape
functions. To model the small-scale mixing of the transported scalar, the
interaction by exchange with the conditional mean model is adopted. An adaptive
algorithm that computes the velocity-conditioned scalar mean is proposed that
homogenizes the statistical error over the sample space with no assumption on
the shape of the underlying velocity PDF. Compared to other hybrid
particle-in-cell approaches for the PDF equations, the current methodology is
consistent without the need for consistency conditions. The algorithm is tested
by computing the dispersion of passive scalars released from concentrated
sources in two different turbulent flows: the fully developed turbulent channel
flow and a street canyon (or cavity) flow. Algorithmic details on estimating
conditional and unconditional statistics, particle tracking and particle-number
control are presented in detail. Relevant aspects of performance and
parallelism on cache-based shared memory machines are discussed.Comment: Accepted in Journal of Computational Physics, Feb. 20, 200
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