43 research outputs found
Discontinuous Galerkin method for multifluid Euler equations
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106454/1/AIAA2013-2595.pd
Positivity-preserving discontinuous spectral element methods for compressible multi-species flows
We introduce a novel positivity-preserving, parameter-free numerical
stabilisation approach for high-order discontinuous spectral element
approximations of compressible multi-species flows. The underlying
stabilisation method is the adaptive entropy filtering approach (Dzanic and
Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative
formulation of the multi-species flow equations. We show that the
straightforward enforcement of entropy constraints in the filter yields poor
results around species interfaces and propose an adaptive, parameter-free
switch for the entropy bounds based on the convergence properties of the
pressure field which drastically improves its performance for multi-species
flows. The proposed approach is shown in a variety of numerical experiments
applied to the multi-species Euler and Navier--Stokes equations computed on
unstructured grids, ranging from shock-fluid interaction problems to
three-dimensional viscous flow instabilities. We demonstrate that the approach
can retain the high-order accuracy of the underlying numerical scheme even at
smooth extrema, ensure the positivity of the species density and pressure in
the vicinity of shocks and contact discontinuities, and accurately predict
small-scale flow features with minimal numerical dissipation.Comment: Submitted for revie
Evolution equations in physical chemistry
textWe analyze a number of systems of evolution equations that arise in the study of physical chemistry. First we discuss the well-posedness of a system of mixing compressible barotropic multicomponent flows. We discuss the regularity of these variational solutions, their existence and uniqueness, and we analyze the emergence of a novel type of entropy that is derived for the system of equations.
Next we present a numerical scheme, in the form of a discontinuous Galerkin (DG) finite element method, to model this compressible barotropic multifluid. We find that the DG method provides stable and accurate solutions to our system, and that further, these solutions are energy consistent; which is to say that they satisfy the classical entropy of the system in addition to an additional integral inequality. We discuss the initial-boundary problem and the existence of weak entropy at the boundaries. Next we extend these results to include more complicated transport properties (i.e. mass diffusion), where exotic acoustic and chemical inlets are explicitly shown.
We continue by developing a mixed method discontinuous Galerkin finite element method to model quantum hydrodynamic fluids, which emerge in the study of chemical and molecular dynamics. These solutions are solved in the conservation form, or Eulerian frame, and show a notable scale invariance which makes them particularly attractive for high dimensional calculations.
Finally we implement a wide class of chemical reactors using an adapted discontinuous Galerkin finite element scheme, where reaction terms are analytically integrated locally in time. We show that these solutions, both in stationary and in flow reactors, show remarkable stability, accuracy and consistency.Chemistry and Biochemistr
Dynamic p-enrichment schemes for multicomponent reactive flows
We present a family of p-enrichment schemes. These schemes may be separated
into two basic classes: the first, called \emph{fixed tolerance schemes}, rely
on setting global scalar tolerances on the local regularity of the solution,
and the second, called \emph{dioristic schemes}, rely on time-evolving bounds
on the local variation in the solution. Each class of -enrichment scheme is
further divided into two basic types. The first type (the Type I schemes)
enrich along lines of maximal variation, striving to enhance stable solutions
in "areas of highest interest." The second type (the Type II schemes) enrich
along lines of maximal regularity in order to maximize the stability of the
enrichment process. Each of these schemes are tested over a pair of model
problems arising in coastal hydrology. The first is a contaminant transport
model, which addresses a declinature problem for a contaminant plume with
respect to a bay inlet setting. The second is a multicomponent chemically
reactive flow model of estuary eutrophication arising in the Gulf of Mexico.Comment: 29 pages, 7 figures, 3 table
Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations
This article concerns the development of a fully conservative,
positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for
simulating the multicomponent, chemically reacting, compressible Navier-Stokes
equations with complex thermodynamics. In particular, we extend to viscous
flows the fully conservative, positivity-preserving, and entropy-bounded
discontinuous Galerkin method for the chemically reacting Euler equations that
we previously introduced. An important component of the formulation is the
positivity-preserving Lax-Friedrichs-type viscous flux function devised by
Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to
multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548]
in a manner that treats the inviscid and viscous fluxes as a single flux. Here,
we similarly extend the aforementioned flux function to multicomponent flows
but separate the inviscid and viscous fluxes. This separation of the fluxes
allows for use of other inviscid flux functions, as well as enforcement of
entropy boundedness on only the convective contribution to the evolved state,
as motivated by physical and mathematical principles. We also discuss in detail
how to account for boundary conditions and incorporate previously developed
pressure-equilibrium-preserving techniques into the positivity-preserving
framework. Comparisons between the Lax-Friedrichs-type viscous flux function
and more conventional flux functions are provided, the results of which
motivate an adaptive solution procedure that employs the former only when the
element-local solution average has negative species concentrations, nonpositive
density, or nonpositive pressure. A variety of multicomponent, viscous flows is
computed, ranging from a one-dimensional shock tube problem to multidimensional
detonation waves and shock/mixing-layer interaction
A quasi-conservative discontinuous Galerkin method for multi-component flows using the non-oscillatory kinetic flux
In this paper, a high order quasi-conservative discontinuous Galerkin (DG)
method using the non-oscillatory kinetic flux is proposed for the 5-equation
model of compressible multi-component flows with Mie-Gr\"uneisen equation of
state. The method mainly consists of three steps: firstly, the DG method with
the non-oscillatory kinetic flux is used to solve the conservative equations of
the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the
volume fraction equation which can avoid the unphysical oscillations near the
material interfaces; finally, a multi-resolution WENO limiter and a
maximum-principle-satisfying limiter are employed to ensure oscillation-free
near the discontinuities, and preserve the physical bounds for the volume
fraction, respectively. Numerical tests show that the method can achieve high
order for smooth solutions and keep non-oscillatory at discontinuities.
Moreover, the velocity and pressure are oscillation-free at the interface and
the volume fraction can stay in the interval [0,1].Comment: 41 pages, 70 figure