959 research outputs found
Large Eddy Simulations in Astrophysics
In this review, the methodology of large eddy simulations (LES) is introduced
and applications in astrophysics are discussed. As theoretical framework, the
scale decomposition of the dynamical equations for neutral fluids by means of
spatial filtering is explained. For cosmological applications, the filtered
equations in comoving coordinates are also presented. To obtain a closed set of
equations that can be evolved in LES, several subgrid scale models for the
interactions between numerically resolved and unresolved scales are discussed,
in particular the subgrid scale turbulence energy equation model. It is then
shown how model coefficients can be calculated, either by dynamical procedures
or, a priori, from high-resolution data. For astrophysical applications,
adaptive mesh refinement is often indispensable. It is shown that the subgrid
scale turbulence energy model allows for a particularly elegant and physically
well motivated way of preserving momentum and energy conservation in AMR
simulations. Moreover, the notion of shear-improved models for inhomogeneous
and non-stationary turbulence is introduced. Finally, applications of LES to
turbulent combustion in thermonuclear supernovae, star formation and feedback
in galaxies, and cosmological structure formation are reviewed.Comment: 64 pages, 23 figures, submitted to Living Reviews in Computational
Astrophysic
Modeling the Pollution of Pristine Gas in the Early Universe
We conduct a comprehensive theoretical and numerical investigation of the
pollution of pristine gas in turbulent flows, designed to provide new tools for
modeling the evolution of the first generation of stars. The properties of such
Population III (Pop III) stars are thought to be very different than later
generations, because cooling is dramatically different in gas with a
metallicity below a critical value Z_c, which lies between ~10^-6 and 10^-3
solar value. Z_c is much smaller than the typical average metallicity, , and
thus the mixing efficiency of the pristine gas in the interstellar medium plays
a crucial role in the transition from Pop III to normal star formation. The
small critical value, Z_c, corresponds to the far left tail of the probability
distribution function (PDF) of the metallicity. Based on closure models for the
PDF formulation of turbulent mixing, we derive equations for the fraction of
gas, P, lying below Z_c, in compressible turbulence. Our simulation data shows
that the evolution of the fraction P can be well approximated by a generalized
self-convolution model, which predicts dP/dt = -n/tau_con P (1-P^(1/n)), where
n is a measure of the locality of the PDF convolution and the timescale tau_con
is determined by the rate at which turbulence stretches the pollutants. Using a
suite of simulations with Mach numbers ranging from M = 0.9 to 6.2, we provide
accurate fits to n and tau_con as a function of M, Z_c/, and the scale, L_p,
at which pollutants are added to the flow. For P>0.9, mixing occurs only in the
regions surrounding the pollutants, such that n=1. For smaller P, n is larger
as mixing becomes more global. We show how the results can be used to construct
one-zone models for the evolution of Pop III stars in a single high-redshift
galaxy, as well as subgrid models for tracking the evolution of the first stars
in large cosmological simulations.Comment: 37 pages, accepted by Ap
Toward an equivalence criterion for Hybrid RANS/LES methods
International audienceA criterion is established to assess the equivalence between hybrid RANS/LES methods, called H-equivalence, based on the modeled energy of the unresolved scales, which leads to similar low-order statistics of the resolved motion. Different equilibrium conditions are considered, and perturbation analyses about the equilibrium states are performed. The procedure is applied to demonstrate the equivalence between two particular hybrid methods, and leads to relationships between hybrid method parameters that control the partitioning of energy between the resolved and unresolved scales of motion. This equivalence is validated by numerical results obtained for the cases of plane and periodically constricted channel flows. This concept of H-equivalence makes it possible to view different hybrid methods as models for the same system of equations: as a consequence, detached-eddy simulation (DES), which is shown to be H-equivalent to the temporal partially integrated transport model (T-PITM) in inhomogeneous, stationary situations, can be interpreted as a model for the subfilter stress involved in the temporally filtered Navier–Stokes equations
Interacting errors in large-eddy simulation: a review of recent developments
The accuracy of large-eddy simulations is limited, among others, by the quality of the subgrid parameterisation and the numerical contamination of the smaller retained flow structures. We review the effects of discretisation and modelling errors from two different perspectives. We first show that spatial discretisation induces its own filter and compare the dynamic importance of this numerical filter to the basic large-eddy filter. The spatial discretisation modifies the large-eddy closure problem as is expressed by the difference between the discrete 'numerical stress tensor' and the continuous 'turbulent stress tensor'. This difference consists of a high-pass contribution associated with the specific numerical filter. Several central differencing methods are analysed and the importance of the subgrid resolution is established. Second, we review a database approach to assess the total simulation error and its numerical and modelling contributions. The interaction between the different sources of error is shown to lead to their partial cancellation. From this analysis one may identify an 'optimal refinement strategy' for a given subgrid model, discretisation method and flow conditions, leading to minimal total simulation error at a given computational cost. We provide full detail for homogeneous decaying turbulence in a 'Smagorinsky fluid'. The optimal refinement strategy is compared with the error reduction that arises from grid refinement of the dynamic eddy-viscosity model. The main trends of the optimal refinement strategy as a function of resolution and Reynolds number are found to be adequately followed by the dynamic model. This yields significant error reduction upon grid refinement although at coarse resolutions significant error levels remain. To address this deficiency, a new successive inverse polynomial interpolation procedure is proposed with which the optimal Smagorinsky constant may be efficiently approximated at a given resolution. The computational overhead of this optimisation procedure is shown to be well justified in view of the achieved reduction of the error level relative to the 'no-model' and dynamic model predictions
Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling
We explore the potential of a formulation of the Navier-Stokes equations
incorporating a random description of the small-scale velocity component. This
model, established from a version of the Reynolds transport theorem adapted to
a stochastic representation of the flow, gives rise to a large-scale
description of the flow dynamics in which emerges an anisotropic subgrid
tensor, reminiscent to the Reynolds stress tensor, together with a drift
correction due to an inhomogeneous turbulence. The corresponding subgrid model,
which depends on the small scales velocity variance, generalizes the Boussinesq
eddy viscosity assumption. However, it is not anymore obtained from an analogy
with molecular dissipation but ensues rigorously from the random modeling of
the flow. This principle allows us to propose several subgrid models defined
directly on the resolved flow component. We assess and compare numerically
those models on a standard Green-Taylor vortex flow at Reynolds 1600. The
numerical simulations, carried out with an accurate divergence-free scheme,
outperform classical large-eddies formulations and provides a simple
demonstration of the pertinence of the proposed large-scale modeling
Stochastic climate theory and modeling
Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models
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