scholarly journals A numerical approach to the non-uniqueness problem of cosmic ray two-fluid equations at shocks

2021 ◽  
Vol 502 (2) ◽  
pp. 2733-2749
Author(s):  
Siddhartha Gupta ◽  
Prateek Sharma ◽  
Andrea Mignone
Keyword(s):  
Large Scale ◽  
Fluid Model ◽  
Cosmic Ray ◽  
Numerical Approach ◽  
Uniqueness Problem ◽  
Test Problems ◽  
Fluid Approximation ◽  
Fluid Equations ◽  
Two Fluid Model ◽  
Two Fluid

ABSTRACT Cosmic rays (CRs) are frequently modelled as an additional fluid in hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of astrophysical flows. The standard CR two-fluid model is described in terms of three conservation laws (expressing conservation of mass, momentum, and total energy) and one additional equation (for the CR pressure) that cannot be cast in a satisfactory conservative form. The presence of non-conservative terms with spatial derivatives in the model equations prevents a unique weak solution behind a shock. We investigate a number of methods for the numerical solution of the two-fluid equations and find that, in the presence of shock waves, the results generally depend on the numerical details (spatial reconstruction, time stepping, the CFL number, and the adopted discretization). All methods converge to a unique result if the energy partition between the thermal and non-thermal fluids at the shock is prescribed using a subgrid prescription. This highlights the non-uniqueness problem of the two-fluid equations at shocks. From our numerical investigations, we report a robust method for which the solutions are insensitive to the numerical details even in absence of a subgrid prescription, although we recommend a subgrid closure at shocks using results from kinetic theory. The subgrid closure is crucial for a reliable post-shock solution and also its impact on large-scale flows because the shock microphysics that determines CR acceleration is not accurately captured in a fluid approximation. Critical test problems, limitations of fluid modelling, and future directions are discussed.

scholarly journals Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

1995 ◽  
Vol 441 ◽  
pp. 629 ◽  
Author(s):  
Adam Frank ◽  
T. W. Jones ◽  
Dongsu Ryu
Keyword(s):  
Fluid Model ◽  
Cosmic Ray ◽  
Time Dependent ◽  
Two Fluid Model ◽  
Two Fluid

A spatially-averaged two-fluid model for dense large-scale gas-solid flows

AIChE Journal ◽  
2017 ◽  
Vol 63 (8) ◽  
pp. 3544-3562 ◽  
Author(s):  
Simon Schneiderbauer
Keyword(s):  
Large Scale ◽  
Fluid Model ◽  
Two Fluid Model ◽  
Two Fluid

Large Interface Simulation in Multiphase Flow Phenomena

2006 ◽  
Author(s):  
Aparicio Henriques ◽  
Pierre Coste ◽  
Sylvain Pigny ◽  
Jacques Magnaudet
Keyword(s):  
Free Surface ◽  
Large Scale ◽  
Fluid Model ◽  
Surface Tension Force ◽  
Recognition Algorithm ◽  
Small Scale ◽  
Rising Bubble ◽  
Flow Phenomena ◽  
Two Fluid Model ◽  
Two Fluid

An attempt to represent multiphase multiscale flow, filling the gap between Direct Numerical Simulation (DNS) and averaged approaches, is the purpose of this paper. We present a kind of Large Interface (LI) simulation formalism obtained after a filtering process on local instantaneous conservation equations of the two-fluid model which distinguishes between small scales and large scales contributions. LI surface tension force is also taken into account. Small scale dynamics call for modelisation and large scale for simulation. Joined to this formalism, a criterion to recognize LI’s is developped. It is used in an interface recognition algorithm which is qualified on a sloshing case and a bubble oscillation under zero-gravity. This method is applied to a rising bubble in a pool that collapses at a free surface and to a square-base basin experiment where splashing and sloshing at the free surface are the main break-up phenomena.

Numerical Modeling of Quench Cooling Using Eulerian Two-Fluid Method

2002 ◽  
Author(s):  
De Ming Wang ◽  
Ales Alajbegovic ◽  
Xuming Su ◽  
James Jan
Keyword(s):  
Heat Transfer ◽  
Fluid Model ◽  
Numerical Approach ◽  
Temperature History ◽  
Modeling Approach ◽  
Simple Geometry ◽  
Two Fluid Model ◽  
Quench Cooling ◽  
Strong Body ◽  
Two Fluid

A numerical approach is presented for modeling quenching cooling of solid metal parts. The approach is based on the Eulerian two-fluid model with modifications toward specific needs of boiling heat transfer. Numerical aspects are discussed on handling high phase change rate of boiling, flux formulation in the presence of strong body force, and conjugate heat transfer of solid immersed in a liquid bath. The modeling approach is assessed for a quenching cooling problem with a simple geometry. Comparisons are made between the computed temperature history within the solid and that of measured data using thermocouples. Reasonably well agreement is observed. The study demonstrated that the present modeling approach is able to meet the numerical challenges entailed in quenching cooling and able to predict the quenching cooling phenomena with satisfactory accuracy. Areas for further improvement are also noted.

The Modeling of Lift and Dispersion Forces in Two-Fluid Model Simulations: Part II — Boundary Layer Flows

2003 ◽  
Author(s):  
F. J. Moraga ◽  
M. Lopez de Bertodano ◽  
D. A. Drew ◽  
R. T. Lahey
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Liquid Velocity ◽  
Dispersion Model ◽  
Fluid Model ◽  
Turbulence Models ◽  
Model Simulations ◽  
Point Injection ◽  
Two Fluid Model ◽  
Two Fluid

Two-fluid model simulations of a bubbly vertical boundary layer with point injection are presented. A new bubble turbulence dispersion model, designed to be used with RANS type turbulence models, was formulated and compared with recent data of [1] and [2]. These data showed that bubble migration toward the wall is controlled by the coherent large scale liquid structures within the boundary layer. The model is based on the application of a kinetic transport equation, similar to Boltzmann’s equation, and the idea that by selectively removing bubbles from the liquid eddies within the boundary layer, bubble capture at the wall introduces a preferential direction of migration and/or nonhomogeneous, anisotropic dispersion. This is the first model capable of predicting all the types of void fraction profiles observed experimentally for point injection. It is shown that without this new model, two-fluid model simulations fail to predict the experimental data. In addition, a new physical interpretation of the data of [1] and [2] is presented, which strongly suggests that the quantity controlling bubble migration toward the wall and bubble dispersion, is the boundary layer drift parameter (i.e., the ratio of the bubble’s terminal velocity to the free-stream liquid velocity).

The Effect of Closure Laws on the Simulation Results of Two-Fluid Model of Gas-Solid Flows

2015 ◽  
Author(s):  
Yifei Duan ◽  
Zhi-Gang Feng
Keyword(s):  
Heat Transfer ◽  
Large Scale ◽  
Fluid Model ◽  
Transfer Model ◽  
Hill Model ◽  
Simulation Results ◽  
Two Fluid Model ◽  
Drag Models ◽  
Closure Laws ◽  
Two Fluid

There are two primary approaches in modeling fluid-solid flows based on the method of treating particles suspended in flows. The first approach is the Eulerian-Lagrangian or Discrete Element Method (DEM) approach that tracks individual particles by solving the equations of motion of these particles. The second approach is the Eulerian-Eulerian approach or two-fluid model (TFM) that considers particles as another continuum phase or fluid. The TFM is preferred in modeling and predicting gas-solid flow behaviors in many engineering applications because of its efficiency in handling large-scale complex systems with large number of particles. However, one of the challenges in TFM is the uncertainty related to the selection of closure laws and transport properties of solid phases. In this study we employ the MFIX code, a general-purpose TFM computer code developed at the National Energy Technology Laboratory, to investigate the effect of different drag models and heat transfer models on the simulation results on the flow hydrodynamics and heat transfer of gas-solid fluidized beds. Three drag models (Gidspow model, Syamlal-O’Brien model, and Koch-Hill model) and two heat transfer model (Gunn model and a recently developed model by Sun et al., 2015) are tested. Simulation results from these models are compared with experimental measurements. The accuracy and applicability of these models are assessed and discussed.

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