Multidimensional Gas Dynamics Research Articles

VERIFICATION OF AN ENTROPY DISSIPATIVE QGD-SCHEME

An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme.

A variational time discretization for compressible Euler equations

We introduce a variational time discretization for the multidimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each time step requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.

Supernova Explosion Mechanism with the Neutrinos and the Collapse of the Rotation Core

Most of the energy released in the gravitational collapse of the cores of massive stars is carried away by neutrinos. Neutrinos play a pivotal role in explaining core-collapse supernovae. In this work the multidimensional gas dynamics is used with neutrino transport in the flux-limited diffusion approximation to study the role of multi-dimensional effects. The possibility of large-scale convection is discussed, which is interesting both for explaining SNII and for setting up observations to register possible high-energy (≲10 MeV) neutrinos from the supernova. In compare with the previous work describing a new multidimensional gas dynamics method with neutrino transport we investigate the role of the rotation in the convection.

Large-scale Instability during Gravitational Collapse with Neutrino Transport and a Core-Collapse Supernova

Most of the energy released in the gravitational collapse of the cores of massive stars is carried away by neutrinos. Neutrinos play a pivotal role in explaining core-collape supernovae. Currently, mathematical models of the gravitational collapse are based on multi-dimensional gas dynamics and thermonuclear reactions, while neutrino transport is considered in a simplified way. Multidimensional gas dynamics is used with neutrino transport in the flux-limited diffusion approximation to study the role of multi-dimensional effects. The possibility of large-scale convection is discussed, which is interesting both for explaining SN II and for setting up observations to register possible high-energy (≳10MeV) neutrinos from the supernova. A new multi-dimensional, multi-temperature gas dynamics method with neutrino transport is presented.

Supernova-Explosion Mechanism Involving Neutrinos

A major part of the energy released upon the gravitational collapse of massive-star cores is carried away by neutrinos. Neutrinos play a crucial role in collapsing supernovae (SNe). At the present time, mathematical models of core-collapse SNe are based on multidimensional gas dynamics and thermonuclear reactions, whereas the neutrino transport is frequently treated in simplified way. An accurate analysis of neutrinos in a spherically symmetric gravitational collapse is performed on the basis of Boltzmann kinetic equations including all weak-interaction reactions with exact quantum-mechanical matrix elements. The role of multidimensional effects is studied bymeans of multidimensional gas dynamics allowing for the neutrino transport via diffusion treated by employing flux limiters. The possibility of largescale convection, which is of interest both from the point of view of explaining a type II supernova (SN) and from the point of view of implementing an experiment aimed at detecting possible energetic (≳10 MeV) neutrinos from an SN, is discussed. Thermonuclear burning leads to the explosion of a type I SN. A hot central region and the subsequent large-scale convection may also play an important role in the SN mechanism. If neutrinos and convection play a key role for a type II SN, then, in order to explain gamma radiation from product radioactive elements, convection is of importance in the case of SNe belonging to both types. In addition, convection may be important for bright type I SNe. Original methods are presented for multidimensional gas dynamics involving thermonuclear burning and for multitemperature gas dynamics involving radiative transfer.

Parallelizaton of visual magneto-hydrodynamics code based on fluctuation distribution scheme on triangular grids

A two-dimensional (2D) magneto-hydrodynamics (MHD) code which has visualization and parallel processing capability is presented in this paper. The code utilizes a fluctuation splitting (FS) scheme that runs on structured or unstructured triangular meshes. First FS scheme which included the wave model: Model-A had been developed by Roe [Roe PL. Discrete models for the numerical analysis of time-dependent multi-dimensional gas dynamics. J Comp Phys 1986;63:458–76.] for the solutions of Euler’s equations. The first 2D-MHD wave model: MHD-A, was then developed by Balci and Aslan [Balci Ş. The numerical solutions of two dimensional MHD equations by fluctuation splitting scheme on triangular meshes, Ph.D. Thesis, University of Marmara, Science-Art Faculty, Physics Dept Istanbul, Turkey; 2000; Aslan N. MHD-A: A fluctuation splitting wave model for planar magnetohydrodynamics. J Comp Phys 1999;153:437–66.] to solve MHD problems including shocks and discontinuities. It was then shown in [Balci S, Aslan N. Two dimensional MHD solver by fluctuation splitting and dual time stepping. Int J Numer Meth Fluids, in press.] that this code was capable of producing reliable results in compressible and nearly incompressible limits and under the effect of gravitational fields and that it was able to identically reduce to model-A of Roe in Euler limit with no sonic problems at rarefaction fans (Balci and Aslan, in press). An important feature of this code is its ability to run time dependent or steady problems on structured or unstructured triangular meshes that can be generated automatically by the code for specified domains. In order to use the parallel processing capability of the code, the triangular meshes are decomposed into different blocks in order to share the workload among a number of processors (here personal computers) which are connected by Ethernet. Due to the compact nature of the FS scheme, only one set of data transfer is required between neighbor processors. As it will be shown, this phenomenon results in minimum amount of communication loss and makes the scheme rather robust for parallel processing. The other important feature of the new code is its visual capability. As the code is running, colorful images of scalar quantities (density, pressure, Mach number, etc.) or vector graphics of vectoral quantities (velocity, magnetic field, etc.) can be followed on the screen. The extended code, called PV-MHDA, also allows following the trajectories of the particles in time by means of a recently included particle in cell (PIC) algorithm. Because the numerical dissipation embedded in its wave model reflects real physical viscosity and resistivity, it is able to run accurately for compressible flows (including shocks) as well as nearly incompressible flows (e.g., Kelvin–Helmholtz instability). The user-friendly visual and large-scale computation capability of the code allow the user more thorough analysis of MHD problems in two-dimensional complex domains.

Computational Fluid Dynamics Modeling of Solid Oxide Fuel Cells

A comprehensive, multi-physics, multi-dimensional model has been developed to simulate solid oxide fuel cells (SOFCs). The model fully couples electrochemical kinetics with multi-dimensional gas dynamics and multi-component transport of species. The developed model is a full cell model, including all components of SOFC, flow channels, active and inactive gas diffusion electrodes and electrolyte. The present model is implemented in a commercially available CFD software, Fluent, using its customization ability via User Defined Functions (UDF). It is able to predict conventional I-V curve (polarization curve) in addition to details of internal processes, such as flow field, species concentrations, potential and current distributions throughout the cell. It is found that mass transfer limitation plays an important role in SOFC performance, especially under high current density operation.

Finite-difference method-of-characteristics scheme for the equationsof gasdynamics

A finite-difference method for the characteristic equations of gasdynamics is developed. The method is based on a directed discretization of the characteristic compatibility conditions. The choice of wave propagation directions is arbitrary, and there is no need to fit interpolating functions to the initial data. The proposed method is not conservative and therefore must be used with a shock-fitting procedure in regions wherever the flow is discontinuous. Numerical results are presented for sample two-dimensional problems. UMERICAL schemes based on the method of character- istics have been applied successfully to a large number of problems in multidimensio nal gasdynamics. Schemes of this type are attractive because they are typically robust, in no small part because of their direct relationship to the physics of nonlinear wave propagation. The review paper by Roe 1 out- lines a number of characteristi cs-based methods, including algorithms for both the conservative and nonconservative forms of the governing equations. In this paper, we present a nonconservative discretization for the characteristic compati- bility conditions. It is important to keep in mind that al- gorithms of this type must, in general, incorporate a capability to track discontinuities in the flowfield explicitly to ensure that the Rankine-Hugoniot jump conditions are satisfied in these regions. A number of conservative and nonconservative schemes for multidimensional flow are constructed under the assumption that the multidimensional problem can be split into quasi-one- dimensional pieces in which waves travel normal to the mesh lines (this is true of approximate Riemann solvers,2'3 flux vector splitting schemes, 4>5 and also of the nonconservative SCM scheme6). How this assumption affects the quality of the numerical results is not entirely clear, especially if the compar- ison is made keeping the total computational effort fixed. Nevertheless, there is evidence that numerical methods that account for the multidimensional nature of the problem can produce high-resolution solutions. One family of methods of this type has evolved from Colella's conservative unsplit Godunov method:7 multidimensional characteristics informa- tion is propagated to the faces of the computational cell to compute the numerical fluxes required in the solution of the integral form of the conservation equations. Perhaps even more sophisticated is Roe's model,8 for which, unfortunately, little data is available. Several recent papers report research and opinions on this subject.8'9'10 Many nonconservative, truly multidimensional schemes, based on the method of characteristi cs for multidimensional flow, have evolved from the method of Butler.11 According to this method, a number of characteristi c compatibility condi- tions are combined to yield a set of equations in which the spatial derivatives of the dependent variables are evaluated along the projections onto space of the stream- and wave-type characteristic rays. The advantage in this approach is clear: the upwind directions are defined by the characteristic rays.

Multidimensional time dependent method of characteristics

Abstract A numerical technique based on the theory of characteristics in multidimensional gasdynamics is developed. The characteristic equations are recast into a form in which linear combinations of the gasdynamic variables are integrated along initially characteristic rays, and the solution is subsequently obtained by the inversion of a system of linear algebraic equations. This formulation facilitates the imposition of boundary conditions, which are usually algebraic, and can therefore be directly incorporated into the linear system. The choice of boundary treatments, which is guided by the theory of characteristics, is described in detail. The method was successfully applied to sample problems in 2- and 3-D nozzle flow. While familiar, these examples involve a variety of boundary conditions.

Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics

This paper explores a possible technique for extending to multidimensional flows some of the upwind-differencing methods that have proved highly successful in the one-dimensional case. Attention here is concentrated on the two-dimensional case, and the flow domain is supposed to be divided into polygonal computational elements. Inside each element the flow is represented by a local superposition of elementary solutions consisting of plane waves not necessarily aligned with the element boundaries.

Use of characteristics for boundaries in time dependent finite difference analysis of multidimensional gas dynamics

AbstractRelations form the theory of characteristics are used to connect boundary and interior points in the numerical analysis of multidimensional, unsteady gas dynamics.Since specification of all variables on the boundaries does not occur, it is necessary to employ at least some of the differential equations for boundary points. The objective is to avoid the use of non‐central finite differences and the subsequent generation of a pseudo‐boundary layer.Boundaries consisting of a moving shock wave, a solid wall and a surface in a supersonic, downstream region are considered. The numerical procedure for determining variables on boundaries is described in datail for the hypersonic, blunt body problem.