Flux difference splitting for the Euler equations with axial symmetry

An approximate (linearised) Riemann solver is presented for the solution of the Euler equations of gas dynamics for axially symmetric flows. The method is Roe's flux difference splitting with a technique for dealing with source terms and incorporates operator splitting. Results for the problem of a converging spherical shock are presented. This work forms part of the research programme of the Institute for Computational Fluid Dynamics at the Universities of Oxford and Reading and was funded by A.W.R.E., Aldermaston under contract no. NSN/13B/2A88719.     

PVU and wave-particle splitting schemes for Euler equations of gas dynamics

A new way of flux-splitting, termed as the wave-particle splitting is presented for developing upwind methods for solving Euler equations of gas dynamics. Based on this splitting, two new upwind methods termed as Acoustic Flux Vector Splitting (AFVS) and Acoustic Flux Difference Splitting (AFDS) methods are developed. A new Boltzmann scheme, which closely resembles the wave-particle splitting, is developed using the kinetic theory of gases. This method, termed as Peculiar Velocity based Upwind (PVU) method, uses the concept of peculiar velocity for upwinding. A special feature of all these methods is that the unidirectional and multidirectional parts of the flux vector are treated separately. Extensive computations done using these schemes demonstrate the soundness of the ideas.     

Computation of three-dimensional transonic flow using a cell vertex finite volume method for the Euler equations

Summary A cell vertex finite volume method for the solution of the three dimensional Euler equations has been developed. The computations can be carried out block-wise after dividing the computational domain into smaller blocks to reduce the memory requirement for a single processor computer and also to facilitate parallel computing. A five stage Runge-Kutta scheme has been used to advance the solution in time. Enthalpy damping, implicit residual smoothing, local time stepping, and grid sequencing are used for convergence acceleration. The solution procedure has been studied in detail by computing transonic flow over the ONERA M-6 wing, using both C-H and O-H type structured grids. The effects of changing the artificial viscosity parameters and the distance of the far field boundary are also investigated.     

Correlation between the equations of transfer coefficients and the equation of state for liquids

On the basis of test data on p, v, T, thermal conductivity, and dynamic viscosity of toluene and water, it is shown that the equations of transfer coefficients can be represented in a form analogous to the equation of state.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 3, pp. 492–497, September, 1972.     

Stability analysis of the forward Euler scheme for the convection-diffusion equation using the SUPG formulation in space

The stability and accuracy of the forward Euler scheme for the semidiscrete problem arising from the space discretization of the convection-diffusion equation using the SUPG formulation are analysed. Both linear and quadratic finite elements are considered. The stability limits are derived for the one-dimensional problem and a heuristic criterion is proposed for the multidimensional equation. Some numerical experiments are conducted in order to assess the performance of different finite elements.     

On two-parameter equations of state and the limitations of a hard sphere Peng-Robinson equation

Simple two-parameter equations of state are exceptionally effective for calculations on systems of small, uncomplicated molecules. They are therefore extremely useful for vapour-liquid equilibrium calculations in cryogenic and light hydrocarbon process design. In a search for further improvement three two-parameter equations of state with a co-volume repulsion term and three with a hard sphere repulsion term have been investigated. Their characteristic constants at the critical point have been compared. The procedure for fitting the two parameters to empirical data in the subcritical region was analysed. A perturbed hard sphere equation with a Peng-Robinson attraction term was shown to be unsuitable for application over a wide range of p, T conditions. A similar equation with a Redlich-Kwong attraction term gives good service in the cryogenic range.     

N-particle dynamics of the Euler equations for planar diffeomorphisms

The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this article, we consider the dynamics of N point particles or “blobs” moving under the action of the Euler equations associated with the group of diffeomorphisms of the plane in a variety of different metrics. This dynamical system is already in widespread use in the field of image registration, where the point particles correspond to image landmarks, but its dynamical behavior has not previously been studied. The 2-body problem is always integrable, and we analyze its phase portrait under different metrics. In particular, we show that 2-body capturing orbits (in which the distances between the particles tend to 0 as t  → ∞) can occur when the kernel is sufficiently smooth and the relative initial velocity of the particles is sufficiently large. We compute the dynamics of these “dipoles” with respect to other test particles, and supplement the calculations with simulations for larger N that illustrate the different regimes.     

Analysing the shielding effect of a concrete wall reinforced with steel plate using generalised surface integral equations

The electromagnetic shielding effect of a two-dimensional concrete wall reinforced with steel plate is analysed using a generalised surface integral equation (GSIE) method developed from Huygens? equivalence principle. The wall is divided into small blocks. Each block is analysed independently at first, and its electromagnetic characteristics are expressed by an associated generalised transition matrix defined on the reference surface of that block. In order to find the electromagnetic characteristics of the whole system, a GSIE formulation is established on the reference surface of all blocks, which can be solved efficiently using characteristic basis functions.     

The solution of the three-dimensional Navier-Stokes equations using a new finite difference approach

A numerical method for integrating the three-dimensional Navier–Stokes equations for a viscous, incompressible, laminar duct flow is given. A new method is applied to the derivation of stable, and second order accurate difference representations to the momentum transport and diffusion terms. Numerical examples are given.     

Cubic equations of state for transport properties: An equation for the thermal conductivity of oxygen

A scheme for the development of equations for the transport properties in terms of pressure and temperature, so-called transport equations of state, is presented. The surfaces of transport properties and density as a function of pressure and temperature reveal similarities, which become even more evident when the residual transport property as a function of pressure and temperature is considered. Even the spinodals of transport and thermal properties coincide in the p, T plane, as can be shown mathematically and as was already empirically found for water and oxygen. Based on these similarities a cubic transport equation of state is evaluated for the residual thermal conductivity of oxygen. The new equation is only a little less accurate than the already established virial transport equation of state for oxygen. It is, however, much simpler and needs only a few parameters. The accuracy is still good enough for practical applications. The results demonstrate that cubic equations of state can describe transport properties and are a basis for generalized estimation methods for the transport properties of fluids.     

Analysis of explicit difference methods for a diffusion-convection equation

We consider the numerical solution of a model one-dimensional diffusion-convection equation by a variety of explicit finite difference methods including conventional central and upwind replacements of the convection terms. We discuss commonly observed phenomena such as instability, unwanted oscillations in the numerical solution, and numerical diffusion and we present an analysis of these effects by simple mathematical techniques.     

On the optimization of the fixed-stress splitting for Biot's equations

In this work, we are interested in efficiently solving the quasi-static, linear Biot model for poroelasticity. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well known that the convergence properties of the method strongly depend on the applied stabilization/tuning parameter. We show theoretically that, in addition to depending on the mechanical properties of the porous medium and the coupling coefficient, they also depend on the fluid flow and spatial discretization properties. The type of analysis presented in this paper is not restricted to a particular spatial discretization, although it is required to be inf-sup stable with respect to the displacement-pressure formulation. Furthermore, we propose a way to optimize this parameter that relies on the mesh independence of the scheme's optimal stabilization parameter. Illustrative numerical examples show that using the optimized stabilization parameter can significantly reduce the number of iterations.     

On the Euler equation appearing in the synthesis of radiating systems

A problem appearing when the Euler equation is used in the theory of antenna synthesis is considered. Behavior of a solution to this equation at the ends of the synthesis interval contradicts the Meixner conditions, according to which this solution must tend to zero as a square root of the distance to the ends. It is shown that this contradiction arises if we seek solutions in the L ₂ space and disappears if an L ₂ subspace with limited energy norm is selected as the space of solutions. In this case, the Euler equation arises from a variational minimization problem for a functional involving the norm of current determined in the energy space.     

Advanced quadrature methods and splitting extrapolation algorithms for first kind boundary integral equations of Laplace's equation with discontinuity solutions

The Laplace equation can be transformed to the first kind boundary integral equations (BIEs), and the corner and discontinuity singularity of Laplace's equation can be studied by the open arcs of the BIEs. For the first kind BIEs, there exist the Galerkin method (GM) and the collocation method (CM); but they suffer in low convergence. The advanced (i.e., the mechanical) quadrature methods (AQMs) and the splitting extrapolation methods (SEMs) originated in [16], [17] are proposed in Huang et al. [15] for first kind BIEs with open arcs, to achieve O(h³) or even O(h⁵) convergence, and the excellent stability with Cond. = O(h^?1), where h is the uniform meshspacing, accompanied with the strict analysis. Moreover, the algorithms of AQMs and the SEMs are simple without any integration computation. Hence the AQMs and the SEMs are superior to the existing methods, such as GM and CM. A challenging discontinuity model of Laplace's equation is proposed in Li et al. [20], and the collocation Trefftz method (CTM) is used to give highly accurate solutions. For the AQMs and the SEMs, the strict theoretical analysis is given in [15], and the discontinuity model [20] is dealt with in this paper, to also achieve highly accurate solutions. The numerical solutions in this paper display that the AQMs and the SEMs are significant not only to the first kind BIEs with the open arc singularity, but also to Laplace's equation with highly strong singularity such as the discontinuity of solutions. Moreover, the link of the first kind BIEs with open arcs and the Laplace equation with discontinuity solutions are explored clearly, to display the significance of the proposed algorithms; this paper strengthens the first kind BIEs and its engineering applications.     

Note on eigenvector solutions of the Navier equation in cylindrical coordinates

Summary The eigenvector solution of the spectral Navier equation in cylindrical coordinates is constructed by using the Helmholtz decomposition theorem and the method of separation of variables.     

A modified Benedict-Webb-Rubin equation of state for methane using recent experimental data

A 33 term modified Benedict-Webb-Rubin equation of state is presented for methane. The adjustable parameters in the equation of state have been estimated using recent experimental data and least squares techniques which include the thermodynamic equilibrium conditions for the co-existing liquid and vapour phases. Comparisons of the new equation of state and an older modified Benedict-Webb-Rubin equation of state to experimental data are given.     

The prediction of the excess functions of liquid mixtures using a modified hard-sphere equation of state

The two parameter hard-sphere equation of state used by Snider and Herrington to calculate excess functions for mixtures of condensed gases near their boiling points has been modified to take account of the fact that over a range of temperature the parameters are not constant. The parameter which represents the size of the molecules has been treated as a function of volume, while that associated with the intermolecular energy of attraction has been treated as a function of temperature. Excess functions calculated using the modified equation in general show improved agreement with the experimental values. Moreover the new equation has been used to predict the excess functions of binary mixtures of condensed gases over wide ranges of temperature and pressure, both for systems which have already been partially studied experimentally and for others (for example, fluorine + argon) for which no data yet exist.     

Flux nucleation in the current-induced resistive state of a constricted type I superconductor

The current-induced resistive state in a constricted type I superconductor is characterized by a train of flux tubes traversing the sample perpendicular to the direction of the applied current following its nucleation at the sample edge. The temporal structure of the nucleation process can be investigated by attempting to synchronize this process with small periodic current pulses superimposed on the direct bias current. The resistive dc voltage is then to be measured as a function of the pulse parameters such as frequency and width. We have performed such experiments at 4.2 K on constricted Pb films of 6–8 m thickness and 100 m width. Simultaneously with the electrical measurements the dynamic behavior of the flux tubes was directly observed using a stroboscopic magnetooptical method for magnetic flux detection. Our electrical measurements clearly show how the size of the nucleated flux tubes varies with the direct bias current and the nucleation frequency. The positive wall energy in the Pb films results in a lower limit for this size as expected. The influence of the preceding flux tubes still existing within the constriction upon the flux nucleation process is revealed in detail. All observations can be understood from a consideration of the energy balance during the flux nucleation process.Supported by a grant of the Deutsche Forschungsgemeinschaft.