An efficient numerical method for compressible flows of a real gas using arithmetic averaging

An efficient numerical method is presented for the solution of the Euler equations governing the compressible flow of a real gas. The scheme is based on the approximate solution of a specially constructed set of linearised Riemann problems. An average of the flow variables across the interface between cells is required, and this is chosen to be the arithmetic mean for computational efficiency, which is in contrast to the usual square root averaging. The scheme is applied to a test problem for five different equations of state.

An efficient numerical scheme for the Euler equations

A finite difference scheme based on flux difference splitting is presented for the solution of the Euler equations for the compressible flow of an ideal gas. A linearised Riemann problem is defined, and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to the usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. The scheme is applied to a shock tube problem and a blast wave problem. Each approximate solution compares well with those given by other schemes, and for the shock tube problem is in agreement with the exact solution.

A variational method to retrieve the extinction profile in liquid clouds using multiple field-of-view lidar

Liquid clouds play a profound role in the global radiation budget but it is difficult to remotely retrieve their vertical profile. Ordinary narrow field-of-view (FOV) lidars receive a strong return from such clouds but the information is limited to the first few optical depths. Wideangle multiple-FOV lidars can isolate radiation scattered multiple times before returning to the instrument, often penetrating much deeper into the cloud than the singly-scattered signal. These returns potentially contain information on the vertical profile of extinction coefficient, but are challenging to interpret due to the lack of a fast radiative transfer model for simulating them. This paper describes a variational algorithm that incorporates a fast forward model based on the time-dependent two-stream approximation, and its adjoint. Application of the algorithm to simulated data from a hypothetical airborne three-FOV lidar with a maximum footprint width of 600m suggests that this approach should be able to retrieve the extinction structure down to an optical depth of around 6, and total opticaldepth up to at least 35, depending on the maximum lidar FOV. The convergence behavior of Gauss-Newton and quasi-Newton optimization schemes are compared. We then present results from an application of the algorithm to observations of stratocumulus by the 8-FOV airborne “THOR” lidar. It is demonstrated how the averaging kernel can be used to diagnose the effective vertical resolution of the retrieved profile, and therefore the depth to which information on the vertical structure can be recovered. This work enables exploitation of returns from spaceborne lidar and radar subject to multiple scattering more rigorously than previously possible.

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

A METHOD FOR THE STUDY OF ACCRETION DISK EMISSION IN CATACLYSMIC VARIABLES. I. THE MODEL

We have developed a spectrum synthesis method for modeling the ultraviolet (UV) emission from the accretion disk from cataclysmic variables (CVs). The disk is separated into concentric rings, with an internal structure from the Wade & Hubeny disk-atmosphere models. For each ring, a wind atmosphere is calculated in the comoving frame with a vertical velocity structure obtained from a solution of the Euler equation. Using simple assumptions, regarding rotation and the wind streamlines, these one-dimensional models are combined into a single 2.5-dimensional model for which we compute synthetic spectra. We find that the resulting line and continuum behavior as a function of the orbital inclination is consistent with the observations, and verify that the accretion rate affects the wind temperature, leading to corresponding trends in the intensity of UV lines. In general, we also find that the primary mass has a strong effect on the P Cygni absorption profiles, the synthetic emission line profiles are strongly sensitive to the wind temperature structure, and an increase in the mass-loss rate enhances the resonance line intensities. Synthetic spectra were compared with UV data for two high orbital inclination nova-like CVs-RW Tri and V347 Pup. We needed to include disk regions with arbitrary enhanced mass loss to reproduce reasonably well widths and line profiles. This fact and a lack of flux in some high ionization lines may be the signature of the presence of density-enhanced regions in the wind, or alternatively, may result from inadequacies in some of our simplifying assumptions.

Partial spectral projected gradient method with active-set strategy for linearly constrained optimization

A method for linearly constrained optimization which modifies and generalizes recent box-constraint optimization algorithms is introduced. The new algorithm is based on a relaxed form of Spectral Projected Gradient iterations. Intercalated with these projected steps, internal iterations restricted to faces of the polytope are performed, which enhance the efficiency of the algorithm. Convergence proofs are given and numerical experiments are included and commented. Software supporting this paper is available through the Tango Project web page: http://www.ime.usp.br/similar to egbirgin/tango/.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

ENO adaptive method for solving one-dimensional conservation laws

In this work an efficient third order non-linear finite difference scheme for solving adaptively hyperbolic systems of one-dimensional conservation laws is developed. The method is based oil applying to the solution of the differential equation an interpolating wavelet transform at each time step, generating a multilevel representation for the solution, which is thresholded and a sparse point representation is generated. The numerical fluxes obtained by a Lax-Friedrichs flux splitting are evaluated oil the sparse grid by an essentially non-oscillatory (ENO) approximation, which chooses the locally smoothest stencil among all the possibilities for each point of the sparse grid. The time evolution of the differential operator is done on this sparse representation by a total variation diminishing (TVD) Runge-Kutta method. Four classical examples of initial value problems for the Euler equations of gas dynamics are accurately solved and their sparse solutions are analyzed with respect to the threshold parameters, confirming the efficiency of the wavelet transform as an adaptive grid generation technique. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Development of a method to determine Ni and Cd in biodiesel by graphite furnace atomic absorption spectrometry

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Validation of an HPLC Method for quantitative Determination of Benzocaine in PHBV-Microparticles and PLA-Nanoparticles

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A study of a numerical solution of the steady two dimensions Navier-Stokes equations in a constricted channel problem by a compact fourth order method

We present a numerical solution for the steady 2D Navier-Stokes equations using a fourth order compact-type method. The geometry of the problem is a constricted symmetric channel, where the boundary can be varied, via a parameter, from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular. The set of non-linear equations is solved by the Newton method. Results have been obtained for Reynolds number up to 500. Estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method. (C) 2002 Elsevier B.V. All rights reserved.

Bi-Lipschitz G-triviality and Newton polyhedra, G = R, C, K, R-V, C-V, K-V

In this work we provide estimates for the bi-Lipschitz G-triviality, G = C or K, for a family of map germs satisfying a Lojasiewicz condition. We work with two cases: the class of weighted homogeneous map germs and the class of non-degenerate map germs with respect to some Newton polyhedron. We also consider the bi-Lipschitz triviality for families of map germs defined on an analytic variety V. We give estimates for the bi-Lipschitz G(V)-triviality where G = R,C or K in the weighted homogeneous case. Here we assume that the map germ and the analytic variety are both weighted homogeneous with respect to the same weights. The method applied in this paper is based in the construction of controlled vector fields in the presence of a suitable Lojasiewicz condition. In the last section of this work we compare our results with other results related to this work showing tables with all estimates that we know, including ours.

Numerical simulation of viscous three-dimensional flow over aerospace vehicles using Euler/boundary-layer zonal approach

This paper presents a viscous three-dimensional simulations coupling Euler and boundary layer codes for calculating flows over arbitrary surfaces. The governing equations are written in a general non orthogonal coordinate system. The Levy-Lees transformation generalized to three-dimensional flows is utilized. The inviscid properties are obtained from the Euler equations using the Beam and Warming implicit approximate factorization scheme. The resulting equations are discretized and approximated by a two-point fmitedifference numerical scheme. The code developed is validated and applied to the simulation of the flowfield over aerospace vehicle configurations. The results present good correlation with the available data.