Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Improved dust capture methods for crushing plant

The use of compressive crushing equipment such as gyratory crushers within minerals processing plants can potentially generate large quantities of dust. Remedies to this problem include the retrofitting of shrouds, enclosures, local exhaust ventilation (LEV) systems and water suppression systems. The single or combined application of these systems must be optimised to ensure they operate efficiently. It is desirable that the future design planning of such facilities include integrated dust suppression and/or removal systems to ensure material delivery rates are maintained and the welfare of the workforce is protected.     

Electrochemical studies on the composition,stabilities and thermodynamics of Zn+2, Cd+2 and Pb+2 complexes with methylene bis thio acetic acid

Potentiometric and conductometric studies on the zinc, cadmium, lead—methylene bis thio acetic acid system in aqueous media reveal the formation of 1:1 and 1:2 (M:L) complexes in thepH range 3.4–5.1. Their stability constants have been determined at 20°, 30° and 40 °C by applyingCalvin andMelchior's extension ofBjerrum's method. The overall changes in thermodynamic function H, G and S accompanying complexation have been evaluated at 30 °C.

---

Elektrochemische Untersuchungen über Zusammensetzung, Stabilität und Thermodynamik von Zn(II)-, Cd(II)- und Pb(II)-Komplexen mit Methylen-bisthioessigsäure

Zusammenfassung Potentiometrische und konduktometrische Untersuchungen an Zink-, Cadmium- und Blei-Methylen-bis-thioessigsäure-Systemen in wäßrigem Medium zeigten die Bildung von 1:1 und 1:2 (M:L)-Komplexen impH-Bereich 3,4–5,1. Die Stabilitätskonstanten wurden bei 20°, 30° und 40 °C mittels derCalvin-Melchior-Methode (erweiterteBjerrum-Methode) bestimmt. Die thermodynamischen Parameter H, G und S der Komplexierung wurden für 30 °C ermittelt.

     

Three‐dimensional wave impact on a rigid structure using smoothed particle hydrodynamics

Navier–Stokes computations of a wave–structure interaction are performed with the aim of assessing the potential of smoothed particle hydrodynamics to accurately estimate impact loading time history. A three‐dimensional dam‐break flow with a rectangular column located downstream is considered. The net force and impulse exerted on the column is monitored throughout the simulation with the results correlating well with existing experimental data. Initial and boundary conditions and numerical parameters are varied and their effect on the column load investigated. The column load is found to be most sensitive to the choice of boundary treatment. Copyright © 2011 John Wiley & Sons, Ltd.     

Generation of strong, homochiral bases by electrochemical reduction of phenazine derivatives

Electrochemical reduction of enantiomerically pure amino- and alkoxy-phenazine derivatives forms strongly basic radical anions which give asymmetric induction in the conversion of 3,4-epoxytetrahydrothiophene-1,1-dioxide into the allylic ester with facile regeneration of the phenazine.     

The effect of the stability of mixed finite element approximations on the accuracy and rate of convergence of solution when solving incompressible flow problems

The stability of two different mixed finite element methods for incompressible flow problems are theoretically analysed. The effect of the stability of the mixed approximation on the accuracy and the rate of convergence of solution is assessed for two non-trivial problems. The numerical results presented indicate that if the stability of the mixed approximation is not guaranteed then both pressure and velocity solutions are markedly less accurate. In one of the cases considered the ultimate convergence of both the pressure and the velocity solutions is seriously in doubt.     

Potentiometric determination of coextraction constants of potassium salts in ion-selective electrodes utilizing a nitrobenzene liquid membrane phase

A theoretical treatment of potentiometric data is applied to calculate coextraction constants (K_IA) for three potassium salts from water into a liquid nitrobenzene phase. The experiment involves treating nitrobenzene as a membrane and contacting it with two aqueous solutions of different ion activities. In the presence of either a cation or anion exchanger, the ratio of activities of ions in the two aqueous phases gives rise to a potential difference across the membrane that depends upon the nature and charge of the counter ion of the ion-exchanger in excess. Here, the cation exchanger was chosen to be potassium tetrakis(4-chlorophenyl)borate (KTpClPB) and the anion exchanger was tetradodecylammonium chloride (TDDACl). TDDACl was incrementally added to the nitrobenzene phase containing a fixed concentration of KTpClPB, and the corresponding emf was recorded as a function of concentration of TDDACl. The membrane changes from one with cation exchanger properties (excess KTpClPB) to one with anion exchanger properties (excess TDDACl). The potential difference and shape of the titration curve can be predicted by theory based on the phase boundary potential model. Log(K_IA) values calculated for KCl, KNO₃ and KClO₄ in nitrobenzene were found as: −10.53 (± 0.09), −8.16 (± 0.05) and −5.63 (± 0.03) respectively, in accordance with the Hofmeister series of lipophilicity, and similar to those observed in PVC membranes containing other plasticizers. The method presented here offers the advantage over other methods to calculate K_IA, in that it is relatively experimentally simple without compromising the accuracy of the calculated coextraction constants. The ability to titrate directly into the liquid membrane phase affords a higher precision compared to the preparation of a series of PVC/plasticizer membranes with different compositions.     

Fast solvers for discretized Navier-Stokes problems using vector extrapolation

We discuss the design and implementation of a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equation system. We describe a “proof of concept” implementation of vector extrapolation, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package ( http://www.manchester.ac.uk/ifiss).     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

Superorbital expansion tube tests of a caret waverider

A computational and experimental study was conducted to assess the potential of testing waverider configurations in a high-performance, short-duration expansion tube facility. The tests were performed in the newly commissioned X3 superorbital expansion tube and provide the first experimental data of a waverider tested at a stagnation enthalpy and equivalent flight speed exceeding 40 MJ/kg and 9 km/s, respectively. Two simple caret configurations were chosen as benchmark test cases to test the use of the facility, instrumentation and numerical models to investigate these flows. The general performance of the sharp and blunt leading edge waveriders at angles of attack ranging from 0° to 5° were analyzed and compared to CFD and theoretical predictions. For the conditions tested, the presence of a strong viscous interaction caused the shock wave to be detached from the leading edge of the models resulting in a significant loss in performance. An analytical model was developed to account for the strong coupling between the shock wave and boundary layer. Results were shown to be in very good agreement with CFD estimates for both configurations at all angles of attack considered. Finite-rate chemistry CFD simulations indicated that real gas effects other than the residual levels of nonequilibrium freestream dissociation present in the expansion tube flow were negligible for the conditions tested. The study also revealed that a past flow visualization technique gave a false indication of the leading edge shock location. An improved experimental visualization technique was successfully tested with results from these tests correlating well with computational estimates. This study successfully demonstrated the use of the facility to study waverider performance at speeds representative of orbital flight.