Quasi–incompressible Cahn–Hilliard fluids and topological transitions

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

We present a new approach to simulating unsteady fluid flows, with only very few degrees of freedom, by employing directly eigenmodes extracted from digital particle image velocimetry experimental data. In particular, we formulate standard Galerkin and nonlinear Galerkin approximations of the incompressible Navier–Stokes equations using hierarchical empirical eigenfunctions extracted from an ensemble of flow snapshots. We demonstrate that standard Galerkin approaches produce simulations capable of capturing the short–term dynamics of the flow, but nonlinear Galerkin projections are more effective in capturing both the short– and long–term dynamics, leading to bounded solutions. These findings are documented by applying these approaches to flow past a stationary circular cylinder at Reynolds number 610.

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high-speed high-altitude aerodynamics and the development of micrometre-sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non-equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non-equilibrium flows.

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

On the Construction and Comparison of Difference Schemes

Thermal (or electrical) conduction through a stationary, random close packed granular material is studied by numerical simulation. The particles are smooth uniform spheres with either point contact...

Numerical simulation of a square cavity was conducted to validate an implementation of an Immersed Boundary Method (IBM). The cavity consists of two differentially heated side walls and adiabatic top and bottom walls. A Cartesian grid is used with a finite volume, fractional step pressure correction method. Simulations use Dirichlet boundaries for vertical walls and Neumann boundaries for horizontal walls. The Immersed Boundary Method involves modifying the Navier--Stokes equations to include a forcing function in the momentum and energy equations that creates a virtual boundary. This method is useful because the boundary does not necessarily have to coincide with grid points; however, it is much less computationally expensive than other similar methods such as the cut cell method. The IBM is commonly used in simulations involving complex objects and can also accommodate moving boundaries. A standard numerical simulation with grid aligned with the boundary is first compared with previous results. The same geometry is then simulated by tilting the grid at various angles and using the IBM for each of the walls, and comparing these results with those initially obtained. We detail the implementation method and common problems associated with this. Velocity and temperature profiles are presented and the IBM is shown to maintain second order spatial accuracy. References E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161(1):35--60, 2000. doi:10.1006/jcph.2000.6484 T. Gao, Y. H. Tseng, and X. Y. Lu. An improved hybrid cartesian/immersed boundary method for fluid-solid flows. International Journal for Numerical Methods in Fluids, 55(12):1189--1211, 2007. doi:10.1002/fld.1522 M. P. Kirkpatrick and S. W. Armfield. Experimental and large eddy simulation results for the purging of salt water from a cavity by an overflow of fresh water. International Journal of Heat and Mass Transfer, 48(2):341--359, 2005. doi:10.1016/j.ijheatmasstransfer.2004.08.016 M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent. A representation of curved boundaries for the solution of the navier-stokes equations on a staggered three-dimensional cartesian grid. Journal of Computational Physics, 184(1):1--36, 2003. doi:10.1016/S0021-9991(02)00013-X B. P. Leonard and Simin Mokhtari. Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection. International Journal for Numerical Methods in Engineering, 30(4):729--766, 1990. doi:10.1002/nme.1620300412 S. H. Peng and L. Davidson. Large eddy simulation for turbulent buoyant flow in a confined cavity. volume 22, pages 323--331, 2001. doi:10.1016/S0142-727X(01)00095-9 C. S. Peskin. Numerical-analysis of blood-flow in heart. Journal of Computational Physics, 25(3):220--252, 1977. doi:10.1016/0021-9991(77)90100-0 J. Salat, S. Xin, P. Joubert, A. Sergent, F. Penot, and P. Le Quere. Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity. International Journal of Heat and Fluid Flow, 25:824--832, 2004. doi:10.1016/j.ijheatfluidflow.2004.04.003 Y. H. Tseng and J. H. Ferziger. A ghost-cell immersed boundary method for flow in complex geometry. Journal of Computational Physics, 192(2):593--623, 2003. doi:10.1016/j/jcp.2003.07.024 N. Zhang and Z. C. Zheng. An improved direct-forcing immersed-boundary method for finite difference applications. Journal of Computational Physics, 221(1):250--268, 2007. doi:10.1016/j.jcp.2006.06.012 N. Zhang, Z. C. Zheng, and S. Eckels. Study of heat-transfer on the surface of a circular cylinder in flow using an immersed-boundary method. International Journal of Heat and Fluid Flow, 29(6):1558--1566, 2008. doi:10.1016/j.ijheatfluidflow.2008.08.009

The Continuum Surface Force technique is a popular tool used to implement surface tension forces in Eulerian based Computational Fluid Dynamics codes. Under this technique, inaccuracies present in calculating the surface tension force can manifest themselves as `parasitic' currents. We detail a new Combined Level Set and Volume of Fluid implementation of the Continuum Surface Force method. We then develop a correlation for the magnitude of parasitic currents that are generated under this new method, as a function of both the numerical and physical parameters employed in a simulation. A set of numerical experiments validate this correlation and show that, importantly, the magnitude of currents produced by the method decreases as the computational cell size reduced. References J. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modelling surface tension. Journal of Computational Physics, 100:335--354, 1992. Dalton J. E. Harvie, M. R. Davidson, and Murray Rudman. An analysis of parasitic current generation in volume of fluid simulations. Appl. Math Mod., 30(10):1056--1066, 2006. doi:10.1016/j.apm.2005.08.015 D. B. Kothe and R. C. Mjolsness. RIPPLE: A new model for incompressible flows with free surfaces. American Institue of Aeronautics and Astronautics, 30 (11):2694--2700, 1992. Bruno Lafaurie, Carlo Nardone, Ruben Scardovelli, Stephane Zaleski, and Gianluigi Zanetti. Modelling merging and fragmentation in multiphase flows with SURFER. Journal of Computational Physics, 113:134--147, 1994. Murray Rudman. A volume-tracking method for incompressible multifluid flows with large density variations. International Journal for Numerical Methods in Fluids, 28:357--378, 1998. Ruben Scardovelli and Stepahe Zaleski. direct numerical simulation of free-surface and interfacial flow. Annual Review of Fluid Mechanics, 31:567--603, 1999. J. A. Sethian and Peter Smereka. Level set methods for fluid interfaces. Annual Review of Fluid Mechanics, 35:341--372, 2003. doi:10.1146/annurev.fluid.35.101101.161105 Mark Sussman. A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles. Journal of Computational Physics, 187:110--136, 2003. doi:10.1016/S0021-9991(03)00087-1 Mark Sussman and Elbridge Gerry Puckett. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. Journal of Computational Physics, 162:301--337, 2000. doi:10.1006/jcph.2000.6537 Mark Sussman, Peter Smereka, and Stanley Osher. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 114:146--159, 1994. S. P. van der Pijl, A. Segal, C. Vuik, and P. Wesseling. A mass-conserving level-set method for modelling of multi-phase flows. International Journal for Numerical Methods in Fluids, 47:339--361, 2005. doi:10.1002/fld.817

This survey is a review of technical papers in the area of numerical heat transfer (NHT) published in 1990 and 1991 and is a continuation of past literature surveys. The journals surveyed in this article include (1) Numerical Heat Transfer, Part A (2) Numerical Heat Transfer, Part B (3) International Journal for Numerical Methods in Fluids, (4) ASME Journal of Heat Transfer (5) International Journal of Heat and Mass Transfer, (6) Journal of Computational Physics, and (7) AIAA Journal. Several other archival journals, such as Journal of Fluid Mechanics, International Journal for Numerical Methods in Engineering, SIAM Journal of Numerical Analysis, Computers and Fluids, Symposia (International) on Combustion, Combustion and Flames, Computer Methods in Applied Mechanics and Engineering, Communication on Pure and Applied Mathematics, and Applied Mathematical Modeling, occasionally publish works in the area of NHT. However, they are not surveyed. It is hoped that the relevant papers are cited in the references listed in this article. For ease of presentation, the survey is divided into 16 technical categories. Naturally, many papers cover more than one topic. In this case, they are categorized under the main topic and are cross referenced under others. Also, readers may be interested in amore » particular physical phenomenon or a particular method that is described in more than one of these categories. To facilitate the readers' search, the authors list some of these topics with related papers.« less

A computational framework is developed which couples a series of models, each describing vastly different physical events, in order to characterize particle growth (agglomeration) in thermochemically reacting granular flows. The modelling is purposely simplified to expose the dominant mechanisms which control agglomeration. The overall system is comprised of relatively simple coupled submodels describing impact, heat production, bonding and fragmentation, each of which can be replaced by more elaborate descriptions, if and when they are available. Inverse problems, solved with a genetic algorithm, are then constructed to ascertain system parameters which maximize agglomeration likelihood within a range of admissible data.

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Aircraft icing

References ^Zaroodny, S. J. and Greenberg, M. D., On a Sheet Approach to the Numerical Calculation of Water Waves, Journal of Computational Physics, Vol. 11, March 1973, pp. 440-446. Zalosh, R. C., Discretized Simulation of Sheet Evolution with Buoyancy and Surface Tension Effects, AIAA Journal, Vol. 14, Nov. 1976, pp. 1517-1523. Daly, B. J., A Technique for Including Surface Tension Effects in Hydrodynamic Calculations, Journal of Computational Physics, Vol.4, 1969, pp. 97-117. Baker, G. R., Meiron, D. I., and Orszag, S. A., Vortex Simulations of the Rayleigh-Taylor Instability, Physics of Fluids, Vol. 23, Aug. 1980, pp. 1485-1490. Clements, R. R. and Maull, D. J., The Representation of Sheets of Vorticity by Discrete Vortices, Progress in Aerospace Sciences, Vol. 16, 1975, pp. 129-146. Bromilow, I. G. and Clements, R. R., Some Techniques for Extending the Application of the Discrete Method of Flow Simulation, Aeronautical Quarterly, Vol. 33, May 1982, pp. 73-89. Rosenhead, L., The Formation of Vortices from a Surface of Discontinuity, Proceedings of the Royal Society, Ser. A, Vol. 134, 1931, pp. 170-192. Hama, F. R. and Burke, E. R., On the Rolling up of a Sheet, University of Maryland Technical Note BN-220, 1960. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961, pp. 428-435. Drazin, P. G., Kelvin-Helmholtz Instability of Finite Amplitude, Journal of Fluid Mechanics, Vol. 42, June 1970, pp. 321335.

Approximate Integration of Stochastic Differential Equations