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1

Mora Pérez, M., G. López Patiño, M. A. Bengochea Escribano, and P. A. López Jiménez. "Cuantificación de la eficiencia de la fachada cerámica ventilada mediante técnicas de la mecánica de fluidos computacional." Boletín de la Sociedad Española de Cerámica y Vidrio 50, no. 2 (April 30, 2011): 99–108. http://dx.doi.org/10.3989/cyv.142011.

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2

Sandrakov, Gennadiy. "Computational Fluid Mechanics with Phase Transitions by Particle Methods." Modeling Control and Information Technologies, no. 6 (November 22, 2023): 90–91. http://dx.doi.org/10.31713/mcit.2023.025.

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A computational method of simulations for processes of heterogeneous hydrodynamics with take of phase transitions will be discussed. The method is based on relevant approximation of conservation laws for mass, momentum, and energy in integral and differential forms. The time and spatial approximation is natural and numerical simulations are realized as direct computer experiments. It is supposed that the fluids are compressible and non-viscous. Heterogeneities of the fluids are considered as small drops or particles of one fluid within other fluid. Total number of the drops may be large enough and the drops may have phase transitions. Therefore, simulations of the main fluid with small transited drops dynamics are considered. The particle dynamics will be modelled as in the particle-in-cell method, and in the main fluid as in the large particle method. This approach makes it possible to simulate phase transitions under certain assumptions about heterogeneous fluids. The calculation algorithm of this method is implemented as a computer simulation of the dynamics of a multiphase carrier fluid containing particles that can undergo, for example, graphite-diamond phase transitions. Such transitions are modelled on the basis of the theory of phase transformations and the laws of thermodynamics. In fact, the method is a combination of the Harlow's particle-in-cell method, Belotserkovskii's large particles method and Bakhvalov's homogenization method. A modification of this method has also been developed to take into account the effects of viscosity when simulating the dynamics of a multiphase fluid in porous media. A model of the motion of such a liquid in a porous medium is obtained by freezing the motion of particles of the corresponding size in the presented method. The method will certainly be promising for numerical simulations of absorption and diffusion processes in complex fluids with phase transitions.
3

Zamora, Blas, Antonio S. Kaiser, and Pedro G. Vicente. "Improvement in Learning on Fluid Mechanics and Heat Transfer Courses Using Computational Fluid Dynamics." International Journal of Mechanical Engineering Education 38, no. 2 (April 2010): 147–66. http://dx.doi.org/10.7227/ijmee.38.2.6.

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This paper is concerned with the teaching of fluid mechanics and heat transfer on courses for the industrial engineer degree at the Polytechnic University of Cartagena (Spain). In order to improve the engineering education, a pedagogical method that involves project-based learning, using computational fluid dynamics (CFD), was applied. The project-based learning works well for mechanical engineering education, since it prepares students for their later professional training. The courses combined applied and advanced concepts of fluid mechanics with the basic numerical aspects of CFD, including validation of the results obtained. In this approach, the physical understanding of practical problems of fluid mechanics and heat transfer played an important role. Satisfactory numerical results were obtained by using both Phoenics and Fluent finite-volume codes. Some cases were solved using the well known Matlab software. Comparisons were made between the results obtained by analytical solutions (if any) with those reached by CFD general-purpose codes and with those obtained by Matlab. This system provides engineering students with a solid comprehension of several aspects of thermal and fluids engineering.
4

Kim, Youngho, and Sangho Yun. "Fluid Dynamics in an Anatomically Correct Total Cavopulmonary Connection : Flow Visualizations and Computational Fluid Dynamics(Cardiovascular Mechanics)." Proceedings of the Asian Pacific Conference on Biomechanics : emerging science and technology in biomechanics 2004.1 (2004): 57–58. http://dx.doi.org/10.1299/jsmeapbio.2004.1.57.

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Chen, Yinwei, and Suzanne Garcia. "Preface: 1st International Conference on Fluid Mechanics, Computational Mathematics and Physics (FMCMP 2023)." Highlights in Science, Engineering and Technology 77 (November 29, 2023): I. http://dx.doi.org/10.54097/hset.v77i.13926.

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The 1st International Conference on Fluid Mechanics, Computational Mathematics and Physics (FMCMP 2023) was held in Xi'an City, China during November 18-19, 2023, it included original and peer-reviewed research papers. FMCMP is an annual international conference devoted to the discussion of original work on theoretical, computational, and experimental aspects of the mechanics of fluids, with special regards to the Navier-Stokes equations. FMCMP conference covers the field of Fluid Mechanics, Computational and Applied Mathematics, and Physics. FMCMP aims to provide a knowledge exchange and sharing platform of fluid mechanics, computational mathematics and physics for the aviation and aerospace industry, automobile industry, shipbuilding industry, etc. We welcome experts in fields related to fluid mechanics, computational mathematics and physics to join us. We would like to thank all the author submitted papers to this FMCMP 2023 and thank all the reviewers for their time and effort in reviewing articles. Especially we would like to thank the organizing committee for their valuable advices in the organization and helpful peer review of the papers. Organizing Committee of FMCMP 2023 Xi'an City, China
6

Lin, Guang, Xiaoliang Wan, Chau-hsing Su, and George Karniadakis. "Stochastic Computational Fluid Mechanics." Computing in Science and Engineering 9, no. 2 (March 2007): 21–29. http://dx.doi.org/10.1109/mcse.2007.38.

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7

Drikakis, Dimitris, Michael Frank, and Gavin Tabor. "Multiscale Computational Fluid Dynamics." Energies 12, no. 17 (August 25, 2019): 3272. http://dx.doi.org/10.3390/en12173272.

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Computational Fluid Dynamics (CFD) has numerous applications in the field of energy research, in modelling the basic physics of combustion, multiphase flow and heat transfer; and in the simulation of mechanical devices such as turbines, wind wave and tidal devices, and other devices for energy generation. With the constant increase in available computing power, the fidelity and accuracy of CFD simulations have constantly improved, and the technique is now an integral part of research and development. In the past few years, the development of multiscale methods has emerged as a topic of intensive research. The variable scales may be associated with scales of turbulence, or other physical processes which operate across a range of different scales, and often lead to spatial and temporal scales crossing the boundaries of continuum and molecular mechanics. In this paper, we present a short review of multiscale CFD frameworks with potential applications to energy problems.
8

HALLEZ, YANNICK, and JACQUES MAGNAUDET. "A numerical investigation of horizontal viscous gravity currents." Journal of Fluid Mechanics 630 (July 10, 2009): 71–91. http://dx.doi.org/10.1017/s0022112009006454.

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We study numerically the viscous phase of horizontal gravity currents of immiscible fluids in the lock-exchange configuration. A numerical technique capable of dealing with stiff density gradients is used, allowing us to mimic high-Schmidt-number situations similar to those encountered in most laboratory experiments. Plane two-dimensional computations with no-slip boundary conditions are run so as to compare numerical predictions with the ‘short reservoir’ solution of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), which predicts the front position lf to evolve as t1/5, and the ‘long reservoir’ solution of Gratton & Minotti (J. Fluid Mech., vol. 210, 1990, pp. 155–182) which predicts a diffusive evolution of the distance travelled by the front xf ~ t1/2. In line with dimensional arguments, computations indicate that the self-similar power law governing the front position is selected by the flow Reynolds number and the initial volume of the released heavy fluid. We derive and validate a criterion predicting which type of viscous regime immediately succeeds the slumping phase. The computations also reveal that, under certain conditions, two different viscous regimes may appear successively during the life of a given current. Effects of sidewalls are examined through three-dimensional computations and are found to affect the transition time between the slumping phase and the viscous regime. In the various situations we consider, we make use of a force balance to estimate the transition time at which the viscous regime sets in and show that the corresponding prediction compares well with the computational results.
9

Urreta, Harkaitz, Gorka Aguirre, Pavel Kuzhir, and Luis Norberto Lopez de Lacalle. "Actively lubricated hybrid journal bearings based on magnetic fluids for high-precision spindles of machine tools." Journal of Intelligent Material Systems and Structures 30, no. 15 (July 13, 2019): 2257–71. http://dx.doi.org/10.1177/1045389x19862358.

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The research work reported in this article is focused on the use of magnetic fluids as active lubricant for improving the performance of hybrid journal bearings, with application to high-precision machine tools. Prototype design was optimized following numerical computation of Reynolds equation and computational fluid dynamics calculations, in both cases with Herschel–Bulkley model for the magnetorheological fluid. This fluid (LORD Corp. MRF 122-2ED) was experimentally characterized in detail. The improvement of the hydrodynamic effect in journal bearings was demonstrated with 50% higher load capacity and stiffness, mainly at half of shaft eccentricity 0.4 < ε < 0.7. Active hydrostatic lubrication achieved quasi-infinite stiffness within working limits (load and speed), at low frequencies. For high dynamic response, the active lubrication based on magnetorheological valves did not show good response. The feasibility of using magnetic fluids for developing high performance machine tool spindles and the validity of the simulation models was demonstrated experimentally.
10

Wu, Xiang, and Ling Feng Tang. "Review of Coupled Research for Mechanical Dynamics and Fluid Mechanics of Reciprocating Compressor." Applied Mechanics and Materials 327 (June 2013): 227–32. http://dx.doi.org/10.4028/www.scientific.net/amm.327.227.

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Research statuses of mechanical dynamics and fluid mechanics of a reciprocating compressor are reviewed respectively ,along with the presentation of coupled research for these two disciplines of a reciprocating compressor. Analyses for mechanical dynamics are focused on modal analysis and dynamic response analysis. Three methods can be adopted in dynamic response analysis,which are the combination of the formula derivation and finite element method, the combination of multi-rigid-body dynamics and finite element method , and thecombination of multi-flexible body dynamics and finite element method. Analytical models for fluid dynamics include 1-D computationalfluid dynamics model, 2-D computational fluid dynamics model and 3-D computational fluid dynamics model. In addition, limitations of researches for mechanical dynamics and fluid mechanics in a reciprocating compressor are also presented, as well as the prospect for the coupled research of two disciplines.
11

Newling, B., S. J. Gibbs, J. A. Derbyshire, D. Xing, L. D. Hall, D. E. Haycock, W. J. Frith, and S. Ablett. "Comparisons of Magnetic Resonance Imaging Velocimetry With Computational Fluid Dynamics." Journal of Fluids Engineering 119, no. 1 (March 1, 1997): 103–9. http://dx.doi.org/10.1115/1.2819094.

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The flow of Newtonian liquids through a pipe system comprising of a series of abrupt expansions and contractions has been studied using several magnetic resonance imaging (MRI) techniques, and also by computational fluid dynamics. Agreement between those results validates the assumptions inherent to the computational calculation and gives confidence to extend the work to more complex geometries and more complex fluids, wherein the advantages of MRI (utility in opaque fluids and noninvasiveness) are unique. The fluid in the expansion-contraction system exhibits a broad distribution of velocities and, therefore, presents peculiar challenges to the measurement technique. The MRI protocols employed were a two-dimensional tagging technique, for rapid flow field visualisation, and three-dimensional echo-planar and gradient-echo techniques, for flow field quantification (velocimetry). The Computational work was performed using the FIDAP package to solve the Navier-Stokes equations. The particular choice of parameters for both MRI and computational fluid dynamics, which affect the results and their agreement, have been addressed.
12

Spelce, T. "Finite element computational fluid mechanics." Finite Elements in Analysis and Design 1, no. 4 (December 1985): 389–90. http://dx.doi.org/10.1016/0168-874x(85)90035-6.

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13

Carey, Graham F. "Finite element computational fluid mechanics." Computer Methods in Applied Mechanics and Engineering 49, no. 2 (June 1985): 247–48. http://dx.doi.org/10.1016/0045-7825(85)90062-3.

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14

Shakibaeinia, Ahmad, and Amir Reza Zarrati. "Computational Fluid Mechanics and Hydraulics." Water 14, no. 24 (December 7, 2022): 3985. http://dx.doi.org/10.3390/w14243985.

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15

Liu, Jianing. "Current Situation and Prospect of Computational Fluid Dynamics in Automotive Design." Highlights in Science, Engineering and Technology 37 (March 18, 2023): 392–96. http://dx.doi.org/10.54097/hset.v37i.6103.

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With the continuous innovation and progress of automobile technology, people have put forward higher and stricter requirements on the safety and environmental protection of automobiles. Therefore, the air resistance, surface pressure, aerodynamic lift, aerodynamic side force and other mechanical problems must be considered comprehensively in the design of automobiles. Fluid mechanics is the study of the motion of gases and liquids under the action of various forces and the application of the discipline. All objects moving in the earth's atmosphere are affected by fluid mechanics, so fluid mechanics has a very important guiding significance to the design of vehicle. This paper is mainly aimed at the influence of air resistance on the performance of the car, the fluid mechanics Angle analysis of the role of the car streamlined design, the principle of the car streamlined body, the shape of the streamline several aspects of the comprehensive introduction of fluid mechanics in the car body design application.
16

Takizawa, Kenji, Yuri Bazilevs, and Tayfun E. Tezduyar. "Computational fluid mechanics and fluid–structure interaction." Computational Mechanics 50, no. 6 (September 18, 2012): 665. http://dx.doi.org/10.1007/s00466-012-0793-8.

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17

Kendall, S. R., and H. V. Rao. "Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 214, no. 11 (November 1, 2000): 1401–7. http://dx.doi.org/10.1243/0954406001523371.

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Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.
18

Illidge-Araujo, JorgeMario, Jorge Luis Chacon Velasco, Angel José Chacon Velasco, and Carlos A. Romero Piehadraita. "Diseño y simulación de un sistema pico-hydro para la generación de energía eléctrica en zonas rurales, mediante un software de mecánica de fluidos computacional." Revista UIS Ingenierías 19, no. 1 (January 1, 2020): 155–70. http://dx.doi.org/10.18273/revuin.v19n1-2020015.

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En este artículo se presenta un procedimiento para el diseño de todos los componentes de un sistema pico-hydro, a partir de una turbina hidráulica tipo hélice de acuerdo a las condiciones específicas del potencial del agua para un sitio estimado de operación basado en un análisis teórico y técnico. Para este fin, las principales características del rodete se determinan por medio de correlaciones estadísticas de diferentes autores que han estudiado turbinas instaladas alrededor del mundo, y definiendorestricciones para el diseño tales como el salto de la turbina, el caudal nominal y la potencia requerida, a partir de los datos mencionados anteriormente, se establece el valor de todas las variables relacionadas con el comportamiento del fluido en su paso por el rodete y a partir del valor de dichas variables y de la geometría establecida para el rodete, se procede a determinar la geometría y las especificaciones de los demás componentes del sistema pico-hydro tales como la cámara espiral, el tubo de aspiración, el generador y el distribuidor, para el cual se estudiaron dos tipos que son un distribuidor de entrada axial del fluido y otro de entrada radial del fluido con respecto al eje de rotación de la turbina. Para la verificación del diseño y de los resultados esperados, se utiliza una herramienta moderna de ingeniería como lo es la dinámica de fluidos computacional (CFD), en especial el componente (CFX) para predecir el flujo y el rendimiento que puede arrojar el sistema diseñado. Por último,se procede a realizar un análisis técnico-económico para estudiar la viabilidad de implementar este tipo de sistemas en una zona rural.El diseño del presente sistema pico-hydro y el aporte de la dinámica de fluidos computacional CFD, puede ser una alternativa viable para suplir las demandas de energía eléctrica de una zona rural en Colombia que no cuente con este servicio por parte de las redes de suministro encargadas de esta función
19

TAKIZAWA, KENJI, and TAYFUN E. TEZDUYAR. "SPACE–TIME FLUID–STRUCTURE INTERACTION METHODS." Mathematical Models and Methods in Applied Sciences 22, supp02 (July 25, 2012): 1230001. http://dx.doi.org/10.1142/s0218202512300013.

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Since its introduction in 1991 for computation of flow problems with moving boundaries and interfaces, the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation has been applied to a diverse set of challenging problems. The classes of problems computed include free-surface and two-fluid flows, fluid–object, fluid–particle and fluid–structure interaction (FSI), and flows with mechanical components in fast, linear or rotational relative motion. The DSD/SST formulation, as a core technology, is being used for some of the most challenging FSI problems, including parachute modeling and arterial FSI. Versions of the DSD/SST formulation introduced in recent years serve as lower-cost alternatives. More recent variational multiscale (VMS) version, which is called DSD/SST-VMST (and also ST-VMS), has brought better computational accuracy and serves as a reliable turbulence model. Special space–time FSI techniques introduced for specific classes of problems, such as parachute modeling and arterial FSI, have increased the scope and accuracy of the FSI modeling in those classes of computations. This paper provides an overview of the core space–time FSI technique, its recent versions, and the special space–time FSI techniques. The paper includes test computations with the DSD/SST-VMST technique.
20

Agrawal, Manoj Kumar, T. Saritha Kumari, Preeti Maan, Bhishm Pratap, Muthana Saleh Mashkour, and Vishal Sharma. "Coupled Multiphysics Simulation using FEA for Complex Fluid-Structure Interaction Problems." E3S Web of Conferences 430 (2023): 01116. http://dx.doi.org/10.1051/e3sconf/202343001116.

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In the realm of mechanical engineering, the accurate prediction of fluid-structure interaction (FSI) is paramount for the design and analysis of systems where fluids and structures coexist and interact. This research paper presents a novel approach to address complex FSI problems using coupled multiphysics simulation through Finite Element Analysis (FEA). The proposed methodology integrates advanced computational algorithms to capture the intricate interplay between fluid dynamics and structural mechanics, ensuring a more holistic representation of real-world scenarios. The developed framework was tested on a variety of benchmark problems, ranging from aeroelastic flutter in aircraft wings to blood flow-induced stresses in arterial walls. Results indicate a significant enhancement in prediction accuracy and computational efficiency compared to traditional decoupled methods. Furthermore, the study delves into the challenges faced during the coupling process, offering solutions to mitigate numerical instabilities and enhance convergence rates. The findings of this research not only pave the way for improved design and safety protocols in industries such as aerospace, biomedical, and civil engineering but also underscore the potential of Multiphysics simulation in unravelling the complexities of the natural world.
21

Olejnik, Aleksander, Łukasz Kiszkowiak, and Adam Dziubiński. "Aerodynamic analysis of General Aviation airplanes using computational fluid dynamics methods." Mechanik 90, no. 8-9 (September 11, 2017): 802–4. http://dx.doi.org/10.17814/mechanik.2017.8-9.118.

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The problems of an aircraft aerodynamic analysis based on the example of Very Light Aeroplanes and Very Light Jet category airplanes have been presented. A numerical calculations using finite volume method implemented in specialized software were performed. A method of preparing a numerical model of an airplane and the aerodynamic analysis methodology have been presented. An influence of an airplane propulsion on aerodynamic characteristics have been analyzed. A results have been shown in the graphs form of aerodynamic force and moment components as function of angle of attack.
22

Kanai, Taro, Kenji Takizawa, Tayfun E. Tezduyar, Kenji Komiya, Masayuki Kaneko, Kyohei Hirota, Motohiko Nohmi, Tomoki Tsuneda, Masahito Kawai, and Miho Isono. "Methods for computation of flow-driven string dynamics in a pump and residence time." Mathematical Models and Methods in Applied Sciences 29, no. 05 (May 2019): 839–70. http://dx.doi.org/10.1142/s021820251941001x.

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We present methods for computation of flow-driven string dynamics in a pump and related residence time. The string dynamics computations help us understand how the strings carried by a fluid interact with the pump surfaces, including the blades, and get stuck on or around those surfaces. The residence time computations help us to have a simplified but quick understanding of the string behavior. The core computational method is the Space–Time Variational Multiscale (ST-VMS) method, and the other key methods are the ST Isogeometric Analysis (ST-IGA), ST Slip Interface (ST-SI) method, ST/NURBS Mesh Update Method (STNMUM), a general-purpose NURBS mesh generation method for complex geometries, and a one-way-dependence model for the string dynamics. The ST-IGA with NURBS basis functions in space is used in both fluid mechanics and string structural dynamics. The ST framework provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the turbulent nature of the unsteady flow, and the moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the pump geometry and increased accuracy in the flow solution. The IGA discretization also enables increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and fluid dynamics forces computed on the string. The STNMUM enables exact representation of the mesh rotation. The general-purpose NURBS mesh generation method makes it easier to deal with the complex geometry we have here. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small.
23

Mazumder, Sandip. "Modeling Full-Scale Monolithic Catalytic Converters: Challenges and Possible Solutions." Journal of Heat Transfer 129, no. 4 (July 24, 2006): 526–35. http://dx.doi.org/10.1115/1.2709655.

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Modeling full-scale monolithic catalytic converters using state-of-the-art computational fluid dynamics algorithms and techniques encounters a classical multiscale problem: the channels within the monolith have length scales that are ∼1–2 mm, while the converter itself has a length scale that is ∼5–10 cm. This necessitates very fine grids to resolve all the length scales, resulting in few million computational cells. When complex heterogeneous chemistry is included, the computational problem becomes all but intractable unless massively parallel computation is employed. Two approaches to address this difficulty are reviewed, and their effectiveness demonstrated for the computation of full-scale catalytic converters with complex chemistry. The first approach is one where only the larger scales are resolved by a grid, while the physics at the smallest scale (channel scale) are modeled using subgrid scale models whose development entails detailed flux balances at the “imaginary” fluid–solid interfaces within each computational cell. The second approach makes use of the in situ adaptive tabulation algorithm, after significant reformulation of the underlying mathematics, to accelerate computation of the surface reaction boundary conditions. Preliminary results shown here for a catalytic combustion application involving 19 species and 24 reactions indicate that both methods have the potential of improving computational efficiency by several orders of magnitude.
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Douglass, R. W., and J. D. Ramshaw. "Perspective: Future Research Directions in Computational Fluid Dynamics." Journal of Fluids Engineering 116, no. 2 (June 1, 1994): 212–15. http://dx.doi.org/10.1115/1.2910256.

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The current state of computational fluid dynamics (CFD) has yet to reach its full promise as a general tool for engineering design and simulation. Research in the areas of code robustness, complex flows of real fluids, and numerical errors and resolution are proposed as directions aiming toward that goal. We illustrate some of the current CFD challenges using selected applications.
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Battaglia, Laura, Jorge D’Elía, Mario Storti, and Norberto Nigro. "Numerical Simulation of Transient Free Surface Flows Using a Moving Mesh Technique." Journal of Applied Mechanics 73, no. 6 (February 28, 2006): 1017–25. http://dx.doi.org/10.1115/1.2198246.

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In this work, transient free surface flows of a viscous incompressible fluid are numerically solved through parallel computation. Transient free surface flows are boundary-value problems of the moving type that involve geometrical nonlinearities. In contrast to more conventional computational fluid dynamics problems, the computational flow domain is partially bounded by a free surface which is not known a priori, since its shape must be computed as part of the solution. In steady flow the free surface is obtained by an iterative process, but when the free surface evolves with time the problem is more difficult as it generates large distortions in the computational flow domain. The incompressible Navier-Stokes numerical solver is based on the finite element method with equal order elements for pressure and velocity (linear elements), and it uses a streamline upwind/Petrov-Galerkin (SUPG) scheme (Hughes, T. J. R., and Brooks, A. N., 1979, “A Multidimensional Upwind Scheme With no Crosswind Diffusion,” in Finite Element Methods for Convection Dominated Flows, ASME ed., 34. AMD, New York, pp. 19–35, and Brooks, A. N., and Hughes, T. J. R., 1982, “Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations,” Comput. Methods Appl. Mech. Eng., 32, pp. 199–259) combined with a Pressure-Stabilizing/Petrov-Galerkin (PSPG) one (Tezduyar, T. E., 1992, “Stablized Finite Element Formulations for Incompressible Flow Computations,” Adv. Appl. Mech., 28, pp. 1–44, and Tezduyar, T. E., Mittal, S., Ray, S. E., and Shih, R., 1992, “Incompressible Flow Computations With Stabilized Bilinear and Linear Equal Order Interpolation Velocity-Pressure Elements,” Comput. Methods Appl. Mech. Eng., 95, pp. 221–242). At each time step, the fluid equations are solved with constant pressure and null viscous traction conditions at the free surface and the velocities obtained in this way are used for updating the positions of the surface nodes. Then, a pseudo elastic problem is solved in the fluid domain in order to relocate the interior nodes so as to keep mesh distortion controlled. This has been implemented in the PETSc-FEM code (PETSc-FEM: a general purpose, parallel, multi-physics FEM program. GNU general public license (GPL), http://www.cimec.org.ar/petscfem) by running two parallel instances of the code and exchanging information between them. Some numerical examples are presented.
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Toman, Usama T., Abdel-Karim SO Hassan, Farouk M. Owis, and Ahmed SA Mohamed. "Blade shape optimization of an aircraft propeller using space mapping surrogates." Advances in Mechanical Engineering 11, no. 7 (July 2019): 168781401986507. http://dx.doi.org/10.1177/1687814019865071.

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Propeller performance greatly influences the overall efficiency of the turboprop engines. The aim of this study is to perform a propeller blade shape optimization for maximum aerodynamic efficiency with a minimal number of high-fidelity model evaluations. A physics-based surrogate approach exploiting space mapping is employed for the design process. A space mapping algorithm is utilized, for the first time in the field of propeller design, to link two of the most common propeller analysis models: the classical blade-element momentum theory to be the coarse model; and the high-fidelity computational fluid dynamics tool as the fine model. The numerical computational fluid dynamics simulations are performed using the finite-volume discretization of the Reynolds-averaged Navier–Stokes equations on an adaptive unstructured grid. The optimum design is obtained after few iterations with only 56 computationally expensive computational fluid dynamics simulations. Furthermore, an optimization method based on design of experiments and kriging response surface is used to validate the results and compare the computational efficiency of the two techniques. The results show that space mapping is more computationally efficient.
27

Benra, Friedrich-Karl, Hans Josef Dohmen, Ji Pei, Sebastian Schuster, and Bo Wan. "A Comparison of One-Way and Two-Way Coupling Methods for Numerical Analysis of Fluid-Structure Interactions." Journal of Applied Mathematics 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/853560.

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The interaction between fluid and structure occurs in a wide range of engineering problems. The solution for such problems is based on the relations of continuum mechanics and is mostly solved with numerical methods. It is a computational challenge to solve such problems because of the complex geometries, intricate physics of fluids, and complicated fluid-structure interactions. The way in which the interaction between fluid and solid is described gives the largest opportunity for reducing the computational effort. One possibility for reducing the computational effort of fluid-structure simulations is the use of one-way coupled simulations. In this paper, different problems are investigated with one-way and two-way coupled methods. After an explanation of the solution strategy for both models, a closer look at the differences between these methods will be provided, and it will be shown under what conditions a one-way coupling solution gives plausible results.
28

Kolditz,, O., and LA Glenn,. "Computational Methods in Environmental Fluid Mechanics." Applied Mechanics Reviews 55, no. 6 (October 16, 2002): B117—B118. http://dx.doi.org/10.1115/1.1508157.

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Verzicco, R. "Computational Methods for Environmental Fluid Mechanics." European Journal of Mechanics - B/Fluids 21, no. 4 (January 2002): 493–94. http://dx.doi.org/10.1016/s0997-7546(02)01194-9.

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30

Mohammadi, B., and G. Puigt. "Wall functions in computational fluid mechanics." Computers & Fluids 35, no. 10 (December 2006): 1108–15. http://dx.doi.org/10.1016/j.compfluid.2005.02.009.

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31

I., E., Dale A. Anderson, John C. Tannehill, and Richard H. Pletcher. "Computational Fluid Mechanics and Heat Transfer." Mathematics of Computation 46, no. 174 (April 1986): 764. http://dx.doi.org/10.2307/2008017.

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32

Schmidt, Frank W. "Computational fluid mechanics and heat transfer." International Journal of Heat and Fluid Flow 7, no. 3 (September 1986): 239. http://dx.doi.org/10.1016/0142-727x(86)90028-7.

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Schmidt, Frank W. "Computational fluid mechanics and heat transfer." International Journal of Heat and Fluid Flow 7, no. 1 (March 1986): 27. http://dx.doi.org/10.1016/0142-727x(86)90038-x.

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34

Nguyen, Vinh-Tan, Jason Yu Chuan Leong, Satoshi Watanabe, Toshimitsu Morooka, and Takayuki Shimizu. "A Multi-Fidelity Model for Simulations and Sensitivity Analysis of Piezoelectric Inkjet Printheads." Micromachines 12, no. 9 (August 29, 2021): 1038. http://dx.doi.org/10.3390/mi12091038.

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The ink drop generation process in piezoelectric droplet-on-demand devices is a complex multiphysics process. A fully resolved simulation of such a system involves a coupled fluid–structure interaction approach employing both computational fluid dynamics (CFD) and computational structural mechanics (CSM) models; thus, it is computationally expensive for engineering design and analysis. In this work, a simplified lumped element model (LEM) is proposed for the simulation of piezoelectric inkjet printheads using the analogy of equivalent electrical circuits. The model’s parameters are computed from three-dimensional fluid and structural simulations, taking into account the detailed geometrical features of the inkjet printhead. Inherently, this multifidelity LEM approach is much faster in simulations of the whole inkjet printhead, while it ably captures fundamental electro-mechanical coupling effects. The approach is validated with experimental data for an existing commercial inkjet printhead with good agreement in droplet speed prediction and frequency responses. The sensitivity analysis of droplet generation conducted for the variation of ink channel geometrical parameters shows the importance of different design variables on the performance of inkjet printheads. It further illustrates the effectiveness of the proposed approach in practical engineering usage.
35

Yang, Fan, Zhufeng Yue, and Lei Li. "The aeroelastic characteristics of high aspect ratio wing." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 230, no. 14 (August 6, 2016): 2543–56. http://dx.doi.org/10.1177/0954410016629497.

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Owing to the elasticity, the large deformation was brought in the high aspect ratio wing in the flight. The large deformation had a great influence on the flight performance. In this paper, the loosely coupled method was used for the research of high aspect ratio wing aeroelastic problems. The Navier–Stokes equations were solved for fluid domain computation, and the nonlinear finite element method was adopted for solid domain computation. The data exchange program and mesh regeneration progress were adopted for fluid–structure interface problem. Finally, the aerodynamic characteristics of high aspect ratio wing were obtained under different fly conditions. In addition, to validate the proposed method, the flutter analysis of AGARD 445.6 wing is carried out and compared with the experimental data. The numerical result validates the proposed computational fluid dynamics/computational structural mechanics method.
36

Zlenkiewicz, O. C. "ADAPTIVITY-FLUIDS-LOCALIZATION: THE CHALLENGE TO COMPUTATIONAL MECHANICS." Transactions of the Canadian Society for Mechanical Engineering 15, no. 2 (June 1991): 137–45. http://dx.doi.org/10.1139/tcsme-1991-0008.

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37

Evstigneev, Nikolay M., and Oleg I. Ryabkov. "Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs." Mathematics 11, no. 20 (October 18, 2023): 4336. http://dx.doi.org/10.3390/math11204336.

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In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.
38

Grimberg, Sebastian, and Charbel Farhat. "Fast computation of the wall distance in unsteady Eulerian fluid-structure computations." International Journal for Numerical Methods in Fluids 89, no. 4-5 (October 11, 2018): 143–61. http://dx.doi.org/10.1002/fld.4686.

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39

Win, Ko Ko, and A. N. Temnov. "A THEORETICAL STUDY OF OSCILLATIONS OF TWO IMMISCIBLE FLUIDS IN A LIMITED TANK." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 69 (2021): 97–113. http://dx.doi.org/10.17223/19988621/69/8.

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In the paper, the nonlinear oscillations of a two-layer fluid that completely fills a limited tank are theoretically studied. To determine any smooth function on the deflected interface, the Taylor series expansions are considered using the values of the function and its normal derivatives on the undisturbed interface of the fluids. Using two fundamental asymmetric harmonics, which are generated in two mutually perpendicular planes, the differential equations of nonlinear oscillations of the two-layer fluid interface are investigated. As a result, the frequency-response characteristics are presented and the instability regions of the forced oscillations of the two-layer fluid in the cylindrical tank are plotted, as well as the parametric resonance regions for different densities of the upper and lower fluids. The Bubnov-Galerkin method is used to plot instability regions for the approximate solution to nonlinear differential equations. At the final stage of the work, the nonlinear effects resulting from the interaction of fluids with a rigid tank that executes harmonic oscillations at the interface of the fluids are theoretically studied.
40

Oldenburg, Jan, Julian Renkewitz, Michael Stiehm, and Klaus-Peter Schmitz. "Contributions towards Data driven Deep Learning methods to predict Steady State Fluid Flow in mechanical Heart Valves." Current Directions in Biomedical Engineering 7, no. 2 (October 1, 2021): 625–28. http://dx.doi.org/10.1515/cdbme-2021-2159.

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Abstract It is commonly accepted that hemodynamic situation is related with cardiovascular diseases as well as clinical post-procedural outcome. In particular, aortic valve stenosis and insufficiency are associated with high shear flow and increased pressure loss. Furthermore, regurgitation, high shear stress and regions of stagnant blood flow are presumed to have an impact on clinical result. Therefore, flow field assessment to characterize the hemodynamic situation is necessary for device evaluation and further design optimization. In-vitro as well as in-silico fluid mechanics methods can be used to investigate the flow through prostheses. In-silico solutions are based on mathematical equitation’s which need to be solved numerically (Computational Fluid Dynamics - CFD). Fundamentally, the flow is physically described by Navier-Stokes. CFD often requires high computational cost resulting in long computation time. Techniques based on deep-learning are under research to overcome this problem. In this study, we applied a deep-learning strategy to estimate fluid flows during peak systolic steady-state blood flows through mechanical aortic valves with varying opening angles in randomly generated aortic root geometries. We used a data driven approach by running 3,500 two dimensional simulations (CFD). The simulation data serves as training data in a supervised deep learning framework based on convolutional neural networks analogous to the U-net architecture. We were able to successfully train the neural network using the supervised data driven approach. The results showing that it is feasible to use a neural network to estimate physiological flow fields in the vicinity of prosthetic heart valves (Validation error below 0.06), by only giving geometry data (Image) into the Network. The neural network generates flow field prediction in real time, which is more than 2500 times faster compared to CFD simulation. Accordingly, there is tremendous potential in the use of AIbased approaches predicting blood flows through heart valves on the basis of geometry data, especially in applications where fast fluid mechanic predictions are desired.
41

Benmansour, Abdeljalil, and Hacène Hamoudi. "MAGNETOHYDRODYNAMICS AND POWER-LAW FLUIDS IN DOUBLE LID-DRIVEN CAVITY WITH SEMI-CIRCULAR BODIES." Journal of the Serbian Society for Computational Mechanics 17, no. 2 (December 1, 2023): 125–41. http://dx.doi.org/10.24874/jsscm.2023.17.02.09.

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The use of complex fluids is one of the modern techniques used in small devices in order to enhance their thermal performance. This paper is a numerical study of a complex fluid imprisoned in a chamber with two hot bodies exposed to a magnetic field of constant intensity. The upper and lower walls move horizontally at a constant velocity, while the lateral sides are thermally insulated. The numerical simulation of the system was achieved based on the finite volume method that solves the differential equations of fluid mechanics and heat transfer. Simulations were carried out under the following conditions: Re = 1 to 40, Ri = 0 to 100, n = 0.6 to 1.4 and Ha = 0 to 100. The study showed that the thermal activity of the two bodies is different and related to initial condition. Also, the effect of the magnetic field is strong in the case of shear-thinning fluids, while its effect is diminished in the case of shear-thickening fluids.
42

MOEENDARBARY, E., T. Y. NG, and M. ZANGENEH. "DISSIPATIVE PARTICLE DYNAMICS: INTRODUCTION, METHODOLOGY AND COMPLEX FLUID APPLICATIONS — A REVIEW." International Journal of Applied Mechanics 01, no. 04 (December 2009): 737–63. http://dx.doi.org/10.1142/s1758825109000381.

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The dissipative particle dynamics (DPD) technique is a relatively new mesoscale technique which was initially developed to simulate hydrodynamic behavior in mesoscopic complex fluids. It is essentially a particle technique in which molecules are clustered into the said particles, and this coarse graining is a very important aspect of the DPD as it allows significant computational speed-up. This increased computational efficiency, coupled with the recent advent of high performance computing, has subsequently enabled researchers to numerically study a host of complex fluid applications at a refined level. In this review, we trace the developments of various important aspects of the DPD methodology since it was first proposed in the in the early 1990's. In addition, we review notable published works which employed DPD simulation for complex fluid applications.
43

Nakamura, Masanori, Shigeo Wada, Daisuke Mori, Ken-ichi Tsubota, and Takami Yamaguchi. "Computational Fluid Dynamics Study of the Effect of the Left Ventricular Flow Ejection on the Intraaortic Flow(Cardiovascular Mechanics)." Proceedings of the Asian Pacific Conference on Biomechanics : emerging science and technology in biomechanics 2004.1 (2004): 61–62. http://dx.doi.org/10.1299/jsmeapbio.2004.1.61.

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44

Akbarzadeh, Pooria, Mahmood Norouzi, Reza Ghasemi, and Seyed Zia Daghighi. "Experimental study on the entry of solid spheres into Newtonian and non-Newtonian fluids." Physics of Fluids 34, no. 3 (March 2022): 033111. http://dx.doi.org/10.1063/5.0081002.

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This study experimentally investigates the entry of hydrophobic/hydrophilic spheres into Newtonian and Boger fluids. By considering solution of 82% glycerin and 18% water and solution of 80% glycerin, 20% water and 100 ppm polyacrylamide, Newtonian and Boger fluids are made, respectively. It has been tried that liquids' surface tension, density, and viscosity are almost the same. Thus, all dimensionless numbers are approximately the same at a similar impact velocity except for the elasticity number. A PcoDimaxS highspeed camera captures the spheres' trajectory from the impact to the end of the path. Regarding the range of released height ([Formula: see text]), the impact velocities are approximately in the range of [Formula: see text]. The role of fluid elasticity in combination with the sphere surface wettability on the air cavity formation/evolution/collapse is mainly studied. Also, the kinetics of the sphere motion (velocity, acceleration, and hydrodynamic force coefficient) is studied. The results show that air drawn due to the sphere's impact with the Newtonian liquid is more, and the pinch-off takes place later. Also, shedding bubbles are cusped-shaped in the Boger fluid, while in the Newtonian fluid, they are elliptical. In addition, the most significant impact of surface wettability is observed in the Newtonian fluid. Finally, the results reveal that the sphere in the Newtonian fluid can move faster and travel a longer distance in a specific time interval. The differences observed are closely related to the viscoelastic fluid's elasticity property and extensional viscosity.
45

Longatte, E., Z. Bendjeddou, and M. Souli. "Application of Arbitrary Lagrange Euler Formulations to Flow-Induced Vibration Problems." Journal of Pressure Vessel Technology 125, no. 4 (November 1, 2003): 411–17. http://dx.doi.org/10.1115/1.1613950.

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Most classical fluid force identification methods rely on mechanical structure response measurements associated with convenient data processes providing turbulent and fluid-elastic forces responsible for possible vibrations and damage. These techniques provide good results; however, they often involve high costs as they rely on specific modelings fitted with experimental data. Owing to recent improvements in computational fluid dynamics, numerical simulation of flow-induced structure vibration problems is now practicable for industrial purposes. As far as flow structure interactions are concerned, the main difficulty consists in estimating numerically fluid-elastic forces acting on mechanical components submitted to turbulent flows. The point is to take into account both fluid effects on structure motion and conversely dynamic motion effects on local flow patterns. This requires a code coupling to solve fluid and structure problems in the same time. This ability is out of limit of most classical fluid dynamics codes. That is the reason why recently an improved numerical approach has been developed and applied to the fully numerical prediction of a flexible tube dynamic response belonging to a fixed tube bundle submitted to cross flows. The methodology consists in simulating at the same time thermo-hydraulics and mechanics problems by using an Arbitrary Lagrange Euler (ALE) formulation for the fluid computation. Numerical results turn out to be consistent with available experimental data and calculations tend to show that it is now possible to simulate numerically tube bundle vibrations in presence of cross flows. Thus a new possible application for ALE methods is the prediction of flow-induced vibration problems. The full computational process is described in the first section. Classical and improved ALE formulations are presented in the second part. Main numerical results are compared to available experimental data in section 3. Code performances are pointed out in terms of mesh generation process and code coupling method.
46

Adami, P., and F. Martelli. "Three-dimensional unsteady investigation of HP turbine stages." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 220, no. 2 (March 1, 2006): 155–67. http://dx.doi.org/10.1243/095765005x69189.

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This article deals with a three-dimensional unsteady numerical simulation of the unsteady rotor—stator interaction in a HP turbine stage. The numerical approach consists of a computational fluid dynamics (CFD) parallel code, based on an upwind total variation diminishing finite volume approach. The computation has been carried out using a sliding plane approach with hybrid unstructured meshes and a two-equation turbulent closure. The turbine rig under investigation is representative of the first stage of aeronautic gas turbine engines. A brief description of the cascade, the experimental setup, and the measuring technique is provided. Time accurate CFD computations of pressure fluctuations and Nusselt number are discussed against the experimental data.
47

Fisher, E. H., and N. Rhodes. "Uncertainty in Computational Fluid Dynamics." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 209, no. 2 (May 1995): 155–58. http://dx.doi.org/10.1243/pime_proc_1995_209_026_02.

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The Fifth Joint Engineering and Physical Sciences Research Council and Institution of Mechanical Engineers Expert Meeting was held in Bournemouth on 27-29 November 1994. The Fifth Joint Engineering and Physical Sciences Research Council and Institution of Mechanical Engineers Expert Meeting was held in Bournemouth on 27–29 November 1994.
48

Knight, Earl E., Esteban Rougier, Zhou Lei, Bryan Euser, Viet Chau, Samuel H. Boyce, Ke Gao, Kurama Okubo, and Marouchka Froment. "HOSS: an implementation of the combined finite-discrete element method." Computational Particle Mechanics 7, no. 5 (July 31, 2020): 765–87. http://dx.doi.org/10.1007/s40571-020-00349-y.

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Abstract Nearly thirty years since its inception, the combined finite-discrete element method (FDEM) has made remarkable strides in becoming a mainstream analysis tool within the field of Computational Mechanics. FDEM was developed to effectively “bridge the gap” between two disparate Computational Mechanics approaches known as the finite and discrete element methods. At Los Alamos National Laboratory (LANL) researchers developed the Hybrid Optimization Software Suite (HOSS) as a hybrid multi-physics platform, based on FDEM, for the simulation of solid material behavior complemented with the latest technological enhancements for full fluid–solid interaction. In HOSS, several newly developed FDEM algorithms have been implemented that yield more accurate material deformation formulations, inter-particle interaction solvers, and fracture and fragmentation solutions. In addition, an explicit computational fluid dynamics solver and a novel fluid–solid interaction algorithms have been fully integrated (as opposed to coupled) into the HOSS’ solid mechanical solver, allowing for the study of an even wider range of problems. Advancements such as this are leading HOSS to become a tool of choice for multi-physics problems. HOSS has been successfully applied by a myriad of researchers for analysis in rock mechanics, oil and gas industries, engineering application (structural, mechanical and biomedical engineering), mining, blast loading, high velocity impact, as well as seismic and acoustic analysis. This paper intends to summarize the latest development and application efforts for HOSS.
49

Wang, Q., and D. T. Papageorgiou. "Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities." Journal of Fluid Mechanics 683 (August 2, 2011): 27–56. http://dx.doi.org/10.1017/jfm.2011.247.

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AbstractThe nonlinear dynamics of a viscous filament surrounded by a second viscous fluid arranged in a core-annular configuration when a radial electric field acts in the annular region, are studied analytically and computationally using boundary element methods. The flow is characterized by the viscosity ratio, an electric Weber number measuring the strength of the electric field, a geometrical parameter measuring the thickness of the undisturbed annular region, as well as a computational parameter that fixes the wavenumber of the undulations. Axisymmetric solutions are computed by direct numerical simulations in the Stokes limit for general values of the parameters when the two fluids have equal viscosities, and an asymptotic theory is carried out to produce a novel evolution equation for thin film dynamics valid when the undisturbed annular thickness is small and the viscosity ratio is of order one. It is established (in agreement with previous computations in the absence of electric fields) that a sufficiently thick annulus enables thread breakup while a sufficiently thin one (approximately one fifth of the undisturbed thread radius for the case of equal viscosities, for instance) suppresses pinching and drives the interface to approach the tube wall asymptotically without actually touching it. The present simulations show that the electric field affects the dynamics drastically in several ways. First, it promotes interfacial wall touchdown in finite time and a comparison between direct simulations and the asymptotic solutions are in fair agreement. Second, the electric field acts to suppress pinching in the sense that solutions that lead to jet breakup due to a thick enough viscous annulus are driven to wall touchdown. When pinching takes place we find that the ultimate pinching solutions are self-similar and recover the non-electrified ones to leading order for the range of parameters studied.
50

Aznavourian, Ronald, Sébastien Guenneau, Bogdan Ungureanu, and Julien Marot. "Morphing for faster computations with finite difference time domain algorithms." EPJ Applied Metamaterials 9 (2022): 2. http://dx.doi.org/10.1051/epjam/2021011.

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In the framework of wave propagation, finite difference time domain (FDTD) algorithms, yield high computational time. We propose to use morphing algorithms to deduce some approximate wave pictures of their interactions with fluid-solid structures of various shapes and different sizes deduced from FDTD computations of scattering by solids of three given shapes: triangular, circular and elliptic ones. The error in the L2 norm between the FDTD solution and approximate solution deduced via morphing from the source and destination images are typically less than 1% if control points are judiciously chosen. We thus propose to use a morphing algorithm to deduce approximate wave pictures: at intermediate time steps from the FDTD computation of wave pictures at a time step before and after this event, and at the same time step, but for an average frequency signal between FDTD computation of wave pictures with two different signal frequencies. We stress that our approach might greatly accelerate FDTD computations as discretizations in space and time are inherently linked via the Courant–Friedrichs–Lewy stability condition. Our approach requires some human intervention since the accuracy of morphing highly depends upon control points, but compared to the direct computational method our approach is much faster and requires fewer resources. We also compared our approach to some neural style transfer (NST) algorithm, which is an image transformation method based on a neural network. Our approach outperforms NST in terms of the L2 norm, Mean Structural SIMilarity, expected signal to error ratio.

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