Numerical methods in fluid mechanics – an overview
Wojciech Sobieski
University of Warmia and MazuryResearch fields:
• applications and development of numerical methods of mechanics
• investigations on the spatial structure of granular porous media
• investigations on fluid flows through porous media
• investigations on dynamics of fluidized beds
• investigations on the cavitation phenomenon in hydraulic systems
• investigations on water hammer in water rams
• investigations on bifurcation phenomena in flow systems
• investigations on heat diffusion in heterogeneous materials
• sensitivity analysis of mathematical models
https://orcid.org/0000-0003-1434-5520
Božidar Šarler
University of Ljubljana, Ljubljana, Sloveniahttps://orcid.org/0000-0003-0618-6873
Abstract
The article presents in a review way the most important numerical methods used in modern fluid mechanics. The individual chapters discuss Finite Difference Method, Finite Volume Method, Lattice Boltzmann Method, Discrete Element Method and Smoothed Particle Hydrodynamics. The aim of the article is to familiarize the reader with the most important concepts, features and mathematical equations used in particular methods. The article is intended mainly for people who want to get acquainted with the current possibilities of numerical modelling in the field of broadly understood fluid mechanics. The material is intended to facilitate the decision on how to implement the planned play research.
Keywords:
CFD, numerical modelling, mesh methods, meshless methodsSupporting Agencies
References
ABAS A., MOKHTAR N.H., ISHAK H.H.H., ABDULLAH M.Z., TIAN A.H. 2016. Lattice Boltzmann Model of 3D Multiphase Flow in Artery Bifurcation Aneurysm Problem. Computational and Mathematical Methods in Medicine, 2016, article ID 6143126. https://doi.org/10.1155/2016/6143126
Crossref
Google Scholar
ARCHAMBEAU P., PIROTTON M., DEWALS B., DUCHÊNE L., MOUZELARD T. 2013. Development of a didactic SPH model. Université de Liège – Faculté des Sciences Appliquées. Google Scholar
BHATNAGAR P.L., GROSS E.P., KROOK M. 1954. A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems. Physical Review, 94: 511-525.
Crossref
Google Scholar
BUI H.H., NGUYEN G.D. 2021. Smoothed particle hydrodynamics (SPH) and its applications in geomechanics: From solid fracture to granular behaviour and multiphase flows in porous media. Computers and Geotechnics, 138: 104315.
Crossref
Google Scholar
BURDEN R.L., FAIRES J.D. 2011. Numerical Analysis. 9th Edition. Brooks/Cole, Pacific Grove. Google Scholar
CAUSON D.M., MINGHAM C.G. 2010. Introductory Finite Difference Methods for PDEs. Ventus Publishing ApS. Google Scholar
CHU K. 2010. CFD-DEM simulation of complex particle-fluid flows. PhD Thesis. The University of New South Wales. Google Scholar
COURANT R., FRIEDRICHS K., LEWY H. 1928. On the Partial Difference Equations of Mathematical Physics. Mathematische Annalen, 100: 32-74.
Crossref
Google Scholar
CUNDALL P., STRACK O. 1979. A discrete element model for granular assemblies. Géotechnique, 29: 47-65.
Crossref
Google Scholar
DOUILLET-GRELLIER T., JONES B.D, PRAMANIK R., PAN K., ALBAIZ A., WILLIAMS J.R. 2016. Mixed- Google Scholar
-mode fracture modeling with smoothed particle hydrodynamics. Computers and Geotechnics, 79: 73-85.
Crossref
Google Scholar
FULCHINI F., GHADIRI M., BORISSOVA A., AMBLARD B., BERTHOLIN S., CLOUPET A., YAZDANPANAH M. 2019. Development of a methodology for predicting particle attrition in a cyclone by CFD-DEM. Powder Technology, 357: 21-32.
Crossref
Google Scholar
GAO Y., CHENG Y, CHEN J. 2023. Experimental Study and 3-D Meso-Scale Discrete Element Modeling on the Compressive Behavior of Foamed Concrete. Buildings, 13: 674.
Crossref
Google Scholar
GESTEIRA M.G., ROGERS B.B, DALRYMPLE R.A., CRESPO A.J.C., NARAYANASWAMY M. 2010. User Guide for the SPHysics code. Retrieved from https://wiki.manchester.ac.uk/sphysics/index.php/SPHYSICS_Home_Page (20.05.2023). Google Scholar
GOFFIN L. 2013. Development of a didactic SPH model. MSc Thesis, Université de Ličge (Belgium). Google Scholar
HERTZ H. 1881. Über die berührung fester elastischer Körper. Journal für die Reine und Angewandte Mathematik, 92: 156-171.
Crossref
Google Scholar
HUGHES T.J.R., ZIENKIEWICZ O.C. 1979. Finite elements in fluid mechanics: an introduction to the Galerkin method. Pergamon Press, Oxford. Google Scholar
IYENGAR S.R.K., JAIN R.K. 2009. Numerical Methods. New Age International Publishers, New Delhi. Google Scholar
KHAN S.A., KOÇ M. 2022. Numerical modelling and simulation for extrusion-based 3D concrete printing: The underlying physics, potential, and challenges. Results in Materials, 16: 100337.
Crossref
Google Scholar
KORN Ch., HERLITZIUS T. 2012. Coupled CFD-DEM simulation of separation process in combine harvester cleaning devices. Landtechnik, 72(5): 247-261. Google Scholar
KOUKOUVINIS P., KYRIAZIS N., GAVAISES M. 2018. Smoothed particle hydrodynamics simulation of a laser pulse impact onto a liquid metal droplet. PLoS ONE, 13(9): e0204125.
Crossref
Google Scholar
KRAUSE M.J. 2010. Fluid Flow Simulation and Optimisation with Lattice Boltzmann Methods on High Performance Computers – Application to the Human Respiratory System. PhD Thesis. Karlsruhe Institute of Technology. Google Scholar
KRÜGER T., KUSUMAATMAJA H., KUZMIN A., SHARDT O., SILVA G., VIGGEN E.M. 2017. The Lattice Boltzmann Method – Principles and Practice. Springer, Berlin.
Crossref
Google Scholar
LABRA C.A, ONATE E., ROJEK J. 2012. Advances in the development of the discrete element method for excavation processes. Monograph CIMNE, Nº-132: 1-213. Google Scholar
LANGTANGEN H.P., LINGE S. 2017. Finite Difference Computing with PDEs – A Modern Software Approach. Springer Open, Berlin.
Crossref
Google Scholar
LI S., LIU W.K. 2007. Meshfree Particle Methods. Springer, Berlin. Google Scholar
LIND S.J., ROGERS B.D., STANSBY P.K. 2020. Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476(2241).
Crossref
Google Scholar
LIU G.R., LIU M.B. 2003. Smoothed Particle Hydrodynamics. Word Scientific Publishing, Singapore.
Crossref
Google Scholar
LIU Z., MA H., ZHAO Y. 2021. CFD-DEM Simulation of Fluidization of Polyhedral Particles in a Fluidized Bed. Energies, 14: 4939.
Crossref
Google Scholar
MACCORMACK R.W., PAULLAY A.J. 1972. Computational efficiency achieved by time splitting of finite difference operators. AIAA Paper, 72-154, San Diego.
Crossref
Google Scholar
MCDONALD P.W. 1971. The Computation of Transonic Flow Through Two-Dimensional Gas Turbine Cascades. Proceedings of the ASME 1971 International Gas Turbine Conference and Products Show. ASME 1971 International Gas Turbine Conference and Products Show, Houston, Texas, USA. March 28-April 1.
Crossref
Google Scholar
MENTER F.R. 1993. Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows. AIAA Paper, 93-2906.
Crossref
Google Scholar
MOHAMAD A.A. 2011. Lattice Boltzmann Method. Fundamentals and Engineering Applications with Computer Codes. Springer, Berlin.
Crossref
Google Scholar
MOUKALLED F., MANGANI F., DARWISH M. 2016. The Finite Volume Method in Computational Fluid Dynamics. Springer, Berlin.
Crossref
Google Scholar
NGUYEN T.T.T., HOJNY M. 2022. Application of Smoothed Particle Hydrodynamics Method in Metal Processing: An Overview. Archives of Foundry Engineering, 3: 67-80. Google Scholar
NIEDŹWIEDZKA A., SCHNERR G.H., SOBIESKI W. 2016. Review of numerical models of cavitating flows with the use of the homogeneous approach. Archives of Thermodynamics, 37(2): 71-88.
Crossref
Google Scholar
ORSZAG S.A. 1970. Analytical Theories of Turbulence. Journal of Fluid Mechanics, 41: 363-386.
Crossref
Google Scholar
PESKIN C.S. 1977. Numerical Analysis of Blood Flow in the Heart. Journal of Computational Physics, 25: 220-252.
Crossref
Google Scholar
PLATZER F., FIMBINGER E. 2021. Modelling Pasty Material Behaviour Using the Discrete Element Method. Multiscale Science and Engineering, 3: 119-128.
Crossref
Google Scholar
SCHUBIGER A., BARBER S., NORDBORG H. 2020. Evaluation of the lattice Boltzmann method for wind modelling in complex terrain. Wind Energy Science, 5: 1507-1519.
Crossref
Google Scholar
SOBIESKI W. 2009. Switch function and sphericity coefficient in the Gidaspow drag model for modeling solid-fluid systems. Drying Technology, 27(2): 267-280.
Crossref
Google Scholar
SOBIESKI W., GRYGO D. 2019. Fluid flow in the impulse valve of a hydraulic ram. Technical Sciences, 22(3): 205-220.
Crossref
Google Scholar
SOBIESKI W., MATYKA M., GOŁEMBIEWSKI J., LIPIŃSKI S. 2018. The Path Tracking Method as an alternative for tortuosity determination in granular beds. Granular Matter, 20: 72.
Crossref
Google Scholar
SMAGORINSKY J. 1963. General circulation experiments with the primitive equations I. the basic experiment. Monthly Weather Review, 91(3): 99-164.
Crossref
Google Scholar
SPALART P.R. 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In Proceedings of first AFOSR international conference on DNS/LES, Greyden Press. Google Scholar
SUCCI S. 2001. The Latttice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon Press, Oxford.
Crossref
Google Scholar
SUKOP M.C., THORNE D.T. 2006. Lattice Boltzmann Modeling – An Introduction for Geoscientists and Engineers. Springer, Berlin.
Crossref
Google Scholar
The YADE code. Retrieved from https://yade-dem.org/doc/ (20.05.2023). Google Scholar
TRUSHNIKOV D.N., KOLEVA E.G., DAVLYATSHIN R.P., GERASIMOV R.M., BAYANDIN Y.V. 2019. Mathematical modeling of the electronbeam wire deposition additive manufacturing by the smoothed particle hydrodynamics method. Mechanics of Advanced Materials and Modern Processes, 5: 4.
Crossref
Google Scholar
URLICH Ch. 2013. Smoothed-Particle-Hydrodynamics Simulation of Port Hydrodynamic Problems. TUHH, Hamburg. Google Scholar
WAGNER A.J. 2008. A Practical Introduction to the Lattice Boltzmann Method. North Dakota State University, Fargo. Google Scholar
WEINAN E., LIU J.-G. 2000. Gauge finite element method for incompressible flows. International Journal for Numerical Methods in Fluids, 24(8): 701-710.
Crossref
Google Scholar
XIE Y., LIU Y., LI L., XU Ch., LI B. 2018. Simulation of different gas-solid flow regimes using a drag law derived from lattice Boltzmann simulations. The Journal of Computational Multiphase Flows, 10(4): 202-2014.
Crossref
Google Scholar
ZIENKIEWICZ O.C., TAYLOR R.L., NITHIARASU P. 2005. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford (UK). Google Scholar
ZOU Y., CHEN Ch., ZHANG L. 2020. Simulating Progression of Internal Erosion in Gap-Graded Sandy Gravels Using Coupled CFD-DEM. International Journal of Geomechanics, 20(1): 04019135-1.
Crossref
Google Scholar
University of Warmia and Mazury
<p> <span style="font-size: small;"><u>Research fields</u>:</span></p> <p>• applications and development of numerical methods of mechanics<br> • investigations on the spatial structure of granular porous media<br> • investigations on fluid flows through porous media<br> • investigations on dynamics of fluidized beds<br> • investigations on the cavitation phenomenon in hydraulic systems<br> • investigations on water hammer in water rams<br> • investigations on bifurcation phenomena in flow systems<br> • investigations on heat diffusion in heterogeneous materials<br> • sensitivity analysis of mathematical models</p> Poland
https://orcid.org/0000-0003-1434-5520
Research fields:
• applications and development of numerical methods of mechanics
• investigations on the spatial structure of granular porous media
• investigations on fluid flows through porous media
• investigations on dynamics of fluidized beds
• investigations on the cavitation phenomenon in hydraulic systems
• investigations on water hammer in water rams
• investigations on bifurcation phenomena in flow systems
• investigations on heat diffusion in heterogeneous materials
• sensitivity analysis of mathematical models