Numerical methods in fluid mechanics – an overview

Wojciech Sobieski

University of Warmia and Mazury

 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


https://orcid.org/0000-0003-1434-5520

Božidar Šarler

University of Ljubljana, Ljubljana, Slovenia
https://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 methods

Supporting Agencies

The publication was written as a result of the author's internship in the Ljubljana University in Slovenia, co-financed by the European Union under the European Social Fund (Operational Program Knowledge Education Development), carried out in the project Development Program at the University of Warmia and Mazury in Olsztyn (POWR.03.05. 00-00-Z310/17).


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

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Published
2023-11-10

Cited by

Sobieski, W., & Šarler, B. (2023). Numerical methods in fluid mechanics – an overview. Technical Sciences, 26(26), 185–218. https://doi.org/10.31648/ts.9212

Wojciech Sobieski 
University of Warmia and Mazury
<p>&nbsp;<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


Božidar Šarler 
University of Ljubljana, Ljubljana, Slovenia
https://orcid.org/0000-0003-0618-6873



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