Continuum Mechanics and Thermodynamics

, Volume 30, Issue 4, pp 943–952 | Cite as

A note on a derivative scheme for the finite volume method applied to incompressible viscous fluid

  • M. M. Martins
  • M. VazJr.
  • P. S. B. Zdanski
Short Communication


In the last two decades, there have been significant advances in application of the finite volume method to a wide spectrum of physical phenomena, ranging from heat transfer and compressible/incompressible fluid flow to solid mechanics. The finite volume method requires for different applications similar approximations of derivatives at the control surface. Within this framework, this study discusses a derivative scheme used in the finite volume method for incompressible viscous fluids. The numerical scheme is based on an implicit technique associated with the SIMPLE method to attain pressure–velocity coupling. The present work addresses simulation of two-dimensional flows in plane channels with and without contractions. The results show velocities and pressure fields with good agreement when compared to analytical results.


Finite volume method Incompressible fluid flow Derivative scheme 


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  1. 1.
    Khan, M.M.K., Hassan, N.M.S.: Thermofluid Modelling for Energy Efficiency Applications. Academic, London (2016)Google Scholar
  2. 2.
    Yu, H.-Y., Zhang, H.-C., Guo, Y.-Y., Tan, H.-P., Li, Y., Xie, G.-N.: Thermodynamic analysis of shark skin texture surfaces for microchannel flow. Contin. Mech. Thermodyn. 28(5), 1361–1371 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hong, B.Z., Keong, L.K., Shariff, A.M.: CFD modelling of most probable bubble nucleation rate from binary mixture with estimation of components’ mole fraction in critical cluster. Contin. Mech. Thermodyn. 28(3), 655–668 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ahn, J., Sparrow, E.M., Gorman, J.M., Minkowycz, W.J.: Investigation of coupled systems consisting of fluid movers and heat-exchange devices. Numer. Heat Transf. Part A Appl. 70(9), 964–979 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Perão, L.H., Zdanski, P.S.B., Vaz Jr., M.: Conjugate heat transfer in channels with heat conducting inclined fins. Numer. Heat Transf. Part A Appl. 73(2), 75–93 (2018)ADSCrossRefGoogle Scholar
  6. 6.
    Hirsh, C.: Numerical Computation of Internal and External Flows. Butterworth-Heinemann, Oxford (2007)Google Scholar
  7. 7.
    Minkowycz, W.J., Sparrow, E.M., Schneider, G.E., Pletcher, R.H.: Handbook of Numerical Heat Transfer. Wiley, Chichester (1988)Google Scholar
  8. 8.
    Filippini, G., Maliska, C.R., Vaz Jr., M.: A physical perspective of the element-based finite volume method and FEM-Galerkin methods within the framework of the space of finite elements. Int. J. Numer. Methods Eng. 98(1), 24–43 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pan, D.: A high-order finite volume method for solving one-dimensional convection and diffusion equations. Numer. Heat Transf. Part B Fund. 71(6), 533–548 (2017)ADSCrossRefGoogle Scholar
  10. 10.
    Demirdžić, I.: On the discretization of the diffusion term in finite-volume continuum Mechanics. Numer. Heat Transf. Part B Fund. 68(1), 1–10 (2015)ADSGoogle Scholar
  11. 11.
    Prestini, D., Filippini, G., Zdanski, P.S.B., Vaz Jr., M.: Fundamental approach to anisotropic heat conduction using the element-based finite volume method. Numer. Heat Transf. Part B Fund. 71(4), 327–345 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    Tannehill, J.C., Anderson, D.A., Pletcher, R.H.: Computational Fluid Mechanics and Heat Transfer. CRC Press, Boca Raton (1997)zbMATHGoogle Scholar
  13. 13.
    Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, New York (1980)zbMATHGoogle Scholar
  14. 14.
    Chorin, A.J.: A numerical method for solving incompressible fluid flow problems. J. Comput. Phys. 135, 118–125 (1997)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Salari, M., Rezvani, A., Mohammadtabar, A., Mohammadtabar, M.: Numerical study of entropy generation for natural convection in rectangular cavity with circular corners. Heat Transf. Eng. 36(2), 186–199 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    Ferziger, J.H., Peric, M.: Computational Methods for Fluid Dynamics, 2nd edn. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Deng, Q.-H., Tang, G.-F.: Special treatment of pressure correction based on continuity conservation in a pressure-based algorithm. Numer. Heat Transf. Part B Fund. 42(1), 73–92 (2002)ADSCrossRefGoogle Scholar
  18. 18.
    Shamadhani Begum, A., Nithyadevi, N.: Effect of moving flat plate on hydro-magnetic mixed convection in a porous enclosure with sinusoidal heating side walls and internal heat generation. Comput. Therm. Sci. 9(1), 13–28 (2017)CrossRefGoogle Scholar
  19. 19.
    Asako, Y., Faghri, M.: Modification of SIMPLE algorithm to handle natural convection flows with zero-isothermal compressibility. Int. J. Heat Mass Transf. 122, 290–305 (2018)CrossRefGoogle Scholar
  20. 20.
    Natesan, S., Arumugam, S.K., Murugesan, S., Chamkha, A.J.: Heat transfer enhancement of uniformly/linearly heated side wall in a square enclosure utilizing alumina water nanofluid. Comput. Therm. Sci. 9(3), 227–241 (2017)CrossRefGoogle Scholar
  21. 21.
    Hu, W., Wang, L., Guan, Y., Hu, W.: The effect of shape of winglet vortex generator on the thermal hydrodynamic performance of a circular tube bank fin heat exchanger. Heat Mass Transf. 53(9), 2961–2973 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    White, F.M.: Fluid Mechanics, 4th edn. McGraw-Hill, New York (1998)Google Scholar
  23. 23.
    Rhie, C.M., Chow, W.L.: A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA J. 21, 1525–1532 (1983)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University Centre Catholic of Santa CatarinaJoinvilleBrazil
  2. 2.State University of Santa CatarinaJoinvilleBrazil

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