Computation of Incompressible Flows Using Higher Order Divergence-free Elements

고차의 무발산 요소를 이용한 비압축성 유동계산

Kim, Jin-Whan

  • Received : 2011.07.11
  • Accepted : 2011.10.17
  • Published : 2011.10.31


The divergence-free finite elements introduced in this paper are derived from Hermite functions, which interpolate stream functions. Velocity bases are derived from the curl of the Hermite functions. These velocity basis functions constitute a solenoidal function space, and the gradient of the Hermite functions constitute an irrotational function space. The incompressible Navier-Stokes equation is orthogonally decomposed into its solenoidal and irrotational parts, and the decoupled Navier-Stokes equations are then projected onto their corresponding spaces to form appropriate variational formulations. The degrees of the Hermite functions we introduce in this paper are bi-cubis, quartic, and quintic. To verify the accuracy and convergence of the present method, three well-known benchmark problems are chosen. These are lid-driven cavity flow, flow over a backward facing step, and buoyancy-driven flow within a square enclosure. The numerical results show good agreement with the previously published results in all cases.


Divergence-free element;Incompressible flow;Vector potential;Solenoidal basis function;Irrotational basis function


  1. 김진환 (2008). "이차원 비압축성 유동계산을 위한 Hermite 겹 3차 유동함수법," 한국전산유체공학회지, 제13권, 제4호, pp 13-23.
  2. 김진환 (2009). "Hermite 유동함수법에 의한 자연대류 유동계산", 한국해양공학회지, 제23권, 제5호, pp 1-8.
  3. Botella, O. and Peyret, R. (1998). "Benchmark Spectral results on the Lid-driven Cavity Flow", Computers & Fluids, Vol 27, pp 421-433.
  4. Christon, M.A., Gresho, P.M. and Sutton, S.B. (2002). "Computational Predictability of Time-dependent Natural Convection Flows in Enclosures (Including a Benchmark Solution)", Int. J. for Numer. Methods in Fluids, Vol 40, pp 953-980.
  5. De Vahl Davis, G. (1983). "Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution", Int. J. for Numer. methods in Fluids, Vol 3, pp 249-264.
  6. Gartling, D.K. (1990). "A Test Problem for Outflow Boundary Conditions-Flow over a Backward-facing Step," Int. J. for Numer. Methods in Fluids, Vol 11, pp 953-967.
  7. Ghia, U., Ghia, K.N., and Shin, C.T. (1982). "High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method," J. of Comp. Physics, Vol 48, pp 387-411.
  8. Gopalacharyulu, S. (1973). "A Higher Order Conforming Rectangular Plate Element," Int. J. for Numer. Methods in Engr., Vol 6, pp 305-308.
  9. Griffiths, D.F. (1981). "An Approximately Divergence-Free 9-Node Velocity Element (with Variationa) for Incompressible Flows", Int. J. for Numer. Meth. in Fluids, Vol 1, pp 323-346.
  10. Holdeman, J.T. (2002). "Recent Advances in the Finite Element Method for Incompressible Flow," USNCTAM14 Conference, Blacksburg, VA.
  11. Shu, C. and Wee, K.H.A. (2002). "Numerical Simulation of Natural Convection in a Square Cavity by SIMPLE-generalized Differential Quadrature Method", Computers & Fluids, Vol 31, pp 209-226.
  12. Watkins, D.S. (1976). "On the Construction of Conforming Rectangular Plate Element," Int. J. for Numer. Methods in Engr., Vol 10, pp 925-933.


Supported by : 동의대학교