Abstract
New integral equation formulations for steady and unsteady flow problems of an incompressible viscous fluid are presented. The so-called direct approach in which the velocity vector and the pressure are inclued as unknowns is employed in this paper. The nonlinear boundary value, and the initial-boundary value problems described with the Navier-Stokes equations are transformed into integral equations by the method of weighted residuals. Fundamental solutions of the Stokes approximate equations are used as the weight function. The fundamental solution tensors are presented for the steady-state and unsteady-state problems. For the unsteady-state problem, we derive not only the time-dependent fundamental solution tensor but also the one using the finite difference approximation for the time derivative. A numerical example of the two-dimensional driven cavity flow is given to show the validity and effectiveness of the method.
Similar content being viewed by others
References
Banerjee, P. K.; Butterfield, R. (1981): Boundary element methods in engineering science. New York: McGraw-Hill
Brebbia, C. A.; Walker, S. (1980): Boundary element techniques in engineering. Newnes-Butterworths
Brebbia, C. A.; Telles, T. C. F.; Wrobel, L. C. (1984): Boundary element techniques.Berlin, Heidelberg, New York: Springer
Ghia, U.; Ghia, K. N.; Shin, C. T. (1982): High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Physics 48, 387–411
Hörmander, L. (1964): Linear partial differential operators. Second revised printing. Berlin, Göttingen, Heidelberg: Springer
Imamura, T. (1978): Physics and Green function (in Japanese). Tokyo: Iwanami-Shoten
Kakuda, K.; Tosaka, N. (1984): Analysis of a viscous fluid flow by boundary element method (in Japanese). Proc. 5th Symposium on Finite Element Methods in Flow Problems (JUSE, Tokyo), 201–208
Onishi, K.; Kuroki, T.; Tanaka, M. (1984): An application of boundary element method to incompressible laminar viscous flows. Engineering Analysis 1, 122–127
Skerget, P.; Alujevic, A.; Brebbia, C.A. (1984): The solution of Navier-Stokes equations in terms of vorticity-velocity variables by boundary elements. In: Brebbia, C. A. (ed.): Boundary elements VI, pp. 4/41–4/56. Berlin, Heidelberg, New York: Springer
Skerget, P.; Alujevic, A.; Brebbia, C. A. (1985): Analysis of laminar flows with separation using BEM. In: Brebbia, C. A., Maier, G. (eds.): Boundary elements VII, pp. 9/23–9/36. Berlin, Heidelberg, New York: Springer
Thomasset, F. (1981): Implementation of finite elements methods for Navier-Stokes equations. Berlin, Heidelberg, New York: Springer
Tosaka, N. (1986): Numerical methods for viscous flow problems using an integral equation. In: Wang, S. Y., Shen, H. W., Ding, L. Z. (eds.): River sedimentation, pp. 1514–1525: The University of Mississippi
Tosaka, N.; Fukushima, N. (1986): Integral equation analysis of laminar natural convection problems. In: Tanaka, M. and C. A. Brebbia (eds.): Boundary Elements VIII, Vol. II, pp. 803–812. Berlin, Heidelberg, New York: Springer
Tosaka, N.; Kakuda, K. (1986a): Numerical solutions of steady incompressible viscous flow problems by the integral equation method. In: Shaw, R. P., Periaux, J., Chaudouet, A., Wu, J., Marino, C., Brebbia, C. A. (eds.): Innovative numerical methods in engineering, pp. 211–222. Berlin, Heidelberg, New York: Springer
Tosaka, N.; Kakuda, K. (1986b): Numerical simulations for incompressible viscous flow problems using the integral equation methods. In: Tanaka, M. and C. A. Brebbia (eds.): Boundary Elements VIII, Vol. 11, pp. 813–822. Berlin, Heidelberg, New York: Springer
Tosaka, N.; Onishi, K. (1985): Boundary integral equation formulation for steady Navier-Stokes equations using the Stokes fundamental solutions. Eng. Analysis 2, 128–132
Tosaka, N.; Onishi, K. (1986a): Boundary integral equation formulations for unsteady incompressible viscous fluid flow by time-differencing. Eng. Analysis 3, 101–104
Tosaka, N.; Onishi, K. (1986b): Integral equation method for thermal fluid flow problems. In: Yagawa, G., Atluri, S. N. (eds.): Computational mechanics '86, vol. 2, pp. XI–103-XI–108. Berlin, Heidelberg, New York: Springer
Tosaka, N.; Kakuda, K.; Onishi, K. (1985): Boundary element analysis of steady viscous flows based on P-U-V formulation. In: Brebbia, C. A., Maier, G. (eds.): Boundary elements VII, Vol. II, pp. 9/71–9/80. Berlin, Heidelberg; New York: Springer
Wu, J. C.; Rizk, Y. M. (1978): Integral-representation approach for time-dependet viscous flows, in Lecture Notes in physics, Vol. 90, pp. 558–564. Berlin, Heidelberg, New York: Springer
Wu, J. C.; Thompson, J. F. (1973): Numerical solution of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation. Comput. Fluids 1, 197–215
Wu, J. C.; Wahbah, M. M. (1976): Numerical solution of viscous flow equations using integral representation. In: Lecture Notes in physics, Vol. 59, pp. 448–453. Berlin, Heidelberg, New York: Springer
Author information
Authors and Affiliations
Additional information
Communicated by G. Yagawa, June 12, 1987
Rights and permissions
About this article
Cite this article
Tosaka, N. Integral equation formulations with the primitive variables for incompressible viscous fluid flow problems. Computational Mechanics 4, 89–103 (1988). https://doi.org/10.1007/BF00282412
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00282412