### Solution of Viscous Flow around a Circular Cylinder by a New Integral Representation Method (NIRM)

H. Isshiki, S. Nagata, Y. Imai

#### Abstract

New nonlinear integral representations (NIRM) are derived from a nonlinear differential-type boundary value problem using a fundamental solution of the primary space-differential operator of the differential equation. Integral representations are equivalent to differential equations. A set of integral representations is an integral-type boundary value problem. Unknown variables of a boundary value problem can be determined by solving a set of integral equations obtained from a set of integral representations. In the present paper, a set of integral representations using the fundamental solution of the primary space-differential operator is derived for viscous flows. The velocity, vorticity, and pressure of the Navier-Stokes equation can be determined by solving a set of integral equations obtained from a set of integral representations. A new numerical solution of the Navier-Stokes equation is proposed based on integral representations. The integral representation method was used to obtain the numerical results of low-Reynolds-number laminar flows around a circular cylinder. The narrower regions and coarser meshes are used in the numerical calculations using the integral representation method than in those using the ordinary FEM. The numerical results correctly reflect the experimental ones. Unlike FEM, as seen from that constant distribution of the unknown variables is possible, NIRM may not assume the continuity of the unknown variables between elements from the beginning. It would be safe to say this is a big advantage of NIRM.

#### Keywords

New integral representation method (NIRM), Primary space-differential operator, Vorticity, Navier-Stokes equation

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#### References

J.C. Wu, J.F. Thompson, “Numerical solutions of time-dependent incompressible Navier-Stokes equations using an

integro-differential formulations”, Computers & Fluids, (1973), 1, pp. 197-215.

J.C. Wu, “Chap. 4 Problems of general viscous flow”, Developments in Boundary Element Methods-2, edited by P. K.

Banerjee and R. P. Shaw, Applied Science Publishers, (1982), pp. 69-109.

J.C. Wu, “Fundamental solutions and boundary element method”, Engineering Analysis, (1987), 4(1), pp. 2-6.

J.C. Wu, C.M. Wang, “Chap. 7 Recent advances in solution methods for unsteady viscous flows”, Boundary Element

Methods in Nonlinear Fluid Dynamics, Developments in Boundary Element Methods-6, edited by P. K. Banerjee and

L. Morino, Elsvier Applied Science, (1990), 247-283.

J.S. Uhlman, “An integral equation formulation of the equations of motion of an incompressible fluid”, NUWC-NPT

Technical Report 10,086, 15 July, (1992).

A.J. Chorin, “Numerical study of slightly viscous flow”, J, Fuid Mech., (1973), 57, pp. 785-796.

A.N. Brooks, T.R. Hughes, “Streamline Upwind/Petrov-Galerkin formulations for Convection dominated Flows with

particular emphasis on the incompressible Navier-Stokes equations”, Computer Methods in Applied Mechanics and

Engineering, (1982), 32, pp. 199-259.

T.E. Tezduyar, J. Liou, D.K. Ganjoo, “Incompressible flow computations based on the vorticity-stream function and

velocity-pressure formulations, Computers & Structures”, (1990), 35(4), pp. 445-472.

M. Vinokur, “On one-dimensional stretching functions for finite-difference calculations”, NASA Contractor Report

, (1980).

A.S. Grove, F.H. Shair, E.E. Petersen, A. Acrivos, “An experimengtal investigation of the steady separated flow past

a circular cylinder”, J. Fluid Mech., (1964), 33, 60-80 .

J.D. Anderson, Fundamentals of Aerodynamics, 4th ed. McGraw-Hill, (2005), pp.228–236.

C.Y. Choi, E. Balaras, “A dual reciprocity boundary element formulation using the fractional step method for the

incompressible navier-Stokes equations”, Engineering Analysis with Boundary Elements 33(6), June 2009, Pages

–749

J. Matsunashi, N. Okamoto, T. Futagami, “Boundary element analysis of Navier-Stokes equations”, Engineering

Analysis with Boundary Elements, (2012), 36, pp.471-476.

G.B.L. Ying, D. Zorin, “The embedded boundary integral method (EBI) for the incompressible Navier-Stokes

equations, UT Austin, TX, USA, May 28-30, (2002), pp. 1-12.

N. Tosaka, K. Onishi, “Boundary integral equation formulations for steady Navier-Stokes equations using the

Stokes-fundamental solutions”, Engineering Analysis, (1985), 2(3), pp. 128-132.

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