Some uses of Green's theorem in solving the Navier–Stokes equations

Abstract

SUMMARYThis paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier-Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier-Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.

Vorticity dynamics and numerical Resolution of Navier-Stokes Equations

“…We refer to [7,8,14,19] for further details. In the context of the present vorticity-streamfunction formulation, the problem of "vorticity boundary conditions" has been studied by Anderson [1], Quartapelle et al [10,29,30] and E and Liu [11][12][13]. Roughly speaking, the basic idea is to derive linear ("orthogonality") conditions that must be satisfied by the vorticity, as is seen from (2.9).…”

“…This is a set of linear equations corresponding to the discrete analog of the subspace K 0 (Ω) of harmonic functions. However, "unfortunately, this full system of equations has a rather cumbersome profile since the equations expressing the integral conditions have almost all coefficients different from zero" [10] p. 878. This is remedied and simplified in the method proposed by Dean, Glowinski and Pironneau [9] by splitting equations (2.2)-(2.4) into two separate Poisson equations.…”

Differentially heated flow from a rotating sphere

“…Various methods have been advanced to deal with this. One involves the use of so-called integral conditions which are explained in [4].…”

A numerical method for studying impulsively generated convection from heated tubes

“…Later we will discuss a method to prescribe the surface vorticity. In [2], the vorticity field is shown to satisfy integral constraints. These can be derived from the no-slip boundary conditions using Green's second identity and are given by:…”