A Numerical Method for Calculating the Starting and Perturbation of a Two-dimensional Jet at Low Reynolds Number

Abstract

The number of available exact solutions of the full equations of motion of an incompressible viscous fluid is remarkably few. Those that exist are mostly limited to steady flows. Where a steady state solution does exist, one may be able to obtain a little information about the corresponding unsteady flow by the method of small perturbations. However, in the interesting case of instability, this only shows how a small disturbance behaves initially. The subsequent stages of chaotic motion, as the laminar flow 'breaks up', have attracted the attention of many but still largely remain a source of fascination rather than a field for fruitful research. Only when the flow becomes completely turbulent can the theories of turbulence be applied. These theories do not discuss the origin of turbulence, still leaving the gap in the present state of knowledge between small perturbation theory and turbulence. There is, therefore, a great need for a method of attacking directly the full equations of motion of an unsteady viscous flow. Recent advances in high speed electronic computers make available a powerful device for performing the computations. The lack of some such calculating robot has no doubt discouraged earlier attempts to adopt this approach. Since with any electronic computer one has available only a finite storage space, the possibility of solving a completely three-dimensional problem is perhaps a little ambitious at present. Further, from turbulence theory it is known that large eddies have a tendency to break into smaller eddies limited only by viscosity. Hence, in order to follow numerically a turbulent flow, a large number of closely spaced mesh points would be required to include both the large scale and small scale effects. It is therefore necessary to confine the range of eddy sizes, so that a suitably low Reynolds number must be chosen.