Is a Direct Numerical Simulation of Chaos or Turbulence Possible: A Study of a Model Non-Linearity

Abstract

There are many subtle issues associated with solving the Navier-Stokes equations. In this paper, several of these issues, which have been observed previously in research involving the Navier-Stokes equations, are studied within the framework of the one-dimensional Kuramoto-Sivashinsky (KS) equation, a model nonlinear partial-differential equation. This alternative approach is expected to more easily expose major points and hopefully identify open questions that are related to the Navier-Stokes equations. In particular, four interesting issues are discussed. The first is related to the difficulty in defining regions of linear stability and instability for a time-dependent governing parameter; this is equivalent to a time-dependent base flow for the Navier-Stokes equations. The next two issues are consequences of nonlinear interactions. These include the evolution of the solution by exciting its harmonics or sub-harmonics (Fourier components) simultaneously in the presence of a constant or a time-dependent governing parameter; and the sensitivity of the long-time solution to initial conditions. The final issue is concerned with the lack of convergent numerical chaotic solutions, an issue that has not been previously studied for the Navier-Stokes equations. The last two issues, consequences of nonlinear interactions, are not commonly known. Conclusions and questions uncovered by the numerical results are discussed. The reasons behind each issue are provided with the expectation that they will stimulate interest in further study.