On the complexity of a locally perturbed liquid flow

Abstract

A system-theoretic point of view on the increase in the complexity of a laminar flow, as the result of a spatially local, strongly time-dependent perturbation, is suggested. The basic point is that, on the one hand, in the hydrodynamic reality the influence of the perturbation propagated with the flow with a more or less known delay to a point of observation, and, on the other hand, the system-theoretic point of view requires that for realization of the prescribed delay of a function, the structure of the realizing system must be dependent on its inputs and inevitably will be very complicated for the inputs of a not very complicated waveform. It is explained that it is reasonable to use some concepts of the theory of electronic systems, and an estimation of the degree of complexity of the system, composed from lumped elements, which provides a similar input/output map with the prescribed delay, is suggested, intended to lead to a criterion for the occurrence of turbulence in a given hydrodynamic situation. According to the system-theoretical point of view the velocity vector field is considered macroscopically, assuming, as the zero approximation to the solution of the Navier-Stokes (NS) equation, that the perturbation (whatever its details) is carried, as it is, by the main flow, and finding then at least one more approximation including the viscosity. It is shown that the Laplace transformation of the time-dependent part of the solution of the NS equation, written in a relevant approximation, is a well known approximation to the precisely nonrealizable exponential “transfer function” which corresponds to the pure delay. This leads to the connection of the complexity of the velocity vector field with the parameter which characterizes the precision of the approximate realization of the exponent. The number of elements of the “realizing system”, is found to be equal to ν 0X 2ν , where ν 0 is the average velocity, ν is the kinematic viscosity and x is the Cartesian coordinate, or the point of observation, in the direction of the vector ν 0.