TABLE OF CONTENTS Turbulence is an irregular fluctuating flow field both spatially and temporally. It is a fundamental problem to be answered whether the fluid mechanics of continuum media can capture the turbulence.
In the past, there have been a number of discussions about how such turbulence is generated. On the one hand, turbulence is considered to be generated by spatially random initial conditions, or temporally random boundary conditions. This is a view of turbulence generated passively. In the modern view, however, turbulence is considered to be an irregularly fluctuating flow field which develops autonomously by a nonlinear mechanism of field dynamics. This change in view on turbulence is largely due to the recent development of the theory of chaos considered in the last chapter (Chapter 9).
Chaos is studied mostly for nonlinear dynamical systems of low dimensions. In a chaotic state, an initial slight difference of state grows exponentially with time. Despite the fact that the system evolves according to a governing equation, the state of the dynamical system becomes unpredictable after a certain time. This is a property termed deterministic chaos considered in Chapter 9.
Turbulence is a dynamical system of considerably many degrees of freedom. It is not surprising that the turbulence exhibits much more complex behaviors. In such a (turbulence) field, it would not be realistic to specify the initial condition accurately, and a statistical consideration must be made. There are some intrinsic difficulties in statistics: that is, statistical distributions of velocity or velocity difference at two...
