3D Homogeneous Turbulence

In this chapter we focus on what can be called the purest turbulence problem, as well as the classical one. It is in fact so pure that, strictly speaking, it cannot exist in nature, although experience shows that natural turbulence often has behavior remarkably close to the pure form. This is the case where rotational and gravitational forces are negligible (i.e., g = f = 0); b or c can be considered to represent a generic passive tracer (or neglected altogether); all boundary influences are ignored; there is no mean flow; and the statistics are spatially isotropic and homogeneous and temporally either stationary (in equilibrium) or simply decaying from a specified initial condition. In a strict sense, therefore, the domain must be infinite in extent and the time span infinite in duration, but it is often assumed to be spatially periodic on a scale large compared to the motions of interest, and the integration or sampling interval must be large enough (e.g., compared to eddy turnover time) to achieve satisfactory statistical accuracy. With periodic boundary conditions the rotation symmetry is only approximately valid, and presumably more so on the smaller scales than the larger ones. But a periodic domain is more readily computable than an infinite one, and in laboratory experiments domains are both finite and have non-periodic boundaries. The essential governing equations are