a nonlinear tield equation (such as the Navier-Stokes equation) must have a
number of moments greater than the number of equations (“the closure
problem,” a phrase introduced to turbulence possibly by Kraichnan), the
object of most theoretical basic research and analytical practical technology
in the area is to discover/devise whatever number of additional relations is
needed to render the problem determinate. For technological purposes (including meteorological and oceanographic forecasting), we want primarily
some additional assumptions which are self-consistent. violate no scientific
principles. and “work *’ ; for research, we want assumptions which not only
work but also have some a priori plausibility (even if discovered a posteriori), and contribute enough explanation to be called a ” theory.”
Simple gradient transport assumptions are virtually the only kind used for
practical calculations in turbulent transport in 1973, in spite of the fact that
it has long been realized (see. e.g, the textbook of Bosworth, 1952) that a
gradient transport model requires (among other things) that the characteristic scale of the transporting mechanism [mean free path in gas kinetics:
Lagrangian velocity integral time scale multiplied by root-mean-square
velocity, in turbulence (Taylor, 1921, 1938b)], must be small compared with
the distance over which the mean gradient of the transported property
changes appreciably. From time to time (e.g., Batchelor, 1950; Corrsin,
1957) it has been pointed out that nearly all traditional turbulent transport
problems violate this requirement, yet papers introducing or using gradient
transport models proliferate, and contain no hint that they must often be
wrong in principle. Surprisingly, they sometimes can be made to yield good
agreement with experiment, although this “SUCIXSS” may be attributable to
the number of empirical constants (sometimes an empirical function) available. There has been an astonishing lack of amazement at the partial success
of these models.
The purpose of this paper is to identify some conditions which are necessary (though not sufficient) for the validity ofa gradient transport model in a
simple mean-free-path type of random transport process. essentially a
random walk. Presumably, more complex random convective transport
phenomena are subject to analogous necessary conditions in order that
gradient trnnsport be a plausible approximation. and we shall see how well
Iliese conditions are fulfilled in some standard turbulent transport problems.
I t will bc coticluded that the partial succcss ofgradient transport models in
turbulcncc is largely fortuitous, and certainly surprising.
Doubtless the ” generalized ” effects of inhomogeneity and nonstationarity
indicated in this account have been presented in the kinetic theory literature
many decades ago. and in considerably more rigorous fashion. Unfortunately, there has not been sufficient time to search for them, so I have been
unable to ascribe proper credit to writers who deserve it. The monograph of
number of moments greater than the number of equations (“the closure
problem,” a phrase introduced to turbulence possibly by Kraichnan), the
object of most theoretical basic research and analytical practical technology
in the area is to discover/devise whatever number of additional relations is
needed to render the problem determinate. For technological purposes (including meteorological and oceanographic forecasting), we want primarily
some additional assumptions which are self-consistent. violate no scientific
principles. and “work *’ ; for research, we want assumptions which not only
work but also have some a priori plausibility (even if discovered a posteriori), and contribute enough explanation to be called a ” theory.”
Simple gradient transport assumptions are virtually the only kind used for
practical calculations in turbulent transport in 1973, in spite of the fact that
it has long been realized (see. e.g, the textbook of Bosworth, 1952) that a
gradient transport model requires (among other things) that the characteristic scale of the transporting mechanism [mean free path in gas kinetics:
Lagrangian velocity integral time scale multiplied by root-mean-square
velocity, in turbulence (Taylor, 1921, 1938b)], must be small compared with
the distance over which the mean gradient of the transported property
changes appreciably. From time to time (e.g., Batchelor, 1950; Corrsin,
1957) it has been pointed out that nearly all traditional turbulent transport
problems violate this requirement, yet papers introducing or using gradient
transport models proliferate, and contain no hint that they must often be
wrong in principle. Surprisingly, they sometimes can be made to yield good
agreement with experiment, although this “SUCIXSS” may be attributable to
the number of empirical constants (sometimes an empirical function) available. There has been an astonishing lack of amazement at the partial success
of these models.
The purpose of this paper is to identify some conditions which are necessary (though not sufficient) for the validity ofa gradient transport model in a
simple mean-free-path type of random transport process. essentially a
random walk. Presumably, more complex random convective transport
phenomena are subject to analogous necessary conditions in order that
gradient trnnsport be a plausible approximation. and we shall see how well
Iliese conditions are fulfilled in some standard turbulent transport problems.
I t will bc coticluded that the partial succcss ofgradient transport models in
turbulcncc is largely fortuitous, and certainly surprising.
Doubtless the ” generalized ” effects of inhomogeneity and nonstationarity
indicated in this account have been presented in the kinetic theory literature
many decades ago. and in considerably more rigorous fashion. Unfortunately, there has not been sufficient time to search for them, so I have been
unable to ascribe proper credit to writers who deserve it. The monograph of