54
STAX LEY CORRSIN
models in turbulence. These conditions stipulate degrees of homogeneity
itnd stationarity of thc mcan field being transported and of the turbulence
propertics central to the transport mechanism. One or more of these conditions appcar to be violated in each of the traditional turbulent flow boundary value problems, such as boundary layer and jet. The largest error in
principle would secm to arise from the effects of inhomogeneity in transporting mechanism. e.g., the turbulence scale and rms velocity. as illustrated by
Eq. (64).
Is there a simple generaliiation which may suffice for interim applications
while basic research proceeds? Perhaps correction terms analogous to those
in Eq. ( 2 5 ) could be added. A second-order correction term like that in
Eq. (9) may not be appropriate unkss the multitude of other second-order
corrections are included as well. These can be inferred in a straightforward
(but tedious) Taylor series expansion in space and time, allowing mean
concentration f. length scale I, and transport velocity scale V all to vary
with position and time. This would be an extension of the calculation in the
Appendix.
It is also interesting to ask whether a model for turbulent transport can be
approached from the other length-scale limit, ie., analogous to radiative
transport in an "optically thin" medium-a direction which has also occurred to Spalding." The difficulties are, ofcoune, enormously greater than
in the radiation problem, the most obvious reasons being the continuous
interaction in turbulent motion, and the fact that the instantaneous speed of
random turbulent transport is not orders of magnitude larger than the mean
motion of the medium; it is, in fact, often smalkr. Thc formulation which
most nearly resembles radiative transport is the *backward diffusion *'
approach of Corrsin (1952, 1972; see also Batchelor, 1949), in which statistical momcnts at a point in space-time (x, c) are expressed as integrals over
thc (dispersed) prior locat ions of the ensemble of fluid particles which arrive
a t x at timc t (one particle in each "realization").
Finally, it must be remembered that there are many alternative
approaches to turbulent transport which are related to neither a gradient
transport approximation (and its direct generalizations) nor to a hypothetical, long-path, radiative transport analogy. "Closure" schemes for hierarchics of turbulence moment equations have been based on a variety of
truncated expansion procedures which often invoke no explicit physical or
phenomenological images, but are more or less ad hoc, based sometimes on
the intuitive belief that higher order correlation coefficients tend to be very
much smaller than lowcr order ones. It is possible that one of these more
formal closure methods will eventually succeed.
' ' I). U. Spulding, cotnmeni in NASA (197.1).
STAX LEY CORRSIN
models in turbulence. These conditions stipulate degrees of homogeneity
itnd stationarity of thc mcan field being transported and of the turbulence
propertics central to the transport mechanism. One or more of these conditions appcar to be violated in each of the traditional turbulent flow boundary value problems, such as boundary layer and jet. The largest error in
principle would secm to arise from the effects of inhomogeneity in transporting mechanism. e.g., the turbulence scale and rms velocity. as illustrated by
Eq. (64).
Is there a simple generaliiation which may suffice for interim applications
while basic research proceeds? Perhaps correction terms analogous to those
in Eq. ( 2 5 ) could be added. A second-order correction term like that in
Eq. (9) may not be appropriate unkss the multitude of other second-order
corrections are included as well. These can be inferred in a straightforward
(but tedious) Taylor series expansion in space and time, allowing mean
concentration f. length scale I, and transport velocity scale V all to vary
with position and time. This would be an extension of the calculation in the
Appendix.
It is also interesting to ask whether a model for turbulent transport can be
approached from the other length-scale limit, ie., analogous to radiative
transport in an "optically thin" medium-a direction which has also occurred to Spalding." The difficulties are, ofcoune, enormously greater than
in the radiation problem, the most obvious reasons being the continuous
interaction in turbulent motion, and the fact that the instantaneous speed of
random turbulent transport is not orders of magnitude larger than the mean
motion of the medium; it is, in fact, often smalkr. Thc formulation which
most nearly resembles radiative transport is the *backward diffusion *'
approach of Corrsin (1952, 1972; see also Batchelor, 1949), in which statistical momcnts at a point in space-time (x, c) are expressed as integrals over
thc (dispersed) prior locat ions of the ensemble of fluid particles which arrive
a t x at timc t (one particle in each "realization").
Finally, it must be remembered that there are many alternative
approaches to turbulent transport which are related to neither a gradient
transport approximation (and its direct generalizations) nor to a hypothetical, long-path, radiative transport analogy. "Closure" schemes for hierarchics of turbulence moment equations have been based on a variety of
truncated expansion procedures which often invoke no explicit physical or
phenomenological images, but are more or less ad hoc, based sometimes on
the intuitive belief that higher order correlation coefficients tend to be very
much smaller than lowcr order ones. It is possible that one of these more
formal closure methods will eventually succeed.
' ' I). U. Spulding, cotnmeni in NASA (197.1).