Geoscience Reference
In-Depth Information
I was not satisfied with the mixture-length theory, because the idea that a fluid mass would
go a certain distance unchanged and then deliver up its transferable property, and become
identical with the mean condition at that point, is not a realistic picture of a physical process.
He credited his misgivings about mixture-length theory to his development of an
exact solution for the growth of the mean plume from a continuous point source
(
Section 4.3.2
):
While thinking of [analytical “fall out” from attempts to rationalize a mixture-length theory]
I became interested in the formwhich a smoke trail takes after leaving a chimney. Any theory
of diffusion which is based on a virtual coefficient of diffusion must predict a mean shape
for a smoke plume which is paraboloidal, and it was quite clear to me that near the emitting
source the mean outline of a smoke plume is pointed. This led me to think of other ways than
mixture-length theory to describe turbulent diffusion. The result was my paper “Diffusion
by continuous movements,” (
1921
) in which the idea of correlation was introduced into the
subject, I think for the first time.
Taylor's “pointed mean outline of a smoke plume” is the linear limit of
Eq. (4.29)
.
Perhaps Taylor's (
1921
) paper, which presented a formal solution to dispersion
from a continuous point source in steady, homogeneous turbulence, dampened
interest in the much less elegant mixing-length approaches. Indeed
Lumley
(
1989
),
in commenting on the mixing-length (as it is now usually called) assumption, wrote:
When I was a graduate student, I did not know that the mixing length assumption existed….
It was only later that I learned about it.
Taylor subsequently published a four-part series on a statistical approach to
turbulence (
Taylor
,
1935
), which has heavily influenced our present perspectives.
4.6 Summary
We have deduced the behavior of turbulent fluxes in three problems. In each case
they were much larger than the corresponding molecular fluxes, except at a solid
surface. We also found evidence that the turbulent flux, mean-gradient relation need
not be as simple and direct as that in molecular diffusion; we'll see that it rarely is.
We discussed Taylor's early experience with observations of turbulent flows, his
evolving physical notions about turbulent fluxes and their maintenance, and his
attempts to develop a simple “mixture-length” theory for turbulence. We discussed
his misgivings about that theory and his continuing efforts to find better alternatives.
We'll take that path further in
Chapter 5
by examining the evolution equations for
turbulent fluxes.
Questions on key concepts
4.1
Show that the use of the mean-plus-fluctuating decomposition produces sev-
eral types of conserved-scalar fluxes in turbulent flow. Which one is the
turbulent flux
?
