IO. IMPI I( A I R I M OI ‘rtw F
The iypcs o f conditions imentioned as necessary for the gradient transport
model or approximation can bc listed here, omitting the rather general
discussion of Section 2:
(a) the transport mechanism length scale must be much smaller than the
distance over which the curvature of the mean transported field gradient
changes appreciably [Eq. ( 13)];
(b) the transport mechanism time scale must be much smaller tha~i the
time during which the mean transported field gradient changes appreciably
[Eq. ( X ) ] :
(c) the transport mechanism length scale must be essentially constant
over a distance of a length scale (merely for the concept to be meaningful),
and over a distance for which the mean transported field changes appreciably [Eq. (26b)l;
(ci) the transport mechanism velocity must be appreciably more uniform
than the length scale [Eq. (26a)l.
A rather complex condition involving nonuniformities of mean field,
lcngth scale and velocity [Eq. (26c)l is not as simple to interpret.
(e) the restrictions on time dependences of mean field, length scak. and
velocity combine (in the random walk) to a collective condition, obvious
from the form of Eq. (28). which can be described as follows: the relative
local accumulation of f must be very small during one time scale of the
transport mechanism. In a flowing turbulence, for exampk. this should be
established in a frame traveling with the mean flow.
Although we have discussed and illustrated these necessary conditions in
the context of mean free path kinetic theory or random walk problems, it
seems likely that qualitatively analogous forms apply to general random
convective transport as well, including turbulence. Therefore, we shall estimate analogous expressions for a turbulent boundary layer, as an
illustration.
Earlier. more casual, studies gave grounds for pessimism about the validity in principle of gradient transport models. It has been pointed out before
(Corrsin. 1957) that a characteristic length of the turbulent momentum
transfer in a boundary layer is twice as large as the “momentum thickness”
of thc Iaycr. and it has been known for decades that the two-point velocity
correlation function has mcasurable value across nearly the full extent of a
p i p flow (Taylor, 1936). ajet (Corrsin, 1943), and a wake (Townsend, 1949).
The archival literature is replete with data showing, either directly or indirectly. for both scalar and momentum transport, that the mean gradients
The iypcs o f conditions imentioned as necessary for the gradient transport
model or approximation can bc listed here, omitting the rather general
discussion of Section 2:
(a) the transport mechanism length scale must be much smaller than the
distance over which the curvature of the mean transported field gradient
changes appreciably [Eq. ( 13)];
(b) the transport mechanism time scale must be much smaller tha~i the
time during which the mean transported field gradient changes appreciably
[Eq. ( X ) ] :
(c) the transport mechanism length scale must be essentially constant
over a distance of a length scale (merely for the concept to be meaningful),
and over a distance for which the mean transported field changes appreciably [Eq. (26b)l;
(ci) the transport mechanism velocity must be appreciably more uniform
than the length scale [Eq. (26a)l.
A rather complex condition involving nonuniformities of mean field,
lcngth scale and velocity [Eq. (26c)l is not as simple to interpret.
(e) the restrictions on time dependences of mean field, length scak. and
velocity combine (in the random walk) to a collective condition, obvious
from the form of Eq. (28). which can be described as follows: the relative
local accumulation of f must be very small during one time scale of the
transport mechanism. In a flowing turbulence, for exampk. this should be
established in a frame traveling with the mean flow.
Although we have discussed and illustrated these necessary conditions in
the context of mean free path kinetic theory or random walk problems, it
seems likely that qualitatively analogous forms apply to general random
convective transport as well, including turbulence. Therefore, we shall estimate analogous expressions for a turbulent boundary layer, as an
illustration.
Earlier. more casual, studies gave grounds for pessimism about the validity in principle of gradient transport models. It has been pointed out before
(Corrsin. 1957) that a characteristic length of the turbulent momentum
transfer in a boundary layer is twice as large as the “momentum thickness”
of thc Iaycr. and it has been known for decades that the two-point velocity
correlation function has mcasurable value across nearly the full extent of a
p i p flow (Taylor, 1936). ajet (Corrsin, 1943), and a wake (Townsend, 1949).
The archival literature is replete with data showing, either directly or indirectly. for both scalar and momentum transport, that the mean gradients