2n
SrANI.I:Y COHHSIN
v, hcro hl'"' rcprcwits karious moments of velocity and concentration fluctuations. In Eq. (3) 'Dr IS ; I function of r and the M'"). expressing the appropriate consequences of thc physical (or mathematical) mechanism which
causes the transport. F IS a functional of Or.
In very general cases, U+ itself may be a functional of r and the M(% For
example, if random convection is dominant (as in some turbulent cases), U+
must depend on the history of the material point displacements A (or
velocities):
(4)
with - m 5 Z" 5 a, - rn S z'" 5 mi, - act s t" 5 t , - a0 5 t"' s t. Mt)
denotes the nth moment of displacements.
If the transporting mechanism is fairly local in space and time, we can
expect the much simpler possibility that F(z, r ) be expressible as an ordinary
function of the concentration at the same point in space-time, and of its first
few derivatives:
The z, r dependence is included to reflect possible dependence on moments
of displacement and concentration fluctuation fields (the latter generally
unknown, of course).
If the mechanism is further restricted to be independent of the value of r
itself, to be homogeneous and stationary, and t o be a linear function of the
derivatives.
(a)
qZ, t ) = K , r r + K2r, + ~~r~~ + K4r,, + KSrl, + .-.
The Ki are constants which are to be evaluated from detailed information on
the mechanism. e.g1 the displacement field and y-field statistics. The letter
subscripts on variables denote partial differentiation.
Equation (6) simplifies still further if we rquire the character of mechanism to be symmetric in space, so that only odd order r spatial derivatives are
allowed.' I f we restrict to random convective processes with large enough
transporting speeds, all time derivative effects arc negligible, and Eq. (6)
reduces to
(7)
' Thiz d m not mean that the actual transport events are necessarily symmetric; i t means
merely that the proass is invariant under a refleaion of the E axis. An even order derivative
would not reverbe aipn under such a rekction. yet f mwt rcvew its sign.
SrANI.I:Y COHHSIN
v, hcro hl'"' rcprcwits karious moments of velocity and concentration fluctuations. In Eq. (3) 'Dr IS ; I function of r and the M'"). expressing the appropriate consequences of thc physical (or mathematical) mechanism which
causes the transport. F IS a functional of Or.
In very general cases, U+ itself may be a functional of r and the M(% For
example, if random convection is dominant (as in some turbulent cases), U+
must depend on the history of the material point displacements A (or
velocities):
(4)
with - m 5 Z" 5 a, - rn S z'" 5 mi, - act s t" 5 t , - a0 5 t"' s t. Mt)
denotes the nth moment of displacements.
If the transporting mechanism is fairly local in space and time, we can
expect the much simpler possibility that F(z, r ) be expressible as an ordinary
function of the concentration at the same point in space-time, and of its first
few derivatives:
The z, r dependence is included to reflect possible dependence on moments
of displacement and concentration fluctuation fields (the latter generally
unknown, of course).
If the mechanism is further restricted to be independent of the value of r
itself, to be homogeneous and stationary, and t o be a linear function of the
derivatives.
(a)
qZ, t ) = K , r r + K2r, + ~~r~~ + K4r,, + KSrl, + .-.
The Ki are constants which are to be evaluated from detailed information on
the mechanism. e.g1 the displacement field and y-field statistics. The letter
subscripts on variables denote partial differentiation.
Equation (6) simplifies still further if we rquire the character of mechanism to be symmetric in space, so that only odd order r spatial derivatives are
allowed.' I f we restrict to random convective processes with large enough
transporting speeds, all time derivative effects arc negligible, and Eq. (6)
reduces to
(7)
' Thiz d m not mean that the actual transport events are necessarily symmetric; i t means
merely that the proass is invariant under a refleaion of the E axis. An even order derivative
would not reverbe aipn under such a rekction. yet f mwt rcvew its sign.