TRANSPORT MODEL 1 IMITAI'IONS IN TURBULENCE
49
F'urthemorc. as4uming that the hypothetical configuration = f'(s,) in a
hounddry layer would yield ;ui / ,'u; z p3 /g'u', , an experimental result for
f = I?(u,). hc offered the conjecture
(76)
D3I = -3D11.
If we speculate that DI, z 45D3, in a boundary layer [based on the fact
that at x3/h = 0.45, the Blackwelder/Kovasznay (1972) data give
I,{ T, z 4.53T3,, where these 7"s arc companent Eulerian time scales in
the frame convectcd with the correlation peak], then we find that Yaglom's
suggest ions correspond to
(77)
D 3 , 5 +Dl3 *
Latcr in this section we shall estimate somc D components from data taken
under better defined conditions, in the laboratory.
First, it may be helpful to recall Yaglom's (1969) qualitative explanation
of the offdiagonal components of D a .
The I-component of the mean flux veclor, Eq. (70), is
(78)
E , = - D ~ ~
c!r/axl - D , ~
ariax,,
where we restrict to two dimensions for simplicity. At first glance it may
seem paradoxical that transport of r in the xI direction can be proportional
to the mean gradient along x3, the axis perpendicular to xI. A simple
example can illustrate the possibility. Suppose that at an arbitrary time the r
field has a constant mean gredicnt (Fig. 5a). The material field undergoes
random displacements in the plane. It is essentially obvious that the random
particle displacements will convect r down the gadicnt on tb average. In
particular, the random xI displacements will tramport r down the dr/ax,
gradient, and the random x3 displacements win transport r down the
i)r/Jx3 gradient. If X I and X 3 (the xI and x3 displacements) are uncorrelated. that is the only kind of transport which will occur. However, if
X , X , # 0, v3 displacements, which by definition are along the d r p u ,
component direction, on the average will transport f parallel to the x, axis
as wcll ;LI: down the X / d y 3 gradient.
Eipurc Sb clearly illustrates the crosseffect. Here dr/L)x; = 0. yet there
will be mean trcinsport of parallel to the xi axis. It is also clear from
Fig. Sb Ihat if X i X 3 > 0, D 3 , > 0, and if XIX3 c 0. D,, < 0 [Eq. (M)].
II' a cross-correlated random walk example were worked out with
X : = X i , it would give D31 = D 1 3 . This is obvi6us from a comparison of
Fig. 5b with the same case with x i and x i axes interchanged.
Having seen why the offdiagonal terms of the difisivity tensor arise, we
may turn to the question of their experimental determination in a turbulent
flow for which the gradient transpart model is aiebumad to be satisfactory.
49
F'urthemorc. as4uming that the hypothetical configuration = f'(s,) in a
hounddry layer would yield ;ui / ,'u; z p3 /g'u', , an experimental result for
f = I?(u,). hc offered the conjecture
(76)
D3I = -3D11.
If we speculate that DI, z 45D3, in a boundary layer [based on the fact
that at x3/h = 0.45, the Blackwelder/Kovasznay (1972) data give
I,{ T, z 4.53T3,, where these 7"s arc companent Eulerian time scales in
the frame convectcd with the correlation peak], then we find that Yaglom's
suggest ions correspond to
(77)
D 3 , 5 +Dl3 *
Latcr in this section we shall estimate somc D components from data taken
under better defined conditions, in the laboratory.
First, it may be helpful to recall Yaglom's (1969) qualitative explanation
of the offdiagonal components of D a .
The I-component of the mean flux veclor, Eq. (70), is
(78)
E , = - D ~ ~
c!r/axl - D , ~
ariax,,
where we restrict to two dimensions for simplicity. At first glance it may
seem paradoxical that transport of r in the xI direction can be proportional
to the mean gradient along x3, the axis perpendicular to xI. A simple
example can illustrate the possibility. Suppose that at an arbitrary time the r
field has a constant mean gredicnt (Fig. 5a). The material field undergoes
random displacements in the plane. It is essentially obvious that the random
particle displacements will convect r down the gadicnt on tb average. In
particular, the random xI displacements will tramport r down the dr/ax,
gradient, and the random x3 displacements win transport r down the
i)r/Jx3 gradient. If X I and X 3 (the xI and x3 displacements) are uncorrelated. that is the only kind of transport which will occur. However, if
X , X , # 0, v3 displacements, which by definition are along the d r p u ,
component direction, on the average will transport f parallel to the x, axis
as wcll ;LI: down the X / d y 3 gradient.
Eipurc Sb clearly illustrates the crosseffect. Here dr/L)x; = 0. yet there
will be mean trcinsport of parallel to the xi axis. It is also clear from
Fig. Sb Ihat if X i X 3 > 0, D 3 , > 0, and if XIX3 c 0. D,, < 0 [Eq. (M)].
II' a cross-correlated random walk example were worked out with
X : = X i , it would give D31 = D 1 3 . This is obvi6us from a comparison of
Fig. 5b with the same case with x i and x i axes interchanged.
Having seen why the offdiagonal terms of the difisivity tensor arise, we
may turn to the question of their experimental determination in a turbulent
flow for which the gradient transpart model is aiebumad to be satisfactory.