I KANSPORT MODEL LIMI’TA’TIONS IN TlIRBL’I-ENCE
47
hand side of the lattcr inequality comes out to 0.15. We infer that the
transport is quasi-stationary in a frame moving with the mean speed.
In summary of this scction, we have found that, of several homogeneity
and (convected frame) stationarity conditions which are necessary for the
applicability of a gradient transport model in turbulence, one is strongly
violated in the central region of a turbulent boundary layer: that requiring
cross-stream uniformity of the length scale and rms velocity fluctuations.
Other homogeneity conditions are moderately well satisfied for t/6 2 0.2.
Thc stationarity conditions are fairly well satisfied.
The reason that the relatively large main gradient difference across one
characteristic transport length [70 7;; see Eq. (6111 does not cause a major
violation. is that the second derivative is excluded by mechanism symmetry
requircmcnts, so the “correction” term is proportional to the mean third
derivative and the cube of the length scale ratio.
1 I , DIRF.(TIC)NAL CONSTRAINT IN GRADIENT TRANSPORT:
THE TURBULENT DIFFUSIWY TENSOR
The simplest ridimensional generalization of Eq. (8). the mean flux of a
transported property which undergoes simple gradient transport, is
(69 1
F(x. t ) = -D vr.
The scalar character of the diffusivity obviously implies an isotropic transport mechanism.
The well-known, lionisotropic generalization of Eq. (69) requires that the
difrusivity be ;I second rank tensor (rather than a vector), in order that F
remain :I vector:
(70)
F = - D.(VI”).
It was notcd quite ii while ago (e.g., by Richardson, 1920) that the turbulciil dilrusivily (or heat. for example) has a different value in different directions. in turbulent shear flow. Richardson suggested, in effect. that
(71) Fl =. - Dl i ? r / ? . ~ l ; Fz = -Dz aF/{?xz; Ps = - D J ?f/?.x3.
A rcview of the applications of this form as applied to a turbulent diffusivity
has been given by Monin and Yaglom (1965, 1971), and it need not be
pursucd here. An obvious difficulty of Eq. (71) appears when it is written in
correct form, i.e. Eq. (70). which recognizes that a nonisotropic diffusivity
must he 3 tensor of at least second order. Recognition that Eq. (71) is actually an inner product of a second rank tensor and a vector, in a particular
47
hand side of the lattcr inequality comes out to 0.15. We infer that the
transport is quasi-stationary in a frame moving with the mean speed.
In summary of this scction, we have found that, of several homogeneity
and (convected frame) stationarity conditions which are necessary for the
applicability of a gradient transport model in turbulence, one is strongly
violated in the central region of a turbulent boundary layer: that requiring
cross-stream uniformity of the length scale and rms velocity fluctuations.
Other homogeneity conditions are moderately well satisfied for t/6 2 0.2.
Thc stationarity conditions are fairly well satisfied.
The reason that the relatively large main gradient difference across one
characteristic transport length [70 7;; see Eq. (6111 does not cause a major
violation. is that the second derivative is excluded by mechanism symmetry
requircmcnts, so the “correction” term is proportional to the mean third
derivative and the cube of the length scale ratio.
1 I , DIRF.(TIC)NAL CONSTRAINT IN GRADIENT TRANSPORT:
THE TURBULENT DIFFUSIWY TENSOR
The simplest ridimensional generalization of Eq. (8). the mean flux of a
transported property which undergoes simple gradient transport, is
(69 1
F(x. t ) = -D vr.
The scalar character of the diffusivity obviously implies an isotropic transport mechanism.
The well-known, lionisotropic generalization of Eq. (69) requires that the
difrusivity be ;I second rank tensor (rather than a vector), in order that F
remain :I vector:
(70)
F = - D.(VI”).
It was notcd quite ii while ago (e.g., by Richardson, 1920) that the turbulciil dilrusivily (or heat. for example) has a different value in different directions. in turbulent shear flow. Richardson suggested, in effect. that
(71) Fl =. - Dl i ? r / ? . ~ l ; Fz = -Dz aF/{?xz; Ps = - D J ?f/?.x3.
A rcview of the applications of this form as applied to a turbulent diffusivity
has been given by Monin and Yaglom (1965, 1971), and it need not be
pursucd here. An obvious difficulty of Eq. (71) appears when it is written in
correct form, i.e. Eq. (70). which recognizes that a nonisotropic diffusivity
must he 3 tensor of at least second order. Recognition that Eq. (71) is actually an inner product of a second rank tensor and a vector, in a particular