212
H. A. ANTONIA AND R. E. LUXTON
4. DIFFUSION VELWITUS OF u AND w
Bradshaw (1967a) defines a velocity of diffusion W, of turbulent
energy in the z direction as W = (ifi + G)/ig, where is the turedge of a self-preserving boundary layer W, h equal to the entrainment rate
provided that advwtion and difhrsion arc aqua1 and that the shear stress,
and therefore the production and dissipatioa d turbulent energy. is small.
Turbulent energy balm- in both the smooth and rough wall boundary
layers show that, near the outer edge of the laycrs, the diffusion of turbulent
energy by i f i is very marly equal to the advection tern, so that, fortunately, it would appear that the pressure ditlusion term is small and the
above definition of W, can be simplified to W, = fi/?, at least well away
from the waN5 Near the edge of the internal layer, the diFfusion by f i is
again approximately equal to the advcction but the contributions from the
productiop ;(8U/&) and the dissipation are appreciably larger (see Fig. 19
of I and Fig. 12 of 11). One cannot then simply equate W, to the rate of
increase of the internal layer thickness b,, but it is useful to compare the
measured values of W, for the two internal layers considered here. The
distributions of W,, normalized by the local mean velocity U, are shown in
Fig6 as a function of z/S,, where 6, is the mtmnal layer thickness, at two
streamwise stations for the smooth to rough cbange and four x stations for
the rough to smooth chan e. In evaluating W, the intensity a was approxWJU reaches a maximum well inside the edge of the internal layer and,
although not shown in Fig. 6, has another maximum near the outer edge of
the external layer. Thia maximum in W, well within the internal layer, when
taken in conjunction with the broad peak in the flatness factor, suggests that
most of the *entrainment” of ‘I old *’ turbuknck occurs near the troughs
in the interface, i.e., at around z/b, ‘Y 0.5. For the rough to smooth change,
on the other hand, WJU first decreases at small values of z/6, but then rises
to a maximum near z/d, = 1.0. Thc value of thie maximum is only about half
that of the maximum for the smooth to roush change which seems to be in
qualitative agreement with the different ratas of propagation of 6, reported
in 11. The location of this peak is consistent with an “upsidedown” intermittency with “upside-down” entrainment occurring at the crests of the
lower energy internal layer, i.e., at the troughs of the higher energy external
layer.
Bradshaw (1967a) has also suggested that the energy flux velocity defined
bulence intensity (2 + 2 + ip w ) and p is the pressure fluctuation. At the
imated by 46’ + 2) as 9 u was not measured. in the smooth to rough case,
’ On the rough wall the prmure difhsion term may ba important in the inner 20 % of the
layer (see Antonia and Luxton. 1971b).
H. A. ANTONIA AND R. E. LUXTON
4. DIFFUSION VELWITUS OF u AND w
Bradshaw (1967a) defines a velocity of diffusion W, of turbulent
energy in the z direction as W = (ifi + G)/ig, where is the turedge of a self-preserving boundary layer W, h equal to the entrainment rate
provided that advwtion and difhrsion arc aqua1 and that the shear stress,
and therefore the production and dissipatioa d turbulent energy. is small.
Turbulent energy balm- in both the smooth and rough wall boundary
layers show that, near the outer edge of the laycrs, the diffusion of turbulent
energy by i f i is very marly equal to the advection tern, so that, fortunately, it would appear that the pressure ditlusion term is small and the
above definition of W, can be simplified to W, = fi/?, at least well away
from the waN5 Near the edge of the internal layer, the diFfusion by f i is
again approximately equal to the advcction but the contributions from the
productiop ;(8U/&) and the dissipation are appreciably larger (see Fig. 19
of I and Fig. 12 of 11). One cannot then simply equate W, to the rate of
increase of the internal layer thickness b,, but it is useful to compare the
measured values of W, for the two internal layers considered here. The
distributions of W,, normalized by the local mean velocity U, are shown in
Fig6 as a function of z/S,, where 6, is the mtmnal layer thickness, at two
streamwise stations for the smooth to rough cbange and four x stations for
the rough to smooth chan e. In evaluating W, the intensity a was approxWJU reaches a maximum well inside the edge of the internal layer and,
although not shown in Fig. 6, has another maximum near the outer edge of
the external layer. Thia maximum in W, well within the internal layer, when
taken in conjunction with the broad peak in the flatness factor, suggests that
most of the *entrainment” of ‘I old *’ turbuknck occurs near the troughs
in the interface, i.e., at around z/b, ‘Y 0.5. For the rough to smooth change,
on the other hand, WJU first decreases at small values of z/6, but then rises
to a maximum near z/d, = 1.0. Thc value of thie maximum is only about half
that of the maximum for the smooth to roush change which seems to be in
qualitative agreement with the different ratas of propagation of 6, reported
in 11. The location of this peak is consistent with an “upsidedown” intermittency with “upside-down” entrainment occurring at the crests of the
lower energy internal layer, i.e., at the troughs of the higher energy external
layer.
Bradshaw (1967a) has also suggested that the energy flux velocity defined
bulence intensity (2 + 2 + ip w ) and p is the pressure fluctuation. At the
imated by 46’ + 2) as 9 u was not measured. in the smooth to rough case,
’ On the rough wall the prmure difhsion term may ba important in the inner 20 % of the
layer (see Antonia and Luxton. 1971b).