TURBULENCE IN AN INTERNAL BOUNDARY LAYER
275
as (4zV + 42; + @)/iqT may be a plausible measure of the convection
velocity in the streamwise direction. The difference u, between the energy
flux velocity and the local mean velocity U is plotted in Fig. 7 neglecting the
contribution from the p cornlation. In the smooth to rough case the energy
flux convection velocity well within the internal layer c&n evidently be as
much as 14 % below the mean velocity. For the rough to smooth case, u,/U
values are close to zero in the central region of the internal layer which
suggests a relatively passive convection of the turbulence field by the mean
field. In contrast to this, the convection velocities deduced from space-time
correlations were approximately the same as the local mean velocity
throughout both the smooth to rough and the rough to smooth internal
layers.
5. SPECTRA OF u AND w
For the inner part of a layer which is in energy equilibrium, Bradshaw
(1967b) has suggested that the use of oil2 and z as, respectively, velocity and
length scales when normalizing spectral densities of u and w and the uw
cospectrum, should collapse the high frequency cmds of the u and w spectra
and the whole of the uw cospcctrum. The low frequency mds of the u and w
spectra receive contributions from the large-scale imctive” motion (Townsend, 1961; Bradshaw, 1967b) which does not contribute to r in the inner
part of the layer and hence the r112, z scaling will not collapse the low
frequencies.
In the present experiments one would not nccesrarily expect the r, z
scaling to collapse spectral densities for the “active” (shear stress producing) motion. The shear stress r varies appreciably a c m the internal layer
and the energy budgets for the smooth to rough ( I ) d the rough to smooth
(11) surface changes reveal that the internal layers are not in energy equilibrium as evidenced by significant advection and diffusion. The inadequacy
of the r, z scaling in these circumstances is clearly shown in Figs. 8-10 for
two streamwise stations in the smooth to rough case and one only in the
rough to smooth case. These figurn are plotted in the b m +(k* versus &*
k* is the dimensionless wave number z = oz/U. Thc cospectrum t$,(k*)
has the obvious constraining property that
where +,(k*), for example, is the dimensionless spectral density -1 u (&*)/I and
\ )#)
dk’ = 1.
The general trend of the results i that for relatively high frequencies (&* > 3,
say) the spectral density increases with increasing z. The opposite trend is
observed for k* < 3.
275
as (4zV + 42; + @)/iqT may be a plausible measure of the convection
velocity in the streamwise direction. The difference u, between the energy
flux velocity and the local mean velocity U is plotted in Fig. 7 neglecting the
contribution from the p cornlation. In the smooth to rough case the energy
flux convection velocity well within the internal layer c&n evidently be as
much as 14 % below the mean velocity. For the rough to smooth case, u,/U
values are close to zero in the central region of the internal layer which
suggests a relatively passive convection of the turbulence field by the mean
field. In contrast to this, the convection velocities deduced from space-time
correlations were approximately the same as the local mean velocity
throughout both the smooth to rough and the rough to smooth internal
layers.
5. SPECTRA OF u AND w
For the inner part of a layer which is in energy equilibrium, Bradshaw
(1967b) has suggested that the use of oil2 and z as, respectively, velocity and
length scales when normalizing spectral densities of u and w and the uw
cospectrum, should collapse the high frequency cmds of the u and w spectra
and the whole of the uw cospcctrum. The low frequency mds of the u and w
spectra receive contributions from the large-scale imctive” motion (Townsend, 1961; Bradshaw, 1967b) which does not contribute to r in the inner
part of the layer and hence the r112, z scaling will not collapse the low
frequencies.
In the present experiments one would not nccesrarily expect the r, z
scaling to collapse spectral densities for the “active” (shear stress producing) motion. The shear stress r varies appreciably a c m the internal layer
and the energy budgets for the smooth to rough ( I ) d the rough to smooth
(11) surface changes reveal that the internal layers are not in energy equilibrium as evidenced by significant advection and diffusion. The inadequacy
of the r, z scaling in these circumstances is clearly shown in Figs. 8-10 for
two streamwise stations in the smooth to rough case and one only in the
rough to smooth case. These figurn are plotted in the b m +(k* versus &*
k* is the dimensionless wave number z = oz/U. Thc cospectrum t$,(k*)
has the obvious constraining property that
where +,(k*), for example, is the dimensionless spectral density -1 u (&*)/I and
\ )#)
dk’ = 1.
The general trend of the results i that for relatively high frequencies (&* > 3,
say) the spectral density increases with increasing z. The opposite trend is
observed for k* < 3.