5.3 Statistical properties of the flow
5.3.1 Single point statistics
To validate the flow characteristics and the performance and accuracy of the double stereo-scopic PIV system, the main statistical properties of the turbulent boundary-layer flow, mea-sured 18 m behind the leading edge of the flat plate in the flow facility displayed on page 8, were calculated from the velocity data and compared with the theoretical results presented in section 5.1 and the hot-wire anemometry measurements in [11, 21]. The hot-wire mea-surement was performed by Bruns, Dengel and Fernholz in 1992 in the large wind-tunnel at the Hermann F¨ottinger Institute (HFI) in Berlin at a similar Reynolds number but different9
and õ> . The measurement position wasßÁÉÎ m behind a V-dymotape tripping device and the
thickness of the boundary layer at the measurement location wasÜ
W
¾Á`ÆÏÎ[Ë m and ¾Á`ÆÏÈ¿Â m at free stream velocities ofõ>
W
ÆÆ m/s and¾ m/s. The corresponding Reynolds number based on the momentum thickness was comparable with the PIV experiments listed in table 5.2,
5 Investigation of the xy-plane namely &~
W
ÎÆË0¾ and 16080. The friction velocity
QP
Øõ> R ]
fàÛ was determined from the parameter R Figure 5.2 reveals the mean velocity profile of the turbulent boundary layer flow in outer- and inner-law scaling. The agreement between the stereoscopic PIV results with the analytical law-of-the-wall and the log-law, derived on page 71, is excellent and also the hot-wire mea-surements match nicely. Only for: T I
Þ¾¾ a progressing departure of the HWA results from the present PIV investigation can be observed in the semilogarithmic representation. This effect is caused by the fact that the boundary layers thickness at HFI is only half of the thick-ness obtained at LML. However, this is not relevant for the present investigation, because here only the near-wall phenomena below :à¿Ü
¾Á2ÆÏÈ will be considered and not the outer-flow
effects caused by intermittency. Remarkable is the average dynamic which could be reached with the PIV-system and the spatial resolution in wall-normal direction. The location of the first measurement point, located at : T W
ÛÁÃÂ , is comparable with the nearest wall location
0 0.1 0.2
FIGURE5.2: Comparison of the measured mean velocity profiles in outer- (left) and inner-law scaling (right) with the hot-wire results by Bruns et al [11, 21] for two Reynolds numbers.
5.3 Statistical properties of the flow for the stream-wise velocity in [55] but here the velocity could be determined directly with-out a sophisticated calibration procedure as required in case of the hot-wire investigation to compensate the additional heat-loss induced by the solid wall. When the wall-normal velocity component is considered, the first measurement point obtained with the hot-wire technique is farther away from the wall by one order of magnitude, compared with the present PIV inves-tigation, due to the orientation and size of the sensor applied in [55]. The domain covered by the PIV investigation is nearly constant when considered in outer variables (
:à[Ü
4, ) because of the weak dependence of the boundary-layer thickness Ü on the Reynolds number.
However, when considered in inner variables on the other hand, it can be seen that the inves-tigation at&~ رοÞz roughly covers the domain from
: T
, because the log-law region increases in thickness with increas-ing Reynolds number when considered in wall-units. This allows to examine the near-wall structure at &~ Ø Î¿Þz , Reynolds number effects within the logarithmic region between
: T h
and large-scale flow structures at
&~
Ø,z . The size of the region, where
10 0 10 1 10 2 10 3 10 4
FIGURE 5.3: Left: Non-dimensional rms-profiles of the three velocity fluctuations for ¡N¢
¤£¦¥§
ss
(top) and ¡N¢
U£©¨
vsss (bottom). Right: Dependence of the anisotropy parameter ª«Z¬® ¯4¬®°²±
ò ¬ on the wall coordinate in inner-law scaling for both Reynolds numbers.
5 Investigation of the xy-plane
the log-law holds, is quite large for both cases so that a direct interaction between the near-wall flow structure with the intermittent flow region can not take place. Thus the turbulent flow state can be considered as fully developed. Fully developed flows are characterised by a Kolmogorov cascade including an inertial range over at least one order of magnitude in the wavenumber space [78]. This condition implies unsteadiness, rotational, three-dimensionality, non-deterministic, diffusion and dissipation, e.g. attributes which are generally referred to characterise turbulent flows. The dynamic of the velocity signal as a function of the wall dis-tance can be estimated from figure 5.3 which shows the normalised rms values of the three velocity fluctuations for both flow cases. It can be seen that the maximum can be properly resolved with state-of-the-art PIV systems, provided the magnification of the imaging system is well adjusted relative to the flow structures and the problems associated with the strong wall reflections are solved. The low Reynolds number results agree quite well with the HWA mea-surements kindly provided by Prof. Dr. Fernholz. Only the first two measurement points near the wall are overestimated. This is natural due to the finite size of the measurement volume.
When the high Reynolds number results are considered the deviation increases but the increase of the second peak with increasing Reynolds number is clearly visible and in agreement with the literature [21]. To validate the degree of anisotropy between different velocity compo-nents in turbulent shear flows, the ratio between orthogonal velocity fluctuations is shown in the right graphs of the same figure. As the energy from the mean motion is first transferred into the stream-wise velocity fluctuation before the transfer into the³ and´ component takes place by means of pressure fluctuations, the value of the anisotropy parameter is usually around 0.6 for large wall distances. In addition, it can be seen that µ³ ¬'¶ ·E¬'¸ ±º¹ ¬ increases gradually with the wall distance. Figure 5.4 shows the turbulence-level defined as µ· ¬ ¸ ±º¹ ¬ ¶ » for both flow cases. The agreement with the hot-wire measurements performed at ¼ ~½¿¾7À
and ÁÂ
is excellent. Only the first PIV measurement point in the left graph is far away from the HWA result, which is around à ·
¾
4
for Ä4Å<ÆÇ . However, this is associated with the
increas-ing measurement error when the displacement of the particle images becomes very small, see chapter 2 for details.
10È 0 10È 1 10È 2 10È 3 10È 4 y+
0.0 0.2 0.4 0.6 0.8
urms / U
Reθ=7800 (PIV) Reθ=7140 (HWA)
10È 0 10È 1 10È 2 10È 3 10È 4 y+
0.0 0.2 0.4 0.6 0.8
urms / U
Reθ=15000 (PIV) Reθ=16080 (HWA)
FIGURE5.4: Turbulence-levelÉʯ1Ë!Ì ¯4¬® Í measured at two Reynolds numbers.
5.3 Statistical properties of the flow