6.2 Statistical properties of the buffer layer
6.2.3 Spatial cross-correlation functions
For phenomena associated with the production of turbulence the cross correlationÃË Ñ is more important than the primary components of the correlation tensor, as ÃË Ñ reflects the size and shape of the structures being responsible for the transport of relatively low-momentum fluid outwards into higher speed regions and for the movement of high-momentum fluid toward the wall and into lower speed regions. This correlation was first studied by Tritton in 1967 by using a pair of hot-wire probes 2.1 m behind a trip-wire at free-stream velocities between
Þ
È Ç
m/s and È ª m/s [100]. The boundary layer thickness wasÞ À mm and the Reynolds num-ber ÃÄ ¿ ÀÀÀ . Due to technical reasons his results were only measured along the three principal axes and at larger wall-locations, so that a direct comparison with the work presented here is questionable. In the following, it is important to keep in mind that the cross-correlation
∆x+ ∆x+
∆+ z∆+ z∆+ z
R (y =30)vu +
vu
+
vu
+
(-v)(+u)
+
(-v)(+u)
R (y =10) +
R (y =20)
(-v)(+u)
R (y =30)+
R (y =20)
R (y =10)
FIGURE6.8: Two-dimensional spatial cross-correlation function of fluctuating stream-wise with wall-normal velocity componentsÐ Ë Ñ (left) and conditional cross-correlation of negative wall-normal with positive stream-wise velocityÐ êJî Ë íê® Ñí (right) measured atÓ ® Ô×ÖØÙÛÚØ and Ü Ø (top to bottom).
6 Investigation of the xz-plane
function is not necessarily an even function with a maximum at Í5Î ¿ À in contrast to the primary correlations, but there exists still an important symmetry property when the random variables § and · are interchanged, namely Ã Ñ Ë ¹ Í5Î|¼ ¿ ÃË Ñ ¹ Í5Î|¼ . The left column in figure 6.8 shows the ÃË Ñ correlation function with the · component fixed and§ shifted in the two homogeneous directions. First of all it should be noted that the sign of ÃË Ñ again indicates that the transport of relatively low-momentum fluid outward into higher speed re-gions (§ßºöÀ and · ½þÀ ) and the movement of high-momentum fluid toward the wall into lower speed regions (§N½¤À and· º¤À ) are the predominant processes in the near-wall region.
In addition, the strong elliptical shape implies that the turbulent mixing in the wall-normal direction is related to the low-speed structures represented by à êJî ÑíêJî Ñí in figure 6.7. But ob-viously only a small part of the low-momentum structures shows a correlated motion in both
−300 −200 −100 0ã 100ã 200ã 300ã
FIGURE 6.9: One-dimensional spatial cross-correlation function of Reynolds stress components mea-sured at different wall-normal location (Solid graph:Ó ® Ô Ü Ø . Dotted graph: Ó ® Ô×ÚØ . Dashed graph:
Ó
®NÔõÖØ
) as a function of ® and ® . The symbols indicate the maximum of correlation and the legend the distance from the minimum to the origin.
6.2 Statistical properties of the buffer layer stream-wise and wall-normal direction with sufficient strength (lower curve of figure 6.8) as the total length of the low-speed structures are several thousand wall-units in length according to the size of theà ÑÑ correlation, shown in figure 6.5. This is fully consistent with the size of the ÃËÌË correlation and the results in [44]. The span-wise extent of the flow-structures asso-ciated with the production of turbulence can be estimated best from the upper right graphs of figure 6.9 and the phase relation between both orthogonal fluctuations from the location of the maximum shown in the upper left graphs of the same figure. Apart from the structural details of the flow regions associated with the turbulent exchange in wall-normal and stream-wise direction, it is important to consider the cross-correlation with the span-wise velocity fluctu-ations because any outflow of fluid induces an organised span-wise flow motion towards the lifting structure due to continuity. In addition, as in isotropic turbulence this quantity must be zero, these correlations indicate any departure from this ideal situation which is
mathemati-∆x+ ∆x+
∆+ z∆+ z∆+ z
R (y =30)uw + R (y =30)vw +
uw
+
uw
+
R (y =10) R (y =20)
+
vw
+
R (y =20)vw
R (y =10)
FIGURE 6.10: Two-dimensional spatial cross-correlation function of fluctuating stream-wise with span-wise velocity componentsÐÒÑ (left) and correlation of wall-normal with span-wise components
Ð Ë (right) measured atÓ ®xÔzÖØÙÛÚØ andÜ Ø (top to bottom).
6 Investigation of the xz-plane
cally accessible. Figure 6.10 shows the two-dimensional spatial cross-correlation function of fluctuating stream-wise with span-wise velocity components Ã Ñ ¹ Í5Î ® Á .® ÁÍ ® Ï ) (left) and correlation of wall-normal with span-wise componentsÃË ¹ Í5Î ® Á .® Á ÍlÏ ® ¼ (right) measured atÃÄÅ Ê ÈÉÀÀ and ® Ê «ÀÁ ÀÁªÀ (top to bottom). First of all, it can be concluded from the general size, shape and height of both correlations, that a very well organised span-wise mo-tion exists, especially at .®pÊ ªÀ where the height of ÃË is above theÃË Ñ correlation. This organised motion implies that the concept of local isotropy, which links the turbulent-energy dissipation in shear flows with the dissipation of isotropic turbulence, is inadequate in near-wall turbulence. Secondly, it should be noted that the degree of organisation increases with decreasing wall distance while the opposite holds for the ÃË Ñ correlation as can be estimated from the intensity of the maximum. Thirdly, the spacing between the extrema is increasing with larger wall distances but also, it can be seen that the location of these particular points
∆x+ ∆x+
∆+ z∆+ z∆+ z
(-u)(w)
R (y =10)+ R (y =10)(+u)(w) +
(+u)(w)
R (y =20)+
(-u)(w)
R (y =20)+
(+u)(w)
R (y =30)+
(-u)(w)
R (y =30)+
FIGURE 6.11: Conditional cross-correlation of negative stream-wise fluctuations with the span-wise velocity component Ð êÑë3ìÌíêí (left) and correlation positive stream-wise fluctuations with span-wise componentsÐ êÑ3ìÌíêí (right) measured atÐÅ ûØØ andÓ ® ÖØÙÛÚØÙ Ü Ø (top to bottom).
6.2 Statistical properties of the buffer layer moves to negative stream-wise displacements, particularly if the ÃË correlation is consid-ered. For clarity, the two lower rows of figure 6.9 reveal the functional dependence of the correlations along the axis of symmetry. This representation allows to compare the height of the correlation and the location of the maximum in detail. Figure 6.11 shows the two-dimensional spatial cross-correlation function of negative stream-wise fluctuations with the span-wise velocity component à î Ñ ¹ Í5Î ® Á .® Á ÍlÏ ® ¼ (left) and correlation positive stream-wise fluctuations with span-stream-wise components à ® Ñ ¹ Í Î ® Á ® ÁÍeÏ ® ¼ (right). Figure 6.12 dis-plays the corresponding function of positive wall-normal fluctuations (extracted from flow regions where§ôºóÀ ) with the span-wise velocity componentà ® Ë ¹Í5Î ® Á ® ÁÍeÏ ® ¼ (left) and correlation of negative wall-normal fluctuations (extracted from flow regions where §!½þÀ ) with span-wise components ÃË with ¹|· º»À¼ (right). These structures permit to resolve the
∆x+
∆x+
∆+ z∆+ z∆+ z
(+v)(w)
R (y =10)+ R (y =10)(-v)(w) +
(-v)(w)
R (y =20)+
(+v)(w)
R (y =20)+
(-v)(w) +
R (y =30)
(+v)(w) +
R (y =30)
FIGURE 6.12: Conditional cross-correlation of positive wall-normal fluctuations (extracted from flow regions where Ø ) with the span-wise velocity componentÐ êË 3ìÌíêí (left) and correlation of neg-ative wall-normal fluctuations (extracted from flow regions where"! Ø ) with span-wise components
Ð êË ë3ìÌíêí (right) measured atÐÅ# ûØØ andÓ ® ÖØÙÛÚØÙ Ü Ø (top to bottom).
6 Investigation of the xz-plane
processes associated with the organised motion in span-wise direction in more detail. Espe-cially the value of maximum correlation should be noted for different wall distances and with respect to à Ñ$ and ÃË . In order to interprete the various correlation patterns qualitatively, it is necessary to assume a pure flow motion indicated by the first subscript (motion towards the wall in case of à êË ë3ìÌíêí) at the origin of the correlation plane. Using this method it be-comes evident that a motion towards the wall is on average associated with a span-wise motion away from the structure while the opposite holds for an organised motion away from the wall (compare the sign of the structures). The location of the maxima, on the other hand, implies that a phase relation exists between the two motions. When the flow direction is taken into account, it becomes apparent that the span-wise motion can be considered as the footprint of the wall-ward motion. Thus, the span-wise motion can be considered as a secondary motion.
It is clear that the processes described are associated with two stream-wise vortices whose length is determined by the flow region moving in -direction. For clarity the main features of the non-conditional functions have been summarised in figure 6.9 in form of one-dimensional graphs extracted at the location of the maximum of correlation in Î - and Ï -direction. This representation allows to compare quantitatively the height of the correlation and the location of the maximum in wall-units.