7.2 Statistical properties of the log-law region
7.2.2 Spatial cross-correlations with
The left column in figure 7.8 shows the ªSÔ· cross-correlation function measured at ¹»5¬
 Ö
°
Ö
¿
c (top to bottom) with theã component fixed while theä signal was shifted in the¹ andà directions. The right column reveals the opposite case where theä component was fixed while ã was shifted. The negative sign of the cross-correlations, represented by the dashed lines, indicates again that the transport of relatively low-momentum fluid outward into higher
7 Investigation of the yz-plane
∆z+ ∆z+
∆+ y + y+ ∆+ y + y+ ∆+ y + y+
vu + R (y =30)
vu + R (y =50)
uv + R (y =30)
uv + R (y =50)
uv + R (y =100)
vu + R (y =100)
FIGURE7.8: Ô·<Ä© » Æ © » È+É © » ÆËÉÌ » Í and ·ÔcÄ© » Æ © » È+É © » ÆËÉÌ » Í correlation measured at© »
Î ÆËÏ ÆÐ .
speed regions (ä Û andã ¡ ) and the movement of high-momentum fluid toward the wall into lower speed regions (ä ¡ and ã Û ) are the predominant processes in the near-wall region. In addition it can be seen from the different size of the correlation functions that in the left case the region, where a correlated motion between both velocity components can be observed, is significantly larger with respect to the case shown in the right column. This implies that the wall-normal motion is strongly affected by the stream-wise fluctuations while the effect of the wall-normal fluctuations on the stream-wise motion is quite limited. This is evident from the equation of motion which states that the energy from the mean motion is transferred into the stream-wise fluctuation at first before a transfer into theã andå component takes place. In addition, it becomes visible from the location of the maximum in figure 7.8 that in the case where theã component is fixed, the maximum of correlation appears at larger wall
7.2 Statistical properties of the log-law region
∆z+ ∆z+
∆+ y + y+ ∆+ y + y+ ∆+ y + y+
vw +
R (y =30) wv +
wv +
vw + R (y =50)
wv +
vw +
R (y =100) R (y =100)
R (y =50) R (y =30)
FIGURE7.9: ÔËÙÇÄ© » Æ © » È_É © » ÆËÉÌ » Í and ÙÔwÄ© » Æ © » È_É © » ÆËÉÌ » Í correlation measured at© »
Î ÆËÏ ÆÐ .
distance with respect to the location of the fixed point, while in the case where theä component was fixed the maximum occurs closer to the wall. The same behaviour can be observed when the cross-correlation between the wall-normal and span-wise component of the velocity is considered. The left column in figure 7.9 displays the ª^ÔËÙ correlation function measured at
¹&»\¬æ Ö
°
&Ö
¿
(top to bottom). Here the ã component was fixed while å was shifted in the two homogeneous directions and the right column reveals the opposite case where the å component was fixed whileã was shifted. It can be seen that the structures presented in the left column are larger for all wall locations. In addition, a significant change in the organisation of the correlation can be observed when the location of the fixed point is altered. In case of the correlation presented in the right column the structural features remain constant with increasing wall distance and only the size varies. The value of the maximum is comparable for
7 Investigation of the yz-plane
all cross-correlations. In order to examine the dynamical properties of the flow represented by the correlations in the left column, a positive span-wise velocity fluctuation which is located at the fixed point will be assumed first. It can be seen from the sign of the correlation that positiveå fluctuation induces a vertical motion towards the wall, which is located upstream of the horizontal velocity structure (negativeà values), and a motion away from the wall located downstream of the coherent structure. The argumentation can be reversed when a motion with å Û is considered. At larger wall distance it can be deduced from the secondary peaks near the wall that a horizontal motion in positive à direction (å ¡ ) induces also a quite local downstream motion, but the location of this coherent motion is relatively close to the location of the fixed point with respect to the position where an upward motion can be observed. In addition, it can be concluded from the increasing height of the correlation value that the coherence of this near-wall motion increases if the location of the fixed point is farther away from the wall. When the correlations shown in the right column of figure 7.9 are analysed in the same way, it can be seen that a vertical motion at the fixed point location with ã ¡ induces a horizontal motion towards ×pûܬ2 in the near-wall region and away from the centre at higher wall locations. A vertical motion towards the wall, on the other hand, induces a horizontal motion away from the centreline at×pà » ¬\ due to continuity. As the correlations do not allow to differentiate between primary and secondary motion it is also conceivable to interprete the results in a different way. For example, it can also be stated that a horizontal motion towards the centreline induces a vertical motion away from the wall at the location of the fixed point. This argumentation was usually applied for the interpretation of the results in chapter 6. The basis for this interpretation is the assumption that the kinetic energy of the vertical motion is not sufficient to alter the turbulence structure to a large extent, especially near the wall. This can be deduced from the rms-profiles shown in figure 7.3. Anyway, the three dimensional size, shape and interaction of the flow structures is now fully determined if the results presented in chapter 5 and 6 are taken into account.