Sweeps - Spatio-temporal buffer layer statistics

6.3 Spatio-temporal buffer layer statistics

6.4.2 Sweeps

Q2 Q2

FIGURE6.22: Velocity fluctuations measured at^Ó ^® ^ÚØ (red: ^C_ ^Ø ; blue: ^CG! ^Ø ).

Ã Ñ Ë correlation in figure 6.8 and 6.9 and implies that no pairs of stream-wise counter-rotating vortices flank the low-speed regions over their total length, as proposed by some authors [54].

Otherwise, one would expect to detect a sign change in the out-of-plane motion on both sides of the streaks or at least a large^· variation over the length of the streaks.

6.4.2 Sweeps

The production of turbulence caused by the lifting of the low-speed streaks is one of the basic processes identified in near-wall turbulence as already mentioned. However, due to the

6.4 Properties of coherent velocity structures

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x [???]

y [???]

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0 50 100 150 200 250 300

x [???]

y [???]

FIGURE6.23: Velocity fluctuations measured at^Ó ^® ^óÜ ^Ø .

complexity of the turbulent motion and the limitations of the measurement techniques the cause and dynamic is still the subject of controversial discussions. In order to investigate this exchange processes in detail, the interaction of the streaks with the surrounding fluid will be investigated next. Especially the interaction regarding the high momentum flow structures which move towards the wall will be considered, as only these possess the energy required to alter the turbulence structure of the flow field effectively.

It is evident that the high-momentum flow structures must be visible in nearly each veloc-ity field in order to compensate the low-momentum movement of the streaks, but in contrast to the low-speed streaks the variety of these structures is much larger. Usually they appear

6 Investigation of the xz-plane

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x [???]

y [???]

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x [???]

y [???]

Q2 Q4

Q2 Q2

Q4 Q4

secondary flow

FIGURE6.24: Same as figure 6.23 but measured at^Ó ^® ^ÚØ (red: ^CG ^Ø ; blue: ^C_! ^Ø ).

less elongated and broader than the low-speed regions because these flow structures originate statistically from flow regions which are further away from the wall, see section 5.4. This can be estimated from the sign of the wall-normal velocity component which is represented by the contours in figure 6.22. At ^® ^Ê ^ªÀ these high momentum structures look sometimes similar to the streaks with regard to size and shape according to figure 6.21 but more frequently they resemble small elliptically shaped islands, approximately 200 wall units in length and roughly 50 to 100 wall-units in width according to figure 6.23. It can be seen from the out-of-plane ve-locity component, displayed in figure 6.24, that these structures, which are labelled as sweeps, transfer momentum towards the wall. Thus, these structures are statistically represented by

6.4 Properties of coherent velocity structures

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Sweep Streak

secondary motion

secondary

Q2 Q2

Q4 Q4 Q2 Q4

Q2 Q2

Q4 Q4

FIGURE 6.25: Velocity fluctuations measured at ^Ó ^® ^ÚØ (top) and Reynolds stress component ^&C (bottom).

the correlations shown in figure 6.8. If such a structure moves towards a low-speed streak, as visible in figure 6.24 for example, an interaction takes place and parts of the streaks, which are directly affected by the sweeps, are forced to move away from the wall due to continuity (see blue region in ellipse). On average, this process is represented by the spatial correlation functions shown in figure 6.10 to 6.12. However, it can be seen from the samples shown in fig-ure 6.24 that the effect on the streaks is quite small in the present case. This can be explained by the fact that the momentum is insufficient to create a strong lifting of the streak. Figure 6.25 shows the same interaction but the momentum transferred by the sweeps is much larger. In

6 Investigation of the xz-plane

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x [???]

y [???]

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x [???]

y [???]

x z

+ +

Q4 Q4

FIGURE6.26: Velocity fluctuations measured simultaneously at^Ó ^® ^ÚØ (top) and^Ó ^® ^óÜ ^Ø .

effect, the streaks move away from the wall, indicated by the blue contours, and it is visi-ble from the lower image that this is associated with the production of Reynolds shear-stress (dashed lines indicate ^§ ^· ^ºñÀ ). Though, it should be noted that the production of Reynolds shear stress induced by the sweeps is quite large as can be estimated from regions denoted by Q4 in the lower contour plot. Based on this experimental result it can be concluded that, in general, the lifting of low-speed streaks can be considered as a secondary motion which is induced by an interaction between a low-speed streak with a sweep, and the size and strength of the region, which moves away from the wall, is related to the momentum of the sweep. In addition it can be stated that the lifting of low-speed fluid into higher momentum flow regions

6.4 Properties of coherent velocity structures is accompanied by two weak stream-wise vortices because any local motion away from the wall is associated with a stream-wise vortex pair, but these vortices are produced locally and can not be considered as primary vortex structures. The similar stream-wise extent of sweeps and regions of significant outward motion supports this conjecture. The average length of these vortices can be estimated from figure 6.10 to 6.12.

Another effect associated with the movement of the sweeps is the creation of counter ro-tating vortex pairs, as indicated by the red and blue circles in figure 6.25. First of all, it should be noted that these structures are different from the vortex models presented in the figure 1.3 as these vortical structures pump high-speed fluid towards the wall and not low-speed away as implied by the vortex models on page 6. Besides, it seems likely that these vortices are generated when the sweeps interact directly with the low-speed regions. Another remarkable effect which can be observed quite frequently is the generation of hairpin like structures when a sweep interacts directly with the back of a streak, see red circles in figure 6.26 for example.

However, it can be seen from the lower image of the same figure that these vortices do not extend into the near-wall region down to^.® ^¿ ^ªÀ in the present case. It can be speculated that this might be related to the strength of the sweep or that the sweep streak interaction did not last long enough at the time the image was acquired to form a clear vortex pair.

6.4.3 Ejection

Figure 6.27 does not reveal any extended streak pattern but the intensity of the Reynolds stress component is very large as can be estimated from the lower image. However, by visual inspec-tion of the velocity field it can be seen that the regions of strong producinspec-tion are associated with counter rotating vortex pairs, denoted by the circles, which indicate the presence of a hairpin like structures, [44]. Moreover it seems that these structures are convecting as a package as described in [112]. It should be kept in mind that according to figure 6.20 the occurrence of such arrangements of hairpin like structures is very low in the near-wall region as only a minority of the fluctuations exhibits an intensity which is typically associated with such struc-tures. Figure 6.28 shows another pair of velocity fields measured simultaneously at^® ^¿` ^À (top) and 10. Clearly visible are the different structures discussed above and their interaction.

In this representation the vortical flow structures are inclined in stream-wise direction as the structures in the lower representation appear slightly upstream with respect to the results in the upper image. Since the position and intensity may be affected by the particular convection velocity subtracted, the wall-normal vorticity component^a ^- is shown in figure 6.29 for both cases, calculated from the instantaneous velocity fields. Whereas the shear layers beneath the streaks appear clearly in this representation, the vortices are less pronounced. However, the relatively sudden change in the vorticity component at the beginning of a streak might indicate the presence of hairpin like structures.

In the last decades of the previous century, detailed hot wire investigations were performed in order to obtain quantitative information about the bursting phenomena. By using different pattern recognition techniques, mentioned on page 90, a quite regular normalised velocity signal could be extracted and it was assumed that this pattern is the signature of a coher-ent structure which is associated with the bursting phenomena [102]. In the near-wall region below ^.® ^¿ ^«À the pattern could be observed in nearly 65 % of the total samples, and by analysing the individual velocity signal it was found that the length of the pattern varied over quite a wide range (1:25). To identify the structures which are responsible for the

characteris-6 Investigation of the xz-plane

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x [???]

y [???]

z z

+ +

Vortex pair Vortex pair

Vortex pair

Q4 Q2

FIGURE 6.27: Velocity fluctuations measured at ^Ó ^® ^ÚØ (top) and Reynolds stress component ^&C (bottom).

tic velocity pattern, the velocity structure of the PIV measurements is analysed in stream-wise direction for various span-wise locations. For the comparison it is important to keep in mind that in the PIV experiment the spatial variation of the velocity is considered at a fixed time while the time signal of the velocity is considered at a fixed point in the hot wire investiga-tion. The left image of figure 6.30 shows three typical instantaneous signals of the^§ -velocity component which nicely matches with the ensemble averaged pattern deduced from the hot wire measurements. The signals were extracted from figure 6.28 along the red lines located at ^Ï ^® ^¿ ^ÈÀ , 120 and 220. It can be clearly seen that the characteristic velocity pattern is

6.4 Properties of coherent velocity structures

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Q2 Q4

Q4 Q4

Vortex pair Vortex pair

Vortex pair

FIGURE6.28: Velocity fluctuations measured simultaneously at^Ó ^® ^ÖØ (top) and^Ó ^® ^ÚØ .

directly associated with the low-speed streaks. As these flow structures appear frequently in the near-wall region according to figure 6.21 to 6.30, the high detection rate in the hot wire results is not surprising and also the strong variation of the pattern length can be explained.

The maximum length is given by the extension of the streaks, which can be longer than 1000 wall-units, and the lower limit of the length appears when only the cross-section of a narrow streak convects along the probe. Since the streaks are only slightly twisted, the projection of the width of the cross-section in stream-wise direction is the relevant parameter for the lower limit of the pattern length. As the minimum width is approximately 30 wall units, according to section 6.2.2, and the maximum angle between the stream-wise coordinate and the streak

6 Investigation of the xz-plane

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y [???]

FIGURE6.29: Vorticity fluctuation calculated from figure 6.28.

is approximately^© ^Çcb , according to the previous figures, the projection of the cross-section in stream-wise direction is roughly 42 wall-units. Thus, the variation of the length can be esti-mated to ^ªÀÀÀ ^¸ ^© ^Ê ^© which nicely corresponds with the result deduced from the hot wire investigation. However, in contrast to the hot wire results, the maximum of the graphs shown in the left image of figure 6.30 always appears on the left hand side from the minimum while in the results presented in [102] the maximum appears on the right hand side. This can be explained by the different representations. While in [102] the pattern is displayed against the time and here against the spatial coordinate, a small time in [102] correspond to a large down-stream position. So when the time axis of the hot wire results is reversed for comparison, the

6.4 Properties of coherent velocity structures

d 100^d 200^d 300^d 400^d 500^d 600^d

x⁺

−6

−4

−2 0 2 4

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−6

−4

−2 0 2 4

e +

=70 z

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=120 z

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=220

FIGURE 6.30: Profiles of ^® and^C ^® velocity components along red lines in figure 6.28.

functional dependence of the graphs agrees nicely. From the analysis of the velocity fields in this section, it has already been seen that the production of turbulence is quite often associated with the streaks. Nevertheless, it is obvious that the velocity pattern, shown in figure 6.30, can be only considered as a necessary condition for the production of turbulence by means of low-speed streaks. This becomes evident when the variation of the wall-normal velocity component, shown in the right image of figure 6.30, is compared with the left image. From the sign of the fluctuations it can be estimated that a significant amount of turbulence is only produced in case of the graphs with the circular and squared symbols as only in this case the wall-normal velocity component is positive while the stream-wise velocity is negative. At the end of this chapter it should be mentioned that the interpretation of the results presented in figure 6.30 requires that the flow structures keep their spatial organisation while travelling down-stream. The validity of this assumption, which is called frozen pattern hypothesis, will be proven in the next chapter.

6 Investigation of the xz-plane

7 Investigation of the yz-plane

The interpretation of the results presented in figure 6.30 on page 133 was based on the as-sumption that the spatial variation of the instantaneous turbulent velocity signal^§ ^¹^Î ^Á ^¼ can be transformed into a time dependent velocity signal^§ ^¹| ^Á:(¼ at a fixed point according to

'§

'Î

¿ ª

'§

')(

(7.1) It is obvious that this so called Taylor hypothesis is certainly valid when the local convec-tion velocity ^¦ is large relative to the turbulent fluctuations ^§ . However, the statistical results shown in figure 5.3 on page 77 indicate that the maximum of the non-dimensional stream-wise velocity fluctuation is around 3 at ^® ^Ê ^ª ^Ç and the average velocity is roughly 10 at the same wall location according to figure 5.2. Although the ratio between both values is only 0.3 at the measurement location where figure 6.30 was recorded, further evidence is re-quired to justify the assumption made because it is not evident that^¦ is the correct convection velocity of the flow structures. For this reason the dependency of the spatio-temporal corre-lations of the velocity fluctuations will be investigated in this chapter. In addition, various spatial correlation and cross-correlation functions of the velocity fluctuations will be investi-gated to validate the concepts proposed in chapter 6 to explain the turbulent mixing by means of coherent structures. Furthermore the dimensions of the shear-layers in the ^Ï -plane will be considered and the characteristic features of the stream-wise vortices will be examined as well as their significance for the turbulent mixing. This is of great interest as pointed out in the introduction, because it is generally assumed that these vortices play a dominant role for the momentum exchange in wall bounded flows, as indicated in figure 1.3. This work can be seen as a completion of the conventional PIV investigation described in [13] and the flow visualisation described in [28]. Here only the main results will be analysed in detail. The ad-ditional information presented in the following may serve for comparison with the predictions of fundamental turbulence models and for the validation of numerical flow simulations in the future.

7.1 Experimental set-up

The multiplane stereo PIV technique in the configuration utilised for this investigation consists of four pulsed Nd:YAG lasers (BMI) each with an output energy of 255 mJ per pulse at ^¿ 532 nm, two optical benches for the generation of independent light-sheets (one for each state of polarisation), and four Peltier cooled high resolution cameras (PCO) with 1280 by 1024 pixel resolution and 12 Bit dynamic range. The schematic arrangement of the equipment with respect to the test-section is outlined in figure 7.1. This arrangement follows directly from the experimental setup, described in section 6.1, after rotating the orientation of the

7 Investigation of the yz-plane

light-sheet by turning the cylindrical lens in the light-sheet optic. The observation angle for each camera in angular imaging configuration with Scheimpflug correction is shown in the following table along with the exact positions of the master cameras (1 and 2 in figure 7.1) with respect to the centre of the field of view (all asymmetries have been taken into account for the calculation of the three velocity components). Four 180 mm lenses (Carl Zeiss)

camera ^Î [mm] ^Ï [mm] ^f@g [mm] ^¯ [deg]

1 -1570 1464 2143 42.9

2 -1560 -1463 2140 43.2

TABLE7.1: Position and observation distance of the master cameras with respect to the centre of each field of view (meeting point of optical axis) and corresponding observation angles.

z x

test-section 3

1 2

4 5

7 8

a b

side window

~86°

main flow-direction

40 mm separation

FIGURE 7.1: Schematic set-up of the recording system and light-sheet position for both experiments (different scales). 1-4 digital cameras, 5 lens, 6 mirror, 7 polarising beam-splitter cube, 8 absorbing material, a measurement location for 1st investigation (40 mm separation between both measurement planes in stream-wise direction), b measurement location for 2nd investigation (all measurement posi-tions at the same location). Different light ray colours indicate different states of polarisation.

were used for the measurements with an aperture of 8 and a magnification of 1/6 along the principal axis of the lens. Due to the strong out-of-plane velocity component, as a result of the light-sheet orientation relative to the main flow direction, the light-sheet pairs with equal polarisation have been shifted in stream-wise direction as indicated in figure 7.2. This improves the signal to noise ratio, as the loss-of-correlation due to unpaired particle images is minimised without reducing the dynamic range and the spatial resolution. Decreasing ^Í_] or increasing the magnification of the imaging system would reduce the dynamic range or spatial resolution.

7.1 Experimental set-up For the evaluation of the stereo-scopic images the second order warping technique was applied again, along with the calibration validation procedure described in section 3.3. This ensures that the interrogation spots from each of a pair of stereo-scopic images correspond to the same region of the flow. The interrogation of the data was performed with the FFT-based free shape cross-correlation outlined in section 2.4, and for the determination of the signal-peak with sub-pixel accuracy, the two-dimensional Gaussian fit using the Levenberg-Marquardt method has been applied. This peak finding method is less sensitive to sub-pixel displacements compared with the three point Gaussian peak fit, see figure 2.14 and table 2.1 for details. For the calculation of the velocity vectors^h ^ji ^h pixel interrogation windows were used for both Reynolds number investigations. The bandwidth of particle images displace-ments varies between zero and 8 pixel (zero and -8 for the left camera system in figure 7.1) for the light pulse delay listed in table 7.2, and the number of spurious vectors was on average be-low^À ^lknm by applying the following set of band-pass and gradient filter ( ^hpo¥ÍeÏpo ^k ^À pixel;

oüÍrq8o

Ç pixel and^Í5Î ^Í5ÎEs ^> ^out pixel). The basic details about the recording and evaluation are summarised in table 7.2.

ÃÄv ÈÉÀÀ

k Ç

ÀÀÀ [1]

ÃÄ ÈwtÀÀÀ 160000 [1]

ÃÄyx h

:Þ i k

Àcz É t i k

Àwz [1]

{ ÿ 3 7 [ m/s]

| 0.121 0.263 [ m/s]

0.37 0.34 [m]

% s

3000 5980 [1]

field of view ^Þ ^h ^ijÆ ^É ^Þ ^h ^iNÆ ^É [ mm^á ] field of view ^À ^}k ^È ⁱ ^À ^Þ ^À ^lk ^Æji ^À ^$Æ [

% á ]

field of view ^Ç ^kk ⁱ ^È ^Æ ^Þ ^kk ^ÀÈ ⁱ ^k ^È [^Írq ^s ⁱ ^ÍlÏ ^s ] spatial resolution ^À ^:Þ ^i~ ^}k ^h ^i~ ^}k ^h ^lk ^h ⁱ ^lk ^h [ mm^B ]

spatial resolution ^Ç ^À ⁱ ^k ^È ^h ⁱ ^k ^È ^h ^hÈ ^t ⁱ ^hÈ ^t [^Í5Î ^s ^Írq ^s ^ÍeÏ ^s ]

pulse separation 200 100 s

dynamic range ^À ^h to ^k ^À ^É ^À ^:Þ to ^kk^É [ pixel]

vectors per sample 13113 13113

number of samples 2975 2100

TABLE7.2: Relevant parameters for the characterisation of the experiment performed 18 m behind the leading edge of the flat plate in the^Óc -plane of the turbulent boundary layer flow.

Two similar experiments have been performed independently. In the first experiment the spatial location of all light-sheet planes was identical and the time separation between a pair of velocity fields being acquired was varied (see left plot of figure 7.2). This allows to study the changes of the velocity structures with time. In the second experiment the time-separation between a pair of measured velocity fields was altered as before but, in addition, the mea-surement planes were spatially shifted in stream-wise direction by ^tÀ mm (^Í5Î ^s ^Ê ^hÀÀ ) as indicated in figure 7.1 and in the right plot of figure 7.2. Thus it was possible to select a flow

Im Dokument The significance of coherent flow structures for the turbulent mixing in wall-bounded flows (Seite 124-138)