Tomographic-PIV measurement of the flow around a zigzag boundary layer trip

The average flow is obtained by averaging the 1,000 snapshots of the velocity field, and the result for U _e  = 0.21 m/s is displayed in the Figs. 2, 3, 4. The downstream development of the flow in terms of its velocity components is firstly shown in a wall-parallel plane at y/δ ₀  = 0.28 (Fig. 2). All components reveal streak-like structures directly behind the trip, extending to about x/δ ₀  = 5. At the present distance from the wall, they are characterized by backflow (as evidenced by a negative u-component of the velocity) and a flow away from the wall (as evidenced by a positive v-component of the velocity) inside the streaks that start from the downstream pointing tips of the zigzag trip. Associated with the upstream pointing tips is a region of positive, but small, u with v < 0. The spanwise velocity component, w, is negative everywhere, but varies in magnitude, being largest inside the streaks starting from the downstream pointing tips. Following the streaks is a region where the average flow regains spanwise uniformity, accelerates spatially (i.e., ∂u/∂x > 0), and is directed toward the wall (i.e., v < 0). This region covers approximately x/δ ₀  = 5–15. Then, starting from x/δ ₀  = 15, the average flow is parallel to the wall (i.e., v ≈ 0) with little variation in both the spanwise and the streamwise directions, as expected for regular (turbulent) boundary flow. As mentioned, the trip is not perfectly aligned normal to the free-stream flow, but rather is slightly tilted by about 5 degrees. As a result, some cross-flow (w < 0) is found, even far downstream of the trip. Moreover, the streaks are also tilted with respect to the x-axis, which has been defined based on the orientation of the trip and not the free-stream velocity (which differ by 5 degrees).

A three-dimensional rendering of the average velocity field near the trip is presented in Fig. 3a. The blue iso-surface marks a region of backflow (u < 0) immediately following the trip, which varies both in height and in length in the spanwise direction following the periodicity of the zigzag. This variation is visible in Fig. 2 as the streaks. Roughly above each streak is a streamwise vortex as detected by the Q-criterion (Hunt et al. 1988), shown in green in Fig. 2. These vortices are co-rotating and are associated with a swirling motion in the spanwise–wall-normal plane (Fig. 3b) that brings high-speed fluid toward the wall and transports low-speed (and backflow) fluid away from the wall. The lack of symmetry, in the sense that there are no counter-rotating vortex pairs, is again attributed to the 5-degree tilt of the trip causing a unidirectional cross-flow in the boundary layer behind it, which is known to contain co-rotating streamwise vortices (e.g., White and Saric 2005). As mentioned above, further downstream of the separated flow region, the average flow is uniform in the spanwise direction and therefore no longer contains three-dimensional features.

The average velocity vectors in the streamwise–wall-normal plane (Fig. 4) show a shear layer, probably emanating from the trailing edge of the trip, separating the reversed flow near the wall from the high-speed outer flow. In the velocity profiles, this introduces an inflection point, which is well known to be unstable and a source for creating turbulence. Due to the limited spatial resolution in the present experiments, it is very likely that the shear layer is smeared out and appears thicker than it actually is. Yet, its presence can be detected. The shear layer subsequently increases in thickness downstream, while the average flow reattaches and a turbulent boundary layer profiles starts to develop. At other spanwise locations, the flow development appears very similar.

The Reynolds normal stresses 〈u′u′〉, 〈v′v′〉 and 〈w′w′〉 and the shear stress 〈u′v′〉 are presented in a streamwise–wall-normal plane through a downstream pointing tip (Fig. 5). The overall patterns and stress levels are similar at other spanwise positions. As expected, the shear layer separating from the zigzag trip is associated with high levels of 〈u′u′〉 and 〈u′v′〉. The peak normal stress 〈u′u′〉 progressively moves closer to the wall, while the peak shear stress 〈u′v′〉 remains at approximately the same height. The wall-normal stress 〈v′v′〉 increases strongly after the average flow reattaches (x/δ ₀  = 5) with the peak values first at about the height of the trip and then slowly moving away from the wall. Similarly, the spanwise stress 〈w′w′〉 increases rapidly near the wall after average reattachment with the peak then moving away from the wall, but at a lower rate compared to 〈v′v′〉. From these results, it is clear that the redistribution of turbulent kinetic energy of the different components takes place in the region near the point of average reattachment, where the turbulence is strained by the flow moving toward the wall.

The spanwise stress 〈w′w′〉 also shows high values above the shear layer directly behind the trip (〈w′w′〉/U _e ²  ≈ 5·10⁻³). This is associated with the unsteadiness in the separated flow region behind the trip, which displays large-scale streamwise and spanwise modes in a POD analysis, as will be demonstrated in Sect. 3.5 (the streamwise and spanwise flow is coupled to some extend due to the present cross-flow).

The present Reynolds stress levels after average reattachment are clearly much higher than those in a canonical developed turbulent boundary layer, which are approximately 2.5·10⁻³, 1.1·10⁻³, 1.6·10⁻³, and −0.8·10⁻³ for 〈u′u′〉/U _e ² , 〈v′v′〉/U _e ² , 〈w′w′〉/U _e ² , and 〈u′v′〉/U _e ² , respectively, at y/δ = 0.5 (Klebanoff 1955). For example, 〈u′u′〉 and 〈u′v′〉 are amplified by a factor 7–10 at a distance of three boundary layer thicknesses downstream of average reattachment (x/δ ₀  ≈ 8 in Fig. 5), which is in agreement with the enhancement of the Reynolds stresses downstream of a laminar separation, as reported by Castro and Epik (1998). Moreover, the order of magnitudes of the Reynolds stresses is consistent with results obtained just behind a turbulent separation (Alving and Fernholz 1996).

The fact that the overall level of each Reynolds stress is higher than expected for a canonical turbulent boundary layer, and that these levels are still decreasing at the downstream edge of the measurement domain (x/δ ₀  = 28), indicates that the boundary layer remains affected by the trip at least up to there. How this is reflected in the instantaneous turbulent structures will be shown below.

Two typical results for the case with a free-stream velocity of U _e  = 0.21 m/s are shown in Fig. 6, displaying regions (in green) of vortical motion as identified by means of the Q-criterion (Hunt et al. 1988) and negative streamwise velocity (in blue). Immediately behind the trip, the flow is separated and a shear layer forms from the top of the trip similar to what has been observed for the average flow (Fig. 7). The instantaneous results (Figs. 6, 7, 8, 9) reveal also a very important difference with respect to the mean. Within the shear layer, vorticity rolls up into spanwise swirling motions rather than the streamwise vortices in the average velocity field. Due to the zigzag shape of the trip, these vortices undulate in the spanwise direction and have a small streamwise component to it (compare the contour levels of swirling strength computed from the in-plane velocities in Figs. 7 and 8). The latter component builds up coherently when averaging due to streamwise vortices traveling in the streamwise direction. On the other hand, the wall-normal velocity fluctuations, associated with the spanwise vortices in a shear layer, tend to cancel each other when they convect downstream. The positive wall-normal fluctuation on one side of the vortex core cancels the negative fluctuation on the other side when it has moved by the vortex diameter. Averaging the velocity fields actually obscures the true nature of the instantaneous vortices and the turbulent production processes that can be associated with them. Therefore, having instantaneous 3-D measurements of the flow structure proves vital here, and we may speculate that this applies to other transition types as well.

At a downstream location of around x/δ ₀  = 5 (see Fig. 6), the spanwise vortices have broken up into individual arches that remain aligned, to some degree, in the spanwise direction (see also the detail in Fig. 9a). Because the necks of these structures are inclined at about 45 degrees with the wall, this process is associated with increasing wall-normal velocity fluctuations, visible as increasing levels of wall-normal Reynolds stress 〈v′v′〉 and Reynolds shear stress 〈u′v′〉 in the same region of the flow (Fig. 5).

Then, from the arches, hairpin-like (or cane-like) structures are seen to develop near x/δ ₀  = 10 (Fig. 9b), meaning that the arches now show legs, which are quasi-streamwise vortices connected to the necks of the arches (Robinson 1991). They are related to the stretching of streamwise vorticity due to the accelerating flow in that region (∂u/∂x > 0, see Sect. 3.1). Streamwise oriented vortices correspond to spanwise and wall-normal velocity fluctuations, and as a result, both the Reynolds normal stresses 〈v′v′〉 and 〈w′w′〉 grow (Fig. 5). Also visible here is the streamwise alignment of a number hairpins forming a so-called packet (Adrian et al. 2000). One example of which is indicated in Fig. 9b. It is noted that other type (undefined) vortex structures are observed as well (Fig. 9b).

The range of flow scales is limited at this stage (i.e., the vortices are all of similar size), but this changes downstream toward the end of the present measurement domain (x/δ ₀  > 18, Fig. 9c). There, structures covering the complete height of the measurement volume (likely extending beyond it) are seen, as well as new small-scale near-wall features. It is speculated that these small scales are linked to the well-known near-wall low-speed streaks in fully developed turbulent boundary layers (Kline et al. 1967), which form in this region and which explain the increasing value of 〈u′u′〉 very close to the wall that can be seen in Fig. 5 for x/δ ₀  > 22. The characteristics of small-scale near-wall turbulence developing well downstream of reattachment is consistent with observations made in a transitional separation bubble (Alam and Sandham 2000) and the slow redevelopment of the inner layer after a fully turbulent separation bubble (Alving and Fernholz 1996).

Some of the features in Fig. 9 are reminiscent of previously proposed models describing the generation of vortical structures in wall-bounded turbulence. In particular, the example of a larger hairpin leg with associated small-scale vortices (Fig. 9c) may be associated with the generation of vortices via surface interaction as proposed by Smith et al. (1991). In brief, the leg induces a low-speed region near the wall with an adverse pressure gradient, which in turn becomes unstable and forms a sheet that rolls up into a new near-wall hairpin adjacent to the existing leg. Furthermore, the packet of similar size hairpins in Fig. 9b is suggestive of the auto-generation model (Zhou et al. 1999), where the primary hairpin of sufficient strength generates a secondary upstream, which grows to a similar size as the primary thereby forming a coherent vortex packet. The visualizations give some support to the idea that these mechanisms may be at work here, although temporal information would be needed to confirm this.

From the visualization of vortical structures, it may appear that the original spanwise alignment of arches (Fig. 9a) gradually disappears further downstream (Fig. 9c). While the individual structures may be characteristic of a developed turbulent boundary layer (that is, hairpins and packets), there remains an important spanwise coherence in the flow, which must be ascribed to the tripping. It was already noted that the Reynolds stresses (Sect. 3.2) indicate that the flow is still affected by the transition at x/δ ₀  = 28.

The spanwise coherence of the flow can be demonstrated by considering the spanwise spatial autocorrelation of the fluctuating u-component of the velocity taken at a height y/δ ₀  = 0.39. Here, the fluctuations are taken with respect to the average over the ensemble of snapshots and over the spanwise direction. The additional averaging over the spanwise direction only affects the results up to x/δ ₀  = 7, after which the ensemble average is uniform in that direction (Fig. 2a). The resulting profiles for the autocorrelation coefficient at various x locations are presented in Fig. 10. The first profile (x/δ ₀  = 2) clearly shows the dominating spanwise oscillations corresponding to the streaks directly behind the trip. The height of the correlation peaks drops slowly, suggesting a large spanwise coherence between the streaks as also observed in the instantaneous snapshots (Figs. 6 and 9a). The spanwise oscillation is then seen to reduce in amplitude with increasing x, while at the same time a positive plateau in the autocorrelation coefficient gradually forms for Δz/δ ₀  > 2. The plateau and the fact that the autocorrelation coefficient does not drop below zero is evidence for a large spanwise coherence or spanwise alignment of structures in the flow, which exists in the flow at least up to x/δ ₀  = 27 where the plateau autocorrelation coefficient is still relatively high at 0.15. It is regarded as the reminiscent of the spanwise vortices in the shear layer coming off the trip at the early stages of transition (Sect. 3.3) that is also responsible for the relatively high values for the Reynolds stresses throughout the current measurement domain (Sect. 3.2).

Before reaching a plateau, the autocorrelation coefficient has a minimum value with a negative correlation coefficient near Δz/δ ₀  = 1, which is associated with the typical size of the spanwise variations in the streamwise velocity, similar to the high- and low-momentum zones in the outer layer of developed turbulent boundary layers (e.g., Hutchins and Marusic 2007, Elsinga et al. 2010). However, in comparison, the autocorrelation profile expected for a developed turbulent boundary layer (the red dashed-dot line in Fig. 10) reaches a distinctly different plateau at zero. The developed turbulent boundary layer profile used here has been obtained 2 m (~500 δ ₀ ) downstream of the trip in the same flow facility, where velocity data were acquired by the same 4-camera tomographic-PIV system. The measured autocorrelation profile agrees well with those available in the literature for other developed turbulent boundary layers (e.g., Hutchins and Marusic 2007), from which it can be concluded that the transition structures no longer affect the boundary layer at a distance of 500 δ ₀ downstream of the trip. Moreover, at that location the profiles of the Reynolds stresses have returned to the shape and amplitudes expected for a developed turbulent boundary layer (Schröder et al. 2008). It is important to realize that the transitional structures are no longer affecting the boundary layer characteristics at the aforementioned location far downstream of the trip, as this was used in some earlier experimental studies of time-resolved 3-D velocity data (Schröder et al. 2008, Elsinga and Marusic 2010). The transitional structures are likely to disappear at shorter distances from the trip, but at present no measurements have been performed at intermediate locations, so this point remains open for now.

Further investigation into the large-scale modes in the flow is performed using proper orthogonal decomposition (POD, Berkooz et al. 1993). This technique decomposes the ensemble of fluctuating velocity fields into a set of uncorrelated basis functions, or modes, which do need to be specified beforehand and are ordered according to their turbulent kinetic energy content. It can be shown that such decomposition is the most efficient; meaning that for a given number of modes POD captures more of the turbulent signal’s energy than any other set of basis functions in fixed coordinates. That is, a limited number of modes are needed to describe the flow up to a certain energy content. The particular scheme employed here is the method of snapshots (Sirovich 1987; Humble et al. 2007).

The resulting first two POD modes appear relatively strong containing 13 and 9 percent of the total turbulent kinetic energy (Fig. 11) compared to only one percent for mode 5 and higher. The fluctuating velocity field of the first POD mode (Fig. 12) reveals elongated regions of spanwise alternating positive and negative u’ close to the trip, which are associated with negative and positive wall-normal velocity fluctuations, respectively. These elongated regions are, in the spanwise direction, located adjacent to the central axis of each streak observed in the average velocity field (Fig. 2). Furthermore, the u′ peak value is attained not directly behind the trip, but rather somewhat downstream at Δx/δ ₀  = 4. Hence, adding or subtracting this mode from the average flow field has the effect of redirecting/tilting the streaks behind the trip. Because this tilting is approximately the same for all streaks, it can be considered a large-scale unsteadiness of the flow.

The second POD mode represents a nearly spanwise uniform transport of high momentum fluid toward the wall. The fluctuating velocity of this mode (Fig. 13) shows fluid moving toward the wall (v′ < 0) around Δx/δ ₀  = 6, which coincides with the location of average flow reattachment (Sect. 3.1). The result is an increase in the streamwise velocity downstream at Δx/δ ₀  = 10 (u′ > 0) and negative u′ in the streaks near the wall upstream at Δx/δ ₀  = 4. The latter causes an increasing backflow velocity in these streaks with respect to the average flow. Furthermore, the positive u′ fluctuating velocity decreases toward the downstream edge of the measurement domain, but remains significant. The second POD mode, like the first, represents an energetic and very large-scale coherent fluctuation, which is associated with the dynamics of the flow features introduced by the transition trip, because of the location of the fluctuating velocity peaks. In particular, the second mode, when combined with the average flow, changes the size of the separation bubble and the point of reattachment of the separated shear layer. Furthermore, the downstream extend of significant u’ in the second mode shows that these features of the trip transition affect the flow even beyond the current measurement domain.

Although less energetic, the higher modes contain also large scales. For example, the third POD mode contains nearly spanwise uniform fluctuations near the point of average flow reattachment, similar to the second mode, but not extending as far downstream. Then, the fourth and fifth modes are again linked to the streaks near the trip, as in the first mode.

The results presented here are consistent with those of Erm and Joubert (1991). They also found that downstream of the trip the large flow scales are affected the most, although they attributed the differences to low Reynolds number effects. In particular, the velocity power spectra taken at their measurement location closest to the trip, yet far downstream compared to where the present data were taken, showed no collapse and indicated increased energy content at low wave numbers, i.e., large scales. This effect was strongest for the lowest velocity considered, corresponding to the lowest Reynolds number based on trip height. Farther downstream, the power spectra again gave a reasonable collapse, suggesting the flow features introduced by the trip had diminished or disappeared. This occurred after the boundary layer reached a Reynolds number based on momentum thickness of 2,175, which, for the present conditions (U _e  = 0.21 m/s), is estimated to be 4.6 m (~1,000δ ₀ ) downstream of the trip. Similarly, the dominance of energetic and very slowly decaying large-scale flow structures was also observed from the spectra taken downstream of the turbulent reattachment of a laminar separation bubble (Castro and Epik 1998).