2.1 Stereoscopic particle image velocimetry system set-up

Stereoscopic particle image velocimetry was the flow measurement technique used in this investigation. The S-PIV system available consists of a double-pulsed New Wave Research Solo PIV 120 Nd:YAG laser, two LaVision Imager Intense cameras ( $1376\times 1040~\text{px}^{2}$ , pixel size: $6.45~\unicode[STIX]{x03BC}\text{m}$ ) equipped with 50 mm lenses and the LaVision DaVis 7.1 software used for the post-processing. The particles can be delivered through both the injection port of the strut and the injection system mounted in the supersonic wind-tunnel reservoir to seed the entire flow field. In some of the surveys (i.e. cases I and II) it was chosen to deliver particles from both locations to better mark the plume issuing from the injection port and enable plume morphology studies. To do so, a very small amount of helium (fuel-to-free stream momentum flux ratio $J=1\times 10^{-3}$ ) was delivered through the injection slit of the strut to ensure the proper entrainment of the tracers yet limiting as much as possible the momentum added into the flow.

Figures 3(a) and 3(b) show the schematic of the PIV set-up and the seeding injection system adopted during the experiments.

Figure 3.

Schematics of (a) the PIV system set-up used in the surveys and (b) the flow-seeding strategy adopted in the supersonic wind tunnel.

$\text{TiO}_{2}$ nano-particles were chosen for their size and limited agglomeration (nominal diameter: 20 nm; cluster diameter: $0.25~\unicode[STIX]{x03BC}\text{m}$ ; density: $4130~\text{kg}~\text{m}^{-3}$ ). The assessment of the particle response downstream of a shock performed by the authors (Vergine & Maddalena Reference Vergine and Maddalena2014) confirmed that the $\text{TiO}_{2}$ particles can closely track the imposed vortex field.

The cameras were positioned so as to achieve approximately the same resolution in all the experiments (i.e. $0.06~\text{mm}~\text{px}^{-1}$ ) resulting in an interrogation area ( $32\times 32~\text{px}^{2}$ ) of approximately 1.86 mm and a grid spacing of 0.93 mm due to the $50\,\%$ overlap among neighbouring cells. Based on the size of the interrogation area, it is possible to calculate a cutoff wavenumber $k_{cut}=1.69~\text{rad}~\text{mm}^{-1}$ , which must be considered the upper bound of the resolved turbulence spectrum in these measurements.

As the turbulence time scale was estimated to be $4~\unicode[STIX]{x03BC}\text{s}$ in this class of flows, the pulse separation time $\unicode[STIX]{x0394}t$ chosen for the experiments was $1.2~\unicode[STIX]{x03BC}\text{s}$ , to permit to freeze the motion of the fastest-decaying turbulent eddies while still allowing for measurable in-plane displacements (Vergine & Maddalena Reference Vergine and Maddalena2014). The upper bound of the pulse separation time was constrained by the necessity of keeping the out-of-plane motion around $1/4$ of the laser sheet thickness which, therefore, was set to its maximum nominal value of 2 mm.

Table 1.

Test matrix.

4.1 Cases I and II: mean flow data

Figure 4 shows the velocity vector fields and the average streamwise vorticity contour plots normalized respect to the maximum value obtained in the S-PIV surveys conducted at station $10h$ , $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ . The central vortices shed from the internal edges of the ramps resulted already merged at station $10h$ , forming an elongated vorticity patch which evolved and rotated up to station $32h$ , assuming a bean-like shape. By visual inspection, the size of the merger is approximately double that of the external vortices. Note that the contour plots have been re-scaled to better highlight the geometrical features of the streamwise vorticity field. A comprehensive analysis of the plume and the mean flow fields of this case can be found in Maddalena et al. (Reference Maddalena, Vergine and Crisanti2014).

Figure 4.

Case I: evolution of the streamwise vorticity normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ . Stations (a)  $10h$ , (b) $13h$ , (c) $16h$ , (d) $20h$ and (e) $32h$ .

The average flow fields obtained for case II are shown in figure 5 for all the streamwise stations considered. As seen in this case, the central vortices did not fully merge throughout the entire survey and their vorticity peaks remained well separated up to station $32h$ .

Figure 5.

Case II: evolution of the streamwise vorticity normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ and (d) $32h$ .

4.2 Circulation of the vortices

The analysis from this point will be carried out by referring to the different vorticity patches with acronyms. Images in figure 6 presents the nomenclature adopted to identify the structures in the two cases inspected. Specifically: ELV and ERV indicate the external-left vortex and external-right vortex, respectively; ML and MR stand for median-left and median-right vortices; M indicates the merger.

Figure 6.

Nomenclature of the structures in the flow fields: (a) case I; (b) case II. The streamwise vorticity contour is shown normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ .

Note that the presentation of the results in this manuscript slightly differs from that provided by Vergine & Maddalena (Reference Vergine and Maddalena2015) due to: (i) updated datasets for the $13h$ station of case I and the $10h$ station of case II and (ii) the addition of the $13h$ station in case II.

The circulation of the structures was evaluated as the flux of streamwise vorticity which was above its maximum statistical uncertainty among all the surveyed stations, in this case $\pm 5000~\text{s}^{-1}$ (i.e. less than $10\,\%$ of the maximum measured value). In figure 7 the actual cells over which the streamwise vorticity was averaged are presented for each structure for case I, station $13h$ : a location where the vorticity patches lay in close proximity. In this way, only the vorticity that can be confidently measured contributes to the calculation of the circulation. Note that the area used to evaluate the circulation coincides with the area over which the $TKE$ is averaged in the calculation of $TKE_{AVG}$ .

Figure 7.

Case I, station $13h$ . Streamwise vorticity magnitude exceeding $5000~\text{s}^{-1}$ .

Figures 8(a) and 8(b) present the evolution of the circulation of each structure calculated as the flux of streamwise vorticity across the patch and normalized with respect to $\unicode[STIX]{x1D6E4}_{ref}$ for case I and case II, respectively. It is noteworthy that the circulation of the merger in case I is essentially double that of the external structures, while the circulation is quite similar for all the vortices in case II.

The measurement error on each velocity component $e_{v}$ is evaluated as the combination of the systematic interpolation error $e_{s}$ and the random error $e_{rand}$ affecting the measured in-plane velocity components: $e_{v}=\sqrt{e_{s}^{2}+e_{rand}^{2}}$ . The systematic error $e_{s}$ is here made to coincide with the intrinsic uncertainty given by the use of a three-point Gaussian interpolator for the evaluation of the fractional displacement within the interrogation area and typically of the order of one-tenth of a pixel. This error has been estimated to be $\pm 4.8~\text{m}~\text{s}^{-1}$ for the experimental conditions attained. Although the random error $e_{rand}$ is usually very small, its evaluation in each interrogation area was necessary due to its strong dependence on the seeding density which could not necessarily be the same in every realization. The analysis of the random error allowed us to consider a maximum of $\pm 1.5~\text{m}~\text{s}^{-1}$ , which was assumed constant to produce conservative estimates. The total error on the measurement of each velocity component is, therefore, $e_{v}=5~\text{m}~\text{s}^{-1}$ . The error on the circulation can be evaluated as follows (4.1):

Where the summation is taken across the area occupied by the streamwise vorticity of each structure above the cutoff, $e_{\unicode[STIX]{x1D714}_{x}}=2\sqrt{e_{s}^{2}+e_{rand}^{2}}$ is the error on the streamwise vorticity (i.e. twice as much as $e_{v}$ ) and $A_{i}$ is the area of the $i$ th cell. Although the error on the velocity components is measurable, that on the circulation resulted always of the order of $10^{-4}~\text{m}^{2}~\text{s}^{-1}$ for each vorticity patch and, therefore, negligible.

Figure 8.

Circulation of the vortices along the probed flow field. (a) Case I and (b) case II. The values are normalized by $\unicode[STIX]{x1D6E4}_{ref}=3.47~\text{m}^{2}~\text{s}^{-1}$ .

4.3 Fluctuating flow measurements: methodology, convergence and uncertainty

Single point values of the $TKE$ are calculated in the location of its peak within the vorticity patches. On the other hand, area-averaged values of the $TKE$ are calculated by means of the following equation:

where $A$ is the area of the vorticity patch remaining above the chosen cutoff and $TKE_{i}$ is the turbulent kinetic energy. The average $TKE$ content across the vorticity patch is in this work a quantity of interest as it is less prone to be affected by possible inexact single-point measurements.

The $TKE$ is calculated as half the sum of the squares of the velocity fluctuations in the three directions across the entire range of instantaneous flow fields (4.3), namely:

Each term in (4.3), in turn, is affected by statistical uncertainty $\unicode[STIX]{x1D716}_{stat}$ , due to the limited number of instantaneous flow fields collected, and a convergence error which was found negligible for the cases presented here.

To assess the degree of convergence of the data, the peak $TKE$ variation with the number of realizations $N$ was analysed. The same steps were repeated for all the surveyed stations, however, figure 9 only shows the $TKE_{P}$ against $N$ at stations $16h$ for both cases I and II and which, for different reasons, could have been seriously affected by convergence problems. In fact, in case I the only meaningful portion of the aforementioned measurement was limited to the plume (due to seeding from the injector only), while in case II only a limited number of realizations could be available. All the results were declared converged if the per cent $TKE$ change respect to the measured final value against the number of realizations was within $\pm 1\,\%$ .

Figure 9.

Peak $TKE$ trends with the number of realizations of the structures present in the flow field at station $16h$ . (a) Case I and (b) case II.

Because of the negligible convergence error, the total uncertainty on the turbulent kinetic energy can be calculated as follows in (4.4)

The statistical uncertainty $\unicode[STIX]{x1D716}_{stat}$ depends on the fourth and the second statistical moments along with the number of realizations collected in each point and it is evaluated for a $95\,\%$ confidence interval (Benedict & Gould Reference Benedict and Gould1996). Specifically, for one of the squares of the fluctuating components of the velocity (namely, $\langle {u^{\prime }}^{2}\rangle$ ), the statistical error can be written as:

Where $N$ is the number of realization in each point of the flow field.

The magnitude of each velocity fluctuation and the fluctuating component of the streamwise vorticity will be extracted from the point of peak $TKE$ to understand their relative contribution. For the uncertainty on the single fluctuations (i.e. velocity and vorticity) the formulation in (4.6) was used. Note that (4.6) is written for the streamwise velocity fluctuations, but can be applied to the remaining components as well as to the fluctuating vorticity by substituting the respective statistical moments:

The error on the $TKE$ measurement is overlapped to the statistical uncertainty. The aforementioned error depends on the square of the fluctuations in the interpolation error $\langle e_{s}\rangle ^{2}$ introduced by the peak-detection algorithm employed by the PIV software (see § 4.2) and on the square of the fluctuations in the random error $\langle e_{rand}\rangle ^{2}$ . In this case, the respective contributions of the aforementioned errors were found to be approximately $\langle e_{s}\rangle ^{2}=23~\text{m}^{2}~\text{s}^{-2}$ and $\langle e_{rand}\rangle ^{2}=0.4~\text{m}^{2}~\text{s}^{-2}$ , therefore, negligible for the $TKE$ values presented in this manuscript.

Figure 10.

Case I, station $10h$ . (a,b) Instantaneous images from one of the PIV cameras in use; (c,d) instantaneous flow fields resulting from the same shots.

In order to show the quality of the recorded images (see figures 10 a,b and 11 a,b) and the number of valid the realizations contributing to both data and uncertainty, two representative instantaneous flow fields were extracted from the data collected at station $10h$ for case I (figure 10 c,d) and at station $13h$ for case II (figure 11 c,d), respectively. The selected locations represent worst cases scenarios as they could be poisoned with reflections and uneven seeding which often result in lack of correlation and, therefore, an empty space in the PIV grid. The two instantaneous flow fields have been chosen to show a typical realization with an almost full vector field and another with fewer vectors.

Figure 11.

Case II, station $13h$ . (a,b) Instantaneous images from one of the PIV cameras in use; (c,d) instantaneous flow fields resulting from the same shots.

Grey vectors in figures 10 and 11 represent the valid realizations calculated by means of the PIV software in use (DaVis 7.1). The software was set-up to remove as many spurious vectors as possible from each instantaneous flow field and attempting to keep the largest number of ‘good’ vectors. However, after visual inspection, a few invalid vectors were still present and an internally developed post-processing code was used to apply a median filter for the removal of the remaining false vectors, which were highlighted in red in figures 10 and 11. Note that the median filter compares the in-plane magnitude of a velocity vector with the mean of its neighbours and eliminates it if the per cent difference is above $75\,\%$ . As can be seen, empty spots were present in the measurements, however, they are mostly confined outside the area of interest in all the instantaneous realizations.

Although any median filter inevitably removes also good vectors, the benefit of getting rid of spurious data largely overcomes the negative effect of removing good data. Figures 12(a) and 12(b) show the total amount of valid realizations $N$ per spatial location; although a slight lack of data is present within the central vorticity patch for case I (much milder is the loss of data in case II), the amount of realizations was still sufficient to attain convergence of the results.

Figure 12.

Number of realizations per point. (a) Case I, station $10h$ ; (b) case II, station $13h$ .

4.4 Case I: fluctuating flow data

Figures 13–15 present the fluctuating components of the velocity across the flow fields for case I, normalized by the nominal free stream velocity (i.e. $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ ). As can be seen, the magnitude of the velocity fluctuations remains of the order of one-tenth of $V_{\infty }$ all along the surveyed flow field. The image in figure 16 shows the normalized fluctuating component of the streamwise vorticity: turbulence imposes a very large fluctuation of the aforementioned quantity, especially inside the vorticity patches where it is actually comparable with its peak value. Finally, figure 17 shows the $TKE$ – normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ – as calculated by means of (4.3) at the different stations. Note that the measurement at station $16h$ was originally conducted by seeding the injectant only; therefore, the abrupt variation of the aforementioned quantities between the wake and the external flow is an artefact of the data reduction code due to the different number of valid realizations in those areas. The data that should be considered valid are those within the dashed line only.

Figure 13.

Case I: evolution of the fluctuating component of the streamwise velocity normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ , (d) $20h$ and (e)  $32h$ .

Figure 14.

Case I: evolution of the fluctuating component of the velocity along the $y$ axis normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ , (d) $20h$ and (e) $32h$ .

Figure 15.

Case I: evolution of the fluctuating component of the velocity along the $z$ axis normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ , (d) $20h$ and (e) $32h$ .

Figure 16.

Case I: evolution of the fluctuating component of the streamwise vorticity normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ . Stations (a)  $10h$ , (b)  $13h$ , (c)  $16h$ , (d)  $20h$ and (e)  $32h$ .

Figure 17.

Case I: evolution of the $TKE$ normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ . Stations (a)  $10h$ , (b)  $13h$ , (c)  $16h$ , (d)  $20h$ and (e)  $32h$ .

The plots in figure 18 show the trends of the fluctuating components of the velocity (figure 18 a–c) and the vorticity (figure 18 d) in the point where the turbulent kinetic energy of each structure reaches a peak. The first main difference between them is the magnitude recorded in the closest streamwise station (i.e. $10h$ ): it appears that the axial fluctuations in that location are approximately double those along $y$ and $z$ for each structure, with higher values in the merger when compared with the external vorticity patches. Additionally, the decay of the in-plane fluctuations is much slower than that in the streamwise direction, suggesting that the strong large-scale stirring motion of the vortices is sustaining them. On the other hand, the fluctuations of the streamwise velocity component measurably decrease, being, therefore, the main cause of the turbulence decay shown in figure 19. As far as the vorticity fluctuations, it is noteworthy that they remain always comparable with the mean peak vorticity (the weaker fluctuations in the point of maximum turbulence at $32h$ are still 35 %–40 % of $\unicode[STIX]{x1D714}_{x,ref}$ ) but, similarly to the other fluctuating quantities, seem to increase with the strength and the size of the structures.

Figure 18.

Case I: trends of the normalized fluctuating components of (a) the streamwise velocity, (b) the $y$ -component of the velocity, (c) the $z$ -component of the velocity and (d) the streamwise vorticity in the point of peak $TKE$ .

Figure 19(a,b) show the evolution of the normalized peak $TKE$ (denoted by the subscript $P$ ) and that of the average $TKE$ (i.e. $TKE_{AVG}$ ) across each vortex, respectively. As might be expected due to the large number of cells constituting the vorticity patch, the total error for $TKE_{AVG}$ is so much smaller than that attained in the single point measurement of $TKE_{P}$ that the error bars are almost completely absent.

Figure 19.

Case I: trends of $TKE$ normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ along the streamwise direction. (a) Peak $TKE$ and (b) average $TKE$ across the vorticity patch.

There are two important considerations to be made based on the plots shown in figure 19: (i) the intensity of the $TKE$ (both peak and average) in the merger is very close to being double that of the isolated vortices at stations $10h$ and $13h$ and (ii) $TKE$ is still being produced between stations $10h$ and $20h$ . Regardless of the uncertainty, in fact, a clear inversion of the trend of the curves in each patch is noticeable, thus suggesting that the $TKE$ production overcame its decay at station $16h$ , especially when the average taken across the mean streamwise vorticity patch is considered.

The first, in particular, is notable result in light of the fact that it is legitimate to assume – at least in the absence of vortex stretching phenomena – that the initial values of the fluctuating components of the vortices should have been the same for all the structures before merging took place, as will also be shown for case II. Therefore, these measurements seem to suggest the presence of a proportionality law between circulation and $TKE$ , de facto linking mean and fluctuating flow fields.

4.5 Case II: fluctuating flow data

Figures 20–23 show the normalized fluctuating components of the velocity and the streamwise vorticity across the flow field imposed by the vortical structures present in case II. Even for this non-merging scenario, the magnitude of the fluctuations remains of the order of one tenth of the free stream velocity all along the surveyed flow field (see figures 20–22), although the spatial distribution of the fluctuations is very different from those of case I (compare with figures 13–15). Also, the morphology of the normalized $TKE$ scalar field, presented in figure 24, is much different from that of the merging case in figure 17. Compared to case I, where the merger induced a rotation in the clockwise direction of both the whole system of vortices and the $TKE$ scalar field, the structures in this case do not rotate around a common centre, thus preventing the redistribution of the turbulent kinetic energy in the direction of rotation. The only notable similarity with case I are the turbulence levels which are not too different and can be presented with the same scale. The major modification that the flow field undergoes is due to diffusion and the natural turbulence decay noticeable at downstream stations.

Figure 20.

Case II: evolution of the fluctuating component of the streamwise velocity normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a) $10h$ , (b)  $13h$ , (c) $16h$ and (d) $32h$ .

Figure 21.

Case II: evolution of the fluctuating component of the velocity along the $y$ axis normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a)  $10h$ , (b) $13h$ , (c) $16h$ and (d) $32h$ .

Figure 22.

Case II: evolution of the fluctuating component of the velocity along the $z$ axis normalized by $V_{\infty }=572.89~\text{m}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ and (d) $32h$ .

Figure 23.

Case II: evolution of the fluctuating component of the streamwise vorticity normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ and (d) $32h$ .

Figure 24.

Case II: evolution of the $TKE$ normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ . Stations (a) $10h$ , (b) $13h$ , (c) $16h$ and (d) $32h$ .

Figure 25 shows the trends of the fluctuating components of the velocity and streamwise vorticity in the point of peak turbulence. Also in the non-merging scenario, the fluctuations along the $y$ and $z$ axes not only are not as strong as those along $x$ , but their decay proceeds in a slower fashion for every vorticity patch from which they have been extracted. In this case, however, the levels of the fluctuations do not differ much although higher values are extracted from the middle patches (i.e. ML, MR and CV), attesting the presence of an area in which the very large shear due to the close vicinity of strong co-rotating vortices contributes to produce turbulence. The vorticity fluctuations, on the other hand, seem to decay in a very similar way for each vortex and, analogously with the findings of case I, their magnitude remains always comparable to $\unicode[STIX]{x1D714}_{x,ref}$ .

Figure 25.

Case II: trends of the normalized fluctuating components of (a) the streamwise velocity, (b) the $y$ -component of the velocity, (c) the $z$ -component of the velocity and (d) the streamwise vorticity in the point of peak $TKE$ .

As can be seen in the plots of figure 26, the intensities of the $TKE$ are very similar for each structure at each streamwise station, with slightly higher values in the central vorticity patches. This result builds on the findings presented in § 4.4, where it was postulated that the peak $TKE$ may scale with the vortex’s size and strength. Interestingly, this is not the case for the central vorticity patch which is characterized by fairly low circulation, a very large value of peak $TKE$ , and which is not a structure generated by the ramps. Also in this case, a measurable inversion of the $TKE$ trend (both peak and average) can be recognized between $10h$ and $16h$ , indicating that turbulence production overcomes the decay in this area.

Figure 26.

Case II: trends of $TKE$ normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ along the streamwise direction. (a) Peak $TKE$ and (b) average $TKE$ across the vorticity patch.

5.1 Cases III–V: mean and fluctuating flow data

Figure 27 shows the normalized streamwise vorticity and the $TKE$ for cases III–V at selected streamwise stations to show the wide variety of streamwise vorticity and turbulent kinetic energy distributions possible with just a slight change in the mutual distance of the ramps in use. Although additional stations were probed for these cases, they are not shown as this section is mainly focused on the extraction of vortex Reynolds numbers as well as peak and average values of the turbulent kinetic energy.

The vortices shed from the internal edges of the ramps merged in all these additional cases to create a structure whose strength was roughly the sum of the circulation of the isolated ones. It is exactly this amalgamation of streamwise vorticity that enabled the study of the evolution of many different categories of supersonic vortical structures, as the merger has actually been considered a single, isolated large vortex in the following calculations.

As the $TKE$ contour plots show, the stronger shock generated by the taller ramps (see the contours corresponding to case IV and V in figure 27) did not noticeably change the background turbulence levels in the shock-free region in which the streamwise vortices evolve (approximately $150{-}200~\text{m}^{2}~\text{s}^{-2}$ in cases I–IV, while $250~\text{m}^{2}~\text{s}^{-2}$ in case V).

Figure 27.

Streamwise vorticity normalized by $\unicode[STIX]{x1D714}_{x,ref}=5.21\times 10^{4}~\text{s}^{-1}$ (a,c,e) and $TKE$ normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ (b,d,f) contours for (a,b) case III, station $32h$ ; (c,d) case IV, station $13h$ ; (e,f) case V, station $13h$ .

The plot in figure 28 shows the vortex Reynolds number evolution of some of the structures (i.e. the isolated vortices and the merger) along the probed flow field for the cases considered here; data from other cases have been omitted but it was verified that the same conclusions apply to all the vortices considered. A decay can be inferred in all the cases, although marginal when compared to the actual values of the Reynolds numbers attained.

Figure 28.

Vortex Reynolds numbers of some of the structures surveyed in cases I–IV.

Figure 29.

Ratios between the fluctuating velocity components: (a,d) $\sqrt{\langle u_{P}^{\prime 2}\rangle /\langle v_{P}^{\prime 2}\rangle }$ , (b,e) $\sqrt{\langle u_{P}^{\prime 2}\rangle /\langle w_{P}^{\prime 2}\rangle }$ and (c,f) $\sqrt{\langle v_{P}^{\prime 2}\rangle /\langle w_{P}^{\prime 2}\rangle }$ . Note that panels (a,b,c) pertain to case I, while (d,e,f) pertain to case II.

5.2 Turbulence characterization

In order to characterize the turbulence, three properties have been analysed: homogeneity, isotropy and compressibility.

The $TKE$ contour plots shown in figures 17, 24 and 27 clearly show the high inhomogeneity of the turbulence under consideration, highlighting how the streamwise structures act not only to produce, but also to transport, turbulence from one region to another in space.

The degree of isotropy of the turbulence was analysed in the point of peak $TKE$ for cases I and II (see figure 29). In case I the ratios between the fluctuations of the components of the velocity seem to lie close to 1 (i.e. local isotropic turbulence), essentially all along the flow field within the external vortices (ELV and ERV), while noticeable differences between the remaining ratios can be noticed in the merger. Especially $u^{\prime }/w^{\prime }$ and $v^{\prime }/w^{\prime }$ in the merger tend to reach a constant value of approximately 1.5 downstream. A situation which is similar to that of the external vortices of case I can be recognized in the same structures of case II, where all the fluctuating components in the point of peak turbulence tend to equalize downstream. The internal vortices (ML and MR), on the other hand, present a much more erratic behaviour, showing relatively high peaks of the quantities $u^{\prime }/v^{\prime }$ and $u^{\prime }/w^{\prime }$ at stations $10h$ and $16h$ . The non-isotropic nature of the turbulence in the same point has been verified for the whole set of cases analysed in the manuscript although the corresponding plots have been omitted.

The magnitude of the fluctuating velocity was used to estimate the fluctuating Mach number in order to discover whether or not the turbulence can be considered incompressible. It stayed below 0.3 across the probed sections remaining, therefore, within the incompressible regime.

For this reason, the data under consideration belong to an incompressible, non-homogeneous and globally anisotropic turbulence, at least in the single point corresponding to the location of the peak turbulence.

Figure 30.

$TKE$ peak (a,c) and average (b,d) – normalized by $TKE_{ref}=3.86\times 10^{3}~\text{m}^{2}~\text{s}^{-2}$ – of some of the structures as a function of (a,b) the vortex Reynolds number at different stations and (c,d) the normalized streamwise distance for different inspected cases.

5.3 Dimensional analysis

The preliminary observation of the multiple turbulence data allowed us to infer that turbulent kinetic energy, both peak and average, increases with the increasing strength of the streamwise structures (i.e. circulation) and, as expected, decays with the streamwise distance $x$ originating at the injection point (see figure 30). For this reason, the functional relationship between the $TKE$ and the aforementioned parameters is postulated to be in the form shown in (5.1)

Assuming valid the relation in (5.1), three non-dimensional groups that regulate the phenomenon can be retrieved, specifically: $\unicode[STIX]{x1D6F1}_{1}=TKE/V_{\infty }^{2}$ , $\unicode[STIX]{x1D6F1}_{2}=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6F1}_{3}=x/h$ ; note that the group $\unicode[STIX]{x1D6F1}_{2}$ is the inverse of the vortex Reynolds number of each structure.

Equation (5.2) shows the form of the final equation relating the dimensionless parameters. In the relation, $\unicode[STIX]{x1D6F1}_{2}$ has been inverted to explicitly show the dependence on the vortex Reynolds number, $Re_{\unicode[STIX]{x1D6E4}}$

A constant Reynolds number to assign to each structure was needed to carry out the dimensional analysis calculations. For this purpose, the Reynolds number of each structure evaluated at the closest possible station per case inspected was assigned (i.e. considering the circulation at station $10h$ for cases I to IV and the circulation at $13h$ for case V).

Figures 30(a) and 30(b) show, respectively, the normalized peak and average $TKE$ as a function of $Re_{\unicode[STIX]{x1D6E4}}$ , measured at different stations. Figures 30(c) and 30(d) show the change in normalized peak and average $TKE$ as a function of the downstream distance normalized by the ramp’s height $h=5.8$  mm. Recall that the average $TKE$ is necessarily lower than the peak $TKE$ due to the averaging process applied to calculate it. Note that in figure 30 simple curve fits of the type: $y=a(x-b)^{n}$ have been included only to highlight possible qualitative trends of the turbulent kinetic energy with the vortex Reynolds number and the normalized streamwise coordinate. Specifically, the exponent $n=0.5$ was used to fit ( $Re_{\unicode[STIX]{x1D6E4}},TKE/TKE_{ref}$ ), while $n=-1$ for ( $x/h,TKE/TKE_{ref}$ ); as will be shown later these exponents are quite close to the optimal $A$ and $B$ coefficients in (5.2) calculated through dimensional analysis.

Figures 30(a,b) and 30(c,d) clearly show that the $TKE$ increases with increasing vortex Reynolds numbers and decays with the streamwise distance. Furthermore, very similar trends exist for both $TKE_{P}$ and $TKE_{AVG}$ , thus indicating that all the points belonging to the vortices behave in a similar manner. However, as the trends of cases I and II show some production of $TKE$ still occurring between $10h$ and $16h$ , the data belonging to this interval in these two cases have been neglected as we are interested in describing the turbulence decay only. In any case, a calculation performed utilizing the entire set of data (i.e. including those neglected in this work) has shown a minimal sensitivity of the final relation to the presence of this ‘production zone’ likely due to its mild nature.

To find the aforementioned relation, an optimization algorithm calculates the best combination of $A$ and $B$ to yield the highest coefficient of determination of the linear fit $R^{2}$ between $TKE/V_{\infty }^{2}$ and $Re_{\unicode[STIX]{x1D6E4}}^{A}\left(x/h\right)^{B}$ . The peak in the map of coefficients $R^{2}=f(A,B)$ corresponds to the optimal combination of $A$ and $B$ . Based on the analysis of the trends obtained from experimental results, the calculation was performed assuming a range of $B$ coefficients between 0 and $-3$ , and a range between 0 and 3 for the $A$ coefficients.

The corresponding maximum $R^{2}$ is 0.97 for the correlation involving $TKE_{P}$ , with the optimal combination of coefficients being: $A=0.48$ and $B=-1.38$ . The $R^{2}$ is instead 0.93 for the correlation on $TKE_{AVG}$ , with corresponding coefficients: $A=0.48$ and $B=-1.35$ .

By plotting the experimental data combined as in the right-hand side of (5.2), the coefficient $C$ in (5.2) can be retrieved by means of a simple linear fit of the data (see 31). The fits in 31 (solid line) include the $95\,\%$ confidence bounds (the dashed lines), indicating that the $95\,\%$ of future realizations are expected to be included in that range. Note that although all the data come from vortices produced and evolving in different ways (i.e. some are shed from the ramps some other are mergers), they all closely follow the proposed power laws. Therefore, the relationships cannot be used to explain how the vortices are formed or evolve, an information that only the analysis of their interaction can provide.

Figure 31.

Correlations and fits for (a) peak $TKE$ and (b) average $TKE$ .

The final functional relationships (i.e. the slope of the solid line in figure 31) that describe the decay of the peak and average $TKE$ across the vortices are presented in (5.3) and (5.4)

At first glance, equation (5.2) is similar to the relation for the grid turbulence decay in (1.3) and reworked in the form given in (5.5)

Particularly, when the vortex Reynolds number is taken in the place of the grid Reynolds number the structure of the equations is the same. Although the data presented § 4 clearly showed a non-isotropic nature of the turbulence under consideration, the inspection of the fluctuating Mach numbers across each flow field did reveal that the turbulence considered here is essentially incompressible in nature, thus partially justifying the aforementioned similarity.

The differences between the isotropic incompressible turbulence and that introduced by the streamwise vortices under consideration are noteworthy. By considering $\left(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D706}V_{\infty }\right)^{2}$ a constant, the exponent corresponding to $A$ in (5.5) is 1, while that corresponding to $B$ is $-0.275$ ; in fact, $m=0.35$ and $n=1.25$ for grid turbulence (Valente & Vassilicos Reference Valente and Vassilicos2011). This indicates that in the subsonic, isotropic turbulence a stronger dependence of the $TKE$ on the Reynolds number and a much slower decay is found when compared to the turbulence data provided in this manuscript.