Combining simultaneous density and velocity measurements of rotor blade tip vortices under cyclic pitch conditions

The measurements were taken at the rotor test stand Göttingen (RTG). The facility features a rotor with a radius of \(R={0.65}\hbox { m}\) and a horizontal axis (see Fig. 1). The purpose of the facility is to enable aerodynamic testing with optical measurement techniques at known conditions and a minimum blockage of the rotor downwash. A wind tunnel provides a defined axial inflow of \(V_{\infty }= {2.6}\hbox { m/s}\), which leads to well-known and reproducible conditions during the measurements and convects the turbulent wake downstream, avoiding recirculation.

The four blades have a DSA-9A airfoil with a chord length of \(c={72}\,\hbox {mm}\), a parabolic ONERA SPP8 blade tip planform and a negative linear twist of − 9.3\(^\circ\) along the aerodynamically relevant span between \(0.25< r/R < 1.00\). The profile and planform of the blades are shown in Fig. 2. The aspect ratio of the blades is \(\varLambda = R/c \approx 9\), and the solidity \(\sigma = N_b A_b/(\pi R^2) = 0.1025\), with the number of blades \(N_b = 4\) and the area of a single blade \(A_b = {0.034}\,\hbox {m}^2\).

A balance, measuring all three forces and moments of the rotor, Hall-effect sensors measuring the blade pitch angle, and temperature and humidity sensors have been used to quantify the operating conditions of the rotor. A detailed description of the RTG can be found in (Schwermer et al. 2016).

The current investigation is based on measurements at a rotational frequency of 23.6 Hz. The resulting Mach- and Reynolds numbers, and the reduced frequency at \(r/R = 0.75\) are given in Table 1.

Figure 3 shows the static thrust polar (black dots) by means of the blade loading \(C_{\text {T}}/\sigma = F_z/(\rho _{\infty }\pi R^2 \varOmega ^2 R^2 \sigma )\), with the thrust \(F_z\), and the ambient density \(\rho _{\infty }\). The pitch angle is given at a radius of \(r/R = 0.75\). The expected linear trend for attached flow conditions in Fig. 3 is met below \(\varTheta _{75} = {22}^\circ\). Above this value, static stall occurs. The maximum blade loading is 0.216. The values obtained from CFD simulations by Goerttler et al. (2020) are added as orange squares. The simulation underpredicts the thrust coefficient compared to the experimental data, depending on the pitch angle.

The current investigation is based on a cyclic test case with a pitch cycle defined byThe range of the cyclic pitch variation is indicated as a blue dashed line in Fig. 3. The chosen pitch setting provides a wide range of lift production: The flow over the blades is fully attached during the entire dynamic pitch cycle, even though the maximum pitch angle of 22.9\(^\circ\) exceeds the static stall angle. This is due to the fact that dynamic stall is delayed by unsteady aerodynamic effects (Mulleners and Raffel 2011). The mean thrust of this pitch setting is \(F_z = {236}\,N\) and the resulting mean blade loading \(C_{\text {T}}/\sigma = 0.157\).

The cyclic pitch setting leads to an azimuthal dependency of the blade loading. Low pitch angles, which go along with a low blade loading, result in a low induced velocity and, thus, a reduced convection rate of weak blade tip vortices compared to high pitch angles. The resulting rotor wake is also azimuth-dependent as well as asymmetric. For the current investigation, three snapshots of the resulting rotor wake were taken with optical measurement systems. Since the field of view (FOV) of these systems was not large enough to capture the entire rotor wake at once, a scanning technique was applied: A key feature of the RTG is the ability to slowly rotate the usually stationary part of the swash plate. With this feature, it is possible to vary the phase of the pitch cycle at a certain position, e.g., relative to a fixed FOV of an optical measurement system. With this approach, it is possible to scan through the pitch cycle of the rotor without moving the optical measurement system. By additionally setting the captured vortex age according to the corresponding phase, snapshots of the rotor wake resulting from this cyclic pitch setting can be reconstructed. In the current investigation, the scan comprises recordings from 20 discrete phase positions. For each measurement, 200 images have been taken with each optical system. Based on this data, the mean shape and the aperiodic behavior of the rotor wake can be studied.

The axial inflow at the RTG represents slow climb conditions, whereas a cyclic pitch relates to maneuver or forward flight. A change in dynamic pressure, as it occurs in forward flight, is not reproduced. Therefore, the measurement conditions at the RTG represent generic test cases, which are suitable to study the influence of different pitch settings and flow conditions with a reduced measurement effort, and to compare the results from different measurement techniques as well as from corresponding numerical simulations.

For the current investigation, both a stereoscopic background oriented schlieren (BOS) setup and a stereoscopic particle image velocimetry (PIV) setup have been used to simultaneously capture the tip vortices of the RTG. Based on these two techniques, it is possible to connect the density-related BOS shift to the velocities, that were measured with the PIV system, to gain an increased understanding of the vortices. In addition to these two optical systems, a camera focused on the blade tip is used. With this additional camera, it is possible to evaluate lead/lag and flap movements of the tip and the blade tip pitch angle. The obtained data of the vortex positions can, therefore, be derived with respect to the blade tip position.

An overview of all optical measurement systems is shown in Fig. 4. The two PIV cameras are positioned below the rotor. The PIV laser with the laser optics is found far downstream of the rotor. The wind tunnel nozzle is marked with a white box upstream of the rotor. On both sides of the nozzle, a BOS camera is placed. On the right side of Fig. 4, the blade tip camera can be seen.

The CFD simulations are taken from Goerttler et al. (2020) and were performed in the department of helicopter aerodynamics at the DLR Göttingen. The numerical computations were performed using DLR’s finite volume solver TAU. Four identical grids with 9.5 million points have been used for the blades. The blade grids are embedded in a background grid with a total size of 48 million points. The first 80\(^\circ\) of one blade tip vortex is refined with a curved, structured region, following the expected path of the blade-tip vortex. The normal spacing to the vortex core propagation direction is \(s_{\text {c}}/c=0.0034\) at a very young wake age and increases up to \(s_{\text {c}}/c=0.0091\) at a vortex age of \(\varPsi _{{\text {v}}}= {90}^\circ\). A zonal approach with a large-eddy simulations (LES) model (Travin et al. 2002) is employed on the four-bladed configuration to reduce numerical dissipation. The LES model is used around the fine structured grid, in all other regions, the Menter shear-stress transport (SST) model (Menter 1993) is used. Therefore, smaller scales which trigger viscous dissipation are not modeled but resolved. This leads to less dissipating and more accurate solutions than pure Reynolds-Averaged Navier Stokes (RANS) approaches (Goerttler et al. 2020). The chimera method is used to carry out the data exchange between the pitching blade grids and the background grid. The results of the computations were published by Goerttler et al. (2020) and are a valuable addition to a discussion of the experimental data presented in this paper.

A reconstructed snapshot of the azimuth-dependent wake of the RTG rotor under cyclic pitch conditions is shown in Fig. 11. A 3D model of the rotor is added to ease the interpretation of the downwash structure. The four characteristic phases of the blade pitch cycle (\(t/T = 0.00\), 0.25, 0.50, and 0.75) are marked in the four corners of the image. Each of the four blades is located 25\(^\circ\) behind one of these phases. Behind each blade, a vortex is present, which was produced at the pitch angles stated around the structure. As a rule of thumb, the vortices on the left side of Fig. 11 were shed from the blades during upstroke and the ones on the right side during the downstroke.

The vortices are represented by averaged and convolution filtered out-of-plane vorticity distributions from PIV, in all 20 measurement planes. For a better visibility, values close to zero (between \(-0.035\) and 0.035) are set to transparent. The vortices in the PIV planes are linked by colored dots, which state the average center position of the vortices as derived from the BOS data. The dots are colored with respect to the blade of origin of each vortex.

The BOS-detected vortex centers are matched by circular spots of positive vorticity, as seen by PIV. The tip vortices are fed by the blade shear layers, which also appear in the vorticity distribution as linearly elongated structures pointing toward the hub. In some cases, the shear layer’s sense of rotation changes further inboard, marked by a transition from red to blue coloring. This point marks the local maximum in the radial lift distribution. Similar findings were described by Milluzzo and Leishman (2016).

Depending on the pitch phase, one or two vortices are present in the FOV of PIV. In cases with a low pitch angle, the downwash velocity is reduced, which leads to a reduced convection rate of the vortices and, thus, two vortices inside the limited FOV of PIV. The FOV of the BOS system on the other hand contains one to three vortices, leading to lines that continue below the vorticity planes from PIV. The oldest vortex shown in the snapshot of Fig. 11 and captured by BOS is around 200\(^\circ\) old and evolved from Blade 4. The reconstructed overview of the vortex system provides a good impression of the asymmetry and azimuthal dependency of the captured rotor wake resulting from the cyclic pitch setting.

Figure 13 shows the vortex locations obtained by PIV, shed by a blade at the minimum pitch angle at \(t/T = 0.00\) (Fig. 13a) and at the maximum pitch angle at \(t/T = 0.51\) (Fig. 13b). Both figures show the locations at different vortex ages \(\varPsi _{{\text {v}}}\). The trailing edge of the blade is included as black line at \(z_{\text {r}}/R = 0\). The colored dots represent the 200 individual center locations from the PIV measurement. The locations are corrected for the blade bending based on the data from the blade tip camera.

The ellipse around each group of locations represents a confidence interval of \(95\%\) (two times the standard deviation) of the individual results. It can be seen that the scatter of the vortex locations has a dominant axis for all cases. This major axis appears to be approximately perpendicular to the mean convection trajectory. This finding hints at a variation of the individual trajectories rather than a change in the convection velocity along a steady trajectory. Similar results were discussed by Mula et al. (2011), who also found increasing scatter ellipses with rising vortex age. The cases at \(\varPsi _{{\text {v}}}= {25}^\circ\) and \(\varPsi _{{\text {v}}}= {55}^\circ\) in Fig. 13a shows an exception to this trend, although the reason for this finding remains unknown.

By comparing Fig. 13a and b, the influence of the blade pitch can be studied. At the minimum pitch angle, the downwash velocity is smaller compared to the case with maximum pitch, leading to a reduced convection rate of the tip vortices. Also, the aperiodic scattering is dependent on the pitch angle.

In addition to the PIV results, the locations computed by the CFD simulation are shown as colored spots. The computations and the experiment show comparable results regarding the convection direction and the influence of the pitch angle. The convection rate in the CFD is higher compared to the PIV results.

The vortex scatter shown in Fig. 13 is based on the PIV data. The same values can be obtained from the 3D vortex positions from BOS. The vortex locations from BOS, extracted at the PIV plane, as well as the comparable values from PIV are plotted in Fig. 14 for two sample cases with very different scatter intensities, both at the same scale. To focus on the representation of the aperiodic vortex wandering, the mean position of each group was subtracted from the individual positions. This step removes a constant offset between the different techniques, which will be discussed in Fig. 15. The comparison of Fig. 14a, b gives an impression of the range of scatter observed: The two major scatter axes vary by a factor of ten within the data captured in the current investigation.

Comparing the results from PIV and BOS in Fig. 14b, a good agreement between the individual vortex locations estimated with two very different techniques is evident. The individual positions and, as a result, also the scatter ellipses agree well. The agreement can be quantified by the correlation coefficient of the PIV and BOS positions, which calculates to 0.998 for the case shown in Fig. 14b. The correlation for the case in Fig. 14a is 0.855, which is significantly smaller. This finding can be explained by two causes: (1) In case of a small scatter, the error between PIV and BOS (e.g., due to noise) is relatively large, which leads to a decrease in the correlation. (2) In case of the small scatter in Fig. 14a, the resolution limit of the BOS technique is reached. Since the vortex positions in the gradient of the shift are found using only the accuracy of integer pixel, the resolution limit in the BOS results leads to a stepping in the BOS positions. The step size can be approximated to be \(7.5\cdot 10^{-4}R\) (\(\approx {0.5}\,\hbox {mm}\)) in the case presented in Fig. 14a, which is 1/8 of the major axis length of the resulting scatter ellipse.

Figure 15 shows the absolute difference between the mean vortex locations obtained by PIV and by BOS. The two graphs represent the differences in \(r_{\text {r}}\)- and \(z_{\text {r}}\)-direction over the vortex age for a sample case during the upstroke (shed from Blade 4, \(t/T = 0.30\) − 0.13). With rising vortex age, the offset in radial direction (\(r_{\text {r}}\)) reduces to zero, while the offset in \(z_{\text {r}}\)-direction increases. The variation of the offsets with the vortex age can be ascribed to a misalignment between the individual calibrations of both systems. The absolute difference between PIV and BOS is always below \(0.0023\,R\) (1.5 mm, or \(\approx 0.5r_{\text {c}}\)) for the shown cases. The agreement of both techniques, despite the different approaches and sources of error, is satisfactory and sufficient for the interpretation of the vortex system presented in this paper.

Figure 16 quantifies the rotor wake given in Fig. 11 over the pitch phase t/T. Figure 16a and b states the pitch angle at the vortex origin \(\varTheta _{75}\) and the vortex age at the recording of the vortex \(\varPsi _{{\text {v}}}\), respectively. These quantities ease the interpretation of the results in Fig. 16c–f. The combination of vortex age and pitch phase correspond to moving along a vortex up to an age of 70\(^\circ\) in Fig. 11. The data in both figures are colored with respect to the blade of origin of each vortex.

The normalized vortex circulation \(\Gamma _{0.5c}\) is given in Fig. 16c. The values follow the expected trend according to the changing pitch angle over the phase. The circulation changes by a factor of three during the pitch cycle. The hysteresis between up- and downstroke observed in Fig. 12 is present as horizontal offset between the curve of the circulation and the pitch angle in Fig. 16a.

Figure 16d shows the length of the major axis of the scatter ellipses a introduced in Fig. 13, normalized with the rotor radius R. The values obtained by both PIV and BOS are given. The data indicate a strong influence of the vortex age on the aperiodic vortex scatter. This trend is superimposed by an influence of the pitch phase. The major scatter length increases faster during the downstroke motion of the blades (\(t/T>0.5\)), hinting at an increased sensitivity to perturbations as also found by a previous investigation at the RTG (Braukmann et al. 2020). A good agreement between PIV and BOS can be seen during the entire pitch cycle.

The aspect ratio of the scatter ellipses is plotted in Fig. 16e. For \(b/a = 1\), the confidence region is perfectly circular, whereas for smaller values, the major axis dominates and the ellipse is elongated along this axis. For most pitch phases, the aspect ratio is found between 0.2 and 0.6. The young vortex evolving from Blade 3 shows an exception to this trend, with aspect ratios up to 0.9, indicating a rather circular region of scatter. The vortex shed from Blade 2 during the downstroke, shows a comparably constant aspect ratio of the confidence ellipses, although the major axis length strongly depends on the vortex age and the phase of origin. This states a symmetric growth of the scatter region along the vortex.

The results obtained by PIV and BOS are in good agreement for most comparisons in Figs. 14, 15, and 16d and e. The agreement can be quantified by the Pearson correlation coefficient, which gives a measure of the linear dependence of the 200 individual positions measured with both methods. The resulting values for the vortices of the snapshot shown in Fig. 11 are plotted in Fig. 16f against the pitch phase. The resulting values are above 0.9 for most of the cases. The youngest vortices show a strongly decreased correlation for the vortices shed by Blade 1, 3 and 4. The reduced correlation for these cases can be explained by the small area of the position scatter for young vortices, as discussed in Fig. 14.

The scatter ellipses obtained by PIV are limited to the 20 discrete measurement planes. Since BOS enables the reconstruction of the vortex position in a measurement volume, it is possible to obtain the scatter ellipses denoting two times the standard deviation of the vortex scatter over the entire rotor azimuth. The resulting structure is shown in Fig. 17. The elliptic tubes are colored with respect to the blade of origin and represent the volume in which the vortex center appears with a confidence of \(95\%\).

Figure 17 clearly shows the growth of the scatter region with increasing vortex age for all four vortices. Additionally to the data plotted in Fig. 16, vortex ages over \(\varPsi _{{\text {v}}}= {70}^\circ\) are also represented.

In addition to the vortex growth with rising age, a stepping in the size of the ellipses is apparent. The snapshot in Fig. 17 is the result of 20 individual measurements, that were all performed in the BOS measurement volume shown in Fig. 10. Due to the rectangular shape of the wind tunnel nozzle and the position of the BOS measurement volume, the wind tunnel shear layer has an increased influence on the upper end of the volume. A close wind tunnel shear layer increases the perturbations, which leads to an increase in the vortex scatter and to increased scatter ellipses. Therefore, the scatter within each single measurement increases stronger than expected with rising vortex age. The combination of the individual measurements into one snapshot, thus, leads to the stepping seen in Fig. 17.

Despite this influence, the varying size of the scatter ellipse due to the varying pitch angle during the vortex creation is visible in Fig. 17. Vortices during the downstroke (around \(t/T = 0.75\)) show an increased position scatter. This finding hints at an increased instability of the rotor wake during this phase of the pitch cycle. Figure 17 gives a good impression of the extent of the vortex scatter in relation to the rotor. The varying vortex positions affect the vortex-vortex as well as the blade-vortex distances, which is relevant when predicting interactions.

The PIV data do not provide reliable information in the region of the particle void. A common method to overcome this deficit is the use of analytical vortex models. Vatistas et al. (1991) formulated a generalized vortex model representing the swirl velocity distribution \(V_{{{\theta }}}\) over the vortex radius \(r\):Different, previously described vortex models can be obtained by choosing the integer factor n: For \(n = 1\), the model by Kaufmann (Kaufmann 1962) or Scully (Scully and Sullivan 1972) and for \(n = 2\), the model by Bagai-Leishman (Bagai and Leishman 1995) is retrieved. The vortex model by Vatistas from 1991 (Eq. (5)) is further referred to as Vatistas1991.

In 2015, Vatistas introduced a modified vortex model, featuring a turbulence factor \(\beta\):This new, extended model provides an additional factor, which can be varied during the fit to experimental data. An increased \(\beta\) represents a higher turbulence in the vortex and leads to a broadened swirl velocity distribution. Following Vatistas (Vatistas et al. 2015), for \(\beta =1\), the swirl velocity distribution of the new model compares to the laminar formulation. The newer model from 2015 is referred to as Vatistas2015 in the following discussion.

The swirl velocity calculated on the basis of the PIV data is plotted in Fig. 18 as black circles. The two figures show the swirl velocity of vortices at a vortex age of 25\(^\circ\) in the up- (Fig. 18a) and downstroke (Fig. 18b). The distribution is taken from one of the 200 instantaneous measurements of that test case. In addition to the experimental data, best fits of the four previously described vortex models (Vatistas1991 and 2015, each for \(n=1\) and \(n=2\)) are plotted. From both figures, it can be seen that it is possible to fit the analytical models to instantaneous swirl velocity distributions, based on only the data outside the particle void.

The variables optimized during the vortex fit are the maximum swirl velocity \(V_{{\theta , \text {max}}}\), the vortex core radius \(r_{\text {c}}\), and, in case of the Vatistas2015 model, also the turbulence factor \(\beta\). From the comparison in Fig. 18, it can be seen that the best fit of the Vatistas1991 model underrepresents the swirl velocity at high radii for both choices of n. This leads to an underrepresentation of the vortex circulation. The model also leads to a reduction of \(V_{{\theta , \text {max}}}\) and an increased vortex core radius compared to the PIV data. The best fit of the newer Vatistas2015 model, featuring the turbulence correction, shows a better agreement with the data of the presented cases. For both the up- and downstroke case and for both factors n, the profiles represent the key features more closely to the experiment.

A measure to quantify the goodness of a fit for a range of vortices is the sum of squared residuals. The average value of the 200 instantaneous results for both models, both factors n as well as for two vortex ages in each the up- and downstroke is shown in Fig. 19. It can be seen that the newer Vatistas2015 model leads to lower residuals compared to its predecessor. This can be ascribed to the additional fit parameter representing the turbulence within the vortex. The Vatistas2015 model with \(n=1\) leads to the lowest residuals and is therefore chosen for further discussions of the results.

A previous investigation of blade tip vortices at the RTG by Wolf et al. (Wolf et al. 2019) used a fit to the average profile of multiple PIV images instead of fits to the individual PIV measurements. Furthermore, the current investigation features an increased spatial resolution, leading to an increase in the maximum measured swirl velocities by PIV (Wolf et al. 2019). Wolf et al. compared several vortex models and found the Vatistas2015 model with \(n=2\) and \(\beta =1.3\) to be the best fit to the obtained swirl velocity profiles at a constant pitch angle of \(\varTheta _{75} = {17.1}^\circ\) and for several vortex ages up to 325\(^\circ\).

The application of the Vatistas2015 model to the data of two test cases, both at \(\varPsi _{{\text {v}}}= {25}^\circ\) in the up- and downstroke, is shown in Fig. 20. Both vortices were shed from a blade at comparable pitch angles of \(\varTheta _{75} = {17.0}^\circ\) and \({16.6}^\circ\), respectively. The gray lines represent all 200 individual profiles from PIV. The black line is the respective average. Only data outside the particle void are provided. The orange dotted line shows the average of the individual fits of the Vatistas2015 model with \(n=1\).

Despite the strong aperiodicity in the downstroke, the average fit represents both average swirl velocity profiles comparably well (see Fig. 20). This is achieved by different turbulence factors \(\beta\). The mean \(\beta\) in the upstroke case is 1.25, whereas the mean \(\beta\) in the downstroke case is 1.07. Contrary to the expected result of a higher turbulence in the downstroke, the fitted \(\beta\) suggests a reduced vortex turbulence in the downstroke compared to the upstroke. In addition to the data based on the PIV measurement, the profile obtained by the corresponding CFD simulation is shown as blue graph. In the upstroke, CFD and PIV agree well for radii above 0.5c. In the vortex core, CFD underpredicts the PIV swirl velocities. The deviation in the downstroke is more pronounced, with agreement in the core region but an offset outside the vortex core. The resulting vortex circulation \(\Gamma _{0.5c}\) from CFD will show lower values in the downstroke than obtained by PIV.

Figure 21 displays the 200 individual values of the turbulence factor \(\beta\) obtained from the fit to the PIV data. Figure 21a provides the values for the upstroke and Fig. 21b for the downstroke, respectively. Each figure provides the values for two different vortex ages, \(\varPsi _{{\text {v}}}= {25}^\circ\) and \({55}^\circ\). It can be seen that the turbulence factor shows only a few occasions of increased deviations from the mean value of \(\beta = 1.25\).

In the downstroke, the variation is twice as high across the entire 200 measurements, although the mean value is lower than in the upstroke. Thus, a higher aperiodicity in the turbulence factor goes along with a lower turbulence as given by the turbulence factor \(\beta\). This is another representation of the finding of a vortex that appears to have a more laminar velocity profile yet being affected by an unsteady downwash topology, as observed in Fig. 20b.

PIV measures the velocity components of a flow. BOS, on the other hand, detects a shift related to the gradient of the density. Bagai and Leishman (Bagai and Leishman 1993) provided a formulation to link these two quantities based on the Navier Stokes equations for an inviscid, stationary, and two-dimensional vortex. Furthermore, the ideal gas law for isentropic flowwith \({n_{\text {p}}}=\gamma =1.4\) was used.

Based on these two assumptions, they stated:which links the radial swirl velocity profile of a vortex to its radial density distribution. Bagai and Leishman solved the integral \(\int {V_{{{\theta }}}^2}/{r}\,\mathrm{d}r\) for the Vatistas1991 model (Vatistas et al. 1991) for \(n=\infty\), \(n=1\), and \(n=2\) and provided the equations for the respective density distributions according to Eq. (8).

Since the more recent Vatistas2015 model shows a better agreement with the PIV data of the current study, the respective integrals \(\int {V_{{{\theta }}}^2}/{r}\,\mathrm{d}r\) have been derived for \(n=1\) and \(n=2\), following the approach of Bagai and Leishman:

Vatistas2015, \(n=1\):Vatistas2015, \(n=2\):The solution for \(n=2\) uses the hypergeometric function (₂F₁) since a closed analytical solution is non-existent.

The two integrals can be used in combination with Eq. (8) to obtain the density distributions. The constants of integration can then be found using the boundary condition \(\rho (r) \rightarrow \rho _{\infty }\) for \(r\rightarrow \infty\), as also stated by Bagai and Leishman (Bagai and Leishman 1993).

The integral \(\int {V_{{{\theta }}}^2}/{r}\,\mathrm{d}r\) can also be computed numerically based on the PIV swirl velocity. An advantage of using the analytical integral of the best fitting vortex model is evident in the core region. A suitable fit based on the velocity outside of the particle void also provides reliable information inside the vortex core, despite a lack of sufficient particle images in this region.

To link the PIV velocities to the shift measured with BOS, two steps are necessary:

The shift obtained from one of the two BOS cameras for a test case at a vortex age of \(\varPsi _{{\text {v}}}= {25}^\circ\) during upstroke (\(t/T=0.42\)) is shown in Fig. 22 over the vortex radius \(r\), normalized with the vortex core radius \(r_{\text {c}}\). The gray lines represent the 200 instantaneous measurements and the black line indicates the resulting average shift. The theoretical shift, calculated based on the analytical integral of the vortex model Vatistas2015 (\(n=1\), see Eq. (9)), is shown as dotted blue graph. The orange dotted graph represents the theoretical shift calculated with the numerical integral of the PIV data. Both curves follow the assumption of isentropic flow in the vortex core, as proposed by Bagai and Leishman (Bagai and Leishman 1993). The expected agreement between the numerical and the analytical integrals can be seen. Due to the particle void, which has the size of \(r_{\text {void}}=0.71r_{\text {c}}\) in the chosen case, no data for the numerical integration are available below this radius.

The comparison of the predicted shift (dotted lines) and the measured data from BOS reveals an underprediction of the measured data, which is observed for all measured cases in the current study. By using only the experimental data, it is not clear whether the underprediction is caused by the calculation of the density distribution based on the approach by Bagai and Leishman (Bagai and Leishman 1993) (step 1) or by the calculation of the shift utilizing the geometric sensitivity factor S (step 2). To fill this gap, corresponding CFD data are used.

The pressure distribution inside a vortex from CFD is depicted in Fig. 23a over the vortex radius normalized with the vortex core radius \(r/r_{\text {c}}\). The expected pressure deficit in the vortex core is present. Figure 23b shows the resulting polytropic exponent \({n_{\text {p}}}\) derived by Eq. (7), based on the pressure and the density from the simulation (in black). It can be seen that the polytropic exponent in the vortex core is well below the isentropic case (\({n_{\text {p}}}= \gamma = 1.4\)) and tends toward \({n_{\text {p}}}= 1\) (the isothermal case) and even a bit lower. The temperature distribution from CFD was calculated based on the pressure and the density from CFD, following the ideal gas law. The data given in Fig. 23c indicate an increase in temperature inside the vortex core (black line). The respective density of the simulation is plotted in Fig. 23d, also in black.

Following the assumption of isentropic flow by Bagai and Leishman (1993), the blue curves in Fig. 23 are obtained. An exponent of \({n_{\text {p}}}=\gamma =1.4\) leads to a reduction of the temperature in the vortex core, based on the pressure distribution from CFD. Furthermore, the density deficit is underpredicted. Assuming isothermal flow (\({n_{\text {p}}}=1\)) instead, the temperature is constant by definition, and the density deficit, as predicted by the CFD, is met more closely. The assumption of isothermal flow, thus, represents the CFD simulation better than isentropic flow. For the computation of a vortex core, Yildirim and Hillier (2012) also assumed isothermal conditions in the vortex core and switched to isentropic flow outside the core.

Due to the finite accuracy of the CFD simulations, the small deviations in the polytropic exponent \({n_{\text {p}}}\), as seen in Fig. 23b, cannot be interpreted. Nonetheless, the tendency toward \({n_{\text {p}}}= 1\) inside the vortex and the expected converging toward \({n_{\text {p}}}= \gamma = 1.4\) outside the vortex are observed for all cases.

When applying the assumption of isothermal flow to the relation between velocity and density by Bagai and Leishman (Eq. (8)) and deriving the theoretical BOS shift from the PIV data in the same way as for the data shown as dotted lines in Fig. 22, the shift distribution indicated as dashed lines in the same figure is obtained. It can be seen that the maximum shift is matched more closely compared to the isentropic case, which supports the argument that the assumption of isothermal flow is transferable from the simulation to the measured data.

The vortex core radius \(r_{\text {c}}\) is defined as the radial location of the maximum swirl velocity. The core radius can also be assessed using the shift distributions measured with BOS. This method is independent from a particle void, unlike PIV, and can be performed with only one camera when the position of the vortex is known.

Figure 24 shows the radial locations of the maximum shift \(r_{\text {max shift}}\) for all measured cases over the core radius \(r_{\text {c}}\) extracted from the PIV swirl velocity distributions. Figure 24a shows the values for \(r_{\text {max shift}}\) obtained by Camera 1 of the BOS system and Fig. 24b the values obtained by Camera 2.

In a numerical assessment of the shift and velocity distributions associated with both models (Vatistas1991 and 2015) and both n values, factors between 0.69 and 0.76 were found to link the core radius \(r_{\text {c}}\) and the radius of maximum shift \(r_{\text {max shift}}\). In case of the Vatistas2015 model with \(n=1\), the relation between the core radius \(r_{\text {c}}\) and the radius of maximum shift \(r_{\text {max shift}}\) is found to beTherefore, a linear trend in Fig. 24 with a slope of 0.70 is expected and indicated as gray dashed line. It can be seen that for \(r_{\text {c}}>{5}\,\hbox {mm}\), the values for \(r_{\text {max shift}}\) are found to be above the indicated line, but following the expected slope. For smaller \(r_{\text {c}}\), the observed radial location of the maximum shift \(r_{\text {max shift}}\) remains constant and, thus, deviates from the expected trend. Around this value, the resolution limit of BOS is reached.

In addition to the observed data, the estimated limits of \({\text {CoC}}/2\) and the feature size, which led to zero contrast during the evaluation of the 1951 USAF resolution test chart (see Fig. 7), are also shown in both figures. A relation between the observed resolution limits of both cameras and the deviation of the measured sizes with BOS from the linear trend is evident.