2.1. PIV

The axial and radial velocities in the vertical centreplane behind the afterbodies were measured with a two-dimensional, two component PIV method. Scarano & van Oudheusden (Reference Scarano and van Oudheusden2003) have shown that PIV methods – due to their nearly non-intrusive character (Tropea, Yarin & Foss Reference Tropea, Yarin and Foss2007) – are suitable for determining the velocities in a supersonic wake without significant alteration of the wake properties.

To seed the flow, a mixture of water and propylene glycol was evaporated and injected into the free stream upstream of the nozzle supply chamber. In order to increase the number of seeding particles in the flow region directly behind the afterbodies, additional seeding was needed. Therefore, a 2 mm-diameter central orifice in the model base was connected with a tube to the outside of the wind tunnel. The pressure difference between ambient conditions and the model base led to a mass flux into the recirculation region. This central mass flux was seeded with tobacco smoke primarily consisting of vaporized water, glycerol and glycol as well as solid particles (Rodgman & Perfetti Reference Rodgman and Perfetti2013). The effect of the seeding at the base centre on the measurement results is described in Appendix A.1.

The seeding particles in the flow were illuminated by a double-pulse frequency-doubled Nd:YAG laser system. The laser system was working with a repetition rate of 7 Hz and the time delay between the two successive pulses was 1 $\mathrm {\mu }$s. Each of the pulses had a duration of 10 ns, thus allowing the recording of instantaneous particle positions. The laser sheet in the vertical centreplane behind the afterbodies had an approximate thickness of 500 $\mathrm {\mu }$m. The scattering of the light by the particles in the flow was recorded by a camera positioned perpendicular to the laser sheet, recording an individual image of $2048 \, \textrm {pixel}\times 2048 \, \textrm {pixel}$ resolution for each of the laser pulses.

During each wind-tunnel blow-down, 100 image pairs were recorded. The axial and radial velocities were calculated for each image pair with LaVision DaVis 10.1 (2020) using an equidistant evaluation grid with a node distance of 16 pixels and a spatial resolution of 38 evaluation nodes per base diameter $D$. After two initial passes using correlation windows with an edge length of 48 pixels, four iterative passes using correlation windows with an edge length of 32 pixels and a Gaussian weighting function were performed.

Figure 5 shows the time-average axial and $y$-Cartesian velocities, $\overline {U_x}$ and $\overline {U_y}$, normalized with the free stream velocity $U_\infty$ to

(2.1a,b)\begin{equation} U_{x,PIV}^\star{=}\overline{U_x}/U_\infty\quad\text{and}\quad U_{y,PIV}^\star{=}\overline{U_y}/U_\infty \end{equation}

for the cylindrical afterbody without fins. The 95 % confidence uncertainty of the measured average flow velocities was calculated as described in Appendix A.1 to 1 % and 2 % of the free stream velocity in the outer and the inner wake, respectively.

Figure 5.

Time-averaged relative velocities in the wake of the non-finned reference model resulting from PIV measurements – also plotted are the time-averaged streamlines, the expansion at the base corner (black dashed line), the recompression region (black dash–dotted line), the shock originating at the nozzle/test-section junction (black dotted line) and the zero-axial-velocity line $\overline {U_x}=0$ (white solid line). (a) Axial velocities and (b) $y$-Cartesian velocities.

The PIV measurements for the non-finned cylinder are compared in table 2 with the experimental results of Leopold (Reference Leopold1993), Herrin & Dutton (Reference Herrin and Dutton1994a) and Kirchner et al. (Reference Kirchner, Favale, Elliot and Dutton2019). The axial position $x_{RSP}$ of the RSP at $x/R\approx 2.86\pm 0.04$ was determined by the change of sign of the axial velocities along the central axis. The present experimental results for the normalized length of the recirculation region $x_{RSP}/R$, defined by the position of the RSP and the base radius $R$, agree with the results of Leopold (Reference Leopold1993) within their uncertainties. In contrast to that, the results of Herrin & Dutton (Reference Herrin and Dutton1994a) and Kirchner et al. (Reference Kirchner, Favale, Elliot and Dutton2019) show a shorter length of the recirculation region since they had been carried out at a higher free stream Mach number of 2.46 and 2.49, respectively (Murthy & Osborn Reference Murthy and Osborn1976). The normalized positions of the maximum upstream velocity, $x|_{\min{U_x^\star}}/x_{RSP}$, however, are in good agreement with the PIV measurements of the present study. Additionally, the maximum reverse flow velocity, $\min {U_x^\star }$, of $-0.257\pm 0.007$ measured for the non-finned afterbody agrees well with the experimental data of Leopold (Reference Leopold1993), Herrin & Dutton (Reference Herrin and Dutton1994a) and Kirchner et al. (Reference Kirchner, Favale, Elliot and Dutton2019).

Table 2.

Comparison of the RSP position, the maximum, relative upstream velocity and its relative position determined by experimental means in axisymmetric wake flows at free stream Mach numbers of 2.0 and 2.5.

The wake pattern in figure 5 is of comparable symmetry as the results of Kirchner et al. (Reference Kirchner, Favale, Elliot and Dutton2019). This was achieved by a careful alignment of the model with the free stream using surface oil flow visualizations. For the finned models, the symmetry of the flow was of similar quality. Therefore, the resulting velocities for each image pair of the finned models were transformed from Cartesian coordinates $(x,y)$ to cylindrical coordinates $(x,r=|y|)$. Thus, an average of up to 800 instantaneous measurements was calculated from 400 image pairs recorded during four blow-downs for each of the finned models. For these finned models, the uncertainties of the average velocities – also based on a confidence level of 95 % – were determined according to Appendix A.1 to typically less than 1 % and 4 % of the free stream velocity in the outer and the inner wake, respectively.

2.2. Pressure measurements

2.2.1. PSP

The static pressure at the model surface was measured by applying PSP. This method is capable of determining the pressure distribution with a high spatial resolution, thus rendering it suitable to visualize the shock footprints caused by the fins.

For the present PSP measurements, platinum porphyrin molecules (PtTFPP) within an oxygen-permeable matrix were applied to the model surface. The electrons of the porphyrin complexes were excited to a higher energy level by a continuous light source of $(400\pm 15)$ nm wavelength. After the excitation, the electrons returned to their original energy level by either emitting a photon of a wavelength between 620 and 750 nm, or by transferring the excess energy to an oxygen molecule of the surrounding air flow. Hence, the intensity of the emitted light is dependent on the local oxygen concentration, which itself is proportional to the local pressure. Higher pressures result in a higher oxygen concentration, thus leading to a lower intensity of the emitted fluorescence. Therefore, the emitted light was imaged with a camera using a bandpass filter of $(650\pm 10)$ nm blocking the scattered light resulting from the excitation of the porphyrin molecules. The local pressure on the model surface was then evaluated using the Stern–Volmer relation

(2.2)\begin{equation} \frac{I_{ref}}{I}=A+B\frac{p}{p_{ref}}, \end{equation}

with the coefficients $A$ and $B$ being dependent on the present temperature and the used PtTFPP molecules (Stern & Volmer Reference Stern and Volmer1919). As a reference, the intensity $I_{ref}$ of the emitted light was recorded directly after each blow-down since the model temperature was then similar to the conditions during the blow-down. The reference pressure $p_{ref}$ at the model surface was then equal to the ambient pressure. The pressures $p$ during each wind-tunnel blow-down were evaluated from the measured fluorescence intensities $I$ during the individual wind-tunnel runs and the corresponding reference measurements (Martinez Reference Martinez2007). The temperature gradients due to the shock waves caused by the fins, and the changing excitability of the molecules caused by the ambient-air humidity (Tropea et al. Reference Tropea, Yarin and Foss2007) to which the PtTFPP molecules were exposed between the individual blow-downs, however, limited the usage of the obtained results to qualitative comparisons.

Figure 6 shows the time-average pressure distribution at the surface of the non-finned afterbody with a mean value of $0.66\times 10^5$ Pa. This value is 8 % higher compared with the previously mentioned static pressure measurements using pressure transducers at $x/R=-1.925$ and different azimuthal positions (Weidner et al. Reference Weidner, Hruschka and Leopold2019b) which are in good agreement with the expected pressures resulting from isentropic flow theory.

Figure 6.

Pressure-sensitive paint measurement at the model surface upstream of the base corner of the non-finned reference model.

2.2.2. Base pressure

The pressure on the model base was measured with miniature pressure transducers (Kulite 2014) integrated in the model base. The transducers were mounted at different radial positions behind 3 mm-deep orifices having a diameter of 1 mm. The orifices were distributed on the base as indicated in figure 7. One transducer was mounted on the central orifice with a larger diameter of 2 mm, also used for the particle seeding during the PIV measurements.

Figure 7.

Positions of the pressure measurement orifices at the model base (dimensions given in mm).

The pressure at the base $p_{b}$ was normalized with the free stream static pressure $p_\infty$, resulting in the relative base pressure $p_{b}^\star$. This way, the effect on the base pressure caused by the increase of the free stream pressure of up to 2 % during the typical blow-down duration of 40 s was corrected for.

Figure 8 shows the average of the measured normalized base pressures at various radial positions for the non-finned afterbody. The error bars indicate the uncertainty interval with a confidence level of 95 % as calculated in Appendix A.2. The measurement results show an average deviation of 3.5 % from the measurements of Leopold (Reference Leopold1993) using the same wind-tunnel set-up and a similar afterbody with a smaller diameter of 38.66 mm. The deviations between the present measurements and the results of Leopold (Reference Leopold1993), however, are of the order of magnitude of the average experimental uncertainty of 3.4 %. Since the deviations primarily originate from the temperature dependency of the transducers mounted in the model, the central base pressure was additionally measured with a transducer (GE Sensing 2007) mounted outside of the wind tunnel and connected to the orifice at the base centre using the tubing of the PIV seeding. The measurements of the external transducer and the model-integrated central transducer deviated by 1.6 %, thus corroborating the estimated error margin for the results obtained with the built-in transducers.

Figure 8.

Measured, time-averaged, radial base pressure profile of the non-finned reference model in comparison with the experimental results of Leopold (Reference Leopold1993).

3.1. Wall-shear stress visualizations

Figure 9 shows the wall-shear stress visualizations (Maltby Reference Maltby1962) of the finned afterbodies also used for the alignment of the models with the free stream flow. The flow pattern at the base of the afterbody with a fin-cant angle $\lambda$ of 0$^\circ$, shown in figure 9(a) and shown in the supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.465, is nearly axisymmetric, thus showing that the asymmetries in the free stream due to the rectangular nozzle shape had no relevance for the present study. The radial flow direction from the base centre to the base corner is typical for the classical supersonic turbulent wake flow (Herrin & Dutton Reference Herrin and Dutton1994a).

Figure 9.

Oil flow visualizations of the velocity field in the vicinity of the surface of afterbody models with different fin-cant angles $\lambda$ (Weidner et al. Reference Weidner, Hruschka, Rey, Leopold, Frohnapfel and Seiler2017): (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

The zones around the fins into which the oil could not intrude coincide with the footprints of the horseshoe vortices (Dolling & Bogdonoff Reference Dolling and Bogdonoff1980) illustrated in figure 10. Moreover, it is shown in figure 10 how the shock waves caused by neighbouring fin leading edges intersect between the fins. Afterward, the shock waves are interacting with the boundary layer at the fin surface, resulting in a local separation of the flow visible in figure 9(a) and movie 1.

Figure 10.

Illustration of the horseshoe vortices generated at the fin leading edges.

Figure 9(b) and movie 2 show a helical pattern on the base of the afterbody having fins with a cant angle, $\lambda$, of 16$^\circ$. The azimuthal deflection of the oil toward the axis is a result of the swirling flow motion introduced by the canted fins and transported into the flow region directly downstream of the afterbody. In addition to the azimuthal deflection, the radial direction of the flow adjacent to the base has changed as it is indicated by the oil accumulation at the base centre. Hence, figure 9(b) and movie 2 show a first experimental evidence of the changed wake flow structure due to swirl as it has been predicted by the numerical simulations of Hruschka & Leopold (Reference Hruschka and Leopold2015).

For the afterbody having a fin-cant angle, $\lambda$, of 32$^\circ$, the wall-shear stress visualization in figure 9(c) and movie 3 show an annular oil accumulation with $r/R\approx 0.8$ at the base. At the position of the oil accumulation, the flow adjacent to the model base separates from the base, resulting in a wake flow structure not yet described in the literature. Despite the fin-cant angle of 32$^\circ$, the oil pattern at the base does not show any circumferential deflection of the oil flow, thus suggesting low circumferential flow velocities in the wake compared with the afterbody with $\lambda =16^\circ$. This becomes even more evident when observing the displacement of the oil over time in the video recordings in movies 2 and 3.

3.2. Axial and radial flow velocities in the wake

Figures 11 and 12 show the measured time-average axial and radial flow velocities normalized by the free stream velocity in the wake of the finned afterbodies. The axial velocity downstream of the fins is of the order of magnitude of the free stream velocity, thus indicating that the fins do not choke the flow. The axial velocity downstream of the fins of the afterbody with $\lambda =32^\circ$ shown in figure 11(c), however, is decreased by approximately 10 % compared with the afterbody with non-canted fins.

Figure 11.

Time-averaged, relative, axial velocities in the wake of the finned afterbody models resulting from PIV measurements – also plotted are the time-averaged streamlines, the fin leading-edge shocks (black solid line), the expansion at the base corner (black dashed line), the recompression region (black dash–dotted line), the shock originating at the nozzle/test-section junction (black dotted line), the zero-axial-velocity line $\overline {U_x}=0$ (white solid line), the counter-rotating vortices ((1), (2)) and the downstream-directed vortex tube (3) (Weidner et al. Reference Weidner, Hruschka, Rey, Leopold, Frohnapfel and Seiler2017). Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

Figure 12.

Time-averaged, relative, radial velocities in the wake of the finned afterbody models resulting from PIV measurements – also plotted are the time-averaged streamlines, the fin leading-edge shocks (black solid line), the expansion at the base corner (black dashed line), the recompression region (black dash–dotted line), the shock originating at the nozzle/test-section junction (black dotted line), the zero-axial-velocity line $\overline {U_x}=0$ (white solid line), the counter-rotating vortices ((1), (2)) and the downstream-directed vortex tube (3) (Weidner et al. Reference Weidner, Hruschka, Rey, Leopold, Frohnapfel and Seiler2017). Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

The velocity fields of figures 11(a) and 12(a) downstream of the afterbody with non-canted fins show a recirculation region that is typical for turbulent supersonic wakes. In comparison with the non-finned afterbody, oblique shock waves are present in the outer wake, resulting from the shock waves caused by the fin leading edges. The RSP for the afterbody with non-canted fins is located at $x/R=2.88\pm 0.02$. Hence, the non-canted fins result in a slightly larger recirculation region compared with the non-finned cylinder. This is due to the reduced Mach number at the base shoulder caused by the compression of the flow due to the presence of the fins, thus resulting in a higher relative base pressure (Lamb & Oberkampf Reference Lamb and Oberkampf1995). Hence, the expansion at the base corner is weaker and the deflection of the flow toward the central axis is less pronounced.

The maximum upstream velocity for the afterbody with non-canted fins is located on the central axis at $x/R=1.46\pm 0.05$, and thus 13 % closer to the base than for the non-finned afterbody. In addition, the magnitude of the normalized maximum upstream velocity of $0.262\pm 0.007$ measured for the afterbody with non-canted fins is 2 % higher than for the non-finned afterbody. The resulting increase of the upstream mass flux is due to the horseshoe vortices enhancing the entrainment of the fluid in the recirculation region by the shear layer (Bourdon & Dutton Reference Bourdon and Dutton2001; Janssen & Dutton Reference Janssen and Dutton2005).

Despite the change of up to 13 % in the position of the maximum upstream velocity, the flow fields of the non-finned afterbody and the afterbody with the non-canted fins are generally similar. Hence, the presence of the non-canted fins does not result in a substantial change of the wake structure.

For the afterbody with 16$^\circ$-canted fins, figure 11(b) shows that the flow detaching from the base corner does not reach the central axis. The outer flow becomes realigned with the axis of the flow field at $x/R\approx 1.5$ and $r/R\approx 0.6$. Downstream of the recompression the shear layer is, however, closer to the central axis than it has been when separating at the base corner. Hence, the conservation of angular momentum results in high azimuthal velocities of the fluid, decreasing the number of seeding particles close to the axis. Thus, it has not been possible to obtain reliable PIV results close to the axis for $x/R>3$, as is shown by the regions without measurement data in figures 11(b) and 12(b).

The adverse pressure gradient caused by the recompression shock redirects parts of the shear layer toward the base of the afterbody, as shown in figure 11(b). Thus, an upstream flow establishes in the wake of the afterbody with $16^\circ$-canted fins having a maximum normalized velocity of $-0.157\pm 0.007$ at $x/R=0.34\pm 0.05$. In contrast to the non-finned afterbody and the afterbody with non-canted fins, the maximum upstream velocity is not located at the axis but at a radial position of $r/R=0.66\pm 0.05$. Neglecting the azimuthal velocity component that could not be measured with the used PIV method, the upstream flow together with the downstream directed shear layer forms an outer toric vortex surrounding a counter-rotating inner toric vortex shown in the time-averaged flow fields of the figures 11(b) and 12(b). As it has also been shown in figure 9(b), the inner toric vortex results in a reversal of the radial flow direction at the base compared with the classical wake flow. As a result, the flow separates at the base centre leading to a central downstream-directed flow . The downstream-directed flow at the axis thus forms the vortex tube simulated by Hruschka & Leopold (Reference Hruschka and Leopold2015) and in § 5 of the present paper.

In contrast to the afterbody with $16^\circ$-canted fins, a fin-cant angle of $32^\circ$ as shown in figure 11(c) results in an upstream directed flow at the central axis. The normalized maximum upstream velocity of $-0.32\pm 0.01$ at $x/R=1.25\pm 0.10$ is again located on the axis. As shown later, this flow is at low density, thus limiting its importance. Only in the vicinity of the base a local region with a downstream-directed flow is present close to the axis caused by the local effects of the PIV seeding at the base centre. The shear layer downstream of the afterbody with $32^\circ$-canted fins is only weakly deflected toward the central axis and the recompression realigns the flow to the axis already at $x/R\approx 1$ and $r/R\approx 0.8$. Due to the small deflection angle toward the axis, the Mach number perpendicular to the recompression shock is lower than for the afterbody with $16^\circ$-canted fins, resulting in a lower adverse pressure gradient in the shear layer. Hence, most of the fluid in the shear layer is capable of overcoming the pressure gradient and hardly any fluid is directed back to the base. Thus, the shear layer convects downstream forming a vortex tube enclosing a low-momentum inner flow region. Similar to the classical wake, the upstream flow at the central axis is a result of the mass entrainment by the shear layer and the conservation of the mass in the inner-wake region. The highly swirling outer flow in conjunction with the low-momentum inner-wake region having a near-zero azimuthal velocity component bears similarity to the interaction of vortex tubes with shock waves in supersonic flow fields (Délery et al. Reference Délery, Horowitz, Leuchter and Solignac1984; Settles & Cattafesta Reference Settles and Cattafesta1993), as is further discussed in Appendix B.

In addition, figure 12 shows that the compression shocks caused by the flow deflection at the fins result in a stronger radial deflection of the outer flow with increasing fin-cant angle. For a fin-cant angle of $32^\circ$, the compression shocks themselves are not visible in figures 11(c) and 12(c) since the location of the detached shock is farther upstream for higher cant angles, as has been shown in figure 9.

3.3. Pressure measurements

3.3.1. Static pressure at the model surface

Figure 13(a–c) show the static pressure $p$ on the model surface normalized with the free stream pressure $p_\infty$ for the different fin-cant angles $\lambda$ measured with the PSP technique. With increasing fin-cant angle $\lambda$, the detached shocks upstream of the fin leading edges result in a higher pressure on the model surface and an upstream shift of the shock position, which is also visible in figures 9 and 12.

Figure 13.

Time-averaged pressure distribution, normalized with the free stream pressure $p_\infty$, at the cylindrical model surface and the fin surfaces of the afterbody models with different fin-cant angles obtained with the PSP method – $\unicode{x24B6}$ interaction of the fin leading-edge shock with the fin boundary layer, $\unicode{x24B7}$ detachment shock, $\unicode{x24B8}$ right-leg shock of the shock wave–boundary layer interaction caused by the fin leading-edge shock, $\unicode{x24B9}$ expansion due to the fin wake, $\unicode{x24BA}$ recompression due to the fin wake, $\unicode{x24BB}$ compression due to the fin leading edge, $\unicode{x24BC}$ expansion due to the fin leading edge. Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$; (d) $\lambda =0^\circ$ with cylindrical model extension; (e) $\lambda =16^\circ$ with cylindrical model extension; (f) $\lambda =32^\circ$ with cylindrical model extension.

The high pressure immediately upstream of the fin leading edge in figure 13(a) is due to the right-leg shock of the shock wave–boundary layer interaction (Dolling & Bogdonoff Reference Dolling and Bogdonoff1980). On the fin surface, the footprint of another shock wave–boundary layer interaction is visible resulting from the interaction of the fin leading-edge shock with the fin boundary layer. Downstream of this interaction, the boundary layer re-establishes, thus matching the pressure directly upstream of the base corner to the free stream pressure level.

In contrast to the model with $\lambda =0^\circ$ with its symmetrical flow around the fins, the cant angle of the fins results in an increased pressure on the windward side and a decreased pressure on the leeward side of the fins as shown in figure 13(b,c).

For the model with $\lambda =16^\circ$, the expansion and compression forming at the leading edge of two neighbouring fins interact with each other upstream of the fin trailing edges. Hence, the flow on the suction side of the fins is compressed and the flow on the pressure side is expanded. Thus, the pressure differences level off upstream of the base corner.

Since the fin-cant angle $\lambda =32^\circ$ is larger than the Prandtl–Meyer angle of $26^\circ$ for a flow Mach number of $2$, the flow separates on the suction side of the fins directly at the leading edge. Hence, the flow cross-section between the fins is reduced compared with the model with $\lambda =16^\circ$ and the detached shocks upstream of the fin leading edges increase in strength. This, in turn, results in the upstream shift of the separation shock footprint on the model surface shown in figure 13(c).

To investigate the effect of the flow between the fins on the pressure immediately upstream of the flow separation at the base corner, a 30 mm long cylindrical extension was attached to the different models. The extension had the same diameter as the models. As it is shown in figure 13(d–f), the model extension does not alter the flow between the fins.

In figure 13(d), the measured surface pressure distribution for the model with $\lambda =0^\circ$ with extension is shown. The local expansion downstream of the fin trailing edges is caused by the finite thickness of the fins. The average pressure downstream of the fin trailing edges ($0< x/R<0.5$) is 10 % lower than upstream of the fins.

For a fin-cant angle of $16^\circ$, the average pressure downstream of the fin trailing edges is 30 % lower than upstream of the fins, as shown in figure 13(e). This is due to the stronger detached shock upstream of the fins as well as due to azimuthal flow velocities resulting in a radial flow deflection, hence reducing the surface pressure. The shocks and expansions originating from the fin leading edges, which are reflected from the fins, result in the characteristic pressure pattern on the cylindrical model extension.

For the model with $\lambda =32^\circ$, figure 13(f) shows a decrease of the average pressure downstream of the fins of 50 % compared with the free stream pressure. The larger pressure decrease compared with the model with $\lambda =16^\circ$ results from the flow separation at the fin leading edges amplifying the detached shock upstream of the fins. In addition, the increase of the azimuthal velocities results in an enhanced radial flow deflection compared with the model with $\lambda =16^\circ$ which further decreases the pressure downstream of the fins.

Overall, the surface pressure measurements show that the canted fins decrease the static pressure in the plane of the fin trailing edges.

3.3.2. Base pressure

In figure 14, the measured radial base pressure profiles for the finned models are shown and compared with the non-finned model. The absolute measurement uncertainties – calculated as described in Appendix A.2 – have the same order of magnitude as for the non-finned model.

Figure 14.

Measured, time-averaged, radial base pressure profile of the finned afterbody models in comparison with the non-finned reference model (Weidner et al. Reference Weidner, Hruschka, Rey, Leopold, Frohnapfel and Seiler2017).

For the model with non-canted fins, the pressure decrease of 10 % in the fin trailing edge plane, shown in figure 13(d), is counteracted by the reduced Mach number at the base corner due to the presence of the fins. The lower Mach number at the base corner results in a higher base pressure compared with the pressure at the base corner (Lamb & Oberkampf Reference Lamb and Oberkampf1995). The effects of the decreased pressure at the base corner and the horseshoe vortices (Bourdon & Dutton Reference Bourdon and Dutton2001) are, however, larger, thus resulting in an overall base pressure decrease of 6 % compared with the non-finned model. Hence, the alteration of the base pressure is within the range of 5 % to 10 % given by Moore, Hymer & Wilcox (Reference Moore, Hymer and Wilcox1992) for the fins of the present study with a length-to-thickness ratio of 20 and the present free stream Mach number of 2.

The radial pressure profile of the model with $\lambda =16^\circ$ shows for $r/R>0.5$ a pressure decrease of 50 % compared with the model with non-canted fins, resulting from the decreased pressure upstream of the base corner. In addition, the base pressure profile for the model with $\lambda =16^\circ$ shows a radial pressure gradient at $r/R<0.5$ caused by the azimuthal velocities $U_\varphi$ in the wake, visualized in figure 9(b), and hence resulting in a pressure minimum at the base centre. This pressure distribution is similar to those found in gas centrifuges (Kemp Reference Kemp2009), which can be described by

(3.1)\begin{equation} \frac{\partial p}{\partial r}=\rho \frac{U_\varphi^2}{r}. \end{equation}

Compared with the model with $\lambda =16^\circ$, the average base pressure for the model with a fin-cant angle of 32$^\circ$ has the same order of magnitude. The base pressure is decreased by 50 % compared with the model with non-canted fins – in comparison with the model with $\lambda =16^\circ$, however, there is no distinct radial pressure gradient. This is in agreement with the observations in figure 9(c) indicating low azimuthal velocities in the wake compared with the pronounced helical pattern for the model with $\lambda =16^\circ$.

3.4. Summary of experimental results

In the present experiments, the three distinctively different wake flow structures, as sketched in figure 15, were observed. The schematics are constructed by a synthesis of the individual experimental results and will be backed up by the results of the numerical simulations in §§ 5 and 6. The movies 4 to 6 show animations of the simulated, time-averaged flow fields for the different model configurations.

Figure 15.

Illustrations of the time-averaged wake structures for the different fin cant angles $\lambda$, as observed in the experiments and backed up by the time-averaged results of the numerical simulations shown in § 4 and animated in the movies 4 to 6; the colour of the ribbons indicates the flow velocity. Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

For a fin-cant angle $\lambda$ of $0^\circ$, the time-average flow field illustrated in figure 15(a) corresponds to the classical supersonic wake flow of a non-finned axisymmetric afterbody. In comparison with the experimental data for the non-finned afterbody, the base pressure is decreased by 6 % due to the presence of the fins.

Figure 15(b) shows a sketch of the wake structure observed for the model with $\lambda =16^\circ$. The swirling flow detaches at the base corner and is deflected toward the central axis due to the lower pressure at the base compared with the free stream. The centrifugal forces, present in a non-inertial reference frame rotating with the fluid around the central axis, increase when the flow approaches the central axis. Thus, the streamwise pressure gradient resulting from the radially converging flow also increases due to the centrifugal forces. As a result of this streamwise pressure gradient, a part of the shear layer is redirected toward the base before reaching the central axis. This, in turn, results in the outer toric vortex shown in figure 11(b). The counter-rotating, inner toric vortex results in the observed change of the radial flow direction at the base compared with the classical wake flow due to the vanishing centrifugal forces in the vicinity of the base surface. The additional circumferential flow velocity leads to the helical pattern on the base visualized in figure 9(b) and illustrated in figure 15(b). The flow detaches at the base centre, thus resulting in a downstream-directed central vortex tube with low axial momentum.

The sketched flow field for $\lambda =32^\circ$ in figure 15(c) shows a flow detaching at the base corner which is only marginally deflected toward the central axis as observed in the PIV measurements. In contrast to lower fin-cant angles, the streamwise pressure gradients are hence negligible and most of the fluid is capable of flowing downstream without redirection to the base. The forming shear layer entrains parts of the inner wake resulting in a downstream directed swirling flow. The conservation of mass in the inner wake thus induces a central upstream flow. At the base centre, the upstream flow is redirected toward the base corner. The low interaction between inner and outer wake leads to small circumferential flow velocities at the base. This, in turn, explains the barely observable radial pressure gradient in comparison with the model with $16^\circ$-canted fins. The radially deflected central upstream flow separates from the base before reaching the base corner resulting in the characteristic annular oil accumulation shown in figure 9(c).

The conducted measurements provide experimental proof for the change of the wake structure dependent on the swirl rate introduced to the flow by the fins with the different cant angles $\lambda$. The change of the wake structure, measured for the model with $\lambda =16^\circ$, has been observed before only in numerical simulations (Hruschka & Leopold Reference Hruschka and Leopold2015). In contrast to that, the change of the wake structure which was experimentally observed for the model with $\lambda =32^\circ$ has not yet been described in the literature. Laminar, incompressible stability theory calculations (Jiménez-González et al. Reference Jiménez-González, Sevilla, Sanmiguel-Rojas and Martínez-Bazán2014) propose a similar change of the wake flow structure, however, referring to the formation of two stagnation points along the central axis. The differences compared with the currently described structure are attributed to the vast importance of compressibility and possibly Reynolds number effects.

5.1. Pressure at the model surface

First, the numerical results for the finned configurations were compared against measured surface-pressure distributions around the fins. The normalized pressure distributions for the different fin-cant angles $\lambda$ in figure 18(a–c) show a good agreement of the simulated positions of the separation shocks in front of the fin leading edges and the measured positions in figures 13 and 18(d–f), respectively. The numerical simulations are also capable of reproducing the interactions of the fin leading-edge shocks with the fin boundary layer as observed experimentally in figure 9. However, quantitative comparisons in the vicinity of the shocks are not feasible due to the uncertainties of the PSP technique discussed in § 2.2.1.

Figure 18.

Numerical results for the time-averaged pressure distribution, normalized with the free-stream pressure $p_\infty$, at the cylindrical model surface and the fin surfaces compared with the PSP results already shown in figure 13 – $\unicode{x24B6}$ interaction of the fin leading-edge shock with the fin boundary layer, $\unicode{x24B7}$ detachment shock, $\unicode{x24B8}$ right-leg shock of the shock wave–boundary layer interaction caused by the fin leading-edge shock, $\unicode{x24B9}$ expansion due to the fin wake, $\unicode{x24BA}$ recompression due to the fin wake, $\unicode{x24BB}$ compression due to the fin leading edge, $\unicode{x24BC}$ expansion due to the fin leading edge. Here (a) $\lambda =0^\circ$ (DES); (b) $\lambda =16^\circ$ (DES); (c) $\lambda =32^\circ$ (DES); (d) $\lambda =0^\circ$ (PSP); (e) $\lambda =16^\circ$ (PSP); (f) $\lambda =32^\circ$ (PSP).

The simulated sting surface pressure of $0.59\times 10^5$ Pa is 3 % lower than the pressure measured at $x/R=-1.925$ at the model surface. Hence, the deviation is within the range of the numerical and experimental uncertainties.

5.2. Axial and radial velocity field

The figures 19 and 20 compare the measured and simulated axial and radial velocity distributions, respectively, in the central plane behind the finned afterbody configurations.

Figure 19.

Comparison of the time-averaged, relative, axial velocities in the wake of the finned afterbody models obtained by numerical simulations with the results of the PIV measurements – also plotted are the time-averaged streamlines, the fin leading-edge shocks (black solid line) as well as their reflections from the wind-tunnel wall (black spaced line), the expansion at the base corner (black dashed line), the recompression region (black dash–dotted line), the shock originating at the nozzle/test-section junction (black dotted line), the zero-axial-velocity line $\overline {U_x}=0$ (white solid line), the counter-rotating vortices ((1),(2)) and the downstream-directed vortex tube (3). Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

Figure 20.

Comparison of the time-averaged, normalized, radial velocities in the wake of the finned afterbody models obtained by numerical simulations with the results of the PIV measurements – also plotted are the time-averaged streamlines, the fin leading-edge shocks (black solid line) as well as their reflections from the wind-tunnel wall (black spaced line), the expansion at the base corner (black dashed line), the recompression region (black dash–dotted line), the shock originating at the nozzle/test-section junction (black dotted line) and the zero-axial-velocity line $\overline {U_x}=0$ (white solid line). Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

For $\lambda =0^\circ$, the numerical and experimental results are in good agreement as shown in figures 19(a) and 20(a). The discrepancy of 3 % in the length of the recirculation region between the numerical and experimental results is within the experimental and numerical uncertainties and the wake structure is unchanged compared with the non-finned afterbody. The numerical results of Forsythe et al. (Reference Forsythe, Hoffmann, Cummings and Squires2002), Kawai & Fujii (Reference Kawai and Fujii2005) as well as Barone & Roy (Reference Barone and Roy2006) for the experimental set-up of Herrin & Dutton (Reference Herrin and Dutton1994a) show the same tendency of overpredicting the length of the recirculation region, which is attributed to the transition between modelled and resolved turbulent structures at the base corner typical for hybrid RANS/LES (where LES is an abbreviation of large eddy simulation) schemes (Forsythe et al. Reference Forsythe, Hoffmann, Cummings and Squires2002; Simon et al. Reference Simon, Deck, Guillen and Sagaut2006, Reference Simon, Deck, Guillen, Sagaut and Merlen2007). Overall, as can be seen in the radial profiles of the flow velocities shown in figure 21(a–e) for different axial positions, the numerical and experimental results for the model with $\lambda =0^\circ$ do not only agree qualitatively but also show a fair quantitative agreement except for minor differences in the shear layer.

Figure 21.

Radial profiles of the axial and radial velocities at the relative axial positions $x/R$ of 0.5, 1.5, 2.5, 3.5 and 4.5 for the different finned afterbody model configurations.

In figures 19(b) and 20(b), it is shown for the afterbody with a fin-cant angle of $16^\circ$ that the experimentally observed changes of the wake flow structure are well reproduced by the numerical simulations. The numerical results also clearly show the counter-rotating toric vortex pair in the near wake. The difference of the measured axial velocities close to the axis and the simulated value visible in figure 21(f) could be attributed to a local effect of the PIV-seeding mass flux injected from the base centre. Farther downstream, the differences are smaller, as shown in figure 21(g), and of the order of the numerical and experimental uncertainties as in figure 21(h).

For $\lambda =32^\circ$, figure 21(k) shows distinct differences between the measured and simulated axial velocities close to the base. As simulations including base seeding have shown (Weidner Reference Weidner2020), these are much larger than expected from bias due to the central seeding at the base. More likely, the still RANS-like behaviour in the simulated shear layer close to the base could have resulted in less mixing between the inner and the outer wake, thus leading to a less homogeneous distribution of the flow quantities. As observed for the lower fin-cant angle, these differences are less pronounced at axial positions farther downstream, where the effects caused by the unphysical transition of the turbulence modelling become less important. Figure 20(c) also shows that the region of radial velocities toward the axis between the expansion and recompression of the free stream is smaller compared with $\lambda =16^\circ$. As will be shown later, this is due to the higher azimuthal velocities directly downstream of the fins, preventing the flow from converging toward the axis immediately after the separation from the base corner. In figure 22, the weak shocks originating from the junction between the nozzle and the test section as well as the stronger shocks generated by the fin leading edges are illustrated. The fin leading-edge shocks are reflected from the wind-tunnel walls, and hence are redirected toward the central stream. The reflected shocks are located outside of the PIV measurement region, and thus can only be observed in the numerical results. The numerical results indicate an upstream shift of the reflected shocks for higher fin-cant angles, since a higher cant angle results in stronger shocks with larger shock-wave angles in the vicinity of the fins. Furthermore, figure 20(c) shows that the reflected shocks interact with the trailing wake for $x/R>8$ resulting in a radial flow deflection toward the central axis. A part of the low-momentum downstream flow close to the axis is not able to overcome the resulting pressure gradient, and thus is reversed back toward the base.

Figure 22.

Schematic of the compression shocks originating at the fin leading edges that are reflected from the wind-tunnel walls as well as of the shocks originating at the junction between the nozzle and the test section.

5.3. Wall-shear stresses

The simulated wall-shear stresses $\tau _{w}$ at the base are shown in figure 23. The surface vectors of the wall-shear stresses agree well with the qualitative oil flow visualizations in figure 9. The time- and azimuthal-average of the radial position of the simulated minimum wall-shear stress at the base of the model with $\lambda =32^\circ$ at $r/R\approx 0.87$ is also in fair agreement with the position of the annular oil accumulation in the experiments.

Figure 23.

Numerical results for the time-averaged wall-shear stresses at the base of the finned afterbody models. Here (a) $\lambda =0^\circ$; (b) $\lambda =16^\circ$; (c) $\lambda =32^\circ$.

Quantitatively, the numerical results show an increase of the wall-shear stresses by an order of a magnitude for the configuration with $\lambda =16^\circ$ compared with both the model with non-canted fins and $32^\circ$-canted fins. This is due to the additionally present circumferential flow velocities in the vicinity of the base, hence imposing additional wall-shear stresses compared with the other configurations.

5.4. Base pressure

In figure 24, the simulated base pressure profiles for the different fin configurations are compared with the experimental data. In comparison with the non-finned configurations, the numerical results for the model with non-canted fins show a pressure decrease of 6 %, in agreement with the experimental findings. For the configurations with fin-cant angles $\lambda$ of $16^\circ$ and $32^\circ$, the base pressure is decreased overall by more than 50 %, also showing good agreement with the experimental data.

Figure 24.

Comparison of the measured, time-averaged, radial base pressure profiles with the simulated, azimuthal and temporal average of the radial pressure profiles.

In addition to the overall base pressure decrease, the numerical results for the model with $\lambda =16^\circ$ indicate qualitatively the same radial base pressure gradient as observed in the experiments. The uncertainties of the numerical simulations $\pm \sqrt {\text {GCI}_{p_{b}}^2+(2\sigma _{\varphi ,p_{b}})^2}$, indicated in figure 24 as shaded areas, are a combination of the estimated discretization error expressed as grid convergence index $\text {GCI}_{p_{b}}$ (Roache Reference Roache1994), discussed in Appendix A.3, and the standard deviation $\sigma _{\varphi ,p_{b}}$ of the simulated azimuthal base pressure distribution. The azimuthal inhomogenities of the flow field at larger radii, visible in figures 9(b) and 23(b), result in an increased standard deviation $\sigma _{\varphi , p_{b}}$, thus locally increasing the uncertainties of the simulated mean base pressure of the $\lambda =16^\circ$ model. At smaller radii, as well as for the other fin configurations, the uncertainty of the numerical results is dominated by the discretization error expressed by the grid convergence index $\text {GCI}_{p_{b}}$ which is described in detail in Appendix A.3.

For $\lambda =32^\circ$, the radial pressure gradient observed for a fin-cant angle of $16^\circ$ vanishes together with the circumferential flow velocities in the vicinity of the base, as has been shown in figure 23(c). The discrepancies between measured and simulated values are attributed to the shortcomings in shear layer modelling described in § 1 and also observed in figure 21.