3.1.1. Set-up

To visualize the flow field, the two spheres are held in position with profiled supports, which do not obstruct the flow. Sphere $S_1$ is maintained with a vertical support fixed in a rotary system to allow the sphere to be removed from the flow in order to record background images. Sphere $S_2$ is maintained with a horizontal support coming from the rear of the sphere so as not to be intrusive. This support is fixed on a triaxial displacement system so that $S_2$ can be moved in the wake of $S_1$, and also removed from the flow for the recording of background images.

The low density of current experimental conditions does not allow the application of common techniques such as schlieren or particle image velocimetry to visualize and analyse the flow field. A technique used less commonly but adapted to our experimental conditions is used in this work. It is the technique of the glow discharge based on a cathode to ionize molecules. In our experiment, the cathode was made of copper and was placed upstream of the models around the exit of the nozzle, as shown in figure 2. It consists in applying an electrical discharge on a cathode placed near the flow. A negative voltage $-$1 kV is applied on the copper ring, inducing current 4 mA. The polarization is low enough not to modify the nature of the flow (Coumar & Lago Reference Coumar and Lago2017). With this method, a local increase in density induces an increase in the local emitted luminous intensity. Since a shock wave is a compression of molecules, the glow discharge makes it possible to determine visually the shock waves of the spheres, by analysing the brighter region.

This method generates a volumetric illumination of the flow. Consequently, all the images taken are the result of luminous integration in the line of sight of the eye or of any recording tool. So even if the glow discharge technique can determine shock waves, the local density cannot be calculated with a simple scaling law.

Images were recorded with a Kuro CMOS camera using back illumination. The camera is equipped with a VUV objective lens that gives a resolution 154 $\mathrm {\mu }$m px$^{-1}$. For each relative positioning of the spheres ($S_1$ and $S_2$ fixed during acquisition), a set of 200 raw images was recorded, with exposure time 60 ms. An equivalent set of images without the models in the flow was recorded as background image. For each position, images of the flow are averaged and divided by the mean background image. This processing allows us to reduce the noise of images and avoid luminous gradient due to the experimental set-up. The resulting normalized image, shown in figure 3(a), presents an improved luminous contrast of images that allows the analysis of shock waves.

Figure 3. Detection of a sphere thick shock wave. (a) Normalised image of the flow around a single sphere; (b) detection method on a pixel line; (c) result of shock-wave detection on the entire image.

3.1.2. Shock-wave detection

Images give important information, in particular, on shock-wave shapes. In our study, the detection of shock waves is a key step to determine the type of shock/shock interferences that occur. This detection is based on the luminous intensity of the images. Kovacs et al. (Reference Kovacs, Passaggia, Mazellier and Lago2022) proposed the Fourier self-deconvolution (FSD) method, which was developed and validated for our specific experimental conditions. This method is applied to the present work. As illustrated in figure 3(b), the luminous intensity in analysed for each horizontal line of pixels, and allows us to define three regions due to the thickness of the shock wave: the maximal intensity, the FSD maximum, and the maximum of the FSD derivative. In what follows, the three regions will be named, from upstream to the surface of the sphere, as: foot of shock (FS), middle of shock (MS), and boundary layer (BL). Concerning this last region, the maximum luminosity is representative of the denser region. Close to the surface of the sphere, particularly at the front half, this denser region corresponds to the boundary layer. Outside this area, this term of BL is inappropriate. However, it will be more convenient to call it BL in the rest of this paper. Note that the region between the FS and the BL is called ‘middle’ but obviously, it is not located at the centre. Applying the method on each horizontal line of pixels of the image, we obtain three well-plotted regions corresponding to the regions described previously, as shown in figure 3(c).

The analysis of the images, and more specifically the intensity values, can also provide important information to locate regions of stronger interactions. Figure 4 shows the four main points that will be used to deepen the knowledge on the effect of shock/shock interferences in supersonic rarefied flows. Thanks to the shock-wave detection, the intersection point $P_i$ between the middle shock of the two spheres is found, where $P_i$ is characterized by $d_i$ and $\varTheta _i$. Throughout this work, $\varTheta _i$ will be used as the reference parameter to reflect the relative positions of the spheres. Concerning the other three points of interest, their locations are based on the luminous intensity of the image. By finding the point of maximum luminosity in the intersection area, $P_{ml}$, the maximum local density can be found. The last two points, $P_{0_1}$ and $P_{0_2}$, concern the maximum luminosity on the horizontal line passing by the nose of the spheres. Here, $\varDelta _{BL_1}$ and $\varDelta _{BL_2}$, respectively, give the boundary layer stand-off distances of $S_1$ and $S_2$. In addition to location information, these three points contain intensity values that make it possible to compare the level of density of each type of shock/shock interference. All the data from $P_{0_1}$ will serve as normalizing parameters.

Figure 4. Detection of points of interest: intersection point between $S_1$ and $S_2$ middle shock ($P_i$), point of maximum luminosity in the intersection area ($P_{ml}$), and points of maximum luminosity along the stagnation line of $S_1$ and $S_2$ ($P_{0_1}$ and $P_{0_2}$).

3.2.1. Aerodynamic balance

The aerodynamic balance used in this investigation was designed and experimented by Noubel & Lago (Reference Noubel and Lago2021). It was developed especially for low-density flow conditions, such as those in the MARHy wind tunnel, where the force values are estimated to be between 1 mN and 1 N. The balance is a sting type and comprises two modules to measure drag and lift forces.

The drag and lift forces are required for the following sphere ($S_2$). It is thus screwed until its centre of gravity on the sting of the aerodynamic balance. The balance is positioned horizontally in the flow and fixed to a triaxial displacement system as presented in figure 5.

Figure 5. Forces balance set-up. Pictures of a measurement of (a) the total forces, and (b) the residual forces.

The balance is positioned horizontally to avoid gravity effects, as explained by Noubel & Lago (Reference Noubel and Lago2021). Thus lift forces are measured in the $\boldsymbol {y}$-direction, so the second sphere will be displaced in the $\boldsymbol {y}$-direction and not in the $\boldsymbol {z}$-direction as for the other experiments. Since the flow in the core is isotropic, the change in direction does not modify the physics of the shock/shock interferences or the physical values. The positioning is controlled with a camera placed on top of the test chamber.

Sphere $S_1$ and a plate are placed in a rotary system so that the sphere is in either the flow (figure 5a) or the plate (figure 5b). The usefulness of this system will be explained in the next paragraph. The first sphere is placed as described in § 3.1 (figure 5a), but this time, to keep the same displacement area for the second sphere, $S_1$ off-centre from the middle of the nozzle is in the $\boldsymbol {y}$-direction ($Z=0$ mm, $Y=-30$ mm).

For each position of $S_2$, two measures have to be realized: with $S_1$ in the flow (figure 5a), and with a plate hiding $S_2$ from the flow (figure 5b). The first measurement records the total drag and lift forces, including residual forces induced by the suction of the pumping group, or vibration from the pumping group. The second measurement records only the residual forces. The acquisition time, both with and without the plate, is 10 s, with acquisition frequency 1000 kHz. These two steps are repeated five times. For each module, drag and lift, values are averaged and the residual force values are subtracted from the total force to obtain the real drag and lift forces.

3.2.2. Swinging sphere method

This method consists in suspending the second sphere by a wire system so that it can move freely according to the surrounding flow. The angle of deviation of the sphere will depend on the force acting on the sphere, the measurement of which allows the aerodynamic force to be deduced (Cardona & Lago Reference Cardona and Lago2022). As shown in figure 6, the swinging sphere method was employed in two different ways. In figure 6(a), the suspended sphere is the second one, in order to measure the impact of $S_1$ over $S_2$. In figure 6(b), on the contrary, the first sphere is suspended, in order to measure the effect of $S_2$ on $S_1$.

Figure 6. Picture of the swinging sphere experiments: (a) method applied on $S_2$ supported by a moving support; (b) method applied with $S_1$ supported by a fixed support, and $S_2$ moving.

In both cases, the suspended sphere is drilled from side to side on an axis passing through its centre. A thin, inelastic wire of 0.07 mm diameter passes through the hole in the sphere and through the hole in a hollow tube that serves as a support to hold it like a swing. This tube is positioned horizontally, parallel to the $y$–$z$ plane so as to give a single degree of freedom along the $x$ axis. In order to avoid any movement in the $\boldsymbol {y}$-direction, the wire is glued to the sphere and to the stem. To obtain precise results, the sphere is positioned so that both sides of the wire have the same length, forming an isosceles triangle with the straight stem as base. This experiment is analysed from images obtained with the Kuro camera equipped with the VUV lens described in § 3.1. Despite the high stability of the flow and the equilibrium state reached by $S_2$, a slight movement of very small amplitude may affect the sphere. To improve the quality image definition, the images were recorded with very short acquisition times since the aim is to detect accurately moving spheres and wires. On the contrary, due to the size of each image, the duration of acquisition must not be too long so that the sets of images generated are not heavy files. A good compromise was to set the acquisition time to 20 ms per image.

For the experiment that consists in studying the movement of the sphere $S_2$ (figure 6a), $S_1$ is set as for the set-up of flow visualization described in § 3.1. The tube that holds the wires of $S_2$ is attached to a triaxial Cartesian displacement system, which aligns $S_2$ with $S_1$ in the same vertical plane that passes through $Y=0$. After the flow has been established and stabilized, $S_2$ is first placed in its initial position, right behind $S_1$ in the $\boldsymbol {x}$-direction. The experiment then consists in moving the tube vertically, or horizontally with the displacement system with constant speed 0.8 mm s$^{-1}$. As $S_2$ is moved in the wake of $S_1$, the sphere progressively balances itself so that it compensates for the forces induced by the flow.

For the second experiment (figure 6b), $S_1$ is suspended by the swinging system. The holding tube of $S_1$ is then fixed to the wall of the test chamber. The second sphere is held by the same moving system as the one described in § 3.1. Both spheres are placed in the same positions as described in § 2.3. In this experiment, the support of $S_1$ does not move, and only $S_1$ responds to the experienced forces. This time, it is $S_2$ that is displaced behind $S_1$ to study the impact of its positioning on the upstream sphere. For this experiment, some positions of $S_2$ were studied. Images were recorded with stationary positions for reasons that are explained in Appendix A.

The determination of the force value of the suspended sphere is based on the measurement of the angle of its wires with respect to the vertical, as detailed in Cardona & Lago (Reference Cardona and Lago2022). Assuming an equilibrium state of the sphere in each image (i.e. acceleration 0 m s$^{-2}$ is considered), Cardona & Lago (Reference Cardona and Lago2022) demonstrated that the drag force $F_x$ of the sphere is given by

(3.1)\begin{equation} F_x=m\, g \sin(\alpha). \end{equation}

With the manual detection of the wire, and automatic detection of the sphere, accuracy $\pm$1 px for both the centre of the sphere and the wire is assumed. Consequently, the real angle is given with estimated accuracy $\pm 0.4^{\circ }$, and the drag force of each position with accuracy $\pm$0.18 mN. This method was validated by comparing the drag forces of a single sphere obtained by Cardona & Lago (Reference Cardona and Lago2022) and literature results. In the present work, the swinging method is applied to an interactive sphere ($S_2$) to study the effects of interaction. Results are compared to those obtained with the aerodynamic balance in the same experimental conditions in Appendix B. These two methods show a good agreement, which allows us to be confident with the results obtained with the swinging method.

4.2.1. Stand-off distances

The shock-wave detection method allowed us to identify three regions of shock. On the horizontal line from the nose of the sphere (see the horizontal lines in the two schemes in figure 4), the stand-off distances ($\varDelta$) of the foot shock (FS), middle shock (MS) and boundary layer (BL) were determined. Figure 11 presents the evolution of these stand-off distances as a function of the angle of intersection $\varTheta _i$, and reveals the deformation of the shock waves on the horizontal line of $S_2$. These results were not observed in Cardona et al. (Reference Cardona, Joussot and Lago2021) where $X_2$, and thus the incident shock angle, varied, suggesting that the incident shock angle tends to influence the SSI, as has been found in many other studies in the continuous regime (Edney Reference Edney1968; Keyes & Hains Reference Keyes and Hains1973; Borovoy et al. Reference Borovoy, Chinilov, Gusev, Struminskaya, Délery and Chanetz1997; Boldyrev et al. Reference Boldyrev, Borovoy, Chinilov, Gusev, Krutiy, Struminskaya, Yakovleva, Délery and Chanetz2001; Peng et al. Reference Peng, Luo, Han, Hu, Han and Jiang2020). Nevertheless, to investigate the influence of the incident shock angle under rarefied conditions, further study would be required. On the three graphs in figure 11, the black lines represent the mean value for stand-off distances of $S_1$, which do not change (with $\pm 0.154$ mm accuracy) whatever the vertical position ($Z_2$-coordinate) of $S_2$. Since $S_2$ has exactly the same diameter as $S_1$, $\varDelta _1$ also represents the stand-off distances that $S_2$ should have without any interactions.

Figure 11. Shock stand-off distances of $S_2$ according to the type of interference: (a) foot shock, (b) middle shock, and (c) boundary layer.

As observed, the stand-off distances for the sphere $S_2$ ($\varDelta _2$) are strongly impacted. Note that $\varDelta _{BL_2}$ is almost constant (figure 11c), except for type VI SSIs, where an increase can be observed. It corresponds to positions where $S_2$ enters the wake of $S_1$ more deeply, i.e. where the local mean free path also increases. This is in agreement with Rembaut, Joussot & Lago (Reference Rembaut, Joussot and Lago2020) and Akhlaghi & Roohi (Reference Akhlaghi and Roohi2021) who showed that an increase in the rarefaction level leads to an increase in $\varDelta$, and therefore to an increase in $\varDelta _2$ for type VI SSIs.

For all the five other types of interference, the analysis is based on $\varDelta _{FS_2}$ and $\varDelta _{MS_2}$. For type I SSIs, the stagnation line of $S_2$ is not impacted. The area of interaction affects only a small area of the lower part of the shock. Then, from type II to type III, $\varDelta _2$ increases strongly. From a detailed analysis of the schematics of SSI in figure 10, it seems that the shock wave of $S_1$ penetrates in between the surface of $S_2$ and its shock wave by going up, pushing it away at the nose. The range of angles of type IV SSIs seems to be a transitional area, where the stand-off distances suddenly decrease. This could be due to the particular impact where the two shock waves meet almost perpendicularly around the nose of $S_2$. This creates a V-shaped deformation of the shock, which seems pushed away from $S_2$ surface above its nose. But at the nose, the strong incident shock wave penetrates deeper into the shock wave of $S_2$, decreasing the stand-off distances of both its FS and MS. Types V and VI SSIs do not impact the shock wave shape much at the stagnation point. As discussed above, the increase in $\varDelta$ is due mostly to the increase in rarefaction level.

4.2.2. Distances between the interaction area and the surface of $S_2$

Figure 12 shows the points of interest $P_i$ and $P_{ml}$ that we identified in figure 4. Distances $d_i$ and $d_{ml}$ were determined on the $S_2$ shock wave for each angle of intersection $\varTheta _i$. Here, $P_i$ is located at the intersection between the middle of the shock waves so it refers to the middle shock region, and $P_{ml}$ is the most luminous point in the intersection area so it refers to the boundary layer region. Thus $d_i$ and $d_{ml}$ are, respectively, compared to $d_{MS_1}$ and to $d_{BL_1}$, the distances of the shock wave without SSI.

Figure 12. Distances of (a) the interaction point and (b) the most luminous point in the shock/shock interaction area from the surface of the sphere, according to the type of interference.

First, it is important to note that $d_i$ remains greater than or equal to its reference value ($d_{{MS}_1}$), while this is not the case for $d_{ml}$ as it is smaller than its reference value ($d_{{BL}_1}$) from types III to VI SSI. This means that in the intersection area, the middle of the shock is mostly pushed away from the surface of $S_2$, while the boundary layer is mostly pushed towards it. Consequently, in this area, shock waves are thicker and more diffuse, increasing the viscous effect at the wall of the sphere.

Also, another remarkable feature of the curves in figure 12 is the V-shape that appears around the interference angle $0^{\circ }$. This area shows a very localized increase in the stand-off distance for both the shock interaction and the point of greatest brightness that reflects the point of greatest density. This feature can also be observed in the specific shape of the type IV SSI shown in figure 10 in a schematic way, and in the image in figure 8. Curiously, it is more marked than any other aspects of SSI shapes. This is consistent with the fact that the type IV SSI is a transition region where the interaction of shocks exhibits compression, as can be seen in figure 11, where the distances of the different shock regions become smaller than the reference distances.

4.2.3. Intensity levels in the intersection area

Another interesting observation that deserves analysis is the distribution of the light intensity in the area of the shock interaction. Even if the intensity is not exactly proportional to the density, it gives information concerning the impact of SSI on the local density. In figure 13, the intensity level of $P_{ml}$ ($I_{ml}$ in figure 13a) and of the most luminous point passing by the nose of $S_2$ ($I_{0_2}$ in figure 13b) are plotted. In order to compare the intensity levels of the different images, their values have been normalized with $I_{0_1}$ of each image. It should be noted that $I_{0_1}$, located at the stagnation line of $S_1$, is the densest point of a single sphere, and its value is the one that $S_2$ should have without any interaction.

Figure 13. Intensity values of the most luminous points (a) in the shock/shock interaction area, and (b) on the stagnation line, according to the type of interference.

As shown in figure 13, the two tendencies are the same: the slopes increase from type I to IV, and then decrease from type IV to VI. Figure 13(b) shows the light intensity at the $S_2$ stagnation point normalized to that of $S_1$. The curve shows a maximum for the type IV SSI, which is representative, for this type of interaction, of an increase in local density in front of $S_2$. Figure 13(a) presents the evolution of $I_{ml}$, showing that whatever the type of SSI, the local density in the intersection zone of the shocks is higher than the maximum density observed at the stagnation point of the $S_1$ sphere taken as reference. On the other hand, the graph in figure 13(b) shows for type I, where the sphere is almost out of the influence of the shock wave $S_1$, that the densities of the two stagnation points are equal. This is consistent, since the free flows seen by $S_1$ and $S_2$ are the same. Then, from type II to type VI, the intensity values vary in the same way as for $P_{ml}$. Approaching type VI, the density at $P_{0_2}$ becomes lower than that at $P_{0_1}$, leading to an increase in separation distances due to a higher level of rarefaction. Also, the mean free path increases, and the free flux density decreases as well as the local density in the shock wave of $S_2$.

4.3.1. Drag and lift analysis

The aerodynamic balance described in § 3.2 was used to measure the drag and lift forces of $S_1$ alone, and of $S_2$ interacting with $S_1$. From the previous images, the six relative positions of the sphere were selected to observe the impact of the different SSI on the aerodynamic forces. Drag ($F_{x_2}$) and lift ($F_{z_2}$) force values of sphere $S_2$ are presented in figures 14(a,b), respectively. Black lines with the values $F_{x_0}$ and $F_{z_0}$ are the drag and lift forces undergone by a single sphere without any interactions. They represent the forces that the sphere $S_2$ would have experienced without any interaction.

Figure 14. (a) Drag and (b) lift forces of $S_2$ measured with thrust device.

The evolution of the drag force shows that for type I, $F_{x_2}$ is almost the same as the reference value. This result was expected since both spheres see almost the same free flow (with the difference of a slight impact of the $S_1$ shock wave on $S_2$). From SSI types I to IV, as $S_2$ goes down and interacts with the $S_1$ shock wave, $F_{x_2}$ increases. It reaches a maximum for a type IV SSI, the transition area highlighted particularly in § 4.2. For types V and VI, the drag force suddenly decreases, reaching values lower than $F_{x_0}$. Again, this is due to the fact that $S_2$ goes deeper into the wake of $S_1$, where the flow is more rarefied, and presumably reaches speeds that are lower than the free-stream flow. As a result, the forces experienced by $S_2$ behind $S_1$ (in the $\boldsymbol {x}$-direction) decrease greatly, suggesting that when a fragmented piece of debris remains behind a parent piece of debris, the latter protects it from being damaged by the free flow. In figure 15, the luminous profile along the vertical axis $X_2/r_1=3$, and the drag force profile, are plotted, both according to $Z_2$. Since the luminous intensity is the reflection of the local density, no recompression area is observed in the wake of $S_1$. Therefore, since the profile of the drag forces follows the profile of the intensity, the decrease of the altitude of the sphere $S_2$ leads to the decrease of the force $F_{x_2}$. This statement will be confirmed later in this work with the oscillating sphere results.

Figure 15. Drag force and luminous intensity profiles along the vertical axis $X_2/r_1=3$.

With respect to lift forces, for types I–III, $S_2$ is mainly above the shock wave of $S_1$, and after, it is mainly below. Therefore, one would expect $S_2$ to be respectively repelled or attracted in the $\boldsymbol {z}$-direction by ricocheting off the shock wave of $S_1$, as can be observed in the continuous regime (Register et al. Reference Register, Aftosmis, Stern, Brock, Seltner, Willems, Guelhan and Mathias2020). But in fact, since the shock wave from $S_1$ is thick and diffuse, it does not represent a clear boundary that would block $S_2$ in the wake of $S_1$. For this reason, the sign of the lift force of $S_2$ does not change as it approaches the shock of $S_1$ from above or below, but simply decreases. Nevertheless, it could undergo a change of sign for a type VI SSI, but this would be due not to shock/impact interference but to the protection created by $S_1$, when $S_2$ is in its wake and close to the axis of $S_1$.

According to the forces undergone by $S_2$ and $S_1$, one can deduce the displacement of $S_2$ toward $S_1$ as follows:

1. (i) if $F_{x_2}< F_{x_0}$, then $S_2$ approaches $S_1$ in the $\boldsymbol {x}$-direction ($\Leftarrow$)

2. (ii) if $F_{x_2}>F_{x_0}$, then $S_2$ moves away from $S_1$ in the $\boldsymbol {x}$-direction ($\Rightarrow$)

3. (iii) if $F_{z_2}< F_{z_0}$, then $S_2$ approaches $S_1$ in the $\boldsymbol {z}$-direction ($\Downarrow$)

4. (iv) if $F_{z_2}>F_{z_0}$, then $S_2$ moves away from $S_1$ in the $\boldsymbol {z}$-direction ($\Uparrow$).

Taking into account both the values and directions of each force, the resulting forces were calculated in the $S_1$ referential, and the results are drawn in figure 16, giving information on the direction of $S_2$ towards $S_1$. As can be seen, when $S_2$ and its shock are enveloped in the shock of $S_1$, it is attracted to the first sphere.

Figure 16. Calculated displacement of $S_2$ in $S_1$ reference frame according to the shock/shock interaction type.

4.3.2. Comparison with the continuum regime

Fisher (Reference Fisher2019) studied the SSIs of Edney in a continuum regime at Mach 5. His experiment consists in releasing a sphere that flies freely in the flow field generated by a primary sphere. This work allows us to compare SSI depending on the level of rarefaction, since drag forces were measured.

In the work of Fisher (Reference Fisher2019), the drag coefficient was calculated with $C_d=\tfrac {1}{2}{F_x}/$ $({\rho _{\infty } V_{\infty }^2 S})$, where $\rho _{\infty }$ and $V_{\infty }$ are the values in the free-stream flow (upstream of $S_1$), and $S={\rm \pi} ({D}/{2})^2$. Note that this method does not take into account the local flow seen by $S_2$ moving in the flow generated by $S_1$. Since free-stream flow conditions impact directly the flow generated downstream of $S_1$, it is questionable if this parameter is effectively representative of the aerodynamics of $S_2$. Nevertheless, it allows us to compare results with the literature, where this method is mostly used. In our conditions, the drag coefficient of a single sphere is 1.35, while Fisher's is about 0.88. For both results, the drag coefficient of $S_2$ ($C_{d_2}$) is normalized by that of a single sphere ($C_{d_o}$). In figure 17, it is clear that the trends of all the curves are similar: for type I, the drag force of $S_2$ is almost that of the sphere alone in the free stream; in the wake of the sphere $S_1$, the dynamic pressure decreases as one moves towards the centre of the flow, i.e. towards a type VI interaction. The flow therefore becomes rarefied; the viscosity increases as well as the mean free path. As the local density and velocity decrease, the forces become smaller. The second sphere will experience weaker forces and, in a way, $S_2$ will be ‘protected’ by remaining in the wake of the first sphere, $S_1$. So the impact of the SSI, in terms of drag coefficients, is similar for both regimes. Nevertheless, this analysis has to be taken with caution for two reasons. On the one hand, the trajectory described by the sphere of Fisher is different from ours. On the other hand, the drag coefficients are given according to a type of SSI, but not in terms of location. As the shock wave of $S_1$ is thicker and more diffuse in a rarefied flow, the area where SSI occurs will be larger than in a continuum regime, expanding the area of high drag forces. This hypothesis seems to be confirmed by the results of Laurence & Deiterding (Reference Laurence and Deiterding2011).

Figure 17. Drag coefficient of the present work on the vertical axis $X_2/r_1=3$, compared with the results of Fisher (Reference Fisher2019), with variable coordinates.

In figures 18(a,b), respectively, the drag and lift coefficients are plotted according to the normalized distance of $S_2$ towards the incident bow shock. Here, $Z_2$ is the altitude of $S_2$ towards $S_1$, $R_s$ is the altitude of the shock wave (middle shock in our case), and $r_2$ is the radius of $S_2$. Both results concern the vertical axis $X_2/D_1=1.5$ ($X_2/r_1=3$). As far as drag is concerned, it can be seen that the maxima are obtained globally at the same location for all cases, but their amplitude varies inversely with the radius ratios of the spheres. The two iso-Mach curves have roughly the same shape, although they were obtained for different sphere radius ratios. This suggests that for ratio 1 , the corresponding curve would also have the same shape but would show a smaller maximum value. The curve obtained in the present work has a slightly different shape from the other two, with a wider curvature around the maximum, which could be due to rarefaction effects. However, this remains a hypothesis because it was performed at a lower Mach number. Regarding the lift coefficients, at iso-Mach, the maxima and minima have the same locations, and again with very similar shapes, as for the drag. Nevertheless, the maximum lift is observed for the same location, the curve does not have a minimum, and especially the shape of the curve around the maximum is less sharp and wider. This last point, observed for both the drag and the lift, might be due to the important decrease in Mach number, but in any case, if they exist, these effects are coupled with the increase in the rarefaction level that thickens the incident shock wave, and thus enlarge the area where the aerodynamic coefficients of $S_2$ are impacted.

Figure 18. Drag and lift coefficients of the present work for a longitudinal distance of 1.5 diameters, compared with the results of Laurence & Deiterding (Reference Laurence and Deiterding2011).

4.4.1. Wall pressure analysis

The pressure distribution at the surface of $S_2$ was measured for the same six locations as those already investigated. However, three other relative positions were explored to better describe the transitions between the different types of SSI, in particular between types III and IV, and between types IV and V. The pressures were also measured for a single sphere, representative of the reference wall pressure distribution. Figure 19 describes, for each type of SSI, the pressure distribution at the surface of $S_2$ in blue, along with the reference case in orange. The wall pressure values ($p_p$) are normalized with the stagnation pressure ($p_0$) which is 56.17 Pa. The wall pressure distributions are given as functions of the angle of the measurement point ($\varTheta _{p_p}$) with respect to the surface of $S_2$, $0^{\circ }$ being taken with respect to the nose of the sphere.

Figure 19. Pressure distributions for the six types of SSIs and for a single sphere (reference case).

The pressure distribution reveals two different behaviours. Concerning types I, II and III, the surface pressure is disturbed only on the lower part of the sphere ($\varTheta _{p_p}<0^{\circ }$). For these three types, the wall pressure is higher than the reference case in the interference area. After this increase, the wall pressure returns to the reference behaviour. Pressure distribution clearly presents two peaks: one which corresponds to the intersection zone of two shocks, and the other which corresponds to the nose of the sphere (at the stagnation point). In the first case, the peak is not, mathematically speaking, a maximum, but it is obtained with the minimum of the pressure derivative according to the angle, corresponding to the interaction area. For types IV, V and VI SSI, the pressure distributions are modified globally, resulting in only one maximum corresponding to the intersection area. Outside the interference area, in the lower part of $S_2$ ($\varTheta _{p_p}<0^{\circ }$), values are lower than those of the reference case. At maximum, a great peak is reached, and then the slope tends to stick to the reference value. For types IV and V, the peak is clearly higher than the reference maximum value, while type VI is lower, which is coherent with the fact that $S_2$ penetrates deeper into the wake of $S_1$, i.e. when the flow is more rarefied.

Pressure peaks in the interference area ($p_{p_{max, interf}}$) are plotted according to $\varTheta _{i}$ in figure 20(a). The angle of these peaks ($\varTheta _{p_{p_{max, interf}}}$) are also plotted according to $\varTheta _{i}$ in figure 20(b). Due to the thickness of the shock wave, the angles given as frontier (figure 10) are not clear delimitations; this is why the three positions of type IV presented in figure 19 are not located in the exact $\varTheta _{i}$ range. Nevertheless, regarding the shape of the pressure distribution of the three points discussed for type IV, they seem to follow the same behaviour. With this assumption, we will henceforth consider them as representative of a type IV SSI. Figure 20(a) shows an increase from I to IV and then a decrease of the maximum pressure in the intersection area. The maximum pressure peak is reached when middle shock waves interact at approximately $-5^{\circ }$ from the horizontal, which corresponds to a type IV SSI. Figure 20(b) plots the location of the pressure peak at the surface of $S_2$ as a function of the angle of the interaction point. As can be observed, the curve presents two distinct regions with two different slopes. The first area corresponds to types I to IV, where the value of $\varTheta _{p_{p_{max, interf}}}$ increases until it reaches the same value as the angle of intersection. From this point, the curve has a gentler slope for types V and VI, since $\varTheta _{p_{p_{max, interf}}}$ increases by $20^{\circ }$ for an increase of $\varTheta _i$ by $60^{\circ }$. One can notice that the maximum pressure is located mostly on the lower part of the sphere and does not seem to go higher than $25^{\circ }$ from the stagnation point, even if the intersection angle is on the higher part of the sphere.

Figure 20. (a) Maximum surface pressure $p_{p_{max, interf}}$ and (b) its angle $\varTheta _{p_{p_{max, interf}}}$ according to the interaction angle $\varTheta _{i}$.

4.4.2. Comparison with the continuum regime

A comparison of the pressure distribution of the present work has been made with the results of Edney (Reference Edney1968) (see Appendix A). His work, based on the interaction of an oblique incident shock with the bow shock of a hemisphere, indicates a similar behaviour for SSI types I to III, with more pronounced pressures values, and thinner pressure peaks. The real difference is observed for SSI types IV to VI. In particular, the specific pressure distribution of the continuum regime seems to vanish completely in our rarefied conditions, appearing a lot like a type V or VI. This might be due to the thicker boundary layer that would prevent a supersonic jet from impinging at the wall of $S_2$. Thus the type IV interference might be disappearing as the level of rarefaction increases, as suggested by Gusev & Erofeev (Reference Gusev and Erofeev2004) or White & Kontis (Reference White and Kontis2018), who studied the impact of an incident shock on a cylinder at Knudsen numbers equivalent to our conditions.

Figure 21 focuses more specifically on pressure peaks. It is obvious, whatever the angle of incident shock, that the continuum case presents higher values of pressure peaks. Their repartition at the wall of $S_2$ is more localized: it concerns a region of about $10^{\circ }$, against $40^{\circ }$ in our case. Also, the maximal pressure peak is located on a lower angle (about $-25^{\circ }$) than the one that we observed in the rarefied regime (about $-10^{\circ }$). These differences are consistent with the increase in the thickness of the shock wave due to an increase in the level of rarefaction: the impact of the SSI is localized not in one point, but in a zone leading to a decrease in the impact of the boundary layer of the second sphere at its surface, and therefore also the decrease in wall pressure.

Figure 21. Pressure peak distribution, comparison with results of Edney (Reference Edney1968) in a continuum regime.

5.2.1. Case $Z_2/r_1=0$

The sphere $S_2$ was first displaced on the horizontal axis ($Z_2/r_1=0$), which corresponds to the study by Golubev (Reference Golubev2012). Four locations of $S_2$ were studied, and the results are given in figure 25. Here, we present the drag force measured for the swinging sphere $S_1$ according to the relative position of $S_2$. Figure 26 shows the normalized enhanced images corresponding to these positions. They allow us to better understand the physics of the flow field, since they are compared to images of a single sphere in the flow.

Figure 25. Drag forces of $S_1$ according to $X_2$, the position of $S_2$ relative to $S_1$ in the $\boldsymbol {x}$-direction ($Z_2/r_1=0$).

Figure 26. Normalized images with jet colour map and detection of shock waves. Top images of (a–d) show $X_2/r_1=7, 5, 4, 3$ mm, respectively, and $Z_2/r_1=0$. Bottom images show single sphere.

As observed in figure 25, when $S_2$ is far behind $S_1$, the latter is confronted by the same drag force as when it is alone in the flow. As $S_2$ gets closer, it reaches a point (around $X_2/r_1=4.6$) where it begins to affect the aerodynamics of $S_1$. Gradually, the $S_1$ drag force decreases. By observing the images of figure 26, a few things are noticeable. When $S_2$ is far enough from $S_1$ (figures 26a,b) not to disturb the $S_1$ drag force, it seems that the wake $S_1$ is not modified in at least the first diameter behind it. The $S_2$ compression wave is visible and shows the sign of a highly rarefied flow, with a great thickening of the wave and a great stand-off distance: 6.7 mm for figure 26(a), and 5.8 mm for figure 26(b). As $S_2$ gets closer to $S_1$ and enters a more rarefied flow with a lower velocity, the shock wave of $S_2$ has difficulty developing, but a compression is still visible upstream of $S_2$. The few molecules that constitute this compression might create a viscous layer that pushes $S_1$ upstream. Another point that might induce the variation of drag force is the merging of the incident shock with the compression wave induced by $S_2$.

Golubev (Reference Golubev2012) calculated the $S_1$ drag force in a continuum regime. His results, shown and compared to ours in figure 27, were obtained for different Mach numbers but the same Reynolds number. In the continuum regime, the inter-sphere distance for which the drag coefficient of $S_1$ has a value equal to the reference is the same, whatever the Mach number. These three curves also show that the influence of the presence of $S_2$ is all the more important as the Mach number decreases. The curve obtained in this work, at Mach 4 in the rarefied regime, shows that the distance for which the drag coefficient of $S_1$ has a value equal to that of the reference drag is greater than that obtained at the same Mach number in the continuum regime. Comparison of the iso-Mach curves also shows that the slopes show that the influence of $S_2$ is substantially the same; meanwhile, the value of the normalized drag coefficient of $S_1$ is greater in the continuum regime.

Figure 27. Dimensionless drag force of $S_1$ according the dimensionless distance between spheres $l$.

5.2.2. Case $X_2/r_1=3$

In the same manner as for the previous case, $S_2$ is moved behind $S_1$, this time vertically on the axis $X_2/r_1=3$. Results of the measured $S_1$ drag force are plotted in figure 28 according the position of $S_2$. First, $S_2$ is placed so that it does not act on $S_1$, which does not displace itself. As $S_2$ is moved down, and gets closer to $S_1$, the swinging sphere moves ahead, as it does when $S_2$ is close behind it on the axis $Z_2/r_1=0$. With the normalized enhanced images of figure 29, an analysis equivalent to that from the case $Z_2/r_1=0$ can be made. As $S_2$ gets down, the lower part of the $S_2$ shock wave modifies the wake of $S_1$ by intensifying the density in its wake.

Figure 28. Drag forces of $S_1$ according to the position of $S_2$ in the $\boldsymbol {z}$-direction ($X_2/r_1=3$).

Figure 29. Normalized images with jet colour map and detection of shock waves. Top images in (a–d) show $Z_2/r_1=3.77, 3.25, 2.59, 1.74$ mm, respectively, and $X_2/r_1=3$. Bottom images show single sphere.

The cases where $S_2$ locations generate SSIs do not seem to affect the drag of $S_1$. Indeed, we observe 3 % of drag variation whether for displacement in the axis $Z_2/r_1=0$, where there is no SSI, or in the axis $X_2/r_1=3$, where there is. This variation is not really significant, but is still interesting. In the case of space debris re-entry, it would mean that the aerodynamics of a parent piece of debris, and thus its trajectory, depends on the location of the fragments debris. These fragments can be a single object, but more often they are clouds of fragments for which the modification of $S_1$ aerodynamics might not be negligible.