Aerodynamics of inclined cylindrical bodies free-flying in a hypersonic flowfield

Figure 1 shows the two right circular cylindrical bodies that serve as models for this investigation having a diameter of 0.05 m and a length of 0.1 m. The solid line shows a simple constant radius cylinder, while the dashed line indicates the concave or “dogbone” model. This body shape is a generic geometry inspired by rocket engines that belong to the rocket-mission-related objects of space debris representing a significant amount of reentered objects in 2020 (ESA Space Debris Office 2020). An axially symmetrical cavity with a parabolic shape is defined for the dogbone, whereby its local radius r depending on the axial coordinate x is given in Eq. (1).

This geometry features a minimum diameter \(\textit{d}_{c}\) of 0.02 m and a cavity length \(\textit{l}_{c}\) of 0.06 m. For the experimental part, the models of both geometries were manufactured of structural steel with the standardized material number 1.0037. The mass was measured with a precision scale, and the moment of inertia was computated based on the actual mass and length. In doing so, the test articles with cylinder geometry have a mass of 1.54 kg and a transverse inertia of \(1.53\cdot 10^{-3}\,{\mathrm{kg}}\,{\mathrm{m}}^{2}\), while the test objects with dogbone geometry exhibit a mass of 0.97 kg and a transverse inertia of \(1.30\cdot 10^{-3}\,{\mathrm{kg}}\,{\mathrm{m}}^{2}\).

Figure 2 depicts the reference frame of the entirely free-moving object, where x, y and z are the Cartesian coordinates as well as g is the gravitational acceleration. The definition of the reference frame is needed to determine its aerodynamic coefficients of lift force \({C}_{L}\), drag force \({C}_{D}\) and pitching moment \({C}_{M}\). The origin of the right-handed global coordinate system (GCS) here is spatially fixed in the center of nozzle’s exit plane. It has the positive x-axis pointing downstream, the y-axis in the transverse direction and the positive z-axis pointing vertically up. A body-fixed local coordinate system (LCS) is defined in the center of gravity (CoG) of the free-flight object that x-axis is congruent with the symmetry axis of the cylinder. Unlike, the aerodynamic coefficients of this object are specified in the spatially fixed reference frame of the GCS. The influence factor of interest in this study is the pitch angle \(\vartheta \) (shown in Fig. 2), which designates the inclination between the global x-axis and cylinder’s symmetry axis. The positive direction of rotation is counter-clockwise when viewed to the y-axis. Hence, \(\vartheta = 0^{\circ }\) means that the base area is exposed to the flow. This configuration is called base-exposed in the following. A further appearing orientation designation in this article is side-exposed, whereby the symmetry axis of the cylindrical body is orthogonal to the flow direction.

For the flow conditions (FC) used in this study, the main parameters are listed in Table 1, whereby the numerical investigations were performed with a theoretical reservoir pressure of 5.2 bar and the wind tunnel tests were run at 4.8 bar.

The experimental setup and test procedure were largely similar to that in previous work with free-flying cubes Seltner et al. (2019). The following section briefly describes the main details, minor modifications as well as the model release mechanism, measurement technique and post-processing system.

Two flow solvers commonly used by NASA were applied in the present work containing the inviscid Cart3D code (see Sect. 2.3.1) and the viscous US3D code (see Sect. 2.3.2).

The primarily examined parameter in this research is the pitch angle \(\vartheta \). A value of \(\vartheta =0^{\circ }\) here signifies that the base of cylinder is orthogonal to the inflow as described in Sect. 2.1. A visualization of the density gradients in x-direction of this base-exposed attitude under the equal flow conditions is depicted in Fig. 4 presenting the flow structures around the body. The upper part of Fig. 4 shows the integrated density gradient through the fluid domain in the schlieren image from free-flight tests, whereas the lower part depicts the computational image of the gradient merely on the central plane coming from US3D simulations. This qualitative comparison of the experimental and computational flowfields shows a very good agreement, whereby the bow shock shape and stand-off distance seem to be identical. However, an expansion fan on the trailing edge followed by flow separation as well as shocks at the body’s shell surface downstream the leading edge and also at the shear layer of the wake region is presented just in the computational image, which are not visible on the schlieren image due to the low sensitivity of the schlieren system to free-stream density.

A sketch of the supersonic flow phenomena with the corresponding experimentally determined schlieren photograph in the background is shown in Fig. 5, which is consistent with the flow topology of blunt bodies as in the established literature (Hoerner 1965) as well as in a previous study of the authors with a cube (Seltner et al. 2019). Most notable here is the occurrence of a weak reattachment shock as a result of a tiny separation bubble immediately downstream the leading edge. This type of separation is called shoulder separation as described in Kaufman II et al. (1966), which is caused by an adverse pressure gradient because of recompressions of the flow after a strong expansion. Such a separation on cylinder’s shell surface was also observed in Matthews and Eaves Jr. (1967).

For a quantitative analysis of the gas dynamic flow structures, CFD simulations were performed using US3D considering viscous effects. Figure 6 illustrates the flowfield around a base-exposed cylinder with isothermal boundary condition of their walls. In Fig. 6a, the Mach number distribution with streamlines is shown. This color contour plot confirms the observations from schlieren images that the Mach number falls below one in front of the windward surface. Downstream of both body edges, a rise of the Mach number is apparently being caused by expansion fans, whereby an increase in Mach number due to the previously mentioned recompression is visible subsequent to the expansion on the leading edge. The recirculation zone in the wake region of the body has a length equal to the cylinder’s diameter, whereby this part is subsonic. In Fig. 6b, the distribution of the flowfield temperature as well as the surface pressure is depicted. The bow shock here effects a sudden increase in the flow temperature due to the compression, whereby the highest values with over 550 K are reached in the subsonic region between bow shock and cylinder’s front face. Furthermore, a second region of high-temperature occurs within the wake region having temperatures up to 400 K. Interestingly, the highest values are reached in the supersonic part instead of the subsonic part. Also, a thin boundary layer with higher flow temperatures than in the free stream is visible in Fig. 6b around cylinder’s shell. The surface pressure immediately decreases downstream the leading edge, rises afterward until a third of body’s length and finally decreases until the trailing edge. Thus, an adverse pressure gradient occurs in the first third of cylinder’s shell surface provoking a boundary layer separation as shown in the schlieren images (see Fig. 4).

As soon as the cylinder is inclined, the flowfield around the body becomes asymmetric as well as aerodynamic phenomena become more complex, which is illustrated by means of US3D’s numerical schlieren images in Fig. 7. All of these flow structures contain detached bow shocks, whereas the shock shape and stand-off distance depend on the flow angle relatively to the body. The more a body edge protrudes into the flow, the more the shock stand-off distance is reduced, and the shock curvature is greater in the surrounding of the edge. At the leading edges, expansion fans rise that encounter with the bow shock resulting in an aerodynamic interaction associated with a bending of the shock wave over the length of the interaction. In Fig. 7d, a smearing out of the bow shock further downstream is just visible in the numerical schlieren image, which is an artifact due to the reduction of mesh refinement away from the object. Moreover, it seems here that two separate wake regions appear, whereby the flow attaches after the shoulder separation on the leeward shell surface before it detaches again on the furthest edge. Experimental schlieren images from present free-flight tests are presented in Fig. 8. Qualitatively, the stand-off distances and shapes of the bow shocks show a good agreement, and the experimental flowfield is accurately reproduced by the numerical computation (see image of a superposed flowfield as an example in the online supplementary Information).

Furthermore, Fig. 8 qualitatively depicts the motion behavior of several drops with different initial pitch angles, whereby the duration between the first and last image of each sequence is 60 ms. The cylinders here experience streamwise, vertical and pitching displacements. Regarding the pitching, the bodies rotate counter-clockwise except for \(\vartheta _{0} = 0^{\circ }\) and \(\vartheta _{0} = 10^{\circ }\) (shown in Fig. 8b and c). The reason for this counter-clockwise rotation is a negative pitching moment caused by an asymmetric surface pressure distribution during the passing through the upper shear layer of the free jet as described in Sect. 2.2.2. For \(\vartheta _{0} = 0^{\circ }\) and \(\vartheta _{0} = 10^{\circ }\), the flight attitude does not change, because it seems that a moment, effected by cylinder’s flowfield at the current flight attitude, counteracts the rotational motion caused by the free jet’s shear layer.

By the use of the stereo-tracking system, the motion of the rigid body is reconstructed, whereby each three components of position and orientation are determined. Figure 9 shows an example for the measured displacement of a cylinder configuration with an initial pitch angle of \(90^{\circ }\), whereby Fig. 9a depicts the translational displacement (streamwise, spanwise and vertical) and Fig. 9b depicts the angular displacement (roll, pitch and yaw). It is expected that the motion is two-dimensional due to the rotational symmetry of the model as well as the alignment of cylinder’s axis and nozzle’s axis. This assumption can be confirmed with a view to the displacement data (shown in Fig. 9), whereby the roll, yaw and spanwise motion components are almost zero in contrast to the streamwise, vertical and pitch motion components over the entire free-flight time. Thereby, the changes in streamwise position (\(\varDelta x=161.3\) mm), vertical position (\(\varDelta {z}=-339.3\) mm) and pitch angle (\(\varDelta \vartheta =-30.6^{\circ }\)) are about two orders of magnitude greater than the changes in spanwise position (\(\varDelta {y}=3.0\) mm), roll angle (\(\varDelta \varphi = -0.6^{\circ }\)) and yaw angle (\(\varDelta \psi = -0.2^{\circ }\)). Hence, the roll, yaw and spanwise displacements can be neglected.

Figure 10 illustrates the two-dimensional trajectories of all runs in the xz-plane with respect to the center of the nozzle exit plane showing also the ram pressure distribution of the free jet from experimental flow characterizations as a color contour layer. The red area represents the core flow with a homogenous ram pressure distribution. It is apparent that the flight trajectories do not overlap. Unlike the free-flight technique in Seltner et al. (2019), two release mechanisms at two different positions were employed for tests with inclined cylinders, whereby the initial positions in x- and z-direction vary due to the model alignment. Broadly speaking, a significant streamwise acceleration of all free-flight tests is visible in Fig. 10 as well as the majority of measured data points are within the homogenous flow.

Six-DoF velocities and accelerations of the same exemplary run (as of Fig. 9) are depicted in Fig. 11. As previously reported about the neglect of three motion components, the derivatives of roll, yaw and spanwise motion also confirm the previous observations, because these quantities are roughly zero over time. As a result, the streamwise, vertical and pitching components are only considered for all experimental results presented hereinafter and the motion analysis is 3DoF. The velocities of the streamwise, vertical and pitch motion components in Figs. 11a and 11 b clearly differ from zero at \({t}= 0\,{\mathrm{s}}\), when the entire body is within core flow. These deviations from zero demonstrate the impact of the free jet’s shear layer on the body motion causing a strong pitch rotation as well as a significant streamwise acceleration. It is notable that the pitch acceleration in Fig. 11d is negative at the beginning \(({t} = 0\,{\mathrm{s}})\) and turns to positive values shortly after. The reason for this change of sign seems to be that the first values are affected by adjacent data points during the filtering, for which the body experience a negative pitch acceleration due to the shear layer. For the determination of accelerations (second-order derivatives), the data are filtered thrice by use of 100 adjacent data points before and after each value for all filtering steps. Thus, the first and also the last values in Fig. 11d are influenced by measurement points captured within the shear layer, which explains the strong deviations at the edges of the plot and can also be observed for the pitching moment coefficients. In contrast to the angular derivatives, the translational acceleration of cylinders in crossflow seems to be less sensitive to inhomogenous distributions of the ram pressure than the pitch acceleration. Inside the core flow, the pitch acceleration in Fig. 11d has significant positive values, which cause a counter-rotation referred to the motion of the model within the upper shear layer.

The resulting aerodynamic coefficients derived from the motion derivatives in streamwise (drag force) and vertical (lift force) direction as well as around the spanwise axis (pitching moment) are presented in Fig. 12 as a function of pitch angle. In addition to the experimental data of several runs with different initial pitch angles, trigonometric fitting and hypersonic approximation curves are shown in the diagrams, as well. Aerodynamic coefficients based on two different reference areas are presented here. On the one hand, a constant circular base area (Fig. 12a) and, on the other hands, a pitch-angle-dependent projected frontal area (Fig. 12b) are shown here. The second reference quantity is calculated by means of Eq. (2). Figure 12 reports derived measurement data just in a positive range of pitch angle, where open symbols indicate mirrored values. In doing so, the absolute pitch angles are applied here as well as the lift force and pitching moment coefficients experience a change of sign for the mirrored values. In general, the aerodynamic coefficients reveal a significant dependence on the body orientation, which confirms previous findings of cubes in the literature (Seltner et al. 2019; Hansche and Rinehart 1952).

Due to the mirror symmetry of cylinders, the correlations between aerodynamic coefficients and pitch angle have a period of 180\(^{\circ }\). Thus, the results of pitch angles in the range from 0\(^{\circ }\) to 180\(^{\circ }\) can be extrapolated on its full-rotation aerodynamic characteristics. Trigonometric functions by means of Fourier series with period \(\pi \) are used for the curve fitting on the experimental data. This means that the angular frequency of each Fourier term is an integer multiple of 2. For high-order terms, the Fourier coefficients are very small excepting for the pitching moment coefficient, which is why these terms are neglected for drag and lift coefficients. The resulting functions of drag, lift and pitching moment coefficient are given in Eqs. (3–5) being subject to the pitch angle. The overall uncertainties as described in Sect. 2.2.4 are 2% in \({C}_{D}\), 11% in \({C}_{L}\) and 25% in \({C}_{M}\) referred to the peak value.The drag coefficient of cylinders based on circular reference area (see Fig. 12a) exhibits a much greater influence of the inclination angle than of cubes as shown in Seltner et al. (2018). Increasing the pitch angle of the cylindrical body causes first a slight decrease in drag coefficient until \(13^{\circ }\) before it begins to increase significantly reaching the peak value at roughly \(90^{\circ }\). At a pitch angle of \(0^{\circ }\), the drag coefficient is 1.747, which differs by 5.9% from the value (\({C}_{D}=1.65\)) determined by Hoerner (1965). This underestimation of the base-exposed cylinder’s drag can be explained by the neglect of three-dimensional flowfield phenomena, and hence, the aerodynamic influence of the body ends. As opposed to this, the experimental drag coefficient at a pitch angle of \(90^{\circ }\) is identical to Hoerner ’s value (\({C}_{D}=3.08\) referred to the base area), whereas it is overestimated by Penland (1954) (\({C}_{D}=3.16\)) having a relative deviation of 2.5%. In comparison, the drag coefficient determined by the modified Newtonian theory decreases more strongly with increasing pitch angle reaching the minimum at a higher pitch angle of \(31^{\circ }\) before it begins to increase until \(90^{\circ }\), as well. Moreover, the base-exposed drag coefficient is overestimated with a relative deviation of 4.4% compared to the experimental value, while the side-exposed drag coefficient is almost identical with a relative deviation of 0.5%. Notwithstanding, the hypersonic approximation shows a fairly good agreement with the experimental results.

Asymmetric flowfields in consequence of the body inclination effects the development of lift forces and pitching moments, whose function graphs behave sinusoidally. In Fig. 12a, lift coefficient’s maximum of 0.740 appears at a pitch angle of \(63^{\circ }\) and its minimum of \(-0.161\) occurs at a pitch angle of \(16^{\circ }\). The maximum arises at an angle, where the space diagonal of the cylinder is orthogonal to the inflow direction, and thus, the projected area is close to highest value. Referring to the circular base area, the maximum lift coefficient amounts to 24.0% of the maximum drag coefficient at \(\vartheta =90^{\circ }\). The modified Newtonian theory reveals a good agreement with the experimental results on lift coefficients, whereby the maximum absolute deviation is 0.086 (\(\vartheta =12^{\circ }\)). The graph of pitching moment coefficient has three zero-crossings within the half period at \(0^{\circ }\), \(20^{\circ }\) and \(38^{\circ }\), while its maximum occurs at \(54^{\circ }\). Surprisingly, positive pitching moment coefficients occur for small positive pitch angles, even though negative values are expected due to a shifting of the stagnation point in counterwise direction. However, the surface pressures of the bottom shell surface are much higher than of the upper side due to a higher recompression of the flow in combination with a stronger reattachment shock (see Fig. 7b), whereby its center of pressure (CoP) lies ahead of the CoG close to the leading edge. This aerodynamic moment acts in opposite direction of rotation than the moment of the stagnation region, whereby the lever arm of the shell surface’s CoP is longer than of the stagnation point. By rising the pitch angle of the cylinder, the CoP shifts backward causing a decrease of its lever arm length and aerodynamic moment, whereas both increase for the stagnation point. At a pitch angle of \(20^{\circ }\), the moment proportions are in equilibrium (\(C_{M}=0\)), which is not expected for an asymmetric flowfield. Thus, the aerodynamic moment proportion of the base behaves contrary to the one of the shell surface explaining zero-crossings for inclined configurations. Analytical outcomes of \({C}_{M}\) are not presented in Fig. 12, because the pitching moment is constant zero owing to the symmetry of the surface pressure distribution with respect to the xy-, xz- and yz-plane. These results offer vital evidence for the occurrence of significant lift forces and pitching moments of inclined cylinders in hypersonic flowfield, which are remarkable that the impact on rigid body motion should be considered. In addition, there is a good probability that the zero-crossings slightly depend on cylinder’s aspect ratio (l/d).

It was shown previously in this analysis that an increasing pitch angle toward \(90^{\circ }\) results in an increase in the drag coefficient when using a constant reference area for normalization. However, the frontal area also increases, which has a major influence on the rise of drag coefficient, in fact. By referring to an inclination-dependent reference area as in Fig. 12b, the drag coefficient decreases between a pitch angle of \(0^{\circ }\) and \(37^{\circ }\) before it begins to increase with lower gradients reaching a local maximum of 1.210 at \(90^{\circ }\). This drag coefficient based on the projected reference area amounts just to 39.3% of the value with constant base reference area, which is why the larger proportion is attributed to the increase in the frontal area. The global maximum occurs at a pitch angle of \(0^{\circ }\) and is significantly higher with a relative difference of 44.4% than the local maximum at roughly \(90^{\circ }\). The minimum is reached with a value of 0.831 at an inclination, where the stagnation point is close to cylinder’s leading edge. It follows that the part of the body, which protrudes into the flow, primarily determines the magnitude of drag coefficient, whereby the flat-faced configuration (e.g., \(\vartheta =0^{\circ }\)) exhibits a higher effective drag coefficient than the round-curved configuration (e.g., \(\vartheta =90^{\circ }\)) and a much higher effective drag coefficient than the leading-edge configuration (e.g., \(\vartheta =37^{\circ }\)). Thereby, a lower drag coefficient correlates with a lower stand-off distance of the bow-shock (see Fig. 7). As for the lift force and pitching moment coefficients, the consideration of the pitch-angle-dependent reference area effects a decrease of its values with remaining angles for zero-crossings, in general.

In addition to experimental and analytical results, Fig. 13 depicts the numerically determined aerodynamic coefficients by viscous US3D code and inviscid Cart3D code for comparison purpose. It is notable that both numerical solutions have a very good agreement to each other with a relative deviation in drag coefficient of 1.1% for the base-exposed cylinder and 0.3% for the side-exposed cylinder, although the Cart3D solver neglects viscous effects in contrast to US3D. This is consistent with the well-established theory like Hoerner (1965) that the viscous effects of blunt bodies in hypersonic flows are subordinate in contrast to compressibility effects. In general, all three aerodynamic coefficients of both numerical solutions match well with the experimental curves. Yet, the numerical data of the drag coefficient are lower over the almost entire range of pitch angle (except for \(90^{\circ }\)) than the experimental data having a deviation based on the US3D data of \(-0.064\) at a pitch angle of \(0^{\circ }\) and 0.012 at \(90^{\circ }\). It is not surprising here that the numerical and experimental drag coefficients feature a very small deviation, because both flowfields with identical shape and stand-off distance of the bow shock perfectly match with each other at a zero pitch angle (see Fig. 4). As for the lift force coefficient, the results of experiments and corresponding numerical simulations show a very good agreement with zero-crossings at the same pitch angle, where the maximal relative deviation with respect to the peak lift coefficient is very small with 1.3%. By contrast, the agreement between experimental and numerical pitching moment coefficient is good, but the maximal relative deviation with respect to the peak value is higher with a value of 16.7%.

Figure 14 summarizes all curves of the curve-fitted aerodynamic coefficients depending on the pitch angle for an entire period of \(180^{\circ }\) as well as it depicts the ranges of static stability and trim points with regard to pitching. As requirements for statically stable trim points, moments equilibrium (\(C_{M}=0\)) and negative stability derivative (\(\partial C_{M}/\partial \vartheta < 0\)) need to be fulfilled. Three stable trimmed flight attitudes exist at roughly \(21^{\circ }\), \(90^{\circ }\) and \(159^{\circ }\) where both conditions are satisfied. Unexpectedly, the base-exposed orientation does not fulfill the stability condition, and in consequence, it is an unstable trimmed attitude, while a flight configuration at a pitch angle of \(21^{\circ }\) is a stable trimmed attitude despite an asymmetric flowfield. Thus, the static stability of a cylinder reveals no dependency on the symmetry of flow topology. Both results of CFD simulations and wind tunnel tests confirm these findings.

The magnitude of stability depends on the amount of the stability derivative. In the neighborhood of \(90^{\circ }\), the highest negative stability derivative is obtained with a peak value of \(-0.260\,\mathrm{rad}^{-1}\), which is the range of highest drag coefficient and hence the highest pressure force. Since one edge of the cylinder moves upstream due to an initiated rotation away from a side-exposed configuration, the stagnation point shifts toward this upstream edge. In consequence, the body part, which protrudes more into the flow, experiences a higher surface pressure causing a reverse pitching moment and hence a counterwise rotation. In addition to the maximal stability at \(90^{\circ }\), the pitch angle is statically stable in a wide range from \(54^{\circ }\) to \(126^{\circ }\), which is why a side-exposed configuration is a probably final flight attitude for randomly tumbling cylinders.

The axially symmetrical cavity as of the dogbone is a significant geometry modification of the shell surface in comparison to the ordinary cylinder, which is why an significant impact on the aerodynamic coefficients is expected. Aerodynamic coefficients based on circular base as well as projected area of the dogbone from present free-flight tests are compared in Fig. 15 with the cylinder’s results under the same test conditions. Great differences between both bodies in drag force, lift force and pitching moment coefficient arise as expected (see Fig. 15a), whereby the differences for low pitch angles are small in contrast to higher angles. At a zero pitch angle, the geometry of the flow-exposed forebody, and hence, the shape of the bow shock is the same, which explains the similarity of aerodynamic force coefficients for low pitch angle. However, the experimentally determined drag coefficients toward \(90^{\circ }\) apparently diverge with a relative difference, for example, of 27.3% at \(\vartheta = 70^{\circ }\) that can be anticipated due to a lower frontal area of the dogbone compared to the cylinder. As for the lift force and pitching moment coefficients, the greatest differences occur in the vicinity of \(45^{\circ }\) that complex unsteady flow structures appear in the cavity (see Fig. 16). In comparison with the cylinder, the same flow features like bow shock, expansion fans and wake region with a recompression shock are visible in Fig. 16. By contrast, SSI appears that is the dominant flow phenomenon in the windward part of the cavity-inducing extremely high pressures and heating rates (as described in Edney (1968b)). The bow shock here consists of two parts, whereby the attached shock impinges almost perpendicularly the downstream detached shock in such a way that a type IV shock–shock interaction as in Edney (1968a) emerges (see Fig. 16a) effecting a supersonic jet impingement on the model surface. At the intersection of both shocks, a short-transmitted shock connects both parts of the detached shock, whereby a supersonic jet as an extension of the impingement shock also forms downstream of this transmitted shock. The supersonic jet flows toward the model surface as it separates two regions of subsonic flow. Close to the impingement point, the supersonic jet is primarily deflected outward and follows the contour until the edge of the transition to the cylindrical part, where it separates and reattaches shortly after. Thus, a separation bubble develops underneath the jet causing a downstream reattachment shock. This shock impinges the detached shock provoking a type VI shock–shock interaction. In conclusion, the region of peak surface pressures due to supersonic jet impingement seems to be reasonable for the shown differences in the aerodynamic coefficients between cylinder and dogbone. On the one hand, this peak pressure on the lower surface causes a great negative pitching moment. On the other hands, the surface’s normal vectors close to the impingement are almost parallel to the inflow direction, which is why the resulting pressure force consists mainly of a drag proportion instead of a lift proportion in comparison to the cylindrical body without cavity.

It is crucial to note that the numerical aerodynamic coefficients of the dogbone determined by US3D reveal a good agreement with the experimental data as shown in Fig. 15. The maximum absolute deviations here are very low with 0.061 in the drag coefficient and with 0.054 in the lift coefficient both at \(\vartheta =45^{\circ }\). Yet, a single significant deviation can be observed in the pitching moment coefficient at a pitch angle of \(30^{\circ }\), which is in the uncertainty margin of both values. For this configuration, the impingement as a result of SSI is close to the interface between cavity and cylindrical part.

Furthermore, dogbone’s hypersonic approximation determined by modified Newtonian flow theory is shown in Fig. 15, as well. At the first glance, the experimental and analytic data of the dogbone reveal a fairly good agreement. In particular, the approximations of lift force and pitching moment coefficient are consistent with the results of the free-flight tests having a maximum absolute deviation in lift coefficient of 0.174 (see Fig. 15a). The analytical drag coefficient shows a good consistency with the order of magnitude compared to the experimental data but a different development with respect to the body inclination. Thereby, the approximated drag coefficient starts to increase at a smaller pitch angle with increasing inclination reaching the maximum at \(19^{\circ }\) before it experiences an unexpected alternation of decreasing and increasing. The larger deviations in the aerodynamic coefficients can be explained by the neglect of SSI effects on surface pressure distribution by the use of the Newtonian flow theory.

Interestingly, the course of pitching moment coefficient in Fig. 15 is considerably different between the cylindrical body with and without cavity, whereby the present cavity causes a continuously negative pitching moment over the entire range of \(90^{\circ }\) with a roughly 5 times higher absolute peak value of the dogbone than of the cylinder. In contrast to two stable plus two unstable trim points of the cylinder in the range between \(0^{\circ }\) and \(90^{\circ }\), the dogbone merely has one stable and one unstable trim point. These two trim points of the dogbone are at the same pitch angle as of the cylinder but with a contrary stability behavior. The most remarkable result from these data is that the present axially symmetrical cavity effects a reversal of the static stability behavior at a zero pitch angle with considerable magnitude by contrast with the body without cavity. Thereby, the dogbone’s stability derivative of \(-1.172\,{\mathrm{rad}}^{-1}\) at a pitch angle of \(0^{\circ }\) is roughly 5 times greater than of the cylinder’s peak stability derivative at \(90^{\circ }\). As a result, it is more likely that a randomly tumbling cylindrical body with the present axially symmetric cavity tends to a base-exposed configuration as a final flight attitude.