3.1.1 Wake sensitivity to the body clearance and inclinations

The four sensitivity maps of the base pressure gradient due to the variation of the ground clearance $c^{\ast }$ are shown in figure 3(a,b) for the Cartesian components and figure 3(c,d) for the polar form. When the body is gradually raised in figure 3(a), the most probable branch for $g_{y}^{\ast }$ observed for $c^{\ast }<0.080$ bifurcates in two symmetrical branches resulting from an instability. This was already fully described by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ), Cadot et al. (Reference Cadot, Evrard and Pastur2015) showing that the ground proximity suppresses the instability towards a symmetric wake through a pitchfork bifurcation. The region we consider is for large values of $c^{\ast }$ for which the wake dynamics is dominated by the stochastic exploration of the two symmetric branches. Independently of the sign of $g_{y}^{\ast }$ (i.e. the random switching dynamics) the permanent asymmetry introduced by the symmetry-breaking (SB) modes can be seen in the modulus $g_{r}^{\ast }$ displayed in figure 3(c). The modulus saturates to a constant value when $c^{\ast }>c_{S}^{\ast }\simeq 0.105$ as shown in the figures. The regime $c^{\ast }>c_{S}^{\ast }$ corresponds to the unstable wake due to the $y$ - or $z$ -instability (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013a ), and will be referred to as the saturated regime throughout. The interesting result is that, whilst the vertical base pressure gradient $g_{z}^{\ast }$ decreases significantly, the gradient modulus $g_{r}^{\ast }$ remains constant and corresponds to a change in the gradient orientation $\unicode[STIX]{x1D711}$ .

Figure 3. Base pressure gradient response to a variation of the ground clearance $c^{\ast }$ for the square-back body. Sensitivity maps (a) $f(c^{\ast },g_{y}^{\ast })$ , (b) $f(c^{\ast },g_{z}^{\ast })$ , (c)  $f(c^{\ast },g_{r}^{\ast })$ and (d) $f(c^{\ast },\unicode[STIX]{x1D711})$ . The clearance $c_{S}^{\ast }\simeq 0.105$ is defined as the threshold from which the instability is saturated. See discussion § 4.1 for white symbols.

Figure 4. Base pressure gradient response to variations of the yaw angle $\unicode[STIX]{x1D6FD}$ for the square-back body. Sensitivity maps (a) $f(\unicode[STIX]{x1D6FD},g_{y}^{\ast })$ , (b) $f(\unicode[STIX]{x1D6FD},g_{z}^{\ast })$ , (c) $f(\unicode[STIX]{x1D6FD},g_{r}^{\ast })$ and (d) $f(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D711})$ . See discussion § 4.1 for white symbols.

Figure 5. Base pressure gradient response to variations of the pitch angle $\unicode[STIX]{x1D6FC}$ for the square-back body. Sensitivity maps (a) $f(\unicode[STIX]{x1D6FC},g_{y}^{\ast })$ , (b) $f(\unicode[STIX]{x1D6FC},g_{z}^{\ast })$ , (c) $f(\unicode[STIX]{x1D6FC},g_{r}^{\ast })$ and (d) $f(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D711})$ . See discussion § 4.1 for white symbols.

Figure 6. Modulus $g_{r\unicode[STIX]{x1D6FC}}^{\ast }(t^{\ast })$ and phase $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D6FC}}(t^{\ast })$ time series (left) of the base pressure gradient for the square-back body. Corresponding probability density functions (PDFs) (right) for (a) nose-down $\unicode[STIX]{x1D6FC}=-1^{\circ }$ , (b) baseline $\unicode[STIX]{x1D6FC}=0^{\circ }$ and (c) nose-up $\unicode[STIX]{x1D6FC}=+1^{\circ }$ . The smooth red lines superimposed on the PDFs ( $g_{r}^{\ast }$ ) in (a), (b) and (c) are best fits of Rigas et al.’s (Reference Rigas, Morgans, Brackston and Morrison2015) PDF model; the three parameters of the fit are not given.

For the next two series of experiments concerning yaw and pitch sensitivities, we will consider small variations of the inclination around the baseline (see table 2). The baseline has most probable gradients that take phase values very close to $\unicode[STIX]{x1D711}\simeq 0$ or $\unicode[STIX]{x03C0}$ and a modulus $g_{r}^{\ast }\simeq 0.187$ obtained in figure 3(c).

The sensitivity analysis to small variations of the yaw angle $\unicode[STIX]{x1D6FD}$ with a fixed ground clearance $c^{\ast }=0.168$ and pitch $\unicode[STIX]{x1D6FC}=0^{\circ }$ is shown in figure 4. The main result observed in figure 4(a) and previously reported in Cadot et al. (Reference Cadot, Evrard and Pastur2015), Volpe et al. (Reference Volpe, Devinant and Kourta2015), Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016) is a discontinuous transition between two opposite branches of $g_{y}^{\ast }$ which implies a phase jump (figure 4 d) between values close to $0$ and $\unicode[STIX]{x03C0}$ . Nevertheless, this transition occurs with a fairly constant modulus $g_{r}^{\ast }$ as shown in figure 4(c). The vertical component $g_{z}^{\ast }$ in figure 4(b) has a slight, unexpected affine variation with the yaw which is likely to be due to an imperfection of the set-up, coming from multiple sources such as wind inhomogeneity, non-zero roll angle and cable passage behind the rear right-hand side cylindrical support. Because of the constant modulus, the set-up imperfections affect slightly the phase $\unicode[STIX]{x1D711}$ (figure 3 d) which slightly deviates from $\unicode[STIX]{x1D711}_{0}=0$ or $\unicode[STIX]{x1D711}_{0}=\unicode[STIX]{x03C0}$ .

We now turn to the sensitivity to the pitch angle $\unicode[STIX]{x1D6FC}$ . The yaw angle is set to $\unicode[STIX]{x1D6FD}=0^{\circ }$ and the front and rear ground clearances are adjusted for a given pitch but keeping $c^{\ast }=(c_{f}^{\ast }+c_{r}^{\ast })/2=0.168$ in order to recover the baseline when $c_{f}^{\ast }=c_{r}^{\ast }$ . The sensitivity results of the base pressure gradient are shown in figure 5. Looking at figure 5(a), which shows the sensitivity of the horizontal gradient component $g_{y}^{\ast }$ , we can distinguish three regimes. At large nose-down, for $\unicode[STIX]{x1D6FC}\lesssim -0.75^{\circ }$ , there is a single branch located around $g_{y}^{\ast }=0$ , which bifurcates in two opposite branches in the range $-0.75^{\circ }\lesssim \unicode[STIX]{x1D6FC}\lesssim 0.5^{\circ }$ . In the last regime with nose-up, for which $\unicode[STIX]{x1D6FC}\gtrsim 0.5^{\circ }$ , the horizontal component varies almost uniformly in a wide range $|g_{y}^{\ast }|\lesssim 0.2$ .

The effect of the pitch on the vertical gradient component $g_{z}^{\ast }$ is shown in figure 5(b). It displays a monotonic variation with the pitch angle in the range $-0.75^{\circ }\lesssim \unicode[STIX]{x1D6FC}\lesssim 0.5^{\circ }$ coincidently with the two branches observed for $g_{y}^{\ast }$ . Apart from this range, the vertical component is saturated to extreme values. Although these three regimes are very different, the modulus shown in figure 5(c) displays an almost symmetric evolution, with decreases on both sides of the maximum. In the third regime, the phase is unlocked in the $[-\unicode[STIX]{x03C0},0]$ interval with a rather uniform exploration (figure 5 d), unlike nose-down regimes. This different behaviour is attributed to the wall proximity at the trailing edge which has a major effect on the base flow in nose-up configurations.

Most importantly, a similar conclusion as for the ground clearance and the yaw experiments can be drawn. The small pitch angle variation produces a component $g_{z}^{\ast }$ of the vertical pressure gradient. The modulus $g_{r}^{\ast }$ is slightly modulated, maximum for horizontal orientation and minimum for vertical one.

Figure 7. Cross-sections of the mean velocity field of the square-back body visualized using streamlines superimposed on the modulus of the components in the vertical plane $(x^{\ast },z^{\ast })$ (a,c,e) and horizontal plane $(x^{\ast },y^{\ast })$ (b,d,f) for (a,b) nose-down $\unicode[STIX]{x1D6FC}=-1^{\circ }$ , (c,d) baseline $\unicode[STIX]{x1D6FC}=0^{\circ }$ ( $P$ state, see text) and (e,f) nose-up $\unicode[STIX]{x1D6FC}=+1^{\circ }$ .

The wake dynamics of the four configurations of the base pressure gradient orientations: vertical positive, horizontal (positive and negative) and vertical negative is now respectively investigated through the pitch angles $\unicode[STIX]{x1D6FC}=-1^{\circ }$ , $\unicode[STIX]{x1D6FC}=0^{\circ }$ and $\unicode[STIX]{x1D6FC}=1^{\circ }$ . We show the corresponding time series of the gradient modulus $g_{r\unicode[STIX]{x1D6FC}}^{\ast }(t^{\ast })$ and phase $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D6FC}}(t^{\ast })$ in figure 6. The nose-down at $\unicode[STIX]{x1D6FC}=-1^{\circ }$ in figure 6(a) clearly depicts a phase lock-in for $\unicode[STIX]{x1D711}_{-1^{\circ }}(t^{\ast })$ at $\unicode[STIX]{x03C0}/2$ with turbulent fluctuations. When the pitch is set to zero (baseline configuration), the phase dynamics in figure 6(b) consists of random jumps of $\unicode[STIX]{x03C0}$ and elsewhere to very long-time duration of typically $\unicode[STIX]{x1D6FF}t^{\ast }=1000$ of phase lock-in at $0$ and $-\unicode[STIX]{x03C0}$ . This long-time bistable dynamics was fully described in Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013b ) using Cartesian coordinates for the gradient. It is worth mentioning that the dynamics is not only mainly made of phase jumps, since some events of the phase $\unicode[STIX]{x1D711}_{0^{\circ }}$ occur around $-\unicode[STIX]{x03C0}/2$ with long-time evolution related to phase drift or wake rotation (see for instance within the time interval $[1;2]\times 10^{3}$ in figure 6 b). For the nose-up case, there is no phase lock-in but large phase fluctuations associated with random wake rotations exploring the range $[-\unicode[STIX]{x03C0},0]$ with some phase jumps like the one observed at $t^{\ast }=4.5\times 10^{3}$ in figure 6(c). There are obviously some similarities with the diffusive dynamics of the turbulent axisymmetric wake of Rigas et al. (Reference Rigas, Oxlade, Morgans and Morrison2014, Reference Rigas, Morgans, Brackston and Morrison2015), the main difference being the random walk of the phase that is bounded. Since the modulus is decoupled from the phase in Rigas et al.’s (Reference Rigas, Morgans, Brackston and Morrison2015) model, it is possible to present a best fit of the statistics of the gradient modulus $g_{r}^{\ast }$ shown as the red lines in figure 6.

In the following, $N$ and $P$ states terminology refers to negative and positive horizontal base pressure gradient as introduced by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013b ). Making use of this definition, the state depicted in figure 1(a) is state $N$ . The mean flows of the wake measured in the two perpendicular planes confirm the four different wake orientations according to the base pressure gradient alignments. For the baseline case, the mean flow has been conditioned by $-\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D711}_{0^{\circ }}<\unicode[STIX]{x03C0}/2$ in order to capture the $P$ state only (i.e. phase lock-in $\unicode[STIX]{x1D711}_{0^{\circ }}(t\ast )\simeq 0$ ) in figure 7(c,d). The $N$ state (i.e. phase lock-in $\unicode[STIX]{x1D711}_{0^{\circ }}(t\ast )\simeq \unicode[STIX]{x03C0}$ ) is not shown. It is the mirror of the $P$ state using the transformation $(y^{\ast },z^{\ast })\rightarrow (-y^{\ast },z^{\ast })$ . We can see in figure 7 that the strong wake asymmetry is successively detected in the $y^{\ast }=0$ plane in figure 7(a), in the $z^{\ast }=0$ plane in figure 7(d) and in the $y^{\ast }=0$ plane in figure 7(e) while the wake in the other perpendicular plane is always more symmetric. They respectively correspond to the phase lock-in $\unicode[STIX]{x1D711}_{-1^{\circ }}\simeq \unicode[STIX]{x03C0}/2$ (figure 7 a,b), $\unicode[STIX]{x1D711}_{0^{\circ }}\simeq 0$ (state $P$ in figure 7 c,d) or equivalently $\unicode[STIX]{x1D711}_{0^{\circ }}\simeq \unicode[STIX]{x03C0}$ (state $N$ , not shown) and finally to the mean orientation (with $-\unicode[STIX]{x03C0}<\unicode[STIX]{x1D711}_{1^{\circ }}(t^{\ast })<0$ ) towards a negative vertical pressure gradient (figure 7 e,f). A supplementary movie available at https://doi.org/10.1017/jfm.2018.630 provides the dynamics of wake pressure imprints at the base of the body for the pitch angles from which pressure gradients shown in figure 6 are extracted.

As a summary of the wake sensitivity to the body orientation in the saturated regime of the instability, it is found that independently of the phase dynamics, the modulus of the base pressure gradient depends upon the vector orientation, maximum for a horizontal alignment with $g_{r}^{\ast }(0\text{ or }\unicode[STIX]{x03C0})\simeq 0.187$ and minimum for a vertical alignment with $g_{r}^{\ast }(\unicode[STIX]{x03C0}/2)\simeq 0.159$ . Ground clearance, yaw and pitch variations produce a vertical component $g_{z}^{\ast }$ of the pressure gradient that constrains the phase dynamics because of the modulus properties. Different phase dynamics scenarios have been identified: phase locks-in, phase jumps and bounded drifts. As depicted in the mean PIV fields, the phase of the base pressure gradient is confirmed to be an accurate indicator of the global wake orientation.

3.1.2 Wake sensitivity with a rear cavity

The sensitivity analyses have been repeated in exactly the same conditions but with the rear cavity. For the sake of brevity, the three sensitivity experiments are grouped in figure 8 for Cartesian coordinates and in figure 9 for the polar form. The ground clearance bifurcation observed in figure 3(a) has been completely suppressed by the rear cavity in figure 8(a) as expected because of the stabilization observed by Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016). For the baseline configuration with the rear cavity, both Cartesian components of the base gradient are approximately zero. From figure 8(c,d,e,f), we see that a slight change of yaw (respectively the pitch) shifts almost linearly the horizontal component $g_{y}^{\ast }$ (respectively vertical component $g_{z}^{\ast }$ ) while the other component $g_{z}^{\ast }$ (respectively $g_{y}^{\ast }$ ) remains constant. However, even with the cavity, a saturation of the varying component is observed for angles, either yaw or pitch, larger than $\pm 1^{\circ }$ in figure 8(c,f).

Figure 8. Base pressure gradient response to ground clearance $c^{\ast }$ (a,b), yaw $\unicode[STIX]{x1D6FD}$ (c,d) and pitch angle $\unicode[STIX]{x1D6FC}$ (e,f) variations for the square-back body with a rear cavity. Sensitivity maps $f$ (same colour bar for $f$ as in figures 3–5) of the horizontal component $g_{y}^{\ast }$ (a,c,e) and vertical component $g_{z}^{\ast }$ (b,d,f).

Figure 9. Base pressure gradient response to ground clearance $c^{\ast }$ (a,b), yaw $\unicode[STIX]{x1D6FD}$ (c,d) and pitch angle $\unicode[STIX]{x1D6FC}$ (e,f) variations for the square-back body with a rear cavity. Sensitivity maps $f$ (same colour bar for $f$ as in figures 3–5) of the gradient modulus $g_{r}^{\ast }$ (a,c,e) and phase $\unicode[STIX]{x1D711}$ (b,d,f).

The modulus and phase sensitivity maps are shown in figure 9. Despite the fact that the modulus is considerably reduced (by more than a factor 2), it is now a function of the yaw and pitch angles that reaches a minimal value around the baseline where the gradient changes sign. The complex phase dynamics observed without the cavity (phase drifts and jumps) is now replaced by trivial permanent lock-in except when the phase is poorly defined for small modulus. This loss of phase dynamics is the consequence of the suppression of the wake instability.

The mean wake of the body with the rear cavity is presented in figure 10 which can be directly compared to the same plane views without the cavity in figure 7. The wake observed for the nose-down configuration at $\unicode[STIX]{x1D6FC}=-1^{\circ }$ in figure 10(a,b) is similar to that of the figure 7(a,b) albeit that with the cavity the separation from the ground is prevented in figure 10(a) and the wake reflectional symmetry is better checked in figure 10(b). For the baseline configuration in figure 10(c,d) the wake is fully symmetrized as in Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016), Lucas et al. (Reference Lucas, Cadot, Herbert, Parpais and Délery2017). The wake for the nose-up configuration at $\unicode[STIX]{x1D6FC}=+1^{\circ }$ in figure 10(e,f) is also quite similar to that of figure 7(e,f) without the cavity. Hence, for the two extreme cases $\unicode[STIX]{x1D6FC}=\pm 1^{\circ }$ , the pitch angle of the body orientates the wake in the vertical direction with a similar strength as without the cavity.

Figure 10. Cross-sections of the mean velocity field of the square-back body with the rear cavity visualized using streamlines superimposed on the modulus of the components in the vertical plane $(x^{\ast },z^{\ast })$ (a,c,e) and horizontal plane $(x^{\ast },y^{\ast })$ (b,d,f) for (a,b) nose-down $\unicode[STIX]{x1D6FC}=-1^{\circ }$ , (c,d) baseline $\unicode[STIX]{x1D6FC}=0^{\circ }$ and (e,f) nose-up $\unicode[STIX]{x1D6FC}=+1^{\circ }$ .

Figure 11. Mean (a–d) and fluctuating (e–h) cross-flow force coefficients versus yaw (a,b,e,f) and pitch angle (c,d,g,h) of the square-back body (black filled circles) and with the rear cavity (empty circles).

Figure 12. Mean side force coefficient versus yaw $\unicode[STIX]{x1D6FD}$ of the square-back body. In (a), measured force coefficient (symbols), basic flow coefficient (red line) and coefficient computed from (3.1) (blue dashed line, see text). In (b), mean contribution of the instability $C_{y}-C_{y}^{B}$ (filled circles) and mean horizontal base pressure gradient $G_{y}^{\ast }$ (crosses).

Figure 13. Components of the fluctuation force coefficients of the square-back body $C_{y}^{\prime }$ and $C_{z}^{\prime }$ (filled circles) compared to the fluctuation base pressure gradient ${G_{y}^{\ast }}^{\prime }$ and ${G_{z}^{\ast }}^{\prime }$ (crosses) versus yaw  $\unicode[STIX]{x1D6FD}$ (a,c) and pitch angle  $\unicode[STIX]{x1D6FC}$ (b,d).

3.1.3 Global force sensitivity versus wake aerodynamic loading

The $y$ - and $z$ -components of the mean force coefficients obtained for the yaw and pitch sensitivity experiments are shown in figure 11(a–d) with and without the rear cavity. A clear stabilization of the unstable wake can be seen in the mean horizontal $y$ -component in figure 11(a) obtained with the yaw variations. The nonlinear behaviour due to the instability (also reported in Perry et al. (Reference Perry, Pavia and Passmore2016b )) is replaced by a linear law with the cavity. For the other mean coefficients, we can hardly distinguish any effect induced by the rear cavity. For the force coefficient fluctuations shown in figure 11(e–h), one can observe fluctuation crisis each time phase unlocking occurs.

Figure 11(a) suggests that the mean side force coefficient $C_{y}^{B}$ obtained without the instability (the superscript  $^{B}$ will refer to this basic flow) should evolve linearly with $\unicode[STIX]{x1D6FD}$ as for the cavity experiment. The force coefficient $C_{y}^{B}(\unicode[STIX]{x1D6FD})=-0.055\times \unicode[STIX]{x1D6FD}$ (where  $\unicode[STIX]{x1D6FD}$ is expressed in degrees), simply obtained from a linear fit of the cavity experiment, is shown in figure 12(a) as the red straight line. We compare the strength of the instability $C_{y}-C_{y}^{B}$ to the mean horizontal base pressure gradient $G_{y}^{\ast }$ in figure 12(b). The two curves are satisfactorily proportional such that the mean side force coefficient can be directly related to the mean base gradient as:

(3.1) $$\begin{eqnarray}\displaystyle C_{y}=C_{y}^{B}-\frac{G_{y}^{\ast }}{10}. & & \displaystyle\end{eqnarray}$$

We can see in figure 12(a) the good agreement of (3.1) (blue dashed line) with the experimental data (symbols). An instantaneous relationship can be speculated from (3.1) to take into account the unsteady loading due to the wake instability through the dynamics of the base gradient:

(3.2) $$\begin{eqnarray}\displaystyle c_{y}(t^{\ast })=C_{y}^{B}-\frac{g_{y}^{\ast }(t^{\ast })}{10} & & \displaystyle\end{eqnarray}$$

implying that $C_{y}^{\prime }={G_{y}^{\ast }}^{\prime }/10$ . The fluctuation force coefficient $C_{y}^{\prime }$ is plotted in figure 13(a) for yaw variation and figure 13(b) for pitch variation and matches satisfactorily with ${G^{\ast }}_{y}^{\prime }/10$ .

Since the mean lift coefficient $C_{z}$ is not affected by the instability (no differences in figure 11(b,d) with or without the rear cavity), we can state that for the vertical force coefficient:

(3.3) $$\begin{eqnarray}\displaystyle C_{z}=C_{z}^{B} & & \displaystyle\end{eqnarray}$$

regardless of the base pressure gradient orientation. Again, an instantaneous relationship but with only the fluctuations of the vertical gradient component can be speculated:

(3.4) $$\begin{eqnarray}\displaystyle c_{z}(t^{\ast })=C_{z}^{B}-\frac{{g^{\ast }}_{z}^{\prime }(t^{\ast })}{10} & & \displaystyle\end{eqnarray}$$

implying that ${C_{z}}^{\prime }={G_{z}^{\ast }}^{\prime }/10$ . Although the fluctuations are much smaller vertically than horizontally, we still see good agreements of (3.4) in figure 13(c,d).

One may wonder if a similar relationship could be obtained from the drag $c_{x}$ and the base suction $c_{b}$ coefficients. Actually, independently of the ground clearance, the yaw or the pitch, the rear cavity has a beneficial effect on the mean base suction or mean drag coefficient that are respectively reduced within the ranges $[23\,\%,25\,\%]$ and $[9.5\,\%,10.5\,\%]$ as can be seen in figure 14(a–d). Comparable drag reduction is also observed below the instability threshold ( $c^{\ast }<0.105$ ). As shown in Lucas et al. (Reference Lucas, Cadot, Herbert, Parpais and Délery2017), a deep rear cavity has a primary effect of considerably lengthening the recirculating region towards the inside of the body. Consequently, low pressure sources therein are reduced independently from the symmetry properties of the wake. It then remains impossible to distinguish the contribution from the wake symmetrization due to the instability suppression to this strong mean flow modification. Hence, from our point of view, the drag of the basic flow $C_{x}^{B}$ is not extractable from the data. Considering all yaw and pitch variations, the fluctuations of both the base suction and drag remain almost identical to those of the baseline given in table 2 as can be seen in figure 14(e–h). There are then no consequences of the unstable wake phase dynamics on these coefficients. The reason is that base suction is more related to the magnitude of the gradient’s modulus that was shown to be almost constant rather than to its orientation. The correlation between base suction and the gradient’s modulus was previously reported in the sensitivity analysis to a disturbance placed in the wake by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014b ) or instantaneously by Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016).

Figure 14. Mean (a–d) and fluctuating (e–h) drag force and base suction coefficients versus yaw (a,b,e,f) and pitch angle (c,d,g,h) of the square-back body (black filled circles) and with the rear cavity (empty circles).

3.2.1 Wake sensitivity to the body clearance and orientations

The four sensitivity maps of the base pressure gradient due to the variation of the ground clearance $c^{\ast }$ are shown in figure 15(a,b) for the Cartesian components and figure 15(c,d) for the polar form. When the ground clearance is gradually increased in figure 15(a), the horizontal component $g_{y}^{\ast }$ remains almost around zero. The vertical component $g_{z}^{\ast }$ (figure 15 b) experiences a jump for $c^{\ast }\simeq 0.080$ from a positive branch ( $g_{z}^{\ast }\simeq 0.1$ ) to a negative one ( $g_{z}^{\ast }\simeq -0.1$ ). A bistable dynamics is then obtained around $c^{\ast }\simeq 0.080$ . In the experiment of Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ), the ground clearance is non-dimensionalized using the body width, $c_{W}^{\ast }=c^{\ast }/W^{\ast }$ , and the $z$ -instability is found to saturate on the positive branch only for a ground clearance $c_{SW}^{\ast }\simeq 0.060$ and then switches to the negative branch at $c_{W}^{\ast }\simeq 0.1$ . Our experiment satisfactorily reproduces the switch from the positive to the negative branch but with smaller ground clearance. Using non-dimensionalization of Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ), the switch we observe at $c^{\ast }\simeq 0.080$ in figure 15 is within the range $0.068\lesssim c_{W}^{\ast }\lesssim 0.085$ when translated using either $W^{\ast }$ or $w_{b}^{\ast }$ of the boat-tailed body. The range remains below the value $c_{W}^{\ast }\simeq 0.1$ of Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ). This can be ascribed to a Reynolds number effect, which is approximately ten times larger in the present study ( $Re\simeq 4.0\times 10^{5}$ ) than in Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ) ( $Re\simeq 4.5\times 10^{4}$ ). It is in accordance with Reynolds number effect studied in Cadot et al. (Reference Cadot, Evrard and Pastur2015) who report a $30\,\%$ decrease of the critical ground clearance for the $y$ -instability when the Reynolds number is increased from $1.7\times 10^{4}$ to $1.6\times 10^{5}$ . The critical ground clearance $c_{S}^{\ast }$ of the $z$ -instability is not observed in our study because it is below the minimal value $c^{\ast }=0.05$ allowed by the set-up. It is also found that the critical ground clearance is larger for the $y$ -instability than for the $z$ -instability as in Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a ). The modulus $g_{r}^{\ast }$ displayed in figure 15(c) has a value close to $0.1$ that is smaller than the modulus found within the range $0.15{-}0.2$ (see figures 3 c, 4 c, 5 c) in § 3.1 for the square-back body. This must be ascribed to the different after-body geometry. The larger fluctuations observed for $c^{\ast }\simeq 0.080$ are a consequence of the bistable dynamics at the branches switch in figure 15(d).

Figure 15. Base pressure gradient response to a variation of the ground clearance $c^{\ast }$ for the boat-tailed body. Sensitivity maps (a) $f(c^{\ast },g_{y}^{\ast })$ , (b) $f(c^{\ast },g_{z}^{\ast })$ , (c)  $f(c^{\ast },g_{r}^{\ast })$ and (d) $f(c^{\ast },\unicode[STIX]{x1D711})$ .

Figure 16. Base pressure gradient response to variations of the yaw angle $\unicode[STIX]{x1D6FD}$ for the boat-tailed body. Sensitivity maps (a) $f(\unicode[STIX]{x1D6FD},g_{y}^{\ast })$ , (b) $f(\unicode[STIX]{x1D6FD},g_{z}^{\ast })$ , (c) $f(\unicode[STIX]{x1D6FD},g_{r}^{\ast })$ and (d) $f(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D711})$ .

Figure 17. Base pressure gradient response to variations of the pitch angle $\unicode[STIX]{x1D6FC}$ for the boat-tailed body. Sensitivity maps (a) $f(\unicode[STIX]{x1D6FC},g_{y}^{\ast })$ , (b) $f(\unicode[STIX]{x1D6FC},g_{z}^{\ast })$ , (c) $f(\unicode[STIX]{x1D6FC},g_{r}^{\ast })$ and (d) $f(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D711})$ .

Figure 18. Modulus $g_{r\unicode[STIX]{x1D6FC}}^{\ast }(t^{\ast })$ and phase $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D6FC}}(t^{\ast })$ time series (left) of the base pressure gradient for the boat-tailed body. Corresponding probability density functions (right) for (a) $\unicode[STIX]{x1D6FD}=-3^{\circ }$ , (b) $\unicode[STIX]{x1D6FD}=-5^{\circ }$ . The smooth red lines superimposed on the PDFs ( $g_{r}^{\ast }$ ) in (a) and (b) are best fits of Rigas et al.’s (Reference Rigas, Morgans, Brackston and Morrison2015) PDF model; the three parameters of the fit are not given.

Figure 19. Cross-sections of the mean velocity field for the boat-tailed body visualized using streamlines superimposed on the modulus of the components in the vertical plane $(x^{\ast },z^{\ast })$ for (a) baseline configuration ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}=0^{\circ }$ , $c^{\ast }=0.168$ ), (b)  $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}=0^{\circ }$ , $c^{\ast }=0.060$ , (c) $\unicode[STIX]{x1D6FC}=0^{\circ }$ , $\unicode[STIX]{x1D6FD}=-5^{\circ }$ , $c^{\ast }=0.168$ , (d) nose-down configuration $\unicode[STIX]{x1D6FC}=-0.6^{\circ }$ ( $\unicode[STIX]{x1D6FD}=0^{\circ }$ ). The associated mean base pressure distributions $C_{p}(y^{\ast },z^{\ast })$ are also provided.

For the next two series of experiments concerning yaw and pitch sensitivities, we will consider small variations of the inclination around the baseline with characteristics given in table 2.

The sensitivity analysis to the yaw angle $\unicode[STIX]{x1D6FD}$ with a fixed pitch $\unicode[STIX]{x1D6FC}=0^{\circ }$ is shown in figure 16. The results being reasonably symmetric with respect to $\unicode[STIX]{x1D6FD}$ , we only comment the part for $\unicode[STIX]{x1D6FD}<0^{\circ }$ . From these figures, four distinctive regions can be observed, successively named $l_{1}$ , $b_{1}$ , $b_{2}$ and $l_{2}$ in figure 16(a). They are delimited in all the figures by vertical dashed lines. In region $l_{1}$ the base pressure gradient remains almost identical to that of the baseline: it is locked with a vertical direction orientated towards the ground corresponding to $\unicode[STIX]{x1D711}\simeq -\unicode[STIX]{x03C0}/2$ in figure 16(d) and a modulus close to $g_{r}^{\ast }\simeq 0.1$ in figure 16(c). When the yaw angle $\unicode[STIX]{x1D6FD}$ is increased, the wake becomes bistable in the region $b_{1}$ and the gradient experiences small phase jumps of amplitude $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}_{1}\simeq \unicode[STIX]{x03C0}/7$ without noticeable changes in the modulus $g_{r}^{\ast }$ . This small lateral bistability has not been observed for yaw experiments at lower ground clearance – for which transitions occur at smaller angles – presented in § 3.2.2. Although not discernible on the side pressure measurements, it is suspected to be caused by premises of separation on the leeward boat tail. For larger yaw angles, another bistable regime in $b_{2}$ is observed with much larger phase jumps of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}_{2}\simeq 3\unicode[STIX]{x03C0}/4$ in figure 16(d) associated with large fluctuations of the modulus. In the last regime $l_{2}$ , the gradient is locked with a vertical direction orientated towards the top ( $\unicode[STIX]{x1D711}\simeq +\unicode[STIX]{x03C0}/2$ ) and a modulus close to $g_{r}^{\ast }\simeq 0.15$ . The main result is that the yaw around $\unicode[STIX]{x1D6FD}=\pm 4.5^{\circ }$ triggers the switch of the negative vertical pressure gradient component of the baseline observed in $l_{1}$ to a positive vertical component in $l_{2}$ as clearly shown in figure 16(b).

The sensitivity towards the pitch angle $\unicode[STIX]{x1D6FC}$ is now given in figure 17. For this experiment, the yaw angle is set to $\unicode[STIX]{x1D6FD}=0^{\circ }$ and, as in § 3.1, the front and rear ground clearances are adjusted for a given pitch but keeping $c^{\ast }=(c_{f}^{\ast }+c_{r}^{\ast })/2=0.168$ . Based on the mechanism identified for the $y$ -instability in § 3.1, we restricted the range of pitch angles to nose-down configurations for which a clear transition is observed. The pitch angle has almost no effect on the horizontal component in figure 17(a) except for a disturbance around $\unicode[STIX]{x1D6FC}=-0.4^{\circ }$ that corresponds to the switching region with a bistable dynamics in figure 17(b) of the vertical component of the gradient.

The successive experiments changing ground clearance, pitch and yaw mainly show that for the boat-tailed body subjected to the $z$ -instability, two principal orientations of the base pressure gradient are observed: almost positive vertical and negative vertical. The phase then remains locked except for the bistable dynamics where the wake switches between opposite orientations. The dynamics of the two bistable cases $b_{1}$ and $b_{2}$ evidenced during the yaw sensitivity in figure 16(c) is respectively presented in figure 18(a) and figure 18(b). The bistable dynamics $b_{2}$ is similar to that obtained in ground clearance sensitivity (figure 15 c) and pitch sensitivity (figure 17 c) analyses. The striking observation is that the rate of random switching between the two most probable phases is substantially larger than that of the bistable wake subjected to the $y$ -instability in figure 6(b). For the most important bistable behaviour that involves vertical wake reversals in figure 18(b), approximately $50$ switches are observed against only $9$ in figure 6(b) for horizontal wake reversals. This trend was checked for all bistable cases, thus confirming that the characteristic time of the random switching is approximately five times smaller for the $z$ -instability than for the $y$ -instability.

The different wake configurations are illustrated by their mean flow in the vertical plane in figure 19 together with their corresponding mean base pressure distribution. Because of the boat tailing, the after-body develops four strong longitudinal vortices at each corner of the base affecting considerably the mean bubble closure in the figure 19 compared to that observed for the square-back after-body in figures 7 and 10. For the baseline in figure 19(a), the negative vertical pressure gradient observed in the pressure distribution is associated with a skewed feedback flow orientated towards the bottom edge of the base that is the best topological indicator of the $z$ -instability. We have identified wake reversals from the baseline for each of the three sensitivity experiments either by decreasing the ground clearance, increasing the yaw angle or by pitching down the body. For a lower ground clearance, the wake has effectively switched to the opposite base pressure gradient in figure 19(b) with a skewed feedback flow orientated to the top edge of the base. For a large yaw in figure 19(c), the switch from the baseline is not clearly observable in the vertical velocity field, while it is clearly established in the corresponding pressure distribution. Finally, for large nose-down pitch, the skewed feedback flow and the pressure distribution are clearly inverted from those of the baseline.

3.2.2 Exploration of the most probable states

We now investigate the wake reversals with yaw for different fixed ground clearances: $c^{\ast }=0.060$ ; $0.080$ ; $0.124$ ; $0.168$ and a pitch angle set at $\unicode[STIX]{x1D6FC}=0^{\circ }$ . With no yaw angle, we can see from figure 15(a,b) that these ground clearances successively correspond to a locked positive vertical gradient for $c^{\ast }=0.060$ , a bistable dynamics for $c^{\ast }=0.080$ and a locked negative vertical gradient for $c^{\ast }=0.124$ and $c^{\ast }=0.168$ . It is remarkable that the ground clearance has almost no effect on the gradient modulus in figure 15(c). So, it seems that the ground orientates the unstable wake with a positive vertical gradient for small $c^{\ast }$ and with a negative vertical gradient for large $c^{\ast }$ . At large ground clearance we would have expected the ground to become irrelevant and that the recovery of the top/down symmetry of the flow would have led to a bistable dynamics independently of the ground proximity. It is then the presence of the four supports that is responsible for the locked wake orientation at large ground clearance.

Figure 20. Most probable pressure gradients for the boat-tailed body $g_{y}^{\ast }$ and $g_{z}^{\ast }$ (a), $g_{r}^{\ast }$ (b) and $\unicode[STIX]{x1D711}$ (c) versus the yaw angle at different ground clearances (with descending values from top to bottom in a). The triangles at $\unicode[STIX]{x1D6FD}=0^{\circ }$ correspond to the experiment with a fifth cylindrical support (see text).

Sensitivity analyses with the yaw angle limited to $\unicode[STIX]{x1D6FD}\in [-6^{\circ },0^{\circ }]$ are presented in figure 20. For clarity, we extract the branches of most probable gradients using the following technique. The location of the maxima of each branch of the sensitivity maps such as the one given in figure 16 for $c^{\ast }=0.168$ are identified for each yaw angle. Due to the normalization, the first maximum is 1, and a second maximum is plotted only if its value is larger than $0.10$ . This means that the second state is considered only if its probability exceeds approximately $10\,\%$ of the observation time. The corresponding states for the vertical gradient component $g_{z}^{\ast }$ are shown in figure 20(a). They form two distinctive branches, branch $P$ for positive gradients, and branch $N$ for negative gradients. Besides, the most probable horizontal component $g_{y}^{\ast }$ of the gradient remains close to zero with little evolution for yaw around the switch for the two largest ground clearances. Except for the smallest ground clearance $c^{\ast }=0.060$ , large yaw angles select the $P$ state. The larger the ground clearance, the larger the yaw angle at which the $P$ state is selected.

In order to evidence the important role of the four supporting cylinders in the state selection mechanism, a fifth identical support is placed between the two front cylinders for $c^{\ast }=0.168$ . This experimental point is indicated by the black triangle in figure 20 that clearly shows a permanent wake reversal from branch $N$ to branch $P$ . On the basis of this simple experiment, one can draw a plausible mechanism for the change of branch as yaw increases. It is likely that in yaw conditions, the wake of the front leeward support that develops under the body gets closer to the mid-track of the base as the yaw increases, resulting in the wake reversal.

For yaw angles $|\unicode[STIX]{x1D6FD}|<3^{\circ }$ , and independently of the ground clearance, the modulus shown in figure 20(b) is found to be almost constant and the gradient has only two opposite phase orientations $+\unicode[STIX]{x03C0}/2$ for branch $P$ and $-\unicode[STIX]{x03C0}/2$ for branch $N$ , as can be seen in figure 20(c). For larger yaw $|\unicode[STIX]{x1D6FD}|>3^{\circ }$ the $N$ state becomes yaw dependent and the modulus not as well defined.

3.2.3 Global force sensitivity versus wake aerodynamic loading

The $y$ - and $z$ -components of the mean force coefficients obtained for small yaw and pitch inclinations around the baseline are shown in figure 21(a–d). With respect to the yaw angle $\unicode[STIX]{x1D6FD}$ , $C_{y}$ in figure 21(a) has a linear evolution showing that it is not influenced by the $z$ -instability. Accordingly, the fluctuations (figure 21 e) are small independently of the angle. On the contrary, the behaviour of the mean lift coefficient $C_{z}$ (figure 21 b) is strongly impacted by the selected wake state which reveals two distinctive levels corresponding each to one wake orientation, either at $C_{z}\simeq -0.14$ for the $P$ state or at $C_{z}\simeq -0.18$ for the $N$ state. As expected, there is a fluctuation crisis during the bistable dynamics at the transitions around $\pm 4^{\circ }$ as shown in figure 21(f).

Figure 21. Mean (a–d) and fluctuating (e–h) cross-flow force coefficients versus yaw (a,b,e,f) and pitch angle (c,d,g,h) for the boat-tailed body.

Figure 22. Mean lift coefficient versus inclination of the boat-tailed body. The measured force coefficient (symbols) are compared to the basic flow coefficient (red line) and coefficient computed from (3.7a ) (blue dashed line, see text). (a) Pitch angle sensitivity, (b) yaw angle sensitivity.

For the pitch experiments with varying $\unicode[STIX]{x1D6FC}$ , neither $C_{y}$ in figure 21(c) nor its fluctuation $C_{y}^{\prime }$ in figure 21(g) are influenced by the $z$ -instability. In contrast, the mean lift coefficient $C_{z}$ in figure 21(d) and its fluctuation $C_{z}^{\prime }$ in figure 21(h) reveal the $z$ -instability. The transition between the two states is clearly observable in both figures around $\unicode[STIX]{x1D6FC}\simeq -0.4^{\circ }$ . Actually the transition in pitch of the $z$ -instability is similar to that in yaw of the $y$ -instability when compared to figure 11(a,e).

We now look for a relationship between the mean gradient $G_{z}^{\ast }$ and $C_{z}$ as we did in figure 12(a) between $G_{y}^{\ast }$ and $C_{y}$ for the $y$ -instability. For pitch sensitivity, $C_{z}$ measurements reported in figure 22(a) suggest that the mean lift coefficient $C_{z}^{B}$ obtained without the instability would be a linear function of the pitch angle $\unicode[STIX]{x1D6FC}$ . The basic flow coefficient $C_{z}^{B}=0.11\unicode[STIX]{x1D6FC}-0.177$ (where  $\unicode[STIX]{x1D6FC}$ is expressed in degrees) is shown in figure 22(a) as the red straight line. We recall that the superscript $^{B}$ refers to as the basic flow that is the flow without the instability as introduced in § 3.1.3. For the yaw sensitivity experiment, $C_{z}$ measurements reported in figure 22(b) have a quadratic dependency that is not present in $g_{z}^{\ast }$ shown in figure 16(b). Thus the quadratic dependency should be ascribed to the basic flow. The basic flow coefficient $C_{z}^{B}(\unicode[STIX]{x1D6FD})=6\times 10^{-4}\unicode[STIX]{x1D6FD}^{2}-0.177$ (where  $\unicode[STIX]{x1D6FD}$ is expressed in degrees) is shown as the red curve in figure 22(b). For both pitch and yaw sensitivities, the blue dashed lines in figure 22 combines the base pressure gradient and the lift of the basic flow and shows satisfactorily a relationship:

(3.5) $$\begin{eqnarray}\displaystyle C_{z}=C_{z}^{B}-\frac{G_{z}^{\ast }}{5}. & & \displaystyle\end{eqnarray}$$

The strength of the $z$ -instability contribution to the lift coefficient, given by $C_{z}-C_{z}^{B}$ for either a pure $P$ state or $N$ state of the wake, is found to be approximately $0.02$ as for the $y$ -instability. Since the side force coefficient $C_{y}$ is not affected by the instability, one should have

(3.6) $$\begin{eqnarray}C_{y}=C_{y}^{B}.\end{eqnarray}$$

Both mean relationships (3.5) and (3.6) suggest the instantaneous expressions:

(3.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle c_{z}(t^{\ast })=C_{z}^{B}-\frac{g_{z}^{\ast }(t^{\ast })}{5}, & \displaystyle\end{eqnarray}$$

(3.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle c_{y}(t^{\ast })=C_{y}^{B}-\frac{{g_{y}^{\ast }}^{\prime }(t^{\ast })}{5}. & \displaystyle\end{eqnarray}$$

$y$

$y$

$z$

Figure 23. Components of the fluctuating force coefficients $C_{y}^{\prime }$ and $C_{z}^{\prime }$ (filled circles) compared to the fluctuation base pressure gradient ${G_{y}^{\ast }}^{\prime }$ and ${G_{z}^{\ast }}^{\prime }$ (crosses) versus yaw $\unicode[STIX]{x1D6FD}$ (a,c) and pitch angle $\unicode[STIX]{x1D6FC}$ (b,d) for the boat-tailed geometry.

Figure 24. Mean (a–d) and fluctuating (e–h) drag force and base suction coefficients versus yaw (a,b,e,f) and pitch angle (c,d,g,h) for the boat-tailed geometry.

As displayed in figure 24(a–d), the wake dynamics associated with the $z$ -instability has almost no influence on either the drag or the base suction. The fluctuations remain one order of magnitude smaller than those of the lateral force coefficients (observed around 0.02 in figure 23). An interesting result is that drag (or equivalently base suction) is found to be the smallest during the bistable dynamics, either around $\unicode[STIX]{x1D6FD}=\pm 4.5^{\circ }$ for the yaw experiment or around $\unicode[STIX]{x1D6FC}=\pm 0.4^{\circ }$ for the pitch experiment. This can be explained by the low-drag events related to wake switching as reported in Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016).