3.1.1 Surface pressure contours

The flow features at the vicinity of the axisymmetric compression corner with and without the microramp array was inspected through the surface pressure contours displayed in Fig. 11(a)-(d). The pressure plateau observed on the cylinder surface in the uncontrolled interaction (Fig. 11(a)) suggested that the shock foot might be bifurcated, which indicates the presence of a separation bubble around the compression corner. Under the influence of microramp array, (Figs. 11(b), 11(c) and 11(d)), corrugated flow patterns that were roughly sinusoidal in nature were found on the cylinder and flare surfaces. A close examination of these patterns indicated that their wavelengths on the cylinder surface were equal to the circumferential inter-device spacing between the microramps. On the flare surface, even as the wavelength increased linearly along the downstream direction, the angle suspended by the wavelength at the model axis remained constant. It can also be noted from the contours that the pressure rise ahead of the compression corner has been reduced due to the presence of the microramps. On the flare surface, higher pressure is found in the region downstream of the gap between the microramps while comparatively lower pressure is found along the device centerline.

Figure 11.

Surface pressures contours depicting the flowfield characteristics in the interaction region. Black line denotes the upstream influence. a) Uncontrolled; b) X_mvg = −5δ (−22.7h); (c) X_mvg = −10δ (−44.4h); (d) X_mvg = −15δ (−68.1h).

Crests and troughs of the sinusoidal pattern on the cylinder surface can be seen to have roughly aligned with the device centerline and the gap between two adjacent devices, respectively. This denotes that the upstream influence length is the longest along the device centerline and it reduces as the spanwise distance from the device centerline increases. Spanwise variation of the interaction length on the flare also seems to be coupled with that of the upstream influence length. As mentioned earlier, the upstream influence is the streamwise distance between the point where the static pressure rises beyond (P_Inf + 6σ) and the compression corner. In Fig. 11, a black line runs on the cylinder surface in the circumferential direction through all the points at which the static pressure value has just crossed (P_Inf + 6σ). The spanwise average of the upstream influence length for all the four configurations was calculated and the same is presented in Table 5. The average upstream influence length reduced when the microramps were placed upstream and the maximum reduction was observed when X_mvg = −10δ (−44.4h) and was closely followed by X_mvg = −5δ (−22.7h) configuration.

Table 5

Spanwise average of upstream influence length

3.1.2 Streamwise pressure contours

Streamwise sections of static pressure contours were used to diagnose the interaction flowfield. Three streamwise sections across the span of the microramp has been chosen as shown in Fig. 12. It is to be noted that, these locations have been selected after closely inspecting the repetitive flow structures observed in the surface pressure contours along the circumferential direction and also by taking into account the symmetrical nature of the microramp geometry about its centerline. The first probe (s1 = 0) was located exactly along the device centerline while other probes were located at a distance of one-fourth (s2 = 1.875h) and half (s3 = 3.75h) of the inter-device spacing (7.5h). It is also worthwhile to mention that the location of the probe s3 corresponds to the maximum distance away from the device centerline.

Figure 12.

Location of the streamwise probes across the MVG span.

The contours obtained from X_mvg = −10δ (−44.4h) configuration (as this position corresponds to maximum reduction in average upstream influence length) are compared with that of the uncontrolled interaction in Fig. 13. The angle of the main shock was found to be approximately equal to the theoretically calculated shock angle of 31.65° (corresponding to the Mach number of 3.96 and a flow deflection angle of 25°) from the conical shock relations and was unaffected by the presence of microramps. In the region close to the solid surface, all four contours show a bifurcation of the main shock into two shocks confirming the presence of reversed flow in the region surrounding the compression corner. This region serves as an obstruction to the incoming flow inducing the formation of the separation shock. The fluid removed from the separation point hits the flare at an angle and deflects in the direction parallel to the surface by creating compression waves that coalesce together to form the reattachment shock. The interaction between the separation and reattachment shock was identified as a Type VI Shock-Shock interaction^{(Reference Edney56)}.

Figure 13.

Streamwise sections of the interaction region depicting static pressure. (a) Uncontrolled Interaction; (b) X_mvg = −10δ (−44.4h), s1 = 0; (c) X_mvg = −10δ (−44.4h), s2 = 1.875h; (d) X_mvg = −10δ (−44.4h), s3 = 3.75h.

The local upstream influence length along s1 and s3 from X_mvg = −10δ (−44.4h) case varied noticeably from that of the uncontrolled interaction (L_UC). As determined earlier from the surface pressure contours of Fig. 11, the upstream influence length was the largest along the device centerline (Fig. 13(b)), but the streamwise section showed that the value of was even larger (by 15%) than what was observed in the uncontrolled interaction (Fig. 13(a)). In the other two locations (Fig. 13(c) and 13(d)), the local upstream influence was found to be smaller than L_UC and the maximum reduction of 18% was achieved at s3 = 3.75h as expected.

The uncontrolled separation and reattachment shock positions are shown as dashed lines in Fig. 13(b-d) for comparison. The separation shock angle was slightly reduced at all sections of X_mvg = −10δ (−44.4h) case resulting in the lower pressure rise that was observed on the cylinder surface in Fig. 11. Interestingly, the reattachment shock angle seemed to have also reduced along the device centerline region (Fig. 13(b)) but it quickly recovers as the spanwise distance from the device centerline increases. The strongest reattachment shock was found along s3 = 3.75h (Fig. 13(d)) where its strength was significantly greater than the reattachment of the uncontrolled interaction.

3.2.1 Surface flow patterns

Skin friction lines are generated on the cylinder and flare surfaces to observe surface flow patterns and to characterize modifications induced in the structure of separation bubble by Microramp arrays. These lines are generated based on wall shear stress distributions on the surface. Patterns formed by skin friction lines under all the four configurations are depicted in Fig. 14. In the uncontrolled interaction (Fig. 14(a)), the separation and reattachment lines were distinct and there was very little change in the distance between them and the compression corner location along the circumferential direction. As Microramps were introduced into the flowfield, substantial modifications were observed in the region downstream of these devices and the separation region in particular (Fig. 14(b-d)). Skin friction lines immediately downstream of the Microramps seemed to converge into the region close to the device centerline. This indicates a spanwise (circumferential) component in flow velocity from the device gap region and towards the device centreline just above the surface.

Figure 14.

Skin friction lines on the cylinder and the flare surfaces. (Flow is from left to right) (a) Uncontrolled interaction; b) X_mvg = −5δ (−22.7h); (c) X_mvg = −10δ (−44.4h); (d) X_mvg = −15δ (−68.1h).

Surface flow patterns in the presence of Microramps illustrated a corrugated separation line and formation of spade shaped patterns on the cylinder surface. These observations were similar to the oil flow patterns reported from previous experimental investigations^{(Reference Bur, Coponet and Carpels30,Reference Babinsky, Li and Pitt Ford32,Reference Verma and Manisankar38)} of normal/oblique shock-boundary layer interaction control using vortex generators. Apexes of these spade structures were roughly aligned along the centrelines of the Microramps. Additionally, striation patterns were observed on the Flare surface beyond reattachment. The number of striations and the circumferential spacing between them are in accordance with those of the Microramp array, indicating a direct association.

Separation topology in the presence of Microramps was interpreted based on the Critical Point Theory (CPT) that is widely employed in the characterization of three dimensional flow separations^{(Reference Delery57)}. The Poincare’s formula [Equation (1)] that relates the number of critical points on a two dimensional vector field to the ‘complexity’ of the surface was also used to understand the patterns formed by the skin friction lines and check the consistency of the narrative.

(1) \begin{equation}\sum\left({Nodes + Foci} \right) - \sum\ {Saddle\ points} = 2 - 2p \label{eqn1}\end{equation}

where ‘p’ is the complexity of the surface.

As per CPT, the complexity of the surface (p) is zero, if a closed curve on the surface could be converged into a point without departing the surface. In the present study, neither the cylinder nor the flare surfaces have any discontinuities such as holes or gaps that may obstruct the shrinkage of any closed curve on it and hence throughout this investigation, the complexity term in the above equation vanishes as shown in Equation (2).

(2) \begin{equation} \sum\left( {Nodes + Foci} \right) - \sum\ Saddle\ points = 2\label{eqn2}\end{equation}

According to CPT, flow over a body is said to have been separated if there exists more than two nodes and at least one saddle point on the surface flow pattern. All skin friction lines that make up the surface flow pattern must originate and terminate from a node or a focus. A node (N) has a single tangent line passing through it while a focus (F) has no common tangent. Based on whether the skin friction lines are directed inwards (converging) or outwards (diverging), the nodes and foci can be classified as separation type or attachment type respectively. Saddle points (SP) are critical points through which only two skin friction lines (hereafter called as separatrices) pass through, out of which one performs the function of preventing all other skin friction lines from reaching certain parts of the surface.

Figure 15 gives a closer look at the separation region and also depicts the wall shear stress vector fields along with the skin friction lines. The separation and reattachment lines observed in the uncontrolled interaction (Fig. 15(a)) are a series of adjacent node – saddle point combinations^{(Reference Zheltovodov, Knight, Babinsky and Harvey58 – Reference Lighthill and Rosenhead60)}. This denotes the presence of Goertler vortices which has been reported in previous compression corner interaction studies^{(Reference Heffener, Chpoun and Lengrand8,Reference Zheltovodov, Schulein and Yakovlev61,Reference Grilli, Hickel and Adams62)} . Two separatrices lines pass through each saddle point — one that prevents the incoming skin friction lines from moving further downstream and the other one that separates the skin friction lines that converge into the nodes located on either side of the saddle point. Structure of the reattachment line is also similar to that of the separation line except that the skin friction lines diverge from the attachment nodes.

Figure 15.

Critical points at the interaction region (flow is from left to right). a) XMVG = −5δ (−22.7h); (b) XMVG = −10δ (−44.4h); (c) XMVG = −15δ (−68.1h).

Figure 15(b-d) depicts the critical points at the interaction region. Each spade had a saddle point (SP₁) at its apex (along the device centerline) and housed two separation foci (F₁ & F₂) inside it. Along the device gap region a pair of separation (N₁) and attachment (N₂) nodes were observed close to the compression corner on the cylinder and flare surfaces respectively. A saddle point (SP₄) was also observed on the flare surface along the device centerline. A total of 14 Microramps were placed inside the computational domain and each of them could be very clearly associated with two saddle points (SP₁ and SP₄), two separation foci (F₁ and F₂), a separation node (N₁) and a reattachment node (N₂). Including the attachment and separation nodes at the beginning and end of the body along with the observed critical points, the Poincare’s formula [Equation (2)] is not satisfied [2 + 14(2) + 14(2) – 14(2) ≠ 2]. It is inferred from the math that, two other saddle points need to be associated with each microramp device to satisfy Equation (2). It is postulated that these saddle points (SP₂ and SP₃) would be on either shoulders of the spade (illustrated in Fig. 16) although the present visualizations could not exactly pin point their location.

Figure 16.

Schematic diagram depicting surface topological features in the separation region under the influence of microramp array.

A line diagram illustrating the various critical points and separatrices in the presence of microramps is presented in Fig. 16. The separatrix S₁ arrives along the microramp centerline and reaches the first saddle point, SP₁. The two separatrices (S₂ and S₃) leaving SP₁ prevent the adjoining skin friction lines from proceeding downstream. These two separatrices, along with the other skin friction lines, terminate into the two separation foci (F₁ and F₂). The separatrices (S₅ and S₆) that pass through SP₂ and SP₃ would be the last of the ones to terminate at the foci. Beyond, SP₂ and SP₃, they are termed as S₉ and S₁₀ respectively in Fig. 16. S₇ and S₈ from SP₂ and SP₃ terminate at the separation node, N₁. These two separatrices prevent all skin friction lines in between S₅ and S₆ from travelling any further downstream and they are all forced to converge at N₁. Two separatrices (S₁₁ and S₁₂) that originate at N₂ join these saddle points and along with S₅ and S₆, perform the function of isolating the flow region in between two adjacent spades. On the flare surface, the skin friction lines are seen to diverge from N₂ and some of these lines terminate at the separation foci, while some others terminate at N₁ on the cylinder surface. The separatrix S_4, that separates the two foci also originates from N₂ and reaches SP₁. The saddle point SP₄ on the flare surface roughly aligns with the microramp centerline and is connected to the attachment nodes on either side by S₁₃ and S₁₆. One of the diverging separatrix (S₁₅) from SP₄ could be seen moving into the separation region and terminating at one of the foci (F₁ in Fig. 16). The other diverging separatrix (S₁₄) performs the function of separating the skin friction lines that originate from adjacent attachment nodes (N₂).

The average separation bubble length along the device centerline (between SP₁ and SP₄) and the device gap region (between N₁ to N₂) is presented in Table 6. These two locations are chosen because they correspond to the maximum (SBL_max) and minimum (SBL_min) values of separation bubble length (SBL) in the microramp-controlled flowfield. Although, a glance at Fig. 14 may indicate that the separation point along the device centerline has moved upstream, a closer observation will show that the reattachment point also has moved upstream. Measurements presented in Table 6 indicated that, even SBL_max of the controlled interaction, was less than the SBL observed in the uncontrolled interaction. Therefore, it can be shown that, at all locations across the microramp span, a reduction in SBL was observed. The maximum reduction (89%) in SBL occurred between N₁ and N₂ (along the device gap region) when X_mvg = −5δ (−22.7h) and X_mvg = −10δ (−44.4h). But, the gap between two adjacent spades (across the device gap region) was slightly larger when X_mvg = −5δ (−22.7h).

Table 6

Average separation length along device centreline and device gap regions

3.2.2 Separation bubble height

Streamwise velocity contours at the vicinity of the compression corner along s1 and s3 locations (Fig. 12) are displayed in Fig. 17 along with a contour from the uncontrolled interaction. It is also to be noted that, these locations correspond to the spanwise positions where maximum (SBL_max) and minimum (SBL_min) separation lengths were observed as shown in the previous section. A quick glance at the contours would convey that along s1, the separation bubble is thicker than the uncontrolled interaction while along s3, the bubble height has been drastically brought down. A closer look at the contours from s3 will also make it clear that, the least bubble height was observed when the microramps are placed at X_mvg = −10δ (−44.4h).

Figure 17.

Streamwise velocity contours at the vicinity of compression corner.

A quantitative evaluation of the bubble height at all three spanwise locations, (viz. s1, s2 and s3) was conducted from the boundary layer profiles at the compression corner location (X = 0) and a comparison is presented in Table 7. Both largest and smallest height of the bubble occurred when the microramps were placed at X_mvg = −10δ (−44.4h). The bubble grew 53% taller than the uncontrolled interaction along s1 when the devices where placed at that streamwise position. The bubble height reduced by 85%, 92% and 74% when the microramps were placed at −5δ (−22.7h), −10δ (−44.4h) and −15δ (−68.1h), respectively.

Table 7

Height of the separation bubble at the compression corner (X = 0)

3.3.1 Cross sectional velocity contours

As discussed in the introduction section, the MVGs control shock-induced flow separations by modifying the incoming boundary layer characteristics. In this section, these changes inflicted on the boundary layer by the microramps is analyzed through cross-sectional velocity contours (Fig. 18) at the vicinity of the interaction region. A glance at these contours suggested that the boundary layer experienced a significant velocity redistribution across the span of the microramps. The alternating bands of upwash and downwash regions that can be observed in Fig. 18(e-p) align perfectly along the device centerline and device gap regions, respectively. The downwash causes the higher momentum fluid at upper parts of the boundary layer to entrain into the near wall region, which makes the boundary layer fuller and more resilient towards the adverse pressure gradient offered by the shock. The converse happens in upwash regions along the device centerlines wherein the low momentum fluid in the near wall region is pushed upwards in the direction normal to the surface, making the boundary layer in that region more susceptible to separation.

Figure 18.

Cross-sectional velocity contours at various streamwise positions.

In all the four streamwise locations in which the contours were analyzed, the greatest entrainment was observed when X_mvg = −10δ (−44.4h) (Fig. 18((i-l)). When X_mvg = −5δ (−22.7h), the entrainment just outside the separation region (Fig. 18(e and f)) was noticeably lower than what was observed when X_mvg = −10δ (−44.4h), but inside the separation region (Fig. 18(g and h)) the entrainment was on par with what was observed when X_mvg = −10δ (−44.4h). When the microramps were moved to X_mvg = −15δ (−68.1h), both the upwash and downwash were consistently lower at the vicinity of the interaction region than what was observed when they were placed in the other two positions (Fig. 18(m through p)). A sole exception was when X = −4δ, where the entrainment under X_mvg = −15δ (−68.1h) was more than what was observed when X_mvg = −5δ (−22.7h) (Fig. 18(e and m)). This may be because, this location is very close to the microramp (at a distance of δ) when X_mvg = −5δ (−22.7h) and the changes had just started to appear.

3.3.2 Incompressible shape factor

Health of the incoming boundary layer and fullness of the boundary layer velocity profile is generally quantified based on the incompressible shape factor (H_i). It is defined [Equation (3)] as the ratio of the incompressible displacement thickness ( $\delta^{\ast}_{i}$ ) [Equation (4)] to the incompressible momentum thickness ( $\delta^{\ast\ast}_{i}$ ) [Equation (5)]. A higher shape factor indicates that the boundary layer is more likely to separate in the event of experiencing an adverse pressure gradient.

(3) \begin{equation}{H_i} = \frac{\delta^{*}_{i}}{\delta^{**}_{i}}\label{eqn3}\end{equation}

(4) \begin{equation}{\mathop \delta \nolimits^* _i} = \int\limits_{\mathop R\nolimits_X }^{\mathop r\nolimits_{\max } } {\left[ {1 - {V \over {\mathop V\nolimits_{Inf} }}} \right]} dr\label{eqn4}\end{equation}

(5) \begin{equation} \delta^{**}_{i} = \int\limits_{R_X }^{r_{\max}} \frac{V}{V_{Inf}} \left[ {1 - \frac{V}{V_{Inf}}}\right]dr\label{eqn5}\end{equation}

where, R_X is the radius of the cylinder/flare body at a given axial position and r_max is the radial distance from the axis of the body to the end of the computational domain. It is to be noted that, because the domain had a square cross-section, the r_max value varied slightly across different locations of microramp span. These variations do not reflect on the results as r_max is extremely far away from the interaction region.

The incompressible shape factor (H_i) of the undisturbed incoming boundary layer at x_c (location of the compression corner) was found to be 1.27 which denotes that the boundary layer is turbulent. The value of H_i was obtained for the uncontrolled and controlled interactions at five different streamwise positions at the vicinity of the interaction region viz. X = −4δ, −2δ, 0, 2δ and 4δ. The changes in the flow characteristics across the micoramp span was also documented by acquiring these values along s1, s2 and s3, and the comparative plots are provided in Fig. 19.

Figure 19.

Variation in incompressible shape factor across the microramp span. a) X = −4δ; b) X = −2δ; c) X = 0; d) X = 2δ; e) X = 4δ.

It can be clearly observed from these plots that in the controlled cases, the boundary layer health improves as the spanwise distance from the microramp centerline increases. The value of H_i for the controlled interactions has consistently been less than that of the uncontrolled interaction along s2 and s3 locations indicating that the boundary layer has become more robust in these regions due to the downwash induced by the microramps. Along the microramp centerline, the value of H_i was greater for the controlled interactions until the compression corner (Figs. 19(a), 19(b) and 19(c)) indicating that the boundary layer in this region is more likely to separate. But at the same time, on the flare surface it can be seen that, even along s1, the value of H_i in the presence of microramps has dropped significantly from that of the uncontrolled interaction (Figs. 19(d) and 19(e)). This suggests that the presence of microramps resulted in an overall stronger and sturdier boundary layer in the post interaction region.

With respect to the effect of streamwise positioning of the microramps, it can be seen that, along s3, in all the five axial positions the lowest value of the incompressible shape factor was recorded when X_mvg = −10δ (−44.4h). This is quite comprehensible as it directly aligns with the observations made during flow separation characterizations presented in the previous section that showed a maximum reduction in separation bubble size (both in terms of length and height) when the microramp array was placed in this streamwise position. Also, the position X_mvg = −15δ (−68.1h) was found to be the least effective position among the three positions that were investigated.