3 Results

As shown in the schlieren images in figure 3, the reference (undisturbed) laminar interaction is characterised by the presence of a separation and a reattachment shock while the roughness wake in the fully turbulent interaction results in a single shock attached to the corner (intrinsic to the present low-deflection ramp angle). The gradual transition from the laminar to the fully turbulent state as roughness height is increased is captured in the plots of average heat transfer in figure 4, which correspond to the diamond element cases for the complete range of heights investigated ( $0.03\leqslant k/\unicode[STIX]{x1D6FF}_{k}\leqslant 1.71$ ). The figures also include the experimental and numerical fully laminar trends for reference.

With an element height of $k/\unicode[STIX]{x1D6FF}_{k}=0.03$ (figure 4 a), the mean heat transfer $q$ along the interaction remains practically undisturbed and exhibits a reduction characteristic of laminar separation but with a slight overshoot as the element wake starts transitioning upon reaching the corner. For $k/\unicode[STIX]{x1D6FF}_{k}=0.06$ , 0.09 and 0.12 (figure 4 a), $q$ overshoots progressively at a higher rate over the ramp as roughness-induced disturbances gain strength with increasing height. A similar effect is also observed for the $k/\unicode[STIX]{x1D6FF}_{k}=0.15$ element (figure 4 b), and a subsequent increase in height to $k/\unicode[STIX]{x1D6FF}_{k}=0.17$ then leads to disturbances gaining sufficient strength so as to alter the laminar separation upstream of the flare and rise towards the turbulent state. For the present purposes (noting the influence of the corner flow), the $k/\unicode[STIX]{x1D6FF}_{k}=0.15$ case is hence identified as the critical roughness condition. As roughness height is further increased to $k/\unicode[STIX]{x1D6FF}_{k}=0.20$ and 0.23, transition onset continues to shift closer to the element. The mean level upstream of the interaction is thus gradually elevated and leads to an offset of the separation trend ahead of the corner; these effects are consistent with observations in the aforementioned numerical studies which described the mechanisms of disturbance near-wall penetration and breakdown as common features associated to the onset of transition. In Choudhari et al. (Reference Choudhari, LI, Chang and Edwards2009), a nearly monotonic decay in the amplitude of subcritical stationary streaks was also shown to amplify across the compression corner and enhance the growth of non-stationary instabilities. As per their findings, the concave streamline curvature near the corner is believed to also contribute to the reinforcement of streak amplitudes and to have a destabilising influence.

Figure 4.

Time-averaged axial heat transfer distributions for different diamond roughness element heights: (a) $k/\unicode[STIX]{x1D6FF}_{k}=0.03$ , 0.06, 0.09, 0.12; (b) $k/\unicode[STIX]{x1D6FF}_{k}=0.15$ , 0.17, 0.20, 0.23; (c) $k/\unicode[STIX]{x1D6FF}_{k}=0.26$ , 0.29, 0.32, 0.35; (d) $k/\unicode[STIX]{x1D6FF}_{k}=0.38$ , 0.41, 0.44, 0.47; (e) $k/\unicode[STIX]{x1D6FF}_{k}=0.52$ , 0.58, 0.69, 0.80; (f) $k/\unicode[STIX]{x1D6FF}_{k}=1.03$ , 1.26, 1.49, 1.71. From lower to higher $k/\unicode[STIX]{x1D6FF}_{k}$ for each subfigure: grey ○, blue ♢, green △ and red ◻ . Reference fully laminar results ( $k/\unicode[STIX]{x1D6FF}_{k}=0$ ) are indicated as black ♢ in each plot, along with the laminar CFD for the cases with and without the flare respectively as solid and dashed lines. Element location $x_{k}$ indicated as a small square for reference. Right axis indicates respective levels normalised by local laminar heating at roughness element location $q/q_{L}$ (see table 2).

Figure 5.

Effect of isolated diamond roughness element height: (a) transition length from roughness trailing edge $x_{tr}$ as a function of $k/\unicode[STIX]{x1D6FF}_{k}$ compared with related studies in the literature and present heat transfer $q$ at $x/L=-0.1$ (close ahead of the corner), and (b) power spectral density $G_{xx}$ based on heat transfer signal at $x/L=-1$ (freq. resolution $\unicode[STIX]{x0394}f=0.78$  kHz) for cases $k/\unicode[STIX]{x1D6FF}_{k}=0$ (black), $k/\unicode[STIX]{x1D6FF}_{k}=0.23$ (blue), $k/\unicode[STIX]{x1D6FF}_{k}=0.32$ (green) and $k/\unicode[STIX]{x1D6FF}_{k}=1.0$ (red). Flow conditions in reference studies are: $M_{\infty }=3.5$ , $Re_{\infty }=9.84\times 10^{6}~\text{m}^{-1}$ (CFD by Choudhari et al. Reference Choudhari, Li, Chang, Norris and Edwards2013, flat plate) and $M_{\infty }=3.5$ , $Re_{\infty }=10.8\times 10^{6}~\text{m}^{-1}$ (experiments by Kegerise et al. Reference Kegerise, King, Owens, Choudhari, Norris, Li and Chang2012, flat plate) both with $\unicode[STIX]{x1D6FF}_{k}=0.7\pm 0.01$  mm; and $M_{\infty }=6$ , $Re_{\infty }=14.1\times 10^{6}~\text{m}^{-1}$ (experiments by Berry & Horvath Reference Berry and Horvath2008 on zero-incidence $5^{\circ }$ sharp cone), with $\unicode[STIX]{x1D6FF}_{k}=0.4\pm 0.02$  mm. Dashed lines in the first figure assume extrapolation towards critical height conditions and eventual natural transition below that. $^{\ast }$ Noise levels in the latter study reported to fall between the indicated ‘noisy’ and ‘quiet’ thresholds.

As element height increases to $k/\unicode[STIX]{x1D6FF}_{k}=0.26$ , 0.29 and 0.32 (figure 4 c), the heat transfer rise across the corner becomes abrupt as the roughness wake reaches a fully turbulent state and the attached corner shock becomes established. The $k/\unicode[STIX]{x1D6FF}_{k}=0.32$ case shows early evidence of the monotonic decrease in fully turbulent heat transfer level over the flare and is identified as the effective condition. Further increases in element height, and up to $k/\unicode[STIX]{x1D6FF}_{k}\approx 0.5$ (figure 4 d), produce little change in the attached flow downstream of the corner but induce increased heating ahead of it. The heat transfer augmentation close to the element is further enhanced for heights up to the order of the boundary layer thickness $0.5<k/\unicode[STIX]{x1D6FF}_{k}<1$ (figure 4 e), however, without significantly altering the downstream SBLI, which remains effectively at the turbulent attached condition. The local heating effects continue to be enhanced for taller elements ( $k/\unicode[STIX]{x1D6FF}_{k}>1$ ) yet with minor influence on the downstream corner flow (figure 4 f).

Accordingly, in figure 5(a) the increase in roughness height is seen to lead to an upstream shift of the transition onset location $x_{tr}$ , defined here as distance from element trailing edge (with associated uncertainty below 5 $\,\%$ ) and to follow an inverse trend with respect to local heat transfer, $x_{tr}$ being thus deemed to correlate with ${\sim}q^{-1}$ (e.g. here shown at a sample location $x/L=-0.1$ and compared with previous studies; with reference values needed for further normalisation given in the figure caption and in tables 1 and 2). Below $k/\unicode[STIX]{x1D6FF}_{k}=0.15$ , $x_{tr}$ is fixed at the corner due to the local destabilising effects and it is then shortened with increasing height, eventually asymptoting towards $x_{tr}\approx 21\pm 1$  mm ( ${\sim}12\unicode[STIX]{x1D6FF}_{k}$ ). The effective transition length is of a similar order as that found under ‘noisy’ (high-disturbance) environmental conditions in Kegerise et al. (Reference Kegerise, King, Owens, Choudhari, Norris, Li and Chang2012) and Choudhari et al. (Reference Choudhari, Li, Chang, Norris and Edwards2013). In their studies, the effective transition length induced by diamond elements was shown to be over 2.5 times longer under ‘quiet’ (low-noise) conditions, where the stability analysis of the element wake was based on N-factor calculations; the effect of environmental disturbance levels on $x_{tr}$ was further found to lead to even greater differences between ‘quiet’ and ‘noisy’ conditions for shorter elements, with the breakdown of the dominant modes associated to transition onset exhibiting a strongly nonlinear dependence on roughness height. The results by Berry & Horvath (Reference Berry and Horvath2008) on a $5^{\circ }$ sharp cone – in great part relying on surface heat transfer – also reported some degree of nonlinearity in transition length as a function of roughness height. Nonetheless, following an extensive parametric analysis, a substantial effective transition dataset was shown to still correlate as $(Re_{\unicode[STIX]{x1D703}}/M_{e})(k/\unicode[STIX]{x1D6FF}_{k})=70\pm 20$  % – so that the assumption of linear height effect could be found plausible with sufficient error margins by accounting for the more immediate effects of Reynolds number (based on displacement thickness $Re_{\unicode[STIX]{x1D703}}$ ) and edge Mach number $M_{e}$ . It may then be noted that the different effective conditions in the reference studies, in comparison to the present case, are in part related to pertinent variations in the parameters above.

As a sensitive measure of practical impact, we thus discuss the effects on heat transfer. The respective power spectra in figure 5(b) (again shown for $q$ at $x/L=-0.1$ ) exhibit a broadband distribution without clear dominant frequencies and at levels above the undisturbed laminar state. The spectra remain similar at the effective height $k/\unicode[STIX]{x1D6FF}_{k}=0.32$ and above (e.g. here shown for $k/\unicode[STIX]{x1D6FF}_{k}=1$ ) with a slight enhancement in the lower and upper frequency ranges found with $k/\unicode[STIX]{x1D6FF}_{k}=0.23$ . The present bandwidth, with upper bound considered safely reliable up to at least $f_{max}=42$  kHz (corresponding to a Strouhal number $S_{\unicode[STIX]{x1D6FF}}=f_{max}\unicode[STIX]{x1D6FF}_{k}/U_{e}$ of approximately 0.057, where edge velocity is $U_{e}=1295~\text{m}~\text{s}^{-1}$ ), however proves insufficient to capture dominant frequencies characteristic of the transitional wake as well as to enabling a broader stability analysis for frequencies in excess of this threshold.

Figure 6.

Normalised standard deviation $\unicode[STIX]{x1D70E}_{q}/q$ contours in the ( $x,k$ )-plane for (a) diamond roughness element and (b) square roughness element, and corresponding profile over $k/\unicode[STIX]{x1D6FF}_{k}=0.03{-}1.71$ range for axial locations $x/L=-1.0$ (black ○), $x/L=-0.1$ (blue ♢), $x/L=+0.1$ (green △ ), $x/L=+1.0$ (red ◻ ) respectively for (c) diamond roughness element and (d) square roughness element, with corresponding schlieren for $k/\unicode[STIX]{x1D6FF}_{k}=1$ cases. Horizontal and vertical dashed lines respectively mark effective height and $\unicode[STIX]{x1D70E}_{q}/q=0.1$ . Schematics indicate test geometry for reference.

Alternatively, we thus refer to the contour maps in figure 6(a,b), which summarise the normalised standard deviation $\unicode[STIX]{x1D70E}_{q}/q$ over the interaction length for both the diamond and square arrangements and for the complete range of heights. The effective height, $(k/\unicode[STIX]{x1D6FF}_{k})_{eff}=0.32$ , presents a clear threshold above which the turbulent wake remains fully attached and induces a rapid rise in heat transfer at the corner. The respective profiles in figure 6(c,d) show that $\unicode[STIX]{x1D70E}_{q}/q$ remains below ${<}0.1$ above this height for both the diamond and square cases, i.e. $(k/\unicode[STIX]{x1D6FF}_{k})_{eff}$ is practically common to both – within the $\pm \unicode[STIX]{x0394}(k/\unicode[STIX]{x1D6FF}_{k})=0.03$ resolution. In the sample schlieren images for the respective $(k/\unicode[STIX]{x1D6FF}_{k})=1$ cases, the corner shock is seen to remain attached at the corner $x_{c}$ and two further oblique shocks are induced near the roughness element. For the diamond, the foremost shock appears attached to the element’s leading edge and the downstream shock has its origin at a location close behind the element. For the square, a local upstream separation of length ${\sim}5\unicode[STIX]{x1D6FF}_{k}$ can be seen and the downstream shock appears relatively weaker.

Figure 7.

Normalised heat transfer $q/q_{L}$ contours in the ( $x,k$ )-plane for (a) diamond roughness element and (b) square roughness element, and corresponding profile over $k/\unicode[STIX]{x1D6FF}_{k}=0.03$ –1.71 range for axial locations $x/L=-1.0$ (black ○), $x/L=-0.1$ (blue ♢), $x/L=+0.1$ (green △ ), $x/L=+1.0$ (red ◻ ) respectively for (c) diamond roughness element and (d) square roughness element. Regions below and exceeding $q/q_{L}$ contour levels are respectively delimited with white and black lines. Horizontal dashed line marks effective height. Schematics indicate test geometry for reference.

The effects of roughness height for the diamond and square arrangements are further summarised in the form of $q/q_{L}$ contour maps in figure 7(a,b). The heat transfer amplification close behind the element (with respect to $q_{L}$ ) is significantly enhanced for the diamond cases (figure 7 a). In contrast, for the square configuration, the local heating effects within the region downstream of the element are relatively weaker (figure 7 b), in this instance with only a slight amplification found for elements taller than the boundary layer thickness $k/\unicode[STIX]{x1D6FF}_{k}>1$ (cf.  $k/\unicode[STIX]{x1D6FF}_{k}>0.5$ for the diamond). The corresponding heat transfer profiles in figures 7(c) and 7(d) offer further insight into the effect of element height. As more clearly seen from its subsequent enhancement downstream of the interaction ( $x/L=1.0$ ), an effectively linear increase in $q/q_{L}$ levels with roughness height is documented in both cases below subcritical conditions ( $k/\unicode[STIX]{x1D6FF}_{k}\leqslant 0.15$ ). Heat transfer levels continue to increase with height between $k/\unicode[STIX]{x1D6FF}_{k}=0.15$ and $k/\unicode[STIX]{x1D6FF}_{k}=0.23$ (see $x/L=-1.0$ , $-0.1$ and 0.1). For the square, the surface heat transfer is found to increase with element height and to then reverse its trend between 0.23 ${\leqslant}k/\unicode[STIX]{x1D6FF}_{k}\leqslant 0.32$ . The peak is interpreted as indicative of a reversal in the effect of roughness height on roughness-induced disturbance levels between $k/\unicode[STIX]{x1D6FF}_{k}=0.23$ and effective conditions ( $k/\unicode[STIX]{x1D6FF}_{k}=0.32$ ). The reversal is more clearly evidenced from the heat transfer trends downstream of the element ( $x/L=-1$ ) and close ahead of the corner ( $x/L=-0.1$ ), with heat transfer levels consistently decreasing across cases $k/\unicode[STIX]{x1D6FF}_{k}=0.23$ , 0.26, 0.29 and 0.32. Above $k/\unicode[STIX]{x1D6FF}_{k}\geqslant 0.58$ , the flow remains insensitive to further increases in element height independently of the increasing strength in the local effects in the vicinity of tall elements. The trend reversal between 0.23 ${\leqslant}k/\unicode[STIX]{x1D6FF}_{k}\leqslant 0.32$ therefore coincides with the peak at $y/\unicode[STIX]{x1D6FF}_{k}=0.23$ within the incoming laminar boundary layer profile. With reference to figure 2(a,b), this range corresponds to locations within the supersonic portion of the boundary layer, where the Mach number increases from $M_{k}=1.52$ to 1.96 ( $M_{k}/M_{e}=0.44$ to 0.57). Farther downstream over the ramp ( $x/L=1.0$ ), the heat transfer reaches a similar level of order ${\sim}10q_{L}$ for both configurations once the element wake has attained a fully turbulent state – where $\unicode[STIX]{x1D70E}_{q}/q<0.1$ , as the heat transfer signal no longer oscillates between the base laminar and turbulent levels (figure 4), and the shock remains attached at the corner.

For further reference, the undisturbed laminar flow conditions at the roughness element location $x_{k}$ and for the different reference heights are presented in table 3. The table includes some of the commonly used reference parameters in high-speed transition studies: (i) roughness Reynolds number with all values evaluated at roughness height $Re_{kk}=U_{k}k/\unicode[STIX]{x1D708}_{k}$ , (ii) a similar definition but with flow properties evaluated at boundary layer edge $Re_{ke}=U_{e}k/\unicode[STIX]{x1D708}_{e}$ , (iii) Reynolds number with kinematic viscosity and velocity evaluated at roughness height and with boundary layer thickness as length scale $Re_{\unicode[STIX]{x1D6FF},k}=U_{k}\unicode[STIX]{x1D6FF}_{k}/\unicode[STIX]{x1D708}_{k}$ , often using displacement $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ or momentum thickness $\unicode[STIX]{x1D703}_{k}$ as alternatives, and (iv) the Reynolds to Mach number ratio with reference conditions at roughness height, $Re_{\unicode[STIX]{x1D6FF},k}/M_{k}$ , where $M_{k}=U_{k}/\sqrt{\unicode[STIX]{x1D6FE}RT_{k}}$ . Recent numerical studies have also proposed mixed roughness Reynolds number definitions with both density and viscosity evaluated at the wall instead of at roughness height, $\unicode[STIX]{x1D70C}_{w}U_{k}k/\unicode[STIX]{x1D707}_{w}$ (Bernardini, Pirozzoli & Orlandi Reference Bernardini, Pirozzoli and Orlandi2012). As per the review in Schneider (Reference Schneider2008), the first parameter ( $Re_{kk}$ ) is more widely used in high-speed transition studies while $Re_{ke}$ and $Re_{\unicode[STIX]{x1D6FF},k}$ , which avoid the calculation of conditions at roughness height, are also proven useful through real flight correlations (Reshotko and Tumin Reference Reshotko and Tumin2004). The complex nature of such flows and sensitivity to a wide array of variables renders these simpler parameters, and variances such as $Re_{\unicode[STIX]{x1D703},e}/M_{e}$ , practical for atmospheric re-entry analysis given the overwhelming effect of density variations with altitude, yet subject to limited physical interpretation (Reda Reference Reda2002; Reshotko Reference Reshotko2007).

Table 3.

Flow properties at reference conditions ranging from wall to boundary layer edge.

Reference to flow properties at roughness height thus appears relevant to different degrees in the characterisation of roughness-induced transition as a function of the local flow topology, and hence of roughness element geometry. As shown in the schematic in figure 8(a), the more slender geometry of diamond elements reduces the extent of local separation near the roughness element and induces enhanced spanwise gradients, in turn leading to increased local heating effects (e.g. Choudhari et al. Reference Choudhari, Li, Chang, Norris and Edwards2013). On the other hand, for square elements (figure 8 b), the local separation effects induced by the blunt geometry are relatively stronger and often result in an upstream horseshoe vortex which convects further downstream (e.g. De Tullio et al. Reference De Tullio, Paredes, Sandham and Theofilis2013), with the actual topology often being highly complex (note only some of the most distinctive features are highlighted here). While sharp obstacles such as the diamond element would be expected to induce significantly higher local vorticity due to enhanced lateral mixing, upstream separation effects for short blunt elements (e.g. square case) may exhibit strong upstream feedback (Estruch-Samper Reference Estruch-Samper2016) and be dominated by local shear effects which are bound to influence the roughness-induced transition phenomenon.

Figure 8.

Characteristic flow topology for (a) diamond and (b) square roughness elements (simplified to highlight relevant features), and plots of $Re_{\unicode[STIX]{x1D6FF}_{k},k}=U_{k}\unicode[STIX]{x1D6FF}_{k}/\unicode[STIX]{x1D708}_{k}$ (red line) and $(Re_{\unicode[STIX]{x1D6FF}_{k},k}/M_{k})^{-1}=(\sqrt{\unicode[STIX]{x1D6FE}RT_{k}}\unicode[STIX]{x1D6FF}_{k}/\unicode[STIX]{x1D708}_{k})^{-1}$ (green line) at roughness location compared with heat transfer $q$ at $x/L=-0.1$ (from figure 5) and $(Re_{x,tr})^{-1}=(U_{e}x_{tr}/\unicode[STIX]{x1D708}_{e})^{-1}$ (green square symbols), for (c) diamond and (d) square roughness elements. $Re_{x,tr}$ and $q$ are scaled for comparison purposes.

Further evaluation of the $Re_{\unicode[STIX]{x1D6FF},k}$ and $Re_{\unicode[STIX]{x1D6FF},k}/M_{k}$ profiles in figures 8(c) and 8(d), respectively against $q-k$ profiles for the diamond and square cases ( $x/L=-0.1$ ) and corresponding transition Reynolds number $Re_{x,tr}=U_{e}x_{tr}/\unicode[STIX]{x1D708}_{e}$ , highlights the higher suitability of $Re_{\unicode[STIX]{x1D6FF},k}/M_{k}$ to characterise roughness-induced transition between critical and effective conditions for the square configuration, whereby local separation effects are significant. The $Re_{\unicode[STIX]{x1D6FF},k}/M_{k}$ ratio captures the coupled effect of the shear and compressibility effects at roughness height while cancelling out the velocity term, and proves to correlate directly with roughness-induced disturbance levels. Inertial terms become more dominant above the effective height, rendering $Re_{\unicode[STIX]{x1D6FF},k}$ itself as a suitable parameter for characterisation of such flows, and hence $Re_{kk}$ through accounting for $k/\unicode[STIX]{x1D6FF}_{k}$ variations, $Re_{kk}=Re_{\unicode[STIX]{x1D6FF},k}(k/\unicode[STIX]{x1D6FF}_{k})$ . Results thus partly attribute the suitability of transition parameters of the type $Re_{\unicode[STIX]{x1D703},e}/M_{e}\sim (k/\unicode[STIX]{x1D6FF}_{k})$ , as developed from Hyper-X and Shuttle transition data (Berry & Horvath Reference Berry and Horvath2008, Horvath et al. Reference Horvath, Zalameda, Wood, Berry, Schwartz, Dantowitz, Spisz and Taylor2012), to the reduced influence of inertial terms within the $Re/M$ ratio. Instead, correlations accounting for wall temperature effects such as $Re_{\unicode[STIX]{x1D703},e}\sim (k/\unicode[STIX]{x1D703})^{-1}(T_{e}/T_{w})^{-n}$ in Reshotko and Tumin (Reference Reshotko and Tumin2004), or effectively $Re_{ke}\sim (T_{e}/T_{w})^{-n}$ , may prime inertial terms and not fully encompass some of the mechanisms influencing the transition process (e.g. shear layer strength). Prediction accuracy may therefore be potentially enhanced by carefully accounting for local roughness effects. While far from a ‘universal’ (nor practical) criterion, a roughness Reynolds number to Mach number ratio with flow properties evaluated at roughness height $Re_{\unicode[STIX]{x1D6FF},k}/M_{k}=\sqrt{\unicode[STIX]{x1D6FE}RT_{k}}\unicode[STIX]{x1D6FF}_{k}/\unicode[STIX]{x1D708}_{k}$ would be expected to enhance accuracy in the prediction of roughness-induced transition. Accounting for $k/\unicode[STIX]{x1D6FF}_{k}$ variations, this can in turn be expressed as $Re_{kk}/M_{k}$ .

It is worth noting again that the local flow topology around roughness elements can be highly complex and this study limits itself to assessing the sensitivity documented at the present base flow conditions (§ 2). Such a sensitivity appears to be manifested as a result of the $T_{k}$ (hence $\unicode[STIX]{x1D708}_{k}$ ) maxima in the incoming boundary layer falling between the critical and effective heights. Further understanding of the roughness-induced transition process of high-speed boundary layers could gain from the investigation of canonical roughness effects such as the square element and use numerical methods and comprehensive empirical analysis to develop deeper insights into the influence of the local (to roughness) flow effects.