2.1 Test section and model details

Experiments were performed in the subsonic Eiffel wind tunnel of PRISME Laboratory, at University of Orléans, France. The facility has a $2~\text{m}$ long test section with a square cross-section of width $w=0.5~\text{m}$ , preceded by a settling chamber connected to the test section by a 16 : 1 contraction. It can reach a maximum free-stream velocity of $50~\text{m}~\text{s}^{-1}$ , with residual turbulence intensity of $0.5\,\%$ . The experimental model (figure 2) on which separation/reattachment occurs is composed of four parts: (i) a fifth-order polynomial contraction (Bell & Mehta Reference Bell and Mehta1988; Brassard & Ferchichi Reference Brassard and Ferchichi2005); (ii) a flat plate where turbulent boundary layer grows; (iii) a descending ramp of slope $\unicode[STIX]{x1D6FC}=25^{\circ }$ , with salient edge and step height $h=30~\text{mm}$ . The resulting expansion ratio $ER=w/(w-h)$ is equal to 1.064; (iv) a second flat plate on which the flow relaxes. The aspect ratio $w/h$ is approximately 17, which guarantees that the mean flow is quasi-bidimensional (see Eaton & Johnston Reference Eaton and Johnston1981 and § 3.3). We define the Cartesian coordinate system ( $X;Y;Z$ ), centred at midspan on the upper ramp edge (see figure 2). Following the suggestions of Kourta et al. (Reference Kourta, Thacker and Joussot2015), transition of the boundary layer was triggered with a zigzag tripper of height $0.5~\text{mm}$ , placed at $x/h\approx -48.5$ . The model replaced the floor of the test section, so that the junctions with the convergent and the diffuser of the wind tunnel were smooth. The reference free-stream velocity $U_{ref}$ and free-stream pressure $p_{ref}$ were measured with a Pitot tube (see figure 3) placed at midspan, above the upper edge of the ramp ( $x/h=0$ ; $y/h=0$ ; $z/h=9$ ). Five values of $U_{ref}$ were tested, ranging from $U_{ref}=15~\text{m}~\text{s}^{-1}$ to $U_{ref}=35~\text{m}~\text{s}^{-1}$ . One straightforward choice for a parameter of similitude is the Reynolds number based on step height $Re_{h}=U_{ref}h/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}=1.5\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ is the kinematic viscosity for air. In a way, $Re_{h}$ relates viscous scales to the scale of the macroscopic forcing of the flow. Table 1 lists the values of $Re_{h}$ for all tested velocities.

Figure 2.

Dimensions of the experimental model. Dots represent pressure taps. The boundary layer measured at $x/h=-9$ (red line) is used as a reference for scaling parameters.

Figure 3.

Integration of the experimental model into the wind tunnel and PIV set-ups. Letters indicate the three PIV fields. (a) Large field; (b) detailed field; (c) auxiliary field.

Table 1.

Boundary layer properties, measured at the reference section $x/h=-9$ . The full boundary layer thickness $\unicode[STIX]{x1D6FF}$ and the friction velocity $u_{\unicode[STIX]{x1D70F}}$ are obtained with the composite profile of Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009). It is $Re_{h}=U_{\infty }h/\unicode[STIX]{x1D708}$ , $Re_{\unicode[STIX]{x1D6FF}_{1}}=U_{\infty }\unicode[STIX]{x1D6FF}_{1}/\unicode[STIX]{x1D708}$ , $Re_{\unicode[STIX]{x1D703}}=U_{\infty }\unicode[STIX]{x1D703}/\unicode[STIX]{x1D708}$ and $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ . $H_{12}$ is the shape factor ( $\equiv \unicode[STIX]{x1D6FF}_{1}/\unicode[STIX]{x1D703}$ ). The Kolmogorov length scale, $\unicode[STIX]{x1D702}$ , is estimated at the height from the wall where the mean TNTI is located.

2.2 Measurement devices

3.1 Incoming boundary layer

Table 1 summarises the main properties of the incoming boundary layer for all available $Re_{h}$ . $Re_{\unicode[STIX]{x1D70F}}$ is computed as $\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ , where boundary layer thickness $\unicode[STIX]{x1D6FF}$ and friction velocity $u_{\unicode[STIX]{x1D70F}}$ were computed with the composite boundary layer profile proposed by Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009). The displacement thickness $\unicode[STIX]{x1D6FF}_{1}$ and the momentum thickness $\unicode[STIX]{x1D703}$ are classically defined as follows:

According to Song & Eaton (Reference Song and Eaton2003), the values of $Re_{\unicode[STIX]{x1D703}}$ are high enough to expect a fully turbulent boundary layer, with the possible exception of $Re_{\unicode[STIX]{x1D703}}=2006$ ( $Re_{h}=3\times 10^{4}$ ). To provide further insight into the state of the boundary layer, figure 4 presents the profiles of $U^{+}$ and of $\sqrt{\langle (u^{\prime })^{2}\rangle }^{\,+}$ , for the incoming boundary layer at $Re_{\unicode[STIX]{x1D70F}}\approx 1310$ ( $Re_{h}=4\times 10^{4}$ , $Re_{\unicode[STIX]{x1D703}}=3262$ ). Both curves are normalised with inner scales. For comparison purposes, the velocity profiles at $Re_{\unicode[STIX]{x1D70F}}=1043$ ( $Re_{\unicode[STIX]{x1D703}}=3270$ ) extracted from the DNS of a zero-pressure gradient (ZPG) boundary layer (Schlatter & Örlü Reference Schlatter and Örlü2010) are also reported. One can notice a fairly good agreement for $U^{+}$ , while $\sqrt{\langle (u^{\prime })^{2}\rangle }^{\,+}$ collapses well on the DNS for $y^{+}>100$ only. The strong attenuation of the inner peak of $\sqrt{\langle (u^{\prime })^{2}\rangle }^{\,+}$ may be mainly attributed to spatial integration over the sensing length of the hot-wire probe (see for example Wyngaard Reference Wyngaard1968; Ligrani & Bradshaw Reference Ligrani and Bradshaw1987; Örlü & Alfredsson Reference Örlü and Alfredsson2010; Philip et al. Reference Philip, Hutchins, Monty and Marusic2013) but also to various effects such as pressure gradient and wall interferences. Since the RRI is not defined in the boundary layer and the TNTI is rather located in its outer region, probe resolution is good enough for the purposes of this work. Figure 4 also reports the boundary layer profiles computed from PIV data of the auxiliary field, at $Re_{h}=4\times 10^{4}$ . Due to wall reflections, data are exploitable only for $y^{+}>500$ , which corresponds approximately to $y/\unicode[STIX]{x1D6FF}>0.38$ . According to Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014), this extent is sufficient to cover almost the entire distribution of the TNTI.

Figure 4.

Normalised boundary layer velocity profiles at $Re_{\unicode[STIX]{x1D70F}}=1310$ ( $Re_{\unicode[STIX]{x1D703}}=3262$ ). (a) $U^{+}$ ; (b) $\sqrt{\langle (u^{\prime })^{2}\rangle }^{+}$ . + Hot-wire measurements at $x/h=-9$ ; $\circ$ average along the PIV auxiliary field. —— (shown in red online) DNS at $Re_{\unicode[STIX]{x1D70F}}=1043$ ( $Re_{\unicode[STIX]{x1D703}}=3270$ ) as given in Schlatter & Örlü (Reference Schlatter and Örlü2010). - - - Log law, $\unicode[STIX]{x1D705}=0.39$ and $B=4.3$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013).

Figure 5.

Mean separated flow at $Re_{h}=5\times 10^{4}$ . - - - Mean TNTI. —— Mean RRI. The shaded area marks the control volume ${\mathcal{V}}_{c}$ .

Table 2.

Evolution of the recirculation length $L_{R}$ with $Re_{h}$ .

3.2 The separated shear layer

Figure 5 shows the mean separated flow for $Re_{h}=5\times 10^{4}$ . The separation of the incoming boundary layer is induced by the sharp expansion of the test section. Downstream of the separation point, the flow is marked by a large mean recirculation region where $U<0$ , which extends up to the reattachment point where the separated shear layer hits the wall. The RRI is the external boundary of the recirculation region. In this study, it is defined either by the isoline $U=0$ on the mean streamwise velocity field, or by the set of points where the backflow coefficient $\unicode[STIX]{x1D712}$ is equal to 0.5, $\unicode[STIX]{x1D712}$ being defined as the fraction of time that the flow moves downstream (Simpson Reference Simpson1989; Cuvier Reference Cuvier2012). The recirculation region is usually characterised by its length $L_{R}$ , which corresponds to the streamwise distance between the separation point (i.e. the first point of the RRI) and the reattachment point (i.e. the last point of the RRI). Although not shown here, topologies that are comparable to that displayed in figure 5 were observed for each $Re_{h}$ , the main difference being the location of the mean reattachment point, which moves upstream for increasing $Re_{h}$ . Based on PIV data, in this work the ratio $L_{R}/h$ ranges from 5.16 to 4.11 (table 2). This trend is consistent with data reported by Kourta et al. (Reference Kourta, Thacker and Joussot2015) and those observed in other massively separated turbulent flows, e.g. downstream of a backward-facing step (Eaton & Johnston Reference Eaton and Johnston1981). It seems important to stress that $L_{R}$ can be interpreted as the streamwise scale of shear layer development. The latter is often characterised with the streamwise evolution of either a generalised momentum thickness $\unicode[STIX]{x1D703}_{SL}$ or of the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ . According to Dandois et al. (Reference Dandois, Garnier and Sagaut2007), $\unicode[STIX]{x1D703}_{SL}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ are defined as follows:

where $U_{min}(x)$ is a local minimum streamwise velocity. Figure 6 shows clearly that $L_{R}$ and $h$ are appropriate scaling parameters for both $\unicode[STIX]{x1D703}_{SL}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ . In principle, the development of the shear layer may be governed by at least two factors: entrainment of external fluid, that drives the spreading rate of free shear layers (see Pope Reference Pope2000) and $ER$ . Adams & Johnston (Reference Adams and Johnston1988a ) show that for high values of $\unicode[STIX]{x1D6FF}/h$ the influence of $ER$ is sizeable at shear layer reattachment. To have some insights into the relative weight of these two mechanisms, let us consider spreading rates typical of free turbulent mixing layers, modelled by Browand & Troutt (Reference Browand and Troutt1985) as follows: For $x/L_{R}<0.5$ , the velocity ratio in (3.3) ranges between 1.15 and 1.3. Then, equations (3.3) yield $\text{d}\unicode[STIX]{x1D703}_{SL}/\text{d}x\approx 0.042\pm 0.003$ and $\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x\approx 0.205\pm 0.015$ , compared to measured values of $0.05\pm 0.006$ and $0.22\pm 0.03$ , respectively. The relatively good agreement of estimated and observed values suggests that the separated shear layer behaves similarly to a free shear layer in a large region downstream of the upper edge of the ramp (Eaton & Johnston Reference Eaton and Johnston1981). This means that, in this region, the growth of the separated shear layer is dominated by entrainment. Interestingly, the growth rates of $\unicode[STIX]{x1D703}_{SL}/h$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/h$ decrease for $x/L_{R}>0.5$ and (3.3) are no longer acceptable approximations. A similar slope change at a similar relative position was also observed by Dandois et al. (Reference Dandois, Garnier and Sagaut2007). Although the interactions between $ER$ and shear layer impingement at reattachment are complex (Adams & Johnston Reference Adams and Johnston1988a ), pressure data shown at § 3.3 illustrate that lower spreading rates are correlated with strong pressure recovery: this suggests that $ER$ might indeed play a role in the second half of the flow. However, entrainment appears to be the main contributor to spreading of the separated shear layer as a whole, since 75 % of total shear layer growth takes place within $x/L_{R}<0.5$ . Finally, provided that the TNTI and the RRI are well approximated by straight lines (see figure 5), it is easy to show that $L_{R}/h$ evolves as $(\text{d}\unicode[STIX]{x1D703}_{SL}/\text{d}x)^{-1}$ or $(\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x)^{-1}$ equivalently, i.e. the faster the separated shear layer grows, the smaller is the recirculation region (also see Adams & Johnston (Reference Adams and Johnston1988a ) on this matter). This is well supported by the inserts in figure 6.

Figure 6.

Shear layer growth. (a) Generalised momentum thickness $\unicode[STIX]{x1D703}_{SL}$ ; (b) vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ . Symbols: ○  $Re_{h}=3\times 10^{4}$ ;  

  $Re_{h}=4\times 10^{4}$ ;  

  $Re_{h}=5\times 10^{4}$ ;  

$Re_{h}=6\times 10^{4}$ ; ▴  $Re_{h}=7\times 10^{4}$ . The inserts show the $L_{R}$ dependency on shear layer growth rate. - - -  $-1$ power law.

Figure 7.

Streamwise evolution of (a) the pressure coefficient and (b) its dimensionless derivative. Symbols as in figure 6. —— (red online) Pressure data from Kourta et al. (Reference Kourta, Thacker and Joussot2015).

Figure 8.

Spanwise pressure distributions. (a) $x/h=-7.67$ ; (b) $x/h=-0.17$ ; (c) $x/h=0.36$ . Symbols as in figure 6.

3.3 Wall pressure distribution

Separation has a direct effect on wall pressure distribution and accordingly on entrainment (see e.g. Eaton & Johnston Reference Eaton and Johnston1981). Figure 7 shows the streamwise evolution of the pressure coefficient $C_{p}$ , defined as follows:

where $p$ is wall pressure. The streamwise coordinate is normalised on $h$ . The $C_{p}$ distribution appears to be a relatively weak function of $Re_{h}$ . Downstream of the contraction, $C_{p}$ stays roughly constant until the flow approaches the ramp. Although not perfectly zero, the dimensionless pressure gradient $C_{px}=h\,\text{d}C_{p}/\text{d}x$ in this region is small (figure 7 b), so that its influence on boundary layer development is assumed to be mild. The onset of separation at the upper edge of the ramp ( $x/h=0$ ) induces a strong decrease of pressure. The recirculation region is characterised by a low pressure plateau followed by a rapid pressure rise that reaches a maximum at $x/h\approx 7$ . In the following, the plateau and the peak $C_{p}$ values will be referred to as base $C_{p}$ (noted $C_{p,b}$ ) and peak reattachment $C_{p}$ (noted $C_{p,r}$ ), respectively. Spanwise pressure distributions (figure 8) confirm that the mean flow over the model is nearly bidimensional and that the mean separation is homogeneous at least over 60 % of tunnel width. Using $h$ to normalise the streamwise distance, $C_{p}$ and $C_{px}$ are in good agreement with Kourta et al. (Reference Kourta, Thacker and Joussot2015) in all those regions where the geometries of the two experiments are comparable, that is downstream of the separation point and over a large fraction of the flat plate upstream of the edge of the ramp (figure 7). This simply indicates that, for ramps with the same geometry, the pressure distribution approximately scales with step height. According to Adams & Johnston (Reference Adams and Johnston1988a ), the difference in observed $C_{p,r}$ can be explained by the different $\unicode[STIX]{x1D6FF}/h$ ratios, that are of the order of 1 for the present study and 0.18 for Kourta et al. (Reference Kourta, Thacker and Joussot2015). In this respect, let us consider the reduced pressure coefficient $C_{p}^{\ast }$ , defined by Chapman et al. (Reference Chapman, Kuehn and Larson1958) as:

Figure 9 shows that the values of $C_{p,r}^{\ast }$ from the present study and from Kourta et al. (Reference Kourta, Thacker and Joussot2015) agree acceptably well with the $\unicode[STIX]{x1D6FF}/h$ correlation observed by Adams & Johnston (Reference Adams and Johnston1988a ) in BFS flows ( $ER=1.25$ , $Re_{h}=3.6\times 10^{4}$ ). Deviations might be due to $Re_{h}$ effects or to geometric factors (e.g. the slope of the ramp or the different value of $ER$ ). Figure 10 presents the streamwise $C_{p}^{\ast }$ distribution, normalised on $L_{R}$ as suggested by Roshko & Lau (Reference Roshko and Lau1965). For $x/L_{R}$ between 0 and 1, $C_{p}^{\ast }$ data from the present work and from Kourta et al. (Reference Kourta, Thacker and Joussot2015) collapse well on the curve observed by Roshko & Lau (Reference Roshko and Lau1965). As already mentioned at § 3.2, the $C_{p}^{\ast }$ distribution appears to be correlated to the development of the shear layer shown in figure 6: the region where entrainment drives shear layer growth ( $x/L_{R}<0.5$ ) corresponds to the $C_{p,b}^{\ast }$ plateau, while for $x/L_{R}>0.5$ lower spreading rates seem to be linked with pressure recovery. Interestingly, the collapse of $C_{p}^{\ast }$ distributions is less satisfactory for $x/L_{R}>1$ , as if after reattachment $L_{R}$ was no longer the (unique) flow-organising scale.

Figure 9.

$\unicode[STIX]{x1D6FF}/h$ dependency of $C_{p,r}^{\ast }$ . Symbols: ○ present work; $\blacklozenge$ Kourta et al. (Reference Kourta, Thacker and Joussot2015),  $Re_{h}=2\times 10^{5}$ and $ER=1.1$ ;  

 BFS data by Adams & Johnston (Reference Adams and Johnston1988a ),  $Re_{h}=3.6\times 10^{4}$ and $ER=1.25$ .

Figure 10.

Streamwise distributions of reduced pressure coefficients. Symbols: ○  $Re_{h}=3\times 10^{4}$ ;  

  $Re_{h}=4\times 10^{4}$ ;  

  $Re_{h}=5\times 10^{4}$ ;  

  $Re_{h}=6\times 10^{4}$ ; ▴  $Re_{h}=7\times 10^{4}$ . —— (red online) Reduced pressure data from Kourta et al. (Reference Kourta, Thacker and Joussot2015). – – – (black online) Roshko & Lau (Reference Roshko and Lau1965) series A. $\cdots \cdots$  (magenta online) Roshko & Lau (Reference Roshko and Lau1965) series D. — ⋅ ⋅ — (green online) BFS data by Adams & Johnston (Reference Adams and Johnston1988a ).

4.1 Dependencies on the incoming boundary layer

The work of Adams & Johnston (Reference Adams and Johnston1988a ) motivated us at investigating the dependencies of shear layer properties on those of the incoming boundary layer. It is important to notice that in this work the possible influence of several parameters such as ramp characteristics (e.g. $\unicode[STIX]{x1D6FC}$ or salient/rounded edge profile) and free-stream turbulence (see for example Adams & Johnston Reference Adams and Johnston1988b ) were not considered. Future works will address this issue. To begin with, figure 9 already provides such a correlation in terms of $\unicode[STIX]{x1D6FF}/h$ and $C_{p}^{\ast }$ . However the overlapping effects of $\unicode[STIX]{x1D6FF}/h$ and $Re_{\unicode[STIX]{x1D703}}$ (which are inseparable in our experimental set-up) make its interpretation complex. Furthermore, the dimensionless length $\unicode[STIX]{x1D6FF}/h$ provides a comparison between the incoming flow and the perturbation induced by the expansion, but it tells nothing about turbulence in the incoming flow, which is known to have a huge influence on shear layer development. $Re_{\unicode[STIX]{x1D703}}$ and $L_{R}/h$ form an alterative set of parameters that could characterise the upstream dependencies of the shear layer, since $Re_{\unicode[STIX]{x1D703}}$ is intended to include some information about the state of the incoming boundary layer whereas $L_{R}/h$ relates to shear layer development. It is found that $L_{R}/h$ evolves as $Re_{\unicode[STIX]{x1D703}}^{m}$ (shown in figure 11 a), where $m$ is a negative power, at least on the available range of $Re_{\unicode[STIX]{x1D703}}$ . This relationship seems to be robust to $\unicode[STIX]{x1D6FF}/h$ , since the point reported by Kourta et al. (Reference Kourta, Thacker and Joussot2015) ( $Re_{\unicode[STIX]{x1D703}}=2983$ , $L_{R}/h=5$ ) falls well within the trend defined by measurements taken in this study. Surprisingly, our measurements reveal that $m$ changes abruptly from $-0.1$ to $-0.55$ around a critical value $Re_{\unicode[STIX]{x1D703}c}=4100$ . This seems to be linked to the evolution of the turbulent state of the flow. Figure 12 presents the vertical profiles of Reynolds stresses at $x/L_{R}\approx 0.1$ and $x/L_{R}\approx 0.8$ for all available $Re$ . The curves are normalised on $\unicode[STIX]{x1D703}_{SL}$ and $U_{\infty }$ . Let us use $Re_{\unicode[STIX]{x1D703}c}$ to discriminate low $Re_{\unicode[STIX]{x1D703}}$ from high $Re_{\unicode[STIX]{x1D703}}$ . Figure 12(a–c) shows that at $x/L_{R}\approx 0.1$ Reynolds stresses scale relatively well with $\unicode[STIX]{x1D703}_{SL}$ and $U_{\infty }$ both at low and high $Re_{\unicode[STIX]{x1D703}}$ , separately. However, we observe that peak values of $\langle u^{\prime 2}\rangle$ and $-\langle u^{\prime }v^{\prime }\rangle$ for high and low $Re_{\unicode[STIX]{x1D703}}$ do not collapse, with much more intense turbulent levels at high $Re_{\unicode[STIX]{x1D703}}$ . This difference seems consistent with the change of power law found in figure 11. The turbulent level of the flow changes as one moves downstream of the mean separation point. Figure 12(d–f) highlights that, at $x/L_{R}\approx 0.8$ , Reynolds stress profiles are a more progressive function of $Re_{\unicode[STIX]{x1D703}}$ . Similar $Re_{\unicode[STIX]{x1D703}}$ dependencies of turbulent quantities have already been observed in separating/reattaching flows, for example by Song & Eaton (Reference Song and Eaton2003). However, peak values of Reynolds stresses also evolve as $Re_{\unicode[STIX]{x1D703}}^{m}$ , where $m$ is now positive. Once again $m$ changes abruptly around $Re_{\unicode[STIX]{x1D703}c}$ . The values of $m$ appear to change in the streamwise direction also, but the $Re_{\unicode[STIX]{x1D703}}$ trends shown in the inserts in figure 12(d–f) are representative. This further supports the interpretation suggested by figure 11, i.e. that $L_{R}$ decreases with increasing turbulence levels.

Figure 11.

Log–log plot in function of $Re_{\unicode[STIX]{x1D703}}$ of (a) $L_{R}/h$ ; (b) the parameter $C_{SL}=\langle -\unicode[STIX]{x1D703}_{SL}/(2hR_{uv}^{\ast })\rangle$ ; (c) $(v_{E}^{A}|_{RRI}+v_{E}^{A}|_{TNTI})/U_{\infty }$ on $x/L_{R}\in (0,0.5)$ . Symbols: ○ present work; – – – Power-law fits (in the sense of least-mean square); $\blacklozenge$  Kourta et al. (Reference Kourta, Thacker and Joussot2015), data at $Re_{h}=2\times 10^{5}$ .

Figure 12.

Reynold stresses $\langle u^{\prime 2}\rangle$ , $\langle v^{\prime 2}\rangle$ and $-\langle u^{\prime }v^{\prime }\rangle$ in the separated shear layer at (a–c) $x/L_{R}\approx$ 0.1 and (d–f) $x/L_{R}\approx$ 0.8. Inserts in (d–f) show the $Re_{\unicode[STIX]{x1D703}}^{m}$ evolution of the peak of $\langle u^{\prime 2}\rangle$ , $\langle v^{\prime 2}\rangle$ and $-\langle u^{\prime }v^{\prime }\rangle$ , respectively. At $x/L_{R}\approx$ 0.8, the exponent $m$ passes from 0.05–0.1 to 0.65–0.80 (0.22 for $\langle v^{\prime 2}\rangle$ ). Other symbols as in figure 6.

4.2 RANS equation in the separated shear layer

It seems now wise to give some physical underpinning to these observations. To begin with, let us consider the transport equation for mean momentum. Putting in evidence the pressure term, one can write:

where repeated indexes indicate summation. PIV data give access to all quantities on the right-hand side of (4.1) (see appendix A). The results presented at § 3.3 prove that the development of the shear layer is correlated to the longitudinal pressure gradient. Then, we first focus on the mean streamwise momentum transport equation, which by dropping negligible terms is rewritten as:

The relative weight of the terms on the right-hand side of (4.2) evolves in the streamwise direction. In particular, figure 13(a) shows their evolution along the RRI. It appears that the convective term $V\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y$ , related to the main component of mean shear, is dominant in the neighbourhood of the separation point. This is no longer the case in the neighbourhood of the reattachment point ( $x/L_{R}\approx 0.7$ ), where $\unicode[STIX]{x2202}\langle u^{\prime }v^{\prime }\rangle /\unicode[STIX]{x2202}y$ dominates the other terms and causes $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x$ to reach its maximum.

Figure 13.

Longitudinal pressure gradient at $Re_{h}=5\times 10^{4}$ (a) along the RRI; (b) along the TNTI. Symbols: ○  $\unicode[STIX]{x2202}\langle u^{\prime 2}\rangle /\unicode[STIX]{x2202}x$ ;  

  $\unicode[STIX]{x2202}\langle u^{\prime }v^{\prime }\rangle /\unicode[STIX]{x2202}y$ ; ▴ $\unicode[STIX]{x2202}UV/\unicode[STIX]{x2202}y$ ;  

$\unicode[STIX]{x2202}\langle v^{\prime 2}\rangle /\unicode[STIX]{x2202}x$ ;  - - - - -  $\unicode[STIX]{x2202}\langle u^{\prime }v^{\prime }\rangle /\unicode[STIX]{x2202}y+\unicode[STIX]{x2202}\langle u^{\prime 2}\rangle /\unicode[STIX]{x2202}x$ ; —— (red online) $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x$ , computed as in (4.1);  

  $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x$ at the TNTI, computed as in (4.6). All curves are normalised on dynamic pressure and step height $h$ .

We now investigate the RANS equation for the vertical velocity component $v$ . Neglecting smaller terms listed in appendix A, equation (4.1) becomes:

It is observed that $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}y$ always favours an injection of mass into ${\mathcal{V}}_{c}$ , because here $\langle v^{\prime 2}\rangle$ is higher than inside the recirculation region or in the external flow (Cuvier Reference Cuvier2012; Kourta et al. Reference Kourta, Thacker and Joussot2015). However, on most of the large field $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}y$ is not well correlated to local mass fluxes. Following Pope (Reference Pope2000), equation (4.3) can be integrated to:

where $p_{0}$ is the pressure in the free flow. The streamwise pressure gradient can then be computed by taking the derivative of (4.4), which gives:

By plugging into (4.2), this yields for the mean RRI:

Now, if one considers that the mean TNTI bounds the free flow, equation (4.6) evaluated at the mean RRI should at least approximate the evolution of the pressure gradient along the TNTI. Figure 13(b) reports the longitudinal pressure gradient along the TNTI at $Re_{h}=5\times 10^{4}$ , computed with both (4.1) (evaluated at the TNTI) and (4.6) (evaluated at the RRI). Similar results are obtained at all $Re_{h}$ and are not shown for sake of simplicity. The agreement of the two curves is not completely satisfactory in the first part of the separated region, but tendencies are clearly the same throughout the flow. This should confirm, at least qualitatively, that the mean pressure distribution connects the two mean interfaces on scales of the order of $L_{R}$ .

4.3 On the role of turbulent shear stresses

The prominent role of $\unicode[STIX]{x2202}\langle u^{\prime }v^{\prime }\rangle /\unicode[STIX]{x2202}x$ at reattachment is consistent with important past results, that can be invoked to develop the present analysis. In particular, close to reattachment, Chapman et al. (Reference Chapman, Kuehn and Larson1958) wrote:

Figure 12 suggests that $\langle u^{\prime }v^{\prime }\rangle \sim R_{uv}^{\ast }U_{\infty }^{2}$ , where $R_{uv}^{\ast }$ is a local turbulent shear-stress intensity, and that $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y\sim 1/\unicode[STIX]{x1D703}_{SL}$ within the shear layer. It is pointed out that for similar considerations Chapman et al. (Reference Chapman, Kuehn and Larson1958) suggested to scale the $y$ axis with a displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ . The two scalings are equivalent, at least for dimensional analysis. Normalising all terms of (4.7) gives:

where the superscript $^{\ast }$ indicates normalisation and $X$ is a characteristic streamwise scale. On dimensional grounds, equation (4.8) is a valid approximation of the RANS equation if:

where the empirical values of $\unicode[STIX]{x1D703}_{SL,r}$ and $R_{uv,r}^{\ast }$ were calculated at the mean reattachment point. Equation (4.9) shows that $L_{R}$ depends on the interaction of the separated shear layer with the wall. Also, the agreement between the numerical value of $X$ (7.7 $h$ ) and the position of $C_{p,r}$ ( ${\approx}7h$ ) is impressive, if one considers the relative simplicity of the premises to (4.7). This confirms the idea that the pressure rise is one of the main effects of the interaction of the separated shear layer with the wall, at least for the present value of $ER$ . In the framework of this section, however, the most important suggestion given by (4.9) is that $L_{R}$ should be a function of shear layer properties in a neighbourhood of reattachment. Hence, taking into account that $h$ is constant in this study, one should verify that at least in the mean:

In (4.10), the overline symbol indicates the average over a streamwise domain within $L_{R}$ where (4.7) is valid, i.e. $x/L_{R}\in \left[0.7,1\right]$ . The $Re_{\unicode[STIX]{x1D703}}$ trend of $C_{SL}$ is shown in figure 11(b). Very good agreement with figure 11(a) is obtained both at low and high $Re_{\unicode[STIX]{x1D703}}$ . We stress the importance of this latter result, that provides new insight in the functioning and possible control of separating/reattaching shear layers, as follows. The size of the recirculation region relative to step height (and more in general those features of the flow that scale with it) is strongly influenced by the interactions of the shear layer with the wall. The latter depend on the turbulent properties of the shear layer in the neighbourhood of reattachment, which are in turn correlated to the turbulent state of the incoming boundary layer, represented by $Re_{\unicode[STIX]{x1D703}}$ . Since the pressure rise at reattachment seems to be one of the consequences of impingement, in the mean the fluidic system composed by the incoming boundary layer, the separated shear layer and the wall might behave like a closed loop, in which the (favourable) pressure gradient induced upstream of separation acts as a feedback branch, influencing properties of the incoming boundary layer. This might explain why boundary layer quantities (e.g. $\unicode[STIX]{x1D6FF}/h$ and $Re_{\unicode[STIX]{x1D703}}$ ) estimated far upstream of separation show some degree of correlation with properties of the separated shear layer (see Neumann & Wengle Reference Neumann and Wengle2003 and § 2.2.1). Also, it appears that controlling turbulent shear-stress intensity within the shear layer or possibly within the incoming boundary layer could be an efficient strategy to reduce $L_{R}$ and eventually reattach the shear layer to the ramp.

Further work is needed to better assess the validity of both correlations presented in figure 11(a,b). In particular, more precise data are required to understand the local behaviour of the exponent $m$ : the good collapse with $h$ shown by $\unicode[STIX]{x1D703}_{SL}$ (figure 6) suggests that $\langle R_{uv}^{\ast }\rangle \sim 1/L_{R}$ , but in principle the relationship between $\unicode[STIX]{x1D703}_{SL}$ and $R_{uv}^{\ast }$ might itself be a function of $Re_{\unicode[STIX]{x1D703}}$ . In spite of these open questions, $\unicode[STIX]{x1D703}$ and $L_{R}$ will be used as scaling parameters in the following sections.

4.4 On the role of mean shear

Thanks to available PIV data, the approach of Chapman et al. (Reference Chapman, Kuehn and Larson1958) can be extended to the investigation of a neighbourhood of the separation point with very little effort. Figure 13 shows that the convective term $\unicode[STIX]{x2202}UV/\unicode[STIX]{x2202}y$ determines the pressure gradient in this region, since Reynolds stresses balance each other. Then, equation (4.2) can be rewritten as:

Of course, one can put $U\sim U_{\infty }$ . Since the mean flow is almost bidimensional (see § 3.3), by applying the continuity equation to the recirculation region one obtains $V\sim U_{\infty }Y/L_{R}$ , where $Y$ is a suitable vertical scale of the recirculation region. Then, normalising (4.11) gives:

where the same notation as in (4.7) applies. Dimensional analysis then yields:

If it is accepted that (4.11) represents at least the dominant feature of a free shear layer, equation (4.13) states that in presence of a wall a free shear layer approximation holds just up to $x/L_{R}\approx 0.5$ . This is pleasingly consistent with the considerations of § 3.2. However, it must be stressed that the growth rate of the shear layer is influenced by the interaction with the wall even for $x/L_{R}<0.5$ : since $L_{R}/h\sim \left(\text{d}\unicode[STIX]{x1D703}_{SL}/\text{d}x\right)^{-1}$ (§ 3.2), it follows immediately that $\left(\text{d}\unicode[STIX]{x1D703}_{SL}/\text{d}x\right)\sim C_{SL}^{-1}$ . The presence of the wall might then explain why measured growth rates do not match free shear layer models perfectly.

6.1 Mean mass balance

To begin with, a mean mass balance is computed over the fixed control volume ${\mathcal{V}}_{c}$ shown in figure 5. ${\mathcal{V}}_{c}$ is delimited by the mean TNTI, the mean RRI and two vertical sections, placed at the position of the mean separation and reattachment points. Note that the volume ${\mathcal{V}}_{c}$ encompasses the mean separated shear layer. The total mass flux per spanwise unit length through one boundary of ${\mathcal{V}}_{c}$ is given by:

where $\unicode[STIX]{x1D70C}$ is air density, $L_{i}$ is the length of one boundary, $l$ a curvilinear abscissa, $\boldsymbol{n}(l)$ is the local normal to the boundary (pointing outward of ${\mathcal{V}}_{c}$ ) and $\unicode[STIX]{x1D719}$ is the angle between $\boldsymbol{n}(l)$ and the $Y$ axis. Due to the different definitions of the two interfaces, it is $\boldsymbol{n}(l)_{TNTI}=-\unicode[STIX]{x1D735}\tilde{k}/|\unicode[STIX]{x1D735}\tilde{k}|$ and $\boldsymbol{n}(l)_{RRI}=-\unicode[STIX]{x1D735}U/|\unicode[STIX]{x1D735}U|$ . It is worth noticing that ${\dot{m}}_{2}$ must be zero, since the RRI is the only permeable boundary of the recirculation bubble, which is, on average, two-dimensional and stationary. This is consistent with the idea that backflow and shear layer entrainment through the RRI must balance out (Chapman et al. Reference Chapman, Kuehn and Larson1958; Adams & Johnston Reference Adams and Johnston1988b ). The topology of the flow also suggests that ${\dot{m}}_{1}$ (at the mean separation point) will be positive and ${\dot{m}}_{3}$ (at the mean reattachment point) negative. Since in general the TNTI is not a streamline, it will be ${\dot{m}}_{4}={\dot{m}}_{3}-{\dot{m}}_{1}\neq 0$ . The scaling of ${\dot{m}}_{4}$ can be predicted on dimensional grounds. With good approximation, the right-hand side of (1.1) can be rewritten as:

It is straightforward to see from figure 5 that the length of the mean TNTI above the recirculation region scales with $L_{R}$ . Following § 4.1, it is then $L^{A}\sim L_{R}\sim h\,Re_{\unicode[STIX]{x1D703}}^{m}$ . Since $\unicode[STIX]{x1D719}$ is small along both the mean RRI and the mean TNTI, from (6.1) $v_{E}^{A}$ could be approximated as follows:

The geometry of the problem suggests that $\unicode[STIX]{x1D719}\sim h/L_{R}$ , at least for the present values of $h$ and $L_{R}$ . This comes to say that the slope of the TNTI relates to shear layer development, as in the case of the RRI (§ 3.2). The continuity equation can be invoked to obtain the scaling of $V$ (see § 4.4). We put $Y=h$ , so that $V\sim U_{\infty }h/L_{R}$ . This relationship can be interpreted as a measure of streamline deviation due to the ramp: the steeper is the deviation, the higher will be the vertical velocity component acquired by the flow. Finally, it is $v_{E}^{A}\sim U_{\infty }h/L_{R}$ , which implies that:

Minding that $h$ is constant in this study, ${\dot{m}}_{4}$ should then scale with $U_{\infty }$ . Measured mean mass fluxes are reported in table 5. Uncertainties on the mass balance appear to be mainly due to laser reflections introducing corrupted velocity vectors near the wall. In any case, our results evidence that mass is not conserved within ${\mathcal{V}}_{c}$ : the TNTI contributes to the mass balance with at least 30 % of the mass injected into ${\mathcal{V}}_{c}$ by the separating boundary layer, which is by far not negligible. This confirms that entrainment of irrotational fluid plays an important role in the functioning of the separated shear layer. Equation (6.4) is also verified relatively well. Considering that ${\dot{m}}_{1}\sim U_{\infty }\unicode[STIX]{x1D6FF}$ and that $\unicode[STIX]{x1D6FF}\approx h$ in this study, it is possible to write:

which is consistent with the prediction provided by (6.4). Taking into account all dependencies, this can be interpreted geometrically as follows: while the mean slope of the TNTI increases with $Re_{\unicode[STIX]{x1D703}}$ , the reduction (as $L_{R}$ ) of mean TNTI length is compensated for by the higher contribution of velocity (due to increased streamline deviation).

6.2 Local mean mass fluxes

To gather information of a finer grain, let us now consider the normalised local mean fluxes per unit surface through the mean RRI and the mean TNTI. This extends the 0-dimensional analysis of § 6.1 by including the streamwise evolution of local mass fluxes. According to (6.1), the normalised local flux at each point of any of the two boundaries can be computed as:

7.1 Fractal analysis

For simplicity, we begin by characterising the term $L(\unicode[STIX]{x1D6E5})$ appearing in (1.1). $L(\unicode[STIX]{x1D6E5})$ can be assessed easily, at least compared to $v_{E}(\unicode[STIX]{x1D6E5})$ , in the well-known framework of fractal analysis. This technique provides information on the range of scales on which a turbulent interface develops and on the complexity of its convolutions (i.e. its roughness). These are key parameters for mass entrainment as for any other transfer problem (e.g. combustion at a turbulent flame or diffusion of chemicals), because the more convoluted is the interface, the larger is the available exchange surface. It is known since the seminal work of Sreenivasan & Meneveau (Reference Sreenivasan and Meneveau1986) that the contributions of wrinkles of a turbulent interface to its arclength $L$ are well fitted by a self-similar expression $L(r)\sim r^{-\unicode[STIX]{x1D6FD}}$ , where $r$ is the scale of wrinkles and $\unicode[STIX]{x1D6FD}$ is the fractal codimension, at least between an inner scale $\unicode[STIX]{x1D702}_{i}$ and an outer scale $\unicode[STIX]{x1D702}_{o}$ . This is to say that small-scale wrinkles contribute much more to total interface length than large bulges and valleys. Also, the higher is $\unicode[STIX]{x1D6FD}$ , the more the interface is wrinkled. For $r\gg \unicode[STIX]{x1D702}_{o}$ , $L(r)$ is equal to $L_{0}$ , which is close to the length of the mean interface. For $r\ll \unicode[STIX]{x1D702}_{i}$ , it is instead $L(r)/L_{0}=(\unicode[STIX]{x1D702}_{o}/\unicode[STIX]{x1D702}_{i})^{\unicode[STIX]{x1D6FD}}$ (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989). The recent work of Thiesset et al. (Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2016) matches these threes scalings into the following expression.

Table 7 summarises the values of $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D702}_{i}$ , $\unicode[STIX]{x1D702}_{o}$ , $L_{0}$ and $L_{tot}$ , the total average length of the TNTI, estimated from data collected on the detailed field. To this purpose, each instantaneous interface was processed with the calliper technique (see Cintosun, Smallwood & Gülder Reference Cintosun, Smallwood and Gülder2007 among others), which associates with a scale $r$ the length $n\times r$ , where $n$ is the number of $r$ -long segments needed to approximate the entire interface. The operation is repeated for a wide enough range of $r$ . The ensemble-averaged, log–log plot of $r$ versus $n\times r$ is known as the Richardson plot, which gives the cumulative contribution of all scales larger than $r$ to total interface length (figure 18). Then, equation (7.1) is fitted (in a least mean square sense) onto the Richardson plot. This approach has the advantage of yielding unambiguous values of the fractal parameters even at low and moderate $Re$ numbers (see Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2016). It is found that $\unicode[STIX]{x1D6FD}$ agrees with previous observations in TNTIs (see Sreenivasan & Meneveau Reference Sreenivasan and Meneveau1986; Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014) and in other turbulent interfaces (Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2016 and references therein). The range of surface-producing scales spans almost 1.5 decades, bounded by $\unicode[STIX]{x1D702}_{o}\approx 0.2h$ and $\unicode[STIX]{x1D702}_{i}\sim \unicode[STIX]{x1D702}$ . This latter result should be considered qualitative, since neither PIV nor hot-wire data are resolved enough to accurately assess the smallest scales of the flow. It is observed that $L(\unicode[STIX]{x1D6E5})\approx L_{tot}$ for the present value of $\unicode[STIX]{x1D6E5}$ . As for $L_{0}$ , it does not scale with $Re_{\unicode[STIX]{x1D703}}$ (as expected for the mean TNTI) because it is imposed by the extent of the detailed field. The ratio $L_{tot}/L_{0}$ should then be considered as a surface density, i.e. the amount of developed surface per mean unit TNTI length. $L_{tot}/L_{0}$ seems to evolve with $Re_{\unicode[STIX]{x1D703}}$ (see insert in figure 18). For convenience, the hypothesis is made that $L_{tot}/L_{0}$ also scales with a power law $Re_{\unicode[STIX]{x1D703}}^{p}$ , with $p$ changing from ${\approx}0.17$ for $Re_{\unicode[STIX]{x1D703}}<Re_{\unicode[STIX]{x1D703}c}$ to ${\approx}0.46\,Re_{\unicode[STIX]{x1D703}}>Re_{\unicode[STIX]{x1D703}c}$ .

Table 7.

Fractal parameters of the TNTI for all available $Re_{h}$ , computed on the detailed field. $\unicode[STIX]{x1D6FD}$ is the fractal dimension, $L_{tot}$ is the total average length of the TNTI, $L_{0}$ relates to the length of the visible section of the mean TNTI and $\unicode[STIX]{x1D702}_{i}$ and $\unicode[STIX]{x1D702}_{o}$ are the inner and the outer scale, respectively. Low values of $L_{0}/h$ reflect the small size of the detailed field.

Figure 18.

Richardson plot of the TNTI, computed on the detailed field. (a) Comparison of data at $Re_{h}=5\times 10^{4}$ with the best fit of (7.1) (—— shown in red online). (b) Comparison of Richardson plots at all available $Re_{h}$ , with symbols as in figure 6. The best fit of (7.1) is also reported for each dataset (– – –). Note that (7.1) attains $L_{tot}$ for values of $r$ that are not resolved in this experiment. Accordingly, Richardson plots are cutoff at a scale $r_{c}$ , associated with PIV resolution $\unicode[STIX]{x1D6E5}$ . $r_{c}$ is computed following the considerations of § 7.2 as $2/k_{c}$ . Values of $\unicode[STIX]{x1D702}_{i}/\unicode[STIX]{x1D702}_{o}$ are reported in table 7. The insert shows the evolution of $L_{tot}/L_{0}$ in function of $Re_{\unicode[STIX]{x1D703}}$ .