3.1. Axial symmetry, impact of tethers and trip height

The axial symmetry of the flow was examined at two axial stations, in different flow quantities; upstream, on the nose ($x/D = 0.5$), mean surface pressure was examined; downstream, at the BOR tail ($x/D = 3.172$) stagnation pressure, mean velocity and turbulence intensity were examined. The circumferential ring of surface pressure taps on the nose ($x/D = 0.5$) suggested a residual ${\pm } 0.25^\circ$ angle of attack. Contours of the stagnation pressure coefficient ($C_{p_o}$) at the BOR tail are shown in figure 6(a,b). Here, $C_{p_o}=(p_o - p_\infty )/(p_{o,\infty } - p_\infty )$ where $p_o$ is the stagnation pressure in the wake, $p_{o,\infty }$ is the stagnation pressure of the ambient free stream, $p_\infty$ is the static pressure of the tunnel ambient. Outside the wakes from the upstream tethers $C_{p_o}$ is axisymmetric, varying within 9 % from the circumferential average, with a standard deviation of 5 %.

Figure 6. (a) Schematic showing the location of stagnation pressure cross-section measured to verify circumferential uniformity. (b) Contours of stagnation pressure coefficient ($C_{p_o}$) at BOR tail verifying circumferential uniformity; green dashed line represents the turbulence measurement plane. (c) Contours of streamwise mean velocity over a quadrant at the BOR tail revealing the axisymmetry. (d) Contours of streamwise turbulence intensity on a grid similar to that of mean velocity in (c).

Similarly, uniformity in both the mean velocity and turbulence intensity at the BOR tail (figure 6c,d) were examined with a single hot-wire, over a 200 point-grid spread across 15 radial profiles covering a quadrant $(+y, -z)$. The mean velocity was axisymmetric to within 2 % of the circumferential mean, and the turbulence intensity was axisymmetric to within 7 %. Note that the tether wakes shown in figure 6 correspond to the original 1.6 mm tethers which were upgraded to 0.9 mm over the course of the experiments. The wakes of upgraded tethers, measured outside the BOR boundary layer at $x/D =$ 3.172, were found to be approximately 10$^\circ$ wide and were mild, with a $0.05U_\infty$ peak velocity deficit, and $0.015U_\infty$ peak turbulence intensity. Additionally, the impact of these tethers on the BOR boundary layer seems constrained, if not negligible, since the boundary layer velocity and turbulent intensity at the BOR tail, directly downstream of the tethers, indicated no explicit variation from the other circumferential stations. Regardless, all turbulence measurements, discussed in subsequent sections were made at the plane furthest away from the tethers ($\theta =3{\rm \pi} /2$ see figure 6b).

The sensitivity of the flow to trip height was examined, by replacing the original 0.8 mm one with a trip double the height, and comparing the stagnation pressure profiles at the BOR tail. The resultant wake was slightly stronger, with roughly 9 % lower $C_{p_o}$, suggesting the turbulence structure is not overly sensitive to the trip.

3.2. Characteristics of the inflow to the APG ramp

The streamwise variation of mean pressure along the body is shown in figure 7. Estimates of the static-pressure coefficient ($C_p$) are consistent with potential flow calculations (using a doublet panel method for a BOR) and in turn with numerical simulations (wall-modelled large eddy simulation) from Zhou, Wang & Wang (Reference Zhou, Wang and Wang2020). The flow accelerates over the nose, passing the trip ring sandwiched between the nose and the constant diameter mid-body. Further downstream, the sharp corner between the mid-body and ramp generates an intense local acceleration as the flow enters the ramp. Hereafter, the flow decelerates rapidly over the 20$^\circ$ tail cone, with the boundary layer resisting a strong APG.

Figure 7. Streamwise pressure distribution on the BOR; (black circle) measurement, (blue dash line) potential flow simulation, (black line) large eddy simulation (LES) from Zhou et al. (Reference Zhou, Wang and Wang2020).

The boundary layer approaching the ramp has been documented to understand the initial conditions for the APG region. The mean velocity and turbulence intensity approximately 10 mm upstream of the corner ($x/D = 1.977$), sampled by a single hot-wire, are shown in figure 8(a,b). Here, the vertical axis represents the position ($z$) relative to the surface ($z_s$), scaled on the BOR diameter, while the horizontal axis reveals the corresponding mean velocity (figure 8a) and turbulence intensity (figure 8b). Generally, the mean velocity ($U_s$) is higher compared with the tunnel inlet velocity due to the acceleration past the nose, and does not fully asymptote to a constant due to the local acceleration and curvature induced by the downstream corner. The boundary layer thickness – defined throughout this paper as the location from the surface with a turbulence intensity of 2 % – is 8.4 mm, and is thin relative to the local radius ($\delta /r_s=0.04$). The peak measured turbulence intensity is approximately 9 %, occurring at 0.12$\delta$ from the surface, and decays on moving further away. Additional characteristics of the layer such as the integral parameters are shown in table 1. The displacement and momentum thicknesses, $\delta _1$ and $\delta _2$, respectively, have been estimated using the planar definitions (3.1), where the missing data very near the wall have been extrapolated by a cubic spline fit, constrained by a no-slip condition at the wall. The results were only mildly sensitive to the extrapolation process, varying within 5 % of the spline estimates, when tried with linear, cubic, and quadratic extrapolations.

(3.1) \begin{equation} \left.\begin{array}{c@{}} \displaystyle\delta_1 =\int_{0}^{\delta}(1-{U_s}/{U_e})\,\textrm{d}(|z - z_s|) , \\[4pt] \displaystyle\delta_2 =\int_{0}^{\delta}({U_s}/{U_e})(1-{U_s}/{U_e})\,\textrm{d}(|z - z_s|). \end{array}\right\} \end{equation}

Figure 8. (a) Mean-velocity profile upstream of corner ($x/D = 1.977$), $U_\infty$ is the tunnel free-stream velocity (see table 1). (b) Turbulence intensity profile at $x/D = 1.977$.

Table 1. Boundary layer characteristics at inflow ($x/D = 1.977$) to the BOR ramp.

3.3. Mean flow characteristics on the ramp

Downstream of the corner, the mean flow decelerates significantly on the ramp with the boundary layer thickening, as seen from the streamwise mean-velocity contours in figure 9. At the BOR tail ($x/D = 3.172$) the boundary layer is 79.5 mm thick; growing approximately 10 times the thickness just upstream of the corner, over a distance of 1.2$D$. The corresponding velocity at the edge of the boundary layer ($U_e$) decreases by over 40 %, from $1.27U_\infty$ upstream to $0.89U_\infty$ at the tail. Despite such a strong deceleration, the flow is well behaved, as seen from the velocity vectors in figure 9, diverging away from the wall and increasingly aligned with the BOR axis. The circumferential velocity measured by a quadwire, reliable only in the outer 60 %, was less than 8 % $U_s$. While the inner 40 % of the boundary layer saw a stronger circumferential component that varied along the tail, it is strongly corrupted by significant bias errors due to the highly turbulent flow probed with a sensor not aligned with the streamwise direction. However, auxiliary measurements discussed in § 3.1 suggest that it is unlikely that the flow is significantly three-dimensional.

Figure 9. Contours of streamwise mean velocity ($U_s$) on the ramp from single hot-wire measurements, scaled on the tunnel reference velocity ($U_\infty$). Arrows at the measurement stations reveal the flow orientation as measured by quadwire; black dashed line identifies the edge of the boundary layer. Inset to the top right shows the global position of the measurement.

The flow appears to be out of equilibrium, as seen from the downstream evolution of boundary layer characteristics, shown partially in figure 10(a–c) with full details in table 2. Note that all integral parameters have been estimated by integrating radially outward (due to the orientation of the profiles) instead of normal to the surface; however, it was found that the resulting parameters are consistent to within 5 % of their counterparts estimated in the conventional boundary layer coordinates (see Appendix B). The displacement thickness, shown in figure 10(a), (estimated according to (3.1)), increases relative to the boundary layer thickness from 0.2$\delta$ to 0.5$\delta$, suggesting a stronger wake component downstream and the corresponding shape factor $H$ ($=\delta _1/\delta _2$), shown in figure 10(b), increases by over 50 %, from 2.14 upstream to 3.24 at the tail. The associated momentum thickness based Reynolds number $Re_{\delta _2}$ (table 2) rises by an order of magnitude, from some 2000 upstream to approximately 16 000 at the ramp tail. While the shear stress at the wall was not measured, and since no universally accepted hypothesis exists to empirically estimate the value, we infer the trends in wall friction from Reynolds number matched LES of the BOR flow, borrowed from Zhou et al. (Reference Zhou, Wang and Wang2020). Even after confirming the agreement in the mean surface pressure (figure 7), in the mean velocity as well as auto-spectra of unsteady surface pressure (refer to Zhou et al. Reference Zhou, Wang and Wang2020), we rely on the estimates only for qualitative conclusions: here, that the skin friction is not constant across the ramp (figure 10c). Therefore, a strongly varying $H$, $Re_{\delta _2}$ and $C_f$ suggests the non-equilibrium character of the flow.

Figure 10. Characteristics of boundary layer on ramp: (a) displacement thickness ($\delta _1$) as a fraction of the velocity thickness ($\delta$), (b) shape factor $H={\delta _1}/\delta _2$, (c) skin-friction coefficient ($C_f$) from LES (Zhou et al. Reference Zhou, Wang and Wang2020).

Table 2. Boundary layer characteristics on the ramp. Here, $C_f$, $U_\tau$ are obtained from large eddy simulations on the BOR at matched Reynolds number (Zhou et al. Reference Zhou, Wang and Wang2020).

The strength of the pressure gradient has been previously inferred from various parameters, with the most common being Clauser's $\beta _{C} = (\delta _1/\tau _w) \,\textrm {d}p/{\textrm {d}\kern0.7pt x}$ (Clauser Reference Clauser1954), where $\tau _w$ is the shear stress at the wall; in our case, $\beta _{C}$ (table 2) varies between 5 and 18, except at the tail ($\beta _{C} = -14.7$) due to the flow accelerating onto the support shaft (see figure 1). However, $\beta _C$ is not universally applicable, as it tends to infinity for separating boundary layers ($\tau _w\rightarrow 0$). More importantly, Maciel et al. (Reference Maciel, Tie, Gungor and Simens2018) argued that $\beta _C$ and $Re_{\delta _2}$ do not form a consistent set of parameters that describe the flow; specifically, these parameters have not been derived from the boundary layer equations using a consistent choice of length and velocity scales. As a remedy to this problem, they developed a set of parameters from the non-dimensional boundary layer equations, using a consistent choice of length and velocity scales, say $L_o$ and $U_o$. The resulting parameters: $\beta _{o} = L_o/(\rho U_o^2)\,\textrm {d}p_e/{\textrm {d}\kern0.7pt x}$, $\alpha _o = U_eV_e/U_o^2$ and $Re_o = U_oL_o/\nu (U_o/U_e)$ were found to have a direct physical interpretation: they represented the ratio of the order of magnitude of the forces, with the apparent turbulent force (Reynolds shear-stress gradient) as the reference force; $\beta _o$, $\alpha _o$ and $Re_o,$ represented the strength of the pressure gradient force, inertial and viscous forces, respectively, relative to the apparent turbulence force. Furthermore, on examining their DNS datasets, they observed that these parameters, with velocity scale $U_o = U_{zs} = U_e\delta _1/\delta$ and length scale $L_o = \delta$, accurately tracked the ratio of the forces. Here, $U_{zs}$ represents the velocity defect of the bulk flow, first proposed by Zagarola & Smits (Reference Zagarola and Smits1998) for the pipe flows, and later examined for APG layers first by Castillo & George (Reference Castillo and George2001). The parameters $\beta _{zs}$ and $Re_{zs}$ on the BOR ramp, based on the characteristics in table 2, are shown in figure 11(a,b). For reference, $\beta _{zs}$ is co-presented with Castillo's pressure gradient parameter, $\varLambda = (\delta /({U_e\,\textrm {d}\delta /{\textrm {d}\kern0.7pt x}}))\,\textrm {d}U_e/{\textrm {d}\kern0.7pt x}$ in figure 11(a), while $Re_{zs}$ is shown with $Re_{\delta _2}$ in figure 11(b). Three observations can be made; first, the flow is confirmed to be out of dynamic equilibrium since the parameters, representing the ratios of the fluid forces, vary strongly across the ramp; second, the pressure gradient is strong relative to the turbulent force ((pressure gradient)/(turbulent force) ${\approx }10\beta _{ZS}$) and decays downstream on the tail; third, the turbulent force is stronger than the viscous force, even more so downstream, implying that the pressure gradient dominates both the turbulent and therefore the viscous forces.

Figure 11. Flow parameters in APG region. (a) Pressure gradient parameters, $\varLambda = (\delta /({U_e\,\textrm {d}\delta /{\textrm {d}\kern0.7pt x}}))\,\textrm {d}U_e/{\textrm {d}\kern0.7pt x}$ and $\beta _{ZS}=(\delta /{{U^2}_{zs}}) U_e \,\textrm {d}U_e/{\textrm {d}\kern0.7pt x}$ following the work of Castillo & George (Reference Castillo and George2001) and Maciel et al. (Reference Maciel, Tie, Gungor and Simens2018), respectively. (b) Reynolds numbers, $Re_{\delta _2} = \delta _2U_e/\nu$, and $Re_{zs}=U_{zs}\delta /\nu (U_{zs}/U_e)$.

While this non-equilibrium boundary layer over the ramp suffers a strong APG, the effect of transverse curvature appears to be mild. The curvature parameter, $\delta /r_s$ is mostly less than 1 (table 2), implying the boundary layer is thinner than the local radius of curvature ($r_s$) except near the BOR tail. The radius based Reynolds number, $r_s^+ = r_su_\tau /\nu$, although initially high at approximately 11 000, decreases to 477 at the BOR tail. Following the observations of Piquet & Patel (Reference Piquet and Patel1999) and Snarski & Lueptow (Reference Snarski and Lueptow1995) such a range of parameters may affect the mean flow and turbulence statistics moderately, but is not expected to strongly influence the fundamental turbulence mechanisms. The flow, therefore, corresponds to that of a high Reynolds number, strongly decelerating flow over a large cylinder, subjected to an axially varying lateral curvature and pressure gradient.

3.4. Turbulence statistics on the APG ramp

The streamwise evolution of Reynolds normal stress ($\overline {u_s^2}/U_\infty ^2$), measured with a single hot-wire, is contoured in figure 12. Consistent with previous studies, $\overline {u_s^2}$ develop an ‘outer’ peak as the flow decelerates, which is centred initially about 0.4$\delta$ from the surface, drifting further away downstream and reaching ${\sim }0.55\delta$ at the tail. However, contrary to most studies on planar APG boundary layers where the peak intensified with the pressure gradient, we observe the peak to relax downstream, even if normalized by the edge velocity ($U_e$). This can be viewed as a response to transverse curvature combined with a decreasing local pressure gradient (see figure 11a). In an LES study of a BOR flow, Kumar & Mahesh (Reference Kumar and Mahesh2018a) observed the turbulence intensity away from the wall to decay faster than in a flat-plate flow under similar conditions, even for a modest $\delta /r_s$ of 0.3. This is also consistent with earlier work by Piquet & Patel (Reference Piquet and Patel1999), who showed that the transverse curvature does not alter the turbulence production mechanism itself, but the reduction of turbulence activity compared with a planar boundary layer is merely due to a smaller surface area where the production occurs, and the corresponding vorticity per unit volume of the flow, introduced by this surface, is lower than in the planar boundary layer. However, if the stress profiles are scaled with the friction velocity, the expected trend of a magnifying peak is observed, suggesting that the turbulent motions responsible for the outer peak are not dictated by the near-wall shear-stress inducing motions.

Figure 12. Contours of streamwise Reynolds normal stress on the ramp from single hot-wire measurements, scaled on the tunnel reference velocity. White dashed line indicates the boundary layer edge. Inset to the top right shows the global position of the measurement.

The structure of the other in-plane Reynolds stresses, $u_n^2$ (resolved orthogonal to the streamline) and $u_su_n$, is similar to that of the streamwise Reynolds stress discussed above. In fact, this is not just superficially true, since the various Reynolds stresses were found to be directly proportional to the in-plane turbulent kinetic energy (TKE), $E=0.5(\overline {u_s^2} + \overline {u_n^2})$ through precise constants over the measured domain, as shown in figure 13. While the contours of $E$ in figure 13(a) – obtained from planar PIV over the rear third of the ramp – paint a picture consistent with the hot-wire estimates of $\overline {u_s^2}$ stress in figure 12, the ratios of $\overline {u_s^2}$, $\overline {u_n^2}$ and $\overline {u_su_n}$ to $E$, shown in figure 13(b–d), are invariant over the measured domain; $\overline {u_s^2}$ and $\overline {u_n^2}$ are consistently approximately 1.4$E$ and 0.6$E$, and the Reynolds shear stress $u_su_n$ is invariant at $-0.45E$. This behaviour deviates only near the boundary layer edge, as seen about the white dashed line in figure 13(b,c), where $\overline {u_n^2}$ dominates, as one might expect due to the oscillating turbulent–non-turbulent interface. In any case, apart from being a pleasant simplification in extracting the different components from just the TKE through simple RANS calculations, this is an interesting observation. While we do not have a definite explanation, there appears to be some sort of self-preserved structure of the turbulent motions, with a dominant, organized mode through which the streamwise turbulent energy is transferred into the other components. As a first step towards understanding this, we examine the mean flow and turbulence statistics for self-similarity in the following section, deriving the associated length and velocity scales, which will be further examined on the turbulence and correlation structure in subsequent sections.

Figure 13. (a) Turbulence kinetic energy based on in-plane Reynolds normal stresses ($E/U_\infty ^2 = 0.5(\overline {u_s^2} + \overline {u_n^2})/U_\infty ^2$) from PIV; (b) ratio of streamwise normal stress to kinetic energy $(\overline {u_s^2}/E)\approx 1.4$; (c) $\overline {u_n^2}/E \approx 0.6$; (d) $\overline {u_su_n}/E \approx 0.45$.

3.5. Self-similarity in the outer region along the ramp

Figure 14(a,b) shows the profiles of streamwise velocity and Reynolds normal stress, obtained from a single hot-wire, that correspond to the contours in figures 9 and 12 discussed earlier. While the position from the wall is scaled with the boundary layer thickness, the mean velocity, figure 14(a), and turbulence stress, figure 14(b), are scaled with the edge velocity $U_e(x)$. Supporting the preliminary observations in §§ 3.3 and 3.4, the character of the profile changes significantly as the flow decelerates downstream. The velocity deficit increases across the boundary layer and the outer peak in the Reynolds stress weakens as it drifts higher in the boundary layer, reaching 0.55$\delta$ at the tail. Much of this streamwise variation can be accounted for, when $\delta$ is replaced by $\delta _1$ as the length scale, as shown in figure 15(a,b). In general, the mean-velocity profiles form a tighter collapse while the Reynolds stress profiles realign such that the functional form, especially near the peak, is somewhat consistent. Furthermore, the position of the outer peak is almost invariant at $(1.2\pm 0.06)\delta _1$ (figure 15c) and we find that the mean-velocity profiles exhibit an inflection point at this position. This is consistent with Kitsios et al.'s (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017) observation of an inflection point collocated with the outer turbulence peak, occurring at $1.3\delta _1$ and $1\delta _1$ for their mild ($\beta _{C}=1$) and strong ($\beta _{C}=39$) pressure gradient cases. Further confirming this, Maciel et al. (Reference Maciel, Tie, Gungor and Simens2018) observed the peak in the range $1.0\text {--}1.3\delta _1$ for a diverse range of numerical and experimental datasets of non-equilibrium layers. While the relationship between the outer turbulence peak and $\delta _1$ is worth exploring, the success of the $U_e-\delta _1$ scaling is limited only to the vicinity of the Reynolds stress peaks, as evident from the spread in the profiles at positions further away. On that note, by examining several scalings in the literature Maciel et al. argued that it is impossible to produce a complete collapse of the profiles, especially of the second-order statistics, and suggested the success of a scaling, over a broad range of APG, can only be judged by the order of magnitude of the resulting collapse. With such a view, they found $U_{zs} = U_e\delta _1/\delta$ and $\delta$ was the most successful scaling. Commonly referred to as the Zagarola–Smits scale, this scaling was originally proposed as an outer velocity scale for turbulent pipe flow by Zagarola & Smits (Reference Zagarola and Smits1998) and represents the bulk velocity defect of the flow. While Zagarola–Smits scaling may well be successful when considering a wide range of pressure gradients, for our relatively narrow spectrum of strong APG flow, the performance of $U_{zs}-\delta$ scaling is inferior to both $U_e-\delta _1$ and $U_e-\delta$ scalings. In any case, focusing on the larger picture, where we observe inflectional velocity profiles collocated with turbulence stress peak in the outer region, combined with a new peak in the turbulence production and transfer observed in other studies (Skåre & Krogstad Reference Skåre and Krogstad1994; Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017), suggests a fundamentally different mechanism for boundary layers under strong APG.

Figure 14. (a) Mean-velocity profiles, with the vertical axis representing the distance from the surface scaled with $\delta$ and the horizontal axis representing the velocity scaled with the edge velocity. Black line indicates change in the profiles along the ramp. (b) Profiles of the streamwise Reynolds stress with the $U_e-\delta$ scaling. Legend to the right shows the streamwise positions.

Figure 15. (a) Mean-velocity profiles with $U_e$ and $\delta _1$ scaling. (b) Profiles of the streamwise Reynolds stress with the $U_e$ and $\delta _1$ scaling. (c) Location of the peak streamwise Reynolds stress scaled on $\delta _1$. See figure 14 for legend.

A promising proposal is that strong APG layers behave more like a free-shear layer, developing inviscid instabilities in the outer regions, while the importance of near-wall turbulence weakens. For example, experimental studies by Elsberry et al. (Reference Elsberry, Loeffler, Zhou and Wygnanski2000), Song et al. (Reference Song, DeGraaff and Eaton2000) and Schatzman & Thomas (Reference Schatzman and Thomas2017) as well as a numerical study by Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017) invoked the similarity with mixing layers. Through detailed investigation, Schatzman & Thomas gathered evidence for coherent spanwise vorticity, centred about the inflection point. Via quadrant analysis of the shear-stress profiles, they observed that the sweeping motions were more frequent above the inflection point, while ejections dominated below. At the inflection point both the motions were equally likely. Invoking the Rayleigh–Fjørtoft theorem they attributed this observation to inviscid instabilities, and hypothesized an ‘embedded shear layer’ (ESL) in the boundary layer, centred about the inflection point. Inspired from self-similarity in free-shear layers, they proposed the ESL scaling, with the length scale as the vorticity thickness

(3.2)\begin{equation} \delta_\omega = (U_e - U_{IP})/{(\textrm{d}U/\textrm{d}z)}_{IP}, \end{equation}

where IP refers to the outer inflection point and ${(\textrm {d}U/\textrm {d}z)}_{IP}$ is the slope of the velocity profile at the inflection point. The associated velocity scale is the velocity defect at the inflection point

(3.3)\begin{equation} U_d = U_e - U_{IP}, \end{equation}

with these length and velocity scales, and a coordinate system centred about the inflection point (3.4), they observed the mean velocity and Reynolds normal and shear stress to collapse, despite not being in equilibrium

(3.4)$$\begin{gather} \eta =(z-z_{IP})/\delta_\omega, \end{gather}$$

(3.5)$$\begin{gather}U^* =(U_e - U)/U_d. \end{gather}$$

For our case, the mean-velocity and turbulence stress profiles with the ESL scaling are shown in figure 16(a,b). In order to minimize the numerical errors related to finding the inflection point, we centred the coordinate system about the location of peak $\overline {u_s^2}$. The velocity defect profiles, figure 16(a), collapse well, and the functional form away from the wall ($\eta >-1$) is accurately described by the complementary error function ($1 - {\rm erf}(\eta )$), similar to the error function (erf()) commonly used for planar mixing layers (Pope Reference Pope2001). But, the collapse in the turbulence stress profiles, figure 16(b), appears no better than the $U_e - \delta _1$ scaling, particularly on the low-speed side ($\eta < 0$, figure 15b). However, a lack of any immediately obvious trend in the spread suggests significant uncertainty, arising from the discretized profiles used to estimate the scaling parameters. Despite a considerable spread, the peak magnitude of the turbulence intensity $\sqrt {\overline {u_s^2}}$ varied between $(0.232\pm 0.014)U_d$, close to the 0.21$U_d$ observed by Schatzman & Thomas. Furthermore, although not shown here, the collapse in the corresponding PIV profiles from the rear third of the ramp was closer to 0.21$U_d$.

Figure 16. ESL scaling for mean flow. (a) Mean-velocity defect profiles, with (black line) representing the complementary error function, $1-{\rm erf}(\eta )$. (b) Streamwise Reynolds normal stress profiles. (c) Streamwise growth of vorticity thickness ($\delta _\omega$) of the shear layer. Slope of the dashed line corresponds to $\textrm {d}\delta _\omega /{\textrm {d}\kern0.7pt x} \approx 0.048$. For legend, refer figure 14.

Examining the shear layer parameters, we observe the vorticity thickness $\delta _\omega$ to grow almost linearly along the ramp (figure 16c), at a rate $\textrm {d}\delta _\omega /{\textrm {d}\kern0.7pt x} \approx 0.048$, within the range commonly observed in free-shear layer studies (such as Oster & Wygnanski Reference Oster and Wygnanski1982). Furthermore, consistent with the condition for similarity, we observe that the ESL length and velocity scales are proportional to the boundary layer thickness and edge velocity respectively as

(3.6)\begin{equation} \delta_\omega/\delta \approx U_d/U_e \approx 0.4 \pm 0.05. \end{equation}

Figure 17 shows comparisons of the BOR results with the axisymmetric APG boundary layer measurements of Dengel & Fernholz (Reference Dengel and Fernholz1990) (red, hereafter known as DF90), in addition to the planar APG flow DNS of Maciel et al. (Reference Maciel, Tie, Gungor and Simens2018) (black dashed lines, hereafter known as MTGS18). The shape factors for each of the cases – $H = 2.35$ for DF90, and $H = 1.76 \text {--} 2.94$ for MTGS18 (see caption) – are within the range observed on the BOR tail. Since a quantitative survey requires a rigorous study with matched flow parameters, which are yet to be established for APG flows, our intention here is to reveal only qualitative but insightful comparisons. For example, DF90 had a significantly different experimental set-up, with a constant-radius circular cylinder where the APG was enforced through flow suction from porous wind-tunnel walls, unlike a conical contraction for the BOR: the success of the ESL scaling suggests its generic applicability for axisymmetric flows. Similarly, comparisons with the planar and strong APG results of MTGS18 suggest that the flow physics of strong APG boundary layers are not very sensitive to the flow history or specific boundary conditions. As a side note, the ESL scaled turbulence intensity profiles have a wider spread, especially near the peak and in the low-speed regions ($\eta < 0$), suggesting that the peak value of $0.21U_d$ reported by Schatzman & Thomas (Reference Schatzman and Thomas2017) is not strictly universal.

Figure 17. Comparison of BOR profiles (from figures 14 and 16) with previous studies from flat-plate and axisymmetric APG boundary layers. (a) ESL scaling for mean velocity; left bottom shows the ESL scaling and right top shows the same data re-plotted with $U_e$ and $\delta$ scaling. (b) Corresponding turbulence intensity profiles; bottom left shows the comparisons in ESL scaling and top right shows the profiles in $U_e$ and $\delta$ scaling. Legend: (redline-redcircle-redline) axisymmetric case, with $H = 2.35$ and $\delta /r_s = 0.2$ ( from Dengel & Fernholz (Reference Dengel and Fernholz1990), see case 2: $x = 1.131$ m). Black dashed lines correspond to flat-plate results from the DNS2017 case of Maciel et al. (Reference Maciel, Tie, Gungor and Simens2018); (orange square) $H = 1.76$, (purple diamond) $H = 1.92$, (magenta circle) $H = 2.11$, (blue triangledown) $H = 2.34$, (black triangleup) $H = 2.57$, (yellow hexagon) $H = 2.78$, (green triangleleft) $H = 2.94$.

While these observations certainly advocate the success of ESL scaling for axisymmetric APG boundary layers, some fundamental questions remain open. The idea that the instantaneous flow can ‘see’ the mean-velocity profile, and therefore the inviscid instabilities from inflection points is debatable. Previous studies in fully turbulent free-shear flows (see Wygnanski, Champagne & Marasli Reference Wygnanski, Champagne and Marasli1986) have observed that the linear stability analysis of inflectional mean-velocity profiles yielded eigen-solutions that resemble the turbulent intensity profiles. Extending the analysis to wall-bounded flows, Michalke (Reference Michalke1991) made similar observations for flows approaching separation, noting that the wall has a stabilizing effect. It certainly seems to be convincing that the inviscid instabilities play an important role. However, Maciel et al. (Reference Maciel, Simens and Gungor2017) were unable to deduce any coherent structures relevant to inviscid instability, in their sharply decelerated boundary layer. Furthermore, the commonly observed outer stress peak, even for milder APG boundary layers without inflectional profiles, cannot be explained by the instability mechanism. One may therefore speculate that the inflectional velocity profiles and the amplified turbulence activity in the outer regions are just correlated without sharing a cause–effect relationship. Of course, further investigation into this aspect is needed before the scaling can be concretely attributed to inviscid instabilities. Subsequently, some work is also needed to develop a rigorous framework for strong APG layers. We must establish the conditions under which the ESL scaling is valid, and how this ties into the layer structure of the boundary layer, including the near-wall layer.

While the questions raised above must certainly be addressed to illuminate the fundamental physics and develop a rigorous framework for strong APG layers, we are interested in examining the turbulence structure and the implications of the shear layer scaling for the length and time scales of the turbulence. This is expected to provide significant input for the aeroacoustic predictions where the integral length scales and the turbulence quantities are a direct input to the estimate the far-field spectrum, for example when a fan is ingesting the boundary layer.

3.6. Turbulence structure of streamwise velocity

The spectral structure of streamwise turbulence along the ramp, measured with a single hot-wire, is shown at representative axial stations in figure 18(a–d). In each figure, contours of the pre-multiplied spectra $f'G_{u_su_s}/U^2_\infty$ are presented, with the vertical axis showing the position in the boundary layer in terms of $\delta$, and the horizontal axis showing the frequency normalized with the reference scale $f'=fD/U_\infty$. In general, the structure of the turbulence resonates with the character of the Reynolds stress profiles discussed in § 3.5. At all locations on the ramp, the most active region, across the frequency range, is centred about the outer turbulence peak that drifts further away from the wall downstream (following the dashed line from figure 18a–d). Consistent with the stress profiles the peak level reduces downstream, by approximately 30 %, from over 0.0025 upstream ($x/D = 2.297$) to just over 0.0018 at the BOR tail ($x/D = 3.172$). Furthermore, the approximate centroid of the active region shifts to lower frequency, sliding from $f'\sim 7$ at $x/D=2.297$, figure 18(a), to $f' \sim 1.8$ at $x/D = 3.172$, figure 18(d), suggesting an amplification in the low-frequency motions in the outer region. This amplification of the large-scale motions can be visualized clearly by comparing the structure at each axial station, with that at a reference station, say at the BOR tail, as shown in figure 19. Here, the turbulence structure at various upstream stations (figure 18a–c) is scaled with that of the BOR tail (figure 18d), with the contour level representing the ratio of the spectra in dB scale. Compared with the BOR tail, the energy is generally weaker at low frequency (blue contours) but stronger at higher frequency (red contours). This difference intensifies upstream, with the maximum difference of the order of 10 dB at $x/D = 2.297$. For example, in figure 19(a), at $z - z_s=0.25\delta$, the energy is lower for $f'<2$ and higher for $f'>2$, by over 10 dB. While this amplification of the large-scale motions is a generic feature of APG flows (Harun et al. Reference Harun, Monty, Mathis and Marusic2013), the presence of lateral curvature is expected to further assist the amplification, as observed by Snarski & Lueptow (Reference Snarski and Lueptow1995).

Figure 18. Contours of the pre-multiplied spectra of the streamwise velocity, $f'G_{u_su_s}/U_\infty ^2$, at representative streamwise stations on the ramp. Frequency $f' = fD/U_\infty$, where $D$ is body diameter ($D = 0.4318$ m) and $U_\infty$ is tunnel free-stream velocity; (a) $x/D = 2.297$, (b), $x/D = 2.615$, (c) $x/D = 2.854$, (d) $x/D = 3.172$. Dashed lines indicate the position of the peak levels, relative to the surface, and the corresponding frequency.

Figure 19. Comparison of the turbulence structure at various upstream stations, with that at the downstream BOR tail ($x/D = 3.172$). Contour levels represent the ratio of the pre-multiplied power spectrum to that at the BOR tail, on a dB scale; (a) $x/D = 2.297$, (b) $x/D = 2.615$, (c) $x/D = 2.854$.

Figure 20(a–d) shows the turbulence structure plotted in terms of the ESL scaling. When the pre-multiplied spectra are plotted in the ESL coordinate system as $f'G_{u_su_s}/U^2_d$ where $f' = f\delta _{\omega }/U_e$, the spectral structure appears similar along the ramp. Here, $U_d$ is chosen as the velocity scale, $\delta _{\omega }/U_e$ is chosen as the time scale since $U_e \propto U_d$ (3.6). The success of the ESL scaling was found to be superior to other related time scales $\delta /U_e$, $\delta _1/U_e$ and $\delta /U_{zs}$. The geometrical features are consistent, with the peak levels centred about $\eta =0$, and about frequency $f'=f\delta _{\omega }/U_e\sim 0.18$. The consistency along the ramp can be visualized better in figure 21. Similar to figure 19, the spectra at each station are normalized with the that of the BOR tail, with the contour levels representing the ratio of the spectra in dB scale. With the ESL scaling, the streamwise variations are typically within $\pm$2 dB as compared with $\pm$10 dB for the original spectra. Furthermore, for frequencies $f'<1$ the streamwise variations are within $\pm$1 dB, corresponding to a $\pm$25 % change. This can be seen in the pre-multiplied line spectra extracted from the contours in figure 20, corresponding to various representative locations in terms of, $\eta$, shown for all streamwise locations on the ramp, in figure 22. While the functional form of the spectra changes with the distance from the wall, consistent with the expected inhomogeneity, the spectra from various streamwise stations are consistent at each $\eta$, particularly for $f'<1$, suggesting that the low-frequency (large-scale) motions are consistent along the APG region. One can expect this from a boundary layer with an ESL, where the large-scale motions primarily driven by the shear layer are superposed on the underlying boundary layer turbulence.

Figure 20. Contours of the pre-multiplied spectra of the streamwise velocity, $f'G_{u_su_s}/U_d^2$, at different streamwise stations on the ramp. Frequency $f' = f\delta _{\omega }/U_e$, where $\delta _{\omega }(x)$ is vorticity thickness and $U_e(x)$ is edge velocity; (a) $x/D = 2.297$, (b) $x/D = 2.615$, (c) $x/D = 2.854$, (d) $x/D = 3.172$. Horizontal dashed lines indicate the position of the peak levels in the shear layer, and vertical dashed lines indicate the corresponding frequency of the peaks.

Figure 21. Contours of the pre-multiplied spectra of the streamwise velocity, $f'G_{u_su_s}/U_d^2$, at different streamwise stations on the ramp. Frequency $f' = f\delta _{\omega }/U_d$, where $\delta _{\omega }(x)$ is vorticity thickness and $U_e(x)$ is edge velocity; (a) $x/D = 2.297$, (b) $x/D = 2.615$, (c) $x/D = 3.172$.

Figure 22. (a) Profiles of Reynolds stress with ESL scaling, showing the locations corresponding to the spectra to the right. (b) Pre-multiplied spectra of the streamwise velocity based on shear layer parameters. Line spectra in each plot are shown for a particular location in the shear layer based coordinate system. Line spectra shown for $\eta = -1$, $-$0.5, 0, 0.5, 1, moving from bottom to top. For legend refer to figure 14.

To summarize, the structure of streamwise turbulence is significantly modified by the strong APG and lateral curvature. The most active region continuously drifts higher in the boundary layer and is centred about the inflection point, consistent with the turbulence stress. As observed in previous studies, the importance of large-scale motions increases as they increasingly energize across the layer while small-scale motions weaken significantly. These large-scale motions appear to be driven by the ESL, with the spectrum below $f\delta _{\omega }/U_e<1$ retaining its functional form along the ramp. Additionally, the performance of other scalings, based on the Zagarola–Smits velocity scale ($U_{zs} -\delta$) and on displacement thickness ($U_e -\delta _1$) was inferior to the ESL scaling. In the next section, the impact of APG on the spatial characteristics of the streamwise turbulence, including the evolution of length scales, two-point correlations and convection velocities are investigated.

3.7. Correlation structure of streamwise velocity

The integral time scale of the streamwise velocity on the ramp $\varGamma _{u_s}$, estimated from single hot-wire results, is shown in figure 23(a). The time scale is obtained by directly integrating the time-delay correlation coefficient ($\rho _{u_su_s}$)

(3.7)\begin{equation} \varGamma_{u_s} (x,z) = \int_{0}^{\infty} \rho_{u_su_s}(x,z,\tau) \,\textrm{d}\tau, \end{equation}

where $\tau$ is the time delay. As seen in figure 23(a) the time scale increases moving into the boundary layer (for $|z-z_s|>0.1\delta$) at all streamwise stations, and is consistent with both ZPG and APG studies in the past (Glegg & Devenport Reference Glegg and Devenport2017; Lee Reference Lee2017). Moving downstream along the ramp, the integral time scales – that represent large-scale motions – elongate as the mean flow expands, and at $|z-z_s|=0.5\delta$ are approximately eight times longer at the BOR tail ($x/D = 3.172$) compared with upstream ($x/D = 2.059$). When normalized with the local boundary layer time scale $\delta /U_s$ (figure 23b) it can be seen that time scales grow roughly proportional to the boundary layer, and the overall form of the profile is somewhat preserved (within the uncertainty). Interestingly, this functional form resembles that of a planar, zero pressure gradient boundary layer at $Re_{\delta _2} \approx 15\,000$ (Morton, Devenport & Glegg Reference Morton, Devenport and Glegg2012). While the organization is similar, the time scales of the BOR flow are approximately four times shorter than the ZPG layer, in proportion to the respective $\delta /U_s$, highlighting the importance of the pressure gradient and flow history. The integral time scales normalized on the ESL time scale ($\delta _{\omega }/U_e$) are shown in figure 23(c). The tighter collapse indicates the importance of the embedded shear layer motions in the boundary layer turbulence.

Figure 23. Integral time scales of the streamwise velocity, on the ramp boundary layer. (a) Integral time scales normalized on constant reference velocity and BOR diameter, black arrow shows the variation along the downstream direction. (b) Integral scales from (a) normalized on $\delta /U_s$; orange curve represents the corresponding integral scales from a planar, zero pressure gradient boundary layer, at a $Re_{\delta _2}=15\,000$ (Morton et al. Reference Morton, Devenport and Glegg2012); corresponding $x$-axis shown on top. (c) Integral time scales normalized on the shear layer time scale ($\delta _\omega /U_e$). For legend see figure 14.

The corresponding streamwise integral length scales can be derived from the normalized time scale $\varGamma _{u_s}U_s/\delta$, shown in figure 23(b). In general, it is very common to estimate the turbulence length scales from single-point measurements such as described above. The typical assumption is to regard the turbulence as if it were frozen and convected by the mean flow, as hypothesized by Taylor (Reference Taylor1938) for homogeneous and low turbulence flows. Several studies have investigated Taylor's hypothesis for wall-bounded flows (see Atkinson, Buchmann & Soria (Reference Atkinson, Buchmann and Soria2015); de Kat & Ganapathisubramani (Reference de Kat and Ganapathisubramani2015) and Renard & Deck (Reference Renard and Deck2015) for ZPG boundary layers and Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) for channel flow); both a scale-independent convection velocity (as a function of wall-normal location) and a comprehensive frequency-dependent velocity were developed and examined by these studies. A common finding for boundary layers has been that the convection velocity, for all scales, is nearly equal to the local mean speed except in the inner region, where the turbulence has been interpreted to travel significantly faster (Atkinson et al. Reference Atkinson, Buchmann and Soria2015; Renard & Deck Reference Renard and Deck2015). This is generally attributed to the amplitude modulation of near-wall turbulence by the outer, larger-scale turbulence (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009, Reference Mathis, Hutchins and Marusic2011). While such observations appear to extend to moderate APG boundary layers over a flat plate (Drozdz & Elsner Reference Drozdz and Elsner2017), we are interested in briefly examining the convection velocity for the strongly decelerating BOR flow, towards the development of turbulence modelling and aeroacoustic predictions for vehicle-relevant configurations.

In order to examine Taylor's hypothesis for the BOR flow, we consider the comparisons between two-point spatial correlation from PIV measurements, and the single-point time-delay correlation from hot-wire measurements. In the following analysis, we will first introduce the two-point correlation function of the streamwise velocity in the BOR boundary layer, followed by a comparison with Taylor's hypothesis estimates of the correlation function, deduced from the single-point measurement. The comparisons will then be used to reveal the validity of Taylor's hypothesis, highlighting the importance of nonlinear effects due to the highly turbulent flow.

The two-point, spatial correlation coefficient for the streamwise velocity is defined as

(3.8)\begin{equation} \rho_{u_su_s}(x, z ; x', z')=\frac{\langle u_s(x,z)u_s(x',z')\rangle}{\sqrt{\langle u_s^2(x,z)\rangle \langle u_s^2(x',z')\rangle}}, \end{equation}

where $(x',z')$ is the reference location with respect to which the correlation is computed by averaging over time. For illustration, figure 24(a) shows the two-point correlation for a reference point at the BOR tail, located $0.56\delta$ from the surface, corresponding to the outer peak in Reynolds stress. The horizontal and vertical axes are to scale in order to ensure accurate interpretation, and the contour lines radiate outward for every 0.1 drop in the correlation coefficient. The average eddy structure appears elliptic, with the major axis (connecting $\rho _{u_su_s}=0.2$) inclined to the surface by 27$^\circ$. While these geometric details are slightly sensitive to streamwise and radial position of the reference point, the structure is generally more compact (in proportion to $\delta$) and further tilted from the horizontal in comparison with high Reynolds number ZPG layers (7$^\circ$–12$^\circ$) (Tutkun et al. Reference Tutkun, George, Delville, Stanislas, Johansson, Foucaut and Coudert2009). This is consistent with our observation earlier in that the streamwise integral time scales of the BOR flow were four times shorter than the ZPG layer, in comparison with the respective $\delta$.

Figure 24. (a) Two-point correlation of the streamwise velocity from PIV, $\rho _{u_su_s}(x',z';x,z)$, with anchor point at $x'/D = 3.172$ and $z'/D = -0.177$ (corresponds to 0.56$\delta$ from surface). (b) Spatial correlation from (a) as a function of streamwise separation from PIV, (black line) $\rho _{T, u_su_s}(x,z,\Delta x_s)$, compared with hot-wire estimates via Taylor's hypothesis (black dash) $\rho _{u_su_s}(x',x'-U_s\tau )$.

A slice of the correlation structure from figure 24(a), drawn in the streamwise direction (in the direction of the local mean velocity) is shown in figure 24(b); this is compared with the single hot-wire estimate obtained through Taylor's hypothesis, defined as

(3.9)\begin{equation} \rho_{T,u_su_s}(x', z'; \Delta x_s)=\frac{\langle u_s(x',z')u_s(x',z', x' - U_s \tau) \rangle}{\sqrt{\langle u_s^2(x',z')\rangle \langle u_s^2(x',z',x' - U_s\tau)\rangle}}, \end{equation}

where $x',z'$ is the reference location (where the probe is positioned), $\Delta x_s (= x'-U_s\tau )$ is the separation along the streamwise direction, assuming that the turbulence is frozen and convecting at the local mean speed $U_s$. Clearly, the dashed line representing the Taylor's hypothesis estimate (3.9) decays faster than the solid line representing the two-point correlation from PIV (3.8), decaying to 10 % at 0.33$\delta$, nearly twice as fast compared with 0.53$\delta$ for the two-point estimate. This inconsistency exists at all other streamwise stations, and there appears to be a definite pattern in the wall-normal direction to the inconsistency, as shown in figure 25. Here, the two-point correlation and Taylor's hypothesis estimate are compared for various locations in the boundary layer at a slightly upstream position ($x/D = 2.933$): while $\rho _{T,u_su_s}( x',z',x'- U_s\tau )$ generally decays faster than the two-point estimates, the discrepancy intensifies further inside the boundary layer. To quantify this systematic deviation, we evaluate the integral length scales by numerically integrating the correlation coefficient (until the first zero crossing, using the trapezoidal rule), with the length scale obtained from PIV defined as,

(3.10)\begin{equation} L(x',z') = \int_{0}^{\infty}{\rho(x',z', \Delta x_s)\,\textrm{d}\kern0.7pt x_s}. \end{equation}

And for the length scale estimate from time-delay correlations (through Taylor's hypothesis)

(3.11)\begin{equation} L_s(x',z') = \int_{0}^{\infty}{\rho_T(x',z';x' - U_s\tau)\,\textrm{d}(x' - U_s\tau)} = \varGamma(x',z') U_s(x',z'). \end{equation}

Figure 25. Two-point correlation from PIV (black line) compared with spatial correlation obtained from single-point hot-wire measurements (black dash) using Taylor's hypothesis, for $x/D=2.933$.

Estimates of the integral length scales at $x/D = 2.933$ obtained through (3.10)–(3.11) are shown in figure 26(a). Generally speaking, Taylor's hypothesis seems to reasonably predict the length scale in the outer half of the boundary layer (${>}0.6\delta$), where we see PIV and hot-wire estimates are in agreement, at approximately 0.2$\delta$. However, in the lower 40 % of the boundary layer, Taylor's hypothesis significantly underpredicts the length scale. Based on the local mean velocity, the length scale estimates are approximately 60 % smaller at $|z-z_s|=0.1\delta$, suggesting that the nonlinear interactions play a significant role over distances much further from the wall than reported by ZPG flow studies. Therefore, significant corrections are required when estimating the length scale from single-point measurements, which is an important input for predictions of the far-field noise spectrum, for example of a rotor ingesting such highly turbulent flows. In this case, although an ingesting rotor sees the turbulence at a fixed location just like the fixed hot-wire, the fact that this inferred length scale is inaccurate may complicate the extrapolation of this estimate into the full multi-dimensional correlation function, increasing the error of using standard forms, such as from the homogeneous turbulence. To contain such a potential error, approximate corrections can be proposed, by assuming that the turbulence is still frozen as it convects, and estimating the apparent convection velocity ($U_{ac}$) of the integral turbulence structures, based on the measured length and time scales,

(3.12)\begin{equation} U_{ac}(x',z')=L_{u_su_s}(x',z')/\varGamma_{u_su_s}(x',z'). \end{equation}

The apparent convection velocity $U_{ac}$ normalized on the local mean velocity, is shown in figure 26(b). Results are shown for available data at various stations along the tail, and the trends are approximately consistent across the stations. The frozen turbulence appears to convect at the local mean speed only in the outer portions of the boundary layer (${>}0.6 \delta$). Below $0.6\delta$, the convection velocity is over 1.4$U_s$, as seen from the apparent convection velocity method. Closer to the wall, even significant corrections are expected. This is an important result for the aeroacoustics community.

Figure 26. (a) Integral length scales in the boundary layer at $x/D = 2.933$; (black circle) $L_{u_su_s}/\delta$ (PIV), (red square) $\varGamma _{u_su_s}U_s/\delta$ (Taylor's hypothesis with local mean velocity). (b) Apparent convection velocity ($U_{ac} = L_{u_su_s}/\varGamma _{u_su_s}$) at representative streamwise stations; (orange triangleleft) $x/D = 2.854$; (orange square) $x/D = 2.933$; (orange diamond) $x/D = 3.092$.

Figure 27. (a) The cross-section at BOR tail ($x/D = 3.172$), showing the measurement grid for circumferential and radial correlation measurements. Circumferential correlations are measured from the vertical axis and radial correlations along the horizontal axis. The contour levels in the background reveal the turbulence intensity. (b) Correlation coefficient of the streamwise velocity for an anchor point at $0.4\delta$ at $x/D = 3.172$: solid lines indicate PIV results, dashed line indicate hot-wire results. Streamwise correlation in black, radial in red, and circumferential in blue.

To fully document the average eddy structure of streamwise velocity in order to provide quantitative inputs for turbulence modelling and aeroacoustic predictions, we measured a subset of the radial and circumferential correlations with a single hot-wire at the BOR tail ($x/D = 3.172$). With the conventional anchor-probe and moving-probe arrangement, the correlations were measured at four anchor points in the boundary layer, represented in figure 27(a) with the turbulence intensity contours in the background; while the circumferential correlations were measured about the vertical axis, the radial correlations were measured the horizontal, anchored at 0.40, 0.65, 0.75 and 0.85$\delta$. As the trends were more or less consistent at all anchor points, results are shown for just the $0.4\delta$ case in figure 27(b), along with the streamwise correlation for comparison. Here, the correlation coefficient $\rho _{u_su_s}$ is shown on the vertical axis, with separation (normalized on $\delta$) on the horizontal axis (separation implies $x' - x_s$ for streamwise, $r'- r$ for radial, $r'(\Delta \theta )$ for circumferential). Furthermore, to cross-validate measurements, the corresponding results from PIV (solid lines) are included with the hot-wire estimates (dashed lines): while the disagreement in the streamwise correlation has been attributed to the inaccuracy of Taylor's hypothesis as discussed above, the agreement in the radial correlations suggests an absence of hot-wire probe interference effects.

Similar to the streamwise correlation, the radial correlation (red) decays monotonically with increasing separation, but decays approximately twice as fast, reaching 10 % level at a separation of 0.2$\delta$. The circumferential correlations (blue) decay even faster, reaching 10 % level in approximately 0.12$\delta$ (depending slightly on the anchor position), and develop a negative tail ($-$5 %) at larger separation. The corresponding integral length scales, calculated by integrating the area under the correlation curve (including the negative excursions), and considered to represent the large-scale features, are shown in table 3 for the four anchor positions. Consistent with the correlations in figure 27(b) the radial length scale is 0.09$\delta$, approximately 40 % of the streamwise length scale. The associated circumferential length scale at this position is even smaller at 0.06$\delta$, roughly a quarter of the streamwise scale. This anisotropy of the length scales is not too different in the outer boundary layer as all the length scales shorten slightly, and are organized at a $1\,{:}\,0.6\,{:}\,0.3$ ratio (between the streamwise, radial and circumferential scales) at 0.85$\delta$.

Table 3. Integral length scales of the streamwise velocity, in the radial and circumferential directions, at the BOR tail ($x/D = 3.172$).