2.1 Wind tunnel

The experiments reported in this paper were carried out in a suction type low-turbulence wind tunnel. It is an open-circuit tunnel, which has a 3000 mm long test section with a square cross-section of $610~\text{mm}\times 610~\text{mm}$ . The settling chamber of the tunnel houses a honeycomb and six screens. The contraction ratio of the tunnel contraction section, which ensures a monotonic velocity increase, is 16:1. The test section of the tunnel was made optically transparent for PIV measurements. The test section is followed by a diffuser of length 3000 mm. The tunnel fan at the diffuser end is driven by an AC motor (14.5 kW), which is controlled by a speed controller. The streamwise turbulence level at the test section, measured following Mandal et al. (Reference Mandal, Venkatakrishnan and Dey2010), is found to be $0.1\,\%$ of the free-stream velocity, $U_{0}$ .

Boundary layer measurements were carried out on a transparent acrylic flat plate that was 12 mm thick, 1200 mm long and 610 mm wide. It was mounted horizontally at the midplane of the test section. The plate had an asymmetrical modified superelliptic leading edge, which was designed based on the guidelines suggested by Hanson, Buckley & Lavoie (Reference Hanson, Buckley and Lavoie2012) with a thickness ratio of $7/24$ between the working and non-working sides. The leading edge was 120 mm long with aspect ratios of 34.3 and 14.1 for the upper and lower ellipses respectively. Hanson et al. (Reference Hanson, Buckley and Lavoie2012) pointed out that a leading edge of this kind reduces the receptivity of the boundary layer by eliminating the discontinuity at the juncture where the curved surface meets the flat surface of the plate. The curvature discontinuity at the juncture can affect the stability properties of the flow (e.g. Saric, Reed & Kerschen Reference Saric, Reed and Kerschen2002). Flat plates with asymmetric leading edges were also utilized in earlier transition studies to reduce the leading edge pressure gradient region (e.g. Klingmann et al. Reference Klingmann, Boiko, Westin, Kozlov and Alfredsson1993; Fransson et al. Reference Fransson, Matsubara and Alfredsson2005; Li & Gaster Reference Li and Gaster2006). Moreover, a trailing edge flap was used to minimize the leading edge suction peak by positioning the stagnation point along the measurement side of the plate.

2.2 Measurement techniques: PIV and hotwire anemometry

We utilized three different PIV configurations, namely the conventional PIV technique, the time-resolved PIV (TR-PIV) technique and the simultaneous orthogonal dual-plane PIV technique, for this experimental work. All of the PIV measurements in the wall-normal ( $x{-}y$ ) plane were carried out at the plate centreline, i.e. $z=0$ , and measurements in the spanwise ( $x{-}z$ ) plane were carried out at $y=2.5$  mm unless otherwise stated. Here, the streamwise, wall-normal and spanwise directions are denoted by $x$ , $y$ and $z$ respectively. The plate leading edge is at $x=0$ , and $y=0$ refers to the wall.

The conventional PIV measurements were carried out using high-resolution CCD cameras (12bit, 8MP Imprex, 10 Hz; 16MP TSI, 2 Hz; 29MP Imperx, 2 Hz) and a Nd:YAG laser (Innolas Spitlight Compact 400 PIV, 180  $\text{mJ}~\text{pulse}^{-1}$ , 10 Hz). The PIV data over a large field of view in bypass boundary layer transition were acquired at a frequency of 2 Hz using the 29MP CCD camera. The measurements in the $x{-}y$ plane were carried out by centring the camera at $x=303$  mm and 542 mm. The field of view for both of these measurements was $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}y\approx 255~\text{mm}\times 170~\text{mm}$ . Hence, the streamwise distance of a particular PIV measurement zone varied from ( $x_{c}-\unicode[STIX]{x0394}x/2$ ) mm to ( $x_{c}+\unicode[STIX]{x0394}x/2$ ) mm, where $x_{c}$ is the location of the camera centre. The measurements in the $x{-}z$ plane were carried out in two separate fields of view, i.e.  $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}z\approx 272~\text{mm}\times 181~\text{mm}$ and $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}z\approx 308~\text{mm}\times 205~\text{mm}$ , centring the camera at $x=300$  mm and 600 mm respectively. The results of the conventional PIV measurements in bypass boundary layer transition in the $x{-}y$ and $x{-}z$ planes are reported in §§ 3 and 6 respectively. The conventional PIV measurements in the Blasius boundary layer, i.e. without using any grid in the test section, and in the free stream, ahead of the plate leading edge, were carried out using the 8MP and 16MP CCD cameras respectively. The fields of view for these measurements were $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}y\approx 100~\text{mm}\times 75~\text{mm}$ and $107~\text{mm}\times 71~\text{mm}$ respectively. These measurements were performed for determination of the virtual origin of the flat-plate boundary layer and for characterization of the grid turbulence respectively.

The TR-PIV measurements were carried out using a high-frequency and high-resolution CMOS camera (resolution: normal mode, 2336 $\times$ 1728 pixels; U-mode, 2560 $\times$ 1920 pixels; IDTpiv, USA, Model Y5.2) and a high-repetition-rate Nd:YLF DPSS laser (Dual head, 30  $\text{mJ}~\text{pulse}^{-1}$ , 1 kHz, Photonics Industries, USA). The camera could capture 730 images per second (i.e. 365 image pairs per second) at full resolution. However, the frame rate could be increased by reducing the active pixels of the sensor. Utilizing the 2560 pixel $\times$ 1080 pixel array of the CMOS camera, all of the TR-PIV data reported here were acquired at 645 Hz (i.e. 1290 images per second). All of the TR-PIV measurements in the $x{-}y$ and $x{-}z$ planes were carried out by centring the camera at $x=700$  mm. The fields of view in the $x{-}y$ and $x{-}z$ planes were $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}y\approx 104~\text{mm}\times 44$  mm and $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}z\approx 140~\text{mm}\times 57$  mm respectively, except in § 5.1.2. The field of view of the TR-PIV measurements presented in § 5.1.2 was reduced to $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}y\approx 66~\text{mm}\times 28$  mm for better spatial resolution in the wall-normal plane.

Figure 1.

A schematic of the simultaneous orthogonal dual-plane PIV measurement set-up in the wind tunnel test section.

The simultaneous orthogonal dual-plane PIV measurements were carried out following the works of Ganapathisubramani et al. (Reference Ganapathisubramani, Longmire, Marusic and Pothos2005) and Hambleton et al. (Reference Hambleton, Hutchins and Marusic2006), who used this technique for measurements in a turbulent boundary layer. This technique is based on the polarization characteristics of scatted light from seeding particles (see Kähler & Kompenhans Reference Kähler and Kompenhans2000, for details). Using this measurement technique, the conventional PIV measurements were carried out in the $x{-}y$ and $x{-}z$ planes simultaneously. Figure 1 shows the present measurement set-up schematically. The $x{-}y$ plane at $z=0$ was illuminated using a Quantel Evergreen Nd:YAG double pulsed laser with an energy of 200 mJ per pulse. The laser beam had been passed through a half-wave plate before it entered into the sheet forming optics. The half-wave plate rotated the polarization of the beam by $90^{\circ }$ . Thus, the initial laser beam, which was vertically polarized, was changed to a horizontally polarized laser beam. On the other hand, the $x{-}z$ plane at $y=2.5$  mm from the plate surface was illuminated using the Innolas Nd:YAG laser, details of which are given above. In this case, the initial vertical polarization of the laser beam was preserved. The illuminated particles in the $x{-}y$ plane were imaged by using a 12 bit 1MP PCO pixelfly CCD camera. Two polarizing filters were mounted in front of this camera lens which could be rotated independently to allow only the horizontally polarized light of the wall-normal plane laser and block the vertically polarized light from the wall-parallel plane laser. The flow field in the $x{-}z$ plane was captured using the CMOS camera, described above, and the two polarizing filters mounted in front of this camera lens were adjusted to allow only the vertically polarized laser light. With these arrangements, major reflections from the plate surface and from the line of intersection of the two perpendicular laser sheets were minimized. Following Hambleton et al. (Reference Hambleton, Hutchins and Marusic2006), a series of image processing steps were also performed in MATLAB to retain the information at the intersection. Centring the CCD camera at $x=700$  mm, the simultaneous PIV measurements were carried out at 5 Hz. The fields of view in the $x{-}y$ and $x{-}z$ planes were $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}y\approx 46~\text{mm}\times 34$  mm and $\unicode[STIX]{x0394}x\times \unicode[STIX]{x0394}z\approx 57~\text{mm}\times 43$  mm respectively. The field of view in the $x{-}y$ plane was chosen to be smaller for better spatial resolution.

All of the cameras used here were equipped with 100 mm focal length lenses (Carl Zeiss Macro-Planar lenses). The laser and the camera were synchronized using pulse generators (BNC Nucleonics, Model 575, and IDTpiv, Motion Pro Timing hub). The flow was seeded with glycol based fog particles with a mean diameter of approximately 1  $\unicode[STIX]{x03BC}\text{m}$ . These particles were generated using a SAFEX fog generator (Dantec Dynamics, Denmark). The acquired images were processed using the package ProVision XS, IDTpiv, with a correlation window of 32 pixel $\times$ 32 pixel. This package is based on the second-order-accurate mesh-free algorithm developed by Lourenco & Krothapalli (Reference Lourenco and Krothapalli2000). This package was also used in our previous work (e.g. Mandal & Dey Reference Mandal and Dey2011; Phani Kumar, Mandal & Dey Reference Phani Kumar, Mandal and Dey2015), and more details on the data processing can be found in Mandal et al. (Reference Mandal, Venkatakrishnan and Dey2010). However, based on the correlation window, the spatial resolutions of the conventional PIV measurements in bypass boundary layer transition were found to be 1.3 mm and 1.5 mm in the $x{-}y$ and $x{-}z$ planes respectively. Similarly, the spatial resolutions of the conventional PIV measurements in the Blasius boundary layer (without using any grid in the test section) and in the free stream (in front of the leading edge) were found to be 0.97 mm and 0.69 mm respectively. The spatial resolutions of the TR-PIV measurements for the two different fields of view in the $x{-}y$ plane, as mentioned above, were found to be 0.8 mm and 1.3 mm, whereas the spatial resolution in the $x{-}z$ plane was found to be 1.75 mm. The spatial resolutions for the simultaneous orthogonal PIV measurements in the $x{-}y$ and $x{-}z$ planes were found to be 1 mm and 0.7 mm respectively. While these values compare well with published work on boundary layers (e.g. Liu, Adrian & Hanratty Reference Liu, Adrian and Hanratty2001; Dennis & Nickels Reference Dennis and Nickels2011), a high-resolution feature is also in-built in this package; details can be found in the literature (Lourenco & Krothapalli Reference Lourenco and Krothapalli2000; Alkislar, Krothapalli, & Lourenco Reference Alkislar, Krothapalli and Lourenco2003). Therefore, the actual spatial resolution of our measurements is expected to be even better than the values mentioned above.

Although the PIV technique is now considered to be a reliable tool for whole-field velocity measurements, one has to consider the uncertainty involved in the measured values. Sources often considered for error/uncertainty analysis are equipment, particle lag, sampling size and the processing algorithm (e.g. Lazar et al. Reference Lazar, DeBlauw, Glumac, Dutton and Elliott2010). Considering equipment error, sampling size (random error) and the processing algorithm (bias error) in the present PIV measurements, the overall/combined standard uncertainty (Coleman & Steele Reference Coleman and Steele2009) was found to be within $\pm 1.3\,\%$ of the free-stream velocity. Uncertainty due to particle lag was neglected as it is often reported to be negligible for glycol based fog particles (e.g. Raghav & Komerath Reference Raghav and Komerath2014). The error due to the processing algorithm was estimated by generating synthetic particle image pairs with known pixel displacements (Raffel et al. Reference Raffel, Willert, Wereley and Kompenhansürgen2013).

A constant-temperature hotwire anemometry system (Dantec Dynamics, Denmark) was also utilized for the present experimental work. A single-wire probe (55P11, Dantec Dynamics), which was connected to a multichannel CTA system (54N81), was used for data acquisition. The sensing element of this probe was a 5  $\unicode[STIX]{x03BC}\text{m}$ diameter tungsten wire, and its length to diameter ratio was 250. The data were acquired using a 16 bit National Instruments data acquisition card (NI-6034E) and LabVIEW software. Setting the overheat ratio at 1.8, data were acquired at a sampling rate of 6 kHz and low-pass filtered at 3 kHz. Without installing any grid in the tunnel test section, the hotwire probe was often calibrated using a Pitot-static tube connected to a digital manometer (Furness Controls, FC012), and the King’s law fit. The hotwire data were used to characterize the free-stream turbulence and the flow intermittency in the boundary layer. A hotwire probe in the plate centreline was often placed at the end of the PIV measurement region at $y=2.5$  mm for simultaneous observation and acquisition of the flow signal during PIV measurements. The uncertainty analysis of the hotwire data was carried out following Yavuzkurt (Reference Yavuzkurt1984), and the overall uncertainty was found to be within $\pm 1.7\,\%$ of the free-stream velocity.

2.3 Free-stream turbulence characteristics in the test section

Turbulence generating grids have served as important experimental devices in the study of bypass boundary layer transition, and square mesh grids are often used to investigate the effect of free-stream turbulence on boundary layer transition (e.g. Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Fransson et al. Reference Fransson, Matsubara and Alfredsson2005; Hernon et al. Reference Hernon, Walsh and Mceligot2007; Nolan et al. Reference Nolan, Walsh and McEligot2010). However, few works have been carried out using parallel rod/bar grids, mainly focusing on streak generation and their initial development in bypass transition (e.g. Kendall Reference Kendall1985; Roach & Brierly Reference Roach and Brierly1992; Pook, Watmuff & Orifici Reference Pook, Watmuff and Orifici2016). Here, we intend to focus on the secondary instability characteristics of the boundary layer streaks generated by a parallel rod grid and a square mesh grid, identified as grid A and grid B respectively, to investigate the effect of grid geometry on the streak secondary instability, if any. The parallel rod grid (grid A) was installed in the tunnel such that the bars were perpendicular to the free stream and parallel to the plate leading edge, in contrast to the work of Pook et al. (Reference Pook, Watmuff and Orifici2016) where a screen model made of parallel bars installed in the settling chamber of a wind tunnel was used such that the bars were perpendicular to the free-stream flow and to the plate leading edge.

Figure 2.

Downstream turbulence decay for grids A and B; symbols: ○, grid A; ▫, grid B. Solid line: curve fitted using (2.1).

Grid A was made of uniformly spaced parallel cylindrical rods with a centre-to-centre distance of 34 mm. Grid B was made of cylindrical rods arranged perpendicularly in two different planes so that they formed a square mesh with a mesh width of 44 mm. Both grid A and grid B were made of cylindrical rods of 10 mm diameter. Grid solidity ( $S_{g}$ ) is defined as $S_{g}=1-\unicode[STIX]{x1D6FD}$ , where $\unicode[STIX]{x1D6FD}$ is the porosity (e.g. Roach Reference Roach1987; Kurian & Fransson Reference Kurian and Fransson2009). In terms of the mesh width, $M$ , and the bar/rod diameter, $d$ , the porosity can be expressed as $\unicode[STIX]{x1D6FD}=(1-(d/M))$ for a parallel rod grid and $\unicode[STIX]{x1D6FD}=(1-(d/M))^{2}$ for a square mesh grid (e.g. Roach Reference Roach1987). Using these expressions, the solidities for grids A and B were found to be 0.294 and 0.403 respectively. However, the grids were placed near the entrance of the test section. The distances of grids A and B from the leading edge ( $x_{grid}$ ) were $-1165$  mm and $-1185$  mm respectively. Using the hotwire anemometry technique, the streamwise turbulent intensity, $Tu$ , was measured at various downstream locations to characterize the free-stream turbulence. The measurements were carried out at $U_{0}=5.8~\text{m}~\text{s}^{-1}$ and $6.4~\text{m}~\text{s}^{-1}$ for grids A and B respectively. For both of the grids, the measured data, as shown in figure 2, follow the power-law decay (e.g. Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010) given by

(2.1) $$\begin{eqnarray}Tu=\frac{u_{rms}}{U_{0}}=C(x-x_{vo})^{-b},\end{eqnarray}$$

where $x_{vo}$ is the virtual origin of a grid, $b$ is the exponential decay rate and $C$ is a constant for a particular grid. The geometric parameters and the numerical values of these constants for these grids are given in table 1. The value of $b$ , which is a measure for isotropic turbulence, was found to be 0.6 for both of the grids, and this value compares well with the values reported in the literature (e.g. Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Fransson et al. Reference Fransson, Matsubara and Alfredsson2005; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010). The streamwise turbulent intensities at the plate leading edge were found to be 2.5 % and 2.6 % of the free-stream velocity for grids A and B respectively. The integral length scale, $\unicode[STIX]{x1D6EC}_{x}$ , and the Taylor microscale, $\unicode[STIX]{x1D70F}_{x}$ , were estimated from the auto-correlation function of the streamwise fluctuating velocity, $R_{uu}$ , at the leading edge (e.g. Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010). The numerical value of the integral length scale was found to be 17 mm, whereas the numerical value of the Taylor microscale was found to be 7( $\pm$ 1) mm, for both of the grids.

Table 1.

Details of grids A and B, and the power-law constants.

For further characterization of the free-stream turbulence, conventional PIV measurements in the $x{-}y$ and $x{-}z$ planes were separately carried out in the free-stream flow. Due to experimental constraints in optical access at the leading edge position, the measurements were performed ahead of the plate leading edge, as schematically shown in figure 3(a). For measurements in the $x{-}y$ and $x{-}z$ planes, the camera was centred at approximately $x=-185$  mm, $y=100$  mm and $z=0$  mm, such that the measurement zone was at least 20 mesh widths downstream of the grid location. The measurement field of view ( $107~\text{mm}\times 71~\text{mm}$ ) and the number of PIV realizations were chosen such that the present spatial resolution, 0.69 mm, and the number of PIV realizations, 1500, were comparable with published PIV measurements on grid generated turbulence (e.g. Gomes-Fernandes, Ganapathisubramani & Vassilicos Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012). Here and in the following, the streamwise, wall-normal and spanwise fluctuating velocities are denoted by $u(=U_{I}-U)$ , $v(=V_{I}-V)$ and $w(=W_{I}-W)$ respectively, the local instantaneous velocities in the $x$ -, $y$ - and $z$ -directions are denoted by $U_{I}$ , $V_{I}$ and $W_{I}$ respectively, and the corresponding local mean velocities in these directions are denoted by $U$ , $V$ and $W$ respectively.

Figure 3.

Conventional PIV measurements in the free stream for characterization of grid generated turbulence. (a) A schematic illustrating the measurement planes. (b,c) Spatial correlations for grids A and B respectively; longitudinal, $f$ (——), and transverse, $g$ (- - -), correlation functions calculated from the isotropic relation (2.3) are also plotted for comparison with the measured data. Symbols in (b):

, $R_{uu}$ versus $\unicode[STIX]{x0394}x$ ; ○, $R_{vv}$ versus $\unicode[STIX]{x0394}x$ ; ●, $R_{ww}$ versus $\unicode[STIX]{x0394}x$ ; $\bigoplus$ , $R_{uu}$ versus $\unicode[STIX]{x0394}y$ ; $\bigotimes$ , $R_{uu}$ versus $\unicode[STIX]{x0394}z$ . Symbols in (c):

, $R_{uu}$ versus $\unicode[STIX]{x0394}x$ ; ▪, $R_{vv}$ versus $\unicode[STIX]{x0394}x$ ; ▵, $R_{ww}$ versus $\unicode[STIX]{x0394}x$ ; ▫, $R_{uu}$ versus $\unicode[STIX]{x0394}y$ ; ♢, $R_{uu}$ versus $\unicode[STIX]{x0394}z$ .

For investigation of the longitudinal and transverse length scales associated with an any turbulent velocity fluctuations, $q$ , the two-point spatial correlations

(2.2) $$\begin{eqnarray}R_{qq}(r)=\frac{\displaystyle \overline{q(r_{i},t)q(r_{i}+r,t)}}{\displaystyle \sqrt{\overline{q^{2}(r_{i},t)}}\sqrt{\overline{q^{2}(r_{i}+r,t)}}}\end{eqnarray}$$

were estimated from the measured data; here, $r_{i}$ denotes a reference point, with $r$ the separation between two points, and an overbar denotes an ensemble average over the number of PIV realizations. The spatial correlation is called the longitudinal correlation when $r$ is parallel to the direction of $q$ and the transverse correlation when $r$ is perpendicular to the direction of $q$ (O’Neill et al. Reference O’Neill, Nicolaides, Honnery and Soria2004). Measured two-point spatial correlations at $x=-200$  mm are shown in figures 3(b) and 3(c) for grids A and B respectively. For comparison, the longitudinal, $f(r)$ , and transverse, $g(r)$ , correlation functions, which are related as

(2.3) $$\begin{eqnarray}f+\frac{1}{2}r\frac{\text{d}f}{\text{d}r}=g\end{eqnarray}$$

for homogeneous isotropic turbulence (e.g. Batchelor Reference Batchelor1982; Kurian & Fransson Reference Kurian and Fransson2009), are also displayed in figure 3(b,c). The longitudinal correlation function in homogeneous isotropic turbulence is often approximated as $f(r)=\text{e}^{-r/\unicode[STIX]{x1D6EC}_{x}}$ , where $\unicode[STIX]{x1D6EC}_{x}$ is the longitudinal integral length scale (e.g. Roach Reference Roach1987; Kurian & Fransson Reference Kurian and Fransson2009). We may note that the longitudinal integral length scale, $\unicode[STIX]{x1D6EC}_{x}$ , is estimated from the hotwire data at $x=-200$  mm and $g(r)$ is obtained from (2.3). The present data in figure 3(b,c) are found to be similar to those reported by Kurian & Fransson (Reference Kurian and Fransson2009). Therefore, the present free-stream turbulence can be considered as nearly isotropic for both of the grids. However, the data for grid A appear to compare better with the calculated $f(r)$ and $g(r)$ functions than the data for grid B. This may be because of the fact that the measurement location is at larger distance in terms of mesh width for grid A than that for grid B, as the mesh width for grid A is less compared with grid B.

The longitudinal ( $\unicode[STIX]{x1D6EC}_{x}$ ) and transverse ( $\unicode[STIX]{x1D6EC}_{y}$ , $\unicode[STIX]{x1D6EC}_{z}$ ) integral length scales and the corresponding Taylor microscales ( $\unicode[STIX]{x1D70F}_{x}$ , $\unicode[STIX]{x1D70F}_{y}$ , $\unicode[STIX]{x1D70F}_{z}$ ) were estimated from the auto-correlation functions of the streamwise velocity fluctuations in the respective directions (e.g. Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001). The transverse integral length scales ( $\unicode[STIX]{x1D6EC}_{y}$ , $\unicode[STIX]{x1D6EC}_{z}$ ) and the corresponding Taylor microscales ( $\unicode[STIX]{x1D70F}_{y}$ , $\unicode[STIX]{x1D70F}_{z}$ ) can also be estimated from the auto-correlation functions of the cross-stream velocity fluctuations in the $x$ -direction. The correlation curves were numerically integrated up to the zero-crossing point (e.g. O’Neill et al. Reference O’Neill, Nicolaides, Honnery and Soria2004) to estimate all of the length scales reported here. The estimated length scales at $x=-200$  mm are provided in table 2; it should be noted that the data given inside parentheses refer to the corresponding transverse scales estimated from the auto-correlation functions of the cross-stream velocity fluctuations in the $x$ -direction. We may note that the value of $\unicode[STIX]{x1D6EC}_{x}$ in table 2 is estimated from the hotwire data at $x=-200$  mm, as $R_{uu}(\unicode[STIX]{x0394}x)$ does not reach zero because of the smaller PIV field of view. The uncertainty in the estimated Taylor microscales is found to be within $\pm 1$  mm.

Table 2.

Longitudinal and transverse integral and Taylor microscales estimated at $x=-200$  mm; the data inside parentheses refer to the corresponding transverse scales estimated from the auto-correlation functions of the cross-stream velocity fluctuations in the $x$ -direction.

2.4 Pressure gradient characteristics in the test section

The streamwise variation of the free-stream pressure coefficient, $C_{p}=1-(U/U_{r})^{2}$ , as shown in figure 4, was determined by measuring the local mean velocity in the free stream (e.g. Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010). Here, $U_{r}$ denotes the reference velocity. The velocity measurements in the free stream were carried out using a hotwire probe, which was first calibrated using a Pitot-static tube without any grid in the test section. Then, the measurements were carried out without installing any grid in the test section for the no-grid case and with the respective grid installed for the grid cases. The probe was kept at $y=25$  mm from the wall, and it was then traversed along the plate centreline for velocity measurements. The reference point was chosen at $x=340$  mm for the no-grid case and at approximately $x=668$  mm for the grid cases. The initial large value of $C_{p}$ near the leading edge is attributed to the leading edge shape. However, with and without a grid in the test section, the variation of $C_{p}$ within the domain of our investigation was approximately $\pm 2\,\%$ . Similar variation of the pressure coefficient is also reported in various works on bypass transition (e.g. Arnal & Juillen Reference Arnal and Juillen1978; Roach & Brierly Reference Roach and Brierly1992).

Figure 4.

Variation of the pressure coefficient, $C_{p}$ , in the streamwise direction for grids A and B. Data without any grid in the test section are also shown. Symbols: ○, grid A; ▫, grid B; ▵, no grid.

5.1 Instability of the lifted-up shear layers in the $x{-}y$ plane

The simultaneous dual-plane PIV measurements presented in figure 12 indicate that the sign of an instability first gets manifested as shear layer oscillation in the $x{-}y$ plane. Therefore, it is interesting to track these shear layer oscillations at different instants of time to investigate their eventual outcome.

Figure 14.

Sequence of TR-PIV snapshots displaying the development of an instability over a lifted-up low-speed streak in the $x{-}y$ plane for grid B. The contours correspond to the normalized spanwise vorticity. The vectors represent the fluctuating velocity vectors. Each snapshot is separated by $\unicode[STIX]{x0394}t=0.00155$  s.

Figure 15.

Sequence of TR-PIV snapshots displaying flow randomization through the propagation of clockwise–anticlockwise vortex pairs over a high-speed streak in the $x{-}y$ plane for grid B. The contours correspond to the normalized spanwise vorticity. The vectors represent the fluctuating velocity vectors. Each snapshot is separated by $\unicode[STIX]{x0394}t=0.00155$  s.

A time sequence of flow features associated with a lifted-up shear layer and its eventual outcome is displayed in figure 14. The arrows in this figure are the fluctuating velocity vectors over the contours of the normalized spanwise instantaneous vorticity. In this figure, one may clearly see the onset of an oscillation in an almost straight shear layer followed by its amplification in space and time. One may also notice that this oscillation is developing near the leading edge of the inclined shear layer, which eventually ends up with rolled-up vortices with clockwise rotation. Several numerical studies on bypass transition (e.g. Brandt et al. Reference Brandt, Schlatter and Henningson2004; Schlatter et al. Reference Schlatter, Brandt, de Lange and Henningson2008; Hack & Zaki Reference Hack and Zaki2014) have revealed the existence of three-dimensional vortical structures over the oscillating low-speed streaks. The present rolled-up vortices may represent the wall-normal view of these three-dimensional vortical structures. However, one may also observe in the last few snapshots of figure 14 that while the clockwise vortices come close to the wall and interact with a high-speed streak, some pairs of clockwise and anticlockwise vortices are formed. These counter-rotating vortex pairs, while propagating over positive $u$ fluctuations, i.e. over a high-speed streak, are found to grow in size and destabilize the underlying high-speed streak and generate smaller scales in the flow, as illustrated in figure 15. The sequence of events from the time-resolved measurements, as shown in figures 14 and 15, therefore, clearly confirms the previous speculation that an incipient spot can be formed from an inclined shear layer oscillation and its associated vortex shedding (Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010).

In contrast to the oscillation in figure 14, which develops at the leading edge of an inclined shear layer, figure 16 displays an oscillation that develops near the trailing edge, i.e. almost at the start of its inclination with the wall. Moreover, these oscillations are found to occur a little closer to the wall compared with those in figure 14, and their amplification eventually generates some smaller inclined shear layers, as seen at the $12\unicode[STIX]{x0394}t$ instant in figure 16. However, these oscillations are also found to take part in generating smaller scales in the flow.

Figure 16.

Sequence of TR-PIV snapshots displaying the development of an instability nearly at the trailing edge of an inclined shear layer in the $x{-}y$ plane for grid B. The contours correspond to the normalized spanwise vorticity. The vectors represent the fluctuating velocity vectors. Each snapshot is separated by $\unicode[STIX]{x0394}t=0.00155$  s.

It would have been interesting if simultaneous spanwise plane views corresponding to the flow features presented in figures 14 and 16 were available to investigate how these time sequences are related to the streak oscillations in the spanwise plane. However, in the absence of such measurements, we refer to our simultaneous measurements for further comprehensive assessment of the data presented in figures 14 and 16. The shear layer oscillations in figure 14 (for example, realizations at $7\unicode[STIX]{x0394}t$ and $8\unicode[STIX]{x0394}t$ instants) are found to be similar to those shown in figure 10(a,b), which are related to the sinuous streak instability in the $x{-}z$ plane. Therefore, if one could have measured the spanwise data corresponding to the time sequence shown in figure 14, one would have seen sinuous instability in the $x{-}z$ plane. Further, the shear layer oscillations in figure 16 (for example, realizations at the $11\unicode[STIX]{x0394}t$ and $12\unicode[STIX]{x0394}t$ instants) are similar to those in figures 10(c,d) and 12(c). Therefore, these oscillations are most likely related to the varicose streak instability in the $x{-}z$ plane.

5.1.1 Linear stability analysis

The time sequences in figures 14 and 16 indicate that an instability develops on a low-speed streak, as the fluctuating velocity vectors associated with the inclined shear layers show large negative $u$ fluctuations. The instantaneous velocity profiles, as seen from the instantaneous velocity vectors associated with these shear layers, are found to be highly inflectional, as shown in figure 17(a,c) at the $\unicode[STIX]{x0394}t$ time instant. This implies that these shear layers may be prone to inflectional instability, and a local linear instability analysis may be useful in predicting the frequency and wavelength of their oscillations. In such an effort, Hack & Zaki (Reference Hack and Zaki2014) utilized the instantaneous velocity data in the $y{-}z$ plane for two-dimensional linear stability analysis (2D-LSA), i.e. $U_{I}(y,z)$ as the base profile that includes the effect of spanwise shear. They found that the theoretical growth rate closely follows the estimated spatio-temporal growth rate from their DNS data. However, their linear stability and statistical analyses indicate that the growth rate of a base streak is well correlated with the wall-normal shear and the base streak amplitude, whereas it is poorly correlated with the spanwise shear. This indicates that the stability analysis of the wall-normal velocity profile itself, i.e. $U(y)$ as the base profile that does not include the effect of spanwise shear, may be useful in providing elementary stability characteristics of these shear layer oscillations.

Figure 17.

(a,c) Illustrations of the inflectional nature of the instantaneous velocity vectors corresponding to the shear layers in figures 14 and 16 at the $\unicode[STIX]{x0394}t$ instant of time respectively. The contours represent the normalized spanwise vorticity. (b,d) Spatially averaged velocity data (symbols) over the white rectangular regions in (a,c) respectively, along with the fitted velocity profiles (solid line) using (5.1); grid B data.

For example, in their experimental study on boundary layer transition, Asai et al. (Reference Asai, Minagawa and Nishioka2002) carried out one-dimensional linear stability analysis (1D-LSA) of the mean velocity profile of a low-speed streak by fitting a tangent-hyperbolic curve. They found that the calculated most amplified frequency and the corresponding wavenumber compared well with their measured counterparts. In this regard, the numerical reproduction of the experimental work of Asai et al. (Reference Asai, Minagawa and Nishioka2002) reported by Brandt (Reference Brandt2007) is worth mentioning. Using the perturbation kinetic energy analysis, he found that both sinuous and varicose instabilities are governed by the work of the Reynolds stress against the wall-normal shear, and this is contrary to the common belief that sinuous and varicose instabilities are driven by the streak spanwise and wall-normal shears respectively. Similarly, Philip, Karp & Cohen (Reference Philip, Karp and Cohen2016) carried out linear stability analysis of a streaky base flow measured in a laminar plane Poiseuille flow. Neglecting the spanwise shear, they carried out 1D-LSA and found that their stability results (i.e. wavelength, eigenfunction shape) compared well with their experimental data. Further, their 2D-LSA and energy balance analysis indicated that the wall-normal shear is more important than the spanwise shear for producing symmetric varicose mode. Moreover, the 1D-LSA in bypass transition may be justified based on the fact that not all of the streaks in the spanwise direction simultaneously develop instability; rather, the instability in a natural bypass transition is localized in nature (e.g. Hack & Zaki Reference Hack and Zaki2014), i.e. it is often seen to develop on a single low-speed streak, in contrast to the collective instability on the entire row of streaks as considered by Andersson et al. (Reference Andersson, Brandt, Bottaro and Henningson2001) and Vaughan & Zaki (Reference Vaughan and Zaki2011).

The boundary layer streaks in bypass transition are highly unsteady in nature compared with the controlled ones considered by Asai et al. (Reference Asai, Minagawa and Nishioka2002) and Philip et al. (Reference Philip, Karp and Cohen2016). Hence, the usual ensemble average or time average data cannot provide a mean velocity profile corresponding to a particular low-speed streak that eventually develops instability. Therefore, a base velocity profile is obtained here by spatial averaging of the instantaneous velocity profiles over a distance of $10\unicode[STIX]{x1D6FF}^{\ast }{-}15\unicode[STIX]{x1D6FF}^{\ast }$ for a single low-speed streak on which instabilities like those in figures 14 and 16 develop. Then, the spatially averaged velocity profile is fitted with a tangent-hyperbolic profile, following Asai et al. (Reference Asai, Minagawa and Nishioka2002), for the local linear stability analysis. The tangent-hyperbolic profile

(5.1) $$\begin{eqnarray}\frac{U\left(y\right)}{U_{L}}=a_{1}\tanh \left(b_{1}\frac{y}{\unicode[STIX]{x1D6FF}_{L}^{\ast }}-c_{1}\right)+a_{2}\tanh \left(b_{2}\frac{y}{\unicode[STIX]{x1D6FF}_{L}^{\ast }}-c_{2}\right)+a_{3}\tanh \left(b_{3}\frac{y}{\unicode[STIX]{x1D6FF}_{L}^{\ast }}-c_{3}\right)+d\end{eqnarray}$$

was found to describe the spatially averaged mean velocity data quite well; here, $a_{1}$ , $a_{2}$ , $a_{3}$ , $b_{1}$ , $b_{2}$ , $b_{3}$ , $c_{1}$ , $c_{2}$ , $c_{3}$ and $d$ are constants; the local free-stream velocity, $U_{L}$ , and the displacement thickness, $\unicode[STIX]{x1D6FF}_{L}^{\ast }$ , are estimated based on the spatially averaged velocity profile.

The above procedure is illustrated in figure 17. All of the instantaneous velocity profiles within the white rectangular regions in figure 17(a,c) are spatially averaged, and (5.1) has been used to fit the respective averaged velocity profiles. The fitted curves are found to closely follow the measured data, as shown in figure 17(b,d). Although there are several constants in (5.1), these constants can easily be determined by a trial and error approach in MATLAB. The numerical values of $\{a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3},d\}$ for the velocity profiles in figure 17(b,d) are found to be $\{0.518,1.698,0.8207,0.1085,-2.4,-4.08,0.175,2.45,6.02,0.4155\}$ and $\{0.1995,3.9,0.1721,0.1155,2.88,2.05,0.2695,2.8,4.25,0.4155\}$ respectively.

For stability analysis on the fitted velocity profiles, the Orr–Sommerfeld equation, following Boutilier & Yarusevych (Reference Boutilier and Yarusevych2012), is written as

(5.2) $$\begin{eqnarray}\left(\unicode[STIX]{x1D6FC}U-\unicode[STIX]{x1D714}\right)\left[\unicode[STIX]{x1D719}^{\prime \prime }-\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D719}\right]-\unicode[STIX]{x1D6FC}U^{\prime \prime }\unicode[STIX]{x1D719}=-\frac{\text{i}U_{L}\unicode[STIX]{x1D6FF}_{L}^{\ast }}{Re_{\unicode[STIX]{x1D6FF}_{L}^{\ast }}}\left[\unicode[STIX]{x1D719}^{\prime \prime \prime \prime }-2\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D719}^{\prime \prime }+\unicode[STIX]{x1D6FC}^{4}\unicode[STIX]{x1D719}\right].\end{eqnarray}$$

Here, $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D719}^{\prime }$ denote the complex amplitude function of the wall-normal perturbation velocity, i.e. $v=\unicode[STIX]{x1D719}(y)\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}x-\unicode[STIX]{x1D714}t)}$ , and its derivative with respect to $y$ respectively. The wavenumber, angular frequency and Reynolds number are denoted by $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D714}$ and $Re_{\unicode[STIX]{x1D6FF}_{L}^{\ast }}(=U_{L}\unicode[STIX]{x1D6FF}_{L}^{\ast }/\unicode[STIX]{x1D708})$ respectively. The above equation (5.2) is solved with the boundary conditions $\unicode[STIX]{x1D719}(0)=\unicode[STIX]{x1D719}(\infty )=0$ and $\unicode[STIX]{x1D719}^{\prime }(0)=\unicode[STIX]{x1D719}^{\prime }(\infty )=0$ . Similarly, for inviscid stability analysis, the Rayleigh equation

(5.3) $$\begin{eqnarray}\left(\unicode[STIX]{x1D6FC}U-\unicode[STIX]{x1D714}\right)\left[\unicode[STIX]{x1D719}^{\prime \prime }-\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D719}\right]-\unicode[STIX]{x1D6FC}U^{\prime \prime }\unicode[STIX]{x1D719}=0\end{eqnarray}$$

is solved with the boundary conditions $\unicode[STIX]{x1D719}(0)=\unicode[STIX]{x1D719}(\infty )=0$ . Both the Orr–Sommerfeld and the Rayleigh equation are posed as spatial eigenvalue problems considering the wavenumber, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{r}+i\unicode[STIX]{x1D6FC}_{i}$ , as complex and the angular frequency, $\unicode[STIX]{x1D714}$ , as real, and these equations are then solved using the spectral method (Schmid & Henningson Reference Schmid and Henningson2001). The details of this method and the solution procedure are given in Dabaria (Reference Dabaria2015).

Figure 18.

(a–d) Spatial growth rate, wavenumber, phase velocity and magnitude of $u$ eigenfunction $|\hat{u} |$ respectively for the velocity profile in figure 17(b). (e–h) Spatial growth rate, wavenumber, phase velocity and magnitude of $u$ eigenfunction $|\hat{u} |$ respectively for the velocity profile in figure 17(d). Solid line, Orr–Sommerfeld calculation; dashed line, Rayleigh calculation.

The growth rate, wavenumber and phase velocity obtained by solving the Orr–Sommerfeld and Rayleigh equations at various circular frequencies are displayed in figures 18(a–c) and 18(e–g) for the velocity profiles in figures 17(b) and 17(d) respectively. The eigenfunctions corresponding to the streamwise velocity perturbation are also shown in figure 18(d) and 18(h), for the velocity profiles in figure 17(b) and 17(d), respectively. One may notice that the stability results for a particular velocity profile calculated based on the Orr–Sommerfeld and Rayleigh equations are comparable except for the growth rates, which are found to be higher for the inviscid stability calculations (see figure 18 a,e). However, the most amplified frequencies calculated based on the Orr–Sommerfeld and Rayleigh equations are found to be nearly equal for each of the velocity profiles considered.

Since the oscillations in the shear layers are prominent, as seen in figures 14 and 16, one can easily estimate an average wavelength, $\unicode[STIX]{x1D706}$ , from the experimental data. Similarly, utilizing the time sequences at different instants of time, an average convection/propagation velocity, $U_{c}$ , of an instability can also be estimated, and the frequency, $f$ , can be estimated from the ratio of the convection velocity and the wavelength, i.e. $U_{c}/\unicode[STIX]{x1D706}$ . Using the contour levels of the normalized instantaneous vorticity for identification of a shear layer oscillation, we estimated the average wavelength and convection velocity of an oscillating shear layer from the experimental data. At different instants of time, we followed the portion of the oscillating shear layer that was initially confined within the rectangular box in figure 17(a,c). Then, considering the wavelength as the peak to peak/trough to trough distance of the oscillating shear layer at different instants of time, an average wavelength and its standard deviation were obtained. Similarly, by tracking a peak/trough of the oscillating shear layer at different instants of time and plotting its relative position with time, the convection velocity could be determined (see Mans et al. Reference Mans, Kadijk, de Lange and van Steenhoven2005, for details). Considering such an estimate for different peaks/troughs, an average convection velocity and its standard deviation were then obtained.

Based on the Orr–Sommerfeld equation, the calculated normalized wavelengths ( $\unicode[STIX]{x1D706}/\unicode[STIX]{x1D6FF}^{\ast }$ ), corresponding to the most amplified frequency, were found to be 5.2 and 8.1 for the cases in figures 17(b) and 17(d) respectively. Following the method discussed above, the estimated wavelengths from the experimental data, for the corresponding cases, were found to be 5 and 8.4 respectively. Similarly, the calculated phase speeds ( $c/U_{0}$ ) at the most amplified frequencies corresponding to the cases in figures 17(b) and 17(d) were found to be 0.83 and 0.63 respectively, and the estimated convection velocities, $U_{c}/U_{0}$ , corresponding to these cases were found to be 0.78 and 0.68 respectively. The calculated wavelengths and phase velocities compare well with the estimated values of the wavelengths and convection velocities, for both of the velocity profiles in figure 17(b,d). The maximum differences between the calculated and estimated values of the wavelength and the phase velocity are found to be 4 % and 8 % respectively. The most amplified frequencies calculated based on the Orr–Sommerfeld equation are found to be 0.16 and 0.078 for the cases in figure 17(b,d), and the estimated frequencies based on the wavelengths and the convection velocities, for the corresponding cases, are found to be 0.16 and 0.08 respectively.

Several cases like those in figures 14 and 16 were identified and linear stability analyses were carried out to investigate whether the above observations are consistent with all such cases. We identified a total of 10 such cases, as we chose only those shear layers that showed development of an instability from its inception within the field of view, and neglected those shear layers that had already developed significant oscillations before reaching the measurement field of view. Comparison of the calculated wavelengths, phase velocities and frequencies with the estimated ones is shown in figure 19, and we find similar results for all of the cases. We may note that the abscissa in figure 19 indicates the case index, $N_{case}$ , and figure 19(a–c) also includes the data corresponding to the velocity profiles in figure 17(b,d) ( $N_{case}=8$ and 7 respectively). Hence, the results in figure 19 indicate that an oscillating shear layer is most likely to be composed of a monochromatic wave that is excited by the linear instability of the velocity profile associated with the inclined shear layer.

Figure 19.

Corresponding to the maximum growth rate, comparison of the calculated wavelength (a), phase velocity (b) and frequency (c) with the estimated values of wavelength, convection velocity and frequency for different cases; $N_{case}$ refers to the case index. Symbols: circle, grid A; square, grid B; open symbols, calculated data; filled symbols with error bars, estimated data.

The numerical value of the calculated spatial growth rate ( $-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FF}^{\ast }$ ) for the most unstable frequency is higher for the velocity profile in figure 17(d) compared with the velocity profile in figure 17(b). On the other hand, the instability in figure 14, corresponding to the velocity profile in figure 17(b), seems to grow temporally faster than the instability in figure 16, corresponding to the velocity profile in figure 17(d). However, this is not inconsistent, as the phase/group velocity of the instability in figure 14 is higher than the phase/group velocity of the instability in figure 16 (see figure 18 c,g). The shapes of the eigenfunctions for both of the velocity profiles are also found to bear some similarity to the $u$ fluctuations at the later stage of the shear layer oscillations (see the fluctuating velocity vectors in figures 14 and 16 at some later stage in time).

5.1.2 Penetration of the free-stream turbulence into the boundary layer

The data presented in the previous sections clearly indicate the existence of an instability near the edge of the boundary layer. It would have been revealing if it were possible to establish a direct role of the free-stream turbulence in intensification of the instability process in the upper part of the boundary layer. However, it appears to be a challenging task at present as far as the measurement techniques and analysis tools are concerned. Nonetheless, using statistical analysis, it is interesting to check whether there exists any relation of the free-stream velocity fluctuations to the velocity fluctuations in the boundary layer edge. Nolan & Walsh (Reference Nolan and Walsh2012) reported that the correlation analysis of the wall-normal perturbation velocity across the boundary layer edge is a more direct representation of the penetration depth of the free-stream disturbance inside the boundary layer. Therefore, following Nolan & Walsh (Reference Nolan and Walsh2012), a spatial correlation analysis of the wall-normal perturbation velocity was carried out to investigate the possibility of an interaction.

Figure 20.

Contours of the spatial correlation coefficient ( $R_{vv}$ ) of the wall-normal velocity fluctuations at $z=0$ ; non-intermittent flow (a) and intermittent flow (b) for grid A; non-intermittent flow (c) and intermittent flow (d) for grid B. Contour levels ${<}0.3$ are omitted. The dashed line indicates the boundary layer edge.

Figure 20 shows the contours of the spatial correlation coefficient of the wall-normal disturbance velocity, defined here as $R_{vv}(\unicode[STIX]{x0394}x,\unicode[STIX]{x0394}y)=\overline{v(x_{0},y_{0},t)v(x_{0}+\unicode[STIX]{x0394}x,y_{0}+\unicode[STIX]{x0394}y,t)}/v_{rms}^{2}$ ; here $x_{0}$ and $y_{0}$ indicate the coordinates of the reference location. In this figure, the data are presented for both of the grids at two different free-stream velocities, which were chosen such that we could measure a non-intermittent flow/pre-transitional flow and an intermittent flow at the same location. The non-intermittent flows for both of the grids were measured at $U_{0}=4.1~\text{m}~\text{s}^{-1}$ and the intermittent flows for grids A and B were measured at the velocities given in table 4. Figures 20(a) and 20(c) show the contours of $R_{vv}$ corresponding to the measured non-intermittent flows for grids A and B respectively, whereas figures 20(b) and 20(d) display those corresponding to the intermittent flows for grids A and B respectively. For non-intermittent flows, it can be seen that the lower portion of the nearly circular correlation contours begins to elongate as the location of the reference point moves inside the boundary layer from the free stream, whereas the nearly circular shapes of the correlation contours are retained until the location of the reference point is moved very closed to the wall for intermittent flows. However, the extensions of the correlation contours across the boundary layer edge (as shown by a dashed line in figure 20) are almost the same for the intermittent and non-intermittent flows. The change of shape of the correlation contours near the boundary layer edge is due to the fact that the shear at the boundary layer edge is comparatively greater for a non-intermittent flow than the shear for an intermittent flow, as the mixing of high-momentum fluid is enhanced due to intermittent bursts of turbulent spots in an intermittent flow, and thereby reduces the velocity gradient near the boundary layer edge.

Figure 21.

(a,b) Contours of $R_{vv}$ for non-intermittent flows at $z=14$  mm and $z=-14$  mm for grids A and B respectively at $U_{0}=4.1~\text{m}~\text{s}^{-1}$ . (c,d) Contours of $R_{vv}$ for non-intermittent flows at $z=0$  mm and $z=-14$  mm for grids A and B respectively at $U_{0}=3~\text{m}~\text{s}^{-1}$ . Contour levels ${<}0.3$ are omitted. The dashed line indicates the boundary layer edge.

Measurements were also carried out at different spanwise locations and at different velocities to investigate the consistency of the above observation. Contours of $R_{vv}$ at different spanwise locations and at different velocities in non-transitional flows for both of the grids are displayed in figure 21. Figure 21(a,b) shows $R_{vv}$ contours at $U_{0}=4.1~\text{m}~\text{s}^{-1}$ and at $z=14$  mm and $z=-14$  mm for grids A and B respectively; it should be noted that the free-stream velocity was kept equal to the one at which the data presented in figure 20(a,c) were measured. Figure 21(c,d) shows $R_{vv}$ contours at $U_{0}=3~\text{m}~\text{s}^{-1}$ and at $z=0$ mm and $z=-14$ mm for grids A and B respectively. Comparing the contours in non-intermittent flows in figures 20 and 21, one may notice a similarity in the correlation contours at different spanwise locations and at different velocities. We may also note that the $R_{vv}$ contours in intermittent flows (not presented here) at different spanwise locations are found to be similar to those presented in figure 20(b,d).

However, the present correlation maps for intermittent flows compare well with the data of Nolan & Walsh (Reference Nolan and Walsh2012). The results for non-intermittent flows are different from those reported by them. They did not find any significant correlation of $v$ fluctuations between the free stream and the boundary layer for non-intermittent flows. The reason for the difference from the findings of Nolan & Walsh (Reference Nolan and Walsh2012) may be because of different transition mechanisms involved in these two works. They reported the interaction between a low-speed streak and a high-speed streak to be a dominant spot precursor mechanism. In their numerical simulation of a bypass transition on a flat plate with a blunt leading edge, Nagarajan, Lele & Ferziger (Reference Nagarajan, Lele and Ferziger2007) reported a wavepacket type flow breakdown, which was found to be similar to the varicose type flow breakdown associated with the near-wall inner-mode mechanism (Vaughan & Zaki Reference Vaughan and Zaki2011; Hack & Zaki Reference Hack and Zaki2014). In this respect, the results of Nolan & Walsh (Reference Nolan and Walsh2012) may correspond to the inner-mode mechanism of Vaughan & Zaki (Reference Vaughan and Zaki2011) and Hack & Zaki (Reference Hack and Zaki2014), and the results found in the present work are perhaps related to the outer-mode mechanism, because our data reveal a greater number of sinuous instabilities compared with varicose instabilities.

5.2 Streak instabilities in the $x{-}z$ plane

Figure 22.

Time sequence of a sinuous instability developing on a low-speed streak in the spanwise plane for grid B. The vectors are the fluctuating velocity vectors over the contours of $u/U_{0}$ . Maintaining the aspect ratio, the region displayed is zoomed in to clearly show the instability development. Each snapshot is separated by $\unicode[STIX]{x0394}t=0.00155$  s.

Time-resolved PIV measurements in the $x{-}z$ plane at $y=2.5$  mm were performed for a detailed quantitative assessment of the streak secondary instability characteristics in bypass transition. Using dye-flow visualizations in a water channel, Mans et al. (Reference Mans, Kadijk, de Lange and van Steenhoven2005) showed the time evolution of the natural unforced sinuous and varicose instabilities in bypass transition. To the best of our knowledge, similar time sequences of quantitative visualizations of these natural unforced instabilities are not available in the experimental literature in bypass transition under a high level of free-stream turbulence. Therefore, time sequences of TR-PIV realizations displaying the evolution of sinuous and varicose instabilities are shown in figures 22 and 23 respectively. Here, the field of view of our measurements was chosen such that the development of these instabilities could be visualized and traced at each instant of time while they propagated in the downstream direction. One may clearly notice that these instabilities are localized and develop on a low-speed streak. The sinuous instability in figure 22 is seen to develop on the low-speed streak located at $z/\unicode[STIX]{x1D6FF}^{\ast }=8$ , whereas the varicose instability in figure 23 develops on the low-speed streak located at $z/\unicode[STIX]{x1D6FF}^{\ast }=0$ . The frequencies of these instabilities in figures 22 and 23 are found to be 281 Hz and 186 Hz respectively.

Figure 23.

Time sequence of a varicose instability developing at the juncture of high- and low-speed streaks in the $x{-}z$ plane for grid A. The vectors are the fluctuating velocity vectors over the contours of $u/U_{0}$ . Maintaining the aspect ratio, the region displayed is zoomed in to clearly show the instability development. Each snapshot is separated by $\unicode[STIX]{x0394}t=0.00155$  s.

Furthermore, the varicose instability that originates at the juncture of an incoming high-speed streak and a downstream low-speed streak eventually generates lambda structures, as can clearly be seen in figure 23. These structures grow in size as time progresses and finally end up with small-scale structures because of their localized inflectional instability, as a lambda structure is generally associated with a highly inflectional velocity profile (e.g. Mandal & Dey Reference Mandal and Dey2011). In their DNS of a bypass transition, Brandt et al. (Reference Brandt, Schlatter and Henningson2004) also found that the varicose instability is generated due to the interaction of an incoming high-speed streak with a downstream low-speed streak. They further reported that the varicose type instability eventually generates lambda structures, which are susceptible to the localized inflectional instability. Recent DNS on bypass transition by Hack & Zaki (Reference Hack and Zaki2014) also indicates the presence of lambda structures at the juncture of an incoming high-speed streak and a downstream low-speed streak. The present results, thus, experimentally confirm these numerical observations in bypass transition under an elevated level of free-stream turbulence.

Previous work on bypass transition has indicated that the growth of a sinuous/varicose instability can lead to a turbulent spot in the flow (e.g. Brandt et al. Reference Brandt, Schlatter and Henningson2004; Mans et al. Reference Mans, Kadijk, de Lange and van Steenhoven2005; Hack & Zaki Reference Hack and Zaki2014). In their experimental work on bypass transition in a water channel, Mans et al. (Reference Mans, de Lange and van Steenhoven2007) carried out combined PIV and dye visualizations in the spanwise plane, and visualized the flow structures using a moving camera, enabled by a traversing system. Identifying the streak geometry from their measured velocity data, they calculated the secondary amplitude at the centreline of an unstable low-speed streak and estimated the growth rate. They found that the normalized temporal growth rate for the sinuous instability is approximately 0.01; the local boundary layer thickness and the free-stream velocity were used for normalization.

Figure 24.

Illustration of the procedure to estimate the growth rate for sinuous and varicose instabilities in the $x{-}z$ plane. (a,b) Tracking of sinuous and varicose instability wavepackets at each instant of time to find the secondary amplitude, $A_{s}$ , for grids B and A respectively. (c,d) Variation of the normalized secondary amplitude, $A_{s}(t+\unicode[STIX]{x0394}t)/A_{s}(t)$ , with the normalized time, $tU_{0}/\unicode[STIX]{x1D6FF}^{\ast }$ , for the sinuous instability in (a) and the varicose instability in (b) respectively. Symbols, estimated amplitude; solid line, linear fit.

Instead of using the streak geometry to estimate the secondary amplitude and the growth rate, here we follow Lundell (Reference Lundell2004) and Hack & Zaki (Reference Hack and Zaki2014), and utilize a quantitative criterion based on the measured velocity data. We estimate the normalized temporal growth rate as

(5.4) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=\frac{\unicode[STIX]{x1D6FF}^{\ast }}{U_{0}\unicode[STIX]{x0394}t}\ln \left[\frac{A_{s}\left(t+\unicode[STIX]{x0394}t\right)}{A_{s}\left(t\right)}\right],\end{eqnarray}$$

where the secondary amplitude, $A_{s}$ , similarly to Hack & Zaki (Reference Hack and Zaki2014), is estimated as $A_{s}(t)=\max w(x,y=2.5~\text{mm},z,t)-\min w(x,y=2.5~\text{mm},z,t)$ . It should be noted that the secondary amplitude in the present work is estimated based on the maximum and minimum spanwise perturbation velocity in the $x{-}z$ plane located at $y=2.5$  mm, in the absence of three-dimensional velocity data. Isolating the localized instability region in a PIV realization, we estimate the temporal growth rate of these instabilities. This procedure is illustrated in detail in figure 24. Time-resolved PIV realizations showing the growth of sinuous and varicose instabilities are tracked using a rectangular region at each instant of time, as shown in figures 24(a) and 24(b) respectively. Then, the secondary amplitude, $A_{s}$ , is estimated based on the maximum and minimum spanwise perturbation velocity within the rectangular region at that instant of time. Finally, the natural logarithm of the normalized secondary amplitude, $A_{s}(t+\unicode[STIX]{x0394}t)/A_{s}(t)$ , is plotted against the normalized time, $U_{0}\unicode[STIX]{x0394}t/\unicode[STIX]{x1D6FF}^{\ast }$ , and the normalized growth rate is estimated from the slope of a straight line fitted to the estimated data, as clearly shown in figures 24(c) and 24(d) for the sinuous and varicose instabilities respectively. The growth rates, as estimated from figures 24(c) and 24(d) for the sinuous and varicose instabilities in figures 24(a) and 24(b), are found to be 0.033 and 0.029 respectively.

Figure 25.

Average growth rates of sinuous and varicose instabilities based on several observations. (a) Sinuous instability for grid A. (b) Varicose instability for grid A. (c) Sinuous instability for grid B. (d) Varicose instability for grid B. Symbols, estimated amplitude; solid line, linear fit to the data.

However, to find the average growth rates for sinuous and varicose instabilities, several instability events, similar to those in figure 24(a,b), were identified from the entire TR-PIV realizations, and the above procedure was followed to estimate the secondary amplitude for each of these instability events. It should be noted that only those instabilities that grow over a minimum of five time steps are considered for estimating the growth rate. The variation of the normalized secondary amplitude versus the normalized time, for 24 sinuous and 12 varicose instabilities, is shown in figures 25(a) and 25(b) respectively for grid A. Similarly, figures 25(c) and 25(d) show the variation of the normalized secondary amplitude versus the normalized time for 24 sinuous and 11 varicose instabilities respectively for grid B. The scatter in the data as seen in these figures was found to be similar to that found by Mans et al. (Reference Mans, de Lange and van Steenhoven2007). However, an average growth rate for the sinuous/varicose instability for a particular grid was estimated based on a straight line fit to all of the data for a particular case (i.e. sinuous/varicose). The sinuous growth rates for grids A and B were found to be 0.022 and 0.025 respectively, whereas the varicose growth rates for grids A and B were found to be 0.035 and 0.04 respectively.

The present growth rate for the sinuous instability is comparable with the theoretical growth rates reported by Andersson et al. (Reference Andersson, Brandt, Bottaro and Henningson2001) and Schlatter et al. (Reference Schlatter, Brandt, de Lange and Henningson2008). Hack & Zaki (Reference Hack and Zaki2014) reported the normalized growth rate based on the length and velocity scales corresponding to their inlet Reynolds number, $Re=800$ . Therefore, to compare our present data with their calculated growth rates, we estimate the equivalent length scale at that Reynolds number based on our present free-stream velocity and kinematic viscosity. According to this equivalent length scale and the free-stream velocity, the present growth rates for the sinuous instability are found to be 0.02 and 0.022 for grids A and B respectively, whereas the growth rates for the varicose instability are found to be 0.031 and 0.035 for grids A and B respectively. Therefore, the present growth rate for the sinuous instability is found to be within the range reported by Hack & Zaki (Reference Hack and Zaki2014). The present growth rate for the varicose instability is also found to be comparable with their growth rate statistics for the inner mode.

Figure 26 shows the histograms of the convection velocity and the wavelength for both sinuous and varicose instabilities; here, white bars refer to the sinuous instability and black bars refer to the varicose instability. It should be noted that these statistics are based on all of the sinuous and varicose instabilities identified for both of the grids. Following a similar methodology to that in § 5.1.1, the propagation/convection velocity is estimated from the space–time shift of these instabilities, and the wavenumbers are estimated from the measured wavelengths. However, we may mention that in contrast to the oscillations in the $x{-}y$ plane, which were visualized using the $\unicode[STIX]{x1D714}_{z}\unicode[STIX]{x1D6FF}^{\ast }/U_{0}$ contours, wavy oscillations (sinuous/varicose) in the $x{-}z$ plane are visualized using the $u/U_{0}$ contours to estimate the wavelength and convection velocity reported in this section. Figure 26(a) shows that the convection velocity of the sinuous instability varies from $0.5U_{0}$ to $0.9U_{0}$ with a peak at approximately $0.7U_{0}$ , and that of the varicose instability varies approximately from $0.5U_{0}$ to $0.7U_{0}$ with a peak at approximately $0.7U_{0}$ . However, the average convection velocities for the sinuous and varicose instabilities are found to be $0.70U_{0}$ and $0.64U_{0}$ respectively. These results are comparable to the numerical simulations of Brandt et al. (Reference Brandt, Schlatter and Henningson2004) and Hack & Zaki (Reference Hack and Zaki2014). Based on their dye-flow visualizations at two different wall-normal heights in a water channel, Mans et al. (Reference Mans, Kadijk, de Lange and van Steenhoven2005) reported a value of $0.75U_{0}$ for a single sinuous instability and $0.47U_{0}$ for a single varicose instability. The convection velocities estimated from the present TR-PIV measurements are thus comparable with their visualization results. Similarly, figure 26(b) shows that the normalized wavelength varies from 5 to 19 for the sinuous instability and from 5 to 14 for the varicose instability. These values are also found to be consistent with the simulated values reported in the literature (e.g. Brandt et al. Reference Brandt, Schlatter and Henningson2004; Schlatter et al. Reference Schlatter, Brandt, de Lange and Henningson2008; Hack & Zaki Reference Hack and Zaki2014). Considering three sinuous instabilities, a similar variation in wavelength was also reported by Mans et al. (Reference Mans, de Lange and van Steenhoven2007) based on their PIV measurements in a water channel flow. One may also notice that the estimated and calculated values of the convection velocity and the wavelength of the oscillating shear layers in the $x{-}y$ plane (figure 23) are found to be within the range seen in figure 26.

Figure 26.

Histograms of the convection velocity (a) and the wavenumber (b) for the sinuous and varicose instabilities. White bars, sinuous instability; black bars, varicose instability.