2.1. Experimental facility and measurement technique

The measurement campaign was performed in the minimum turbulence level (MTL) wind tunnel located at the Royal Institute of Technology (KTH) in Stockholm. Minimum turbulence level is a closed-circuit wind tunnel with a 7 m test section and $0.8\ \textrm {m} \times 1.2\ \textrm {m}$ ($\textrm {height}\times \textrm {width}$) cross-sectional area. An axial fan (DC 85 kW) can produce airflow in the empty test section with a speed up to 69 m s$^{-1}$. The cooling system of the wind tunnel is capable of maintaining a constant temperature in the test section within $\pm 0.05\,^{\circ }\textrm {C}$. At the nominal speed of 25 m s$^{-1}$, the streamwise velocity fluctuation level is less than 0.025 % (Lindgren & Johansson Reference Lindgren and Johansson2002). The experiments were carried out over a 5 m flat plate. To minimize the effect of the leading edge an asymmetric leading edge was used (same as used in Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994), cf. their figure 2). With the installation of a trailing flap along with the adjustable ceiling of the test section, a zero-pressure gradient boundary-layer flow was obtained with a relatively short negative pressure gradient or accelerating flow around the leading-edge region.

Hot-wire anemometry was employed throughout the measurement campaign to measure velocity signals both in the free stream, at the leading edge and along the plate to characterize the inflow condition and the continuous forcing by the decaying turbulence, and inside the boundary layer. All probes were manufactured in-house of Wollaston wire composed of a platinum core. All wires were fully etched before soldered to the hot-wire prongs. Single-wire sensors, used for boundary-layer measurements, had a wire diameter and length of $2.54\ {\rm \mu} \textrm {m}$ and 0.7 mm, respectively. All boundary-layer measurements were performed with two single-wire probes located at the same distance above the wall, one being fixed at a spanwise location while the other one being traversable in the spanwise direction to determine the spanwise scale of the streaky structures through two-point correlation measurements. A dual-wire sensor in the shape of an $X$ was used to measure the cross-flow velocity-fluctuation components in the free stream and had a wire diameter and length of $5.08\ {\rm \mu} \textrm {m}$ and 1.4 mm, respectively. All probes were calibrated against the dynamic pressure using a Prandtl tube, connected to a differential manometer (Furness FC0510), in the free stream. The manometer used external probes for registering the temperature and the total pressure inside the test section for accurate determination of the density. For the single-wire probes, the modified King's law (Johansson & Alfredsson Reference Johansson and Alfredsson1982) was used as a calibration function, which has an extra term compensating for natural heat convection from the wire (important at low velocities). A typical calibration consisted of 15 calibration points in the range $0\text {--}7\ \textrm {m}\ \textrm {s}^{-1}$. The $X$-probe was calibrated at different angles to the flow direction between $-30^{\circ }$ and $+30^{\circ }$ with a step of $5^{\circ }$ and different speeds in the range 5 to 7 m s$^{-1}$. A two-dimensional fifth-order polynomial was fitted to the data and used as a calibration function.

The anemometer system was a DANTEC Dynamics StreamLine 90N10 Frame anemometer at CTA mode and the data acquisition was done with a National Instruments converter board (NI PCI-6259, 16-Bit) at a sampling frequency of 10 kHz.

2.2. Zero-pressure gradient boundary-layer base flow

Here we study the effect of FST on the laminar-turbulent transition in a boundary layer developing under a close to zero-pressure gradient flow. In figure 2 the adjusted base flow over the flat plate is compared with the Blasius boundary-layer solution (solid and dashed lines). In the comparison, a virtual origin has been introduced 25 mm downstream of the leading-edge nose, which compensates for the favourable leading-edge pressure gradient region.

Figure 2.

Base flow over the flat plate boundary layer. (a) Wall-normal mean velocity profiles at different $x$-locations. ($b$) Streamwise skin-friction ($\circ$) and free-stream velocity ($\square$) evolution. ($c$) Integral boundary-layer parameters: shape factor $H_{{12}}$ (${\rm \Delta}$), displacement thickness $\delta _{{1}}$ ($\Diamond$) and momentum thickness $\delta _{\mathrm {2}}$ ($\nabla$). Solid and dashed lines correspond to the Blasius boundary-layer solution.

2.3. Free-stream turbulence conditions

Free-stream turbulence can be characterized by its intensity ($Tu$) and length scales, such as the integral ($\Lambda$), Taylor ($\lambda _x$) and Kolmogrorov length scales. Each combination of these parameters constitutes a unique FST condition, which will be used from here on when referring to variations of these parameters. The $Tu$ and the length scales are independent of $U_\infty$, but the transition location is a function of all parameters. Non-dimensional parameters will be defined later using the integral length scale, $U_\infty$ and $u_{\textit {rms}}$, namely $\mathit {Re}_{{\textit{FST}}}$ and $\mathit {Re}_\Lambda$, which together with $Tu$ can be used to characterize the FST condition. These non-dimensional parameters are related to each other leaving one of the three ($\mathit {Re}_{{\textit{FST}}}$, $Re_\Lambda$, $Tu$) redundant (cf. (3.3)). The integral length scale is the most energetic, the Taylor length scale is the smallest energetic length scale and the Kolmogorov length scale is the smallest viscous scale in a turbulent flow. Both the longitudinal and transverse integral length scales, respectively denoted by $\Lambda _x$ and $\Lambda _z$, are here determined through direct integration of their corresponding correlation functions $f$ (auto-correlation) and $g$ (cross-correlation). All scales were determined using the same techniques as described in Kurian & Fransson (Reference Kurian and Fransson2009), that is, all time scales are converted to spatial scales using Taylor's hypothesis of frozen turbulence. According to the analysis on transition prediction in the present paper, it is shown that the central quantities to characterize the FST at the leading edge are $Tu$ and $\Lambda _x$ and are therefore properly defined below:

(2.1)\begin{equation} \Lambda_x = U_\infty \int_{0}^\infty f(\tau) \,\mathrm{d}\tau, \end{equation}

where $f(\tau )$ is the autocorrelation function of the velocity-time signal at the position of the leading edge of the flat plate, and

(2.2)\begin{equation} Tu = \frac{u_{{\textit{rms}}}}{U_\infty}. \end{equation}

The free-stream turbulence was generated using turbulence generating grids. Different leading-edge FST characteristics can be obtained by changing the relative distance between the grid and the leading edge ($X_{{\textit{grid}}}$) of a particular grid or by changing the solidity of the grid, i.e. the ratio between the solid to the total area. Placing the grid further upstream from the leading edge gives a lower turbulence intensity and a longer integral length scale at the leading edge, and vice versa. The turbulence intensity is proportional to the pressure drop over the grid, which is given by its solidity ($\sigma$). In general, the higher the solidity the higher the turbulence intensity at the leading edge for a given $X_{{\textit{grid}}}$. The FST length scales are functions of the mesh width ($M$) and bar diameter ($d$) of the grid.

Another way to increase the pressure drop and in turn the $Tu$ is to inject a secondary counter-flow, relative to the free stream, using upstream pointing air jets from the grid. This idea was first presented by Gad-El-Hak & Corrsin (Reference Gad-El-Hak and Corrsin1974) and has since been applied successfully in other experiments (Fransson & Alfredsson Reference Fransson and Alfredsson2003; Yoshioka, Fransson & Alfredsson Reference Yoshioka, Fransson and Alfredsson2004; Fransson, Matsubara & Alfredsson Reference Fransson, Matsubara and Alfredsson2005). To broaden the range of FST characteristics in the present experiments, six new active turbulence generating grids, similar to the one described and used in Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005), were manufactured. The new grids were manufactured using copper tubes as grid bars, and the secondary airflow was obtained by pressurized air carried by eight hoses to each grid. This secondary air is injected upstream through small orifices of diameter 1.5 mm. A regulating valve was employed to adjust the pressure inside the grids leading to controlled injection speeds in the range 0–40 m s$^{-1}$. In the experiments three injection rates were typically applied, namely, zero- (0), intermediate- (Mid) and full (Max) injection rate.

In total eight different grids were used. In table 1 the grids G1–G6 are the new active grids, while grids G7 and G8 are regular passive grids used in previous experiments. Four of the grids (G1–G3 and G8) have the same solidity, but different bar diameters and mesh widths, and four have the same mesh width (G2, G4, G5, and G8), but different bar diameters and hence different solidities. Grid G6 is designed to generate low $Tu$, but with larger length scales with respect to G1–G4. Note that, G5 has almost the same solidity as G6 but with the aim to generate smaller scales.

Table 1.

Geometrical data of all grids. Here $d$, $M$ and $\sigma$ are the bar diameter, the mesh width and the solidity, respectively. The symbols are only used for figure 3.

All the generated FST conditions, numbered by its test case (C1–C42), are summarized in table 2. The extensive database contains FST conditions with ranges of $Tu$: 1.81 – 6.19 (%), $\Lambda _x$: $16.05\text {--}25.61$ (mm) and $\lambda _x$: $11.97\text {--}15.02$ (mm). The Kolmogorov length scale is in the range 1.2–3.4 mm at the leading edge but is not shown in the table since it is not believed to be an important scale for the boundary-layer transition process. To assess the spanwise length scales of the unsteady streaks inside the boundary layer, two-point correlation measurements need to be carried out and hence two hot-wire probes are needed. Indexes 1 and 2 in table 2 correspond to the fixed and traversable hot-wire probes, respectively. The root-mean-square value of the mean velocity difference can be calculated, giving a reading difference between the probes of ${\rm \Delta} U \leq \pm 0.029$ m s$^{-1}$, i.e. with an absolute difference of less than 1 % between the two probes. This difference leads to an absolute difference in the measured turbulence intensities of about 0.2 percentage units. In table 3 more non-dimensional leading-edge FST parameters are given, among them the turbulence anisotropy being quantified in terms of fluctuation ratios at the leading edge. As for most grid generated turbulence, the streamwise component has a higher fluctuation level with respect to the cross-flow components, which in turn are about the same. Grid G4, with the highest solidity, is the grid that generates the lowest level of anisotropy and case C24 is the closest to isotropic free-stream turbulence.

Table 2.

Free-stream conditions at the leading edge for cases C1–C42. Indexes 1 and 2 represent hot-wire probes 1 and 2, respectively.

Table 3.

Boundary-layer transition location ($x_{ {\textit{tr}}}$) and averaged spanwise wavelength of the unsteady streamwise streaks ($\lambda _z$) for cases C1–C42. Here $v_{\textit {rms}}/u_{\textit {rms}}$ and $w_{\textit {rms}}/u_{\textit {rms}}$ are anisotropy measures and $\mathit {Re}_{{\textit{FST}}}$ and $\mathit {Re}_{\Lambda }$ are free stream Reynolds numbers, all corresponding to leading-edge data. Indexes 1 and 2 represent hot-wire probes 1 and 2, respectively.

2.4. Method of transition location determination

The transition location is determined as the position where the intermittency factor ($\gamma$) of the velocity signal is 0.5, which is halfway through the transition region. An intermittency factor of 0 and 1 correspond to a completely laminar and fully turbulent flow. The intermittency factor was calculated using the velocity-time signal acquired at the wall-normal location corresponding to the disturbance velocity peak inside the boundary layer (i.e. around $y \approx 1.3\delta _1$), using the method presented in Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005). In figure 4, $\gamma$-distributions of grid G1 are shown for both increasing injection rate ($a$) and decreasing $X_{{\textit{grid}}}$ ($b$). In both cases, $Tu$ is increasing while $\Lambda _x$ is slightly decreasing. Dashed and dotted lines indicate the values of $\gamma = (0.1,0.5,0.9)$ and their intersection values with the $\gamma$-distribution where $\gamma = 0.5$, respectively. The latter being indicated with big square symbols and give locations of transition $x_{\textit{tr}} \equiv x_{\gamma = 0.5}$, from which the transitional Reynolds number ($\mathit {Re}_{{\textit{tr}}}$) is calculated according to

(2.3)\begin{equation} \mathit{Re}_{{\textit{tr}}} = \frac{U_\infty x_{{\textit{tr}}}}{\nu}, \end{equation}

where $U_\infty$ is measured at the leading edge. From these distributions, the transition region is defined for each case as ${\rm \Delta} x_{{\textit{tr}}} = x_{\gamma = 0.9}-x_{\gamma = 0.1}$. For improved accuracy of all $x$-location determinations ($x_{\gamma = 0.1},x_{\gamma = 0.5},x_{\gamma = 0.9}$), a sigmoid curve was first fitted to the data in the least-square fit sense and then used as an analytical function to determine the $x$-locations. The curve fits are shown with solid lines in figure 4 and the $(+)$-symbols in ($b$) are simply added to differentiate the symbols between injection and $X_ {\textit{grid}}$ changes.

Figure 3.

Free-stream turbulence intensity decay ($a$), and longitudinal integral length scale ($b$) in the streamwise direction for cases C1–C42 in table 2. Here $X_{{\textit{grid}}}$ corresponds to the distance between the grid and the leading edge for each case, with the origin at the grid location. For symbol definitions, see table 1. The different streamwise evolution curves have been generated by changing $X_{{\textit{grid}}}$ and the injection rate for the active grids.

Figure 4.

Streamwise intermittency factor distribution of grid G1. ($a$) Effect of increasing the injection rate: (0, Mid, Max) as increased darkness of the symbols, which correspond to (C6, C7, C8), respectively. ($b$) Effect of reducing $X_{{\textit{grid}}}$: (140, 110, 80) mm as increased darkness of the symbols, which correspond to (C6, C4, C1), respectively. See running text for lines and other symbols.

3.1. Transition region and location

The intermittency distribution derived by Johnson & Fashifar (Reference Johnson and Fashifar1994) has previously shown to agree well with experimental boundary-layer FST data (see, e.g., Fransson et al. Reference Fransson, Matsubara and Alfredsson2005). Their $\gamma$-function, given as

(3.1)\begin{equation} \gamma (\xi)=1-{\mathrm{exp}}[-\mathcal{A}_1(\xi+\mathcal{A}_2)^{m}], \end{equation}

with $m = 3$ from the derivation can be seen as a universal distribution since the experimental data closely falls on this curve independent of $Tu$ and $\Lambda _x$. Here, the dimensionless streamwise coordinate $\xi$ is defined as

(3.2)\begin{equation} \xi= \frac{x-x_{{\textit{tr}}}}{{\rm \Delta} x_{{\textit{tr}}}}, \end{equation}

where ‘tr’, in turn, was defined in § 2.4. Note also, that (3.1) is only valid in the range $-A_2 \leq \xi < \infty$. For $\xi < -A_2$, the intermittency is per definition equal to zero since the flow is laminar upstream of the transition zone. In figure 5 all 84 intermittency distributions (from hotwires 1 and 2) are plotted versus $\xi$ and they all fall close to the dashed line corresponding to the universal $\gamma$-function, where the coefficients in (3.1) are the ones determined in the least square sense to the data in Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005), i.e. $(\mathcal {A}_1,\mathcal {A}_2) = (0.60,1.05)$. The inset of figure 5 shows the same data in the dimensional streamwise coordinate $x$. This strengthens the universality of the distribution and serves as a consistency check between different data sets taken during different campaigns in different experimental set-ups. However, it is clear from the figure that the experimental distributions rises and flattens somewhat sharper than (3.1) (which also was the case in Fransson et al. Reference Fransson, Matsubara and Alfredsson2005), suggesting that the exponent $m$ is actually greater than 3 as derived by Johnson & Fashifar (Reference Johnson and Fashifar1994). Performing a straight forward unphysical least-square fit of (3.1) to the present data, by the inclusion of the exponent, gives $(\mathcal {A}_1,\mathcal {A}_2,m) = (0.023,1.87,5.47)$, which corresponds to the solid line in figure 5.

Figure 5.

All 84 intermittency distributions from hot-wires 1 and 2 are shown in the scaled streamwise coordinate $\xi$. Dashed and solid lines correspond to the Johnson & Fashifar (Reference Johnson and Fashifar1994) distribution and the best sigmoid curve fit to the data, respectively. The inset shows all intermittency distributions unscaled, filled and open symbols correspond to hot-wire probes 1 and 2, respectively.

In figure 6 the experimental intermittency distributions versus $\xi$ are plotted for all different grids at 20 different spanwise locations, additionally confirming the universality of (3.1). To make the data accessible all boundary-layer transition results are collected for all FST conditions and summarized in table 3.

Figure 6.

Streamwise intermittency distributions for all different grids spanning an averaged length of about $9\delta _1$ in the spanwise direction. For symbols, see table 1.

The classical way to plot $\mathit {Re}_{\textit {tr}}$ is versus $Tu$, which is known to have a strong influence on the transition location. However, as shown by Jonáš et al. (Reference Jonáš, Mazur and Uruba2000), $Re_{\textit {tr}}$ can change by a factor of 20 by changing the integral length scale of the FST while keeping $Tu$ constant at 3 %. Their result suggests that the integral length scale is of equal importance as $Tu$ when it comes to FST induced transition and, hence, the primary variable should be an FST parameter including both the velocity disturbance level and the integral length scale at the leading edge. Here, we will show that this parameter is an FST Reynolds number defined as

(3.3)\begin{equation} \mathit{Re}_{\textit{FST}} = \frac{u_{{\textit{rms}}} \Lambda_x}{\nu} = Tu \cdot \mathit{Re}_\Lambda, \end{equation}

where

(3.4)\begin{equation} \mathit{Re}_\Lambda = \frac{U_\infty \Lambda_x} {\nu}. \end{equation}

In figure 7(a) and (b) the transitional Reynolds number is plotted versus both $Tu$ and $\mathit {Re}_{{\textit{FST}}}$, respectively. Using the physical reasoning by Andersson et al. (Reference Andersson, Berggren and Henningson1999), we can argue that the curve in figure 7(a) should have the form

(3.5)\begin{equation} (\mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}}^{Tu} = \mathcal{B}_1 \cdot Tu^{-2} + \mathcal{B}_2, \end{equation}

where $\mathcal {B}_1$ and $\mathcal {B}_2$ are constants and become $(\mathcal {B}_1,\mathcal {B}_2) = (148,31\,956)$ when least-square fitted to the data. Here $\mathcal {B}_2$ has been added, with the motivation of an existing minimum Reynolds number for self-sustained turbulence. The subscript ‘cf’ in (3.5) stands for an empirical curve fit to the present data and will be used throughout this paper. The same physical reasoning, based on input energy, as in Andersson et al. (Reference Andersson, Berggren and Henningson1999), can be used for the variable $\mathit {Re}_{\textit{FST}}$ and, hence, the curve in figure 7(b) corresponds to

(3.6)\begin{equation} (\mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}} = \mathcal{C}_1 \cdot \mathit{Re}_{{\textit{FST}}}^{-2} + \mathcal{C}_2, \end{equation}

where $\mathcal {C}_1$ and $\mathcal {C}_2$ are determined to $(\mathcal {C}_1,\mathcal {C}_2) = (7.6961\times 10^{9},62\,538)$. At first glance, the choice of $Tu$ seems to be the better option, since it collects the data points closer to the curve fitted line. However, plotting the data versus the primary variable $\mathit {Re}_{{\textit{FST}}}$ reorders the set of data in a favourable way, such that one can relate the deviation from the curve to the integral length scale through a scale-matching model, which will be introduced later. When comparing the values $\mathcal {B}_2$ with $\mathcal {C}_2$ one surprisingly realizes that they differ by a factor of 2 even though they should represent the same physical asymptotic value, i.e. a minimum $\mathit {Re}_{\textit {tr}}$ below which natural transition cannot take place. It becomes clear that $\mathcal {B}_2$ and $\mathcal {C}_2$ will only correspond to each other if additional data at much higher $Tu$ and $\mathit {Re}_{{\textit{FST}}}$ values are included in the fits.

Figure 7.

Transitional Reynolds number, for all cases C1–C42 (HW1 and HW2), plotted versus $Tu$ and $\mathit {Re}_{{\textit{FST}}}$ in ($a$) and ($b$), respectively.

The present data were collected with the mind of creating a large data set, which was done by changing both $Tu$ and $\Lambda _x$ at the leading edge of the plate, without the effort in trying to change one of them independent of the other. However, there is enough data to group cases together with about the same $Tu$ but with different $\Lambda _x$. If one does that for different $Tu$ levels our data show that, for low turbulence intensities, the transition location moves upstream with increasing $\Lambda _x$ and downstream with increasing $\Lambda _x$ for high turbulence intensities. That is, there is a twofold effect of $\Lambda _x$ on the transition location, which confirms the findings reported both by Jonáš et al. (Reference Jonáš, Mazur and Uruba2000), Brandt et al. (Reference Brandt, Schlatter and Henningson2004), Ovchinnikov et al. (Reference Ovchinnikov, Piomelli and Choudhari2004) (of transition advancement) and Hislop (Reference Hislop1940) (of transition delay). The grouped cases are plotted in figure 8 with open symbols corresponding to the experiments and filled grey symbols to the corresponding predicted values that will be introduced and discussed later. The two trends will be explained by the fact that an optimal ratio between FST length scale and boundary-layer thickness at transition onset exists.

Figure 8.

Twofold effect on $\mathit {Re}_{\textit {tr}}$ by increasing $\Lambda _x$, while keeping $Tu$ close to constant. (a) For low $Tu~(\approx 2.16\,\%$ and 2.45 %$)$, transition moves upstream (i.e. $\mathit {Re}_{\textit {tr}}$ is reduced) for increasing $\Lambda _x$. (b) For high $Tu\ (\approx 2.82\,\%,\,3.37\,\%$ and 4.28 %$)$, transition moves downstream (i.e. $\mathit {Re}_{\textit {tr}}$ is increased) for increasing $\Lambda _x$. For symbols and cases, see tables 1 and 2, respectively. Here open symbols correspond to experimental data (from hot-wire 1) and filled grey symbols to predicted values.

3.2. Streamwise streaks inside the boundary layer

The asymptotic value of the spanwise length scale of the streamwise streaks is here argued not to be important for the transition process, in figure 11 of Matsubara & Alfredsson (Reference Matsubara and Alfredsson2001) the transition takes place farther upstream than the asymptotic limit in most cases, which makes the region where the data differs from each other the most interesting. This means that the cross-sectional aspect ratio of the streamwise structures at transition is such that the streaks are ovally shaped with the major axis being parallel with the surface. Now, the unsteady streamwise boundary-layer streaks, induced by the FST, can only be important for the transition location if their characteristic length scale correlates with the FST leading-edge parameters. The study by Fransson & Alfredsson (Reference Fransson and Alfredsson2003) supports the idea that the spanwise scale is set by the FST condition. In that study it was shown that by reducing the boundary-layer thickness by a factor of two, by means of creating an asymptotic suction boundary layer, the spanwise scale in the boundary layer remained the same, giving structures of an aspect ratio of two instead of one as suggested by the asymptotic limit $\lambda _z/2 \approx 3\delta _1$ from Matsubara & Alfredsson (Reference Matsubara and Alfredsson2001).

In the present measurement campaign, two-point spanwise correlation measurements were performed at five different streamwise distances from the leading edge and at the location above the wall of maximum $u_{\textit {rms}}$. These measurements were carried out at minimum and maximum injection when the grid was located at the closest and farthest downstream distances from the leading edge, resulting in 23 of the total 42 cases from tables 2 and 3. In figure 9 all measured spanwise correlation functions are plotted, where increasing darkness of the curves corresponds to increasing intermittency value. The spanwise distance $({\rm \Delta} z)$ between the two hot-wires where a negative minimum in $R_{uu} ({\rm \Delta} z)$ appears can be interpreted as the averaged half spanwise wavelength of the streaky structures $(\lambda _{z}/2)$. For each function, this minimum is consistently determined by a third-order polynomial fit to the data, and to obtain $\lambda _z$ for a specific intermittency value simple linear interpolation using the five streamwise locations is applied. Figure 10(a) shows $\lambda _z$ unscaled for $\gamma = 0.5$ versus its corresponding $Tu$ & $\Lambda _x$ and it shows that the data is scattered and uncorrelated with $Tu$ and $\Lambda _x$, it appears to be a range of possible $\lambda _z$ values for each $Tu$ and $\Lambda _x$. However, when scaled with the leading-edge FST condition using both $\Lambda _x$ and $Tu$, and plotted versus the primary variable $\mathit {Re}_{{\textit{FST}}}$ as in figure 10(b), the data fall close to a single curve. Here, the spanwise wavelength for $\gamma = 0.1$ has been added with filled square symbols, showing that the spanwise wavelength is constant throughout the transition region. The curve in figure 10(b) has been fitted to the data in the least-square fit sense, and implies that the averaged spanwise wavelength can be estimated from leading-edge FST parameters according to

(3.7)\begin{equation} \mathcal{L} = \Lambda_x \cdot Tu \left( \mathcal{D}_1 \cdot \mathit{Re}_{{\textit{FST}}}^{-1/\sqrt{2}} + \mathcal{D}_2 \right)^2, \end{equation}

with $(\mathcal {D}_1,\mathcal {D}_2) = (186,0.8)$. That is, $\lambda _z$ is concluded to be correlated with the leading-edge FST parameters and, hence, to be an important length scale in the breakdown process to turbulence. From figure 10(c), it is clear that the estimated $\lambda _z$ using (3.7) stays within ${\pm }10\,\%$ of the measured value.

Figure 9.

Two-point spanwise correlation functions for all measured distributions at different intermittency values. The colour bar shows the intermittency $\gamma$.

Figure 10.

Averaged spanwise wavelength of the streamwise streaks. Panels (a) and (b) show unscaled and scaled wavelengths versus $Tu$ and $\Lambda _x$ and $\mathit {Re}_{{\textit{FST}}}$, respectively. Panel (c) indicates the accuracy in estimating $\lambda _z$ using (3.7) for different transitional Reynolds numbers. The solid lines correspond to deviations of ${\pm }10\,\%$ from the dashed line. Circular and square symbols correspond to streamwise locations of $\gamma = 0.5$ and 0.1, respectively.

Another evaluated quantity is the longitudinal integral length scale inside the boundary layer ($\Lambda _{x}^{{\textit{BL}}}$, where ‘BL’ denotes boundary layer), which is a measure of how long the structures are in the streamwise direction. Here $\Lambda _{x}^{\textit {BL}}$ is calculated at the position above the wall corresponding to maximum $u_{\textit {rms}}$. Figure 11(a) shows streamwise $\Lambda _{x}^{\textit {BL}}$ distributions for all cases (see table 1 for symbols). The downstream distance is shown in $\xi$ and the colour brightness corresponds to the turbulence intensity at the leading edge, the darker the higher. A direct conclusion from this figure is that $\Lambda _{x}^{\textit {BL}}$ asymptotes to a constant value (just below 10 mm) in the turbulent region, i.e. for $\xi >2$. It is noteworthy that, regardless of the FST characteristics, the streamwise length scale will eventually be the same, i.e. there is no memory of the initial FST condition in the turbulent region. From the location where the unsteady streaks are created, they will grow in the streamwise direction becoming longer until around $\xi = -0.5$, i.e. around the onset of transition where $\gamma$ is close to 0.10. After this location, the streaks become shorter due to the appearance of turbulent spots and asymptotes to the constant value. The maximum value of $\Lambda _{x}^{\textit {BL}}$ (around $\xi = -0.5$) is plotted in figure 11(b). No satisfactory correlation function was found with $\mathit {Re}_{{\textit{FST}}}$ as the primary variable as was found for a dimensionless $\lambda _z$ (cf. figure 10b), but there is a clear trend of shorter streaks as $\mathit {Re}_{{\textit{FST}}}$ is increased. Instead, in figure 11(c) $\Lambda _{x,\textit {max}}^{\textit {BL}}$ is normalized with $\mathcal {L}$ (3.7) and $Tu$, which makes the data points fall close to a straight line when plotted against $\mathit {Re}_{\textit {tr}}$. The length scale ratio ($\Lambda _{x,\textit {max}}^{\textit {BL}} / \mathcal {L}$) corresponds to the streamwise/spanwise aspect ratio at the maximum streak length and can be estimated from known inlet FST conditions as

(3.8)\begin{equation} \left( \frac{\Lambda_{x,{\textit{max}}}^{{\textit{BL}}}}{\mathcal{L}} \right) = Tu \cdot \left( \mathcal{E}_1\,\mathit{Re}_{{\textit{tr}}} + \mathcal{E}_2 \right), \end{equation}

where $(\mathcal {E}_1,\mathcal {E}_2) = (5.293 \times 10^{-4},-15.75)$ corresponding to the straight line in figure 11(c). The missing parameter to assess the aspect ratio is the parameter $\mathit {Re}_{\textit {tr}}$, which can be predicted for the given leading-edge FST parameters $\mathit {Re}_{{\textit{FST}}}$ and $Re_\Lambda$, as will be shown in § 4.

Figure 11.

(a) Streamwise distribution of the integral length scale inside the boundary layer ($\Lambda _{x}^{\textit {BL}}$) versus the non-dimensional transition coordinate $\xi$. The brightness of the symbols illustrates the turbulence intensity, with darker colours corresponding to higher $Tu$. (b) Maximum values of $\Lambda _{x}^{\textit {BL}}$ for each distribution in (a) versus its corresponding $\mathit {Re}_{{\textit{FST}}}$. Panel (c) indicates a linear correlation between appropriately scaled $\Lambda _{x,\textit {max}}^{\textit {BL}}$ and the transitional Reynolds number.

4.1. Scale-matching model

The present experimental data indicates that there is a change of trend of the influence of $\Lambda _x$ on the transitional Reynolds number. This information suggests that a scale matching between the FST and the boundary layer has to take place at the trend change. The local integral length scale of the FST, which grows with the square root of the downstream distance, is proportional to the integral length scale at the leading edge (see, e.g., Kurian& Fransson (Reference Kurian and Fransson2009), or the present figure 3b) which in turn suggests that the ratio $\Lambda _x / \delta _{\textit{tr}}$ is important for where the transition happens. Here, the following hypothesis is formulated.

Hypothesis.For a given $Tu$, there is an optimal scale ratio $(\Lambda _x / \delta _{\textit{tr}})_\textit {opt}$that promotes transition. The transitional Reynolds number versus the scale ratio has a minimum at the optimal scale ratio. A mismatch from the optimal value will give a negative derivative of $\mathit {Re}_{\textit {tr}}$, with respect to the scale ratio, if the scale ratio is lower than the optimal, and a positive derivative if it is larger.

This hypothesis was first formulated in Fransson (Reference Fransson2017), but there with the scale ratio inverted. A sketch is shown in figure 12, where the curves correspond to a specific $Tu$ value. The higher the $Tu$, the closer the curve is to the abscissa. To confirm the hypothesis new experiments at constant $Tu$ with varying $\Lambda _x$ need to be carried out, the present database is not sufficient since both $Tu$ and $\Lambda _x$ were varied. However, the present data seem to indicate that $(\Lambda _x / \delta _{\textit{tr}})_\textit {opt}$ is close to 15, which would imply that the change of trend appears where $\Lambda _x$ is about three times the boundary-layer thickness, i.e. $\delta _\mathrm {99}$, at transition. This means that if $\Lambda _x < 3\,\delta _{99}$ at transition, the transition location is expected to move upstream, and vice versa, with increasing $\Lambda _x$.

Figure 12.

Sketch shows the scale-matching model, based on the hypothesis that, for each $Tu$, there exists an optimal $\Lambda _x/\delta _{\textit {tr}}$ that promotes transition to a lowest possible $\mathit {Re}_{\textit {tr}}$.

4.2. The equation for the transitional Reynolds number

The effect of $\Lambda _x / \delta _{\textit{tr}}$ on $Re_{\textit {tr}}$ is assumed to enter as a correction to (3.6) and, hence, we make the following ansatz:

(4.2)\begin{equation} \mathit{Re}_{{\textit{tr}}} = (\mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}} + {\rm \Delta} \mathit{Re}_{{\textit{tr}}}, \end{equation}

where

(4.3)\begin{equation} {\rm \Delta} \mathit{Re}_{{\textit{tr}}} = {\rm \Delta} \mathit{Re}_{{\textit{tr}}}(\Lambda_x/\delta_{{\textit{tr}}}), \end{equation}

i.e. we assume that the correction term ${\rm \Delta} \mathit {Re}_{\textit {tr}}$ can be written as a function of the scale ratio, as derived below. If the scale ratio is matched with the one corresponding to the curve of (3.6), ${\rm \Delta} \mathit {Re}_{\textit {tr}}$ will give a zero contribution. On the other hand, the more the scale ratio deviates from the one corresponding to the curve (3.6), the larger the correction becomes. To amplify the sensitivity of the scale ratio, the square of this ratio is introduced in the correction as

(4.4)\begin{equation} {\rm \Delta} \mathit{Re}_{{\textit{tr}}} = \kappa\left[ \left( \frac{\Lambda_x}{\delta_{{\textit{tr}}}} \right)^2 - \left( \frac{\Lambda_x}{\delta_{{\textit{tr}}}} \right)^2_{{\textit{cf}}} \right], \end{equation}

where

(4.5)\begin{equation} \left( \frac{\Lambda_x}{\delta_{{\textit{tr}}}} \right)^2 \equiv \left( \frac{\mathit{Re}_{{\textit{FST}}}}{Tu} \right)^2 \frac{1}{\mathit{Re}_{{\textit{tr}}}}, \end{equation}

and $\kappa$ is a weighting function. Furthermore, by making the above choice of introducing the correction with the square of the scale ratio will simplify the final equation. In analogy with (4.5) we then introduce $( \Lambda _x / \delta _{\textit {tr}} )^2_{\textit{cf}}$ in (4.4) as

(4.6)\begin{equation} \left( \frac{\Lambda_x}{\delta_{{\textit{tr}}}} \right)^2_{\textit{cf}} = \left( \frac{\mathit{Re}_{{\textit{FST}}}}{Tu} \right)^2 \frac{1}{( \mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}}}. \end{equation}

Now, by combining (4.2) and (4.4)–(4.6), and multiplying with $\mathit {Re}_{\textit {tr}}$ the following second-order equation of $\mathit {Re}_{\textit {tr}}$ can readily be derived:

(4.7)\begin{equation} \mathit{Re}_{{\textit{tr}}}^2 +\underbrace{ \left[ \kappa\,\mathit{Re}_\Lambda^2 \frac{1}{(\mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}}} - ( \mathit{Re}_{{\textit{tr}}})_{{\textit{cf}}} \right] }_{ = \, \alpha}\, \mathit{Re}_{{\textit{tr}}}~- \underbrace{\kappa\,\mathit{Re}_\Lambda^2}_{= \, \beta} = 0, \end{equation}

with the solution

(4.8)\begin{equation} \mathit{Re}_{{\textit{tr}}} = -\frac{\alpha}{2} \pm \sqrt{\left( \frac{\alpha}{2} \right)^2 + \beta}. \end{equation}

When the scale ratio $(\Lambda _x / \delta _{\textit {tr}})^2$ corresponds to (4.6), the Reynolds number correction (4.4) is zero and the solution is a double root corresponding to $\mathit {Re}_{\textit {tr}} = (\mathit {Re}_{\textit {tr}})_{{\textit{cf}}}$ (cf. 3.6). If there is a mismatch of the scale ratio with respect to (4.6), there will be two different roots, namely (3.6) and (4.2), which is given by (4.8). Finally, we note that both $\alpha$ and $\beta$ depend on the parameter $\mathit {Re}_\Lambda$ and the weighting function $\kappa$. While $\mathit {Re}_\Lambda$ is considered known from the FST condition at the leading edge, $\kappa$ requires further analysis.

4.3. Weighting function

The weighting function $\kappa$ is introduced in (4.4) and corresponds to

(4.9)\begin{equation} \kappa = \frac{{\rm \Delta} \mathit{Re}_{{\textit{tr}}}}{\left[ \left( \Lambda_x / \delta_{{\textit{tr}}} \right)^2 - \left( \Lambda_x /\delta_{{\textit{tr}}} \right)^2_{{\textit{cf}}} \right]}, \end{equation}

which may be rewritten using (4.5), (4.6) and (3.3) to

(4.10)\begin{equation} \kappa = - \mathit{Re}_\Lambda^{-2}\,\mathit{Re}_{{\textit{tr}}} \cdot ( \mathit{Re}_{{\textit{tr}}})_{\textit{cf}} < 0. \end{equation}

This expression of $\kappa$ needs to be modelled since it includes $\mathit {Re}_{\textit {tr}}$ and renders a trivial solution when inserted to (4.7). Besides, it is noted that $\kappa$ is always negative. For the modelling, $\kappa$ is plotted versus $\mathit {Re}_{{\textit{FST}}}$ in figure 13(a) using the experimental data and (3.6). The model function should have two constraints for an accurate representation, in the limits of $\mathit {Re}_ {\textit{FST}} \rightarrow 0$ and $\mathit {Re}_{\textit{FST}} \rightarrow \infty$ the function should approach zero. The physical reasoning behind this is that ${\rm \Delta} \mathit {Re}_{\textit {tr}}$ should approach zero in these two limits. For $\mathit {Re}_{\textit{FST}} \rightarrow \infty$, the transitional Reynolds number will approach the minimum Reynolds number for self-sustained turbulence (cf. $\mathcal {B}_1$ in (3.6)) and the dependence of $\Lambda _x$ will diminish. For $\mathit {Re}_{\textit{FST}} \rightarrow 0$, the FST transition scenario will eventually be replaced by the T–S wave transition scenario and the effect of $\Lambda _x$ is expected to disappear. However, it is worth pointing out that independent of an accurate representation of $\kappa$ the transition prediction model described by (4.7) is only valid for the transition scenario dominated by FST. Here, the $\kappa$ function is created in order to fulfil the above constraints with a good fit to the data according to

(4.11)\begin{equation} \kappa = -10^3 \times \left[\underbrace{\frac{{G} \cdot \mathit{Re}_{ {\textit{FST}}}}{a + \vert\,(\mathit{Re}_{{\textit{FST}}} - a)^b\,\vert }}_{= f_1} + \underbrace{c\cdot \exp\{ -d(\mathit{Re}_{{\textit{FST}}}-e)^2 \} }_{= f_2} \right]^2. \end{equation}

The five coefficients ($a$–$e$) are determined in the least-square fit to the data and ${G}$ is set to ${G} = 25$, which is the gain to $f_1$ and chosen such that the turnaround of $\kappa$ to fulfil the constraint in the limit for $Re_{{\textit{FST}}} \rightarrow 0$ is around $\mathit {Re}_{{\textit{FST}}} = 73$. This value of $\mathit {Re}_{{\textit{FST}}}$ corresponds to $Tu = 1\,\%$, which is obtained through the relation

(4.12)\begin{equation} \mathit{Re}_{{\textit{FST}}} = \left(\frac{\mathcal{B}_1 \cdot Tu^{-2} + \mathcal{B}_2 - \mathcal{C}_2}{\mathcal{C}_1}\right)^{-1/2}, \end{equation}

derived by combining (3.5) and (3.6). The value of $Tu \approx 1$ % is often used as a threshold from where the transition scenario is dominated by the growth and breakdown of unsteady streamwise streaks (see, e.g., Arnal & Juillen Reference Arnal and Juillen1978). The five coefficients in (4.11) are determined to be

(4.13)\begin{align} \begin{aligned} a &= 7.2279 \times 10^1;\quad b = 1.7574 \times 10^0;\quad c = 3.9519 \times 10^{-2}; \nonumber\\ d &= 5.6340 \times 10^{-5};\quad e = 3.1987 \times 10^2. \end{aligned} \end{align}

In figure 13(a) both parts, $f_1$ and $f_2$, in (4.11) are plotted and it is clear that the sum of the two gives a satisfactory fit to the data. Figure 13(b) shows the full $\kappa$ function. The grey areas in figure 13 indicate $\mathit {Re}_{{\textit{FST}}}$ values being lower than $Tu = 1\,\%$. On the one hand, there are too few data points at low $\mathit {Re}_{{\textit{FST}}}$ to comment on the behaviour of $\kappa$ in the lower range of $\mathit {Re}_{{\textit{FST}}}$, on the other hand, the data do indicate that the effect of $\Lambda _x$ on $Re_{\textit {tr}}$ only becomes greater as $\mathit {Re}_{{\textit{FST}}}$ is reduced. The physical constraint that the effect of $\Lambda _x$ eventually has to diminish as $\mathit {Re}_{\textit {tr}} \rightarrow 0$ gives rise to the sharp turnaround of the function.

Figure 13.

Weighting function $\kappa$ versus $\mathit {Re}_{{\textit{FST}}}$. (a) Close up of the $\kappa$-function according to (4.10), symbols correspond to experimental data. (b) Full $\kappa$-function produced by (4.10) for the range $\mathit {Re}_{{\textit{FST}}} = 0\text {--}700$. Grey regions correspond to $\mathit {Re}_{{\textit{FST}}} \leq 73$, which by combining (3.5) and (3.6) leads to $Tu \leq 1\,\%$.