On the effect of free-stream turbulence on boundary-layer transition

Abstract Free-stream turbulence (FST) and its effect on boundary-layer transition is an intricate problem. Elongated unsteady streamwise streaks of low and high speed are created inside the boundary layer and their amplitude and spanwise wavelength are believed to be important for the onset of transition. The transitional Reynolds number is often simply correlated with the turbulence intensity (${Tu}$), and the characteristic length scales of the FST are often considered to have a small to negligible influence on the transition location. Here, we present new results from a large experimental measurement campaign, where both the ${Tu}$ and the integral length scale ($\Lambda _x$) are varied ($1.8\,\% < {Tu}< 6.2\,\%$; $16\ \textrm {mm}< \Lambda _x < 26\ \textrm {mm}$). In the current experiments it has been noted that on the one hand, for small $Tu$, an increase in $\Lambda _x$ advances transition, which is in agreement with established results. On the other hand, for large $Tu$, an increase in $\Lambda _x$ postpones transition. This trend can be explained by the fact that an optimal ratio between FST length scale and boundary-layer thickness at transition onset exists. Furthermore, our results strengthen the fact that the streaks play a key role in the transition process by showing a clear dependence of the FST characteristics on their spanwise scale. Our measurements show that the aspect ratio of the streaky structures correlates with an FST Reynolds number and that the aspect ratio can change by a factor of two at the location of transition. Finally, we derive a semi-empirical transition prediction model, which is able to predict the influence of $\Lambda _x$ for both small and high values of ${Tu}$.


Introduction
Today, we cannot honestly say that we are capable of accurately predicting the transition location in a boundary layer subject to free-stream turbulence (FST). Not even in the simplest boundary-layer flow, namely the one developing over a flat plate under a zero-pressure gradient condition, are we successful. There are numerous empirical relations for predicting the transition location in the presence of FST, mostly only based on the turbulence intensity (Tu) as input, but none with an accuracy better than typically 65 % for all Tu. Thus, in a sense, we have failed in delivering a simple and reliable transition prediction model for engineering predictions, which takes both Tu and the FST characteristic length scale into account in a physically correct way.
When a two-dimensional boundary-layer flow undergoes laminar to turbulent transition under the presence of FST, the transition scenario is different from the condition with a low background disturbance level. Under the latter condition, the initial phase of velocity disturbance growth is characterized by small-amplitude exponentially growing Tollmien-Schlichting (T-S) waves described by modal stability theory (Tollmien 1929;Schlichting 1933). Under the former condition, the disturbance growth is instead characterized by algebraically growing unsteady streamwise velocity streaks, and is described by non-modal stability theory (Ellingsen & Palm 1975;Landahl 1980;Gustavsson 1991). The designation 'bypass transition' is still often used when referring to FST induced boundary-layer transition, although somewhat incorrectly. The term bypass transition was according to Morkovin coined by himself (cf. Morkovin 1985) in Morkovin (1969) and was introduced to denote any transition process that bypassed common knowledge or existing theories, which at the time was limited to the T-S wave transition scenario described by modal theory. Bypass as a stand-alone term, without transition, is however encountered before the introduction of the term by Morkovin, then referring to artificially forced high amplitude T-S waves, which through nonlinearity advanced the known secondary instability and breakdown to turbulence and, hence, completely bypassed the linear range of the T-S wave (Klebanoff, Tidstrom & Sargent 1962). Going back to the meaning of bypass transition by Morkovin, it was originally used for surface roughness induced transition but became a common notation for FST induced transition. However, after the distinguished amount of work on non-modal theory and FST transition over the last decades, both experimentally and numerically, it is inappropriate to continue terming this transition scenario bypass transition since the basic mechanism for energy growth today is known. This even though there are minor insights missing on the receptivity process and actual triggering of the breakdown to turbulence.
Free-stream turbulence gives, undoubtedly, rise to the most complicated boundary-layer transition scenario. The reason for the complexity is that the boundary-layer thickness grows with the downstream distance at the same time as the turbulence intensity (Tu) of the FST decays and the FST characteristic length scales grow. The FST is present everywhere in the free stream but changes characteristics with the downstream distance. This implies that the actual forcing by the FST on the boundary layer changes gradually, which makes it an intricate receptivity problem. Inside the boundary layer, the disturbance growth and structure are physically described by the lift-up effect (Landahl 1980), which is a vertical exchange of momentum. Here, described by high momentum fluid being pushed towards the wall by the FST-forcing, and low momentum fluid being lifted from the wall as a consequence of continuity. This creates a weak cross-sectional fluid motion, which induces unsteady elongated streamwise streaks of alternating high and low speeds. Unsteady, since the continuous forcing is turbulent and hence random.
The streamwise turbulence intensity, defined as the ratio between the root-mean-square value of the velocity (u rms ) and the free-stream velocity (U ∞ ), i.e. Tu = u rms /U ∞ , is a simple measure to quantify the level of FST. As Tu is increased, there is a gradual shift from the T-S to the FST transition scenario. Arnal & Juillen (1978) noted that, for Tu 1 %, the FST transition scenario with unsteady streamwise streaks dominates the transition process over the T-S wave scenario.
The early experimental studies on FST induced transition originates from before 1940 (see, e.g., Hall & Hislop 1938;Taylor 1939;Hislop 1940), but were mostly carried proportional to the FST energy (in agreement with Andersson, Berggren & Henningson (1999)). They also conclude that, for Tu > 2.5 %, the relative length of the transitional zone increases with increasing Tu.
The algebraic disturbance growth observed in experiments was confirmed theoretically by Andersson et al. (1999), Luchini (2000) by considering the spatial disturbance growth from an initial perturbation calculated using optimal perturbation theory (Butler & Farrell 1992). Theoretically, the optimal perturbation in a shear layer that maximises the disturbance energy at some downstream location takes the form of counter-rotating vortices in the cross-sectional plane (Schmid & Henningson 2001). The energy in the optimal perturbation is concentrated to the cross-flow components, which are fed to the streamwise component as they move downstream leading to passively convected high and low velocity streaks. These streamwise streaks die out in the downstream direction unless they are nonlinearly triggered by some secondary instability. Smoke flow visualizations by Matsubara & Alfredsson show the existence of both sinusoidal and varicose secondary instabilities acting on unsteady streamwise streaks originating from FST (partly published in Matsubara & Alfredsson (2001)). Andersson et al. (2001) calculated streamwise streaks from the optimal initial condition (Andersson et al. 1999) using direct numerical simulations (DNS). They performed inviscid secondary instability calculations using Floquet theory on the obtained streaky base flow and showed that the sinuous mode sets in on the low-speed region at a lower streak amplitude than the varicose mode (26 % and 37 %, respectively). In a later paper, the flow structures associated with the sinuous breakdown were reported (Brandt & Henningson 2002). However, even though the non-modal perturbation theory predicts the generation of streamwise streaks and energy growth, the optimal perturbation as such has never been observed in an experiment. The streak spacing does, however, seem to approach the optimal spanwise wavenumber as Tu is increased (cf. Fransson & Corbett 2003).
In the present brief review we have to mention the work by Jacobs & Durbin (2001), who also performed a DNS of FST induced transition but presents an alternative breakdown process. They found boundary-layer streaks, generated nonlinearly by the penetration of FST, with a spacing that was in agreement with the results by Andersson et al. (1999), Luchini (2000). However, they concluded that the streaks were not undergoing a secondary instability since they did not find any evidence of streak instability. The onset of transition was instead argued to originate from the direct penetration of free-stream disturbances and that the low-speed streak simply provides a receptivity path between the FST and the boundary layer. Similar conclusions on FST induced transition has lately been published by Wu et al. (2017).
Other important works in this context are Nagarajan, Lele & Ferziger (2007) and Ovchinnikov, Choudhari & Piomelli (2008), who both report on a different transition scenario if the turbulence intensity or the length scale exceeds a certain threshold. This scenario is dominated by the growth and breakdown of wavepacket-like disturbances close to the leading edge. Streamwise streaks appear downstream of this region and hence are argued not to be responsible for the transition onset. From here their results differ, the primary disturbance is, for instance, reported to originate from streamwise vorticity through vortex stretching around the leading edge by Nagarajan et al. (2007), while it is reported to originate from spanwise vorticity by Ovchinnikov et al. (2008).

Effect of FST integral length scale on the transition
The shear sheltering concept described by Hunt & Durbin (1999) explains the ability of low-frequency free-stream disturbances to penetrate the boundary layer, while Experimental data by Hislop (1940).
high-frequency disturbances do not. It has been shown using DNS (Zaki & Durbin 2005) that transition to turbulence via streamwise streaks and secondary instabilities can be simulated by only including one high-frequency and one low-frequency mode in the free stream. The latter penetrating the boundary layer, giving rise to streamwise streaks and the former triggering the secondary instability on the streaks leading to breakdown to turbulence. This suggests that the low-frequency disturbances associated with large spatial scales are important for the boundary-layer streaks, being the primary disturbance, and the eventual breakdown to turbulence (see also Wang, Mao & Zaki 2019). A similar conclusion was reached in the analysis by Leib, Wundrow & Goldstein (1999), where the boundary-region equations were solved. They concluded that the streamwise streaks were primarily generated by low-frequency transverse velocity fluctuations and that nonlinear effects played an important role (see also Ricco & Wu (2007), who included the effect of compressibility in their analysis). Already in the doctoral thesis by Hislop (1940) an effect of different FST integral length scales on the laminar-to-turbulent transition location can be pointed out. The transitional Reynolds number for three different turbulence generating grids are tabulated in the thesis of Hislop, and this data is plotted here in a log-log plot in figure 1. Generally, the integral length scale (Λ x ) produced by a turbulence generating grid scales with the mesh width of the grid, i.e. the larger the mesh width the larger the turbulence integral length scale (see, e.g., Kurian & Fransson 2009). From figure 1, using the data from 1940, we may conclude that Hislop was the first one to report that an increase in Λ x moves the transition location farther downstream. The trend is modest, but clearly notable, in particular with the added straight lines in the log-log plot. Note that, the added lines are not curve fits to the data, they are simply added to illustrate the movement of the transition location with the mesh width. More recent investigations have also shown that the level of Tu is not the only dependent variable. An increase in Λ x has shown, both in experiments and numerical simulations (Jonáš, Mazur & Uruba 2000;Brandt, Schlatter & Henningson 2004;Ovchinnikov, Piomelli & Choudhari 2004), to advance the transition location. These results contradict the reported results by Hislop since the opposite effect with respect to the movement of the transition location with increasing Λ x is observed.

Spanwise length scale of the boundary-layer streaks
The mean velocity gradients play a direct role in the production of disturbance kinetic energy, but also for the onset of secondary instabilities. Steady streaks act stabilizing on modal disturbances and can even be used to obtain transition delay; see, e.g., Fransson et al. (2006), Shahinfar et al. (2012), Downs & Fransson (2014), Siconolfi, Camarri & Fransson (2015), Sattarzadeh & Fransson (2016) giving five examples of different ways to set up steady streaks for successful transition delay. Depending on the time scale, even unsteady streaks and their local velocity gradients are likely to influence the stability. Unsteady streaks generated by FST are known to damp the growth of T-S waves (see, e.g., Westin et al. 1994;Liu, Zaki & Durbin 2008) even though an advancement of the transition location has so far only been reported. Apart from T-S waves, the spanwise spreading rate of turbulent spots in an unsteady streaky base flow is also known to be damped (Fransson 2010). Both numerically and experimentally it is known that secondary instabilities are associated with mean velocity gradients, with the spanwise and wall-normal gradient triggering the sinuous and varicose type of secondary instability, respectively (see Andersson et al. (2001), and references therein). Hence, the spanwise scale of the streaks induced by FST is likely to be important for the breakdown to turbulence, since, for a given spanwise scale, the maximum spanwise velocity gradient increases in proportion with u rms in the downstream direction, i.e. with the square root of the downstream distance. If in turn, the spanwise scale depends on the FST condition, the maximum spanwise velocity gradient will change at a constant downstream location and affect the streak instability if the FST condition changes. On the one hand, the spanwise scale of the streaks is often said to adapt to the boundary-layer thickness, giving boundary-layer structures of an aspect ratio of one, after some initial mismatch close to the leading edge (cf. Matsubara & Alfredsson 2001). On the other hand, in Brandt et al. (2004) the authors state that the spanwise streak spacing is only slightly dependent on the FST characteristic. However, in the study by Fransson & Alfredsson (2003) 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 remained the same, giving structures an aspect ratio of two. Besides, it was shown that the spanwise boundary-layer scale of the streaks essentially stays constant for a given FST condition and that the scale is set already in the leading-edge receptivity process. Furthermore, they showed that the spanwise scale of the streaks changed by 60 % between the two grids they used (grids B and E), i.e. that the spanwise boundary-layer scale depends on the FST condition. Referring to these experiments and the experiments by Roach & Brierley (1992) along with their DNS results, Ovchinnikov et al. (2008) suggested that there is no universal value for streak separation in the perturbed boundary layer. Instead, it is rather determined by the FST length scale. A possible reason why many investigators conclude that the spanwise length scale of the streaks is not varying or only slightly varying could be that the spanwise length scale is determined by the FST condition, i.e. both the turbulence intensity and a characteristic length scale in the FST.
The present paper is outlined as follows. In § 2 we present the experimental set-up, measurement technique and base flow. In addition, the new active and old passive turbulence generating grids are characterized and presented, and the transition determination method is outlined. In § 3 we present the experimental results on boundary-layer transition and the streamwise streaks. A new semi-empirical transition prediction model is derived with the usage of a scale-matching hypothesis in § 4. In § 5 we give both quantitative and qualitative comparisons between the experimental data and the new model. In § 6 we conclude the paper. Fransson et al. (2005) used several turbulence-generating grids that gave a wide range of both Tu and Λ x . Besides, their experiments were carried out at different free-stream velocities. The advantage of that reported data is that all data were collected in the same experimental set-up and campaign and that the same transition criterion was applied to the data. Several important conclusions were made, but the influence of Λ x on the transition location remained unclear. The variation discussed in the present § 1.1 was hidden in all other parameters being varied at the same time in that study. One suspected parameter of importance, according to the present authors, is the wall-normal boundary-layer scale, which is inversely proportional to the square root of U ∞ .

Experimental set-up and procedures
The data presented here has been reported in the licentiate thesis, , and presented at the conferences Fransson (2011) andFransson (2017). Various passive and active turbulence generating grids were used here to create different FST conditions. In total 42 unique FST conditions were created and thorough boundary-layer measurements were performed throughout the transitional region. Unlike many other extensive FST induced transition measurements (e.g., Fransson et al. 2005), the free-stream velocity here was kept constant for all cases (U ∞ = 6 m s −1 ), implying that the boundary-layer scale is locked up to transition onset.

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 m × 1.2 m (height × 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 ±0.05 • C. At the nominal speed of 25 m s −1 , the streamwise velocity fluctuation level is less than 0.025 % (Lindgren & Johansson 2002). 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. (1994), 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 μm 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 μm 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 (c) Integral boundary-layer parameters: shape factor H 12 (Δ), displacement thickness δ 1 (♦) and momentum thickness δ 2 (∇). Solid and dashed lines correspond to the Blasius boundary-layer solution.
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 1982) 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 0-7 m s −1 . The X-probe was calibrated at different angles to the flow direction between −30 • and +30 • with a step of 5 • 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.

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.

Free-stream turbulence conditions
Free-stream turbulence can be characterized by its intensity (Tu) and length scales, such as the integral (Λ), Taylor (λ 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 ∞ , but the transition location is a function of all parameters. Non-dimensional parameters will be defined later using the integral length scale, U ∞ and u rms , namely Re FST and Re Λ , 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 (Re FST , Re Λ , 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 Λ x and Λ 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 (2009), 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 Λ x and are therefore properly defined below: is the autocorrelation function of the velocity-time signal at the position of the leading edge of the flat plate, and 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 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 (σ ). In general, the higher the solidity the higher the turbulence intensity at the leading edge for a given X 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 (1974) and has since been applied successfully in other experiments (Fransson & Alfredsson 2003;Yoshioka, Fransson & Alfredsson 2004;Fransson, Matsubara & Alfredsson 2005). 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. (2005), 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 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. 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 (%), Λ x : 16.05-25.61 (mm) and λ x : 11.97-15.02 (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 ΔU ≤ ±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.

Method of transition location determination
The transition location is determined as the position where the intermittency factor (γ ) 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 ≈ 1.3δ 1 ), using the method presented in Fransson et al. (2005). In figure 4, γ -distributions of grid G1 are shown for both increasing injection rate (a) and decreasing X grid (b). In both cases, Tu is increasing while Λ x is slightly decreasing. Dashed and dotted lines indicate the values of γ = (0.1, 0.5, 0.9) and their intersection values with the γ -distribution where γ = 0.5, respectively. The latter being indicated with big square x-X grid (mm) x-X grid (mm)  symbols and give locations of transition x tr ≡ x γ =0.5 , from which the transitional Reynolds number (Re tr ) is calculated according to where U ∞ is measured at the leading edge. From these distributions, the transition region is defined for each case as Δx tr = x γ =0.9 − x γ =0.1 . For improved accuracy of all x-location determinations (x γ =0.1 , x γ =0.5 , x γ =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 grid changes.

Boundary-layer streaks and laminar-turbulent transition
A well-cited result by Matsubara & Alfredsson (2001) is that the averaged spanwise scale of the streaks asymptotes, with the downstream distance, to the value λ z /2 ≈ 3δ 1 . Even if the result is right, it is unfortunate, since it often leads to the conclusion that no matter what the FST condition is, the same spanwise scale is eventually obtained. This is a puzzling result if one believes that the streaks are important for the transition and breakdown to the turbulence process. In this case, the FST characteristics should have an influence on the spanwise scale, since both Tu and Λ x have an influence on the transition location. In this section new boundary-layer transition results are presented with a particular focus on a twofold effect of Λ x on the transition location ( § 3.1). It is also shown that the spanwise scale of the streaks is constant throughout the transition process and that the scale correlates with the FST condition ( § 3.2).

Transition region and location
The intermittency distribution derived by Johnson & Fashifar (1994) has previously shown to agree well with experimental boundary-layer FST data (see, e.g., Fransson et al. 2005). Their γ -function, given as 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 Λ x . Here, the dimensionless streamwise coordinate ξ is defined as where 'tr', in turn, was defined in § 2.4. Note also, that (3.1) is only valid in the range −A 2 ≤ ξ < ∞. 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. 2005), suggesting that the exponent m is actually greater than 3 as derived by Johnson & Fashifar (1994). Performing a straight forward unphysical least-square fit of (3.1) to the present data, by the inclusion of the exponent, gives (A 1 , A 2 , m) = (0.023, 1.87, 5.47), which corresponds to the solid line in figure 5. In figure 6 the experimental intermittency distributions versus ξ 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.
The classical way to plot Re tr is versus Tu, which is known to have a strong influence on the transition location. However, as shown by Jonáš et al. (2000), Re 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 where where B 1 and B 2 are constants and become (B 1 , B 2 ) = (148, 31 956) when least-square fitted to the data. Here 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. (1999), can be used for the variable Re FST and, hence, the curve in figure 7(b) corresponds to where C 1 and C 2 are determined to (C 1 , C 2 ) = (7.6961 × 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. 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 B 2 with 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 Re tr below which natural transition cannot take place. It becomes clear that B 2 and C 2 will only correspond to each other if additional data at much higher Tu and Re FST values are included in the fits. The present data were collected with the mind of creating a large data set, which was done by changing both Tu and Λ 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 Λ x . If one does that for different Tu levels our data show that, for low turbulence intensities, the transition location moves upstream with increasing Λ x and downstream with increasing Λ x for high turbulence intensities. That is, there is a twofold effect of Λ x on the transition location, which confirms the findings reported both by Jonáš et al. (2000), Brandt et al. (2004), Ovchinnikov et al. (2004) (of transition advancement) and Hislop (1940) (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.

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 (2001) 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 (2003) 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 λ z /2 ≈ 3δ 1 from Matsubara & Alfredsson (2001).
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 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 (Δz) between the two hot-wires where a negative minimum in R uu (Δz) appears can be interpreted as the averaged half spanwise wavelength of the streaky structures (λ z /2). For each function, this minimum is consistently determined by a third-order polynomial fit to the data, and to obtain λ z for a specific intermittency value simple linear interpolation using the five streamwise locations is applied. Figure 10(a) shows λ z unscaled for γ = 0.5 versus its corresponding Tu & Λ x and it shows that the data is scattered and uncorrelated with Tu and Λ x , it appears to be a range of possible λ z values for each Tu and Λ x . However, when scaled with the leading-edge FST condition using both Λ x and Tu, and plotted versus the primary variable Re FST as in figure 10(b), the data fall close to a single curve. Here, the spanwise wavelength for γ = 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) with (D 1 , D 2 ) = (186, 0.8). That is, λ 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 λ z using (3.7) stays within ±10 % of the measured value. Another evaluated quantity is the longitudinal integral length scale inside the boundary layer (Λ BL x , where 'BL' denotes boundary layer), which is a measure of how long the structures are in the streamwise direction. Here Λ BL x is calculated at the position above the wall corresponding to maximum u rms . Figure 11 colour brightness corresponds to the turbulence intensity at the leading edge, the darker the higher. A direct conclusion from this figure is that Λ BL x asymptotes to a constant value (just below 10 mm) in the turbulent region, i.e. for ξ > 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 ξ = −0.5, i.e. around the onset of transition where γ 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 Λ BL x (around ξ = −0.5) is plotted in figure 11 (b). No satisfactory correlation function was found with Re FST as the primary variable as was found for a dimensionless λ z (cf. figure 10b), but there is a clear trend of shorter streaks as Re FST is increased. Instead, in figure 11(c) Λ BL x,max is normalized with L (3.7) and Tu, which makes the data points fall close to a straight line when plotted against Re tr . The length scale ratio (Λ BL x,max /L) corresponds to the streamwise/spanwise aspect ratio at the maximum streak length and can be estimated from known inlet FST conditions as Λ BL x,max where (E 1 , E 2 ) = (5.293 × 10 −4 , −15.75) corresponding to the straight line in figure 11(c). The missing parameter to assess the aspect ratio is the parameter Re tr , which can be predicted for the given leading-edge FST parameters Re FST and Re Λ , as will be shown in § 4.

A semi-empirical transition prediction model
Recent results have shown a clear dependence of Λ x on the transition location, moving the transition location upstream with increasing Λ x . The present results confirm this result but also show that the transition location can move downstream with increasing Λ x , in agreement with earlier results by Hislop (1940). That is, the effect of Λ x can be twofold. There are numerous simple empirical correlation functions available that have been published over the years, based on different experimental data sets, but none of them includes Λ x in a way such that the twofold effect can be predicted. The existing experimental data, taken in the same experimental set-up under the same conditions, using the same transition determination method, has so far been too scarce and not enough to find the appropriate behaviour of Λ x when other parameters such as Tu and the vertical boundary-layer scale (i.e. U ∞ ) are varied. Apart from the leading-edge FST parameters Λ x (2.1), Re FST (3.3) and Re Λ (3.4), where the latter two are related to each other via Tu (2.2), there is another important parameter that provides the missing link to the boundary layer, namely the vertical boundary-layer length scale, at transition. The only shortcoming is that this parameter includes x tr , which is the one that is in demand.

Scale-matching model
The present experimental data indicates that there is a change of trend of the influence of Λ 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 (2009), or the present figure 3b) which in turn suggests that the ratio Λ x /δ 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 (Λ x /δ tr ) 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 Re 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 (2017), 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 Λ x need to be carried out, the present database is not sufficient since both Tu and Λ x were varied. However, the present data seem to indicate that (Λ x /δ tr ) opt is close to 15, which would imply that the change of trend appears where Λ x is about three times the boundary-layer thickness, i.e. δ 99 , at transition. This means that if Λ x < 3 δ 99 at transition, the transition location is expected to move upstream, and vice versa, with increasing Λ x . there exists an optimal Λ x /δ tr that promotes transition to a lowest possible Re tr .

The equation for the transitional Reynolds number
The effect of Λ x /δ tr on Re tr is assumed to enter as a correction to (3.6) and, hence, we make the following ansatz: where i.e. we assume that the correction term ΔRe 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), ΔRe 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 Re tr , (4.5) and κ 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 (Λ x /δ tr ) 2 cf in (4.4) as Now, by combining (4.2) and (4.4)-(4.6), and multiplying with Re tr the following second-order equation of Re tr can readily be derived: with the solution When the scale ratio (Λ x /δ tr ) 2 corresponds to (4.6), the Reynolds number correction (4.4) is zero and the solution is a double root corresponding to Re tr = (Re tr ) 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 α and β depend on the parameter Re Λ and the weighting function κ. While Re Λ is considered known from the FST condition at the leading edge, κ requires further analysis.

Weighting function
The weighting function κ is introduced in (4.4) and corresponds to (4.9) which may be rewritten using (4.5), (4.6) and (3.3) to This expression of κ needs to be modelled since it includes Re tr and renders a trivial solution when inserted to (4.7). Besides, it is noted that κ is always negative. For the modelling, κ is plotted versus Re 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 Re FST → 0 and Re FST → ∞ the function should approach zero. The physical reasoning behind this is that ΔRe tr should approach zero in these two limits.
For Re FST → ∞, the transitional Reynolds number will approach the minimum Reynolds number for self-sustained turbulence (cf. B 1 in (3.6)) and the dependence of Λ x will diminish. For Re FST → 0, the FST transition scenario will eventually be replaced by the T-S wave transition scenario and the effect of Λ x is expected to disappear. However, it is worth pointing out that independent of an accurate representation of κ the transition prediction model described by (4.7) is only valid for the transition scenario dominated by FST. Here, the κ function is created in order to fulfil the above constraints with a good fit to the data according to (4.11) 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 κ to fulfil the constraint in the limit for Re FST → 0 is around Re FST = 73. This value of Re FST corresponds to Tu = 1 %, which is obtained through the relation derived by combining (3.5) and (3.6). The value of Tu ≈ 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 1978). The five coefficients in (4.11) are determined to be a = 7.2279 × 10 1 ; b = 1.7574 × 10 0 ; c = 3.9519 × 10 −2 ; d = 5.6340 × 10 −5 ; e = 3.1987 × 10 2 . (4.13) 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 κ function. The grey areas in figure 13 indicate Re FST values being lower than Tu = 1 %. On the one hand, there are too few data points at low Re FST to comment on the behaviour of κ in the lower range of Re FST , on the other hand, the data do indicate that the effect of Λ x on Re tr only becomes greater as Re FST is reduced. The physical constraint that the effect of Λ x eventually has to diminish as Re tr → 0 gives rise to the sharp turnaround of the function. Figure 14 shows a direct comparison between the experimental and predicted Re tr as a function of Re Λ and Tu. Noteworthy is that the only input to the model is leading-edge data with two out of the following three FST parameters: Re Λ , Tu and Re FST (which are related to each other in (3.3)). Practically, the model requires a single-point time-resolved velocity measurement in the free stream at the leading edge. Quantitatively, the agreement is satisfactory between figures 14(a) and 14(b). The circular symbols correspond to the actual input FST data, where the corresponding measured or predicted Re tr are used to produce the contour plots. Even though figure 14(a) shows a quite complex behaviour of Re tr with a clear dependence on both Tu and Λ x , our semi-empirical model clearly captures the essential physics with the introduced scale-matching model, which links the FST to the boundary layer.

Validation of the model
In figure 15 a qualitative comparison between experimental and predicted Re tr values is made. From figure 15(a), it is clear that (4.7) performs very well for Re FST > 325, while in the range below 325 one may foresee improvements given more experimental data in the future to tune the model. However, one may conclude that the model does significantly better than simple empirical correlations based on only Re FST or Tu, which completely excludes the effect of Λ x on Re tr . Figure 15(b) quantifies this statement, by showing that the predicted value using the scale-matching model (4.7) is significantly improved with respect to the usage of (3.6). Using (4.7) for predicting the transitional Reynolds number gives 65 % of the values within ±10 % of the experimental data, while (3.6) gives 40 % within ±10 %. Less than 10 % of the data is outside ±20 % using (4.7), while (3.6) gives 31 % of the data worse than ±20 %. Figure 16 shows a classical scatter plot of the same data as shown in figure 15(b) with the corresponding R 2 coefficients given in the caption.
At last, we turn our attention back to figure 8, where the effect of Λ x on Re tr was illustrated for five different Tu levels. The grey symbols in this figure correspond to the predicted Re tr using (4.7) and again a qualitative comparison is made but with the specific cases denoted in the figure. From this comparison, it is confirmed that the prediction is better for higher Re FST values (i.e. in general for higher Tu), as shown in figure 8(b). However, it is also clear that the twofold effect of Λ x on Re tr is captured by the model, even though the influence of Λ x is slightly underpredicted for low Tu values (cf. figure 8a). Going back to the scale-matching hypothesis (figure 12) it is also clear that if one does not accurately capture (Λ x /δ tr ) opt from the data when tuning the model it is easy to underpredict or overpredict the influence of Λ x . From figure 8, it seems that the behaviour for low Tu levels would be predicted even better if the curve in the sketch (cf. figure 12) was shifted to the right. Now, with this information at hand, having data taken under constant Tu levels while varying Λ x instead of scattered data in the (Re Λ , Tu) plane (as the present data), would be valuable for future tuning of the model.
In figure 17 the semi-empirical model (4.7) has been used to produce the contour plots of Re tr in (a)  range of Re Λ and Tu, figure 17(b) shows Re tr predicted in the Re Λ -Re FST plane. Since the behaviour of Λ x on Re tr has so far not been known, except for the Λ x variation for a particular Tu level that has been published (see, e.g., Jonáš et al. 2000), the figures 17(a) and 17(b) are the first ones ever produced for a larger variable range showing how Re tr is affected by the FST condition based on the present experimental data.

Conclusion
Ideally, we would like to be able to predict the transition location in a boundary layer with the only knowledge being the leading-edge FST condition. However, today no simple correlation function exists that includes the effect of FST length scales on transition in a physically correct way. The underlying problem is that none of the existing prediction functions, so far, links the FST condition to the boundary layer. Reviewing the existing literature on the effect of FST length scales one finds that the FST integral length scale has a twofold effect on transition, i.e. an increase of Λ x can both lead to advancement and delay in transition. It is obvious that empirical correlation functions only including the turbulence intensity, as most past empirical relations, and that relations with a linear Λ x dependence, as more recent published relations, will do a poor job in predicting the transition.
Here, we present data from a large wind tunnel campaign aimed at bringing further insights to the FST induced boundary-layer transition scenario. It is a zero-pressure gradient boundary layer subject to different FST conditions at the leading edge that is in focus. The FST was varied by means of eight different turbulence generating grids of different mesh and bar widths. Six of these grids were active in the sense of having upstream injected jets of high speed to increase the pressure drop over the grids, which in turn increases Tu. In total, data from 42 unique FST conditions have been acquired and documented in the present paper (cf. tables 2 and 3). The FST is known to create unsteady elongated streamwise streaks of alternating high-and low-speed regions inside the boundary layer, via some receptivity mechanism which takes place already at the leading edge. The averaged spanwise wavelength of these structures is likely to be set already in the leading-edge region and to be critical for the breakdown to turbulence process.
Our data confirm the previously reported universal intermittency distribution when the non-dimensional streamwise coordinate ξ is used (cf. figure 5). This distribution as a function of the dimensional streamwise coordinate, x, is used to determine the transition location for each case. Here, x tr is consistently determined as to where the intermittency factor of the velocity signal is equal to 0.5 (measured around the u rms peak above the plate).
The present dataset is the first one indicating the twofold effect of Λ x in the same experimental set-up and campaign. On the one hand, it is shown that by increasing Λ x by 12 % the transitional Reynolds number is reduced by 38 % at Tu ≈ 2.16 % (from case C6 to C33). On the other hand, an 18 % increase of Λ x at Tu ≈ 2.82 % results in an increase of Re tr by 45 % (from case C32 to C25).
To characterize the FST condition we establish three FST parameters, namely Re FST , Re Λ and Tu, which are all related to each other (cf. (3.3)). By introducing a scale-matching model the physically correct behaviour of Λ x on Re tr can be obtained. In this analysis the primary variable in predicting Re tr turns out to be Re FST and not Tu, as used in most correlation functions. Dealing with the data as a function of Re FST completely reorders the data and allows for the introduction of a ΔRe tr (4.2)-(4.4), which is a function of the scale ratio Λ x /δ tr that links the FST length scale with the boundary layer. A similar analysis with Tu as the primary variable was carried out, but without success. The key ingredient in being able to predict Re tr is the scale-matching model, which leads to a second-order equation of Re tr with Re FST being the primary variable and Re Λ being a parameter in the equation (cf. (4.7)). This equation captures the twofold behaviour of Λ x on Re tr , and gives a significantly improved prediction of Re tr with respect to a single variable prediction, be it Re FST or Tu. The scale-matching hypothesis leads to the threshold of (Λ x /δ tr ) opt and the data suggest that this value is close to 15, which in turn leads to the conclusion that when Λ x < 3δ 99 at transition, the transition location will move upstream, and vice versa, with increasing Λ x .
The data also indicate that both the averaged spanwise wavelength of the streaks, λ z , as well as the maximum streamwise length of the unsteady streaks, Λ BL x,max , correlate with the FST parameters. This finding is important since quite often the streak spacing is said to adapt to the boundary-layer thickness giving structures of an aspect ratio of one inside the boundary layer (after Matsubara & Alfredsson 2001), which can be used to diminish the role of the streaks in the transition process. Here, we argue that the asymptotic aspect ratio of the streaks, after the adaptation, is not important for the transition, since for most cases (moderate to high Tu levels) transition occurs upstream of this asymptotic state.
Our data show that λ z is proportional to both Λ x and Tu, with the correlation function being a function of Re FST (cf. (3.7)). The maximum streamwise extent of the streaks (around γ = 0.1), Λ BL x,max , is proportional to Λ x , Tu 2 and Re tr , which implies that the transition location needs to be estimated before the streamwise length of the streaks can be estimated (cf. (3.8)). These length scales are believed to be important for the streak instability. In particular, the former length scale, which together with the streak amplitude gives a measure of the spanwise velocity gradient that drives the sinusoidal secondary instability. Here, believed to be the most common precursor to the local breakdown into turbulent spots inside the boundary layer. With this in mind, it is also likely that a universal transitional number based on local flow properties inside the boundary layer exists, which so far has not been recognized.
Finally, the semi-empirical transition prediction model presented herein captures the twofold effect of Λ x on Re tr in a correct way, and we can, for the first time, get the full behaviour of Re tr in the Tu-Re Λ and Re FST -Re Λ planes based on the model. Theoretically, one would only need to perform a single-point time-resolved velocity measurement in the free stream at the leading edge in order to predict the transition location using the new model. From this velocity signal, U ∞ , u rms and Λ x can be obtained, which are required for calculating the non-dimensional FST parameters Re FST , Re Λ and Tu. In reality, there is primarily one additional parameter that plays a role for the transition location, namely the leading-edge pressure gradient, which apart from the leading-edge geometry is given by the leading-edge stagnation line. However, with increasing turbulence intensity the leading-edge pressure gradient effect is expected to diminish. Other parameters related to the wind tunnel set-up, such as plate joints and background acoustic noise are not expected to play a significant role as long as joints are kept smooth without steps. Acoustic noise can interact with the leading edge and two-dimensional plate joints to generate T-S waves but is only expected to play a role when the turbulence intensity is very low (i.e. Tu 1 %). For moderate to high turbulence intensities, the T-S waves are likely to be damped by the unsteady streamwise streaks and can be considered harmless for the transition since the scenario is dominated by the growth of streamwise streaks. The present authors believe that the two most important reasons for the large discrepancy of experimental data acquired in different facilities, when it comes to Re tr = Re tr (Tu), are as follows: (1) different methods in determining the transition location; and (2) past lack of knowledge regarding the significance of Λ x on the transition location.