3.1 Instantaneous flow structures

Owing to the large growth rates associated with the separated shear layer, the disturbances can be amplified by several orders of magnitude, which leads to the development of large unsteady vortical structures that can persist even in the turbulent flow region downstream of the reattachment location (coherent structures). Furthermore, FST can induce additional disturbance waves inside the boundary layer, which contribute to the laminar–turbulent transition process.

Figure 5.

Instantaneous flow visualizations for the case without FST. Plotted are isosurfaces of $\unicode[STIX]{x1D706}_{2}=-0.75$ coloured by $u$ -velocity in perspective and top-down views. The separation line is indicated by $\unicode[STIX]{x1D714}_{z}=0$ at the wall and $x_{R}$ shows the mean reattachment location. Plotted in the $x$ – $y$ plane are the contours of spanwise vorticity (averaged in spanwise direction).

To gain insight into the nature of the vortical structures and their evolution, the instantaneous isosurfaces are visualized in figure 5 using the $\unicode[STIX]{x1D706}_{2}$ -criterion by Jeong & Hussain (Reference Jeong and Hussain1995) coloured by $u$ -velocity together with the contours of spanwise vorticity (averaged in spanwise direction) for a zero FST. The separation line, $\unicode[STIX]{x1D714}_{z}=0$ , is also indicated at the wall by the solid line. Starting from the left in figure 5, the boundary layer separates at $x_{s}=4.26$ , followed by a very smooth separated shear layer, which has no noticeable dependence in the spanwise direction. The thickness of the separated shear layer increases downstream of the separation location. Through the contours of spanwise vorticity, it is found that the separated shear layer rolls up and grows in intensity due to the KH instability as it is convected downstream. This is followed by shedding of strong coherent (clockwise-rotating) vortices (often referred to as ‘rollers’), and finally leading to turbulent reattachment. It should be noted that in our previous research (see Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2013 a ) for the same simulation set-up, we found that transition to turbulence appeared to be self-sustained for zero FST, i.e. in contrast to zero-pressure-gradient boundary layers, no external forcing was required to initiate or sustain the transition process. It was demonstrated that the development of 3D disturbances is due to an absolute secondary instability.

The $\unicode[STIX]{x1D706}_{2}$ -criterion indicates large spanwise-oriented structures near $x=9.4$ , labelled 1 in figure 5. It corresponds to the spanwise rollers or KH vortices. Previous research (Marxen et al. Reference Marxen, Lang, Rist and Wagner2003; Rist Reference Rist2003; Postl et al. Reference Postl, Balzer and Fasel2011; Hosseinverdi et al. Reference Hosseinverdi, Balzer and Fasel2012) suggests that these structures result from the large-amplitude 2D disturbances associated with the primary (KH) instability of the streamwise mean flow profile. Large-scale vortices facilitate the transfer of high-momentum fluid from the free stream towards the wall and of low-momentum fluid away from the wall, and, as a consequence, accelerate reattachment. In figure 5, the structure labelled 2 is a spanwise dominant vortex that has recently been shed from the separated shear layer. Further downstream, the flow breaks down into smaller structures. It can be seen that the small scales are still organized in a spanwise coherent structure, labelled 3. These coherent structures persist even downstream of reattachment.

The picture changes when isotropic FST is introduced at the inflow boundary. Hydrodynamic instability of laminar separated shear layers makes the LSB highly susceptive to FST. Therefore, the onset of transition is moved upstream compared to the zero FST when the FST level is increased to $Tu_{0}=0.5\,\%$ (see figure 6), which in turn leads to a reduction of the separated flow region. Instantaneous isosurfaces of the $\unicode[STIX]{x1D706}_{2}$ -criterion are presented in figure 6 together with the contours of streamwise and spanwise vorticity at several selected planes. One KH roller is visible before the flow breaks down to smaller scales. For this case, the reattaching flow is organized by the vortices coherent in the spanwise direction. Predominantly spanwise structures are indicative of the dominant 2D waves in the transition region and highlight their role as a viable path to turbulence.

Judging from figures 5 and 6, KH vortices are more 2D for case with $Tu_{0}=0.5\,\%$ than for zero FST simulation. As mentioned earlier, transition to turbulence is due to an absolute secondary instability for the natural LSB (zero FST), implying that the disturbances (due to numerical and round-off errors) propagate in both the downstream and upstream directions while being amplified in time and ultimately result in vortex shedding leading to transition. A similar observation was reported in Jones et al. (Reference Jones, Sandberg and Sandham2008), where ‘rollers’ are not purely 2D for uncontrolled LSB. The onset of unsteadiness was also due to an absolute secondary instability in their work. In contrast to the disturbance-free simulation (zero FST), the amplitude level of the incoming 2D disturbance waves is elevated in the presence of FST (orders of magnitude higher than in the zero FST case). Considering that 2D modes are the most amplified instability waves in low-disturbance environments (Marxen et al. Reference Marxen, Lang, Rist and Wagner2003, Reference Marxen, Rist and Wagner2004), the primary convective instability strongly (by orders of magnitudes) amplifies the disturbance waves within the KH frequency range of the separated shear layer leading to the organized spanwise structures for $Tu_{0}=0.5\,\%$ .

Figure 6.

Isosurfaces of $\unicode[STIX]{x1D706}_{2}=-0.75$ coloured by $u$ -velocity in perspective and top-down views for the case with $Tu_{0}=0.5\,\%$ . Plotted in the $x$ – $y$ plane are the contours of spanwise vorticity (averaged in the spanwise direction). Top-left: contours of streamwise vorticity at $x=3$ and 7 and wall streamwise vorticity (dashed contour lines indicate $\unicode[STIX]{x1D714}_{x}<0$ ). Contour line of $\unicode[STIX]{x1D714}_{z}=0$ corresponds to the instantaneous separation line and $x_{R}$ shows the mean reattachment location.

From experiments and numerical simulations (Klebanoff & Tidstrom Reference Klebanoff and Tidstrom1959; Klebanoff Reference Klebanoff1971; Jacobs & Durbin Reference Jacobs and Durbin2001; Fasel Reference Fasel2002; Goldstein Reference Goldstein2014), it is known that the FST can cause the formation of streamwise-elongated streaks inside the boundary layer that are caused by the so-called Klebanoff modes (K-modes). For this case, the Klebanoff modes have very small amplitudes and can be visualized by looking at the streamwise vorticity component as shown in the top-left plot in figure 6 at two different $x$ locations and on the wall. The contours of streamwise vorticity at $x=3$ show that the vortical disturbances are present in the free stream. The free-stream vortical disturbances permeate into the boundary layer and manifest themselves as the counter-rotating longitudinal vortices as illustrated by the streamwise vorticity at the wall with a preferred spanwise wavelength of $\unicode[STIX]{x1D706}_{z,K}\approx L_{z}/5$ , which can be clearly observed for example at slice $x=7$ . These counter-rotating vortices facilitate the exchange of momentum by transporting the low-momentum fluid away from the wall and pushing the high-momentum fluid towards the wall. Furthermore, the spanwise spacing of Klebanoff modes based on the local boundary-layer thickness, $\unicode[STIX]{x1D6FF}$ , is of the order of $\unicode[STIX]{x1D706}_{z,K}\approx O(2.65\unicode[STIX]{x1D6FF}{-}4\unicode[STIX]{x1D6FF})$ for $3\leqslant x\leqslant x_{s}$ , which is consistent with the findings of $\unicode[STIX]{x1D706}_{z,K}\approx O(2\unicode[STIX]{x1D6FF}{-}4\unicode[STIX]{x1D6FF})$ for attached flat-plate boundary layers (Kendall Reference Kendall1985, Reference Kendall1990; Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994). Contrary to the case with zero FST, the boundary-layer fluctuations associated with the K-mode have a local effect on the instantaneous separation line as it is now modulated in the spanwise direction.

Figure 7.

Isosurfaces of $\unicode[STIX]{x1D706}_{2}=-0.75$ coloured by $u$ -velocity in top-down view for the case with $Tu_{0}=3\,\%$ . The mean reattachment location is depicted by the arrow. The instantaneous separation line is indicated by $\unicode[STIX]{x1D714}_{z}=0$ at the wall.

Figure 8.

Time- and spanwise-averaged (a) wall-skin friction and (b) pressure coefficients for various levels of FST intensity.

From instantaneous flow visualization for the highest FST intensity investigated ( $Tu_{0}=3\,\%$ ) shown in figure 7, it appears that the separated shear layer loses its predominantly 2D character and exhibits oblique spanwise-oriented structures that deteriorate due to the large-amplitude streamwise streaks. This gives room to speculate that high FST intensities enhance the level of 3D disturbances and that these eventually dominate the 2D waves. The traces of K-modes could also be identified in the $\unicode[STIX]{x1D706}_{2}$ structures as the K-modes become stronger for elevated FST levels.

3.2 Time- and spanwise-averaged flow field

In this section, the mean flow results for different FST intensities are compared and discussed. Towards this end, the DNS data were averaged in the spanwise direction as well as in time. Boundary-layer quantities from DNS, such as displacement thickness and momentum thickness, are computed based on a ‘pseudo’-free-stream velocity. The pseudo-free-stream velocity was obtained from a wall-normal integral of the spanwise vorticity, $\unicode[STIX]{x1D714}_{z}$ , as suggested by Spalart & Strelets (Reference Spalart and Strelets2000).

Distributions of time- and spanwise-averaged wall skin-friction coefficients, $c_{f}$ , and wall-pressure coefficients, $c_{p}$ , are plotted in figure 8 for DNS with various FST intensities. With increasing FST levels, the mean reattachment point, at which $c_{f}$ changes from negative to positive, moves upstream. The separation location, where $c_{f}$ changes from positive to negative, remains almost constant, although a general trend can be observed that separation is slightly delayed for increased FST levels (see inset in figure 8 a). The wall-pressure coefficient versus $x$ shows a boundary-layer acceleration up to $x=3.3$ , which is the location of the minimum pressure. There is a plateau downstream of $x=5$ , which is an indication of the ‘dead-air’ region in the separation bubble (corresponding to the low velocities within the bubble). Included in figure 8(b) is the inviscid $c_{p}$ distribution. It is difficult to classify the LSB investigated here as either ‘short’ or ‘long’ according to the Gaster criterion (Gaster Reference Gaster1969), since the momentum thickness Reynolds number at the separation location, $Re_{\unicode[STIX]{x1D703},s}\approx 83$ , is below the minimum $Re_{\unicode[STIX]{x1D703},s}$ used in his investigations. However, from figure 8(b), it is clear that the LSB alters the inviscid pressure distribution significantly. It can therefore be concluded that the LSBs, for all levels of FST examined, can be classified as ‘long’.

Figure 9.

Comparison of boundary-layer parameters (time- and spanwise-averaged) for different FST intensities: (a) displacement thickness; and (b) momentum thickness.

The development of the boundary layer for different flow scenarios is compared in figure 9 by plotting the streamwise variation of the displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }$ , and the momentum thickness, $\unicode[STIX]{x1D703}$ . Downstream of $x=4$ , the rapid growth of the displacement thickness corresponds to the onset of the boundary-layer separation. From the displacement thickness distribution, it is found that, in addition to the reduction in separation bubble length, the height of the separation bubble also decreases with increasing FST intensity. While for the zero and low FST levels, the momentum thickness varies only slowly downstream of separation and across the stagnant flow region of the bubble, for the cases with higher FST levels, there is a rise in $\unicode[STIX]{x1D703}$ right downstream of the separation location. This change in the momentum thickness could be attributed to the spanwise redistribution of momentum within the boundary layer due to the FST. Another observation is the rapid rise in $\unicode[STIX]{x1D703}$ corresponding to the ‘reverse-flow vortex’ region which represents the process of shear-layer roll-up and shedding of vortices (in a time-averaged sense).

Figure 10.

Time- and spanwise-averaged $u$ -velocity compared to the experimental data (Radi & Fasel Reference Radi and Fasel2010; Chetan & Fasel Reference Chetan and Fasel2012) at $y=1$ (a) and several $x$ locations (b).

The time- and spanwise-averages of the streamwise velocity, $u$ , at $y=1$ and selected streamwise locations are compared with the experimental measurements in figure 10. For the comparison, the velocity profiles for cases without FST and FST intensities of 0.1 % and 0.5 % are considered here. It should be noted that the reported level of FST intensity in the experiment is $O(0.1\,\%{-}0.35\,\%)$ . Despite the good agreement for the velocity profiles prior to the separation and the front part of the bubble ( $x\leqslant 7.6$ ), the DNS data for the zero FST simulation indicate that reattachment occurred very far downstream compared to the experiments. Downstream of $x=7.6$ , the computed velocity profiles exhibit a larger distance of the separated shear layer from the wall. The discrepancy between the simulation and the experiments is likely due to an earlier onset of transition in the experiments, which could be attributed to the presence of FST disturbances in the experiments, imperfections of the surface and the like. These disturbances will of course accelerate transition, and, as a consequence, reduce the size of the separation bubble. Moreover, the circulation around the displacement body in the experiments, which could affect LSBs, is not present in the DNS. Such disturbances are not present in the DNS, except for a low level of background disturbances (‘numerical noise’), which can be kept extremely small if, as in our case, high-order-accurate methods are used for solving the Navier–Stokes equations.

For the lowest FST level, $Tu_{0}=0.1\,\%$ , the separation length and the thickness of the separation bubble are already reduced compared to the zero FST case; however, the best match is found for the case with $Tu_{0}=0.5\,\%$ for the profiles at the separation point, in the reverse-flow region and at the reattachment location. It is also worth noting that the discrepancy between the experimental and DNS results could partially be caused by differences in the spectral content of FST fluctuations used in DNS (2.5), as such an FST energy spectrum from the experiments was not available. To reproduce ‘realistic’ separation bubbles as observed in the experiments, it is therefore necessary to increase the background disturbance levels in the DNS to levels that are typical for experiments. As an alternative to introducing specified or random disturbances upstream of the separation, as done, for instance, by Marxen & Henningson (Reference Marxen and Henningson2011), we found it is more physical to introduce a realistic isotropic grid turbulence in the DNS.

The boundary layer downstream of the reattachment location is turbulent. However, it is very different from an equilibrium turbulent boundary layer as observed in the zero-pressure-gradient case. A more revealing look at the redeveloping turbulent boundary layer is provided in figure 11 for cases with $Tu_{0}=0.5\,\%$ and $Tu_{0}=3\,\%$ . Plotted are the time- and spanwise-averaged flow quantities for several downstream locations using the near-wall scaling for turbulent boundary layers. Close to reattachment, the velocity profiles are very different from those of an equilibrium turbulent boundary layer. A slow relaxation of the velocity profiles towards the logarithmic law of the wall (given by the equation $u^{+}=2.44\log (y^{+})+0.5$ ) is observed. It is not until the furthest downstream location that the profiles approach equilibrium. A similar behaviour in the recovery region was reported by Na & Moin (Reference Na and Moin1998) and Alam & Sandham (Reference Alam and Sandham2000). The persistence of the dip below the log law can be linked to the large coherent structures observed downstream of the reattachment location (see figure 6). These structures are embedded in the boundary layer and survive far downstream before breaking up into smaller dissipative eddies. In Alam & Sandham (Reference Alam and Sandham2000) the turbulent boundary layer reached the log law of the wall at $\tilde{x}=24.43$ , while in the present simulations for the case with $Tu_{0}=3\,\%$ , at $\tilde{x}=28$ , the velocity profile is still not in equilibrium. It should be noted that in the investigation of Alam & Sandham (Reference Alam and Sandham2000) the relaxation process took approximately six bubble lengths while in the present simulations for the highest FST level, $Tu_{0}=3\,\%$ (shortest bubble length), the domain downstream of reattachment only extends two-and-a-half times the mean bubble length.

Figure 11.

Streamwise velocity of the turbulent relaxing boundary layer in wall coordinates at various $\tilde{x}$ locations defined as $\tilde{x}=(x-x_{R})/\unicode[STIX]{x1D6FF}_{R}^{\ast }$ where $x_{R}$ and $\unicode[STIX]{x1D6FF}_{R}^{\ast }$ are the mean reattachment location and displacement thickness at $x_{R}$ , respectively: (a)  $Tu_{0}=0.5\,\%$ ; (b) $Tu_{0}=3\,\%$ . Plots are time- and spanwise-averaged results. Dashed lines represent $u^{+}=y^{+}$ and dash-dotted line corresponds to $u^{+}=2.44\log (y^{+})+0.5$ .

The results for mean flow presented in figures 8 and 9 indicated a reduction in separation length for increased levels of the incoming FST. However, the rate at which the separation region decreases as FST intensity increases is different for low and high FST intensities as shown in figure 12. It displays the mean separation length normalized by the momentum thickness at the separation location ( $\unicode[STIX]{x1D703}_{s}$ ) versus the turbulence intensity at the free stream at the separation location ( $Tu_{s}$ ). For the cases with elevated FST, $1\,\%<Tu_{0}\leqslant 3\,\%$ , the mean separation length decreases at a lower rate compared to the cases with low to moderate FST intensities, $0.1\,\%\leqslant Tu_{0}\leqslant 1\,\%$ . Also, included in figure 12 is the variation of the maximum mean bubble height normalized with $\unicode[STIX]{x1D703}_{s}$ as a function of $Tu_{s}$ , which shows a similar behaviour as the mean bubble length versus $Tu_{s}$ .

Figure 12.

Variation of mean separation length, $l$ , and the maximum bubble height, $h$ , as a function of turbulence intensity in the free stream at the separation location, $Tu_{s}$ . Both $l$ and $h$ are normalized by the momentum thickness at the separation location.

From the instantaneous flow visualizations in figures 5–7, it appears that two mechanisms contribute to the exchange of momentum, which therefore play an important role in the reattachment process: (i) large-scale vortices that are being shed from the bubble as a result of the shear-layer roll-up and (ii) small vortical structures that are generated near the wall due to the breakdown of these large-scale structures. The shear-layer roll-up produces large mean tangential momentum ( $\bar{u}\bar{v}$ ) while the breakdown of these large-scale vortical structures (spanwise ‘rollers’), which contributes to the transition/reattachment processes, induces Reynolds shear stress ( $\overline{u^{\prime }v^{\prime }}$ ). According to Satta et al. (Reference Satta, Simoni, Ubaldi, Zunino, Bertini and Spano2007) and Simoni, Ubaldi & Zunino (Reference Simoni, Ubaldi and Zunino2014), the negative values of the wall-normal derivatives of these components will accelerate the flow in the $x$ -direction and therefore facilitate the reattachment process.

Figure 13 displays the contours of $-\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y$ and $-\unicode[STIX]{x2202}(\overline{u^{\prime }v^{\prime }})/\unicode[STIX]{x2202}y$ together with the time- and spanwise-averaged streamlines for different FST levels. It should be noted that contour levels are $0\leqslant -\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y\leqslant 0.4$ and $0\leqslant -\unicode[STIX]{x2202}(\overline{u^{\prime }v^{\prime }})/\unicode[STIX]{x2202}y\leqslant 0.15$ in figure 13. The wall-normal derivative of $\bar{u}\bar{v}$ acts mainly outside the separation bubble and reaches a maximum in the aft part of the bubble, the location where the separated shear layer sheds spanwise vortices (see figure 6). In contrast, an appreciable magnitude of the wall-normal derivative of the Reynolds shear stress is present inside the separated region and attains the maximum value further downstream, near the reattachment location, and is mainly confined close to the wall. Included in figure 13(a) are the loci of the (mean) inflection points. Of particular interest is the observation that the maxima of $-\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y$ occur very close to the mean inflection points, thus supporting the conjecture that the main contribution to this term is the inviscid shear-layer instability that produces the boundary-layer roll-up. Besides the spanwise ‘rollers’, it is important to point out that the term $-\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y$ contains (small) contributions from the fluctuations in the form of the Reynolds stresses as well. While the maximum of $-\unicode[STIX]{x2202}(\overline{u^{\prime }v^{\prime }})/\unicode[STIX]{x2202}y$ remains almost unchanged, the maximum of $-\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y$ is significantly reduced for cases with $Tu_{0}>1\,\%$ . This reduction can be attributed to: (i) a stabilization of the convective instability of the separated boundary layer associated with the smaller mean LSB (due to the faster transition), and (ii) weakening of the spanwise-oriented structures by large-amplitude K-modes as was observed in figure 7, for example.

Figure 13.

Contours of $-\unicode[STIX]{x2202}(\bar{u}\bar{v})/\unicode[STIX]{x2202}y$ (a) and $-\unicode[STIX]{x2202}(\overline{u^{\prime }v^{\prime }})/\unicode[STIX]{x2202}y$ (b) together with time- and spanwise-averaged streamlines. From top to bottom: $Tu_{0}=0$ , 0.1 %, 0.5 %, 1 %, 2 % and 3 %. The symbols in (a) indicate the loci of mean inflection points and the dashed red lines in (b) correspond to the mean displacement thickness.

3.3 Spectral analysis

To analyse the unsteady fluid dynamics and understand the dominant laminar–turbulent transition mechanisms for different FST levels, the Fourier spectra of the time-dependent velocity field in time and spanwise directions are evaluated. Note that the results obtained from the simulations are already in Fourier space in the $z$ -direction. Hence, no transform in the spanwise direction has to be performed.

For zero FST simulation, typical time traces and the respective frequency spectra of the wall-normal disturbance velocity (averaged in spanwise direction) are presented in figure 14 for several streamwise locations within the bubble and downstream of the reattachment location. The data are extracted along the local displacement thickness, which is very close to the inflection point of the streamwise velocity component. Here, the dimensional frequency is non-dimensionalized as follows:

(3.1) $$\begin{eqnarray}F=\frac{2\unicode[STIX]{x03C0}\,f^{\ast }\unicode[STIX]{x1D708}^{\ast }}{{U_{\infty }^{\ast }}^{2}}\times 10^{6}.\end{eqnarray}$$

Downstream of the separation point at $x=8.5$ , the time signals are reminiscent of ‘modulated wave-trains’ qualitatively similar to those observed by Schubauer & Skramstad (Reference Schubauer and Skramstad1948) in their ‘natural’ transition experiments of a flat plate boundary layer. The corresponding frequency spectra (figure 14 b) exhibit a dominant frequency peak, which is caused by the growth of instability waves in the separated region. These instabilities lead to the ‘periodic’ shedding of the separation bubble with fundamental frequency of $F\approx 954$ . Strong higher harmonics that are generated nonlinearly can be observed for $x\leqslant 10$ .

Figure 14.

Temporal evolution (a) and the corresponding frequency spectra (b) of the wall-normal disturbance velocity (averaged in the spanwise direction) at various streamwise locations along the displacement thickness for zero FST simulation. Curves from left to right (bottom to top): $x=8.5$ , 9, 9.5, 10, 10.5, 11, 12 and 13.

Figure 15.

Spectral analysis for $Tu_{0}=0.5\,\%$ . (a) Fourier spectra of the wall-normal disturbance velocity versus downstream direction along the displacement thickness. (b) Wall-normal maxima of Fourier amplitude of the streamwise disturbance velocity inside the boundary layer versus downstream direction.

Contours of the Fourier amplitude of the wall-normal disturbance velocity in the frequency $F$ – $x$ plane are shown in figure 15(a) for the case with FST intensity of 0.5 %. The Fourier amplitudes are averaged in the spanwise direction and the amplitudes are extracted at $y=\unicode[STIX]{x1D6FF}^{\ast }(x)$ . The dominant frequency, when the FST is included, remained close to the natural shedding frequency of the zero FST simulation, i.e.  $F\approx 915$ . However, as the level of FST increases, the frequency spectrum becomes broader and the ‘dominant’ frequency peak, although still identifiable in the spectra, is less pronounced. This observation is consistent with the earlier experimental investigation of Burgmann & Schröder (Reference Burgmann and Schröder2008), who found that the dominant shedding frequency of LSBs associated with the KH vortices are almost unaffected by the FST intensity (up to 1.5 %).

It is important to note that the results from the spectral analysis for the LSBs support the findings of Diwan & Ramesh (Reference Diwan and Ramesh2009) that the non-dimensional dominant shedding frequency, $\overline{F}$ , scales linearly with the modified Reynolds number, $\overline{R}$ , in the linear regime. Here, $\overline{F}=Re\,f_{sh}(y_{in}^{2}+\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}^{2})$ and $\overline{R}=Re\,U_{in}y_{in}\sqrt{y_{in}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}$ , where $f_{sh}$ is the most amplified frequency (shedding frequency), $y_{in}$ is the wall-normal position of the inflection point, $U_{in}$ is the mean streamwise velocity at the inflection point and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ is the vorticity thickness (see Diwan & Ramesh Reference Diwan and Ramesh2009). In particular, for the LSBs without and with FST, $\overline{F}\approx 0.2\overline{R}$ . It is worth noting that this scaling holds also in the APG region prior to the separation location, which further corroborates the conjecture by Diwan & Ramesh (Reference Diwan and Ramesh2009) that the origin of the primary instability mechanism can be traced back to upstream of the separation location.

Note that the spectra shown in figure 15(a) were based on the wall-normal disturbance at the local displacement thickness away from the wall. As discussed in § 1.2, FST should give rise to so-called Klebanoff modes with their characteristic very low frequencies that should already be present in the approach boundary layer upstream of the separation. In figure 15(a) no such low-frequency peaks were observed. Thus, the question arises if the Klebanoff modes are suppressed in the presence of separation or if they were missed by using the wall-normal disturbance velocity at the (local) displacement thickness. To answer this question, the maximum amplitude of the disturbance $u$ -velocity is now used instead. This is motivated by the fact that the primary mechanism of Klebanoff modes is a spanwise (low-frequency) modulation of the streamwise velocity component. Consequently, in figure 15(b) the (wall-normal) maximum amplitude of the $u^{\prime }$ -velocity is plotted in the $F$ – $x$ plane for the case with $Tu_{0}=0.5\,\%$ . Now in addition to the dominant shedding frequency, $F\approx 915$ , that was already found before, a much stronger peak appears near $F\approx 26$ .

The frequency spectra now clearly indicate the generation of low-frequency disturbances, due to the presence of Klebanoff modes, which are strongly amplified downstream of the separation location. These low-frequency modes were analysed in the $m{-}x$ plane spectra as shown in figure 16, where $m$ is the spanwise mode number (e.g.  $m=1$ indicates that the spanwise width of the disturbances is $\unicode[STIX]{x1D706}_{z}=L_{z}$ and for $m=2$ , $\unicode[STIX]{x1D706}_{z}=L_{z}/2$ , and so on). There is a dominant spanwise mode number $m=5$ ( $\unicode[STIX]{x1D706}_{z}=L_{z}/5$ ) for which the disturbances have the maximum amplitude. This agrees very well with the instantaneous flow visualization (see figure 6). Another observation is that increasing FST levels caused higher K-mode amplitudes inside the boundary layer.

Corresponding to the results presented in figures 15 and 16, the spatial development of instability waves associated with the high-frequency 2D mode and low-frequency 3D mode are presented in figure 17 for $Tu_{0}=0.5\,\%$ . Plotted are the Fourier amplitude of the $u^{\prime }$ -velocity together with the mean dividing streamline (dashed lines) and displacement thickness (solid line). Two important observations can be made from figure 17. The low-frequency 3D (K) mode is strongly amplified once embedded in the separated shear layer and exhibits a strong peak away from the wall. The maximum of the disturbance aligns with the displacement thickness and the maximum amplitude is reached in the front part of the bubble. On the other hand, the amplitude profile of the $u^{\prime }$ -velocity for 2D disturbances with the shedding frequency is composed of an inviscid shear layer instability and wall-mode instability where they can be distinguished by the streamwise position and the relative wall normal location of the maxima. While the former reaches the maximum at the maximum bubble height, the latter is confined to the wall in the region so-called reverse-flow vortex. This can be interpreted as being caused by the increasing wall shear in this region.

Figure 16.

Contours of the maximum amplitude inside the boundary layer ( $y<\unicode[STIX]{x1D6FF}$ ) of disturbance streamwise velocity in the spanwise mode number $m$ – $x$ spectrum for the frequency $F\approx 25$ : (a) $Tu_{0}=0.5\,\%$ , (b) $Tu_{0}=3\,\%$ . The mean separation and reattachment locations are marked by down arrows.

Figure 17.

Contours of Fourier amplitude of streamwise velocity disturbance for $Tu_{0}=0.5\,\%$ : (a) plotted at $F=915$ and $m=0$ ; (b) plotted at $F=26$ and $m=5$ . Dashed line and solid line are time- and spanwise-averaged dividing streamline and displacement thickness, respectively.

Figure 18.

Comparison of the downstream development of spectral disturbance kinetic energy between the 2D modes at the fundamental frequency, $(915,0)$ (●), and the 3D low-frequency disturbances, $(26,5)$ (○): $Tu_{0}=$ 0 (a), 0.1 % (b), 0.5 % (c), 1 % (d), 2 % (e), and 3 % (f). The shaded area corresponds to the mean separation bubble and the onset of the APG is represented by dashed lines.

Thus, based on these observations, in the presence of FST two different instability modes can be identified in the laminar–turbulent transition process: (i) the strongly amplified 2D modes with a shedding frequency that is directly related to the KH instability, and (ii) the 3D low-frequency Klebanoff modes caused by the FST. To have a complete picture and assess the contributions of both identified instability mechanisms for different FST intensities, the evolution of the spectral disturbance kinetic energy (SDKE) for each mode was tracked in the downstream direction in figure 18. The SDKE is computed from the wall-normal integral of the Fourier amplitude of the velocity components according to

(3.2) $$\begin{eqnarray}E^{(F,m)}(x)=\frac{1}{2}\int _{0}^{\unicode[STIX]{x1D6FF}(x)}[u_{A}^{(F,m)}(x,y)^{2}+v_{A}^{(F,m)}(x,y)^{2}+w_{A}^{(F,m)}(x,y)^{2}]\,\text{d}y,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}(x)$ is the local boundary-layer thickness. The notation $(F,m)$ is used for a pair of mode numbers where $F$ is the non-dimensional frequency (see (3.1)) and $2\unicode[STIX]{x03C0}m/L_{z}$ is the spanwise wavenumber of a structure. Therefore, mode $(26,5)$ corresponds to the Klebanoff modes and $(915,0)$ represents the 2D disturbances with the fundamental (shedding) frequency. In each of the six panels in figure 18, the mean separation region is indicated by the shaded area and the onset of the APG is denoted by dashed lines.

As the level of FST increases, the 2D and 3D disturbances in the attached laminar boundary layer exhibit higher amplitude levels where the K-mode has initially a much larger amplitude than the 2D mode. Immediately downstream of the inflow location, the K-mode experiences streamwise growth for $x<1.8$ , which is most likely the consequence of transient growth and lift-up mechanisms described in § 1.2. The initial transient growth of the K-mode is alleviated in the region of a strong favourable pressure gradient $1.8<x<3.4$ . Downstream of the APG, the K-mode experiences strong exponential growth until it saturates at a finite nonlinear amplitude. Based on this observation and the contours of the Klebanoff disturbances presented in figure 17, it can be conjectured that, once the K-mode is embedded within the separated region, it exhibits exponential growth instead of algebraic.

Just downstream of the inflow location the 2D mode also undergoes growth, but at a smaller rate compared to the K-mode. The growth could be attributed to the initial adjustment of the artificial inlet turbulence. After this initial phase, the 2D mode remains almost constant without experiencing significant downstream growth. Based on the linear stability analysis from our previous research (see Hosseinverdi et al. Reference Hosseinverdi, Balzer and Fasel2012), the location corresponding to $Re_{cr}$ is between the onset of the APG and the separation location; however, exponential amplification of the 2D mode is initiated further downstream inside the separated region. It appears that the vortical fluctuations due to FST are not as effective in generating TS waves as in inducing the streamwise streaks. This observation can be explained by the so-called ‘shear sheltering’ phenomenon (Hunt et al. Reference Hunt, Durbin, Kevlahan and Fernando1996; Jacobs & Durbin Reference Jacobs and Durbin1998), which describes the effect that the boundary layer acts as a low-pass filter, admitting predominantly low-frequency modes to penetrate into the boundary layer and sheltering the high-frequency range disturbances (like oscillations within the KH frequency range).

The maximum amplitude of the 2D mode slightly increases when the intensity of the FST is increased to $Tu_{0}=0.5\,\%$ . When the FST level is increased further, the maximum amplitude of the 2D mode starts to decrease. On the other hand, higher FST intensity promotes stronger Klebanoff modes. In addition to the disturbance kinetic energy level, 2D KH modes can be distinguished from the K-mode by their exponential growth rate, i.e. larger growth rate for 2D mode. One would expect that, with increasing FST, the dominant structures would lose their spanwise coherence as observed in figure 7 for example. In conclusion, for the lowest level of FST, $Tu_{0}=0.1\,\%$ , the region upstream of the reattachment location, which is at the end of the transition region, is dominated by 2D modes, while for the high levels of FST, $Tu_{0}=2\,\%$ and $Tu_{0}=3\,\%$ , the Klebanoff modes are dominant in the entire transition region. For the FST levels in between 0.1 % and 2 %, the two modes are blended together and contribute to the transition process simultaneously. Downstream of the mean reattachment location, the K-modes are decaying much faster than the 2D modes and thus 2D waves exhibit larger amplitudes.

3.4 Modal decomposition

The detection of dominant vortical flow structures is important since the topology of the fluid motion is closely related to the dynamics of these structures. A transitional flow often consists of a superposition of flow structures with different dominant frequencies and wavelengths. However, the instantaneous flow structures, as shown in figure 6 for example, do not provide a complete understanding regarding the underlying flow physics. Therefore, a detailed analysis of the simulation data using modal decomposition is carried out to provide additional insight into the underlying flow physics for various FST intensities. In particular, proper orthogonal decomposition and dynamic mode decomposition have been carried out to identify dominant structures and dynamics of the evolving flow within the transition region as a consequence of hydrodynamic instabilities.

Proper orthogonal decomposition (POD), first proposed by Lumley (Reference Lumley, Yaglom and Tatarsky1967) in the field of fluid mechanics, is a method for identifying large-scale energetic flow structures in transitional and turbulent flows. For the present results, the snapshot method that was proposed by Sirovich (Reference Sirovich1987) is employed. In the POD method, for a given set of time- ( $t$ ) and space-dependent ( $\boldsymbol{x}$ ) flow data, the flow field can be represented by

(3.3) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t_{n})=\mathop{\sum }_{i=0}^{N-1}a_{i}(t_{n})\unicode[STIX]{x1D753}_{i}(\boldsymbol{x}),\end{eqnarray}$$

where the $\unicode[STIX]{x1D753}_{i}$ are the POD spatial eigenfunctions and the $a_{i}$ are the POD time coefficients. The POD modes are orthogonal and optimized with respect to their energy content, i.e. the largest fraction of the total kinetic energy of the flow field is captured with the smallest number of modes. Therefore, assuming that large-scale coherent structures possess a significant amount of energy, POD can provide valuable information regarding the most relevant flow structures.

The dynamic mode decomposition (DMD), proposed by Schmid (Reference Schmid2010), is a data-driven technique capable of extracting dynamical information from data sequences obtained from DNS or experiments. The DMD decomposition allows for an expansion of the velocity data in the form

(3.4) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t_{n})=\mathop{\sum }_{j=0}^{N-1}\unicode[STIX]{x1D707}_{j}^{n}\unicode[STIX]{x1D74D}_{j}(\boldsymbol{x}),\end{eqnarray}$$

where the $\unicode[STIX]{x1D74D}_{j}$ are the complex-valued spatial DMD modes and the $\unicode[STIX]{x1D707}_{j}$ are called the Ritz eigenvalues associated with the respective DMD mode. It should be noted that the $\unicode[STIX]{x1D707}_{j}$ are complex-valued, giving each mode a corresponding growth rate $\unicode[STIX]{x1D70E}_{j}$ and oscillatory angular frequency $\unicode[STIX]{x1D714}_{j}$ as follows:

(3.5a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{j}^{n}=\text{e}^{(\unicode[STIX]{x1D70E}_{j}+\text{i}\unicode[STIX]{x1D714}_{j})t/\unicode[STIX]{x0394}t},\quad \unicode[STIX]{x1D70E}_{j}=\frac{1}{\unicode[STIX]{x0394}t}\log |\unicode[STIX]{x1D707}_{j}|,\quad \unicode[STIX]{x1D714}_{j}=\frac{1}{\unicode[STIX]{x0394}t}\,a\tan \left(\frac{{\mathcal{I}}(\unicode[STIX]{x1D707}_{j})}{{\mathcal{R}}(\unicode[STIX]{x1D707}_{j})}\right),\end{eqnarray}$$

where $\unicode[STIX]{x0394}t$ is the time interval between two consecutive snapshots. In contrast to POD, where a spatial orthogonality of the eigenfunctions could lead to a broad range of frequencies for each individual POD mode, DMD extracts coherent structures with a single frequency. For details of DMD, see Schmid (Reference Schmid2010).

For the present modal decomposition analysis, 720 snapshots of the velocity field $(u,v,w)^{\text{T}}$ equidistantly distributed over 18 fundamental (shedding) periods are used. The focus here is on identifying and extracting the dominant and coherent structures in the transitional region from the initial linear stage up to the nonlinear development of the instability waves. Therefore, the spatial extent of the snapshots was chosen from $x=3$ (upstream of the separation location) up to the mean reattachment location for each case. The POD energy spectrum for the zero FST case and the cases with 0.5 %, 1 % and 3 % of FST intensity are shown in figure 19. The left vertical axis represents the fraction of energy, $E_{i}$ , and the right vertical axis indicates the cumulative energy sum, $S_{p}$ , which are defined as

(3.6a,b ) $$\begin{eqnarray}E_{i}=\left(\unicode[STIX]{x1D706}_{i}\Bigg/\mathop{\sum }_{j=1}^{N-1}\unicode[STIX]{x1D706}_{j}\right)\times 100,\quad S_{p}=\left(\mathop{\sum }_{i=1}^{p}\unicode[STIX]{x1D706}_{i}\Bigg/\mathop{\sum }_{j=1}^{N-1}\unicode[STIX]{x1D706}_{j}\right)\times 100.\end{eqnarray}$$

In the above equations, $\unicode[STIX]{x1D706}_{i}$ is the magnitude of the eigenvalue of the POD mode, which corresponds to twice the kinetic energy content of the respective POD mode, and $N$ is the total number of POD modes. Since the mean flow was not subtracted from the flow field prior to the decomposition, the zeroth POD mode, which represents the mean flow and therefore contains most of the kinetic energy, is not considered here. Thus, the presented $E_{i}$ and $S_{p}$ correspond to the unsteady POD modes.

Figure 19.

Relative (left axis) and cumulative (right axis) POD eigenvalues (energy) spectrum distribution: (a)  $Tu_{0}=0$ ; (b)  $Tu_{0}=0.5\,\%$ ; (c)  $Tu_{0}=1\,\%$ ; and (d)  $Tu_{0}=3\,\%$ .

The POD eigenvalue spectra, shown in figure 19, reveal that, for all cases, most of the energy is contained in the lower mode numbers, although the distribution of energy among the modes is different for the investigated cases. In particular, for the zero FST simulation and the case with $Tu_{0}=0.5\,\%$ , the most energetic modes (modes 1 and 2) have almost identical magnitudes. For the other cases ( $Tu_{0}=1\,\%{-}3\,\%$ ), however, the energy drops among the dominant modes. The drop-off rate in the magnitude of the eigenvalues with respect to the higher mode numbers is similar for all cases. Furthermore, figure 19 indicates that approximately 4.2 % of the unsteady modes (30 modes) only are needed to reproduce 70 % of the unsteady kinetic energy. In the following, the eigenfunctions of the ‘most energetic’ POD modes, their respective time coefficients and the associated Fourier spectra of the time coefficients, and the most coherent dynamic modes and their corresponding growth rates and frequencies are inspected in detail for different levels of FST. The POD and DMD modes are visualized by isosurfaces of the $u/v$ velocity component. Here, time is non-dimensionalized by the shedding period, $t_{nd}=t/t_{sh}$ , and similarly the frequency is non-dimensionalized by the shedding frequency, $f_{nd}=f/f_{sh}$ .

Figure 20 demonstrates that the dominant POD modes exhibit large spanwise-oriented structures for the case without FST. The phase shift in modes 1 and 2 indicates that these modes form pairs, which physically represent structures that are travelling in the streamwise direction (‘travelling waves’). It can therefore be concluded that the spatially evolving coherent structures, $\unicode[STIX]{x1D6FE}(\boldsymbol{x},t)$ , are represented by pairs of eigenfunctions according to $\unicode[STIX]{x1D6FE}(\boldsymbol{x},t)=a_{1}(t)\unicode[STIX]{x1D753}_{1}(\boldsymbol{x})+a_{2}(t)\unicode[STIX]{x1D753}_{2}(\boldsymbol{x})$ . Detailed inspection of modes 1 and 2 also reveals that these modes are composed of 2D and oblique structures. This observation led to the conjecture that the amplification of weakly oblique 3D disturbances due to the inviscid shear-layer instability mechanism may lead to the formation of oblique coherent structures. The time signals of modes 1 and 2 exhibit nearly sinusoidal behaviour with a dominant frequency matching the shedding frequency, $f_{nd}=1$ , as shown in figure 21. It is worth noting that there is a narrow band around the dominant frequency which could be related to the composite structures of modes 1 and 2. POD mode 3 is oriented in the streamwise direction with the preferred spanwise wavelength $\unicode[STIX]{x1D706}_{z}\approx L_{z}/3$ . The corresponding time signal of this mode has a very long period (low-frequency content). The 3D structures with spanwise wavelength $\unicode[STIX]{x1D706}_{z}=L_{z}$ are identified by modes 4–6. The frequency spectra associated with these modes show two dominant frequency bands close to the shedding frequency, $f_{nd}=0.9$ and 1.06.

Figure 20.

Dominant POD eigenfunctions for the zero FST case. Shown are positive (light grey) and negative (blue) isosurfaces of $|v|=0.1$ .

Figure 21.

(a) POD time coefficients and (b) associated frequency spectra for the zero FST case.

Figure 22.

DMD eigenfunctions for the case with zero FST. Here $\unicode[STIX]{x1D70E}_{j}$ and $f_{j}$ are the corresponding growth rate and frequency, respectively. Shown are isosurfaces of $|v|=0.01$ .

To gain further insight into the coherent structure dynamics, the four least damped DMD modes are illustrated in figure 22. DMD modes 1 and 2 capture purely oblique and 2D structures, respectively. The composite structures (2D and oblique structures) are identified by modes 3 and 4. Another interesting observation is that the oblique structures (mode 1) are more unstable than 2D disturbances (mode 2).

Figure 23.

POD eigenfunctions for the case $Tu_{0}=0.5\,\%$ . Shown are positive (light grey) and negative (blue) isosurfaces of $|u|=0.08$ . Dashed lines indicate the mean separation location.

Figure 24.

(a) POD time coefficients and (b) respective frequency spectra for the case $Tu_{0}=0.5\,\%$ .

From the 3D visualizations of POD modes 1, 2 and 3 for $Tu_{0}=0.5\,\%$ shown in figure 23, two different flow structures can be identified: (i) spanwise structures (2D) as the result of the KH instability mechanism and (ii) ‘streaky’ structures which correspond to the Klebanoff modes caused by FST. In particular, the dominant structures (modes 1, 2 and 3) have initially a streaky pattern, with alternating positive and negative streamwise velocity fluctuations, that later turns into a 2D structure. The spanwise wavelength of the Klebanoff modes is in agreement with the results from instantaneous visualizations and spectral analysis ( $\unicode[STIX]{x1D706}_{z,K}\approx L_{z}/5$ ). Accordingly, the frequency spectra associated with these modes provided in figure 24 show two dominant frequency bands: the high-frequency spectral peak is related to the spanwise structures while the low-frequency peak corresponds to Klebanoff modes. Although the shedding frequency amplitude (high-frequency peak) for modes 1 and 2 is larger than the low-frequency peak amplitude (Klebanoff mode), the opposite is true for mode 3. POD modes 4 and 5 display purely low-frequency streamwise-oriented ‘streaky’ structures. However, the frequency and spanwise spacing are different from those captured by modes 1–3. The structures with chequerboard pattern with $\unicode[STIX]{x1D706}_{z}=L_{z}$ in mode 6 in figure 23 have almost the same dominant frequency as POD modes 4–6 for the zero FST case.

The DMD method is able to capture the dominant modes related to the KH and Klebanoff modes separately as illustrated in figure 25. Unsteady ‘streaky’ and steady structures are identified by DMD modes 3 and 4 in figure 25, which shows that the steady mode is highly damped. In contrast to the zero FST simulation, the 2D structure with the shedding frequency is the most unstable mode among the high-frequency disturbances. POD and DMD investigations for $Tu_{0}=0.5\,\%$ support the fact that 2D disturbances within the flow with frequency close to the KH frequency range are the most amplified instability waves in the presence of a finite low-amplitude disturbance environment.

Figure 26 demonstrates that the transition region is dominated by the K-mode for the case with $Tu_{0}=1\,\%$ as captured by the dominant POD mode (cf. mode 1). A similar shape is observed in mode 2; however, the appearance of spanwise structures occurs further downstream. The drop in the magnitude of the eigenvalues between modes 1 and 2 as observed in figure 19(c) could be explained by the fact that modes 1 and 2 do not represent travelling pairs, and therefore they do not occupy the same energy level as was the case for zero FST and $Tu_{0}=0.5\,\%$ . Fully developed KH structures are present in the higher modes 3–5 and the chequerboard pattern can be observed in mode 6.

Figure 25.

DMD eigenfunctions for the case $Tu_{0}=0.5\,\%$ . Here $\unicode[STIX]{x1D70E}_{j}$ and $f_{j}$ are the corresponding growth rate and frequency, respectively. Dashed lines indicate the mean separation location.

Figure 26.

POD eigenfunctions for the case $Tu_{0}=1\,\%$ . Shown are positive (light grey) and negative (blue) isosurfaces of $|u|=0.08$ . Dashed lines indicate the mean separation location.

Figure 27.

POD eigenfunctions for the case $Tu_{0}=3\,\%$ . Shown are positive (light grey) and negative (blue) isosurfaces of $|u|=0.2$ . Dashed lines indicate the mean separation location.

Figure 28.

DMD eigenfunctions for the 2D mode: (a) $Tu_{0}=1\,\%$ and (b)  $Tu_{0}=3\,\%$ .

This detailed analysis based on modal decomposition supports the results obtained from the spectral analysis that the low-frequency 3D Klebanoff modes and high-frequency 2D disturbances are blended together and contribute to the transition process for cases with the FST intensities of $Tu_{0}=0.5\,\%$ and $Tu_{0}=1\,\%$ . Whereas the large contribution comes from the KH instability mechanism for the $Tu_{0}=0.5\,\%$ case, the Klebanoff modes play a more dominant role in the $Tu_{0}=1\,\%$ case.

The perspective views for the streamwise velocity component in figure 27, for the highest level of FST intensity investigated here ( $Tu_{0}=3\,\%$ ), demonstrate that the most energetic structures are streamwise streaks with a distinct spanwise wavelength. Evidence of spanwise structures is only visible in modes 5 and 6, although they are not fully developed. Finally, figure 28 compares the 2D structures with the shedding frequency obtained from DMD for $Tu_{0}=1\,\%$ and $Tu_{0}=3\,\%$ .

3.5 Linear inviscid instability: linearized Navier–Stokes equations

The results presented thus far indicate that the K-modes dominate the transition region for $Tu_{0}>1\,\%$ . Of particular importance is the observation that the slow streamwise growth of the K-mode was overtaken by much stronger amplification rate downstream of the onset of the APG as the K-mode exhibits exponential growth in the strong APG region. To further corroborate this observation, a linear instability investigation with respect to the low-frequency 3D disturbances was carried out.

To predict the evolution of instabilities in the linear regime, we developed an approach that directly solves the linearized Navier–Stokes equations (LNSE) without using any additional assumptions as required by linear stability theory (LST), i.e. assuming that basic flow is parallel (parallel flow assumption). Thus, all locally non-parallel effects of the basic flow are included in the LNSE approach, which is important as separated flows are considered here. For the linear instability analysis employed here based on LNSE, the total flow field is decomposed into the base flow (or basic flow) $\boldsymbol{u}_{B}(x,y)$ and $\unicode[STIX]{x1D74E}_{B}(x,y)$ and disturbances $\boldsymbol{u}^{\prime }(\boldsymbol{x},t)$ and $\unicode[STIX]{x1D74E}^{\prime }(\boldsymbol{x},t)$ . Substituting these decompositions into the Navier–Stokes equations (2.1) yields the governing equations for the total flow field, which after linearization (by neglecting products of the disturbance quantities) is a linear superposition of the base flow and the disturbance flow. Thus, the governing equations for the base flow can be subtracted from the equations for the total flow in order to obtain the LNSE. For numerically solving the LNSE, the same time integration and spatial discretization schemes are employed as for the nonlinear equations used in the DNS.

Figure 29.

(a) Streamwise development of K-mode, mode ( $26,5$ ), computed with LNSE and DNS with $Tu_{0}=0.5\,\%$ . (b) Wall-normal distributions of the $u^{\prime }$ -velocity Fourier amplitudes at several $x$ -locations obtained from LNSE and DNS with $Tu_{0}=0.5\,\%$ for mode ( $26,5$ ). Disturbance amplitudes are scaled by their respective maxima and dash-dotted lines represent the corresponding time-averaged $u$ -velocity profile ( $u_{B}$ ). Wall-normal coordinates are normalized by the local mean displacement thickness. (c) Comparison of local spatial growth rates obtained from LNSE and DNS with $Tu_{0}=0.1\,\%$ (– – –), 0.5 % (——), 1 % (– $\cdot$ – $\cdot$ ), 2 % (– – –) and 3 % (– $\cdot \cdot$ – $\cdot \cdot$ ).

For generating streamwise vortices outside the boundary layer in the LNSE solver in order to excite Klebanoff modes inside the boundary layer (for example mode ( $26,5$ ), which is associated with the K-mode), an unsteady volume force is used as a source term on the right-hand side of the LNSE. For more details regarding this approach, see Fasel (Reference Fasel2002) and Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2018). In figure 29, the downstream development of the wall-normal maximum of the $u^{\prime }$ -velocity Fourier amplitude ( $A_{u}$ ) and the $u^{\prime }$ -velocity Fourier amplitude profile ( $u_{A}$ ) of the 3D disturbance wave with the spanwise mode number $m=5$ (which corresponds to the lateral wavelength of the K-mode) are plotted as obtained from the LNSE and are compared with the DNS results of an LSB with $Tu_{0}=0.5\,\%$ . For the base flow used in the LNSE investigation, the time- and spanwise-averaged results obtained for $Tu_{0}=0.5\,\%$ are employed. In figure 29(b), the Fourier amplitudes are scaled by their respective maximum amplitudes and $u_{B}$ is the corresponding time-averaged $u$ -velocity profile. The streamwise evolution of the K-modes obtained from the LNSE compares very well with those from the DNS as shown figure 29(a). Of particular interest in figure 29(a) is that the K-mode indeed grows exponentially in the APG region ( $A_{u}\sim \text{e}^{1.284x}$ ). The comparison of the wall-normal distributions of the disturbance amplitude in figure 29(b) further confirms the proper implementation and accuracy of the LNSE method. The spatial growth rates associated with the K-mode as obtained from the LNSE (with the base flow corresponding to $Tu_{0}=0.5\,\%$ ) and DNS with various FST intensities are presented in figure 29(c). The local growth rate is computed based on the wall-normal maxima of $u_{A}$ according to $\unicode[STIX]{x1D70E}(x)=\unicode[STIX]{x2202}\log (u_{A,max})/\unicode[STIX]{x2202}x$ .

Figure 30.

Comparison of wall-normal distribution of $u^{\prime }$ -velocity Fourier amplitudes obtained from LNSE (——) and DNS (○) for 2D disturbances with shedding frequency: (a)  $Tu_{0}=0$ , (b)  $Tu_{0}=0.1\,\%$ , (c)  $Tu_{0}=0.5\,\%$ , (d)  $Tu_{0}=1\,\%$ , (e)  $Tu_{0}=2\,\%$ , (f)  $Tu_{0}=3\,\%$ . Disturbance amplitudes are scaled by their respective maximum value.

It is well known that, for attached boundary layers, the linear stages of the transition process are bypassed when FST intensity exceeds 1 % (Morkovin Reference Morkovin and Sinclair Wells1969; Jacobs & Durbin Reference Jacobs and Durbin2001; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001). The critical FST intensity beyond which ‘bypass’ transition occurs depends of course on the pressure gradient, on the background spectrum of the perturbations and on the receptivity (Abu-Ghannam & Shaw Reference Abu-Ghannam and Shaw1980). Previous research (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000; Yarusevych, Kawall & Sullivan Reference Yarusevych, Kawall and Sullivan2006; Marxen & Henningson Reference Marxen and Henningson2011; Postl et al. Reference Postl, Balzer and Fasel2011) has shown that an effective and efficient laminar separation control is achieved when a linear inviscid shear-layer instability is exploited. Therefore, it is necessary to examine the existence of a linear inviscid instability mechanism for different levels of FST. Next, the hydrodynamic instability with respect to 2D disturbances with the dominant frequency (shedding frequency) found in the spectral analysis as discussed in § 3.3 is investigated for different FST intensities. For each FST intensity, the corresponding time- and spanwise-averaged results obtained from the DNS is used as the base flow. Figure 30 compares the wall-normal Fourier amplitude distributions of the $u^{\prime }$ -velocity between the DNS and the LNSE results for all cases. The amplitude distributions from both LNSE and DNS are normalized by their respective maximum value within the boundary layer. These profiles were extracted from the region of exponential growth up to the saturation location. The amplitude distribution from the DNS data agrees remarkably well with that obtained from LNSE for zero FST simulation and cases with $Tu_{0}\leqslant 2\,\%$ even at the location where the disturbances saturate. For the highest level of FST, $Tu_{0}=3\,\%$ , despite the good agreement at the beginning ( $x=6.3$ ), there are differences in the wall-normal location of the disturbance peaks and the amplitudes near the wall for profiles $x>6.3$ . The good agreement for the cases with $Tu_{0}\leqslant 2\,\%$ confirms that 2D disturbances grow due to the linear, inviscid shear-layer instability mechanism, and that the primary linear stage of the transition process is not bypassed. As a result, unsteady flow control strategies may still be effective means to reduce/prevent separation for these cases by exploiting the inviscid shear-layer instability.

These findings are consistent with those by Balzer & Fasel (Reference Balzer and Fasel2016). It is important to note, however, that disturbances associated with the K-modes are orders of magnitude larger than the 2D KH instability waves for LSBs for high levels of FST, as demonstrated in the present investigations by spectral analysis and POD. Hence, it is postulated that, whereas a significant reduction of the separation length can be obtained with unsteady flow control techniques, which are aimed at exploiting the shear-layer instability for zero and small levels of FST, laminar separation control becomes less effective for high levels of FST. Although flow control will exponentially amplify 2D modes, disturbances due to FST (K-modes) will remain at much larger amplitude levels compared to the 2D disturbances in the entire transition region. Thus, for higher levels of FST, in addition to exploiting the shear-layer instability, an effective and efficient active flow control is required also to reduce the streamwise growth of K-modes.

3.6 Effect of free-stream turbulence energy spectrum

For all cases, the von Kármán energy spectrum is used to distribute the turbulent kinetic energy among the modes with different wavenumbers (see § 2.5). A set of numerical simulations were performed while varying the energy spectrum of the incoming synthetic turbulence to investigate if and how it affects the transition process in LSB. Towards this end, energy spectra proposed by Ossia & Lesieur (Reference Ossia and Lesieur2001) were employed,

(3.7) $$\begin{eqnarray}E(k)=\frac{1}{2}Tu_{0}^{2}\,\frac{A_{s}}{k_{e}}\left(\frac{k}{k_{s}}\right)^{s}\exp \left(-\frac{s}{2}\frac{k^{2}}{k_{e}^{2}}\right),\end{eqnarray}$$

where $A_{s}$ is a normalization constant chosen such that $\int \!E(k)\,\text{d}k=1.5Tu_{0}^{2}$ . The wavenumber $k_{e}$ denotes the wavenumber of the most energetic structures ( $k_{e}\approx 1.157/L_{11}$ ). The parameter $s$ determines the slope of the low-wavenumber range. Figure 31 compares the typical spectrum obtained from the von Kármán energy spectrum (vKES) with that obtained using (3.7) with $s=0.5$ . The Ossia and Lesieur energy spectrum (OLES) decreases rapidly above a prescribed wavenumber  $k_{e}$ .

Simulations using the two different energy spectra, namely vKES and OLES, for the FST intensity of $Tu_{0}=0.1\,\%$ and $Tu_{0}=2\,\%$ were carried out. However, the results for $Tu_{0}=0.1\,\%$ are not reported here since the results obtained from the two spectra were very similar, so only the results for $Tu_{0}=2\,\%$ are discussed below. It should be noted that the wavenumber associated with the maximum energy and the integral of the energy with respect to the wavenumber ( $\int \!E(k)\,\text{d}k$ ) are kept the same for both spectra.

Contours of spanwise, one-dimensional power spectra of the streamwise velocity component as a function of the spanwise mode number $(m)$ are plotted for selected streamwise locations for the two FST spectra in figure 32. The spectra were computed using a spanwise Fourier transform and the relationship

(3.8) $$\begin{eqnarray}E_{uu}(x,y,m)=\overline{u^{m}(x,y,t)u^{m}(x,y,t)},\end{eqnarray}$$

where the overbar indicates a time average and $u^{m}$ symbolizes the Fourier-transformed $u$ -velocity component in the spanwise direction (spanwise mode number $m$ ). Five $x$ -stations were selected: the inflow boundary, $x_{inlet}$ , the onset of the APG, $x_{APG}$ , the separation location, $x_{S}$ , the location of maximum bubble height, $x_{h,max}$ , and the mean reattachment location, $x_{R}$ . The wall-normal coordinate is normalized by the local boundary-layer thickness and the local displacement thickness is indicated by dashed lines in each panel.

Figure 31.

Energy spectra used to generate isotropic FST at the inflow boundary: ○, von Kármán energy spectrum, equation (2.5); ●, spectrum proposed by Ossia & Lesieur (Reference Ossia and Lesieur2001), equation (3.7). The wavenumber of the maximum energy, $k_{e}$ , is represented by the dashed line.

Figure 32.

Contours of spanwise one-dimensional energy spectra of the streamwise velocity at selected streamwise locations obtained for (a) von Kármán energy spectrum and (b) Ossia and Lesieur energy spectrum. The wall-normal direction is scaled by the local boundary-layer thickness. The displacement thickness is marked by dashed red lines.

For both cases, the energy in the power spectrum at the inflow is clearly concentrated in the free stream and the maximum amplitude is at $1\leqslant m\leqslant 2$ (this is related to the chosen turbulent integral length scale $L_{11}$ ). While propagating in the downstream direction, the vortical structures in the free stream enter the boundary layer, as can be clearly observed at $x_{APG}$ . At the onset of the APG, two bands ( $m=2$ , $m=5$ ) are visible for the case with vKES. The maximum amplitude has a spanwise mode number of $m=5$ , which, as already known from spectral analysis, corresponds to the K-mode. For the other case (OLES), however, the spectra are dominated by the disturbances with spanwise mode number of $m=2$ without the appearance of the K-mode (or the K-mode has a smaller amplitude than minimum contour level). Moving further downstream to the separation location, the disturbances are amplified inside the boundary layer, and the trace of the K-mode becomes visible for the case with OLES although the disturbances with $m=2$ are still dominant. Close to the location of maximum bubble height, the K-mode attains a larger amplitude than disturbances with a smaller wavenumber for the case with OLES. For the vKES case, the K-mode remains dominant and the disturbances with a smaller wavenumber have penetrated the boundary layer, which is more pronounced at the reattachment location where the near-wall peaks appear for a lower spanwise mode number, indicating that the boundary layer is starting to redevelop. The case OLES exhibits a similar pattern at the reattachment location.

A closer inspection of (3.7) and figure 31 reveals that the kinetic energy is distributed among the different modes such that the lower-wavenumber modes contain most of the kinetic energy and the spectrum lacks the high-wavenumber contents present in the von Kármán energy spectrum. This kind of distribution biases the energy distribution initially towards the low-wavenumber disturbances. However, the modes with low frequency and higher spanwise wavenumber (K-modes) become visible further downstream. Therefore, it could be speculated that, in the absence of high-wavenumber scales in the FST spectrum, the Klebanoff modes need a larger downstream distance to become dominant.

Figure 33.

Streamwise development of disturbance kinetic energy for different FST energy spectra: (a) not-normalized, (b) normalized with inflow values. Plotted are 2D modes at the fundamental frequency and Klebanoff modes, mode ( $26,5$ ). Included in (a) is the streamwise evolution of mode ( $26,2$ ) as obtained from DNS with the OLES (dot-dashed curve). Onset of APG and separation location are indicated by vertical dashed lines and solid lines, respectively.

Figure 33 shows the downstream development of the SDKE and the energy growth ratio, $E(x)/E(x=1)$ , of the 2D modes with the fundamental frequency and of the low-frequency Klebanoff modes, mode ( $26,5$ ), obtained for the two different FST energy spectra. For the 2D modes, both cases experience the same amplification rate in the region of exponential growth; thus, the higher initial amplitude leads to a larger maximum amplitude at the saturation location for the OLES case. Consequently, when the local energy is normalized by the inflow value, the two curves collapse, indicating that the large initial amplitude is carried all the way in the downstream direction as indicated in figure 33(b). The opposite behaviour is observed for the K-mode, where the initial and maximum amplitudes are the largest for the case with the vKES. While both cases undergo the same transient growth, the exponential growth rate is slightly larger for the OLES case. As a result, the normalized energy curves are only identical for $x<5$ . Also, included in figure 33(a) is the streamwise evolution of low-frequency disturbance waves with the spanwise mode number identified in figure 32, mode ( $26,2$ ), which was obtained from the DNS with the OLES energy spectrum (dot-dashed curve). It is worth noting that, for the DNS with the OLES, $m=2$ is the dominant mode until $x<5.2$ , from where onwards disturbance waves with $m=5$ (K-modes) prevail further downstream and dominate the transition process similar to the DNS with the von Kármán energy spectrum.

Figure 34 compares the time- and spanwise-averaged skin-friction coefficients and displacement thicknesses obtained for the two cases, indicating that the separation region is slightly increased for the OLES case in both the streamwise and wall-normal directions. The transition process is governed mainly by the amplification of the K-mode for $Tu_{0}=2\,\%$ . The K-mode is more energetic in the vKES case, and thus leads to a smaller separation bubble.

Figure 34.

Time- and spanwise-averaged skin-friction coefficients (a) and displacement thickness (b) versus $x$ : ○, von Kármán energy spectrum; ●, Ossia and Lesieur energy spectrum.

All results presented here were obtained for a fixed turbulent integral length scale $L_{11}=1.5\unicode[STIX]{x1D6FF}_{0}$ , where $\unicode[STIX]{x1D6FF}_{0}$ is the boundary-layer thickness at the inflow boundary, while the FST intensity and spectrum content were varied. However, related studies of bypass transition on a zero-pressure-gradient flat-plate boundary layer have shown that the choice of the integral length scale, $L_{11}$ , can have a significant effect on the transition process (Brandt et al. Reference Brandt, Schlatter and Henningson2004). Therefore, simulations were carried out for $Tu_{0}=2.0\,\%$ using different integral length scales $L_{11}=0.9\unicode[STIX]{x1D6FF}_{0}$ , $1.5\unicode[STIX]{x1D6FF}_{0}$ and $3\unicode[STIX]{x1D6FF}_{0}$ . These simulations have shown that, despite the considerable variation of the integral length scales, the size of the separation bubble was largely independent of $L_{11}$ for the range investigated here. The same conclusion was obtained for the case with the lowest FST intensity $Tu=0.1\,\%$ . These results are not presented here since they did not display any particular new features/characteristics.