Instability processes in short and long laminar separation bubbles

Abstract

This work studies the link between the bursting process of a flat plate laminar separation bubble and the modification of the stability characteristics of the separated shear layer due to changes in the flow parameters. A vast population of short and long laminar separation bubbles was surveyed by means of Particle Image Velocimetry instrumentation for different values of the Reynolds number, the free-stream turbulence intensity and the streamwise pressure gradient. A fine-step variation of the free-stream velocity allowed us to determine the critical Reynolds number at which bursting occurs. Successively, the most amplified wavelength and frequency were computed for both the short and the long bubble regimes. Once scaled with the boundary layer displacement thickness at separation, the average wavenumber of the vortices shed by the bubble was found to be constant and equal to about 0.9 in the short regime, accordingly to previous studies. Differently, this quantity reduces to about 0.6 in the long bubble regime, and a marked change in the Strouhal number of vortex shedding occurs. Also, the temporal growth of spanwise vortices was seen to occur in the recirculation region of long type bubbles, being linked to an absolute instability of disturbances. The currently acquired data demonstrate the existing link between the bursting process of a laminar separation bubble and a marked change in the instability mechanisms driving the transition process of the boundary layer. A simplified correlation for the prediction of bursting is provided in this work as a function of the free-stream turbulence intensity and the streamwise pressure gradient.

1 Introduction

In case of adverse pressure gradient, boundary layer (BL) separation often occurs based on the free-stream turbulence level (Tu), the Reynolds number (Burgmann et al. 2006; Tani 1964) and the surface roughness (Wu and Piomelli 2018). Then, a laminar separation bubble (LSB) is formed when the shear layer transitions to turbulence and it successively reattaches to the wall (Horton 1968). Due to an increase in the pressure gradient, and/or a decrease in the Reynolds number and the free-stream turbulence level, the bursting of a laminar separation bubble may occur (Marxen and Henningson 2011), causing substantial changes in the aerodynamic performances of airfoils and blade profiles. For this reason, several literature works have been focused on the understanding of the bursting process (Burgmann et al. 2006; Roberts 1980; Toppings and Yarusevych 2023) and on the construction of empirical correlations aimed at bursting prediction (see, e.g., Von Doenhoff (1938); Owen and Klanfer (1953); Crabtree (1959)). Gaster (1967) has been among the first to perform a detailed and comprehensive analysis of the bursting process of an LSB. Particularly, a two parameters bursting criterion has been provided, which is based on the momentum thickness Reynolds number at separation (${\text{Re}}_{\theta ,s}$) and the pressure gradient parameter $P_G=\theta _s^2/\nu (\Delta U/\Delta X)$. Here, $\theta _s$ is the boundary layer momentum thickness at separation and $\Delta U/\Delta X$ is the velocity gradient across the bubble length with reference to an inviscid-like distribution. According to Gaster (1967), the occurrence of bursting depends on the trajectory described by an LSB in the ${\text{Re}}_{\theta ,s}$-$P_G$ plane. Diwan etal. (2006) and Mitra and Ramesh (2019) have successively disproved the universality of Gaster’s criterion. They proposed the alternative bursting parameter $P_{\text{DCR}}=h^2/\nu (\Delta U/\Delta X)_{act}$, where h is the maximum bubble height and $(\Delta U/\Delta X)_{act}$ is the actual velocity gradient between the separation and the reattachment positions. The condition for bursting is $P_{\text{DCR}}<-28$, as it has been shown also by Serna and Lázaro (2015) and Alferez et al. (2013). Serna and Lázaro (2015) demonstrated experimentally that bursting criteria based on the maximum wall distance at the position where the shear layer instability provokes the shedding of vortex pairs are able to accurately predict the onset of the bursting process. Nevertheless, the above reported criteria require the knowledge of the time-mean shape of an LSB for the definition of its corresponding flow state. Empirical correlations predicting the structure of a laminar separation bubble have been therefore provided in the literature based on boundary layer quantities evaluated at the separation position (Mitra and Ramesh 2019; Serna and Lázaro 2015; Dellacasagrande etal. 2020a). The dynamic properties of a laminar separation bubble change significantly when switching from the short to the long regime (Pauley et al. 1990; Alam and Sandham 2000). Kelvin–Helmholtz inflectional instability is the primary source of disturbance amplification in LSBs at high Reynolds numbers, i.e., in the short regime (Hosseinverdi and Fasel 2019). Near the position of maximum displacement of the bubble, the 2D roll-up of the separated shear layer initiates the shedding process of periodic vortices (Marxen et al. 2013), which increase the momentum transfer toward the wall and force the BL reattachment (Marxen and Henningson 2011). The 3D breakdown of vortex pairs finally leads to the turbulent state of the boundary layer (Kurelek et al. 2021). Non-periodic large amplitude fluctuations with reduced spanwise coherence manifest instead when transitioning from the short to the long state of a laminar separation bubble (Marxen and Henningson 2011; Pauley et al. 1990). Additionally, unsteadiness and three-dimensional breakup may occur in the absence of free-stream turbulence, suggesting the presence of self-sustained modes (Rodríguez etal. 2021; Balzer and Fasel 2016). Self-excited instability of LSBs has been studied using direct numerical simulations (Pauley et al. 1990; Fasel and Postl 2006; Embacher and Fasel 2014) and linear stability analyses (Allen and Riley 1995; Hammond and Redekopp 1998; Rist and Maucher 2002). Regions of absolute instability have been thought to serve as the origin of self-excited modes, which trigger the shedding of spanwise vortices (Huerre and Monkewitz 1990). Their subsequent secondary instabilities result in the transition to turbulence, resembling flow cases with external disturbance waves. Among others, Alam and Sandham (2000); Rist and Maucher (2002), and Diwan and Ramesh (2009) have suggested that absolute instability can occur in an LSB if the reversed flow magnitude is higher than 16-20% of the free-stream velocity (see also Fasel and Postl (2006) and Embacher and Fasel (2014)). Differently, Avanci etal. (2019) have shown the occurrence of absolutely unstable waves in LSBs being dependent on the location of the inflection point within the recirculating flow region, rather than on the reversed flow magnitude. They demonstrated the existence of absolutely unstable modes with reversed flows below 16% of the external velocity. Spalart and Strelets (2000) and Hosseinverdi and Fasel (2013) observed transitional separation bubbles in numerical simulations without continuous external disturbances, where the reversed flow magnitude was below 8%. The role of flow parameter variation in causing the bursting of laminar separation bubbles; and therefore, the eventual modification of the stability characteristics of the separated shear layer, has been investigated in the literature on flat plates and airfoils (Boutilier and Yarusevych 2012; Serna and Lázaro 2015 and Michelis et al. (2017)). Boutilier and Yarusevych (2012) investigated the response of an airfoil LSB to changes in edge velocity and angle of attack, observing a reduction in the shear layer’s most amplified frequency when the airfoil stalled due to bursting. Michelis et al. (2017) demonstrated that the cessation or reduction of free-stream disturbances could lead to LSB bursting using impulsive forcing on an airfoil boundary layer (see also the recent work of Toppings and Yarusevych (2023)). The bursting and flapping mechanisms were found to be governed by the altered stability characteristics of the shear layer due to variations in incoming disturbances. Marxen and Henningson (2011) found the wall distance playing a key role in altering the shear layer stability characteristics, rather than a shift from convective to absolute instability, as previously speculated by Gaster (1967). Although the previously mentioned works provide an exhaustive description of the change in the time-mean structure of LSBs undergoing bursting as well as of the shedding properties of short and long LSBs, the study of the bursting effects on both the statistics and dynamics of an LSB via a unique extensive database has not been already provided in the literature. The present work investigates the existing link between the occurrence of bursting of an LSB and the modification of the stability characteristics of the separated shear layer, being quantified in terms of the most amplified wavelengths and frequencies. To this end, the laminar separation bubble forming on a flat plate geometry was surveyed using particle image velocimetry (PIV) instrumentation for variable Reynolds number, free-stream turbulence intensity and streamwise pressure gradient. Overall, 54 flow cases were acquired showing the effects due to changes in the flow parameters. The paper outline is as follows: Sect. 2 illustrates the experimental setup and the adopted measuring techniques. Sections 3 and 4 describe the time-mean and dynamic response, respectively, of an LSB to changes in the flow parameters. A simplified correlation for the prediction of bursting is presented in Sect. 5; while, the main outcomes of the present investigation are summarized in Sect. 6.

2 Experimental setup

Fig. 1

Test section and PIV instrumentation layout. Green boxes indicate the PIV measuring domain. Top and bottom walls were set symmetrically

Measurements were performed in the open circuit wind tunnel of the Aerodynamic and Turbomachinery laboratory of the University of Genova. A flat plate geometry with 4:1 elliptic leading edge was installed between two adjustable end-walls providing the prescribed pressure gradient to the boundary layer (see Fig. 1). The flow is accelerated from the plate leading edge up to 20% of the plate length using a converging fixed geometry. Downstream of the channel throat, adjustable end-walls were installed to allow the pressure gradient variation. The opening angle of the end-walls ($\alpha$) was set to 9 and 12 degrees for the present experiments. These end-wall settings provide an acceleration parameter $\rm AP=\frac{L}{U_{\infty ,0}}\frac{\Delta U_\infty }{\Delta x}$ equal to $-$0.27 and $-$0.41, respectively. Here $L=0.3$ m indicates the length of the flat portion of the plate, $U_{\infty ,0}$ is the free-stream velocity at the channel throat and $\frac{\Delta U_\infty }{\Delta x}$ is the average velocity gradient over the rear part of the plate. A plate width of 0.3 m guaranteed that 2D time-mean flow occurred at the plate mid-span, where PIV data were collected.

Table 1 Turbulence-generating grids characterization: bars width (d), mesh size (M), free-stream turbulence intensity (Tu) and streamwise integral length scale ($L_x$)

The free-stream turbulence intensity was controlled using turbulence generating grids located 0.5 m upstream of the plate leading edge. Their geometry, Tu level and the associated integral length scale ($L_x$), are reported in Table 1. The lowest Tu level is due to the spontaneous turbulence of the tunnel (No-Grid case, NG). Two different grids were instead adopted for the low-Tu (LTU) and the high-Tu (HTU) cases. The root mean square of velocity fluctuations was measured at the entrance of the test section using a laser Doppler velocimetry (LDV) instrumentation. LDV data were acquired in the free-stream region at the same streamwise coordinate of the plate leading edge. Data were recorded at an average sampling rate of about 10kHz for an acquisition period of 120 s. The uncertainty in the evaluation of the Tu level was estimated to be ± 2% of the measured value (Canepa et al. 2018). The streamwise integral length scale of free-stream turbulence, $L_x$, was computed by means of the auto-correlation function of LDV data, with the Taylor’s hypothesis of frozen turbulence (Aleiferis et al. 2017). For each combination of Tu level and AP, 9 different Reynolds numbers (${\text{Re}}_L$) were tested. The Reynolds number is defined based on L and the free-stream velocity at the plate leading edge. The range of Reynolds numbers tested was chosen to acquire both short and long LSBs for different combinations of free-stream turbulence and pressure gradients. Additionally, a fine-step Reynolds number variation was considered to capture the critical value of this parameter for which the change in the state of the LSB occurs. The overall test matrix consists of 54 flow conditions concerning the systematic variation of the most influencing parameters.

Fig. 2

a Normalized time-mean velocity ($u/U_e$) for variable Reynolds number at fixed Tu=1.5% and AP=$-$0.41; b non-dimensional LSB length ($L_b/L$) and height ($H_b/L$) against the Reynolds number for all combinations of Tu level and pressure gradient. Shaded areas represent the $\pm 5\%$ uncertainty bound

LSBs evolving on the flat plate were surveyed by means of a dual-cavity ND:YAG pulsed laser Litron LDY 300 PIV system (energy 30 mJ per pulse at 1000 Hz repetition rate, 527 nm wavelength). The flow in the wind tunnel was seeded with vaseline oil droplets with a mean diameter of $1.5\mu$m. To mitigate laser reflections, the plate was black painted and the laser optic was positioned behind the plate trailing edge to minimize the light incidence angle. Two high sensitive SpeedSense M340 digital cameras with a cooled 2560x1600 pixels CMOS matrix and overlapping fields of view were used for the acquisition of the particle images. The adoption of two cameras guaranteed that the entire evolution of the laminar separation bubbles was recorded with adequate spatial resolution. Data measured in the overlap region were also used to correct small differences in the setup of the cameras. Based on the distance between the cameras and the PIV measuring plane, a scale factor of 5.11 was set for the present experiments. The adaptive cross-correlation of PIV paired-images was computed considering an initial interrogation area of 32x32 pixels and a successive refinement to 16x16 pixels with 50% overlap. This provided a vector spacing of 0.41 mm. Invalid vectors were detected by means of peak validation methods and a sub-pixel recognition accuracy of particle displacement equal to 0.1 pixel was obtained using a Gaussian fitting procedure. The time between laser pulses was chosen in order to obtain a mean particle displacement of 4 pixels in the free-stream region, which corresponds to 1/4 of the PIV interrogation region size. The uncertainty in the instantaneous velocity measurement is 3% in the free-stream region and 6% in the boundary layer. The uncertainty magnitude was computed by means of the peak-ratio method proposed by Charonko and Vlachos (2013). The adopted procedure provides the particle displacement uncertainty as a function of the dominant peak prominence of the cross-correlation function (Sciacchitano et al. 2015). In the near wall region, the particle displacement variation (Wilson and Smith 2013) was verified to not exceed 5% of the final size of the PIV interrogation region. With this setup, a marked peak in the correlation function of PIV images was consistently observed for all cases (Westerweel 2008). For each combination of the flow parameters, two sets of 6000 PIV snapshots were acquired at a sampling rate of 1000 Hz. To ensure the statistical convergence of data, temporal auto-correlation functions of the PIV fluctuating velocity fields were computed at several measuring points (Dellacasagrande etal. 2023). The time-scale of the largest coherent fluctuations was verified to be captured considering only 3000 over 6000 PIV snapshots. Also, about 100 to 500 shedding cycles were acquired at a sampling rate of 1 kHz across the range of Reynolds numbers tested. For the highest Reynolds numbers, the shedding period was discretized with around 20 sample points, which increased to approximately 70 for the lowest ${\text{Re}}_L$. For more comprehensive information about the experimental setup and measurement techniques, the reader is referred to the previous authors’ works Dellacasagrande et al. (2021); Dellacasagrande etal. (2023).

3 Time-mean response of laminar separation bubbles

Figure 2a shows the effects of ${\text{Re}}_L$ variation on the time-mean shape of an LSB for AP=-0.41 and Tu=1.5% (${\text{Re}}_L$ increases from top to bottom). The streamwise velocity u is scaled with the external velocity $U_e$, which was defined based on the local wall-normal velocity gradient. A remarkable variation of the extension of the recirculating flow region is observed when changing the Reynolds number. Since the BL separates at about $x/L=0.42$ for all cases, the enlargement of the bubble observed when lowering the Reynolds number is primarily due to the delayed reattachment. The BL separation position was determined considering the extension of the zero velocity line to the wall. At least three measuring points were used to estimate the slope of the zero velocity line near the wall and its successive extrapolation toward the plate surface. It has to be noted that the growth rate of the bubble dimensions increases for ${\text{Re}}_L<40600$. This is shown in Fig. 2b, where the bubble length ($L_b$) and height ($H_b$) are plotted against the Reynolds number for different Tu levels and pressure gradients. $H_b$ was computed as the maximum height of the separating streamline of the LSB; while, $L_b$ is the distance between the separation and reattachment positions. Both $L_b$ and $H_b$ increase when reducing ${\text{Re}}_L$, as expected. Interestingly, a change in the growth rate of the bubble height is observed below a certain Reynolds number that is not clearly observable in the streamwise length. As shown in the following, the change in the time-mean response of the bubble to the variation of ${\text{Re}}_L$ is linked to the occurrence of bursting and the consequent modification of the bubble state. Also, the different response observe for $H_b$ and $L_b$ to ${\text{Re}}_L$ variation implies that a modification of the scaling law of an LSB occurs when passing from the short to the long bubble regime. In order to verify the occurrence of short and long LSBs within the present database, the bursting parameter $P_{\text{DCR}}=h^2/\nu (\Delta U/\Delta X)_{act}$ (Diwan etal. 2006) was computed. As previously mentioned, h is the maximum height of the laminar separation bubble, and $(\Delta U/\Delta X)_{act}$ is the actual velocity gradient across the bubble length. Figure 3 reports $P_{\text{DCR}}$ against the momentum thickness Reynolds number computed at the separation position (${\text{Re}}_{\theta ,s}$). As ${\text{Re}}_{\theta ,s}$ reduces at fixed Tu and AP, $P_{\text{DCR}}$ falls below the bursting threshold of -28 (Diwan etal. 2006), which indicates the existence of a critical Reynolds number for bursting. It has to be noted that such critical value of ${\text{Re}}_{\theta ,s}$ is influenced by the free-stream turbulence level and, to a smaller extent, by the change in the pressure gradient. The higher the Tu level, the lower the critical ${\text{Re}}_{\theta ,s}$ value. Indeed, a higher level of free-stream disturbances promotes the occurrence of transition, thus shortening the length of an LSB with respect to a low turbulence environment (Istvan and Yarusevych 2018). On the other hand, the increment of the opening angle of the channel causes the thickening of the LSB with the shear layer moving far from the wall. Consequently, bursting occurs for higher values of the flow Reynolds number. Data reported in Sect. 4 will provide evidence of the link between the change in the bubble state observed here and the modification of the instability processes occurring in a laminar separation bubble.

4 Instability properties of short and long laminar separation bubbles

Figure 4 shows exemplary PIV snapshots of the fluctuating velocity fields for the highest (left) and the lowest (right) Reynolds numbers at Tu=1.5% and AP=$-$0.41. These flow cases are representative of a short and a long bubble case, respectively. At the high Reynolds number, trains of spanwise vortices typical of the inviscid Kelvin–Helmholtz instability are observed near the bubble maximum displacement (see the red box in the first snapshot). K–H rolls are formed in the region of maximum shear due to convective instability of disturbances, and they grow accordingly in the streamwise direction (Hosseinverdi and Fasel 2019). Further downstream, the 2D K–H rolls break up to turbulence and new rolls originate from the shear layer roll-up (see e.g., the fifth and the eighth plots). Features typical of the K–H instability, such as those highlighted here, were observed for all the acquired short laminar separation bubbles. As the Reynolds number reduces below the bursting threshold (right plots of Fig. 4), vortical structures are seen to form within the recirculation region (see the red circles within the blue contoured region). Such structures were found to grow primarily in time at almost fixed position, suggesting the occurrence of an absolute type instability instead of a convective one (Pauley et al. 1990). Also, their size is typically smaller than the height of the recirculation region; thus, their effectiveness in promoting the turbulent reattachment is reduced compared to the larger K–H rolls (Serna and Lázaro 2015). Nevertheless, vorticity cores forming in the region of negative streamwise velocity seem to promote the shrinkage of the overlaying shear layer and the successive formation of spanwise oriented vortices near the maximum height of the bubble (see e.g., plots from 1 to 6). These latter structures resemble the convective 2D vortices occurring at higher ${\text{Re}}_L$ values. Features typical of the convective K–H instability are therefore observed in long LSB together with the temporal growth of fluctuations in the dead air region. Data presented in the following will show that a marked change in the scaling of the mean vortex size with the shear layer thickness at the separation position occurs due to the bursting process of an LSB.

Fig. 3

Bursting parameter ($P_{\text{DCR}}$) as a function of the momentum thickness Reynolds number computed at separation position (${\text{Re}}_{\theta ,s}$). Bursting threshold is highlighted with black dashed line. Shaded areas represent the $\pm 10\%$ uncertainty bound

Fig. 4

Vector plots of fluctuating velocity fields for ${\text{Re}}_L$=66,200 (left) and Re=15,300 (right) for Tu=1.5% and AP=$-$0.41. The contour plot of the time-mean streamwise velocity is superimposed to the plots. Data are shown from x/L=0.5 to improve the readability of the plots

Fig. 5

Vector plots of POD modes 1, 2, 3 and 4 for ${\text{Re}}_L$=66,200 (left) and 15,300 (right) at Tu=1.5% and AP=$-$0.41. The contour plot of the time-mean streamwise velocity is superimposed to the plots. Data are plotted from x/L=0.5 to improve the readability of the plots

Fig. 6

Energy of leading POD modes for ${\text{Re}}_L$=15,300 and 66,200 at Tu=1.5% and AP=$-$0.41. Modes were computed from the POD total kernel ($\lambda _{uv}$) and the wall-normal fluctuating velocity only ($\lambda _{v}$)

Fig. 7

Non-dimensional wavenumber ($k\delta ^*_s$) and Strouhal number (St) against ${\text{Re}}_L$ for different Tu levels and pressure gradients. Shaded areas represent the $\pm 10\%$ uncertainty bound

Fig. 8

Amplitude of 2D Fourier transform of wall-normal fluctuations. Data are plotted at fixed wall distance equal to half of the height of the recirculation region. Amplitude of the Fourier transform is scaled with the root mean square of velocity fluctuations at the same wall-normal position at which the Fourier transform is computed

Fig. 9

a Non-dimensional height of the recirculation region ($h_r/\delta ^*$) and b turbulent time-scale ($T_v$) as a function of the momentum thickness Reynolds number at the point of separation (${\text{Re}}_{\theta ,s}$). Threshold for absolute instability is highlighted with black dashed line in (a). Shaded areas represent the $\pm 5\%$ uncertainty bound

To provide the statistical characterization of the coherent structures forming in the short and the long bubble regimes, the proper orthogonal decomposition (POD, Lumley (1970)) was applied to the fluctuating velocity field for all the combinations of the flow parameters. For a better characterization of the spanwise vortices driving the momentum transfer and reattachment near the bubble maximum displacement, only the wall-normal fluctuations were considered for the education of the POD coefficients. The streamwise fluctuating velocity is indeed mostly affected by shear layer motions and flapping bubble behavior (Dellacasagrande etal. 2020b). The spatial POD modes were then obtained by projection of both velocity components on the so derived coefficients. Fig. 5 shows the first four POD modes for the highest and the lowest ${\text{Re}}_L$ values at Tu=1.5% and AP=$-$0.41 (i.e., for the same cases shown in Fig. 4). For completeness, Fig. 6 reports the relative energy of the leading modes derived from both the streamwise and the wall-normal velocity fluctuations (i.e., from the POD total kernel) as well as for the wall-normal fluctuations only. The POD modes of the high Reynolds number case (left plots of Fig. 5) show counter rotating vortices typical of the Kelvin–Helmholtz instability (Lengani et al. 2014). Vorticity cores are shown near the bubble maximum displacement position; while, smaller fluctuations occur within the recirculating flow region. Differently, POD modes of the low Reynolds number case (right plots) depict velocity fluctuations and vorticity cores in the region of negative streamwise velocity, similarly to what was shown in Fig. 4. Additionally, marked fluctuations are captured by the modes in the fore part of the bubble together with large scale vortices forming near the bubble maximum displacement. The POD spectra reported in Fig. 6 show highly coupled modes of the wall-normal fluctuations (squared symbols) at high Reynolds number (modes 1–2 and 3–4). Mode coupling indicates the occurrence of periodic convective structures (Legrand et al. 2011a, b), which, for the high Reynolds number case are indeed the 2D K–H rolls originating from the inflectional instability of the boundary layer (right plots of Fig. 5). When ${\text{Re}}_L$ falls below the bursting threshold, modes coupling is reduced. Accordingly, the first two POD modes of the wall-normal fluctuations cannot be assumed as representative of the same convective structures, which further highlights the modification of the main dynamics of the LSB due to the bursting occurrence. To further investigate the modification of the shear layer stability properties due to changes in the bubble state, the modes of the wall-normal velocity fluctuations were used for the computation of the spatio-temporal scales of the main convective structures shed by short and long laminar separation bubbles. Their wavenumber (k) and frequency ($f_{\textrm{sh}}$) were computed by means of the Fourier transform of the POD modes and the related temporal coefficients, respectively. Namely, streamwise slices of the first POD mode of v were considered, which retained the periodic signatures of the vortices shed by the LSB. Their corresponding Fourier transform provided the associated wavelength $\lambda$, from which $k=2\pi /\lambda$ was computed. Then, the wavenumber k was non-dimensionalized using the boundary layer displacement thickness at the separation position ($\delta ^*_s$), i.e., $k\delta ^*_s$. It is further mentioned that, since $\lambda$ corresponds to about twice of the mean diameter of the vortices shed by an LSB, the quantity $k\delta ^*_s$ provides the scaling of the dimension of the largest spanwise vortices with the shear layer thickness at the separation position. To investigate also the scaling of their temporal frequency, a Strouhal number of the shedding vortices was computed as $St=f_{sh}\theta _s/U_{e,s}$, where $\theta _{s}$ and $U_{e,s}$ are the momentum thickness and the free-stream velocity at the separation position (Talan and Hourmouziadis 2002). Fig. 7 depicts $k\delta ^*_s$ and St against ${\text{Re}}_L$ for different Tu levels and pressure gradients. Data clearly highlight the existence of two distinct flow regimes depending on the Reynolds number. At high ${\text{Re}}_L$, $k\delta ^*_s$ and St are almost constant and fall within the range 0.81$-$1.05 and 0.009$-$0.014, respectively. According to Simoni et al. (2014) and McAuliffe and Yaras (2010) for instance, these are the typical values characterizing the inviscid K–H instability, which was shown to drive the separated shear layer transition in the short regime. As the Reynolds number reduces, $k\delta ^*_s$ and St decrease to 0.55$-$0.68 and 0.003$-$0.007, respectively. Based on data reported in Fig. 3, these cases are representative of long bubble states; thus, the non-dimensional size of the vortical structures shed by an LSB grows by almost 30% when passing from the short to the long bubble regime, and the characteristic time-scale significantly increases. Figure 8 depicts the 2D spatio-temporal Fourier transform of wall-normal velocity fluctuations computed within the recirculating flow region for the same cases shown in Figs. 4, 5. The Fourier transform was computed at fixed wall-normal position equal to half of the maximum height of the recirculating flow region. Such kind of analysis is aimed at the characterization of the smaller vortices forming in the dead air region. Contrarily, results shown in Fig. 7 relate to the sensibly larger structures forming in the region of maximum shear.

The discrete 2D Fourier transform of the wall-normal fluctuating velocity v(x, t) at a fixed wall-normal position is defined as follows (Andrews 1978):

$$\begin{aligned} \frac{1}{MN}\sum ^M_{x=1}\sum ^N_{t=1} v(x, t)_y e^{-i2\pi (f_xx/M + f t/N)} \end{aligned}$$

(1)

Here, $f_x$ and $f$ are the spatial and temporal frequencies, and M and N are the number of spatial points and time instants. Based on separability property of the 2D Fourier transform (Rabiner and Gold 1975), equation 1 can be factored into the following separable form:

$$\begin{aligned} \frac{1}{MN}\sum ^M_{x=1}e^{-j2\pi (f_xx/M)}\sum ^N_{t=1} v(x, t)_y e^{-j2\pi (f t/N)} \end{aligned}$$

(2)

Non-null Fourier amplitudes for positive $f_x$ and f values indicate temporal oscillations that propagate forward. Contrarily, backward propagating waves are identified by spatio-temporal frequencies of opposite sign. By comparison of Fourier magnitudes, one can identify which propagating structures dominate within the LSB, with their associated speed being proportional to the ratio $f/f_x$ (Peyret 2002). At high ${\text{Re}}_L$ (Fig. 8a), large amplitudes are observed for positive values of $f_x$ and f. As above mentioned, this indicates the occurrence of forward propagating fluctuations (black dashed line), accordingly to the convective growth of disturbances typical of short LSBs. Contrarily, as the Reynolds number reduces below the bursting threshold (Fig. 8b), two distinct lines of propagation are recognizable. Forward propagating fluctuations still occur (positive $f_x$ and f); while, a significant amount of energy of the Fourier transform is also observable in the top-left quadrant, where $f_x>0$ and $f<0$ (dash-dotted line). Therefore, most of disturbances tend to move backward within the recirculation region at sensibly lower speed. Data presented in Figs. 5–7 indicate that a substantial change in the stability properties of the separated shear layer may occur when switching from the short to the long bubble regime. With the aim of assessing whether the highlighted changes in the dynamics of long LSBs were indicative of the occurrence of an absolute type instability, fig. 9a reports the maximum height of the recirculation region ($h_r$) scaled with the local displacement thickness, i.e., $h_r/\delta ^*$. Data are plotted against ${\text{Re}}_{\theta ,s}$. According to Rist and Maucher (1994), absolute instability may occur in an LSB if $h_r/\delta ^*>0.6$. Also, a newly defined turbulent time-scale ($T_v$) is introduced in this work to determine the effectiveness of the coherent structures forming near the bubble maximum displacement in promoting momentum transfer and reattachment (Fig. 9b). Such turbulent scale is defined as the ratio between the maximum bubble height h and the wall-normal turbulence stresses, i.e., $T_{v}=h/\sqrt{\overline{v'^2}}$. Figure 9 shows that $h_r/\delta ^*$ approaches the limit value of 0.6 in correspondence of a noticeable increment of $T_v$. This occurs for the same ${\text{Re}}_{\theta ,s}$ values for which $k\delta ^*$ and St reduce in Fig. 7 and $P_{\text{DCR}}$ falls below the bursting threshold of -28. The occurrence of bursting is therefore associated to a significant change in the stability properties of the separated shear layer, which becomes prone to absolute instability. Moreover, the enlargement of the separated flow region appears to be linked to a reduced effectiveness of the largest vortical structures shed by an LSB in promoting momentum transfer through the maximum bubble height.

5 Prediction of bursting Reynolds number

As shown in the previous sections, ${\text{Re}}_L$, Tu level and AP variations can substantially modify the time-mean structure and stability properties of an LSB. In this section, a correlation for the critical value of ${\text{Re}}_{\theta ,s}$ for which bursting occurs is derived as a function of the Tu level and AP. ${\text{Re}}_{\theta ,s}$ was indeed previously adopted in the two parameters bursting criterion proposed by Gaster (1967); while, Mitra and Ramesh (2019) have successively shown the proportionality between their bursting parameter $P_{\text{DCR}}$ and ${\text{Re}}_{\theta ,s}$ (see also Serna and Lázaro (2015)). The main limitation of bursting criteria such as those proposed by Gaster (1967) and Diwan etal. (2006) for instance, is that the time-mean shape of an LSB must be known in order to classify it as short or long one. The current measured data are used instead to provide a simplified tool for the prediction of the state of an LSB as a function of only the free-stream turbulence and the pressure gradient. AP and the Tu level are indeed easily measurable without knowing a priory the geometry of the laminar separation bubble.

Fig. 10

Critical momentum thickness Reynolds number at separation ($Re^*_{\theta ,s}$) as a function of flow parameters

Figure 10 synthesizes the effects due to the flow parameter variation on the critical ${\text{Re}}_{\theta ,s}$ for which $k\delta _s^*$ and St start to reduce in Fig. 7. The corresponding $P_{\text{DCR}}$ values (Fig. 3) were found to be close to the bursting threshold of -28 proposed by Diwan etal. (2006). The critical ${\text{Re}}_{\theta ,s}$ values shown in Fig. 10 fall below the range of 200–240 typically reported in the literature (Rg et al. 2004; Hatman and Wang 1999; Serna and Lázaro 2015; Mitra and Ramesh 2019). Nevertheless, the current data were acquired at significantly higher free-stream turbulence levels compared to the previous studies. Figure 10a clearly shows that the critical ${\text{Re}}_{\theta ,s}$ reduces when increasing the free-stream turbulence level. Disturbances carried by the free-stream act indeed promoting the BL transition and the consequent reduction of the bubble dimension. Contrarily, the higher the pressure gradient the higher the critical ${\text{Re}}_{\theta ,s}$, due to an increase in the bubble height and higher risk of stall. In order to combine the effects due to the Tu level and the pressure gradient variation, the critical ${\text{Re}}_{\theta ,s}$ value was plotted against the absolute value of the ratio Tu/AP (Fig. 10b), where Tu is expressed as a percentage value. Data shown in Fig. 10b are seen to follow a well-defined trend, and a power fit to the experiments provided the fitting law ${\text{Re}}_{\theta ,s}=251.14\vert \textrm{Tu}/\textrm{AP}\vert ^{-0.305}$. Once combined with laminar separation criteria providing ${\text{Re}}_{\theta ,s}$ as a function of the free-stream turbulence and the streamwise pressure gradient e.g., Curle and Skan (1957), the present correlation can be used to predict the occurrence of the bursting process of an LSB. Furthermore, the results reported in this work indicate that the same correlation can be adopted to predict the switch from a convective to an absolute type instability in a laminar separation bubble.

6 Conclusions

The present work reports the experimental study of short and long laminar separation bubbles forming on a flat plate for different values of the Reynolds number, Tu level and the pressure gradient. The critical Reynolds number causing the change in the bubble state was detected for each combination of Tu and AP. More precisely, the bursting condition was identified in terms of the momentum thickness Reynolds number at the separation position. The occurrence of bursting was shown to cause a marked growth of the bubble height; whereas, a smaller increment of the bubble length was observed. This indicates that a change in the scaling law of an LSB may occur between the short and the long bubble regimes. A substantial change in the dynamics of an LSB was found to occur when switching from the short to the long bubble regime. The K–H convective instability was found to drive the separated shear layer transition in short bubbles. Conversely, absolute instability was seen to occur in long LSBs, driving the temporal growth of spanwise vortices within the recirculating flow region. Also, significantly larger vortices were found to form in the region of maximum shear, resembling K–H rolls. Nevertheless, a marked change in the scaling law of the dominant structures forming in the high shear region was observed when passing from the short to the long bubble regimes. Such structures appeared to be less effective in promoting momentum transfer through the bubble maximum height with the consequent enlargement of the recirculating flow region.

Finally, a correlation for the bursting Reynolds number as a function of the Tu level and the pressure gradient was provided. The acquired data showed that the same correlation can be adopted to predict changes in the stability characteristics of an LSB.