Vortex merging in a laminar separation bubble under natural and forced conditions

Vortex merging in a laminar separation bubble (LSB) is studied experimentally. The bubble is formed on the suction side of a NACA 0018 airfoil at an angle of attack of $4^{\circ }$ and a Reynolds number of 125 000. The merging process in the bubble is manipulated through acoustic forcing applied as a tone at either the fundamental vortex shedding frequency or at the first subharmonic of this frequency. The flow field is assessed using time-resolved, two-component particle image velocimetry. A method for detecting merged structures using wavelet analysis is introduced, allowing for reliable quantification of merging events. The results show that vortex merging occurs naturally in the separation bubble, while forcing at the subharmonic and fundamental frequencies promotes and inhibits merging, respectively. While these trends are similar to those observed for free shear layers, the subharmonic forcing of an LSB is found to directly promote disturbance development at the subharmonic frequency. For all cases, the majority of merging events take place in the aft portion of the bubble, i.e., downstream from the maximum bubble height location and upstream of mean reattachment, with subharmonic forcing causing merging to shift upstream. The merged structures are found to be the most energetic flow features; however, promoting vortex merging through subharmonic forcing does not lead to significant changes in the mean bubble topology. The spanwise behavior of the vortex merging process is studied, revealing that structures merge in a spanwise nonuniform manner, with localized merging occurring away from where forward or rearward streamwise bugles develop in the vortex filaments. Statistical characterization reveals that merging tends to occur more often over some specific spanwise segments, with the number of primary structures that merge varying by as much as 50% between spanwise locations. These findings offer insight into vortex merging in laminar separation bubbles and the attendant influence of forcing, while also highlighting the need to consider spanwise aspects of flow development.

Figure 1

Spectra of fluctuating pressure (${\varphi }_{p^{\prime }{p}^{\prime }}$) measured near the natural separation point for the natural (Nat.), fundamental (F), and subharmonic (SH) cases in (a) quiescent and (b) the investigated flow conditions. Dashed and dotted lines denote ${\text{St}}_0$ and $\frac{1}{2}{\text{St}}_0$, respectively. Gray shaded region indicates frequencies associated with test section standing wave.

Figure 2

Experimental configurations for PIV measurements showing the surface-attached coordinate system, whose origin is located at the leading edge and midspan point.

Figure 3

Mean streamwise ($\overline{u}$), and rms of fluctuating ($u_{\text{rms}}^{\prime }$, $v_{\text{rms}}^{\prime }$) velocity contours. Solid lines mark the dividing streamlines, whose uncertainty limits are indicated by the dotted lines. Triangle and square markers denote mean maximum bubble height and reattachment points, respectively. Dashed lines indicate displacement thickness (${\delta }^{*}$).

Figure 4

Sequences of instantaneous spanwise vorticity ($\omega$) contours. Consecutive frames are separated by $t^{*}=t{U}_0/c=3.7\times {10}^{-2}$. Solid black lines indicate ${\lambda }_2$ contours [78]. Solid gray lines mark the dividing streamlines. Dashed lines trace the same vortices.

Figure 5

Normalized wavelet coefficient ($\mathrm{\Gamma }$) contours computed from wall-normal velocity fluctuations sampled at $y={\delta }^{*}$ and the midpoint between $x_h$ and $x_r$. Dashed regions in (a)–(c) replotted in (d)–(f). Dotted regions in (d)–(f) correspond to intervals shown in Fig. 4. Markers denote local maxima identified at frequencies of interest.

Figure 6

Ratio of detected merged structures to total number of shed primary structures ($R_{\text{merg}}$) determined using wavelet methodology (cf. Fig. 5). Dashed and dotted lines (colored according to legend) mark $x_h/c$ and $x_r/c$, respectively. Shaded regions (colored according to legend) indicate uncertainty bounds.

Figure 7

Frequency [(a)–(c)] and wave-number-frequency [(d)–(f)] spectra of wall-normal velocity fluctuations (${\varphi }_{v^{\prime }{v}^{\prime }}$) measured within the separated shear layer ($y={\delta }^{*}$). Dashed and dotted lines mark $x_h/c$ and $x_r/c$, respectively. Solid line is a linear fit estimating the convective ridge.

Figure 8

Streamwise growth of frequency filtered wall-normal disturbances ($\widetilde{v^{\prime }}$) within the separated shear layer ($y={\delta }^{*}$). Dashed and dotted lines mark $x_h/c$ and $x_r/c$, respectively.

Figure 9

POD (a) relative and (b) cumulative modal energy distributions.

Figure 10

Normalized POD spatial modes colored by the wall-normal component (${\mathrm{\Phi }}_v^{\left(m\right)}$). Solid lines mark the dividing streamlines.

Figure 11

Power spectral density (PSD) of POD temporal coefficients ($A^{\left(m\right)}$). Dashed, dotted, and dash-dotted lines mark ${\text{St}}_0$, $\frac{1}{2}{\text{St}}_0$, and ${\text{St}}_{\text{sw}}$, respectively.

Figure 12

Sequences of instantaneous spanwise vorticity contours constructed from POD reduced order models for the subharmonic excited flow. Models reconstructed used the (a) two and (b) eight most energetic modes. Consecutive frames are separated by $t^{*}=2.5\times {10}^{-2}$. Solid black lines indicate ${\lambda }_2$ contours [78]. Solid gray lines mark the dividing streamlines. Dashed lines trace the same structures.

Figure 13

Sequences of instantaneous streamwise velocity contours. Flow is from top to bottom. Consecutive frames are separated by $t^{*}=2.5\times {10}^{-2}$. Dashed lines indicate smoothed spline fits to the center of selected structures (labeled A–F).

Figure 14

Spanwise wavelength probability distributions ($P_{{\lambda }_z}$) at $x/c=0.55$. Dotted lines indicate standard deviation from the mean (dashed line).

Figure 15

Spanwise variation in ratio of detected merged structures to total number of shed primary structures using wavelet methodology (cf. Fig. 5). Method is evaluated at streamwise locations within $0.545\le x/c\le 0.555$, with results averaged to produce a single curve representative of the region where $R_{\text{merg}}$ plateaus in Fig. 6. Shaded regions (colored according to legend) indicate uncertainty bounds.