Response of a turbulent separation bubble to zero-net-mass-flux jet perturbations

The response of a turbulent separation bubble (TSB) to zero-net-mass-flux actuation is investigated via direct numerical simulations. Rectangular jets with their long axis oriented in the streamwise direction are used to generate unsteady streamwise vortices that mimic the streamwise elongated Görtler vortices found to be associated with the low-frequency unsteadiness of the TSB [Wu et al., J. Fluid Mech., 883, A45 (2019)]. Three sinusoidal actuation frequencies are investigated, corresponding to the two natural frequencies of the undisturbed separation bubble ($f_l$ and $f_h$ with a ratio of $f_h/f_l=2.5$) and a high frequency at $10f_l$ motivated by a harmonic resolvent analysis. The results are compared to the baseline uncontrolled flow. Very-large scale (VLS), spanwise-rotating vortices are formed at $f_l$ and $f_h$, causing a 50% reduction in the mean TSB length. A counter-rotating secondary vortex is induced locally by the VLS vortex in the $f_l$ case and forms a vortex pair with the VLS vortex as they move downstream together. The interaction of the vortex pair facilitates their decay. The VLS vortex generated by the forcing at $f_h$ is not strong enough to produce such a secondary vortex. Spectral analysis of the harmonic resolvent operator is used to quantify the receptivity of the flow to actuation at different frequencies. The perturbations that excite the most energetic response in the flow are indeed in the form of streamwise-elongated structures in the separation region at $f_l$ and $f_h$. Energetic structures corresponding to the temporal mean obtained from the analysis are found to extend to distances far downstream of the separation bubble confirming the great sensitivity of the entire flow to such forcing.

Figure 1

Schematic of a separating turbulent boundary layer due to freestream adverse pressure gradient. The contours show the instantaneous turbulent structures by isosurfaces of the second invariant of the velocity gradient tensor, colored by the distance from the bottom wall. Also shown are the suction causing the imposed APG in the flow, the actuators as well as the reference and recycling/rescaling planes.

Figure 2

Comparison of the ZPG TBL generated by the recycling-and-rescaling method by the reference location with other DNS of ZPG TBLs. Square, Lee and Zaki (2018) at ${\text{Re}}_{\theta }=500$; cross, DNS by Wu, Moin and Hickey (2014) at ${\text{Re}}_{\theta }=670$; triangle, DNS by Schlatter and Örlü (2010) at ${\text{Re}}_{\theta }=670$. The current result is at ${\text{Re}}_{\theta }=490$.

Figure 3

Contours of the mean streamwise velocity. Solid lines are selected streamlines that pass through $x=100{\theta }_o$, $y=[0.2,5,10,15,20,25,30,35]{\theta }_o$. Dashed lines show the contour of $U=0$. From top to bottom: case NA, FL, FH, and 10FL.

Figure 4

Mean skin friction coefficient $C_f$ (a) and pressure coefficient $C_p$ (b). $\text{\_\_\_\_\_\_}$ Case NA, $\text{\_}\text{\_}\text{\_}$ Case FL, $\text{\_\_}\text{\_}\text{\_\_}$ Case FH, $+$ Case 10FL. The thick dot-dashed line with a plateau of $C_p$ at $x>300{\theta }_0$ in panel (b) is the inviscid pressure profile shown as a reference.

Figure 5

Instantaneous separating flow shown by the contours of instantaneous streamwise velocity in the middle spanwise section. (a) NA; (b) FL; (c) FH; (d) 10FL. For each case with the actuation, the selected time frame is when $V_{\text{ac}}=0$ and $d{V}_{\text{ac}}/dt<0$. Also shown are instantaneous velocity vectors on the plane. Circle markers show the locations where the joint-PDFs of velocity fluctuations are examined in Fig. 12.

Figure 6

Instantaneous turbulent structures visualized by isosurfaces of the secondary invariant of the velocity gradient tensor (see text). The isosurface shown is for the threshold $Q=0.0165U_o^2/{\theta }_o^2$ and is colored by the distance from the wall. (a–d) FL; (e–h) FH. Top to bottom correspond to time instants $0^{\circ }$ (a, e), ${90}^{\circ }$ (b, f), ${180}^{\circ }$ (c, g), and ${270}^{\circ }$ (d, h) of the corresponding forcing period in each case.

Figure 7

Contours of instantaneous streamwise velocity on the $\left(x\text{−}z\right)$ plane corresponding to the first DNS grid point away from the wall. (a) NA; (b) FL; (c) FH; (d) 10FL. The time frame shown for each case is the same as the one shown in Fig. 5.

Figure 8

Probability density function of the instantaneous, local friction coefficient as function of downstream distance. The mean $C_f$ of each case and range of its root-mean-square value are also plotted for reference as solid and dashed lines. (a) NA; (b) FL; (c) FH; (d) 10FL. The actuation region is $x\in [50,108]{\theta }_o$.

Figure 9

(a) Probability density function of the instantaneous, local friction coefficient at $x=300{\theta }_o$. $\text{\_\_\_\_\_\_}$, case NA; $\text{\_}\text{\_}\text{\_}$, case FL; $\text{\_\_}\text{\_}\text{\_\_}$, case FH $\text{\_}\text{\_}\text{\_}$, case 10FL. (b, c) visualizations of the instantaneous distribution of $C_f$ when values near the peaks of the PDFs shown in panel (a). (b) Case FL, peak A, red; peak B, blue; peak C, green. (c) Case FH, peak D, red; peak E, blue. $C_{f,\text{peak}\pm 0.5}$ is used as condition near each peak, i.e., filled (FL) and hatched (FH) patches in panel (a).

Figure 10

Time history of the total area of the recirculation region in the $\left(x\text{−}y\right)$ plane between $x=0$ and $x=500{\theta }_o$ (a) and its premultiplied energy spectrum (b). The instant shown as $t=0$ in panel (a) is approximately when a statistically steady state is reached after the transition from the initial undisturbed field. Thin vertical lines indicate the three natural frequencies of the undisturbed TSB [39].

Figure 11

Contours of the turbulent kinetic energy. The locations where the joint probability density function of $u^{\prime }$ and $v^{\prime }$ are examined are marked by $+$. (a) NA; (b) FL; (c) FH; (d) 10FL.

Figure 12

Contours of JPDF of streamwise and wall-normal velocity fluctuations for the four cases (one in each row). The JPDFs are examined at $y/{\theta }_o=25$ and four streamwise locations (one in each column). The bin size is $0.005U_o$. The JPDF is calculated using 500 snapshots and all the DNS grid points in the spanwise direction, at each $x$ location.

Figure 13

Schematics of the vortex-induced velocity fluctuation. Point A (B), a location that is above (underneath) the trajectory of the primary vortex. From (a) to (c), time instants that the vortex is approaching and passing Points A and B.

Figure 14

(a, b) Time-averaged streamwise and wall-normal velocities used as the zero-frequency component of the base flow. (c, d) Streamwise and wall-normal velocities from the Fourier mode ${\widehat{\mathbf{Q}}}_1$ associated with the breathing frequency $f_l$.

Figure 15

(a) Leading $m=10$ singular values of harmonic resolvent operator $\mathbf{H}$. (b) Leading singular value ${\lambda }_{(k,j),1}$ computed for the blocks ${\mathbf{H}}_{k,j}$ of the harmonic resolvent operator.

Figure 16

(a) The optimal streamwise velocity as the control input at frequency $\omega$, extracted from the first right singular vector ${\widehat{\psi }}_1$ of $\mathbf{H}$. (b) the most significant response of the streamwise velocity at frequency 0 (i.e., the mean flow), extracted from the first left singular vector ${\widehat{\varphi }}_1$ of $\mathbf{H}$.