Direct numerical simulation of vortex-induced instability for a zero-pressure-gradient boundary layer

Vortex-induced instability caused by a free-stream vortical excitation is explored here quantitatively with the help of controlled computational results. First, the computed results are compared with experimental results in Lim et al. [Exp. Fluids 37, 47 (2004)] for the purpose of validation of the three-dimensional (3D) computations. Thereafter, the computed results are explained using methods developed to study nonlinear and spatiotemporal aspects of receptivity and instability for incompressible flows. Here a zero-pressure-gradient (ZPG) boundary layer is perturbed by a constant strength vortex traveling at a fixed height, moving with constant speed, as in the cited experiment. The vortex is created by a translating and rotating circular cylinder in the experiment, with absolute control of the physical parameters. The sign of the translating vortex is fixed by the direction of rotation of the translating cylinder. A high accuracy computing method is employed to solve the 3D Navier-Stokes equation (NSE) for different translation speeds and signs of the free-stream vortex. A nonlinear disturbance enstrophy transport equation (DETE) for incompressible flows is used to explain the vortex-induced instability. This equation is exact and explains the instabilities, as governed by the NSE. The DETE approach has been successfully developed to explain two-dimensional (2D) vortex-induced instability in Sengupta et al. [Phys. Fluids 30, 054106 (2018)], to trace the linear and nonlinear stages of disturbance growth. Apart from quantification of vortex-induced instability, another major goal is to show how the disturbance evolves from an initial 2D to a 3D stage. While the sign of the translating vortex is important in creating the response field, we additionally highlight the distinct differences caused by increased translation speed and strength of the free-stream vortex on the overall instability. These explain creation of small-scale vortices via the instability of an equilibrium flow, even though the excitation is 2D only. For some cases, this causes 3D bypass transition. We also show a case which demonstrates strong unsteady separation with inflectional velocity profiles, yet the disturbance flow remains essentially 2D, which can be termed a bypass transition.

Figure 1

(a) Schematic of the problem of the computational domain used to study vortex-induced instability by a translating vortex (speed, $c$, and circulation, $\mathrm{\Gamma }$) in the free stream over a ZPG boundary layer. (b) The plan and side view of the experimental arrangement, with the flat plate fitted with a flap; a rotating and translating circular cylinder causing the free-stream vortex and arrangement for dye visualization are shown. (c) Photographic experimental setup in the water tunnel shown.

Figure 2

Schematic of the experimental setup used by Kendall  [19] for the periodic free-stream vortex excitation problem over a zero-pressure-gradient boundary layer (top). Peak response of streamwise disturbance velocity ($u^{\prime }$) measured for different rotor speeds as obtained by Kendall [19] (bottom).

Figure 3

Staggered grid for velocity and vorticity components used for the 3D velocity-vorticity formulation.

Figure 4

Schematic of streamlines obtained by potential flow model created by a translating and rotating cylinder given by Eq. (2). (a) The case for clockwise rotating vortex, and above it is the spectrum of the induced flow field. (b) The streamline and the corresponding spectrum for the case of a counterclockwise vortex.

Figure 5

(a) Snapshots of the receptivity experiment for $c=0.386$ shown in the left column, with experimental parameters described in the text. (b) Corresponding computational results in the right column, for the counterclockwise rotating vortex case, show strong receptivity for a ZPG boundary layer.

Figure 6

(a) Snapshots of the receptivity experiment for $c=0.77$, with other parameters described in the text. For the counterclockwise rotating vortex case, one notices very weak receptivity for a ZPG boundary layer. (b) Corresponding computed results with same value of $c=0.77$ and $\mathrm{\Gamma }=0.5$ are shown for ${\omega }_3$ isocontours at the indicated times, colored with ${\omega }_2$.

Figure 7

(a) Snapshots of the receptivity experiment for the clockwise rotating vortex case, translating with $c=0.19$ and other parameters described in the text. For the clockwise rotating vortex case, one notices weaker receptivity, with vortex-induced perturbation visible upstream of the translating vortex, for the ZPG boundary layer. (b) Corresponding computed results are shown for this case with spanwise vorticity, ${\omega }_3$, at the indicated times for the indicated contour levels.

Figure 8

Maximum vorticity components in the flow domain shown as function of time, for the cases of $c=0.3$, 0.386, and 0.77, for the counterclockwise vortex, with $\mathrm{\Gamma }=2$.

Figure 9

Isocontours of spanwise disturbance vorticity component shown at $t=16$, 24, and 25, for the counterclockwise vortex case, with $c=0.386$ and $\mathrm{\Gamma }=2$.

Figure 10

Isocontours of spanwise disturbance vorticity component shown at $t=28$, 34, and 38, for the counterclockwise vortex case, with $c=0.386$ and $\mathrm{\Gamma }=2$.

Figure 11

Instantaneous and spanwise-averaged skin friction contours $C_f$ shown at the indicated times for the counterclockwise vortex case with $c=0.386$ and $\mathrm{\Gamma }=2$.

Figure 12

Isocontours of spanwise disturbance vorticity component shown at $t=26$, 28, and 31, for the counterclockwise vortex case with $c=0.3$ and $\mathrm{\Gamma }=2$.

Figure 13

Isocontours of spanwise disturbance vorticity component shown at $t=41$, 42.15, and 43.1, for the counterclockwise vortex case with $c=0.3$ and $\mathrm{\Gamma }=2$.

Figure 14

Instantaneous and spanwise-averaged skin friction contours $C_f$ shown at the indicated times for the counterclockwise vortex case with $c=0.3$ and $\mathrm{\Gamma }=2$.

Figure 15

Contours plotted in $z=0$ plane, for ${\omega }_1$ (left) and ${\omega }_2$ (right) at $t=25$, 30, 34, and 38, for the case of $\mathrm{\Gamma }=2$ and $c=0.386$. The free-stream vortex $x$ location is shown by the vertical arrowhead and value given by $x_{cv}$. Also, indicated are minima and maxima values for each frame for both components.

Figure 16

Perspective plots for the three terms in DETE budget, Eq. (6) at $t=25$, with the top frame showing contribution from vortex stretching (Term1), the second frame from the top showing the contribution originating from enstrophy diffusion (Term2), and the third frame from top showing contribution from vorticity gradient (Term3). In the bottom frame, the dark (blue) regions are where negative ${\mathrm{\Omega }}_d$ become more negative and light (gray) regions are where positive ${\mathrm{\Omega }}_d$ become more positive due to instability given in Eq. (7), for free-stream vortex with $\mathrm{\Gamma }=2$ and $c=0.386$ located at $x_{cv}$.

Figure 17

Perspective plots for the three terms in DETE budget, Eq. (6) at $t=30$, with the top frame showing contribution from vortex stretching (Term1), the second frame from the top showing the contribution originating from enstrophy diffusion (Term2), and the third frame from top showing contribution from vorticity gradient (Term3). In the bottom frame, the dark (blue) regions are where negative ${\mathrm{\Omega }}_d$ become more negative and light (gray) regions are where positive ${\mathrm{\Omega }}_d$ become more positive due to instability given in Eq. (7), for free-stream vortex with $\mathrm{\Gamma }=2$ and $c=0.386$ located at $x_{cv}$.

Figure 18

Perspective plots for the three terms in DETE budget, Eq. (6) at $t=34$, with the top frame showing contribution from vortex stretching (Term1), the second frame from the top showing the contribution originating from enstrophy diffusion (Term2), and the third frame from top showing contribution from vorticity gradient (Term3). In the bottom frame, the dark (blue) regions are where negative ${\mathrm{\Omega }}_d$ become more negative, and light (gray) regions are where positive ${\mathrm{\Omega }}_d$ become more positive due to instability given in Eq. (7), for free-stream vortex with $\mathrm{\Gamma }=2$ and $c=0.386$ located at $x_{cv}$.

Figure 19

Perspective plots for the three terms in DETE budget, Eq. (6) at $t=38$, with the top frame showing contribution from vortex stretching (Term1), the second frame from the top showing the contribution originating from enstrophy diffusion (Term2), and the third frame from top showing contribution from vorticity gradient (Term3). In the bottom frame, the dark (blue) regions are where negative ${\mathrm{\Omega }}_d$ become more negative, and light (gray) regions are where positive ${\mathrm{\Omega }}_d$ become more positive due to instability given in Eq. (7), for free-stream vortex with $\mathrm{\Gamma }=2$ and $c=0.386$ located at $x_{cv}$.

Figure 20

Spanwise distribution of DETE budget terms shown in Fig. 19 ($t=38$). The contribution of terms in Eq (6), are shown in the top three frames: With the stretching term (top) dominating over the terms originating from enstrophy diffusion and vorticity gradient. The data are for $x=13$ and 15, and at a height $y=0.0035$, for $\mathrm{\Gamma }=2$ and the $c=0.386$ case.

Figure 21

Spanwise distribution of DETE budget terms shown in Fig. 19 ($t=38$). The contribution of terms in Eq. (6), are shown in the top three frames: With the stretching term (top) dominating over the terms originating from enstrophy diffusion and vorticity gradient. The data are for $x=13$ and 15, and at a height $y=0.1$, for the $\mathrm{\Gamma }=2$ and $c=0.386$ case.

Figure 22

Fourier transform of spanwise distribution of DETE budget terms shown in Figs. 20 and 21 for $x=13$ and 15 at $t=38$, for the indicated heights $y=0.0035$ and 0.1, for the $\mathrm{\Gamma }=2$ and $c=0.386$ case.

Figure 23

The flow field for $\mathrm{\Gamma }=0.5$ and $c=0.77$ case showing (a) the velocity profiles for $x=15.6738$, 16.9717, and 17.188 at $t=24$ with the free-stream vortex at $x_{cv}=17.48$; (b) stream traces at the midspan $(z=0)$ at $t=22$, 24 and 25, with $x_{cv}=15.94$, $17.48$ and $18.25$, respectively; and (c) spanwise vorticity contours in $(z=0)$ plane.

Figure 24

The flow field for $\mathrm{\Gamma }=2$ and $c=0.77$ case showing (a) the velocity profiles for $x=15.6738$, 16.9717, and 17.188 at $t=24$ with the free-stream vortex at $x_{cv}=17.48$; (b) velocity profile shown at $t=23$, 24, 25 for $x=16.9717$; (c) stream traces at the midspan ($z=0$) at $t=22$, 24, and 26, with $x_{cv}=15.94$, 17.48, and 19.02, respectively; and (d) spanwise vorticity contours in the $(z=0)$ plane.

Figure 25

The perspective plots of disturbance component of spanwise vorticity ${\omega }_3$ for the $\mathrm{\Gamma }=2$ and $c=0.77$ case at the indicated times, showing the absence of three-dimensionality during the violent vortex-induced instability.