Instability waves and transition in adverse-pressure-gradient boundary layers

Transition to turbulence in incompressible adverse-pressure-gradient (APG) boundary layers is investigated by direct numerical simulations. Purely two-dimensional instability waves develop on the inflectional base velocity profile. When the boundary layer is perturbed by isotropic turbulence from the free stream, streamwise elongated streaks form and may interact with the instability waves. Subsequent mechanisms that trigger transition depend on the intensity of the free-stream disturbances. All evidence from the present simulations suggest that the growth rate of instability waves is sufficiently high to couple with the streaks. Under very low levels of free-stream turbulence $\left(\sim 0.1\%\right)$, transition onset is highly sensitive to the inlet disturbance spectrum and is accelerated if the spectrum contains frequency–wave-number combinations that are commensurate with the instability waves. Transition onset and completion in this regime is characterized by formation and breakdown of $\mathrm{\Lambda }$ vortices, but they are more sporadic than in natural transition. Beneath free-stream turbulence with higher intensity (1–$2\%)$, bypass transition mechanisms are dominant, but instability waves are still the most dominant disturbances in wall-normal and spanwise perturbation spectra. Most of the breakdowns were by disturbances with critical layers close to the wall, corresponding to inner modes. On the other hand, the propensity of an outer mode to occur increases with the free-stream turbulence level. Higher intensity free-stream disturbances induce strong streaks that favorably distort the boundary layer and suppress the growth of instability waves. But the upward displacement of high amplitude streaks brings them to the outer edge of the boundary layer and exposes them to ambient turbulence. Consequently, high-amplitude streaks exhibit an outer-mode secondary instability.

Figure 1

Schematic of the setup simulated numerically.

Figure 2

(a) $u/U_{\infty }\left(x\right)$ plotted against the similarity variable $\eta$ at select streamwise stations. (b) Pressure gradient parameter ${\lambda }_{\theta }$ plotted against $x$.

Figure 3

(a) Instantaneous $u^{\prime }$ and $v^{\prime }$ plotted against $x$ at $y={\delta }_0/2$. (b) Contours of the growth rate $\left(k_i\right)$ of the most unstable mode as a function of Reynolds number ${\text{Re}}_{\delta }*$ and frequency ${\omega }_{\delta }*$ for the Falkner-Skan boundary layer with ${\beta }_H=-0.14$. The reference scales are displacement thickness $\delta *$ and local free-stream speed $U_{\infty }$. The vertical dashed line indicates the Reynolds number at the inlet plane. The unstable mode from DNS at $x=210$ is marked by the black dot.

Figure 4

Free-stream decay of rms value of $u^{\prime }$ (solid), $v^{\prime }$ (dashed), and $w^{\prime }$ (dash-dotted) for cases 1a (blue), 2 (green), and 3 (red).

Figure 5

(a) Disturbance kinetic energy at inlet as a function of (a) frequency $\left(\omega \right)$ and (b) spanwise wave number $\left(k_z\right)$ for the cases 1a and 1b (see Table 1).

Figure 6

Energy spectral density, $E\left(\kappa \right)$, of inlet perturbations plotted against $\kappa$ for the cases 1a and 1b. Each symbol represents an inlet disturbance mode. Only half of the total number of modes is plotted for clarity.

Figure 7

(a) Instantaneous skin-friction coefficient $\left(C_f\right)$ at a select spanwise location plotted against ${\text{Re}}_x$ for all cases tabulated in Table 1. An enlarged view of the curves is shown in the inset. (b) Mean skin-friction coefficient $\left(\langle C_f\rangle \right)$ plotted against ${\text{Re}}_x$. The line type and color for individual cases are the same as in Fig. 7. The black dashed lines represent the laminar solution and turbulent correlation.

Figure 8

Instantaneous contours of $-0.001\le u^{\prime }\le 0.001$ in $x\text{−}z$ planes at indicated wall-normal heights for case 1a. The $z$ axis has been enlarged by a factor of two. Dark contours represent negative values.

Figure 9

Instantaneous contours of $-0.4\le u^{\prime }\le 0.4$ (top frame) and $-0.2\le v^{\prime }\le 0.2$ (bottom frame) at a wall-normal height $y={\delta }_0/2$ for the case 1b. The $z$ axis has been enlarged by a factor of two. Dark is negative.

Figure 10

$u_{\mathrm{rms}}^{\prime }$ (thick solid) and $v_{\mathrm{rms}}^{\prime }$ (thin dashed) as a function of $x$ at a wall-normal height $y=0.7{\delta }_0$ for the cases 1a (green) and 1b (red). The shaded portion is the transitional regime for case 1b.

Figure 11

(a) Frequency spectra of $v^{\prime }$ for cases (a) 1a and (b) 1b. For case 1a, the wall-normal location is at $y/{\delta }_{99}\sim 0.33$. For case 1b, the locations at $x=\{50,130,210\}$ are to $y/{\delta }_{99}=\{0.51,0.4,0.33\}$. Black squares mark frequencies that are prescribed at the inlet.

Figure 12

(a) Contours of the linear spatial growth rate $\left(k_i\right)$ plotted in $\left(x,\omega \right)$ plane for case 1b. The frequency peaks indicated in Fig. 11 are also marked on the $\omega$ axis. (b) Downstream amplification of $v^{\prime }$ perturbation energy at dominant frequencies from the spectra in Fig. 11: $\omega =0.15$ (solid red), $\omega =0.173$ (dashed blue), $\omega =0.198$ (dash-dotted green), $\omega =0.3$ (dotted pink), and $\omega =0.357$ (dash-dot-dotted orange). The transition regime is marked by the vertical gray strip.

Figure 13

Mean rms perturbation velocity profiles plotted at selected streamwise stations for the case 1b at streamwise stations: $x=50$ (solid red), $x=130$ (dashed green) and $x=210$ (dash dotted blue).

Figure 14

Isosurfaces of the $Q$ criterion colored by the wall-normal distance $y$ for the case 1b. Here, $Q=0.003$. Also shown are the contours of $-0.3\le v^{\prime }\le 0.3$ in a horizontal plane at a wall-normal height $y=0.7{\delta }_0$. Dark is negative. ${\delta }_{99}$ at $x=250$ is at $y=2.33$.

Figure 15

(a) Contours of $-0.2\le u^{\prime }\le 0.2$ in the $\left(t,z\right)$ plane for the case 1b. The streamwise station is $\left(x=250,y=0.775\right)$. (b) Profile of $u_{\mathrm{rms}}^{\prime }$ at $x=250$. The dashed line indicates the wall-normal location of the station corresponding to panel (a).

Figure 16

Instantaneous contours of $-0.3\le u^{\prime }\le 0.3$ (top frame) and $-0.2\le v^{\prime }\le 0.2$ (bottom frame) at a wall-normal height $y=0.87{\delta }_0$ for the case 2. Dark is negative. Local peak of $u_{\mathrm{rms}}^{\prime }$ is at this wall-normal height at transition onset.

Figure 17

(a) Frequency spectra of $u^{\prime }$ (gray) and $v^{\prime }$ (black) for the case 2 at $x=155$. The wall-normal location is $y/{\delta }_{99}\sim 0.285$. (b) Comparison of mean streamwise velocity profiles for cases 1a (black dashed line) and case 2 (orange solid line) at $x=155$.

Figure 18

Downstream amplification of perturbation energy in $u^{\prime }$ and $v^{\prime }$ at dominant frequencies from frequency spectra for the case 2 in Fig. 17: $\omega =0.0077$ (solid), $\omega =0.17$ (dash-dotted). Thick (blue) and thin (red) lines indicate the mode energy for $u$ and $v$ components, respectively. Onset and completion of transition is distinguished by the vertical gray strip.

Figure 19

Instantaneous contours of $-0.3\le u^{\prime }\le 0.3$ (top frame) and $-0.2\le v^{\prime }\le 0.2$ (bottom frame) at a wall-normal height $y=0.75{\delta }_0$ for the case 3. Dark is negative. The wall-normal peak of $u_{\mathrm{rms}}^{\prime }$ is at $y=0.75{\delta }_0$ at transition onset.

Figure 20

(a) Frequency spectra of $u^{\prime },v^{\prime }$, and 2D component of $v^{\prime }$ for the case 3 at $x=130$. The wall-normal location corresponds to $y/{\delta }_{99}\sim 0.267$. The gray region shows the exponentially unstable frequency range for the mean velocity profile. (b) Downstream amplification of perturbation energy in $v^{\prime }$ at dominant frequencies from spectra plot in Fig. 20: $\omega =0.083$ (solid), $\omega =0.114$ (dashed), $\omega =0.17$ (dash-dotted), $\omega =0.243$ (dash-dot-dotted), and $\omega =0.285$ (dotted). The black lines represent frequencies dominated by 2D modes. Onset and completion of transition is highlighted by the vertical ash-colored strip.

Figure 21

Mean rms perturbation velocity profiles plotted at selected streamwise stations for the case 2 at streamwise stations: $x=100$ (solid red), $x=150$ (dashed green), $x=175$ (dash-dotted blue), and $x=200$ (dash-dot-dotted orange).

Figure 22

Mean rms perturbation velocity profiles plotted at selected streamwise stations for the case 2 at streamwise stations: $x=50$ (solid red), $x=100$ (dashed green), $x=125$ (dash-dotted blue), and $x=150$ (dash-dot-dotted orange).

Figure 23

(a) Isosurfaces of $u^{\prime }=-0.15$ colored by the wall-normal distance $y$ showing a helical secondary instability for case 2 [24]. (b) Isosurfaces of streamwise vorticity ${\omega }_x=\pm 0.1$ are overlaid on the helix. The boundary layer thickness at $x=215$ is ${\delta }_{99}=2.26{\delta }_0$.

Figure 24

Instantaneous isosurfaces of $u^{\prime }=\pm 0.1$ depicting inner mode breakdown and outer sinuous mode spot precursor for the case 3: (a) $u^{\prime }=0.1$ (dark) and $u^{\prime }=-0.1$ (gray), and (b) color contoured by the wall-normal distance $y$. Boundary layer thickness at $x=155$ is ${\delta }_{99}=2.28{\delta }_0$.

Figure 25

Contours of spatial growth rate $\left(k_i\right)$ plotted in the $\left(x,\omega \right)$ plane as per linear stability theory for the time- and span-averaged mean flow for cases 2 (red solid) and 3 (black dashed). Contour levels are between 0 and $-0.04$ with step size $-0.02.$ Minimum $k_i$ was $-0.056$ for case 2 and $-0.04$ for case 3.

Figure 26

Local peak $u_{\mathrm{rms}}^{\prime }$ plotted vs streamwise distance $x$ for cases 1b (dark), 2 (gray), and 3 (light).