Development of Tollmien-Schlichting disturbances in the presence of laminar separation bubbles on an unswept infinite wavy wing

The effect of long-wavelength sinusoidal surface waviness on the development of Tollmien-Schlichting (TS) wave instabilities is investigated. The analysis is based on the compressible flow that forms over an unswept infinite wavy wing with surface variations of variable amplitude, wavelength, and phase. Boundary layer profiles are extracted directly from solutions of a Navier-Stokes solver, which allows a thorough parametric analysis to be undertaken. Many wavy surface configurations are examined that can be sufficient to establish localized pockets of separated flow. Linear stability analysis is undertaken using parabolized stability equations (PSE) and linearized Navier-Stokes (LNS) methods, and surface waviness is generally found to enhance unstable behavior. Results of the two schemes are compared and criteria for PSE to establish accurate solutions in separated flows are determined, which are based on the number of TS waves per wavelength of the surface deformation. Relationships are formulated, relating the instability variations to the surface parameters, which are consistent with previous observations regarding the growth of TS waves on a flat plate. Additionally, some long-wavelength surface deformations are found to stabilize TS disturbances.

Figure 1

Flow chart diagram illustrating the methodology for the boundary layer and stability analysis. CoBL, compressible boundary layer; REBL, RANS extracted boundary layer; PSE, parabolized stability equations; LNS, linearized Navier-Stokes.

Figure 2

Cross-sectional view of the unswept infinite wing model.

Figure 3

Flow characteristics obtained using REBL for the surface configurations $\{\lambda ,H\}=\{0,0\}$ (solid line), $\{0.1,0.0001\}$ (dashed), $\{0.2,0.0002\}$ (chain) and $\{0.4,0.0004\}$ (dotted). (a) Surface pressure coefficient $C_p$; (b) skin friction coefficient $C_f{\text{Re}}_{\infty }^{1/2}$.

Figure 4

Nondimensional base flow $U_B$ obtained using REBL over a nondeformed wing in the $\{x,y\}\mathrm{plane}$.

Figure 5

Nondimensional base flow $U_B$ obtained using REBL in the $\{x,y\}\mathrm{plane}$. (a) $\{\lambda ,H\}=\{0.2,0.0002\}$; (b) $\{0.4,0.0004\}$.

Figure 6

(a) Illustration of the base flow $U_B$ obtained using REBL in the $\{x,y\}\mathrm{plane}$, over a wavy surface with $\{\lambda ,H\}=\{0.1,0.0003\}$. (b) Flow separation depicted within the troughs of the surface.

Figure 7

Minimum value of the base flow $U_B$ against the function ${\text{Re}}_{\infty }^{1/2}{\left(H{M}_{\infty }\right)}^{2}n$.

Figure 8

Flow characteristics for the surface configuration $\{\lambda ,H\}=\{0.1,0.0003\}$ obtained using REBL (dashed) and CoBL (dotted). Cross markers indicate the location where the CoBL method breaks down. (a) Skin friction coefficient $C_f{\text{Re}}_{\infty }^{1/2}$; (b) pressure gradient $P_{B,x}$; (c) surface variation $s$.

Figure 9

Stability comparisons for base flows generated by REBL (dashed) and CoBL (dotted), for $f=34\times {10}^{-6}$. Surface configurations are given as (a) $\{\lambda ,H\}=\{0.1,0.0001\}$; (b) $\{0.2,0.0002\}$; (c) $\{0.4,0.0004\}$. Cross markers indicate the location that the CoBL method breaks down, and the solid curves represent the solutions associated with the nondeformed wing.

Figure 10

Disturbance development $u/{\left|u\right|}_{max}$, in the $\{x,y\}\mathrm{plane}$ for $f=34\times {10}^{-6}$ and $\lambda =0.1$. (a) $H=0.0$; (b) 0.0001; (c) 0.0002; (d) 0.0003; (e) 0.0004. Solid black curves highlight the local regions of separated flow.

Figure 11

Maximum absolute value of the $u$-velocity perturbation as a function of $x$ for $f=34\times {10}^{-6}$ and $\lambda =0.1$. Solutions generated for LNS (dashed lines) and PSE (dotted) methods. (a) $H=0.0001$; (b) 0.0002; (c) 0.0003; (d) 0.0004.

Figure 12

Differences $\epsilon$ against the Li-Malik stability criterion $|{\alpha }_r|\mathrm{\Delta }x$ for those disturbances considered in Figs. 10 and 11.

Figure 13

Maximum absolute value of the $u$-velocity perturbation as a function of $x$. Solutions generated for (a) $\{\lambda ,H\}=\{0.1,0.0001\}$; (b) $\{0.1,0.0004\}$; for LNS (dashed lines) and PSE methods (dotted). (1) $f=31\times {10}^{-6}$; (2) $42\times {10}^{-6}$; (3) $52\times {10}^{-6}$; (4) $63\times {10}^{-6}$.

Figure 14

[(a), (b)] Differences $\epsilon$ against the Li-Malik stability criterion $|{\alpha }_r|\mathrm{\Delta }x$ for $H=0.0001$ and 0.0004. Frequencies $f=31\times {10}^{-6}$ (solid line), $42\times {10}^{-6}$ (dashed), $52\times {10}^{-6}$ (chain), and $63\times {10}^{-6}$ (dotted). (c) Differences $\epsilon$ based on the optimum step-size conditions, as a function of $H^{1/2}{\mathrm{\Lambda }}^2$ for ${\text{Re}}_{\infty }\times {10}^{-6}=2.5$, 5, and 7.5, and those frequencies considered in panels (a) and (b). $\mathrm{\Lambda }=\lambda /{\lambda }_{\mathrm{TS}}$ is the number of TS waves per wavelength $\lambda$. Cross and circle markers respectively represent nonseparated and separated flow systems.

Figure 15

Disturbance development $u/{\left|u\right|}_{max}$, in the $\{x,y\}\mathrm{plane}$ for $\{\lambda ,H\}=\{0.1,0.0004\}$. (a) $f=31\times {10}^{-6}$; (b) $42\times {10}^{-6}$; (c) $52\times {10}^{-6}$; (d) $63\times {10}^{-6}$. Solid black curves highlight the local regions of separated flow.

Figure 16

The disturbance wave number ${\alpha }_r$ for frequencies $f=31\times {10}^{-6}$ (solid thick line), $42\times {10}^{-6}$ (dashed), $52\times {10}^{-6}$ (chain), and $63\times {10}^{-6}$ (dotted). (a) $\{\lambda ,H\}=\{0.1,0.0001\}$; (b) $\{0.2,0.0002\}$; (c) $\{0.4,0.0004\}$. Thinner line types depict the results established on the nondeformed wing.

Figure 17

Stability $N$-factor calculations for perturbations with frequency $f=34\times {10}^{-6}$. Parallel flow results (solid line), nonparallel without curvature effects (dashed), and nonparallel with curvature effects (dotted). (a) $\{\lambda ,H\}=\{0,0\}$; (b) $\{0.1,0.0001\}$; (c) $\{0.2,0.0002\}$; (d) $\{0.4,0.0004\}$.

Figure 18

Neutral stability curves for (a) $\lambda =0.2$; (b) 0.3; (c) 0.4 and variable $H$.

Figure 19

Stability $N$-factor calculations for surface waviness configurations (a) $\lambda =0.2$; (b) $\lambda =0.3$; (c) $\lambda =0.4$ and variable $H$.

Figure 20

Stability $N$-factor calculations for surface waviness configurations (a) $\{\lambda ,H\}=\{0.2,0.0003\}$; (b) $\{0.3,0.0003\}$; (c) $\{0.4,0.0003\}$, where the phase shift $\varphi =0$ (dashed), $\pi /2$ (chain), $\pi$ (dotted), $-\pi /2$ (solid with cross symbols). The solid line represents the solution on the nondeformed wing model.

Figure 21

Stability variations $\mathrm{\Delta }N$ as a function of the phase shift $\varphi$ measured about $x_{\mathrm{ref}}=0.55$. (a) $\lambda =0.2$; (b) $\lambda =0.3$; (c) $\lambda =0.4$. The marker symbols $\times ,◯,\square ,\diamond ,*$ represent solutions of the respective amplitudes $H=0.1,0.2,0.3,0.4,$ and 0.5. Dashed lines are spline fitted curves, linking the solutions of the same $H$.

Figure 22

(a) Stability $N$-factor calculations for ${\text{Re}}_{\infty }\times {10}^{-6}=0.5$ (solid lines), 2.5 (dashed) and 7.5 (chain) with $M_{\infty }=0.7$. The fainter line types represent the solutions on the nonwavy surface and the thicker line types correspond to $\{\lambda ,H,\varphi \}=\{0.2,0.0002,0\}$. (b) Stability variation $\mathrm{\Delta }N$ against ${\text{Re}}_{\infty }$ for $\{\lambda ,H,\varphi \}=\{0.2,0.0002,0\}$ (solid), $\{0.3,0.0003,0\}$ (dashed) and $\{0.4,0.0004,0\}$ (chain). (c) Stability $N$-factor calculations for $M_{\infty }=0.1$ (solid lines), 0.25 (dashed), and 0.5 (chain) and ${\text{Re}}_{\infty }\times {10}^{-6}=5$. Line types are the same as in panel (a). (d) Stability variation $\mathrm{\Delta }N$ against $M_{\infty }$. Line types are the same as in panel (b).

Figure 23

Comparison between the variation $\mathrm{\Delta }N$ (measured about $x_{\mathrm{ref}}$) and $n{H}^2{\text{Re}}_{\infty }/\lambda$, for $f=34\times {10}^{-6}$ ($f^{*}=6.5\mathrm{kHz}$) and $M_{\infty }=0.7$.

Figure 24

The chord location $x_{N=4.5}$ plotted against the frequency $f$ that first establishes $N=4.5$, for variable surface wave configurations. The free-stream conditions $\{M_{\infty },{\text{Re}}_{\infty }\}=\{0.7,5\times {10}^6\}$.

Figure 25

Flow characteristics for the surface configuration $\{\lambda ,H,\varphi \}=\{0.4,0.0006,0\}$ (solid lines) and the nondeformed wing (dashed). (a) Form of the surface deformation $s$; (b) pressure gradient $P_{B,x}$; (c) $N$-factor analysis.