Theoretical model for the separated flow around an accelerating flat plate using time-dependent self-similarity

We present a model appropriate to the initial motion of a flat-plate airfoil accelerating in an inviscid fluid. The model is based on the one presented in Pullin and Wang [J. Fluid Mech. 509, 1 (2004)] and is intended to extend the range of validity to lower angles of attack and longer distances traveled. The separated flow structures are represented as vortex sheets in the conventional manner and similarity expansions locally applicable to the leading and trailing edges of the plate are developed. In our approach, an expansion is applied to the attached outer flow rather than the vortex sheet circulations and positions. This allows the asymmetric effect of the sweeping component of the free-stream flow parallel to the plate to be built in to the same governing equation as the singular-order flow. Additionally, we develop a time-dependent self-similarity procedure that allows the modeling of more complex evolution of the flow structures. This is accomplished through an implicit time variation of the similarity variables. As a collective result, the predicted vortex dynamics and forces on the plate compare favorably to Navier-Stokes simulations. The model is split into high and low angles of attack regimes. The former assumes that the leading-edge and trailing-edge flows evolve independently, while the latter makes a further simplification to couple the two flows.

Figure 1

Example streamlines of attached flows around a wedge of angle $\beta \pi$: (a) singular flow from bottom to top, (b) higher-order regular flow from left to right, and (c) the composite flow of Eq. (2). In (c) the black dot marks the stagnation point $z_s$ on the wedge surface.

Figure 2

Definition of the $x\text{−}y$ coordinate system instantaneously coinciding with the mid-chord of the flat plate that translates with speed $U\left(t\right)$ and angle of attack $\alpha$. The trailing edge is at $x=c/2$, while the leading edge is at $x=-c/2$. The complex positions of the corresponding shed vortex sheets are $Z_{+}$ and $Z_{-}$.

Figure 3

Flow chart detailing the transformation of the self-similar solutions to the physical space for given input data. The similarity parameters are given to the governing equation in (13) with an appropriate initial condition.

Figure 4

Variation of similarity results with ${\eta }_{\alpha }$ for trailing-edge $(+)$ and leading-edge $(-)$ solutions. The parameters are $m=0$ (impulsive acceleration) and $J_o=0$. (a) Nondimensional circulation magnitudes in the sheets and the bound circulation. (b) Horizontal and vertical coordinates of the vortex spiral core locations for each sheet [also see Fig. 5].

Figure 5

Sheet shapes in similarity space for flow near the leading edge $-{\overline{\omega }}_{-}$ (left plots), and flow near the trailing edge ${\omega }_{+}$ (right plots). Solutions with $J_o=0$ and different values of the parameter ${\eta }_{\alpha }=2\epsilon cot\alpha$ are plotted as labeled, where ${\eta }_{\alpha }=0$ is the baseline with no effect of asymmetry. The sweeping motion of the free- stream represented by ${\eta }_{\alpha }$ is from left to right. (a) Impulsive acceleration: $m=0$, and (b) constant acceleration: $m=1$.

Figure 6

Schematic of the low angle of attack range. The separated LEV sheet is neglected, but its circulation ${\mathrm{\Gamma }}_{-}$ is lumped with ${\mathrm{\Gamma }}_b$ to give the effective bound circulation ${\mathrm{\Gamma }}_{\text{eff}}$.

Figure 7

(a) The circulation functions for the low angle of attack model (solid) compared with the theories of Wagner [33] (dashed) and Graham [3] (dashed-dotted) for steady translation $m=0$. (b) The same data on a log-log scale. ${\mathrm{\Gamma }}_{\infty }=\pi cUsin\alpha$ is the theoretical steady-state value.

Figure 8

(a) The lift functions for the low angle of attack model (solid) compared with the theory of Wagner [33] (dashed) and the viscous simulations ($\text{Re}=500$) in Jones and Eldredge [18] (dashed-dotted) for steady translation $m=0$. $L_{\infty }=\rho \pi c{U}^{2}sin\alpha$ is the theoretical steady-state value. (b) The modeled lift coefficient $C_L$ (solid) versus chords traveled $s/c$ for an accelerating plate with $m=0.5$ for $\alpha ={10}^{\circ }$ (red) and $\alpha ={30}^{\circ }$ (blue). The viscous simulations ($\text{Re}=100$) are from Chen, Colonius, and Taira [14] (dashed).

Figure 9

Flow structure comparison for the case of impulsive acceleration $m=0$ at $\alpha ={45}^{\circ }$ at different convective times $Ut/c$ as labeled. (Top) Vorticity contours from the viscous vortex particle method of Wang and Eldredge [7]. (Bottom) Vortex sheet positions predicted by the present theory. Since $m=0$, then $Ut/c$ is equal to the number of chords traveled.

Figure 10

Comparison of lift and drag coefficients vs $Ut/c$ for cases with impulsive acceleration $m=0$ at two angles of attack. (a), (b) $C_L$, $C_D$ at $\alpha ={45}^{\circ }$; (c), (d) $C_L$, $C_D$ at $\alpha ={60}^{\circ }$. The Navier-Stokes simulations are from Ref. [11]. The legend applies to all plots. Since $m=0$ then $Ut/c$ is equal to the number of chords traveled.

Figure 11

Comparison of lift and drag forces vs $t$ for cases with constant acceleration $m=1$ at two angles of attack. (a), (b) $L$, $D$ at $\alpha ={30}^{\circ }$; (c), (d) $L$, $D$, at $\alpha ={60}^{\circ }$. The Navier-Stokes simulations correspond to an ellipse of minor-to-major axis ratio $e=0.125$ from Pullin and Wang [1]; their point vortex model is also shown and labeled as “P&W model.” The legend applies to all plots. Vertical gray lines mark when the airfoil has traveled 1, 2, 3, and 4 chord lengths as labeled.

Figure 12

The modeled vortex structure at the time of near maximum lift for the cases shown in Fig. 11. The stagnation streamline corresponding to the flow induced by just the LEV sheet is also plotted (dashed red lines). The corresponding suction-side stagnation point is marked by the circle symbol. (a) $\alpha ={30}^{\circ }$, chords traveled: $s/c=3$. The TE spiral is out of frame. (b) $\alpha ={60}^{\circ }$, chords traveled: $s/c=2$.

Figure 13

Comparison of the nondimensional circulations as predicted by the vortex sheet and point vortex models of this study, along with the point vortex model of Pullin and Wang [1]. The parameters are $m=1$ (constant acceleration) and $J_o=0$. (a) Leading-edge circulation $J_{-}$. (b) Trailing-edge circulation $J_{+}$.