Recirculation regions in wakes with base bleed

The appearance of detached recirculation regions in wakes with base bleed determines the aerodynamic properties of many natural organisms and technological applications. In this work we introduce an analytical model which captures certain key dimensions of the recirculation region in the wake of porous plates of infinite aspect ratio, along with the porosity range over which it exists when vortex shedding is absent or suppressed. The model is used to interpret why the recirculation region (i) emerges, (ii) migrates away from the body with increasing base bleed, (iii) disappears at a critical bleed, and (iv) is partially insensitive to variations in the Reynolds number. The model predictions show considerable agreement with data from laboratory experiments and numerical simulations.

Figure 1

Steady wake bubble boundary and slipstream of plates of variable porosity $\beta$ as measured using PIV at $Re=6\times {10}^3$. The color bar corresponds to the time-averaged nondimensional streamwise velocity. $X$ is the bubble detachment length and $h/2$ is its maximum half-height. $\beta$ is the open area ratio of the plate. The dashed line signifies the presence of a splitter plate, which was used to stabilize the wake in the cases $\beta =0$, 0.1, and 0.2. For $\beta =0.3$, vortex shedding is already suppressed by the large base bleeding [3], rendering the splitter plate superfluous.

Figure 2

Potential-flow representation of the steady wake by Steiros and Hultmark [20]. $U_{\infty }$ and $p_{\infty }$ denote the free-stream velocity and ambient pressure, respectively. $U_w$ and $p_w$ denote the wake velocity and pressure after the initial wake expansion. $H$ is the wake height after the initial expansion, and $d$ is the plate height. $u$ is the velocity immediately upstream or downstream of the plate.

Figure 3

Extension of the potential-flow representation of the wake of Fig. 2 by addition of a recirculation region. The dashed regions ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ denote control volumes.

Figure 4

Normalized (a) bubble height and (b) detachment length as a function of $\beta$. In the dashed part of the prediction, where $\beta \to 0$, the theoretical analysis is not strictly valid. The hatched areas represent the region where the mass and momentum conservation laws cannot be fulfilled. The dashed vertical line indicates the porosity where the recirculation bubble disappears, according to the numerical simulations. Where available, results from the experiments of Castro [3] are also included. The ratio $h/H$ was measured at the location of maximum bubble height, while the distance $X$ is the minimum distance between the bubble and the plate.

Figure 5

Normalized (a) maximum bubble height and (b) detachment length as a function of $Re$ in the laminar regime for a thin plate where $\beta =20.6\%$. The ratio $h/H$ was measured at the location of maximum bubble height, and the distance $X$ as the minimum distance between the bubble and the plate. Symbols show the direct numerical simulations data. Lines show linear fits.

Figure 6

Time-averaged recirculation bubble for different values of porosity $\beta$, as measured using PIV (see Appendix pp1-s1), at $Re=6\times {10}^3$. The colors correspond to the time-averaged nondimensional streamwise velocity. Vortex shedding makes the wake unsteady for $\beta =0$, 0.1, and 0.2, but is suppressed for $\beta =0.3$ due to the large bleeding [3]. The bubble evolution is qualitatively similar to the case where the wake is stabilized using a splitter plate (Fig. 1).

Figure 7

(a) Side and (b) top view of the experimental configuration, and (c) plate design. All tested plates have the same hole pattern, but different hole diameter.

Figure 8

Computational setup for simulations of the flow past porous plates (domain and plate not drawn to scale).

Figure 9

(a) Streamwise velocity along wake centerline. (b) Reynolds stresses at a cut in the wake at $x/d=5$. Black: $\overline{u^{\prime }{u}^{\prime }}/U_{\infty }^2$, red: $\overline{u^{\prime }{v}^{\prime }}/U_{\infty }^2$, blue: $\overline{v^{\prime }{v}^{\prime }}/U_{\infty }^2$. (c) Black shows the velocity magnitude and red shows the Reynolds stress ($\overline{u^{\prime }{v}^{\prime }}/U_{\infty }^2$) along the slipstream. Time-averaged results. Dashed lines show PIV measurements (from a single plane), solid lines show numerical simulation data (averaged over the spanwise extent of the domain).