Longitudinal development of flow-separation lines on slender bodies in translation

Abstract

This paper examines flow-separation lines on axisymmetric bodies with tapered tails, where the separating flow takes into account the effect of local body radius $r(x)$, incidence angle $\unicode[STIX]{x1D713}$ and the body-length Reynolds number $\mathit{Re}_{L}$. The flow is interpreted as a transient problem which relates the longitudinal distance $x$ to time $t^{\ast }=x\,\text{tan}(\unicode[STIX]{x1D713})/r(x)$, similar to the approach of Jeans & Holloway (J. Aircraft, vol. 47 (6), 2010, pp. 2177–2183) on scaling separation lines. The windward and leeward sides correspond to the body azimuth angles $\unicode[STIX]{x1D6E9}=0$ and $180^{\circ }$, respectively. From China-clay flow visualisation on axisymmetric bodies and from a literature review of slender-body flows, the present study shows three findings. (i) The time scale $t^{\ast }$ provides a collapse of the separation-line data, $\unicode[STIX]{x1D6E9}$, for incidence angles between $6$ and $35^{\circ }$, where the data fall on a power law $\unicode[STIX]{x1D6E9}\sim (t^{\ast })^{k}$. (ii) The data suggest that the separation rate $k$ is independent of the Reynolds number over the range $2.1\times 10^{6}\leqslant \mathit{Re}_{L}\leqslant 23\times 10^{6}$; for a primary separation $k_{1}\simeq -0.190$, and for a secondary separation $k_{2}\simeq 0.045$. (iii) The power-law curve fits trace the primary and secondary lines to a characteristic start time $t_{s}^{\ast }\simeq 1.5$.