Aerodynamics of a dart-shaped projectile at low Reynolds number, An assessment of effective slope as a parameter for turbulent drag prediction over multi-scaled roughness
Authors: Saha Sandeep , Hutchins Nicholas 
Published: October 2024

Aerodynamics of a dart-shaped projectile at low Reynolds numberA sports dart pierces the dartboard because it possesses a remarkable aerodynamic property of ‘self-correcting’ its attitude in flight. This property arises from its aerodynamic design with a long heavy Barrel and large cruciform wings known as flights. We characterize the aerodynamics of dart-shaped projectiles at typical flight Reynolds numbers between 14500 and 20500 using wind tunnel experiments and numerical simulations. Force measurement tests from wind tunnel experiments yield the lift, drag, and pitching moment coefficients over a range of angles of attack; the experimental estimates are in quantitative agreement with those obtained from numerical simulations. Examining the surface pressure distribution, streamlines, and wall shear–stress distribution, along with the skin friction lines obtained from numerical simulations, reveals that the aerodynamics of the dart is governed by an interaction between the Barrel vortex (BV) shed by the cone–cylinder body and the wing leading edge vortex (WLV) over the horizontal flights influenced by solid impediment offered by the vertical flights. Smoke flow visualization images corroborate the vortex–vortex and vortex–wall interactions over the flights found in the numerical simulations. A complex interplay of vortex structures is observed, which depends on the angle of attack. The WLV develops an elliptic instability while exhibiting a partial merger with the Barrel vortex in the presence of secondary vorticity generated by the walls amidst the rapid weakening of the WLV. We conclude that the role of aerodynamics is largely pitch stabilization by means of aerodynamic moment and the normal force generation.An assessment of effective slope as a parameter for turbulent drag prediction over multi-scaled roughnessThe streamwise effective slope (ES_x), which is the mean absolute streamwise gradient of the roughness, is considered to be a key parameter in predicting the drag penalty of rough-wall turbulent boundary-layers. However, many real-world rough surfaces are multi-scaled. For such surfaces, ES_x can be unbounded and its value can be dominated by scales of the topography that are invisible to the flow. To illustrate this, a campaign of drag balance measurements was conducted with a set of machined surfaces. A baseline surface is prepared with fine machining parameters. By coarsening the machining precision, artefacts called ‘scallops’ are introduced which increases the ‘measured’ ES_x without changing other geometrical statistics. The drag of the ‘scalloped’ surfaces is higher than the baseline surface, with the relative drag increase scaling with the viscous scaled scallop height, but only when their height exceeds 2∼3 times the viscous length scale. Further, the drag of one of the ‘scalloped’ cases, even when the scallop height is 𝒪(10) viscous units, is seen to be much lower (∼ 28%) than a case with matched $ES_x$, but where the $ES_x$ results from larger scale features. These findings confirm that for multi-scaled surfaces, $ES_x$ may be a misleading topographical metric for drag (or $k_s$) prediction and one must consider which scales contribute to the average slope of a surface.