Many adaptations of the lifting-line theory have been developed since its conception to aid in preliminary aerodynamic wing design, but they typically fall into two main formulations, named $\alpha $- and $\Gamma $-formulation, which differ in terms of the control points chordwise location and the variable updated during the iterative scheme. This paper assess the advantages and drawbacks of both formulations through the implementation of the respective methods and application of standard verification and validation procedures. Verification showed that the $\Gamma $-method poorly converges for wings with nonstraight quarter-chord lines, while the $\alpha $-method presents adequate convergence rates and uncertainties for all geometries; it also showed that the $\Gamma $-method agrees best with analytic results from the cassic lifting-line theory, indicating that it tends to overpredict wing lift. Validation and comparison to other modern lifting-line methods was done for similar geometries, and not only corroborated the poor converge and lift overprediction of the $\Gamma $-method, but also showed that the $\alpha $-method presented the closest results to experimental data for almost all cases tested, concluding that this formulation is typically superior regardless of the wing geometry. These results indicate that the implemented $\alpha $-method has a greater potential for the extension of the lifting-line theory to more geometrically complex lifting surfaces other than fixed wings with straight quarter-chord lines and wakes constrained to the planform plane.

This paper investigates the origin of flow-induced instabilities and their sensitivities in a flow over a rotationally flexible circular cylinder with a rigid splitter plate. A linear stability and sensitivity problem is formulated in the Eulerian frame by considering the geometric nonlinearity arising from the rotational motion of the cylinder which is not present in the stationary or purely translating stability methodology. This nonlinearity needs careful and consistent treatment in the linearised problem particularly when considering the Eulerian frame or reference adopted in this study that is not so widely considered. Two types of instabilities arising from the fluid–structure interaction are found. The first type of instabilities is the stationary symmetry breaking mode, which was well reported in previous studies. This instability exhibits a strong correlation with the length of the recirculation zone. A detailed analysis of the instability mode and its sensitivity reveals the importance of the flow near the tip region of the plate for the generation and control of this instability mode. The second type is an oscillatory torsional flapping mode, which has not been well reported. This instability typically emerges when the length of the splitter plate is sufficiently long. Unlike the symmetry breaking mode, it is not so closely correlated with the length of the recirculation zone. The sensitivity analysis however also reveals the crucial role played by the flow near the tip region in this instability. Finally, it is found that many physical features of this instability are reminiscent of those of the flapping (or flutter instability) observed in a flow over a flexible plate or a flag, suggesting that these instabilities share the same physical origin.

The development of the flow around a circular cylinder with a smaller diameter control rod in close proximity is the subject of this paper. It has long been known that this is an effective way to attenuate regular vortex shedding leading to reductions in its adverse effects on bluff-body flow. The aim of this study is to improve understanding of the ways the control rod affects the near-wake flow including how it influences the positions of boundary layer separation. Experiments were carried out in a water channel to measure lift and drag forces and particle image velocimetry (PIV) was employed to obtain detailed information on flow structure. The values of important properties were fixed as follows: Reynolds number, 20 000; ratio of cylinder and control rod diameters, 10 : 1; centre-to-centre distance between main cylinder and control rod, 0.7$D$ (where $D$ is the main cylinder diameter). The adjustable parameter was the angular position of the rod, $\theta$, which was varied between $90^{\circ }$ and $180^{\circ }$ from the front stagnation line. Lift and drag forces were measured separately for the main cylinder and the control rod. A new method for identifying flow states is introduced using PIV to interrogate the instantaneous flow velocity in the gap between the main cylinder and the control rod. Similarly to previous studies, three stable flow states were observed together with a bistable state. The bistable state is very sensitive to the control rod angle with a small change of ${\pm }1^{\circ }$ being sufficient to change the flow state.