Reducing the skin-friction drag of a vehicle is an important way to reduce carbon emissions. Previous studies have investigated the drag reduction mechanisms of transverse grooves. However, it is more practical to investigate which groove geometry is optimal at different inflow conditions for engineering. The purpose of this paper is to establish the physical model describing the relationship between the dimensionless depth ($H^{+}=H{u}_{\tau }/\upsilon$) of the transverse groove, the dimensionless inflow velocity ($U_{\infty }^{+}=U_{\infty }/u_{\tau }$), and the drag reduction rate ($\eta$) to quasi-analytically solve the optimal and maximum transverse groove depth according to the Reynolds numbers. Firstly, we use the LES with the dynamic subgrid model to investigate the drag reduction characteristics of transverse V-grooves with different depths (h = 0.05~0.9 mm) at different Reynolds numbers ($1.09\times {10}^4~5.44\times {10}^5$) and find that $H^{+}$ and $U_{\infty }^{+}$ affect the magnitude of slip velocity ($U_s^{+}$), thus driving the variation of the viscous drag reduction rate (${\eta }_{\nu }$) and the increased rate of pressure drag (${\eta }_p$). Moreover, the relationship between $U_s^{+}$, ${\eta }_{\nu }$, and ${\eta }_p$ is established based on the slip theory and the law of pressure distribution. Finally, the quasi-analytical solutions for the optimal and maximum depths are solved by adjusting $U_s^{+}$ to balance the cost (${\eta }_p$) and benefit (${\eta }_{\nu }$). This solution is in good agreement with the present numerical simulations and previous experimental results.

Previous studies have implied that the AR (aspect ratio) of the transverse groove significantly affects the stability of the boundary vortex within the groove and thus drives the variation in the drag-reduction rate. However, there is no theoretical model describing the relationship between the AR and the stability of the boundary vortex, resulting in difficulty in developing a forward method to obtain the optimum AR. In this paper, the velocity potential of the groove sidewalls to the boundary vortex is innovatively described by an image vortex model, thus establishing the relationship between the AR and the induced velocity. Secondly, the velocity profile of the migration flow is obtained by decomposing the total velocity inside the groove, by which the relationship between the AR and the migration velocity is established. Finally, the analytical solution of the optimal AR ($A{R}_{opt}=2.15$) is obtained based on the kinematic condition for boundary vortex stability, i.e., the induced velocity equals the migration velocity, and the forms of boundary vortex motion at other ARs are discussed. Furthermore, the stability of the boundary vortex at the optimal AR and the corresponding optimal drag-reduction rate are verified by the large eddy simulations method. At other ARs, the motion forms of the boundary vortex are characterized by “vortex shedding” and “vortex sloshing,” respectively, and the corresponding drag-reduction rates are smaller than those for vortex stability.

Micro Air Vehicles (MAVs) airfoils usually operate at low Reynolds number conditions, where viscous drag will consume a large amount of propulsion power. Due to the small dimensions, many drag reduction methods have failed, resulting in limited current research. To develop an effective method of reducing viscous drag, transverse grooves were placed on the surface of MAVs airfoils in this study, and a numerical investigation was implemented to uncover the corresponding flow control law as well as the mechanism. Research has shown that transverse grooves have an impact on the drag and lift of airfoils. For drag, properly sized transverse grooves have the effect of reducing drag, but under high adverse pressure gradients or when the continuous arrangement of grooves is excessive, the optimal drag reduction effect achieved by the grooves is weakened, and even the drag increases due to the significant increase in pressure difference. In severe cases, it may also cause strong flow separation, which is not conducive to MAV flight. For lift, the boundary vortex in the groove has the ability to reduce the static pressure near the groove. However, high adverse pressure gradients or too many grooves will thicken the boundary layer and increase the blockage effect, resulting in a large static pressure on the grooved side of the airfoil (with an increase in drag). From the perspective of circulation, the static pressure changes on the suction and pressure surfaces have opposite effects on lift. Considering the comprehensive aerodynamic performance of the airfoil, we designed a high lift-to-drag ratio airfoil with grooves, which increased the lift-to-drag ratio by 33.747% compared to the smooth airfoil. Based on the conclusions, we proposed preliminary design criteria for grooved airfoils, providing guidance for subsequent research and applications.

Airfoil stall at low Reynolds numbers is a complex nonlinear dynamic phenomenon, which is characterized by catastrophe and hysteresis. It is difficult but important to mathematically describe the stall points under different Reynolds numbers. In this paper, taking the clockwise hysteresis as an example, a modeling method is proposed to describe the boundary of static airfoil stall according to the topological properties and physical characteristics of airfoil stall. Through numerical simulations, the lift characteristics of an airfoil at different Reynolds numbers are computed, and it is found that Reynolds number can affect not only the catastrophe and hysteresis of airfoil stall, but also the size of the hysteresis loop. Next, the static stall at low Reynolds numbers and the dynamic behavior described by the cusp-catastrophic model are proved to have a similarity in spatial topology and physical properties. According to the topological invariant rules, the topological-mapping relationship between the catastrophe-point set of the cusp-catastrophic model and the stall points of the airfoil stall is established through the development of an accurate topological-transformation function. Consequently, the catastrophe lines described by the cusp-catastrophic model can be used to represent the static-stall boundary of the airfoil at different Reynolds numbers. The effect of model description is verified by comparing the model-predicted values with the simulated values, and the error is found to be less than 1%.