This study amends the inviscid Wagner lift model for starting flow at relatively large angles of attack to account for the influence of additional leading edge and trailing edge vortices. Two methods are provided for starting flow of a flat plate. The first method is a modified Wagner function, which assumes a planar trajectory of the trailing edge vortex sheet accounting for a temporal offset from the original Wagner function given release of leading edge vortices and a concentrated starting point vortex at the initiation of motion. The second method idealizes the trailing edge sheet as a series of discrete vortices released sequentially. The models presented are shown to be in good agreement with high-fidelity simulations. Through the present theory, a vortex force line map is generated, which clearly indicates lift enhancing and reducing directions and, when coupled with streamlines, allows one to qualitatively interpret the effect of the sign and position of vortices on the lift and to identify the origins of lift oscillations and peaks. It is concluded that leading edge vortices close to the leading edge elevate the Wagner lift curve while a strong leading edge vortex convected to the trailing edge is detrimental to lift production by inducing a strong trailing edge vortex moving in the lift reducing direction. The vortex force line map can be employed to understand the effect of the different vortices in other situations and may be used to improve vortex control to enhance or reduce the lift.

This paper introduces a novel neural network—a flow completion network (FCN)—to infer the fluid dynamics, including the flow field and the force acting on the body, from the incomplete data based on a graph convolution attention network. The FCN is composed of several graph convolution layers and spatial attention layers. It is designed to infer the velocity field and the vortex force contribution of the flow field when combined with the vortex force map method. Compared with other neural networks adopted in fluid dynamics, the FCN is capable of dealing with both structured data and unstructured data. The performance of the proposed FCN is assessed by the computational fluid dynamics (CFD) data on the flow field around a circular cylinder. The force coefficients predicted by our model are validated against those obtained directly from CFD. Moreover, it is shown that our model effectively utilizes the existing flow field information and the gradient information simultaneously, giving better performance than the traditional convolution neural network (CNN)-based and deep neural network (DNN)-based models. Specifically, among all the cases of different Reynolds numbers and different proportions of the training dataset, the results show that the proposed FCN achieves a maximum norm mean square error of 5.86% in the test dataset, which is much lower than those of the traditional CNN-based and DNN-based models (42.32% and 15.63%, respectively).

In a previous paper, an inviscid vortex force map approach was developed for the normal force of a flat plate at arbitrarily high angle of attack and leading/trailing edge force-producing critical regions were identified. In this paper, this vortex force map approach is extended to viscous flows and general airfoils, for both lift and drag forces due to vortices. The vortex force factors for the vortex force map are obtained here by using Howe’s integral force formula. A decomposed form of the force formula, ensuring vortices far away from the body have negligible effect on the force, is also derived. Using Joukowsky and NACA0012 airfoils for illustration, it is found that the vortex force map for general airfoils is similar to that of a flat plate, meaning that force-producing critical regions similar to those of a flat plate also exist for more general airfoils and for viscous flow. The vortex force approach is validated against NACA0012 at several angles of attack and Reynolds numbers, by using computational fluid dynamics.

The vortex force is studied for a flat plate at arbitrarily large angle of attack. A suitable vortex force approach, adapted from a previous work, is used to study the vortex force and to build a vortex force line map to identify the force effect of any potential vortex. This map can be used exactly for a potential point vortex and approximately for a concentrated leading-edge vortex (LEV) or trailing-edge vortex (TEV); the latter are shown to have a non-potential vortex core. By means of this map, we identify a force-producing critical region, due to pressure suction, above the front and rear parts of the plate for an LEV and a TEV, respectively. The impulsively started flow problem is used as an application, with validation by computational fluid dynamics. The force variation in time is decomposed into four repeatable stages (force release, force enhancement, stall and force recovery) in close relation to the individual and combined effect by an LEV and a TEV. A pressure distribution analysis shows that force enhancement is due to pressure suction by an LEV, while stall and force recovery are respectively due to the upwash effect (which reduces the pressure below the plate) of a new TEV right off the plate and the pressure suction of this TEV having now moved above the plate. A viscous effect causes a small-amplitude oscillation on the force curves by promoting multiple small-scale LEVs.

The unsteady forces acting on a body depend strongly on the local flow structures such as vortices. A quantitative understanding of the contribution of these structures to the instantaneous overall force is of fundamental significance. In the present study, a three-dimensional (3-D) vortex force map (VFM) method, extended from a two-dimensional (2-D) one, is used to provide better insight into the complex 3-D flow dynamics. The VFM vectors are obtained from solutions of potential equations and used to build the 3-D VFMs where the critical regions and directions associated with significant positive or negative contributions to the forces are identified. Using the existing velocity/vorticity field near the body, these VFMs can be used to obtain the body forces. A decomposed form of the force formula is also derived to separate the correction term contributed from the uncaptured vortices (close to or far away from the body). The present method is applied to the starting flow of a delta wing at high angle of attack, where LEVs are enhanced and stabilized by an axial flow effect. The analogy between the normal force of a slender delta wing and that of a 2-D flat plate with a steadily growing span is demonstrated via the VFM analysis. We find, for this application, that the force evolution exhibits some similar behaviour to a 2-D airfoil starting flow and, surprisingly, the force contribution mainly comes from the conical vortex sheet rather than the central core. Moreover, a quantitative understanding of the influence of LEVs in different evolution regimes on the body force is demonstrated.