#### 1.1. Motivation

In previous work [

1,

2,

3,

4], the authors developed a time-stepping potential flow method that uses higher-order vorticity elements to represent lifting surfaces and wakes. Although all results discussed in previous work were either steady or quasi-steady and time-averaged, the method is capable of quasi-steady time-accurate analysis. By “quasi-steady”, the authors refer to the enforcement of flow tangency boundary conditions based on the time-accurate flow field at each time step and the subsequent propagation of the resulting vorticity distribution of the lifting surfaces throughout the wake. To determine a fully unsteady lift response, the influence of shed vorticity due to changes in lift (also referred to as “shed circulation” [

5]) and apparent mass effects must also be taken into account [

5,

6,

7].

The use of an unsteady formulation of the Kutta–Joukowski theorem was suggested by Drela [

8] as a fast approach to predict unsteady lift for the purpose of an integrated aerodynamic, structural, and control model simulation of highly-flexible aircraft. This approach has also been referred to as the Joukowski theorem, as by Simpson et al. [

9]. The simplicity of this technique, along with the authors’ previous use of the Kutta–Joukowski theorem for steady and quasi-steady lift prediction, make it an appealing solution for the prediction of unsteady lift within the potential flow method.

Few details are available in the literature regarding the implementation and verification of an unsteady formulation of the Kutta–Joukowski theorem within a potential flow method. The objective of this note is to provide future developers with a comprehensive example of such an implementation within the authors’ higher-order potential flow method. In particular, emphasis is placed on identifying the influence of the numerical discretization approach on lift prediction of classical verification cases. The quasi-steady (kinematics only), shed-circulation, and fully unsteady responses are also provided for each case to elucidate the effect of each term and to provide future developers with information to aid in troubleshooting and debugging.

#### 1.2. Background on Unsteady Lift Prediction

In order to support the use of the unsteady formulation of the Kutta–Joukowski theorem, it is helpful to begin with a broader discussion of approaches to the calculation of unsteady lift. The lifting surfaces under consideration are assumed to be at low Mach number and low to moderate angles of attack. As such, the discussion that follows is limited to unsteady lift predictions for incompressible, attached flow systems under general gust loadings.

In the calculation of unsteady lift, one of two approaches is typically applied—pressure integration [

6] or the indicial response method [

5,

10]. Pressure integration is the determination of the lift force through integration of the local pressures on the surface of the wing. The indicial response method, as is found in rotorcraft analysis methods such as RCAS [

11] and CAMRAD [

12,

13], assumes the form of the response to a step change in a set of aerodynamic boundary conditions. For example, the classical solution for a step change in angle of attack of an airfoil in an incompressible flow is the Wagner function [

14], which assumes the circulatory unsteady lift response (including vorticity on the surface and shed into the wake) based on unsteady two-dimensional thin airfoil theory. This solution can then be superimposed over time in order to estimate the two-dimensional, circulatory, unsteady-lift response that is due to a series of changes in angle of attack using the Duhamel integral. For the case of arbitrary changes in angle of attack, the Duhamel integral must be numerically integrated. A relation exists between the two approaches in that the indicial response can be found via pressure integration, but the actual method itself is independent of the way that the response is developed.

The pressure integration method and the indicial response method vary from one another in terms of the assumptions applied and the requirements with respect to the flow-field modeling. The first notable difference comes with respect to the treatment of shed vorticity resulting from changes in lift. In order to successfully predict unsteady lift through pressure integration, the flow-field model must include the vortices shed into the wake due to changes in lift. Assumptions regarding the motion of these vortices may vary, e.g., the motion can be prescribed as in fixed wakes or be determined at each time step based on locally induced velocities as in relaxed wakes. In either case, the velocities induced by the vortices on the lifting surface are the physical mechanism for the unsteady lift response, thus, it is essential that they are taken into account. Alternatively, to predict the unsteady lift using an indicial response method, the shed vorticity must not be present in the flow-field in order to prevent it from being double counted. The location of the shed vorticity is thus assumed within the indicial response.

The second notable difference is that in most applications, the indicial response approach is in essence a strip method. As such, it assumes that each section of a lifting surface behaves as a two-dimensional airfoil. This is akin to a high aspect-ratio assumption for a steady, fixed-wing analysis. In the case of unsteady analysis, it also indicates that a segment of the shed vorticity at one spanwise location directly aft of the lifting surface has no influence on the lift of a neighboring section. The pressure integration approach does not have this limitation and, as a result, is capable of accounting for three-dimensional effects as long as the underlying model can support such a method.

An exception exists to the limitation on use of indicial responses with respect to capturing three-dimensional effects. It is possible to compute a three-dimensional unsteady solution which is “sliced” into spanwise segments used in order to obtain two-dimensional responses. These responses can then later be combined with local angle of attack history within the Duhamel integral to account for three-dimensional unsteady wake effects. An example of this type of analysis can be found in Kitson et al. [

15] for modeling flexible, high-aspect ratio wings.

A final difference between the two approaches is in their respective reliance on the model fidelity relative to the physical system. Unsteady lift predictions through pressure integration depend directly on the quality of the geometric representation of the lifting surface and the location and strength of the shed vortices in the wake. Alternatively, indicial methods depend only on the time history of the variation in angle of attack. While the fidelity of the model has an influence on the accuracy of this time history, the model fidelity is separated from the calculation of the unsteady lift itself.

To put the unsteady Kutta–Joukowski formulation in context with these methods, it is helpful to consider the formulation. According to Drela [

8], the vector form of the unsteady Kutta–Joukowski theorem to be evaluated at each time step is

where

${\overline{F}}_{L}^{\prime}$ is the lift per unit span,

$\overline{V}$ is the relative velocity evaluated at the quarter chord location, and

$\Gamma $ is the local circulation oriented along the unit vector

$\hat{s}$. The component of the relative velocity that is perpendicular to the circulation,

${\overline{V}}_{\perp}$, is given by

Because the circulation as a function of time is determined through the enforcement of flow tangency on the lifting surfaces based on the time-accurate flow-field, this approach depends both on the geometry of the lifting surfaces and an accurate assessment of the local velocities. As a result, in comparison with pressure integration and indicial methods, the approach is more closely related to a pressure integration due to this reliance on the local velocities. In fact, a comparison of the results of these two methods when applied to a flat plate that is at an angle of attack reveals that the main difference is that the Kutta–Joukowski approach implicitly accounts for the leading-edge suction force, which is omitted in the pressure integration on a flat plate [

3].

In order to improve the computational efficiency, in some cases the Kutta–Joukowski method can be supplemented with elements from the indicial approach. For example, in his application, Drela [

8] accounted for the influence of shed vorticity through an empirical lag term based on thin-airfoil theory results, rather than through a direct calculation of induced velocity. The use of this term is reminiscent of an indicial method, as it omits tracking of the shed vorticity.