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The unsteady evolution of trailing vortex sheets behind a wing in ground effect is simulated using an unsteady discrete vortex panel method. The ground effect is included by image method. The present method is validated by comparing the simulated wake roll-up shapes to published numerical results. When a wing is flying in a very close proximity to the ground, the optimal wing loading is parabolic rather than elliptic. Thus, a theoretical model of wing load distributions is suggested, and unsteady vortex evolutions behind lifting lines with both elliptic and parabolic load distributions are simulated for several ground heights. For a lifting line with elliptic and parabolic loading, the ground has the effect of moving the wingtip vortices laterally outward and suppressing the development of the vortex. When the wing is in a very close proximity to the ground, the types of wing load distributions does not affect much on the overall wake shapes , but parabolic load distributions make the wingtip vortices move more laterally outward than the elliptic load distribu tions.

When a wing is flying near the ground, the aerodynamic characteristics of the wing is changed due to the interaction of the wing and the ground. This phenomenon is called as wing-in-ground (WIG) effect. New conceptual vehicles utilizing the WIG effect has been suggested focusing on fuel efficiency and stability [

Panel methods have been frequently applied to the conceptual design of wing-in-ground effect vehicles [

Han and Cho [

The objective of the present work is to investigate the effect of wing load distributions on the unsteady wake vortex evolution behind a wing in ground effect (IGE) and extreme ground effect (EGE). A new load distribution is suggested in order to match the change of the load distributions due to the change of the ground height. A lifting line with an initial load distribution is discretized with discrete vortex elements. The trailing wake vortices from each vortex element are represented by free vortices that deform freely by the assumption of a force-free position during the simulation.

For a lifting line solution of a symmetrical thin rectangular wing in ground effect, Tan and Plotkin [

where is the local lift curve slope, is the local wing chord, and and are respectively geometric and zero-lift angles of attack at the spanwise location. It can be easily verified that the solution of Equation (1) for an elliptic planform, untwisted wing out of ground effect (OGE) has an elliptic spanwise distribution of bound circulation and the elliptic wing loading is the optimal value for minimum induced drag within the limitation of flat rigid wake assumption. When a wing is flying in close proximity to the ground, the optimal spanwise distribution of bound circulation is parabolic [

Out of Ground Effect: Elliptic Loading,

In Ground Effect:

Extreme Ground Effect: Parabolic Loading:

When applying Equation 2 to the calculation of bound circulation distributions, it is also required to understand the relationship between bound circulation and wing height.

Shigemi et al. [

where is the lift coefficient of a wing in free flight (OGE), is the lift coefficient of a wing in extreme ground effect (EGE) and C is the coefficient that makes Equation (3) matching with the experimental data and should have a positive value. h represents the distance between the mid chord to the ground.

As shown in Equation (3), the lift coefficient is a nonlinear function of wing height, which also results in the change of circulation distribution. Thus, in the present paper, it is assumed that p in Equation (2b) changes as follows.

where is set to 1/2. p has the values of 0 and 1 respectively for h = 1.0 and 0.1.

Present method is well described and validated in Ref. where the effect of smoothing schemes or vortex models on the numerical accuracy is shown by comparing computed results with the published data of Krasny [

the same vorticity. On each line segment, the point vortex is located at the 3/4 point. The trailing wake vortices from each vortex panel element are represented by free point vortices that can deform freely with an assumption of a force-free position.

The distributions of the dimensionless tangential velocities induced by a point vortex are represented using the Lamb (Oseen) model [_{c}, is approximately equal to the radial distance of the point where the maximum velocity is induced. This is expressed

The second term on the right-hand side of Equation (5) provides an estimate for the growth in the core radius of the vortex. The vortex Reynolds number is related to the time scale over which the laminar diffusion processes in the vortex core region become turbulent [_{j}, is as follows.

Several smoothing schemes are used in order to obtain accurate solutions by circumventing the singularity behavior when the distance is very small. Krasny [

where is represented as follows depending on the wing distance to the ground.

In order to enforce the no penetration condition at the ground, an image method is used. Since the wake is force-free, the evolution of each vortex is investigated by moving the positions of point vortices using an Euler convection scheme.

more outward and vertically more upward than those of a wing out of ground effect. When the wing is in ground effect, the desingularized point vortices become confined to a small region and coalesce. Present method does not include the viscous diffusion model of interacting point vortices, but it can be said that the interaction of point vortices while merging will decrease further the viscous diffusion. The total circulation of wingtip vortices behind a wing in ground effect will be decreased.

In the present paper, a new formula that matches elliptic

loading for OGE and parabolic loading for EGE is suggested.

From the computed results on the unsteady wake evolution using the suggested model for wing loading, it was found that the overall wingtip vortex shapes were not affected by the wing load distributions. But, as the wing approaches very close to the ground, the parabolic wing load distribution calculation produced more laterally inward wingtip positions than the elliptic load distribution case.

In future, the present model will be applied to the unsteady wake evolution behind wings in formation where the significant interaction between wingtip vortices is expected.

The research was supported by a grant from the 2011 program for visiting professors overseas in Korea National University of Transportation.