Vortex-Induced Roll Moment Model
For vehicle configurations (airplane, missile, submarine, etc.) that have both forward and aft lifting surfaces, the vorticity shed from the forward surfaces can induce significant loads on the aft fins (and to a lesser degree on the hull or fuselage). In particular, the roll moment is particularly susceptible to induced vortex effects.
As an example, consider the loads induced on a submarine's tail fins (rudders, stern planes, dihedrals) by the tip vortex shed by the sail. The local angles of attack induced by the sail tip vortex on each tail fin panel depends on the tip vortex's strength and location with respect to each panel, and, in turn, the local induced lift caused by the induced angle of attack varies from fin panel to fin panel. The resulting induced roll moment from all the induced loads on the tail fins can be quite significant.

The strength and position of the sail tip vortex (see plot below) depend in a nonlinear fashion, on the motion state of the vehicle and on its motion time history; these vortex-induced effects are generally nonlinear even for steady state motion conditions. A coefficient representation of these vortex-induced roll moment effects (or any of the other vortex-loads such as lift) is not adequate, although various attempts have been made in the past to use such a representation (e.g., 2510 equations of motion).

The roll moment plots below compare the experimental roll moment for a submarine in two steady-state conditions - constant angle of drift beta and constant non-dimensional yaw rate r´ - with the roll moment as computed with both a coefficient hydrodynamic model and a component hydrodynamic model that includes vortex-induced effects. As one can see, the vortex-induced effects are significant and only the component-based model captures the correct variation of roll moment with respect to beta or r´.

These considerations also apply to other types of vehicles such as a missile with canard controls. The particular behavior of the vortex-induced effects depends upon the geometric configuration of the vehicle, but the physics governing these effects is the same across all types of configurations, and it is the physics of vortex motion and vortex-induced loads that is implemented in VCT's component hydrodynamic and aerodynamic models.
 

Vortex Path during Horizontal Overshoot Maneuver

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Roll moment due to sideslip and yaw rate

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Comparison between VCT vortex tracking model and a coefficient based model