### Filtern

#### Schlagworte

- Betz vortex relation (1)
- Theodorsen theory (1)
- Trailing vortex (1)
- Unsteady pitching (1)
- aerodynamics (1)
- general fluid mechanics (1)
- vortex shedding (1)

#### Organisationseinheit der BAM

A combined theoretical and experimental investigation was carried out with the objective of evaluating theoretical predictions relating to a two-dimensional airfoil subjected to high amplitude harmonic oscillation of the free stream at constant angle of attack. Current theoretical approaches were reviewed and extended for the purposes of quantifying the bound, unsteady vortex sheet strength along the airfoil chord. This resulted in a closed form solution that is valid for arbitrary reduced frequencies and amplitudes. In the experiments, the bound, unsteady vortex strength of a symmetric 18 % thick airfoil at low angles of attack was measured in a dedicated unsteady wind tunnel at maximum reduced frequencies of 0.1 and at velocity oscillations less than or equal to 50 %. With the boundary layer tripped near the leading edge and mid-chord, the phase and amplitude variations of the lift coefficient corresponded reasonably well with the theory. Near the maximum lift coefficient overshoot, the data exhibited an additional high-frequency oscillation. Comparisons of the measured and predicted vortex sheet indicated the existence of a recirculation bubble upstream of the trailing edge which sheds into the wake and modifies the Kutta condition.
Without boundary layer tripping, a mid-chord bubble is present that strengthens during flow deceleration and its shedding produces a dramatically different effect.
Instead of a lift coefficient overshoot, as per the theory, the data exhibit a significant undershoot. This undershoot is also accompanied by high-frequency oscillations that are characterized by the bubble shedding. In summary, the location of bubble and ist subsequent shedding play decisive roles in the resulting temporal aerodynamic loads.

Experiments indicate that vortices trailing finite wings can be perturbed by periodic wing pitching, leading to rapid dissipation and bursting. To illustrate the perturbation mechanism, Betz vortex rollup relations are combined with the Theodorsen theory for unsteady lift response. A sinusoidal pitch motion on a rigid elliptic planform wing is computed in this study as one example. Pitching modifies the instantaneous lift due to the planform variations, via the reduced frequency, and the rollup relations are applied to low-frequency experimental conditions. The combined Betz–Theodorsen theory shows that relatively large spanwise perturbations of the vortex centers can be achieved and may accelerate the exponential growth associated with the Crow instability. In fact, 84.2% of the displacement of the
unsteady trailing vortices due to unsteady pitching is oriented in the direction of the Crow instability. Furthermore, the axial velocity in the vortex center, calculated based on the Batchelor method, varies on the order of the flight speed.
This forms two types of stagnation points produced by approaching and retreating axial core velocities; in the former case, conservation of mass leads to observations of “bursting.” This observation can be explained adequately on the basis of quasi-steady considerations.