We study an aerodynamic model describing the interaction between cloth and air, with applications to dynamic textile manipulation by robots. After introducing the model, we investigate its theoretical implications by using an analytically solvable system: the damped pendulum. We deduce that aerodynamic forces in the model manifest themselves as a lifting force, more pronounced when the cloth transitions from rest to dynamic motion. The resulting aerodynamic model is simple, with no additional computational cost. The model is validated by comparing cloth simulations to real-world data as captured by a Motion Capture System: the results demonstrate errors of less than 1 cm even for size A2 textiles. Furthermore, we develop an a priori formula for estimating the parameters of the model for various textiles without further optimization. This formula allows us to present a challenging robotics application: a dynamic flattening motion is designed in simulation and then successfully executed by a robot without any fine-tuning or modification. The outcome, a smooth and rapid laying of the real textiles, demonstrates the minimal sim-to-real gap of our model even when aerodynamics plays a leading role.