An Improved Aerodynamic Model for Quasi‐Steady Simulations of Animal Flight at Moderate Reynolds Numbers

We present experimental and numerical studies aimed at improving models of animal flight at moderate Reynolds numbers ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>20</mml:mn> <mml:mi>k</mml:mi></mml:mrow> <mml:annotation>$20k$</mml:annotation></mml:semantics> </mml:math> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>50</mml:mn> <mml:mi>k</mml:mi></mml:mrow> <mml:annotation>$50k$</mml:annotation></mml:semantics> </mml:math> ). Quasi-steady aerodynamic force and moment data were collected using a rectangular wing across various angles of attack, α. The drag coefficient, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>C</mml:mi> <mml:mi>D</mml:mi></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:annotation>$C_D(\alpha)$</mml:annotation></mml:semantics> </mml:math> , is well described by a simple trigonometric function, while the lift coefficient, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>C</mml:mi> <mml:mi>L</mml:mi></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:annotation>$C_L(\alpha)$</mml:annotation></mml:semantics> </mml:math> , combines trigonometric and exponential terms-the latter capturing the linear behavior at small α predicted by inviscid theory. We also derive an empirical relation for the center of pressure as a function of α, allowing evaluation of the pitching moment coefficient, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>C</mml:mi> <mml:mi>M</mml:mi></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:annotation>$C_M(\alpha)$</mml:annotation></mml:semantics> </mml:math> , about any axis. These formulas are integrated into a dynamic flapping wing model to simulate forward flight of a pigeon and a bat at different speeds. Compared to prior models, our approach yields better agreement with wingbeat frequency data, particularly at high speeds. The small α angle regime proves especially beneficial, offering higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>C</mml:mi> <mml:mi>L</mml:mi></mml:msub> <mml:mo>/</mml:mo> <mml:msub><mml:mi>C</mml:mi> <mml:mi>D</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$C_L/C_D$</mml:annotation></mml:semantics> </mml:math> , which translates to reduced power demands and smaller body pitch variation-key considerations for the design of flapping wing robots.