This post is about a 2D NACA 0012 aerofoil undergoing forced flapping motion for propulsion purposes. Heaving motion is achieved by applying a vertical velocity on the aerofoil based on the Eqn. 1. Similarly the pitching motion is achieved by applying a rotational velocity, governed by Eqn. 2.
vy = 2*π*fh*Ho*sin(2*π*fh*t) Eqn. 1
ω = -2*π*fh*ϑ*sin[(2*π*fh*t) + 1.5708] Eqn. 2
w.r.t. Eqn. 1-2 reduced frequency is defined as (2*π*fh*Ho/U∞)), fh is the frequency of oscillations, while ω, t and ϑo represent rotational velocity, instantaneous time and maximum pitching angle. Ho is the heaving amplitude and U∞ is the free stream velocity.
The flapping motion is achieved by a combination of the heaving and pitching. In this particular simulation, the aerofoil is in the propulsion mode, meaning the feathering parameter χ is less in magnitude than 1.0. Feathering parameter is defined by Eqn. 3.
χ = ϑ/arctan(h0*2*π*fh/U∞) Eqn. 3
The boundary conditions employed for the simulation are at Re 1,000, K = 1.41, Ho = aerofoil chord length, χ = 0.5489 and fh = 0.003391 Hz. The animation of the pressure, vorticity and velocity contours is shown in Fig. 1.
Fig. 1, Flow animation, fluid flow direction is from left to right.
The results of present simulation are compared with [1]. In terms of maximum lift, a maximum deviation of 5% is observed as compared to [1], as shown in Fig. 2. The maximum lift coefficient for available data is ~4.224 while the maximum lift coefficient from the present simulation is ~4.057. The average thrust produced is within 2% of [1]. Average thrust coefficient per cycle from [1] is 0.9957 while the result from present simulation reveals the thrust coefficient to be 1.0098.
Fig. 2, A comparison of coefficient of lift.
References
[1] https://doi.org/10.1017/jfm.2017.508
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