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For airfoils in unsteady motion, the lift coefficient oscillates and the wake is characterized by vortex shedding.

How does vortex shedding affect the lift coefficient for an airfoil?

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Shed vortices from unsteady flow over an airfoil result in decrease and phase lag in lift.

Shed vortices

Ref: https://www.researchgate.net/figure/Vortex-shedding-from-an-airfoil-oscillating-in-pure-heave-2_fig2_251006855

Using thin airfoil theory in incompressible flow, the lift coefficient for an airfoil undergoing cyclical pitching and/or heaving can be expressed as (Ref. Drela, Flight Vehicle Aerodynamics):

$$C_l=\text{Re}\{ C(k)C_{l_Q} + C_{l_A} \} + C_{l_0}$$

$C_{l_Q}$ is the circulatory part of the lift, which, at steady-state, is equal to $2\pi\alpha$. $C_{l_A}$ is the non-circulatory part that have to do with fluid inertia; at steady-state, its contribution would be zero. The last term is the camber contribution. $C(k)$ is the Theodorsen function, which serves as a complex gain on the circulatory part of the lift. $\text{Re}$ here refers to the real-part of the function, not Reynolds number.

The decrease in lift and lag is readily seen in the Theodorsen function, where $C(k)=F(k)+iG(k)$:

Theodorsen function

Ref: https://pdfs.semanticscholar.org/8457/c72b980f2f129a9f211617a5cdee4e162b75.pdf

The reduced frequency, $k=\frac{\omega c}{2V_\infty}$, denotes how many airfoil chord lengths per flow distance traveled in one motion period ($\omega$ is the angular frequency of the cyclic motion), and is a measure of how much the cyclical motion affects the flow on the airfoil. As $k \to 0$, there is no attenuation in lift and no phase lag; this is the quasi-steady aerodynamics. As $k$ increases, there is increasing attenuation in lift, up to 50%. The maximum phase lag occurs at $k \approx 0.25$.

For most rigid-body motions in commercial aircraft, the reduced frequency is low and the quasi-steady assumption is valid.

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  • $\begingroup$ Thanks. in the formula of $C_l$ it is not clear, is that a multiplication between Reynolds number and the other coefficients? $C(k)$ is a complex number, so the $C_l$ is a complex number? how to read that graph of theoderson function? $\endgroup$ – IamNotaMathematician Jun 25 at 17:07
  • $\begingroup$ @IamNotaMathematician Re here refers to the real-part of the overall complex function, thereby making Cl real. The graph shows the real and imaginary components (corresponding to the blue line parametrized by k). The arrow shows the phase. $\endgroup$ – JZYL Jun 25 at 18:05
  • $\begingroup$ Ah thanks for clarifying, Just a last question: what is the phase? $\endgroup$ – IamNotaMathematician Jun 25 at 18:07
  • $\begingroup$ @IamNotaMathematician What do you mean "the phase"? What is phase? Phase is the angular lag. en.wikipedia.org/wiki/Phase_(waves) $\endgroup$ – JZYL Jun 25 at 18:13
  • $\begingroup$ Yes I know what it is, but in this context, I have no idea what does this mean "phase lag in lift" how ? with respect to what? $\endgroup$ – IamNotaMathematician Jun 25 at 18:38

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