# Why can't lifting line model be applied to yawed/skewed/oblique straight wing (low M)

If all the horseshoe vortices have skewed central bound vortices on the same skewed 1/4 chord line, then the trailing vortices from the downwind half span of the wing induce less on the windward halfspan and vice versa. So then the windward halfspan of the wing will have increased angle of attack and lift vs unskewed and the leeward have less.

But I gather this is the wrong result vs experiment? Please don't confuse the issue with symmetrical swept back wings. No, I used Lifting Line to generate results that disagree with experiments for asymmetrical oblique or yawed straight wings? What am I overlooking ?

Lifting Line analysis is not suitable for low aspect ratio wing, highly swept wing or a wing undergoing large sideslip, because each of these would have significant spanwise flow gradient.

To see why, it's worth revisiting the central equation of the Lifting Line, applicable to a planar wing:

$$c_l(y)=a_0 \left[ \alpha - \frac{w(y)}{V_\infty} - \alpha_{L=0} + \theta_t(y) \right]= \frac{2}{V^2_\infty c(y)} \lVert \vec{V}_\infty \times \vec{\Gamma}(y) \rVert$$

where $$y$$ is the spanwise coordinate of the section, $$c_l$$ is the 2D airfoil lift coefficient, $$a_0$$ is the 2D airfoil lift slope, $$\alpha$$ is the free-stream angle of attack against the wing, $$w$$ is the downwash generated by the trailing vortices1, $$\alpha_{L=0}$$ is the 2D airfoil zero-lift angle of attack (or aerodynamic twist), $$\theta_t$$ is the geometric twist of the section, $$c$$ is the section chord length, $$\vec{V}_\infty$$ and $$V_\infty$$ are the vector and magnitude of the free-stream air velocity against the wing, and $$\vec{\Gamma}$$ is the bound vortex strength at the section2.

The left hand side of the central equation is the 2D airfoil lift and the right hand side is the lift we expect from the circulation generated by the bound vortex. Assuming the 2D characteristics are known in advance (i.e. the 2D lift slope, zero lift AOA, etc), and the wing geometry is known (chord length at every section), then we should be able to calculate the unknown, $$\Gamma$$, at every section.

To put it crudely, Lifting Line is cheating: it relies on you knowing the 2D airfoil characteristics apriori, in order to calculate the 3D lift and drag (which is its biggest advantage). The downside is that it inherently assumes the 2D data tells the whole story; all influences from the neighbouring sections are lumped into a single downwash (induced AOA), which remains 2D. In a highly swept or sideslip'ed wing, this assumption breaks down.

1 $$w$$ can be calculated using Biot-Savart from the trailing vortices (which have constant strength, $$d{\Gamma}/dy$$, along the filament), but will need to be slightly modified from the typical result shown for the Classical Lifting Line (which is only applicable for straight tapered wing), since the trailing legs are parallel to the freestream, not the bound vortex.

2 Note that I've used the vectorized version of the Kutta-Joukowski Theorem since in a swept or sideslip'ed wing the bound vortex and freestream aren't perpendicular.