# Load Distribution on a Planar Wing (NACA 0012, AR = 7., Taper Ratio = .4) This outlines the process to represent the lift distribution across a planar (untwisted) wing with the specified characteristics using a four-term series. ## Wing Characteristics - **Airfoil:** NACA 0012 (symmetric, linear lift curve) - **Aspect Ratio (AR):** 7. - **Taper Ratio (\(\lambda\)):** .4 - **Geometric Twist:** None (planar wing) ## Series Representation: Lifting-Line Theory The spanwise circulation (or load) distribution can be represented as a Fourier sine series: \[ \theta = \cos^{-1}\left(\frac{2y}{b}\right) \] \[ \frac{c(y) \, C_l(\theta)}{4b} = \sum_{n=1}^{N} A_n \sin(n\theta) \] where: - \(b\): wingspan - \(y\): spanwise location, \( -b/2 \leq y \leq b/2 \) - \(c(y)\): local chord at \(y\) - \(C_l(\theta)\): local section lift coefficient - \(A_n\): series coefficients For a **4-term series** (\(N = 4\)): \[ C_l(\theta) = \sum_{n=1}^4 A_n \sin(n\theta) \] ## Chord Distribution (Tapered Wing) The chord at any spanwise position is: \[ c(y) = c_{root} \left[1 - (1 - \lambda)\frac{2|y|}{b}\right] \] or, in terms of \(\theta\): \[ c(\theta) = c_{root} \left[1 - (1 - \lambda)\sin\theta\right] \] where: - \(\lambda = .4\): taper ratio ## Boundary Conditions For a planar, untwisted wing with a symmetric airfoil, the angle of attack is constant and there is no geometric twist. ## Discretization Select four values of \(\theta\) for \(N = 4\): \[ \theta_j = \frac{j\pi}{N+1}, \quad j = 1, 2, 3, 4 \] \[ \theta_1 = .2\pi \quad \theta_2 = .4\pi \quad \theta_3 = .6\pi \quad \theta_4 = .8\pi \] ## Solving for Series Coefficients (\(A_n\)) The coefficients \(A_n\) are found by solving the following system at the collocation points (\(\theta_j\)): \[ \alpha_ = \sum_{n=1}^4 A_n \sin(n\theta_j) \left[\frac{4b}{c(\theta_j) a_} + n\frac{\sin\theta_j}{\sin(n\theta_j)}\right] \] Where: - \(\alpha_\): geometric angle of attack (assumed constant) - \(a_\): sectional lift curve slope (for NACA 0012, typically \(2\pi\)) - \(c(\theta_j)\): chord at \(\theta_j\) - \(b\): wingspan Set up the 4 equations (one at each \(\theta_j\)), and solve for \(A_1, A_2, A_3, A_4\). ## Summary of Procedure 1. **Express the chord distribution:** \[ c(\theta) = c_{root} \left[1 - .6 \sin\theta\right] \] 2. **Choose collocation points:** \(\theta_j = .2\pi, .4\pi, .6\pi, .8\pi\) 3. **Set up a system of 4 equations:** For each \(\theta_j\), use the lifting-line equation to relate the \(A_n\). 4. **Solve for \(A_1, A_2, A_3, A_4\):** This gives the spanwise loading as a 4-term series. ## Final Series Form The load distribution is: \[ C_l(\theta) = A_1 \sin\theta + A_2 \sin(2\theta) + A_3 \sin(3\theta) + A_4 \sin(4\theta) \] The coefficients \(A_n\) depend on the wing planform and can be determined numerically from the above system. --- **Note:** - This method gives an approximate spanwise load distribution for a tapered, untwisted wing using a 4-term Fourier sine series. - For actual values of \(A_n\), plug in the numerical values for \(c(\theta_j)\), \(\lambda\), \(AR\), and solve the linear system.