Certainly! Let's go through the problem step-by-step. --- ## **Given Data:** - The NACA 4412 airfoil has a mean camber line \( m = .04 \) (or 4% of chord length). - The length \( c = 1 \) (unit length for simplicity). - The maximum camber occurs at \( x_{camber} = .4c \). - The maximum thickness \( t = .12 \) (or 12% of the chord length). - The thickness distribution \( y_t \) is given by: \[ y_t = 5t \left[ .2969 \sqrt{\frac{x}{c}} - .126 \frac{x}{c} - .3516 \left(\frac{x}{c}\right)^2 + .2843 \left(\frac{x}{c}\right)^3 - .1015 \left(\frac{x}{c}\right)^4 \right] \] --- ## **Step 1: Understanding the Camber Line Equation** The camber line \( y_c \) is defined as: \[ \boxed{ y_c = \begin{cases} \frac{m}{x_{camber}^2} (2x c - x^2), & \leq x \leq x_{camber} \\ \frac{m}{(1 - x_{camber})^2} [(1 - 2x) + x^2], & x_{camber} \leq x \leq c \end{cases} } \] Since the maximum camber occurs at \( x = .4c \), we will consider two regions: - **Region 1:** \( \leq x \leq .4c \) - **Region 2:** \( .4c \leq x \leq c \) --- ## **Step 2: Calculate \( y_c \) at any point \( x \)** ### For \( \leq x \leq .4c \): \[ y_c = \frac{m}{x_{camber}^2} (2x c - x^2) \] Plugging in \( m = .04 \), \( x_{camber} = .4 \), and \( c=1 \): \[ y_c = \frac{.04}{(.4)^2} (2x - x^2) \] \[ = \frac{.04}{.16} (2x - x^2) = .25 (2x - x^2) \] ### For \( .4 \leq x \leq 1 \): \[ y_c = \frac{.04}{(1 - .4)^2} [(1 - 2x) + x^2] \] \[ = \frac{.04}{(.6)^2} [(1 - 2x) + x^2] = \frac{.04}{.36} [(1 - 2x) + x^2] \] \[ = \frac{1}{9} [(1 - 2x) + x^2] \] --- ## **Step 3: Calculate the thickness distribution \( y_t \)** Given: \[ y_t = 5t \left[ .2969 \sqrt{\frac{x}{c}} - .126 \frac{x}{c} - .3516 \left(\frac{x}{c}\right)^2 + .2843 \left(\frac{x}{c}\right)^3 - .1015 \left(\frac{x}{c}\right)^4 \right] \] For \( c=1 \): \[ y_t = 5 \times .12 \left[ .2969 \sqrt{x} - .126 x - .3516 x^2 + .2843 x^3 - .1015 x^4 \right] \] \[ = .6 \left[ .2969 \sqrt{x} - .126 x - .3516 x^2 + .2843 x^3 - .1015 x^4 \right] \] --- ## **Step 4: Find the upper surface coordinates** The upper surface is given by: \[ \boxed{ y_{upper} = y_c + y_t } \] and the lower surface: \[ \boxed{ y_{lower} = y_c - y_t } \] --- ## **Step 5: Summary of the process** - **Calculate \( y_c \)** at the desired point \( x \). - **Calculate \( y_t \)** at the same \( x \). - **Determine upper and lower surface points**: \[ \text{Upper surface} = y_c + y_t \] \[ \text{Lower surface} = y_c - y_t \] --- ## **Final note:** This process allows you to generate the airfoil shape by plotting \( y_{upper} \) and \( y_{lower} \) along the chord length \( c \). Let me know if you need specific values at particular points or further clarification!