# Formation Flight Induced Drag Reduction: Step-by-Step Solution Let's break down this problem about formation flight and induced drag, step by step. --- ## **Background Concepts** - **Induced Drag (\(D_i\))**: Drag due to lift production, related to the strength of trailing vortices (downwash). - **Formation Flight**: Multiple airplanes flying side-by-side can "share" vortices, reducing induced drag compared to flying alone. --- ## **Given** - Each airplane has a span \(b\). - Planes are separated by \(b/2\). - Compare induced drag in formation (2 or 3 planes) to a single, isolated airplane. --- ## **Hint**: Evaluate downwash at wing root of each plane. --- ## **Step 1: Induced Drag for a Single Airplane** For an isolated airplane: \[ D_{i, \text{single}} = \frac{L^2}{\pi \rho V^2 b^2} \] where: - \(L\) = Lift - \(b\) = wingspan - \(\rho\) = air density - \(V\) = velocity --- ## **Step 2: Downwash Calculation** - **Downwash (\(w\))** at the wing root is proportional to induced drag. - In formation, the downwash at each root changes due to the proximity of other aircraft's vortices. --- ## **Step 3: Two Airplanes in Formation (i)** #### **Vortex Arrangement** - Each plane produces a pair of trailing vortices. - The root of each plane is \(b/2\) from the nearest neighbor's vortex. #### **Downwash at the Root:** The downwash at the root due to: - **Own wingtip:** \(w_\) - **Neighbor's wingtip (distance \(b/2\))**: \(w_n\) The total downwash at the root becomes: \[ w_{\text{formation}} = w_ + w_n \] But, \(w_n\) (from the neighbor) acts in the **opposite direction** to \(w_\), reducing the net downwash and induced drag. #### **Reduction Calculation** - For two planes, the induced drag at the root is about **half** as large as when flying alone (from classical lifting line theory). - **Percentage reduction:** \[ \text{Reduction} = \frac{D_{i, \text{single}} - D_{i, \text{formation}}}{D_{i, \text{single}}} \times 100\% \] For two planes: **~50% reduction on average** #### **Edge vs. Inside Plane** - Both planes are at the "edge" (no inside). Both experience the same reduction. --- ## **Step 4: Three Airplanes in Formation (ii)** #### **Arrangement** - Middle plane has neighbors on both sides. - Edge planes have only one neighbor. #### **Downwash at the Root** - **Middle plane:** Experiences reduced downwash from both sides (max reduction). - **Edge planes:** Experience reduction only from one neighbor. #### **Calculation** - **Middle:** Induced drag reduction is greatest—almost **2/3** reduction. - **Edge:** Reduction is less—about **1/2** reduction. #### **Average Reduction:** \[ \text{Average reduction} = \frac{\text{Reduction (middle)} + 2 \times \text{Reduction (edge)}}{3} \] \[ = \frac{(2/3) + 2 \times (1/2)}{3} = \frac{2/3 + 1}{3} = \frac{5/3}{3} = \frac{5}{9} \approx 56\% \] --- ## **Summary Table** | # of Planes | Edge Plane Reduction | Middle Plane Reduction | Average Reduction | |:-----------:|:-------------------:|:---------------------:|:----------------:| | 2 | 50% | — | 50% | | 3 | 50% | 67% | 56% | --- ## **Key Points** - **Inside planes** (middle) get **more reduction** in induced drag than edge planes. - **Average reduction** increases with more planes in formation. - **Formation flight** is highly efficient for induced drag reduction. --- ## **Final Answers** **(i) Two airplanes:** - Each plane has a **50% reduction** in induced drag compared to flying alone. **(ii) Three airplanes:** - **Middle airplane:** ~67% reduction. - **Edge airplanes:** ~50% reduction. - **Average reduction:** ~56%. --- ### **If you need the detailed derivation of the downwash formula or more about the physics, let me know!**