One of the most powerful ideas in potential flow theory is superposition—the ability to combine simple, idealized flows to model more complex patterns.
Because the governing equations of potential flow (Laplace’s equation for the potential function) are linear, solutions can be added together to create new solutions.
This principle is essential in aerodynamics because it lets us build models of flow around objects like cylinders and airfoils by combining elementary flows.
1️⃣ Superposition Principle
Key idea:
If φ₁ and φ₂ are solutions to Laplace’s equation, then φ_total = φ₁ + φ₂ is also a solution.
Similarly for the stream function ψ:
If ψ₁ and ψ₂ are valid stream functions, then ψ_total = ψ₁ + ψ₂ describes the combined flow.
✅ The linearity of Laplace’s equation makes this possible.
✅ Velocity components add linearly.
2️⃣ Common Elementary Flows to Combine
These elementary flows each have simple, well-understood solutions:
- Uniform flow: Constant velocity in one direction.
- Source/Sink: Radial flow from/to a point.
- Doublet: Source and sink close together; models the effect of solid bodies.
- Vortex: Circular flow around a point.
By adding these, we can construct complex flow fields with desired properties.
3️⃣ Mathematical Representation
Suppose we have two potential flows:
Their combination:
Similarly:
Velocity components are then:
Interpretation:
- The resulting velocity at any point is just the vector sum of velocities from each component flow.
4️⃣ Physical Examples of Flow Combinations
✅ Uniform flow + Source
- Models flow past a point of fluid injection in a freestream.
- Streamlines form a stagnation point in front of the source.
✅ Uniform flow + Sink
- Models flow into a drain in a freestream.
✅ Uniform flow + Doublet
- Models flow around a solid cylinder (no circulation yet).
- Streamlines wrap smoothly around the cylinder shape.
✅ Uniform flow + Doublet + Vortex
- Models flow around a lifting cylinder (circulation creates lift).
- Explains asymmetric pressure distribution, basis for Kutta–Joukowski theorem.
5️⃣ Visualization of Flow Nets
✅ Streamlines show flow paths.
✅ Equipotential lines are orthogonal to streamlines.
When flows are combined:
- The patterns warp and bend to satisfy boundary conditions.
- The combination of simple, known patterns helps design and predict real flows.
Engineers and scientists often sketch flow nets showing how streamlines and potential lines intersect after superposition.
6️⃣ Importance in Aerodynamics
Superposition allows us to:
✅ Build analytical solutions to model flow around shapes.
✅ Approximate pressure distributions on bodies.
✅ Predict lift and circulation using elementary solutions.
✅ Design airfoils, wings, and bodies by understanding how flow responds to shape changes.
7️⃣ Foundation for Flow Over a Cylinder
Key motivation:
The classic example of flow combination is flow over a cylinder:
- Constructed by combining uniform flow with a doublet.
- To model lift, add a vortex for circulation.
This combination:
✅ Satisfies impermeability on the cylinder surface (no flow through).
✅ Shows stagnation points, flow wrapping around.
✅ Explains pressure variations and resulting forces.
8️⃣ Summary
✨ Flow combinations use the linearity of potential flow to build realistic, analyzable models.
✨ By understanding and adding elementary solutions, we model real aerodynamic scenarios.
✨ This approach is the backbone for studying flow over cylinders, airfoils, and wings in classical aerodynamics.
In the next section, we’ll see how flow over a cylinder is constructed from these building blocks.
