Source & Sink Flow

By /

In potential flow theory, a source (or sink) represents a point in the flow field where fluid is added (or removed) uniformly in all directions.

These are fundamental elementary flows used in superposition to build more complex solutions, such as flow around bodies.

1️⃣ Physical Concept

  • Source: A point where fluid is generated and flows outward radially.
  • Sink: A point where fluid is absorbed and flows inward radially.

These flows are radially symmetric and steady, with no rotation (irrotational) and no viscosity.

Examples:

  • Flow out of a hole in a tank (source-like).
  • Flow into a drain (sink-like).

2️⃣ Velocity Field in Polar Coordinates

For a point source (or sink) located at the origin in 2D polar coordinates (r, θ):

V_r = \frac{Q}{2 \pi r}, \quad V_\theta = 0

Where:

  • Q = source strength (m²/s)
    • Positive for a source.
    • Negative for a sink.
  • r = radial distance from the source.
  • V_r = radial velocity component.
  • V_\theta = 0 (no tangential component).

Explanation:

  • Velocity decreases inversely with distance.
  • Flow is purely radial.

3️⃣ Continuity Considerations

In 2D incompressible flow, the volume flow rate across a circle of radius r must equal the source strength:

\int_{0}^{2\pi} V_r \cdot r , d\theta = Q

This confirms that fluid added (or removed) at the origin spreads uniformly.

4️⃣ Stream Function

The stream function \psi for a source or sink is given by:

\psi(r, \theta) = \frac{Q}{2\pi} \theta

Properties:

  • Lines of constant \psi are radial lines (constant θ).
  • Streamlines radiate outward from a source and inward to a sink.

5️⃣ Velocity Potential Function

The velocity potential \phi describes the potential flow field:

\phi(r, \theta) = \frac{Q}{2\pi} \ln r

Properties:

  • Lines of constant \phi are circles centered at the origin (equipotential lines).
  • Orthogonal to streamlines.

6️⃣ Flow Field Visualization

  • Streamlines: straight lines radiating from the origin.
  • Equipotential lines: concentric circles around the origin.

This flow net of radial lines and circles is a classic illustration in potential flow theory.

7️⃣ Source and Sink Superposition

Potential flow theory allows superposition of elementary flows:

✅ Combining a source and a sink of equal strength but separated in space yields a doublet.
✅ Source or sink with uniform flow creates stagnation points and models flow around obstacles.

8️⃣ Applications in Aerodynamics

While real fluids don’t have true sources or sinks in the freestream, these idealized flows help model:

✅ Flow around bodies (cylinders, airfoils) using combinations of sources, sinks, and uniform flows.
✅ Intake and exhaust in simplified internal flow models.
✅ Groundwater hydrology analogies.

By understanding source and sink flows, we gain tools for constructing more realistic aerodynamic flow fields through superposition.

In summary, source and sink flows are fundamental solutions in potential flow theory, representing idealized, radially symmetric flows. They serve as building blocks for creating more complex flow patterns around aerodynamic bodies.

Shopping Cart
Scroll to Top