Doublet Flow

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In potential flow theory, a doublet is an elementary flow created by combining a source and a sink of equal strength placed very close together.

As their separation distance approaches zero while their strengths remain equal and opposite, the flow field takes on a distinctive form that’s very useful in aerodynamics.

Doublet flow is fundamental because it models flow around symmetrical bodies, like a cylinder in uniform flow.

1️⃣ Physical Concept

✅ Imagine placing a source and a sink of strength Q a small distance d apart.
✅ Let them approach each other ( d \rightarrow 0 ) while maintaining the product Qd = \kappa constant.

The limiting case defines a doublet of strength \kappa .

Interpretation:

  • No net addition or removal of fluid overall.
  • Flow patterns resemble those around a solid obstacle.

2️⃣ Mathematical Representation

In polar coordinates (r, θ) with the doublet at the origin and aligned along the x-axis:

Velocity potential:

\phi(r, \theta) = -\frac{\kappa}{2\pi} \frac{\cos \theta}{r}

Stream function:

\psi(r, \theta) = -\frac{\kappa}{2\pi} \frac{\sin \theta}{r}

Where:

  • \kappa = doublet strength (m²/s).
  • r , \theta = polar coordinates.

3️⃣ Velocity Field

From the potential function, radial and tangential velocity components are:

V_r = \frac{\partial \phi}{\partial r} = \frac{\kappa}{2\pi} \frac{\cos \theta}{r^2}

V_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta} = \frac{\kappa}{2\pi} \frac{\sin \theta}{r^2}

Key Features:

  • Velocity magnitude decreases as 1/r^2 .
  • Flow is symmetric about the x-axis.

4️⃣ Flow Field Visualization

✅ Streamlines: Banana-shaped closed loops around the origin, approximating the shape of a cylinder.
✅ Equipotential lines: Orthogonal to streamlines.

A doublet alone models the disturbance caused by a solid body in flow but without a net source or sink.

5️⃣ Superposition with Uniform Flow

One of the most important uses of the doublet is in superposition with uniform flow:

Uniform flow in x-direction:

\phi_{uniform} = U x = U r \cos \theta

Total potential (uniform + doublet):

\phi_{total} = U r \cos \theta - \frac{\kappa}{2\pi} \frac{\cos \theta}{r}

Stream function:

\psi_{total} = U r \sin \theta - \frac{\kappa}{2\pi} \frac{\sin \theta}{r}

Result:

  • The combined streamlines form a closed contour approximating a cylinder.
  • Outside the “cylinder,” the flow mimics real potential flow around it.

This approach models flow around a circular cylinder without viscosity.

6️⃣ Physical Interpretation

  • The doublet represents the effect of a solid body displacing fluid without adding or removing it.
  • By combining elementary flows (source, sink, uniform flow, doublet), we can model real-world scenarios.
  • The flow has no net mass addition, only a disturbance that creates pressure variations around the object.

7️⃣ Applications in Aerodynamics

✅ Flow around cylinders: Basic model for bluff bodies.
✅ Airfoil theory: Building blocks for more complex conformal mappings.
✅ Understanding lift: Combined with circulation (vortex) to model lifting flows over airfoils.
✅ Teaching tool: Explains how complex flows are built from simple solutions.

8️⃣ Limitations

❌ Inviscid assumption means no boundary layer or drag.
❌ No flow separation predicted.
❌ Only approximates external flow patterns.

Despite these limits, doublet flow is essential in understanding potential flow theory and forms the basis for analytical methods in early aerodynamic design.

In summary, the doublet is a fundamental potential flow element that models the effect of a solid object in a fluid stream. Through superposition, it helps recreate real aerodynamic flows and understand how pressure and velocity distribute around bodies.

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