Potential flow solution

Theory and Concepts
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1

what is the formula for a potential flow vortex

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This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
Below is the standard description for a 2D irrotational flow vortex—what you might call a potential flow vortex—with the appropriate formulas.

In an irrotational flow vortex the velocity potential and streamfunction are given by

\phi = \frac{\Gamma}{2\pi}\theta,

and

\psi = -\frac{\Gamma}{2\pi}\ln r,

where
• r=\sqrt{x^2+y^2} is the radial distance from the vortex center,
• \theta is the polar angle (which can be computed using a two-argument inverse tangent such as \operatorname{atan2}(y,x)), and
• \Gamma is the circulation, which quantifies the strength of the vortex.

The velocity components in Cartesian coordinates are

u = -\frac{\Gamma}{2\pi}\frac{y}{r^2}, \quad v = \frac{\Gamma}{2\pi}\frac{x}{r^2}.

In polar coordinates the relations are particularly simple:

u_r = 0, \quad u_\theta = \frac{\Gamma}{2\pi r}.

These expressions represent the ideal or irrotational vortex flow. Note that although the flow is irrotational everywhere (i.e., with zero vorticity) in the fluid domain, there is a mathematical singularity at the origin.

If you need any more details on this formula or its derivation, feel free to ask!

While crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Here are the sections I found most beneficial in formulating my answer. I encourage you to explore the pertinent ones directly:

Textbook sections:

  1. Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 2 (RELEVANT)
  2. Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 1 (RELEVANT)
  3. Chapter 8.1: Flow Visualization (RELEVANT)
  4. Chapter 9 Fundamentals: Analogy between potential flow and electrostatics (RELEVANT)
  5. Chapter 11.3: Poiseuille Flow (RELEVANT)
  6. Chapter 11.2: Couette Flow (MARGINALLY RELEVANT)
  7. Chapter 8 Fundamentals: Streamfunction (MARGINALLY RELEVANT)
  8. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 2 (MARGINALLY RELEVANT)
  9. Chapter 8.5: Other Flow Visualization Techniques (MARGINALLY RELEVANT)
  10. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 1 (MARGINALLY RELEVANT)

Disclaimer:

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