Polar Coordinate Given Tangential Velocity Calculator

Formula Used:

\[ \theta = \sin^{-1}\left(\frac{V_\theta}{V_\infty + \frac{\mu}{4\pi r^3}}\right) \]

Tangential Velocity (Vθ):

m/s

Freestream Velocity (V∞):

m/s

Doublet Strength (μ):

m³/s

Radial Coordinate (r):

Polar Angle (θ):

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1. What is the Polar Angle Calculation?

The polar angle calculation determines the angular position of a point in a polar coordinate system based on tangential velocity, freestream velocity, doublet strength, and radial coordinate. This is particularly useful in fluid dynamics and aerodynamics applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \sin^{-1}\left(\frac{V_\theta}{V_\infty + \frac{\mu}{4\pi r^3}}\right) \]

Where:

\( \theta \) — Polar Angle (radians)
\( V_\theta \) — Tangential Velocity (m/s)
\( V_\infty \) — Freestream Velocity (m/s)
\( \mu \) — Doublet Strength (m³/s)
\( r \) — Radial Coordinate (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula calculates the inverse sine of the ratio between tangential velocity and the sum of freestream velocity plus the doublet strength term.

3. Importance of Polar Angle Calculation

Details: Accurate polar angle calculation is crucial for analyzing fluid flow patterns, understanding vortex behavior, and solving aerodynamic problems involving doublet flows and potential flow theory.

4. Using the Calculator

Tips: Enter all values in appropriate units (m/s for velocities, m³/s for doublet strength, m for radial coordinate). Ensure all values are positive and the denominator doesn't become zero.

5. Frequently Asked Questions (FAQ)

Q1: What is tangential velocity in fluid dynamics?
A: Tangential velocity is the component of velocity that is perpendicular to the radial direction in a polar coordinate system.

Q2: What does doublet strength represent?
A: Doublet strength represents the product of distance between a source-sink pair and the strength of either the source or sink in potential flow theory.

Q3: When is this calculation particularly useful?
A: This calculation is particularly useful in analyzing flow around cylinders, spheres, and other bodies in potential flow theory applications.

Q4: What are the limitations of this formula?
A: The formula assumes ideal fluid flow, incompressible flow, and applies specifically to doublet flow fields in potential flow theory.

Q5: Why is the result in degrees?
A: While the mathematical calculation produces radians, the result is converted to degrees for easier interpretation, as degrees are more commonly used in practical applications.