Polar Coordinate Given Surface Velocity For Flow Over Sphere Calculator

Formula Used:

\[ \text{Polar Angle} = \sin^{-1}\left(\frac{2}{3} \times \frac{\text{Tangential Velocity}}{\text{Freestream Velocity}}\right) \] \[ \theta = \sin^{-1}\left(\frac{2}{3} \times \frac{V_\theta}{V_\infty}\right) \]

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 from a reference direction in spherical flow over a sphere. It relates the tangential velocity component to the freestream velocity using inverse trigonometric functions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \sin^{-1}\left(\frac{2}{3} \times \frac{V_\theta}{V_\infty}\right) \]

Where:

\( \theta \) — Polar Angle (degrees)
\( V_\theta \) — Tangential Velocity (m/s)
\( V_\infty \) — Freestream Velocity (m/s)

Explanation: This formula calculates the polar angle based on the ratio of tangential velocity to freestream velocity, scaled by a factor of 2/3.

3. Importance of Polar Angle Calculation

Details: Accurate polar angle calculation is crucial for analyzing flow patterns over spherical surfaces, determining flow separation points, and understanding velocity distributions in aerodynamic applications.

4. Using the Calculator

Tips: Enter tangential velocity and freestream velocity in m/s. Both values must be positive. The ratio (2/3 * Vθ/V∞) must be between -1 and 1 for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the 2/3 factor?
A: The 2/3 factor comes from the theoretical solution for potential flow over a sphere, representing the relationship between surface velocity components.

Q2: What range of values is valid for the polar angle?
A: The polar angle typically ranges from 0° to 180° in spherical coordinates, with 0° at the front stagnation point and 180° at the rear stagnation point.

Q3: When is this calculation most accurate?
A: This calculation is most accurate for inviscid, incompressible flow over a sphere where potential flow theory applies.

Q4: Are there limitations to this equation?
A: This equation assumes ideal flow conditions and may not accurately represent real flows with viscosity, turbulence, or compressibility effects.

Q5: How does this relate to actual flow measurements?
A: While based on theoretical potential flow, this calculation provides a baseline for comparing with experimental measurements and computational fluid dynamics results.