Polar Coordinate Given Radial Velocity Calculator

Formula Used:

\[ \theta = \arccos\left(\frac{V_r}{\frac{\mu}{2\pi r^3} - V_\infty}\right) \]

Radial Velocity (Vr):

m/s

Doublet Strength (μ):

m³/s

Radial Coordinate (r):

Freestream Velocity (V∞):

m/s

Polar Angle (θ):

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1. What is the Polar Coordinate Given Radial Velocity Formula?

The polar coordinate given radial velocity formula calculates the angular position (θ) of a point in a polar coordinate system based on radial velocity, doublet strength, radial coordinate, and freestream velocity. This formula is particularly useful in fluid dynamics and aerodynamics applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \arccos\left(\frac{V_r}{\frac{\mu}{2\pi r^3} - V_\infty}\right) \]

Where:

\( \theta \) — Polar angle (radians/degrees)
\( V_r \) — Radial velocity (m/s)
\( \mu \) — Doublet strength (m³/s)
\( r \) — Radial coordinate (m)
\( V_\infty \) — Freestream velocity (m/s)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula calculates the inverse cosine of the ratio between radial velocity and the difference between doublet-induced velocity and freestream velocity.

3. Importance of Polar Angle Calculation

Details: Accurate polar angle calculation is crucial for analyzing flow fields around aerodynamic bodies, understanding vortex dynamics, and solving potential flow problems in fluid mechanics.

4. Using the Calculator

Tips: Enter all values in appropriate units. Radial coordinate must be greater than zero. The calculated ratio must be between -1 and 1 for valid arccos computation.

5. Frequently Asked Questions (FAQ)

Q1: What is doublet strength in fluid dynamics?
A: Doublet strength represents the product of distance between a source-sink pair and their strength, characterizing the flow field around the doublet.

Q2: Why is the radial coordinate cubed in the denominator?
A: The r³ term comes from the mathematical derivation of velocity potential for a doublet in potential flow theory, where velocity decreases with the cube of distance.

Q3: What are typical units for these parameters?
A: Radial velocity and freestream velocity in m/s, doublet strength in m³/s, radial coordinate in meters, and polar angle in radians or degrees.

Q4: When does this formula give invalid results?
A: The formula becomes invalid when the denominator approaches zero or when the calculated ratio falls outside the range [-1, 1] for the arccos function.

Q5: Can this be used for compressible flow?
A: This formula is derived for incompressible potential flow. For compressible flow, additional factors and equations need to be considered.