Radial Coordinate Given Radial Velocity Calculator

Radial Coordinate Formula:

\[ r = \left( \frac{\mu}{2\pi \left( V_{\infty} + \frac{V_r}{\cos(\theta)} \right)} \right)^{1/3} \]

Doublet Strength (μ):

m³/s

Freestream Velocity (V∞):

m/s

Radial Velocity (Vr):

m/s

Polar Angle (θ):

rad

Radial Coordinate (r):

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1. What is the Radial Coordinate Formula?

The radial coordinate formula calculates the radial position of an object in a flow field using doublet strength, freestream velocity, radial velocity, and polar angle. This formula is particularly useful in fluid dynamics and aerodynamics for analyzing potential flow around objects.

2. How Does the Calculator Work?

The calculator uses the radial coordinate formula:

\[ r = \left( \frac{\mu}{2\pi \left( V_{\infty} + \frac{V_r}{\cos(\theta)} \right)} \right)^{1/3} \]

Where:

\( r \) — Radial coordinate (m)
\( \mu \) — Doublet strength (m³/s)
\( V_{\infty} \) — Freestream velocity (m/s)
\( V_r \) — Radial velocity (m/s)
\( \theta \) — Polar angle (rad)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula calculates the radial position based on the flow characteristics and doublet properties in a potential flow field.

3. Importance of Radial Coordinate Calculation

Details: Accurate radial coordinate calculation is crucial for analyzing flow patterns around aerodynamic bodies, designing efficient airfoils, and understanding fluid behavior in various engineering applications.

4. Using the Calculator

Tips: Enter doublet strength in m³/s, velocities in m/s, and polar angle in radians. Ensure polar angle is between -π/2 and π/2 (exclusive) to avoid division by zero.

5. Frequently Asked Questions (FAQ)

Q1: What is doublet strength?
A: Doublet strength is defined as the product of the distance between a source-sink pair and source or sink strength, measured in cubic meters per second.

Q2: Why is the polar angle limited to ±π/2?
A: The formula contains cos(θ) in the denominator. When θ approaches ±π/2, cos(θ) approaches zero, causing division by zero issues.

Q3: What are typical values for doublet strength?
A: Doublet strength values vary widely depending on the specific application, but typically range from 0.1 to 1000 m³/s in aerodynamic applications.

Q4: Can this formula be used for compressible flow?
A: This specific formula is derived for incompressible potential flow. For compressible flow, additional factors and equations would be needed.

Q5: What if I get a negative result?
A: The radial coordinate should always be positive. A negative result indicates invalid input values or mathematical error in the calculation.