Stream Function for Non-Lifting Flow over Circular Cylinder Calculator

Stream Function Formula:

\[ \psi = V_{\infty} \cdot r \cdot \sin(\theta) \cdot \left(1 - \left(\frac{R}{r}\right)^2\right) \]

Freestream Velocity (V∞):

m/s

Radial Coordinate (r):

Polar Angle (θ):

rad

Cylinder Radius (R):

Stream Function (ψ):

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1. What is the Stream Function for Non-Lifting Flow over Circular Cylinder?

The Stream Function describes the flow patterns around a circular cylinder in a uniform flow field. It represents the flow lines and helps visualize the fluid motion around the obstacle without any lift generation.

2. How Does the Calculator Work?

The calculator uses the stream function formula:

\[ \psi = V_{\infty} \cdot r \cdot \sin(\theta) \cdot \left(1 - \left(\frac{R}{r}\right)^2\right) \]

Where:

\( \psi \) — Stream Function (m²/s)
\( V_{\infty} \) — Freestream Velocity (m/s)
\( r \) — Radial Coordinate (m)
\( \theta \) — Polar Angle (rad)
\( R \) — Cylinder Radius (m)

Explanation: This formula describes the potential flow around a circular cylinder, where the flow is irrotational and incompressible.

3. Importance of Stream Function Calculation

Details: The stream function is crucial for analyzing fluid flow patterns, determining streamlines, and understanding the behavior of flow around obstacles in fluid dynamics applications.

4. Using the Calculator

Tips: Enter freestream velocity in m/s, radial coordinate in meters, polar angle in radians, and cylinder radius in meters. All values must be positive (radial coordinate must be greater than cylinder radius for valid results).

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the stream function?
A: The stream function represents the volume flow rate between streamlines. Constant values of ψ define streamlines in the flow field.

Q2: Why is this called "non-lifting" flow?
A: This flow pattern is symmetric about the horizontal axis, resulting in zero lift force on the cylinder.

Q3: What are the limitations of this formula?
A: This formula assumes ideal, inviscid, irrotational flow and does not account for viscous effects or flow separation that occurs in real fluids.

Q4: How does the cylinder radius affect the flow?
A: The cylinder radius determines the size of the obstacle and influences the distortion of streamlines around the cylinder.

Q5: Can this formula be used for compressible flows?
A: No, this formula is specifically derived for incompressible flow conditions.