Stream Function For Lifting Flow Over Circular Cylinder Calculator

Stream Function Formula:

\[ \psi = V_{\infty} \times r \times \sin(\theta) \times (1 - (R/r)^2) + \frac{\Gamma}{2\pi} \times \ln(r/R) \]

Freestream Velocity (V∞):

m/s

Radial Coordinate (r):

Polar Angle (θ):

rad

Cylinder Radius (R):

Vortex Strength (Γ):

m²/s

Stream Function (ψ):

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

The Stream Function for Lifting Flow over a Circular Cylinder describes the flow pattern around a cylinder with circulation. It combines potential flow around a cylinder with a vortex to model lifting flows, commonly used in aerodynamics and fluid mechanics.

2. How Does the Calculator Work?

The calculator uses the stream function formula:

\[ \psi = V_{\infty} \times r \times \sin(\theta) \times (1 - (R/r)^2) + \frac{\Gamma}{2\pi} \times \ln(r/R) \]

Where:

\( \psi \) — Stream Function (m²/s)
\( V_{\infty} \) — Freestream Velocity (m/s)
\( r \) — Radial Coordinate (m)
\( \theta \) — Polar Angle (rad)
\( R \) — Cylinder Radius (m)
\( \Gamma \) — Vortex Strength (m²/s)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The equation combines the potential flow around a cylinder with the effect of circulation to model lifting flows.

3. Importance of Stream Function Calculation

Details: Stream function calculation is crucial for analyzing fluid flow patterns, determining streamlines, and understanding the behavior of lifting flows around circular cylinders in aerodynamics and hydrodynamics.

4. Using the Calculator

Tips: Enter all values in appropriate units. Freestream velocity, radial coordinate, and cylinder radius must be positive values. Polar angle should be in radians.

5. Frequently Asked Questions (FAQ)

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

Q2: When is this stream function model applicable?
A: This model is applicable for inviscid, incompressible, irrotational flow around a circular cylinder with circulation.

Q3: What does the vortex strength represent?
A: Vortex strength quantifies the circulation around the cylinder, which creates lift in the flow field.

Q4: Are there limitations to this model?
A: This model assumes ideal fluid conditions and doesn't account for viscous effects, separation, or compressibility.

Q5: How is this used in practical applications?
A: This model is used in aerodynamics to study lift generation around cylindrical structures and in educational contexts to demonstrate potential flow theory.