Enter Flow Conditions
Leave unused component strengths at zero. Use one consistent unit system for coordinates, speed, source strength, and circulation.
Formula Used
Combined complex potential: W(z) = Ue−iαz + [Q / (2π)]ln(z − zs) − i[Γ / (2π)]ln(z − zv)
Combined stream function: ψ = U(y cos α − x sin α) + [Q / (2π)]θs − [Γ / (2π)]ln(rv) + ψ₀
Here, r is the distance from a component center, and θ is its polar angle. The calculator adds each component using superposition. It also calculates velocity from dW/dz = u − iv.
How to Use This Calculator
- Enter the coordinate where you want the stream function.
- Set the uniform speed and its direction, when required.
- Enter source strength and location. Use a negative strength for a sink.
- Enter circulation and vortex location. Keep it zero when no vortex exists.
- Add an optional stream offset, then select the calculation button.
- Review ψ, potential, velocity components, and component contributions.
- Download the CSV result or print the page for a PDF record.
Example Data
| Input | Example value | Purpose |
|---|---|---|
| x, y | 2, 1 | Evaluation point |
| U, α | 4, 30° | Uniform flow condition |
| Q, xs, ys | 8, 0, 0 | Source component |
| Γ, xv, yv | 3, 1, 0 | Vortex component |
| ψ₀ | 0 | Reference offset |
Understanding Complex Potential Flow
A complex potential combines velocity potential and stream function information. It is written as W(z) = φ + iψ. The real part, φ, describes potential changes. The imaginary part, ψ, labels streamlines. Flow moves along lines where ψ remains constant. This calculator evaluates common elementary flows through superposition for ideal, incompressible, irrotational two-dimensional flow. It compactly shows how several elements influence a location during quick early engineering design checks.
Why the Stream Function Matters
The stream function gives a visual map of fluid movement. Every selected ψ value identifies one streamline. Different streamlines never cross in a valid steady flow field. Differences between stream function values represent volume flow rate per unit depth. This makes ψ valuable for channels, external flow, and basic hydrodynamic models. The result is not a direct pressure prediction. It is a geometric and kinematic description. Pressure estimates require additional information, such as density, elevation, and a suitable energy relation.
Included Flow Components
Uniform flow represents fluid moving at a constant speed and direction. Its stream function changes linearly with position. A source adds fluid outward from one point. A sink uses a negative source strength. Its stream function depends on the polar angle around that point. A point vortex rotates the fluid around its center. Its stream function depends on the logarithm of radial distance. Combining these components can model flow past simple bodies with suitable images or extra elements.
Interpreting Velocity Results
The calculator also reports horizontal and vertical velocity components. These values come from differentiating the complex potential. The speed is the magnitude of the combined velocity vector. Very high values near a source or vortex center are expected mathematically. Those centers are singular points. The ideal equations do not describe the physical core. Avoid evaluating exactly at a singularity. Move the coordinate slightly away, or use a model that includes a finite core radius when real fluid behavior matters.
Practical Calculation Tips
Use one consistent unit system. Enter coordinates and component locations in the same length unit. Enter uniform speed in length per time. Source strength represents area per time for unit depth. Circulation represents area per time. Keep the chosen angle unit consistent with the supplied direction. A positive source is outward. A negative value behaves as a sink. Positive circulation follows the selected sign convention. Compare several coordinates to trace a streamline pattern and confirm whether the combined field matches expectations.
Model Limits and Good Practice
This tool applies potential-flow assumptions. It does not include viscosity, boundary layers, turbulence, compressibility, or separation. These effects can dominate real systems. Use the results for learning, early checks, and idealized flow studies. Validate engineering designs with appropriate numerical simulation, experiments, or professional analysis. Record every input with its unit and sign. Small sign changes can reverse rotation or flow direction. Clear input records also make it easier to reproduce a result and explain it to others.
Frequently Asked Questions
1. What is a complex potential?
A complex potential is W(z) = φ + iψ. Its real part is the velocity potential. Its imaginary part is the stream function. It provides a compact description of an ideal two-dimensional flow field.
2. What does the stream function represent?
The stream function labels streamlines. Fluid moves along curves with constant ψ. The difference between two stream-function values represents volume flow rate per unit depth between those curves.
3. Can I calculate a sink with this tool?
Yes. Enter a negative value for source strength Q. The same source equation is used, but the negative sign reverses the radial flow direction toward the selected center.
4. What does positive circulation mean?
With this convention, positive circulation creates counterclockwise rotation around the vortex center. Reversing the sign creates clockwise rotation. Confirm the sign convention required by your course or model.
5. Why is my result undefined?
The result is undefined when the evaluation point equals an active source or vortex center. The ideal equations contain a logarithm or division by zero at those singular locations.
6. Are degrees and radians both supported?
Yes. Select degrees or radians before calculating. The flow direction is converted internally for the trigonometric terms. The reported flow direction is shown in degrees for readability.
7. Which units should I use?
Use a consistent system. Coordinates use length, uniform speed uses length per time, and source strength or circulation use area per time for unit depth. The stream function then has area-per-time units.
8. Does the calculator include viscosity?
No. It uses ideal potential-flow equations. Viscosity, turbulence, separation, boundary layers, and compressibility are excluded. Use more detailed methods when those effects are important.
9. Can several components be combined?
Yes. The calculator adds uniform flow, one source or sink, and one vortex. This is based on superposition, which applies because the governing potential-flow equations are linear.
10. Why do velocities become very large near centers?
Point sources and point vortices are mathematical idealizations. Their velocity expressions increase without limit close to the center. Real flows have finite cores and physical limits that the ideal model omits.
11. Can I save my calculation?
Yes. After calculating, use Download CSV for a data file. You can also use Print or Save PDF to preserve the visible inputs, result values, equations, and explanatory sections.