Calculator
Example data table
| Model | Inputs | Expected φ (m²/s) | Notes |
|---|---|---|---|
| Uniform / Linear (Cartesian) | Ux=2, Vy=0, Wz=0, x=3, y=0, z=0, φ0=0 | 6 | φ grows linearly along x for constant Ux. |
| Uniform (Polar) | U=5, r=2, θ=60° , φ0=0 | 5 | φ=U r cosθ = 5×2×0.5. |
| Source / Sink (2D) | Q=1, r=2, rref=1, φ0=0 | 0.1103178 | φ=(Q/2π) ln(r/rref). |
| Vortex (2D) | Γ=2, θ=π (rad), φ0=0 | 1 | φ=(Γ/2π)θ = (2/2π)π. |
Formula used
- Uniform / Linear (Cartesian): φ = Ux + Vy + Wz + φ0
- Uniform (polar): φ = U r cos(θ) + φ0
- Source / Sink (2D): φ = (Q/2π) ln(r/r_ref) + φ0
- Doublet (2D): φ = (κ cosθ)/(2π r) + φ0 (or omit 2π by convention)
- Vortex (2D): φ = (Γ/2π) θ + φ0 (multi-valued; choose an angle branch)
How to use this calculator
- Select a flow model that matches your ideal-flow assumption.
- Enter the required parameters with consistent SI units.
- If your model uses θ, choose degrees or radians first.
- Optionally set φ0 as a reference constant.
- Press Submit to view φ above the form instantly.
- Export the last computed result using CSV or PDF.
Professional article
1) What velocity potential represents
In irrotational flow, the velocity field is the gradient of a scalar potential, v = ∇φ. The potential carries units of m²/s because it integrates velocity (m/s) over distance (m). This calculator returns φ for standard analytic solutions used in potential-flow modeling.
2) Uniform and linear potentials in Cartesian space
The uniform/linear option applies φ = Ux + Vy + Wz + φ0. With Ux = 2 m/s and x = 3 m, the potential becomes 6 m²/s when other components are zero. This model is practical for checking sign conventions, reference offsets, and coordinate scaling.
3) Uniform flow in polar coordinates
For a 2D uniform stream aligned with the x-axis, the calculator uses φ = U r cos(θ) + φ0. At U = 5 m/s, r = 2 m, and θ = 60°, you obtain 5 m²/s because cos(60°)=0.5. Switching degrees/radians changes only the angle interpretation.
4) Source and sink strength with logarithmic growth
A 2D source/sink introduces radial flow and a logarithmic potential: φ = (Q/2π) ln(r/rref) + φ0. Here Q is in m²/s. With Q=1, r=2, rref=1, the result is 0.1103178 m²/s. Small r values approach a singular behavior.
5) Doublet modeling and convention control
A doublet approximates the limiting combination of a source and sink separated by a small distance and is widely used in airfoil theory. References differ by factors of 2π. This calculator provides a convention selector, computing either φ=(κ cosθ)/(2πr)+φ0 or φ=(κ cosθ)/r+φ0, so your outputs match your textbook or lecture notes.
6) Vortex potential and angle branch choice
The vortex model uses φ = (Γ/2π) θ + φ0, where Γ is circulation in m²/s. For Γ = 2 and θ = π (radians), the calculator returns 1 m²/s. Because θ can jump by 2π, the potential is multi-valued; keep a consistent angle branch.
7) Units, stability, and practical validation
Use SI inputs for predictable outputs: velocities in m/s, positions in m, and strengths as labeled. Validate results by rough scaling: uniform flow should vary linearly with distance, while source/sink varies with ln(r). Avoid r ≤ 0 and be cautious near very small radii where ideal solutions diverge.
8) Reporting and exporting results for technical work
After calculation, the result panel lists the selected model, formula, and all inputs used to generate φ. Export functions create a CSV table for spreadsheets and a compact PDF for lab notebooks, design reviews, or coursework submissions. These exports capture the timestamp and parameter set, supporting reproducible modeling and quick peer verification.
FAQs
1) Why does velocity potential have units of m²/s?
Because v = ∇φ. Taking a spatial gradient introduces 1/m, so φ must be (m/s)×m = m²/s to produce velocity in m/s.
2) What does φ0 change?
It shifts the reference level of the potential by a constant. Velocities are unchanged because gradients eliminate constants. Use φ0 when you need a specific baseline for comparisons or reporting.
3) Why is the vortex potential called multi-valued?
The angle θ is defined modulo 2π. If θ jumps across a branch cut, φ changes by (Γ/2π)·2π = Γ. Choose a consistent angular range for stable reporting.
4) How do I choose degrees or radians?
Select the unit that matches your input angle. The calculator converts degrees to radians internally for trigonometric functions. Mixing units will produce incorrect results, especially for θ near π or 180°.
5) What is Q in the source/sink model?
Q is source strength per unit depth in 2D, commonly expressed in m²/s. Positive Q represents a source (outflow), and negative Q represents a sink (inflow).
6) Which doublet convention should I pick?
Use the same convention as your reference formula. If your notes include 2π in the denominator, select “Includes 2π factor.” Otherwise, choose “No 2π factor” to match definitions without that constant.
7) Why does the calculator warn about small r values?
Ideal point singularities diverge as r → 0. Real flows have finite core sizes and viscosity, so the analytic model becomes unrealistic at very small radii. Use small-r results only for qualitative trends.
Compute velocity potentials fast, and verify flow assumptions safely.