Calculator
Example Data Table
| Method | Inputs (sample) | Output |
|---|---|---|
| 2D derivatives | \u2202v/\u2202x=1.5, \u2202u/\u2202y=0.4 | \u03c9z=1.1 1/s |
| 2D central differences | v(x+\u0394x)=2.1, v(x−\u0394x)=1.7, \u0394x=0.5; u(y+\u0394y)=0.9, u(y−\u0394y)=0.5, \u0394y=0.5 | \u03c9z=0.8 1/s |
| 3D curl | \u2202w/\u2202y=0.30, \u2202v/\u2202z=0.05, \u2202u/\u2202z=0.12, \u2202w/\u2202x=0.02, \u2202v/\u2202x=0.40, \u2202u/\u2202y=0.10 | \u03c9=[0.25, 0.10, 0.30] 1/s |
| Circulation | \u0393=0.80 m\u00b2/s, A=0.20 m\u00b2 | \u03c9\u0304=4.0 1/s |
| Rotation | \u03a9=1.20 rad/s (about z) | |\u03c9|=2.4 1/s |
Formula Used
Vorticity is the curl of the velocity field V = (u, v, w):
For a 2D flow in the x–y plane, only the z-component is nonzero:
If you have circulation \u0393 around a small loop of area A:
For solid-body rotation with angular speed \u03a9, the vorticity magnitude is |\u03c9| = 2\u03a9.
How to Use This Calculator
- Select the method that matches your available data.
- Enter partial derivatives, or neighboring values and spacings.
- Keep axes consistent with your coordinate definition.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF after calculating.
- Derivatives should be in consistent units to yield s\u207b\u00b9.
- Central differences work best with small, symmetric steps.
- Use the direction vector to interpret rotation axis.
Professional Notes on Vorticity
1) What vorticity represents
Vorticity (\u03c9) measures the local spinning tendency of a fluid element. In Cartesian form it is the curl of velocity, \u03c9 = \u2207\u00d7V, and its SI unit is s\u207b\u00b9. In rigid-body rotation it equals twice the angular velocity, but it also captures shear-driven spinning. Large magnitudes indicate tight rotation or strong shear.
2) Interpreting sign and axis direction
The sign comes from your coordinate system and the right-hand rule. For 2D x\u2013y flow, \u03c9z = \u2202v/\u2202x \u2212 \u2202u/\u2202y. Positive \u03c9z corresponds to counterclockwise rotation when +z points toward you.
3) 2D gradients versus measured velocities
If you already have derivatives from theory or a solver, the 2D derivative method is direct. When you only have nearby measurements, the central-difference method estimates gradients using symmetric points. This reduces first-order bias compared with one-sided differences.
4) 3D curl components in practice
In 3D, each component uses two cross-derivatives: \u03c9x=\u2202w/\u2202y\u2212\u2202v/\u2202z, \u03c9y=\u2202u/\u2202z\u2212\u2202w/\u2202x, \u03c9z=\u2202v/\u2202x\u2212\u2202u/\u2202y. The direction vector reported here summarizes the dominant rotation axis.
5) Accuracy, spacing, and noise
Finite differences amplify noise because they subtract nearly equal numbers. Keep \u0394x and \u0394y small enough to resolve gradients, but not so small that measurement error dominates. Central differences are second-order for smooth fields, so halving the step reduces truncation error strongly. If \u03c9 changes drastically with spacing, consider smoothing or better resolution.
6) Circulation link and area averaging
For a small loop, Stokes\u2019 theorem connects circulation \u0393=\u222eV\u00b7dl to the surface integral of vorticity. The calculator uses \u03c9\u0304\u2248\u0393/A as an area-averaged estimate, useful for PIV loops or vortex core approximations. Because it is area-averaged, it smooths small-scale fluctuations and is best interpreted as a coarse-grained rotation strength.
7) Solid-body rotation benchmark
Rigid rotation provides a clean reference: |\u03c9|=2\u03a9. For example, \u03a9=1.20 rad/s implies |\u03c9|=2.40 s\u207b\u00b9. If your computed field is close to rigid rotation, components should be nearly uniform and aligned with the rotation axis.
8) Typical magnitudes and reporting
Orders of magnitude vary widely: gentle room-scale air mixing may be \u223c0.01\u20130.1 s\u207b\u00b9, small lab vortices \u223c1\u201310 s\u207b\u00b9, and intense atmospheric vortices can exceed \u223c50 s\u207b\u00b9 near the core. Near a no-slip wall, \u03c9 scales like U/\u03b4; for U=1 m/s and \u03b4=1 cm, \u03c9\u2248100 s\u207b\u00b9. Use the CSV/PDF exports to document units and method choice.
FAQs
1) Is vorticity the same as rotation rate?
Not always. Vorticity measures local spin and shear. For solid-body rotation only, the magnitude equals twice the angular speed: |\u03c9| = 2\u03a9.
2) Why does the 2D formula use \u2202v/\u2202x and \u2202u/\u2202y?
In planar flow, the only nonzero component is \u03c9z. The curl reduces to \u03c9z=\u2202v/\u2202x\u2212\u2202u/\u2202y when w=0 and \u2202/\u2202z terms vanish.
3) What unit should I choose?
Use 1/s for consistent dimensional reporting. Some fields write rad/s informally, but radians are dimensionless, so the physical dimension remains s\u207b\u00b9.
4) My finite-difference result changes a lot with \u0394x. What does that mean?
It often indicates coarse resolution, strong curvature, or noisy measurements. Try smaller symmetric steps, verify sensor precision, and compare with a smoothed velocity field when possible.
5) How do I pick the circulation normal direction?
Choose the axis perpendicular to your loop surface. The sign follows the right-hand rule: curling fingers along the loop direction, your thumb gives the positive normal direction.
6) Can vorticity be negative?
Yes. The sign indicates rotation sense relative to your chosen axes. A negative \u03c9z in x\u2013y flow means clockwise rotation when +z is out of the page.
7) When should I use the 3D curl method?
Use it when you have 3D velocity gradients or simulation outputs. It returns \u03c9x, \u03c9y, and \u03c9z plus magnitude and a direction unit vector.