Vortex Flow Calculator

Tangential velocity, pressure, and surface profile for free, forced, and Rankine combined vortex flows at a specified radius.

Fluid Mechanics II · Specialized

Common fluids — click to auto-fill density

Vortex type:

Query radius r (m)

Fluid density ρ (kg/m³)

Reference radius r₀ (m)

Vθ at r₀ Vθ₀ (m/s)

Pressure is referenced to the entry radius r₀. Free vortex: Γ = 2πr₀V_θ₀ = constant.

Free (irrotational) vortex:

V_θ = Γ / (2πr) (velocity decreases outward)

Γ = 2π × r × V_θ = constant

ΔP = ½ρ(V_θ₀² − V_θ²) (relative to r₀)

Irrotational (zero vorticity) everywhere except r = 0. Pressure is lowest at the centre.

Forced (rigid-body) vortex:

V_θ = ω × r (velocity increases linearly)

ΔP = ½ρω²r² (relative to r = 0)

Surface: z = ω²r² / (2g) (paraboloid)

Rotational — vorticity = 2ω. Free surface forms a paraboloid of revolution.

Rankine combined vortex:

r ≤ rc: V_θ = V_θ,max × r / rc (core — forced)

r > rc: V_θ = V_θ,max × rc / r (outer — free)

ΔP(r ≤ rc) = ½ρV_θ,max² (r/rc)²

ΔP(r > rc) = ½ρV_θ,max² (2 − rc²/r²)

V_θ is maximum at r = rc. The Rankine model avoids the singularity at r = 0 of the pure free vortex.

Comparison:

Property	Free	Forced
V_θ vs r	∝ 1/r (decreases)	∝ r (increases)
Vorticity	0 (irrotational)	2ω (rotational)
Γ = 2πr·V_θ	Constant	∝ r² (varies)
Pressure min	At centre (lowest)	At centre (lowest)
Singularity	At r = 0	None
Energy	Conserved along streamlines	External input required