What is the difference between a free vortex and a forced vortex?

In a free (potential) vortex vθ = Γ/(2πr) — velocity decreases with radius, like water draining from a bathtub. In a forced (rigid-body rotation) vortex vθ = ωr — velocity increases with radius, like coffee stirred in a cup. The Rankine vortex combines both, using forced rotation inside the core radius rc and free vortex behavior outside.

What does circulation Γ (Gamma) represent?

Circulation Γ is the line integral of the velocity field around a closed loop enclosing the vortex center. It quantifies the rotational strength of the flow in m²/s. A higher Γ means stronger rotation, lower pressure at the center, and higher induced velocities. The Kutta-Joukowski theorem relates Γ directly to lift on an airfoil.

What is cavitation and when does it occur?

Cavitation is the formation of vapor bubbles in a liquid when local pressure drops below the vapor pressure (≈2338 Pa for water at 20°C). In a free vortex the pressure decreases sharply toward the center; if it falls below vapor pressure, cavitation can cause pitting damage in pumps, propellers, and turbines. This tool warns you when that threshold is exceeded.

Where does the Rankine vortex model appear in engineering?

The Rankine vortex is used to model tornadoes, hurricanes, tip vortices behind aircraft wings, and swirling flows in cyclone separators. The transition from solid-body rotation inside rc to irrotational flow outside matches observed velocity profiles in real rotating flows better than a pure free vortex.

Fluid Vortex Simulator — Free, Forced & Rankine Vortex Velocity and Pressure

Vortex Model & Parameters

Vortex Type

Circulation Γ (m²/s)5.0

Core radius rᶜ (m)0.5

Angular velocity ω (rad/s)5.0

Reference pressure p∞ (Pa)101325

Fluid density ρ (kg/m³)1000

vθ_max (m/s)

—

p_min (Pa)

—

p at r=rᶜ

—

Cavitation

—

Warning: Cavitation Risk
Minimum pressure is below the vapor pressure of water (2338 Pa at 20°C). Cavitation may occur.

Theory

Free Vortex

$$v_\theta = \frac{\Gamma}{2\pi r},\quad p = p_\infty - \frac{\rho\Gamma^2}{8\pi^2 r^2}$$

Forced Vortex

$$v_\theta = \omega r,\quad p = p_\infty - \tfrac{1}{2}\rho\omega^2 r^2$$

Rankine Vortex

$$v_\theta = \begin{cases}\omega r & r < r_c \\ \dfrac{\omega r_c^2}{r}& r \ge r_c\end{cases}$$

Streamlines & Velocity Vectors

Tangential Velocity vθ(r)

Pressure Distribution p(r)

What is a Fluid Vortex?

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What exactly is the difference between a "free" and a "forced" vortex? They both look like swirling water in the simulator.

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Great question! Basically, the key difference is what causes the spin. A free vortex is like water draining from a bathtub—it spins freely, and its velocity depends on distance from the center. A forced vortex is like a spinning bucket of water—the entire fluid rotates as a solid block. In the simulator, try switching the "Vortex Type" control. You'll see the velocity arrows change pattern completely.

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Wait, really? So in a free vortex, the center spins infinitely fast? That doesn't seem right...

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Exactly! That's a physical impossibility—it's called a singularity. In reality, near the center, viscosity matters. That's why we have the "Rankine vortex" model, which combines both! It has a forced vortex core (solid-body rotation) inside a free vortex outside. Adjust the "Core radius rᶜ" slider to see how the inner forced region grows and smooths out that infinite center.

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Okay, that makes sense. But what is "Circulation Γ" that I'm adjusting? It seems to make the whole vortex stronger.

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In practice, circulation is a measure of the vortex's strength or "swirliness." Think of it as the total amount of rotation bundled up in the flow. For a free vortex, a higher Γ means faster swirling velocities everywhere. A common case is a tornado—higher circulation means a more powerful, destructive vortex. Slide the "Circulation Γ" control and watch how the velocity scale and pressure drop change dramatically.

Physical Model & Key Equations

The velocity distribution for a free (irrotational) vortex, where flow outside the core is frictionless and circulation is constant.

$$v_\theta = \frac{\Gamma}{2\pi r}$$

Here, $v_\theta$ is the tangential velocity (m/s), $\Gamma$ is the circulation (m²/s), and $r$ is the radial distance from the center (m). Velocity decreases with increasing radius.

The corresponding pressure field, derived from Bernoulli's principle, shows how pressure drops in the low-pressure core.

$$p = p_\infty - \frac{\rho\Gamma^2}{8\pi^2 r^2}$$

Here, $p$ is the local pressure (Pa), $p_\infty$ is the far-field reference pressure, and $\rho$ is the fluid density (kg/m³). The intense pressure drop near the center is what can cause structural damage.

For a forced (solid-body) vortex, the velocity increases linearly from the center, like a rotating disk.

$$v_\theta = \omega r$$

Here, $\omega$ is the constant angular velocity (rad/s). This is the model for the inner core of a Rankine vortex or a stirred cup of coffee.

Real-World Applications

Tornado & Hurricane Modeling: Meteorologists use Rankine vortex models to simulate the wind speed and pressure profiles of cyclonic storms. The low-pressure core (eye) and high winds in the eyewall are directly predicted by these equations, helping to estimate potential damage.

Aircraft Wingtip Vortices: The swirling air left behind an airplane's wing is a free vortex. Its circulation is directly related to the lift generated by the wing. Engineers must model these vortices to ensure safe separation distances between aircraft during takeoff and landing.

Mixer & Centrifuge Design: In chemical engineering, forced vortex flow is essential in designing mixing tanks and centrifuges. Predicting the velocity profile ($v_\theta = \omega r$) helps determine mixing efficiency and separation rates for particles or immiscible fluids.

Hydraulic Machinery: The flow in turbines and pumps often contains complex vortex structures. Analyzing free vortex behavior helps in designing efficient blade shapes that minimize energy loss and cavitation (caused by extreme pressure drops).

Common Misconceptions and Points to Note

When you start using this simulator, there are a few points that are easy to misunderstand. First, remember that "a free vortex is an ideal model without viscosity". While a sink vortex is often used as an example, in reality, wall friction leads to significantly different behavior. Consider the simulator's beautiful curves as an "idealized form for understanding the essence". Next is parameter setting. "Circulation Γ" and "angular velocity ω" may seem unrelated, but in a Rankine vortex they are connected via the core radius rc. For instance, if you set ω=5 rad/s and rc=0.1m, the velocity at the core boundary is v_θ=ω*rc=0.5 m/s, and Γ is automatically determined so that this value matches the velocity of the outer free vortex (Γ=2π*rc*v_θ). Be careful, as not being aware of this continuity can create non-physical velocity jumps. Finally, regarding pressure: don't take a calculation result showing pressure going to negative infinity at the center at face value. In reality, when pressure drops to the vapor pressure, cavitation occurs and alters the flow itself; moreover, the mathematical singularity at the center point is a limitation of the model. In practical engineering, it's crucial to quantitatively assess "how low the pressure region is and over what extent it spreads". For example, in pump design, "how much Net Positive Suction Head (NPSH) margin to allow relative to the required head" lies on the extension of such analysis.

Related Engineering Fields

The vortex physics handled by this tool actually forms the foundation for a remarkably wide range of fields. First, "wind and hydroelectric power generation". The vortex street (Kármán vortex street) formed behind wind turbine blades possesses free-vortex-like characteristics and can cause vibration and noise in downstream blades. The internal flow in turbines is also a mass of complex vortex structures. Next, "automotive aerodynamic design". The turbulent wake (trailing vortices) formed behind a vehicle body is a primary cause of drag, and controlling it is key to improving fuel efficiency. Wind noise from side mirrors and pillars also originates from vortices. It's also important in "chemical process engineering". To control flow within reaction or mixing vessels for efficient agitation, it's necessary to intentionally create a forced-vortex-like flow field. Furthermore, in "ocean and river engineering", vortices generated around bridge piers or behind ship propellers cause scouring (erosion of riverbeds/seabeds) and cavitation damage. If you can experience how increasing "circulation" in this simulator lowers pressure, you should find it easier to understand the fundamental mechanical reasons behind these phenomena.

For Further Learning

If you want to delve deeper into the theory behind this tool, try the next steps. First, for the mathematical background, grasp the relationship between "vorticity" and "circulation". The curl of the velocity vector field is vorticity ζ, and its surface integral is circulation Γ ($$ \Gamma = \iint_S \boldsymbol{\omega} \cdot d\boldsymbol{S} $$). A free vortex has the interesting property that vorticity is concentrated at the central singularity (a vortex filament), while being zero (irrotational flow) elsewhere. Next, learning the "Navier-Stokes equations", the fundamental equations of fluid dynamics, will reveal how these vortex models are approximated and simplified. As a concrete learning example, display a Rankine vortex in the simulator and try plotting the maximum velocity and its location while varying the core radius rc. From the theoretical formula, the maximum velocity occurs at rc and its value is v_max = ω*rc = Γ/(2πrc). Verifying this relationship yourself is the best shortcut to internalizing the meaning of the equations. For your next topic, we recommend moving on to "vortex stability" or "cases where multiple vortices interact (vortex pairs)". Real-world turbulence is, in fact, a phenomenon where vortices of various scales intertwine complexly.