Explore free vortex, forced vortex, and Rankine vortex flows. Adjust circulation Γ, core radius, and angular velocity to visualize velocity profiles, pressure distributions, and cavitation risk.
Vortex Model & Parameters
Vortex Type
Circulation Γ (m²/s)
m²/s
Core radius rᶜ (m)
m
Angular velocity ω (rad/s)
rad/s
Reference pressure p∞ (Pa)
Pa
Fluid density ρ (kg/m³)
kg/m³
Warning: Cavitation Risk
Minimum pressure is below the vapor pressure of water (2338 Pa at 20°C). Cavitation may occur.
Results
vθ_max (m/s)
—
p_min (Pa)
—
p at r=rᶜ
—
Cavitation
—
Streamlines & Velocity Vectors
Tangential Velocity vθ(r)
Pressure Distribution p(r)
Pres
Theory & Key Formulas
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
$$v_\theta = \begin{cases}\omega r & r \lt r_c \\ \dfrac{\omega r_c^2}{r}& r \ge r_c\end{cases}$$
What is a Fluid Vortex?
🙋
What exactly is the difference between a "free" and a "forced" vortex? They both look like swirling water in the simulator.
🎓
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.
🙋
Wait, really? So in a free vortex, the center spins infinitely fast? That doesn't seem right...
🎓
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.
🙋
Okay, that makes sense. But what is "Circulation Γ" that I'm adjusting? It seems to make the whole vortex stronger.
🎓
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.
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.
Frequently Asked Questions
A free vortex is an inviscid, irrotational flow where velocity is inversely proportional to the distance from the center. In contrast, a forced vortex rotates like a rigid body, with velocity proportional to the radius. In this simulator, you can switch parameters to compare the velocity and pressure distributions of both in real time.
Cavitation may occur when the pressure distribution p(r) falls below the fluid's saturation vapor pressure. In the simulator, if the pressure drops below the threshold, the affected area is color-coded, allowing you to intuitively explore the conditions for cavitation by adjusting parameters such as circulation and angular velocity.
A Rankine vortex is a composite model that behaves as a forced vortex (rigid body rotation) near the center and as a free vortex (potential vortex) in the outer region. This tool allows you to change the core radius with a slider, reproducing velocity and pressure distributions close to real vortices (e.g., tornadoes or vortices during tank drainage).
Try refreshing the browser or clicking the simulator's reset button. Also, if the values of circulation Γ or angular velocity ω are extremely small, changes may be difficult to see. We recommend moving the slider to around the middle and then fine-tuning. You can also enter values directly into the numerical input fields for precise settings.
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.
Enter circulation (Gamma) in m²/s using valGammaNum—typical values range 0.5 to 50 m²/s for industrial swirl flows.
Set core radius (Rc) in meters via valRcNum—common range 0.01 to 0.5 m depending on pipe diameter or cyclone geometry.
Input angular velocity (Omega) in rad/s using valOmegaNum—forced vortex applications typically use 10 to 1000 rad/s.
Observe the velocity profile update in real-time, showing tangential velocity distribution across radial distance.
Toggle between free vortex (constant Gamma), forced vortex (solid-body rotation), and Rankine vortex (combined core+free regions) modes.
Worked Example
A cyclone separator operates with Gamma = 8.5 m²/s, Rc = 0.045 m, and Omega = 350 rad/s. At radius r = 0.15 m in the free vortex region, tangential velocity v_theta = Gamma/(2π·r) = 8.5/(2π·0.15) ≈ 9.0 m/s. At r = 0.02 m (within the Rankine core), v_theta = Omega·r = 350·0.02 = 7.0 m/s. The simulator plots both profiles, confirming the discontinuity at Rc where the forced-vortex core transitions to irrotational free-vortex flow.
Practical Notes
For centrifugal pumps, increase Omega gradually (100–800 rad/s range) to observe how forced-vortex strength affects pressure distribution via Bernoulli equation.
Rankine vortex models real tornadic flows and dust collectors; use Gamma/Rc ratio of 15–200 s⁻¹ for stable separation performance.
Free-vortex approximation (Gamma only, no Omega) suits atmospheric and oceanic phenomena where viscous core is negligible relative to flow scale.
Monitor tangential velocity singularity at r → 0 in pure free-vortex mode; the simulator should show asymptotic behavior approaching infinity, indicating unrealistic physics requiring core modeling.