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Hydraulics Simulator

Froude Number Simulator — Subcritical and Supercritical Open Channel Flow

Compute Fr = V / sqrt(g y) in real time from mean velocity, water depth, unit-width discharge and gravitational acceleration, and automatically classify the open channel flow into subcritical, critical or supercritical regimes. The rectangular critical depth y_c and the critical velocity V_c are computed at the same time, and a channel side-view canvas plus a V–y regime map make the physics intuitive.

Parameters
Velocity V
m/s
Water depth y
m
Unit-width discharge q
m²/s
Gravitational acceleration g
m/s²

Defaults: V = 2.0 m/s, y = 0.50 m, q = 1.0 m²/s, g = 9.81 m/s². q is used only for the critical-depth calculation, while V and y are used in the Froude number.

Results
—
Froude number Fr
—
Flow regime
—
Critical depth y_c
—
Critical velocity V_c
Channel side view

Blue = water surface and stream drawn at depth y. Brown = riverbed. Yellow arrows = flow direction and velocity vectors. For Fr < 1 the surface is calm, for Fr > 1 a choppy fast sheet with foam appears, and for Fr near 1 the critical wave shape is visualized. The background tint shows the regime (blue = subcritical, green = critical, orange = supercritical).

V–y regime map (critical depth curve)

Horizontal axis = velocity V (m/s). Vertical axis = depth y (m). Green curve = critical depth y_c(V), the Fr = 1 boundary. Area above the curve is subcritical (Fr < 1) and below is supercritical (Fr > 1). Yellow circle = current (V, y) point.

Theory & Key Formulas

The Froude number is the ratio of inertia to gravity (or to the surface wave speed) and separates subcritical from supercritical open channel flow. In a rectangular channel the critical depth and velocity follow directly from the unit-width discharge q.

Definition of the Froude number:

$$\mathrm{Fr} = \frac{V}{\sqrt{g\,y}}$$

Critical depth in a rectangular channel (where Fr = 1):

$$y_c = \left(\frac{q^2}{g}\right)^{1/3}$$

Critical velocity at the critical depth:

$$V_c = \sqrt{g\,y_c}$$

Flow regimes:

$$\mathrm{Fr} < 1\,\,(\text{subcritical}),\quad \mathrm{Fr} = 1\,\,(\text{critical}),\quad \mathrm{Fr} > 1\,\,(\text{supercritical})$$

$V$ is the mean velocity [m/s], $y$ is the water depth [m], $g$ is gravity [m/s²] and $q = V\cdot y$ is the unit-width discharge [m²/s]. $\sqrt{gy}$ is the shallow-water wave celerity, so Fr = 1 is the condition where flow and wave speeds match.

What is the Froude Number Simulator?

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In my hydraulics class I keep seeing the Froude number, but how is it different from the Reynolds number? Both are dimensionless numbers that split flow regimes, right?
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Both are dimensionless, but they measure totally different physics. Reynolds Re = rho V L / mu is the ratio of inertia to viscosity and decides laminar vs turbulent flow. Froude Fr = V / sqrt(g y) is the ratio of inertia to gravity (or to the surface wave speed) and decides whether open channel flow is subcritical or supercritical. In open channel flow the free surface is exposed to air and surface waves dominate, so the Froude number is the lead actor. With this tool, V = 2 m/s and y = 0.5 m gives Fr ~ 0.903 — subcritical.
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What is the practical difference between subcritical and supercritical flow? When I look at a real channel I cannot really tell them apart.
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Roughly, subcritical flow lets waves propagate upstream while supercritical flow does not. Drop a stone in an irrigation channel and a subcritical surface ripples both up- and downstream, but a supercritical sheet just carries it away. The thin fast film just below a drop is the canonical supercritical example, and it must always pass through a hydraulic jump to recover subcritical flow. Try V = 6 m/s and y = 0.3 m in this tool — Fr is about 3.5, deep in the supercritical regime, and the channel canvas shows a choppy fast film with white foam droplets.
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Why do we compute the critical depth y_c? It changes when I move the q slider.
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Good catch. The critical depth y_c = (q^2 / g)^(1/3) is the depth at which Fr = 1 for the given unit-width discharge q, and it is also the depth that minimizes the specific energy. If the actual depth is above y_c the flow is subcritical and if it is below the flow is supercritical, so y_c is a one-shot threshold for classification. The defaults q = 1.0 m^2/s and g = 9.81 m/s^2 give y_c = 0.467 m, slightly less than y = 0.5 m — so the actual flow is just on the subcritical side. Weirs and Parshall flumes deliberately narrow the channel to force critical flow and obtain a unique depth-discharge relation for flow metering.
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I heard the Froude number is used in naval architecture too. Is it the same formula?
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Same concept, but for ships the characteristic length becomes the ship length L: Fr = V / sqrt(g L). The wave-making resistance of a hull is dominated by Fr, with a famous "speed wall" peak around Fr ~ 0.4 to 0.5. Luxury liners cruise at Fr < 0.3, ferries and destroyers operate at Fr ~ 0.3 to 0.5, and planing boats run at Fr > 1 — physically surfing on top of the water. In model testing, Froude similarity (matching Fr between full scale and model, William Froude in the 1870s) is still the basic law of naval architecture. Move V and y independently in the right canvas to see how the Fr = 1 boundary actually runs.

Frequently Asked Questions

At Fr = 1 the mean velocity V equals the shallow-water wave celerity sqrt(g y), so it is the threshold at which waves can or cannot propagate upstream. For Fr < 1 the wave speed exceeds the flow speed and downstream disturbances travel upstream as surface waves, controlling the water surface profile. For Fr > 1 the flow outruns the waves, downstream conditions cannot influence upstream and the surface is set purely by upstream conditions. Physically, in a rectangular channel Fr = 1 also minimizes the specific energy H = y + V^2/(2g), so the same flow is carried with the least head. That is the physical basis for forcing critical flow at weirs and Parshall flumes.
The unit-width discharge q [m^2/s] is the flow per unit channel width, q = Q / B, where Q is the total flow [m^3/s] and B is the width [m]. In a rectangular channel, the critical depth y_c and the specific-energy curve depend only on q, so the channel width drops out — this is convenient. In this tool q is provided as an independent slider and is used to compute the critical depth y_c = (q^2 / g)^(1/3) and the critical velocity. The depth y and velocity V move independently, so you can compare "what would the critical state look like" against "how far is the actual flow from critical". In practice you usually evaluate q = V y from the real flow and convert to y_c to classify the regime.
A transition from supercritical (Fr > 1) to subcritical (Fr < 1) is always accompanied by a hydraulic jump. The conjugate depth ratio follows from momentum conservation as y_2 / y_1 = (sqrt(1 + 8 Fr_1^2) - 1) / 2 (the Belanger equation). When the Froude number reported by this tool is much larger than one, a hydraulic jump occurs downstream, dissipates energy and returns the flow to subcritical. Stilling-basin design below dams typically targets the "steady jump" range Fr_1 ~ 4.5 to 9, which balances jump efficiency and stability. Combining this tool with the hydraulic jump calculator lets you design the incoming supercritical jet and the downstream stilling basin consistently.
The general Froude number is defined as Fr = V / sqrt(g D), where D = A / T is the hydraulic depth (A is the flow area and T the top width). For a rectangular section D = y, which is what this tool assumes. Trapezoidal and circular sections need a different D, for example D = (b y + m y^2) / (b + 2 m y) for a trapezoidal channel with bottom width b and side slope m, and the critical-depth equation also becomes more involved. This tool is an educational simulator for rectangular channels — for complex cross sections use a dedicated solver such as HEC-RAS or iterate y_c numerically. The qualitative rule that Fr = 1 is critical and that higher Fr is more supercritical holds for any cross section.

Real-World Applications

River and irrigation channel design: Stable subcritical flow with Fr ~ 0.3 to 0.7 is the standard design target for rivers and irrigation channels. As Fr approaches 1 the free surface becomes unstable, and Fr > 1 leads to high velocities that risk bed scour and revetment damage. Use this tool to enter realistic design values for V and y and to compare Fr against the critical depth y_c as a safety margin. Where supercritical flow is intentionally created (drops, chutes), a downstream stilling basin must trigger a hydraulic jump to dissipate energy safely.

Spillway chutes and stilling basins: Spillways and side-channel chutes below dams typically reach Fr = 4 to 10. Setting V = 10 m/s and y = 0.4 m in this tool gives Fr ~ 5.05, a textbook upstream condition for the "steady jump" range. The stilling basin connects this supercritical jet to the downstream tailwater so that the jump is captured inside the basin and energy is dissipated according to the Belanger equation. Standard design charts from the U.S. Bureau of Reclamation and USACE are organized by Froude-number jump type (weak, oscillating, steady, strong).

Flow measurement at weirs and flumes: Forcing critical flow at a local narrowing or drop makes the depth y_c uniquely determine the discharge. This is the working principle of Parshall flumes and rectangular weirs, widely used for monitoring agricultural and stormwater channels. Sweeping q in this tool and reading y_c gives a feel for weir-height and contraction-width sizing.

Naval architecture and Froude similarity: The wave-making resistance of a hull is governed by Fr_L = V / sqrt(g L) based on the ship length. Model tests have used Froude similarity (matching Fr_L between full scale and model, William Froude in the 1870s) ever since. For a 200 m ship at 1:50 scale the model speed must be reduced by a factor of 1 / sqrt(50) ~ 0.141. This tool is a useful starting point for naval-architecture students to internalize Froude-number physics.

Common Pitfalls and Notes

The most common pitfall is to confuse the Froude number with the Reynolds number. They describe completely different physics — Froude is the ratio against gravity, Reynolds the ratio against viscosity. Use Reynolds to decide laminar vs turbulent in a pipe, and use Froude to decide subcritical vs supercritical in an open channel. In river flow both matter at the same time: Reynolds tells you whether the flow is turbulent and Froude tells you the shape of the free surface. This tool focuses on the Froude number; pair it with the Reynolds number tool when viscous effects also matter.

The second pitfall is to assume that "Fr < 1 means safe". In the near-critical band Fr ~ 0.8 to 1.0 the free surface is very unstable and small disturbances can drive large depth fluctuations through resonance. River and channel designs usually keep Fr below about 0.9 with margin, and target Fr < 0.7 for comfortable design. As you raise V slowly in this tool the critical depth y_c approaches the actual depth y and the design margin disappears.

The third pitfall is to over-trust the Froude number as a simple ratio that can be guessed by intuition. The denominator is sqrt(g y), so halving the depth only multiplies Fr by sqrt(2), but doubling the velocity doubles Fr exactly. Velocity is the much stronger lever for Fr, while depth changes have a softer effect. This means a steeper bed slope drives the flow into the supercritical regime far faster than a depth change of the same magnitude. Move V and y independently in this tool to feel the asymmetry. Understanding Fr not just as a number ratio but as a velocity-vs-wave-speed comparison is the first step to building open channel design intuition.

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