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Critical Depth & Froude Number

Source file: critical_flow.html

Purpose

Determines the critical flow depth for a trapezoidal channel carrying a given discharge, then compares it against an actual (observed/design) flow depth to classify the flow regime (subcritical, critical, or supercritical) using the Froude number.

Governing Equations

Critical flow condition (minimum specific energy for a given discharge):

Q² / g = A_c³ / T_c

Where at critical depth y_c: - A_c — cross-sectional area at critical depth - T_c — top width at critical depth - g — gravitational acceleration (9.81 m/s²)

Froude Number (flow regime indicator):

Fr = V / √(g × D_h)

Where: - V — actual mean velocity = Q / A - D_h — hydraulic depth = A / T (area / top width)

Trapezoidal Section Geometry

For bed width b, side slope z, depth y: - Area: A = (b + z·y) × y - Top width: T = b + 2z·y

Inputs

Input Symbol
Discharge Q (m³/s)
Bed width b (m)
Side slope z (horizontal per 1 vertical)
Actual water depth y (m) — the depth being evaluated against critical conditions

Solution Method

1. Solving for Critical Depth (y_c)

Since the critical-depth equation cannot be solved algebraically in closed form for a trapezoidal section, it is solved numerically by a coarse iterative search:

Start yc = 0.1, step = 1.0
Repeat 100 times:
    Compute Ac, Tc at current yc
    target = Ac³ / Tc
    lhs = Q² / g
    If target < lhs:  yc += step        (increase depth)
    Else:              yc -= step; step /= 10   (overshoot → back off and refine)

This is a simple "hunt and refine" search (similar in spirit to bisection, but not true bisection) that converges y_c to the point where Ac³/Tc = Q²/g.

2. Critical Velocity

Vc = Q / Ac   (using Ac evaluated at the converged yc)

3. Actual Flow State & Froude Number

Using the user-supplied actual depth y:

A_act = (b + z·y) × y
T_act = b + 2z·y
V_act = Q / A_act
Dh = A_act / T_act
Fr = V_act / √(g × Dh)

4. Regime Classification

Froude Number Regime
Fr < 0.99 Subcritical (tranquil / safe)
0.99 ≤ Fr ≤ 1.01 Critical flow state
Fr > 1.01 Supercritical (rapid / turbulent)

(A small tolerance band around Fr = 1.0 is used to flag "critical" rather than requiring an exact match, since floating-point convergence rarely lands exactly on 1.0.)

Outputs

Limitations

Open Critical Depth & Froude Number Calculator →