What This Solves
Calculates the critical depth in an open channel or pipe — the depth at which specific energy is minimized and the Froude number equals 1.
Best Used When
- You need to determine the control depth at a channel transition, drop, or free overfall
- You are checking whether flow in a channel or pipe is subcritical or supercritical
- You need critical depth as a boundary condition for water surface profile calculations
Do NOT Use When
- You need the depth at which uniform flow occurs (normal depth) rather than critical depth — Use Normal Depth Calculator
- You only need the Froude number to classify the flow regime — Use Froude Number Calculator
Key Assumptions
- Specific energy (E = y + V²/2g) is minimized at critical depth
- Hydrostatic pressure distribution exists at the cross-section
- Channel slope does not affect critical depth (it depends only on discharge and geometry)
- The cross-section is constant at the location of interest
Input Quality Notes
Critical depth depends only on discharge and channel geometry, not on roughness or slope. Ensure the cross-section dimensions accurately represent the channel at the location of interest.
Calculate the critical depth (yc) for circular pipes and rectangular, trapezoidal, or triangular channels — the depth at which specific energy is minimum and the Froude number equals 1, marking the boundary between subcritical and supercritical flow.
Critical Depth Overview
Critical depth is the depth at which specific energy is minimum for a given discharge. At critical depth, the Froude number equals 1 and flow transitions between subcritical and supercritical regimes.
Key relationships:
- Critical condition: Fr = V / sqrt(g * Dh) = 1
- Minimum energy: Emin = yc + Vc2/(2g)
- Section factor: Z = A3/2 / T1/2 = Q / sqrt(g)
For rectangular channels: yc = (Q2 / (g * b2))1/3
Design Considerations
- Near-critical flow is unstable - avoid designing for Fr between 0.9 and 1.1
- Critical depth serves as a hydraulic control for flow calculations
- Used to determine if flow is subcritical (y > yc) or supercritical (y < yc)
- Important for hydraulic jump analysis and energy dissipation design
For educational purposes only. Not a substitute for professional engineering judgment.
How critical depth is calculated
Critical depth occurs where the specific energy of the flow is at a minimum for a fixed discharge. Setting the derivative of specific energy with respect to depth to zero gives the general critical-flow condition, expressed as a section factor:
Z = A3/2 / T1/2 = Q / √g
where, at the critical section:
- Z — section factor (ft5/2 or m5/2)
- A — cross-sectional flow area (ft² or m²)
- T — top width of the water surface (ft or m)
- Q — discharge / design flow rate (cfs or m³/s)
- g — gravitational acceleration: 32.174 ft/s² (US customary) or 9.81 m/s² (metric)
Equivalently, the flow is critical when the Froude number equals 1, where Dh = A / T is the hydraulic depth:
Fr = V / √(g · Dh) = 1
For simple geometries the equation has a direct (closed-form) solution. For circular pipes and trapezoidal channels there is no closed form, so the calculator solves Z(y) = Q/√g iteratively with a Newton-Raphson method and a bisection fallback, converging to a tolerance of 1×10-8.
Closed-form solutions used for the simple shapes:
- Rectangular: yc = (Q² / (g · b²))1/3, with minimum specific energy Emin = 1.5·yc and Vc = √(g·yc)
- Triangular (V-shaped): yc = (2Q² / (g · z²))1/5, where z is the side slope (H:V) and the hydraulic depth Dh = yc/2
Method and equations after Chow, Open-Channel Hydraulics (1959), Chapter 4, and FHWA HEC-22.
Critical depth formulas by channel shape
The critical-flow condition A3/2/T1/2 = Q/√g applies to every section. The table summarises the geometry and the resulting solution method.
| Shape | Area A | Top width T | Critical depth solution |
|---|---|---|---|
| Rectangular | b·y | b | yc = (Q²/(g·b²))1/3 (direct) |
| Triangular | z·y² | 2·z·y | yc = (2Q²/(g·z²))1/5 (direct) |
| Trapezoidal | (b + z·y)·y | b + 2·z·y | Iterative (Newton-Raphson) |
| Circular | Partial-flow area (θ) | Chord width (θ) | Iterative (Newton-Raphson) |
b = bottom width, y = depth, z = side slope (horizontal:vertical), θ = subtended angle of the wetted circular segment. For a rectangular channel Dh = y; for a triangular channel Dh = y/2.
Worked example (rectangular channel)
A rectangular channel with bottom width b = 10 ft carries a discharge of Q = 100 cfs (g = 32.174 ft/s²):
- Critical depth: yc = (100² / (32.174 × 10²))1/3 = 1.46 ft
- Critical velocity: Vc = √(g·yc) = √(32.174 × 1.46) = 6.85 ft/s
- Minimum specific energy: Emin = 1.5 × yc = 2.19 ft
- Froude number: Fr = 1.0 (confirms the critical section)
Verification case from Chow (1959); the calculator reproduces yc ≈ 1.46 ft.
Why critical depth matters
Hydraulic control
Critical depth acts as a control section — at culvert inlets/outlets, weirs, flumes and free overfalls — that sets a known depth-discharge relationship for backwater and water-surface-profile calculations.
Flow regime & transitions
Comparing normal depth to critical depth tells you whether flow is subcritical or supercritical, locates hydraulic jumps, and informs energy-dissipation and stilling-basin design. Avoid designing in the unstable near-critical band (Fr 0.9–1.1).
Frequently asked questions
What is critical depth in open channel flow?
Critical depth (y_c) is the flow depth at which the specific energy is minimum for a given discharge. At this depth the Froude number equals 1, and the flow is right at the transition between subcritical (tranquil, deep, slow) and supercritical (rapid, shallow, fast) flow. It depends only on the discharge and the channel geometry — not on the channel slope or roughness.
How is critical depth calculated?
Critical depth is found by solving the section-factor equation A^(3/2) / T^(1/2) = Q / sqrt(g), where A is flow area, T is top width, Q is discharge and g is gravitational acceleration. For rectangular and triangular channels this has a closed-form solution; for circular pipes and trapezoidal channels it must be solved iteratively. This calculator uses a Newton-Raphson solver with a bisection fallback to converge on y_c.
How do I know if flow is subcritical or supercritical?
Compare the actual (normal) flow depth to the critical depth. If the flow depth is greater than y_c, the Froude number is below 1 and the flow is subcritical. If the flow depth is less than y_c, the Froude number is above 1 and the flow is supercritical. At y_c exactly, Fr = 1. Near-critical flow (roughly Fr between 0.9 and 1.1) is unstable and should be avoided in design.
Why does the calculated Froude number come out as 1.0?
By definition, critical depth is the depth where the Froude number equals 1, so the calculator reports Fr ~ 1.0 as a verification check that it converged on the true critical section. A value of 1.000 confirms the geometry, discharge and solver are all consistent. Small deviations indicate an iterative solution that has not fully converged.
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Last verified: February 2026