Critical Depth in a Rectangular Channel
Critical Depth in a Rectangular Channel
Critical depth is the water depth in an open channel at which flow has the minimum possible specific energy for a given discharge — the dividing line between subcritical (slow, deep) and supercritical (fast, shallow) flow regimes, and it governs where control sections form, such as at a free overfall, a broad-crested weir, or a channel entrance. Because flow always passes through critical depth when transitioning between subcritical and supercritical regimes (for example approaching a steep drop), locating critical depth is a standard step in profiling how water surface elevation varies along a channel and in designing measurement structures like flumes that rely on choking flow to critical conditions.
The critical depth in a rectangular channel is yc = (q^2/g)^(1/3), where q is the discharge per unit channel width. where Q is the total discharge, b is the channel width, q = Q/b is the discharge per unit width, and g is gravitational acceleration.
Inputs
Q_ch := 5 m^3/s=5 m^3/s
b_ch := 3 m=3 m
g_ch := 9.81 m/s^2=9.81 m/s^2
Dividing the total discharge by the channel width gives the discharge per unit width, the quantity the critical-depth formula is actually built from since it normalizes out channel size.
q_unit := Q_ch/b_ch=1.67 m^2/s
This is a genuinely dimensionally homogeneous formula, so the cube root can be taken directly: it gives the depth at which this unit discharge carries the minimum possible specific energy.
y_c := (q_unit^2/g_ch)^(1/3)=656.66 mm
Results
A critical depth near 0.66 m for this discharge and channel width is the reference depth against which the actual flow depth anywhere in the channel should be compared: if the actual depth is greater than yc the flow is subcritical, and if it is less the flow is supercritical. This comparison is exactly how engineers identify whether a mild or steep channel slope classification applies and where hydraulic jumps or drawdown curves will form along a reach. Critical depth depends only on discharge per unit width and gravity — not on channel roughness — which is why it is such a robust reference point for control-section design.
Assumptions & limitations
- Rectangular channel cross-section assumed.
- Hydrostatic pressure distribution and one-dimensional flow assumed.
- Discharge per unit width q treated as uniform across the channel.
- No energy losses considered — this locates the critical depth, not the actual flow profile.