Normal Depth Calculator
Enter discharge, bottom width, Manning roughness and slope to compute the normal depth of a rectangular open channel by inverse Manning's formula.
Input Data
Results
At a glance:Normal depth y_n is the steady depth of uniform flow in an open channel — where gravity's downslope pull exactly balances boundary friction, so depth and velocity are constant along the channel. It is set by Manning's equation Q = (1/n)·A·R^(2/3)·S^(1/2), with Q discharge, n roughness, A flow area, R hydraulic radius and S slope. For a rectangular channel A = b·y and R = A/P = b·y/(b+2y). Because R carries a 2/3 power and depends on y, there is no closed-form solution for y; the depth is found by numerical iteration (bisection in this tool). Example: Q = 5 m³/s, b = 3 m, n = 0.015, S = 0.001 → y_n ≈ 1.079 m. Normal depth is the reference for water-surface profile analysis and distinguishes mild (y_n>y_c) from steep (y_n<y_c) slopes.
Formula
Manning: Q = (1/n)·A·R^(2/3)·S^(1/2), solve for y_n
Rectangular: A = b·y, R = b·y/(b+2y)
Bisection converges to target Q for y_n
$$Q = \dfrac{1}{n}\,A\,R^{2/3}\,S^{1/2},\quad A=by,\ R=\dfrac{by}{b+2y}$$How to Use
- Enter discharge Q and bottom width b.
- Enter Manning n and bed slope S.
- The calculator iterates Manning's equation to find the rectangular normal depth y_n.
Case Studies
Irrigation channel check
Rectangular concrete channel b=3 m, n=0.015, S=0.001, Q=5 m³/s.
y_n ≈ 1.079 m.
If channel depth is 1.5 m, freeboard ≈ 0.42 m — adequate; near the top would need widening/deepening.
Higher discharge recheck
Same channel, Q raised to 12 m³/s.
y_n ≈ 1.921 m.
Original 1.5 m depth would overflow — deepen or widen the channel.
FAQ
How is normal depth different from critical depth?
Normal depth y_n is the uniform-flow depth set by Manning's equation (depends on n and S, reflecting gravity vs friction balance). Critical depth y_c is the minimum-specific-energy depth set by geometry and discharge only; it separates subcritical from supercritical flow. Comparing them classifies the slope: y_n>y_c mild, y_n<y_c steep.
Why must normal depth be found by iteration?
Manning's Q depends on A and R, both functions of y, with R having a 2/3 power and a rational form, making the equation highly non-linear in y — no algebraic solution. Numerical methods (bisection, Newton) are standard for inverse depth.
How sensitive are results to n or S?
Very. y_n rises with roughness n (at fixed Q) and falls with slope S (faster flow carries Q at shallower depth). Q ∝ S^(1/2)/n, so errors propagate directly into depth — use authoritative roughness tables and actual slopes, and check conservatively for post-siltation n.
When does uniform flow not apply?
Uniform flow needs a long, straight, constant-section channel. It fails near inlets/outlets, gates, weirs, drops, bends, or backwater — there the flow is non-uniform and actual depth differs from y_n (though y_n remains a reference asymptote).
Can trapezoidal or circular channels use this tool?
No. This tool hard-codes the rectangular geometry (A=b·y, P=b+2y). Trapezoidal (with side slopes) and circular sections need their own area/wetted-perimeter formulas iterated separately.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.