Engineering tools / Unsteady flow
Flows that change with time. Worked out in seconds.
Four screening calculators for unsteady flow and sediment. Route a storm through a detention pond, see the surge when a valve shuts, follow a dam-break wave down a valley, or estimate how fast a flow scours a sand bed. Change any input and the graphs redraw as you type.
Flood attenuation in a detention pond
A triangular storm hydrograph enters a pond that drains through a bottom orifice and an overflow spillway. The calculator steps through the storm and reports how much the pond trims the peak and how high the water rises.
Inflow and outflow hydrographs
The gap between the curves is water held back in the pond.
Water level in the pond
Depth above the pond floor, against the spillway crest and embankment crest.
Method and assumptions
Level-pool (storage) routing: the pond surface is taken as flat, so outflow depends only on water depth. The continuity equation dS/dt = I(t) − O(h) is solved with a fourth-order Runge-Kutta scheme.
- Inflow is a triangle that rises linearly to the peak and falls to zero at the base time. The 2.67 default comes from the NRCS triangular unit hydrograph (USDA NRCS, National Engineering Handbook Part 630, Chapter 16).
- Orifice:
Q = Cd·a·√(2g(h − D/2))once the opening is submerged, blended to zero as the water drops below its soffit. - Spillway:
Q = Cw·L·(h − crest)1.5. - Pond plan area grows with depth according to the side slope.
What this model leaves out
Tailwater on the outlets, infiltration, evaporation, real storm shapes and rainfall directly on the pond. Ponds that are long or shallow can have a sloping water surface that level-pool routing misses. Embankment overtopping is flagged but not modelled.
Water hammer on valve closure
A pipeline runs from a reservoir to a valve that discharges freely. As the valve closes, a pressure wave travels up and down the pipe. The calculator tracks that wave with the method of characteristics and compares the peak with the pipe's pressure class.
Pressure head over time
At the valve and at the middle of the pipeline. The dashed line is the pipe's pressure class.
Pressure envelope along the pipe
Highest and lowest head reached at each point, with the steady-state line for reference.
Method and assumptions
Wave speed follows the Korteweg relation for a thin-walled pipe: a = √[(K/ρ) / (1 + K·D/(E·e))], with water bulk modulus K = 2.19 GPa and density 998 kg/m³. The pipe is split into 24 reaches and solved with the method of characteristics (Wylie and Streeter, Fluid Transients in Systems), using quasi-steady Darcy friction.
- Upstream boundary: constant-head reservoir.
- Downstream boundary: valve discharging to atmosphere, with effective opening τ = (1 − t/tc)m.
- Closure is rapid when tc ≤ 2L/a. The full Joukowsky rise
ΔH = a·V/gthen reaches the valve. - Heads are pressure heads above a horizontal pipe.
Limits of this screening check
Column separation and vapour cavities, air valves, surge vessels, pump trip, a real elevation profile, unsteady friction and pipe movement at bends are all excluded. When the minimum head falls near vapour pressure, the real peak after cavity collapse can be higher than shown here.
Dam break flood wave
The dam fails instantly and the reservoir surges down a rectangular channel. The model solves the one-dimensional shallow water equations with bed friction and follows the wave as it travels downstream. Drag the time slider or press play to watch it move.
Water depth along the valley
Negative distances are inside the reservoir. The dashed curve is the frictionless Ritter solution for a dry channel, shown while it still applies.
Depth at the gauge
How the flood arrives, peaks and recedes at the chosen point downstream.
Method and assumptions
The one-dimensional shallow water (Saint-Venant) equations are solved with a first-order finite volume scheme on 800 cells, using the HLL approximate Riemann solver (Toro, Shock-Capturing Methods for Free-Surface Shallow Flows). The time step follows a Courant number of 0.9.
- Friction: Manning, applied semi-implicitly so that thin, fast flow at the wave front stays stable.
- Upstream end of the reservoir: closed wall. Downstream end: free outflow.
- Ritter (1892) solution for an instantaneous break onto a dry, frictionless bed: front speed
2√(g·h₀)and dam-section discharge(8/27)·h₀·√(g·h₀)per metre width. With friction, the real front travels slower.
Where a full study goes further
Real breaches open over minutes to hours, valleys widen and narrow, and floodplains store water. A dam safety study uses surveyed terrain, a breach model and usually a two-dimensional hydraulic model. This page assumes a prismatic channel and an instant, complete failure, which tends to give a fast and severe wave.
Sediment transport and bed erosion
How much sand can this flow carry? Four published transport formulas run side by side. When less sediment arrives from upstream than the flow can carry, the difference comes from the bed, so the calculator also gives an average erosion rate over a reach.
Transport capacity against velocity
All four formulas for this sand and depth, up to 1.6 times your velocity. Slope rises with velocity squared, holding flow resistance constant. The dashed line marks your velocity.
Particle size distribution
Percentage finer by mass. Each grid line starts the Wentworth class named beside it. Replace the preset with your own sieve results through the custom option.
Formulas used
- Meyer-Peter & Müller (1948):
Φ = 8(θ′ − 0.047)1.5. Developed from flume tests on coarse sand and gravel. - van Rijn (1984), bedload:
qb = 0.053·√((s−1)g)·d501.5·T2.1/D*0.3, for d50 from 0.2 to 2 mm. - Engelund-Hansen (1967):
f·Φ = 0.1·θ2.5, total load, no threshold. Intended for sand beds with dunes. - Yang (1973): unit stream power total load for sand, giving concentration in ppm from
V·S/ω.
Grain shear uses the Keulegan log law with roughness 3·d90. Total shear is ρ·g·h·S for a wide channel. Settling velocity follows Ferguson and Church (2004). Critical Shields stress follows Soulsby and Whitehouse (1997). Preset gradings are log-normal and illustrative only.
Reading the spread between formulas
Transport formulas commonly disagree by a factor of several, because each was fitted to different data. Treat the range across formulas as part of the answer. Bedload formulas leave out sand carried in suspension, so they sit below total-load formulas in fast flow. Cohesive silts and clays, armouring of graded beds and supply from banks all need site-specific assessment.