Tested against the references you already design with

The discharge coefficient of the standard ogee crest is one of the best-documented numbers in dam engineering. We model a slice through the WES standard crest (downstream face x^1.85 = 2H_d^0.85y with the two-radius upstream curve; design head H_d = 0.4 m, crest height P/H_d = 2, side walls without friction) fed from a reservoir. The discharge coefficient is C = q/H_e^1.5 (m^½/s), with q the steady discharge per metre of crest and H_e the energy head of the approach flow above the crest: in each cross-section of the approach channel, the pressure head, elevation and velocity head of the water are averaged weighted by discharge, and these section values are averaged from 2.5 to 6.5 design heads upstream of the crest. We compare it with the US Army Corps of Engineers design chart for this crest shape (the deep-approach curve, P/H_d 1.33 and above, of Hydraulic Design Chart 122-1) and with the 1949 Waterways Experiment Station laboratory tests of the same crest that the chart is drawn through above 0.4 times the design head.

Bazin's classic experiments (1896–98) on a rectangular weir block 0.75 m high with a 0.40 m flat crest, sharp upstream corner and free overfall, in a 2 m wide channel (series 113, as tabulated by Horton 1907). We model a slice of the channel and compare the discharge coefficient C_d, defined by q = C_d(2/3)^3/2√g H₁^3/2 with H₁ the total head above the crest including velocity head (in the model, measured 1.5 m upstream; Bazin read his head 5 m upstream). Bazin's discharges were derived from a standard weir that he calibrated to within about 1 % (Horton 1907).

Crookston's laboratory labyrinth (Utah Water Research Laboratory, 2010): a trapezoidal labyrinth with 15° sidewalls, a quarter-round crest and a weir height P of 0.30 m, with a free outfall. The laboratory model had two identical cycles across the flume, so we model one cycle between planes of symmetry (side-wall friction neglected), with 60 cells across the cycle (100 at the lowest head) and turbulence modelled with k-ε and wall functions. The coefficient is defined by Q = (2/3) C_d L_c √(2g) H_T^3/2, with L_c the crest length along its centreline and H_T the total head upstream. The comparison is with Crookston & Tullis's design curve for this crest, which is fitted to these measurements. At low heads the nappe springs clear of the downstream face; at heads of 0.4–0.5 times the weir height the space under the nappe fills with water and the sheet falls into the downstream pool.

For a culvert, the design question is how much headwater a given flow needs. We compare the headwater ratio HW/D with the inlet-control equations of FHWA HDS-5 for a circular barrel with a square-edged headwall and a concrete box with a square-edged inlet in a vertical headwall (HDS-5 coefficients for 90° and 15° wingwall flares), on a 3 % slope with a free outfall, across the unsubmerged-inlet range. In every run the barrel flowed part-full downstream of the inlet, so the inlet controlled the flow.

Afflux is the rise in upstream water level that a bridge causes. We use measurements for a single-span arch bridge in a laboratory channel with floodplains (Atabay & Knight 2004). Each discharge is run with the bridge in place. The afflux is the upstream water level with the bridge minus the measured normal depth without it, as in the laboratory; in the model the upstream level is averaged over the main channel from 0.1 m to 1.0 m upstream of the bridge. The water surface is compared with the levels measured along the main channel in the metre upstream of the bridge (as plotted by Kocaman 2014), using the model surface averaged over the last 10 s. Each discharge is run for at least 200 s of flow and until it has settled: the afflux is the mean over the last 60 s, changing by less than 0.02 percentage points per 10 s window over its last six windows, and the water stored in the model changing by less than 0.1 % of the inflow.

Rock groynes, rockfill toes and riprap filters are modelled as porous zones: the app takes the porosity and stone size and applies a resistance that grows with the velocity through the rock (a Darcy–Forchheimer law with published coefficients for crushed rock, Jensen et al. 2014, written in terms of the bulk velocity through the rock).

A square box of fluid whose lid slides at constant velocity is the classic test of a flow solver's core equations. We compare the velocity through the middle of the box, and the positions of the main vortex and the two corner eddies, with the benchmark solution of Ghia, Ghia & Shin (1982) at Reynolds numbers 100, 400 and 1000.

The drag coefficient is C_d = F/(½ρU²·πD²/4), with the drag force F taken from a momentum balance on a control volume around the sphere. One quarter of the sphere is modelled between two symmetry planes.

The Strouhal number is St = fD/U, with f the frequency of the lift force on the cylinder once the shedding is regular. The frictionless side boundaries are 10 diameters either side of the cylinder, a blockage of 5 %.

The film thickness is given as δ/δ_N, the computed thickness over Nusselt's thickness for the same flow rate. The velocity profile is compared with the theoretical u/u_max = 2η − η², with η the height above the bed over the film thickness. The film Reynolds number is q/ν, the flow per metre width over the kinematic viscosity; at about 1 on a 30° slope the film stays flat and laminar, the regime Nusselt's solution describes.

The reattachment length x_r/s is the distance from the step to where the flow reattaches to the bottom wall, over the step height s. The Reynolds number is based on the hydraulic diameter of the inlet channel.

The measurements are the University of Tokyo 2.5 m Wigley model, fixed in sinkage and trim, at Fn 0.267 (Kajitani et al. 1983, Table 1), where the wave elevation is tabulated in nondimensional form; we show it as ζ/L × 10³ (the tabulated value × ½Fn² × 10³, reading it as 2gζ/U², the reading consistent with independent measurements), with L the hull length. The headline is the root-mean-square difference between the computed and the measured wave elevation over the whole hull, in ζ/L × 10³ and in mm at the 3.014 m hull length computed here (scaled by Froude similarity). The measured bow crest is ζ/L × 10³ = 12.9, which is 38.9 mm at that length. Errors are computed from unrounded values.