A rotating lateral waters a circle. Because the area of each ring grows with radius, nozzle flow must rise linearly with radius to give a uniform depth. Set the machine, the weather and the soil, then read the catch cans: Christiansen uniformity, wind and evaporation losses, and whether the application rate beats the soil's infiltration (runoff).
The science
Why flow must rise with radius. The ring between r and r+dr has area 2πr·dr and the arm sweeps it once per revolution (time T = 2π/ω). A nozzle of flow q that serves a ring of width s (the nozzle spacing) at radius r therefore puts down
D(r) = q(r)·T / (2π·r·s) = q(r) / (ω·r·s)
D = const ⇔ q(r) ∝ r (design: q_i = Q·r_i / Σr)
Total flow Q gives a mean depth D = Q·T/(πR²). The end tower moves fastest (v = ωR, 100 % timer = 3.5 m/min here); the depth per pass is set by the percent-timer, not by the nozzles.
Nozzle flow. Orifice law, q = Cd·(π/4)d²·√(2p/ρ) with Cd = 0.9, so flow ∝ √pressure: +21 % pressure gives +10 % flow. Pressure regulators hold pset (138 kPa) at every nozzle while the lateral supplies at least that; without them the pressure at each nozzle is the pivot pressure minus friction minus elevation gain, ρg(hf + s·r).
Friction along the lateral (Hazen-Williams, C = 140, 168 mm pipe): hf = 10.67·L·Q1.852/(C1.852d4.87), summed outlet by outlet as the flow falls; the whole-lateral integral has the closed form k·Q1.852L·(√π/2)Γ(2.852)/Γ(3.352). Nozzles are sized from the design pressure profile and rounded to the 1/64-inch size step, with a minimum bore, so the inner nozzles over-apply slightly (their rings are tiny).
Catch cans. Each nozzle spreads its water over a dome-shaped wetted disc of radius Rw ≈ 0.38√p (p in kPa, so about 4.5 m at 138 kPa). Its radial profile is parabolic, w(x) = 3/(4Rw)·(1 − x²/Rw²), so a can at radius ρ collects Σi qiT/(2πρ)·w(ρ − ri − shift). With wide nozzle spacing the profiles do not overlap and the depth shows stripes (ripple amplitude falls as 1/(Rw/s)²).
Uniformity. Christiansen: CU = 100·(1 − Σ|xi − x̄| / Σxi). On a circle, cans far from the pivot represent more area, so the Heermann-Hein weighting CUH = 100·(1 − ΣSi|Vi − V̄p| / ΣViSi) (Si = distance from the pivot) is the right measure; both are shown. DUlq is the mean of the driest quarter over the overall mean. CU ≥ 90 % is good for a pivot, 80-90 % fair.
Wind drift and evaporation. Median drop size d50 ≈ 1.8·(138/p)0.3 mm; fall speed from the Atlas formula vt = 9.65 − 10.3e−0.6d; flight time t = (h + 0.8)/(0.8vt); downwind displacement = uh·t, projected on the can line (cos(φ − θwind)). Losses (evaporation plus spray carried off the field) use an empirical form, loss = (0.012 + 0.009u + 0.032·VPD)·(p/138)0.25·(0.75 + 0.25h/2.5), tuned to the 2-25 % range reported in field studies; VPD from the Tetens saturation pressure. This part is a calibrated rule of thumb, not first-principles physics.
Runoff. Under a moving sprinkler a point receives a rain pulse of duration tw = 2Rw/(ωr), parabolic in time with peak 1.5·D/tw. Because v = ωr, the pulse is shorter and more intense toward the end of the lateral (peak rate does not depend on the timer; depth does). Infiltration follows Green-Ampt, f = Ks(1 + ψΔθ/F); water above f first fills surface storage and then runs off. The model removes runoff from the field; it does not route it downhill or include crusting, residue or soil cracking, and the Ks values are representative effective numbers (order of magnitude from Rawls et al.), not a soil test.
Honest simplifications: one radial can line (cans at 3.1 m, deliberately not a multiple of the nozzle spacing: otherwise the can grid aliases the nozzle ripple); wind modelled as a mean shift and a loss factor (no turbulence, no pattern distortion); slope is radial (dome or bowl) and affects hydraulics only, no overland flow; no end gun or corner arm; no crop water-use model; machine and field are drawn at full scale but the wheels do not track. Educational model, not irrigation design software.
Baked 3D scene. Pivot, towers, truss spans, drop hoses, nozzle plates and catch cans were modelled in Blender 5.2 with baked albedo, normal, AO, roughness and metallic maps in one atlas; the 1.2 km terrain, the circular irrigated field and the crop, windbreak, tree and rock scatter come from Unity 6 with a baked occlusion and sun-shadow lightmap; Three.js r160 runs the live simulation.