1. Manning (uniform flow)
Speed from the balance of gravity and bed friction in a rectangular reach, width w:
V = (1/n) R^(2/3) S^(1/2), R = A/P = w d / (w + 2 d)
Q = V A = V w d (solve for d: Newton, bisection fallback)
2. Froude number and critical depth
Fr = V / sqrt(g d) d_c = (q^2/g)^(1/3), q = Q/w
Fr < 1 calm (subcritical), about 1 standing waves, > 1 fast
3. The stone row (energy method)
Stones of plan width a and height hi block Ab(d) = Σ a min(hi, d). In the gaps the flow area is Ac = w d - Ab:
E = d + Q^2 / (2 g A_c^2) E_min = min over d of E
if E0 < E_min the level upstream rises until its energy equals E_min (afflux = d1 - d0)
V_gap = Q / A_c(d0)
4. Wading hazard
H = d V (m^2/s) guide about 0.6 for an adult (rule of thumb)
HR = d (V + 0.5) + DF (UK flood-risk-to-people method; shown for Researchers)
5. Drag, sliding and toppling of the test post
F = 1/2 rho Cd A V^2, A = wf min(d, Hf)
W_eff = m g - rho g (m / rho_body) min(d,Hf)/Hf (buoyancy)
SF_slide = mu W_eff / F SF_topple = W_eff c_heel / (F min(d,Hf)/2)
Safe: SF ≥ 1.5 Caution: 1.0-1.5 Unsafe: < 1.0
Simplifications
1-D steady uniform flow (Manning) plus a loss-free 1-D energy balance through the stone row (real afflux is larger). No 3-D turbulence, no wake interaction between stones, no sediment or stone movement (the stones are far too heavy to move), the flow map is a visual field built from continuity, not CFD. The reach is a rectangle 6 m wide; in a flood the picture spreads over the banks but the model does not. Thresholds (0.6 m²/s, Cd 1.2, friction 0.4, body density 1050 kg/m³, Manning n) are rules of thumb and typical table values: verify before relying on them. The test subject is an inanimate post, not a person.
Educational model with simplifications. Never use it to decide whether to cross real water. Fast shallow water can sweep an adult off their feet.