Ship Squat Lab: Shallow-Channel Return Flow, Under-Keel Clearance and Havelock Wake

Setting up the lab bench...

Presets

Live readouts

Graphs

Squat versus speed (blue: 1-D mean sinkage, orange: Barrass maximum, red line: static under-keel clearance, dashed: limiting speed)

Havelock wake half-angle versus depth Froude number

Return-flow speed beside the hull versus speed

Numerical checks

running...

How to use

Drag the 3D view to orbit (the camera buttons jump to bridge, bow, side, under-keel and top views). Pick a preset, then change speed, depth and channel width.

Raise the speed in the Harbour approach preset until the dashed squat curve crosses the red static-clearance line: the keel is on the bottom.
Switch Unrestricted water on and off: the effective channel becomes 7.7 B wide and the squat collapses.
In Near critical speed watch the wake wedge open from 19.5° towards 90° as Fr_h approaches 1.
Compare a full hull (C_b 0.85, bow squat) with a fine hull (C_b 0.6, stern squat).

The science

Why a ship sinks when it speeds up. In the frame of the ship the water flows past. Where the hull blocks part of the channel cross-section the flow beside it must speed up (continuity), and faster flow means lower pressure (Bernoulli). The free surface beside the hull drops by a drawdown z and the ship, which floats on the local surface, settles by about the same amount: squat. With channel width W, depth h, midship section A_s = C_m B T (C_m = 0.98 here), speed V and flow speed u beside the hull:

V W h = u [ W (h - z) - A_s ] (mass)
u² = V² + 2 g z (Bernoulli on the free surface)

Eliminating u gives V as a function of the drawdown, with a = [W(h - z) - A_s] / (W h):

V(z) = a √( 2 g z / (1 - a²) )

V(z) rises, reaches a maximum and falls again. The maximum is Schijf's limiting speed V_cr: for a vanishing blockage it tends to the long-wave speed √(g h), but a large ship in a tight channel is limited far below it. The subcritical root of V(z) = V on the rising branch is the drawdown z used for the mean sinkage. The page solves it by bisection, and the check list verifies the mass residual and the S → 0 limit.

Barrass's empirical maximum squat. Real hulls also trim. Barrass (2004) summarises many ship trials with S_max = C_b S^0.81 V_k^2.08 / 20 metres, with blockage S = A_s/A_c, V_k in knots and, for unrestricted water, an effective channel width of 7.7 B. The maximum occurs at the bow for C_b > 0.7 and at the stern for finer hulls. In this lab the bow/stern split is built so that the average of both ends equals the 1-D mean sinkage and the larger end equals the larger of the two estimates. This is a teaching construction, not a ship-handling tool. Under-keel clearance UKC = h - T - squat is the number pilots watch.

The Havelock wake. A steadily moving disturbance makes waves that are stationary in its own frame. A wave component travelling at angle θ to the track must have phase speed c = V cos θ and obey the finite-depth dispersion relation c² = (g/k) tanh(k h). Its energy moves at the group speed n c with n = ½(1 + 2kh / sinh 2kh), so it ends up behind the ship along the ray

tan ψ = n sin θ cos θ / (1 - n cos² θ)

The largest ψ over all θ is the cusp half-angle of the wake. In deep water n = ½ and the maximum is atan(1/(2√2)) = 19.47° (Kelvin). As the depth Froude number Fr_h = V / √(g h) rises to 1 the wedge opens towards 90°, the transverse waves disappear, and for Fr_h > 1 only a Mach-like wedge asin(1/Fr_h) remains. The waves drawn in 3D use these exact wavelengths and angles; their heights are illustrative.

Honest simplifications. One-dimensional flow with a rectangular channel; no sloping banks, no propeller or hull-form detail, no dynamic trim from the full pressure field, no unsteady acceleration, no currents or wind, no waves approaching the ship. Real squat predictions need model tests or CFD plus a safety margin. Educational model only.