Squat Effect

Squat is the localised reduction in under-keel clearance experienced by a vessel moving through shallow or restricted water, caused by the Bernoulli-effect lowering of the water surface and the corresponding sinkage of the hull as the flow accelerates around the displaced volume. The phenomenon arises because, in restricted water, the cross-section available for return flow around the moving vessel is small relative to the displaced volume; the return flow velocity must therefore be high, the dynamic pressure is correspondingly low (per Bernoulli’s principle), and the water surface around the vessel drops; the vessel sinks deeper to maintain its hydrostatic equilibrium with the depressed surface. Squat magnitudes of 0.5 to 3 m are typical for large merchant ships at typical operating speeds in shallow water, sufficient to cause grounding if not properly accounted for in voyage planning and pilot conduct. Squat is a primary navigation hazard in shallow channels (Suez, Panama, Saint Lawrence Seaway, Mississippi, Elbe, Houston Ship Channel, Plata, Hamburg approach, Rotterdam Maasvlakte), harbour approaches, river ports, and constricted canals. Squat is governed empirically by formulas including Barrass (1979), Tuck (1966), Eryuzlu (1994), ICORELS (1980), Norrbin (1986), Yoshimura (1986) and PIANC (2014, 2019) consensus guidelines; modern vessel-specific analysis is by CFD with free-surface modelling, validated against full-scale measurements. Notable squat-related incidents include the Ever Given grounding in the Suez Canal (March 2021), in which excessive speed for the prevailing wind and shallow-water conditions contributed to the loss of steering control. ShipCalculators.com hosts the principal computational tools: the squat calculator (Barrass), the squat calculator (Tuck), the squat calculator (PIANC), the under-keel clearance margin calculator, the blockage coefficient calculator, the shallow water resistance calculator and the channel transit speed calculator. A full listing is available in the calculator catalogue.

Contents

Background and physics

The squat phenomenon

When a ship moves through deep open water, the water flow around the hull adjusts smoothly with negligible disturbance to the free water surface. In shallow water (where the water depth $h$ is comparable to the vessel’s draught $T$) or in restricted water (where lateral channel boundaries constrain the return flow), the situation changes fundamentally:

The return flow of water displaced by the moving vessel must pass through a reduced cross-sectional area (above and beside the hull).
By continuity, the return flow velocity is increased.
By Bernoulli’s principle, the higher flow velocity is associated with lower static pressure.
The lower static pressure causes the water surface around the vessel to drop (creating a depression in the wave system).
The vessel, in hydrostatic equilibrium with the depressed surface, sinks deeper to balance the buoyancy.
The vessel may also trim (longitudinal pitch) due to the asymmetric distribution of the pressure depression along the length.

The combined effect is a reduction in under-keel clearance equal to the squat (the vertical sinkage at the deepest part of the hull). For a typical large merchant ship at operating speed in shallow channel conditions, the squat is in the range of 0.5 to 3 m.

Bow squat versus stern squat

For full-form merchant ships (VLCC, Capesize bulker, container ship), the maximum squat typically occurs at the bow (“trim by the bow”). For finer-form vessels (cruise ships, ferries, naval vessels, fast displacement ships), the maximum squat may occur at the stern (“trim by the stern”).

The distinction matters because the maximum squat point is the location at which grounding first occurs. For deep-draught vessels with bow-dominated squat, the bow is at greatest risk; for stern-dominated vessels, the stern is at greatest risk.

Speed dependence

Squat is approximately proportional to the square or higher power of the ship speed (depending on the formula). A 50% speed reduction therefore reduces squat by approximately 75%; a 25% reduction reduces squat by approximately 44%. Speed reduction is the primary control measure for managing squat in shallow water.

Critical depth Froude number

A useful dimensionless characterisation of squat is the depth Froude number:

$$ F_{nh} = \frac{V}{\sqrt{g \cdot h}} $$

(distinct from the standard Froude number $F_n$, which is based on length).

For typical merchant ships:

$F_{nh} < 0.4$: subcritical, mild squat, manageable.
$F_{nh} = 0.4$ to $0.8$: transcritical, significant squat increases rapidly.
$F_{nh} = 0.8$ to $1.0$: critical region, very high squat (the vessel is approaching the speed of free-surface gravity waves at the channel depth).
$F_{nh} > 1.0$: supercritical, theoretically possible but operationally uncommon for displacement ships in restricted water.

Most operational guidance limits operation to $F_{nh} < 0.4$ to 0.6, which corresponds to typical channel speed limits.

Empirical squat formulae

Barrass (1979) formula

Barrass developed a widely-used empirical squat formula based on full-scale measurements:

$$ S_{max} = \frac{C_B \cdot V_S^{2.08}}{20} $$

where $S_{max}$ is the maximum squat in metres, $C_B$ is the block coefficient, and $V_S$ is the ship speed in knots. The formula is valid for $V_S < 18$ knots and for typical merchant ships in unrestricted shallow channels.

The Barrass formula is the most commonly cited empirical squat estimate and is used in many ship’s pilot training materials and in commercial navigation software.

Barrass II (1995) formula

A revised Barrass formula incorporates the blockage coefficient $S_C$ (the ratio of the immersed cross-section of the vessel to the cross-section of the channel):

$$ S_{max} = \frac{C_B \cdot S_C^{2/3} \cdot V_K^{2.08}}{30} $$

where $V_K$ is the ship speed in knots. This refinement accounts for the lateral restriction of the channel and produces lower squat estimates for unrestricted-channel operations.

Tuck (1966) formula

Tuck’s analytical formula is based on slender-body theory and is the basis for many subsequent refinements:

$$ S_{Tuck} = \frac{\nabla \cdot F_{nh}^2}{L \cdot W \cdot (1 - F_{nh}^2)^{1/2}} $$

where $\nabla$ is the underwater volume, $L$ is the length, $W$ is the channel width (or effectively infinite in unrestricted water), and $F_{nh}$ is the depth Froude number. Tuck’s formula has solid theoretical foundations and is widely used in academic studies; it is somewhat less convenient than the Barrass formula for quick operational estimates.

Eryuzlu (1994) formula

Eryuzlu developed a regression formula based on extensive experimental data:

$$ S_{max} = 0.298 \frac{T^2}{B^{0.81} \cdot h^{0.19}} \cdot V^{2.289} \cdot \left(\frac{h}{T}\right)^{0.20} $$

with units in metres and metres per second. The Eryuzlu formula is particularly accurate for medium-to-large container ships in deep-water channels.

ICORELS (1980) formula

The ICORELS (International Commission for the Reception of Large Ships) formula is a consensus empirical estimate adopted by several maritime authorities:

$$ S_{max} = 2.4 \cdot \frac{\nabla}{L^2} \cdot \frac{F_{nh}^2}{(1 - F_{nh}^2)^{1/2}} $$

ICORELS is conservative (typically gives larger squat estimates than Barrass) and is favoured by some pilots and by some channel authorities.

Norrbin (1986) formula

Norrbin developed a formula that explicitly accounts for the channel cross-section ratio:

$$ S_{max} = \frac{C_B \cdot V^2 \cdot (T/h)^{1.5}}{30} \cdot (1 + S_C / 0.2)^{0.5} $$

The Norrbin formula is particularly useful for canal transits with significant channel restriction.

Yoshimura (1986) formula

Yoshimura developed a formula focused on the bow squat of full-form vessels:

$$ S_{bow} = 0.7 \cdot C_B \cdot \frac{V^2}{g} \cdot \frac{F_{nh}^2}{1 - F_{nh}^2} $$

The Yoshimura formula is widely used for VLCCs and large bulkers in restricted channels.

PIANC (2014, 2019) consensus

The PIANC (Permanent International Association of Navigation Congresses) has published consensus guidelines on squat calculation in 2014 and updated in 2019. The PIANC method:

Recommends specific empirical formulae depending on the vessel type, speed regime and channel configuration.
Provides safety factor recommendations.
Includes guidance on squat allowance in channel design and operational planning.

PIANC guidelines are the standard reference for ports, channel authorities and pilot organisations worldwide.

CFD analysis

For the most accurate vessel-specific squat prediction, CFD with free-surface modelling (using the volume-of-fluid method) is the modern standard. CFD provides:

Accurate prediction of squat for the specific hull form and operating conditions.
Detailed pressure distribution around the hull, useful for identifying squat-sensitive points.
Sensitivity analysis to channel configuration variations.

CFD studies are typically commissioned by:

Newbuild owners as part of the design optimisation.
Channel authorities as part of channel design or capacity expansion studies.
Pilot organisations for specific vessel-channel combinations.

The principal commercial CFD codes for marine squat are STAR-CCM+ (Siemens), OpenFOAM (open-source), and ANSYS CFX. A typical CFD squat study costs USD 10,000 to USD 50,000 per hull-channel combination.

Operational consequences

Channel transit speed limits

All major shallow channels have operational speed limits to control squat:

Suez Canal: typical 13 to 16 knots, depending on convoy direction and vessel type. Larger vessels have stricter limits.
Panama Canal: typical 8 knots maximum for the original locks, slightly higher in the new Neopanamax locks.
Kiel Canal: typical 8.1 knots maximum.
Saint Lawrence Seaway: 13 knots inbound, slower in restricted sections.
Mississippi River (deep-draft section): typical 8 to 12 knots depending on conditions.
Elbe River (Hamburg approach): typical 13 knots, with reductions in tight bends.
Houston Ship Channel: approximately 12 knots maximum.
Plata River (Buenos Aires approach): approximately 10 knots typical.

Pilots routinely operate well below the speed limit when conditions warrant (high winds, currents, limited visibility, deeper-than-design draughts).

Under-keel clearance margin

Channel authorities and pilot organisations specify minimum under-keel clearance (UKC) margins that must be maintained at all times. Typical UKC margins:

Open sea, deep water: typically 5 to 10% of draught minimum.
Restricted channel, normal weather: 10 to 20% of draught.
Restricted channel, heavy weather: 20 to 30% of draught.
Confined harbour basins: 5 to 15% of draught.

The UKC margin must accommodate the squat plus other allowances (wave-induced sinkage, tide variation, channel survey accuracy, ship motion). The margin is typically calculated as:

$$ \text{UKC}{required} = S{squat} + S_{wave} + \Delta_{tide} + \Delta_{survey} + \Delta_{motion} + \text{safety margin} $$

Tide-restricted operations

For deep-draught vessels operating in shallow channels, the available water depth is often tide-dependent. Operations may be restricted to specific tidal windows when the high tide provides sufficient depth above the squat-adjusted minimum draught requirement. For example, VLCCs entering Rotterdam typically operate only during the highest tide windows.

Channel maintenance dredging

Channel authorities must maintain the channel depth through periodic dredging to ensure that the design under-keel clearance is preserved. Significant channel depth reductions (e.g. due to sediment accumulation, storm-deposited sand, or seismic events) trigger dredging or operational restrictions until the depth is restored.

Voyage planning

Modern voyage planning systems integrate squat calculations with route planning:

The vessel’s CFD-predicted squat curves at various draughts and speeds are loaded into the voyage planning system.
The route is checked for shallow-water sections.
For each shallow section, the maximum permissible speed is calculated to maintain UKC.
The voyage time is estimated using the speed-limited sections.

The principal voyage planning systems (NAPA Voyage Optimisation, Wartsila FOS, Kongsberg Vessel Insight, Furuno Voyage Planner) include integrated squat calculation modules.

Squat in narrow channels

Channel cross-section ratio

The channel cross-section ratio (sometimes called the blockage ratio $S$):

$$ S = \frac{A_{ship}}{A_{channel}} $$

where $A_{ship}$ is the immersed cross-section of the ship and $A_{channel}$ is the cross-section of the navigable channel. In open water, $A_{channel}$ is effectively infinite and $S = 0$; in confined channels (Suez, Panama lock chambers), $S$ can be 0.2 to 0.4.

Squat increases significantly with $S$. Channel design conventions typically aim for $S < 0.25$ at design draught to keep squat manageable.

Bank effects

In addition to depth-induced squat, narrow channels with parallel banks introduce bank effects:

Bank suction: the vessel is drawn toward the bank by the local pressure depression on the bank-side of the hull.
Bank cushion: the vessel is pushed away from the bank by the local pressure rise at the bank near the bow.
Yaw effects: combined bank suction and cushion produce a yaw moment that can be hard to counteract.

Bank effects compound squat in narrow channels and are a major contributor to bank-strike incidents.

Ever Given (Suez Canal, March 2021)

The Ever Given (a 20,000 TEU container ship, 400 m length, 59 m beam) ran aground on the eastern bank of the Suez Canal on 23 March 2021, blocking the canal for six days. The Suez Canal Authority and various accident investigations attributed the grounding to a combination of factors including:

Excessive speed: the vessel was reportedly travelling at approximately 13 knots, near the canal speed limit but at the upper end for the vessel’s size.
High winds: a sandstorm produced gusts of 70+ km/h, generating significant lateral force on the high windage area of the container stack.
Squat-induced steering loss: the combination of speed and shallow-water effects reduced the rudder’s effectiveness, allowing the vessel to deviate from the centreline.
Pilot decisions: the canal pilots made decisions in real-time that, with hindsight, may have aggravated the situation.

The incident triggered a global review of shallow-water navigation practices and accelerated the deployment of vessel-specific squat calculation tools.

Costa Concordia (Italy, January 2012)

The Costa Concordia ran aground off the Italian island of Giglio after the captain ordered a close-shore deviation. While primarily a navigation decision incident, the squat effect contributed to the grounding by reducing the under-keel clearance below the level required for the close-shore manoeuvre.

MSC Napoli (English Channel, January 2007)

The MSC Napoli structural failure off the south coast of England was preceded by significant heel and trim (associated with bow flare slamming and structural fatigue). While not a pure squat incident, the dynamic loads associated with shallow-water operation in heavy seas were a contributing factor.

Various river and channel groundings

Many smaller incidents occur annually in restricted channels worldwide, principally bulk carriers and tankers in approach channels to oil and bulk export terminals. The cumulative cost of squat-related incidents is significant; PIANC estimates approximately USD 200 to USD 400 million per year globally in direct costs (vessel damage, channel damage, environmental remediation).

Vessel-specific factors

Block coefficient

Block coefficient ($C_B$) is the principal vessel-specific factor in squat:

High $C_B$ (full-form merchant ships): more squat at given speed and depth.
Low $C_B$ (fine-form ships): less squat.

The Barrass formula explicitly captures the $C_B$ effect.

Length-to-beam ratio

The length-to-beam ratio ($L/B$) affects the longitudinal distribution of the squat. Higher $L/B$ tends to concentrate the squat at the bow; lower $L/B$ produces a more uniform sinkage.

Draught

Squat is approximately proportional to the draught-to-depth ratio ($T/h$). A vessel operating at deeper draught (closer to the bottom) experiences proportionally more squat than the same vessel at lighter draught.

Bow form

The bow form (bulbous bow shape, fineness, sheer) affects the local squat distribution. Bulbous bow vessels in shallow water can experience particularly high local sinkage at the bulb if the bulb operates near the bottom.

Stern form

The stern form (transom shape, propeller arrangement, rudder design) affects the stern squat. Vessels with full transoms and large propellers in shallow water can experience significant stern sinkage.

Mitigation measures

Speed reduction

The most direct squat mitigation is speed reduction. The squat is approximately proportional to $V^2$, so a 30% speed reduction produces a 50% squat reduction. Channel pilots routinely reduce speed below the regulatory maximum when conditions warrant.

Tidal window operation

For deep-draught vessels in tidal channels, restricting operation to high-tide windows provides additional water depth that compensates for the squat.

Light loading

Operating at lighter draught (less than the design summer load line draught) reduces the absolute squat and also increases the available UKC margin. This is a common approach for vessels approaching shallow ports.

Channel deepening

Long-term, channel dredging to deepen the navigable channel is the principal infrastructure response. Major recent channel deepening projects include:

Panama Canal expansion (Neopanamax locks): completed 2016.
Suez Canal “New Suez Canal” parallel channel: completed 2015.
Houston Ship Channel deepening: ongoing.
Saint Lawrence Seaway depth maintenance: ongoing.
Mississippi River jetty system: ongoing.

Pilot training and decision support

Modern pilot training emphasises integrated decision-making that accounts for vessel characteristics, channel conditions, weather, traffic, and squat calculations. Decision support systems on the bridge integrate:

Real-time GPS and ECDIS position.
Vessel dynamic motion sensors.
Wind, current and tide forecasts.
Channel survey data.
Vessel-specific squat curves.

References

Barrass, C. B. Ship squat: A guide for masters. Marine Engineers Review, 1979.
Barrass, C. B. and Derrett, D. R. Ship Stability for Masters and Mates, 7th edition. Butterworth-Heinemann, 2012.
Tuck, E. O. Shallow water flows past slender bodies. Journal of Fluid Mechanics, 1966.
Eryuzlu, N. E., Cao, Y. L. and D’Agnolo, F. Underkeel requirements for large vessels in shallow waterways. PIANC, 1994.
ICORELS. Report of WG IV. International Commission for the Reception of Large Ships, 1980.
Norrbin, N. H. Bank effect on a ship moving through a short dredged channel. Tenth Symposium on Naval Hydrodynamics, 1986.
Yoshimura, Y. Mathematical model for manoeuvring of ship in inland waterways. Society of Naval Architects of Japan, 1986.
PIANC. Harbour Approach Channels: Design Guidelines. PIANC Working Group 121, Report N° 121, 2014 (updated 2019).
PIANC. Capability of Ship Manoeuvring Simulation Models for Approach Channels and Fairways. PIANC Working Group 171, 2017.
IMO. MSC.1/Circ.1228 Revised guidance to the master for avoiding dangerous situations in adverse weather and sea conditions. International Maritime Organization, 2007.