Morison Equation Wave Loading on Offshore Structures Simulator

Q: How should I pick the drag coefficient C_D and inertia coefficient C_M?

For rough circular cylinders (typical jacket legs) the API RP 2A WSD and DNV-OS-J101 design practice is C_D ≈ 1.0-1.2 and C_M ≈ 2.0. Smooth cylinders drop to roughly C_D ≈ 0.65 and C_M ≈ 1.6. The values are functions of Reynolds number, Keulegan-Carpenter number and surface roughness, calibrated from the experiments of Sarpkaya, Isaacson and others. Marine growth (barnacles, mussels) can double C_D, so an additional aged-load case is normally checked for the second half of design life.

Q: What does the Keulegan-Carpenter (KC) number tell us?

KC = u_max·T/D is the number of cylinder diameters a water particle travels during one wave period, and it indicates whether drag or inertia dominates. For KC 20-30 strong vortex shedding develops and the drag (C_D) term dominates — typical of slender risers and pipelines. The range 5 < KC < 20 is a transitional regime where both terms are of the same order and design care is highest.

Q: When should I use Morison vs. diffraction theory?

The usual yardstick is the diameter-to-wavelength ratio D/λ. If D/λ 0.2 (GBS bases, storage tanks, very large 8-10 m monopiles), the structure perturbs the incident wave field; diffraction and reflection then matter, and you need the MacCamy-Fuchs analytical solution or a panel-method diffraction code such as WAMIT or HydroD.

Morison Equation Wave Loading on Offshore Structures Simulator

Quick answer
The Morison equation gives the wave load per unit length on a slender offshore member: F = ρ·C_M·V·u̇ + ½·ρ·C_D·A·|u|·u (inertia term + drag term, ρ = seawater density 1025 kg/m³). The KC number KC = u_max·T/D characterizes the oscillatory flow; inertia and drag peak 90° out of phase.

A real-time Morison-equation calculator for the wave loads on slender offshore members — jacket platform legs, conductors, risers and offshore wind monopiles. Vary the significant wave height, peak period, water depth, member diameter and C_D / C_M to see wavelength, particle velocity, drag and inertia forces, the Keulegan-Carpenter number, total wave load and overturning moment update instantly.

Parameters

Significant wave height H_s

Spectral characteristic wave height (≈ 1/1.6 of the largest wave)

Peak wave period T_p

Peak period of the JONSWAP spectrum

Water depth d

Member diameter D

Member shape

Typical C_D / C_M ranges depend on the cross-section

Drag coefficient C_D

Rough surface 1.0-1.2, smooth 0.65, marine growth doubles it

Inertia coefficient C_M

2.0 for ideal potential flow, 1.6 for smooth marine

Submerged height z

Effective wetted length over which the load is integrated

Results

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Wavelength λ (m)

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Peak particle velocity u_max (m/s)

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Drag per unit length (kN/m)

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Inertia per unit length (kN/m)

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Total wave load (kN)

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KC number

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Sea surface and submerged member — particle orbits & force vectors

Linear (Airy) wave moving over a submerged cylinder (jacket leg). Blue arrow = inertia force F_i, red arrow = drag force F_d, white dots = water particle orbits. The KC number on top shows the dominant regime.

Force time-history over one wave period — F_drag, F_inertia, F_total

KC number vs member diameter D — regime map

Theory & Key Formulas

$$F = \rho C_M V \dot u + \tfrac{1}{2}\rho C_D A |u|u,\qquad KC = \frac{u_{max} T}{D}$$

First term = inertia force (proportional to flow acceleration), second term = drag force (squared velocity, sign carried by |u|u). ρ = sea-water density 1025 kg/m³, C_M = inertia coefficient (rough cylinder 2.0), C_D = drag coefficient (rough cylinder 1.0-1.2, smooth 0.65), V = displaced volume per unit length, A = projected area per unit length.

$$u_{max}=\omega\,\tfrac{H_s}{2},\qquad \dot u_{max}=\omega^{2}\,\tfrac{H_s}{2},\qquad \lambda=\tfrac{2\pi}{k},\quad k=\tfrac{\omega^{2}}{g}\ (\text{deep water})$$

Linear (Airy) wave kinematics at the still water level (z = 0). Deep-water approximation k = ω²/g, ω = 2π/T. Higher-order theories (Stokes 5th) raise peak values 10-30% but that is small compared with the KC / C_D uncertainty.

$$F_{total}/L = \sqrt{F_{drag}^{2}+F_{inertia}^{2}}\quad (\text{quadrature sum at peak})$$

Inertia and drag are 90° out of phase, so the quadrature sum of their peaks gives a close estimate of the peak combined load. A rigorous design integrates the time-history.

Morison Equation Wave Loading on Offshore Structures

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I keep hearing about the "Morison equation" in offshore wind and platform design. In plain words, what does it actually compute?

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Put simply, it predicts how hard the sea waves push on a slim vertical member submerged in the water. Back in 1950, Morison, O'Brien, Johnson and Schaaf were working on Gulf of Mexico oil platform legs and made a bold move — they just added together a drag term from steady-flow theory and an inertia term from potential-flow added-mass theory. The result is F = ρ·C_M·V·du/dt + ½·ρ·C_D·A·|u|·u. The first term is the inertia of the accelerating water "dragging" the cylinder, the second is the drag from vortex shedding behind it.

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Makes sense, two fluid-mechanics effects bolted together. But on the left, if I crank the "Member diameter D" up to 10 m, the KC number drops a lot and it says "inertia dominated". What does that mean?

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Good catch. KC = u_max·T/D tells you how many diameters a fluid particle travels in one wave period. Make D large and KC shrinks. At KC = 2 a particle only moves twice the diameter, so there is no time for a big separated wake to grow behind the cylinder — drag stays small and inertia (acceleration of the surrounding water) wins. On a slim riser with KC = 50 the particle sweeps fifty diameters back and forth, vortices fully develop and drag dominates.

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So a jacket leg with D ≈ 2 m sits right in the middle, doesn't it?

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Exactly. With the default inputs KC is about 9.4, smack inside the transition regime. Look at the "KC vs D" chart below — the jacket-leg point falls between the KC = 5 and KC = 20 lines. This intermediate range is actually the trickiest one to design for because C_D and C_M are no longer flat constants. Sarpkaya's classic oscillatory-flow tests show C_D spiking up to about 1.4 near KC ≈ 10, then dropping back. That's why API RP 2A recommends a conservative C_D = 1.05 and C_M = 1.2 for marine-grown rough cylinders.

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You mentioned marine growth — how much does it really matter in design?

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It matters a lot. On a North Sea jacket, mussels and hydroids can build up to 50-100 mm thick after twenty years in service. The effective diameter grows and the surface becomes very rough, which roughly doubles C_D from the as-built 0.65 to about 1.2. Designers add a separate aged-load case with "50 mm marine growth, 1325 kg/m³ density, C_D = 1.2" for late-life checks. In Japan, where coastal barnacle growth is fast, the transition-piece zone of an offshore wind monopile is designed assuming periodic cleaning.

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One last question — is the 293 kN "Total wave load" we got with the defaults considered a lot for one jacket leg?

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For a 50 m wetted length on a single leg that's a fairly typical magnitude. On a four-legged jacket two legs take the peak load at the same time, so the base shear ends up around 500-600 kN. With a 30 m lever arm above the wave-load centre you get a 10-15 MN·m overturning moment. Real North Sea jackets sized for the 100-year wave (H_s ≈ 14 m, T_p ≈ 15 s) end up with design base shears of 50-100 MN and overturning moments of 3-5 GN·m. This tool is for order-of-magnitude work — production design uses time-domain non-linear solvers like SACS or USFOS.

Frequently asked questions

The Morison equation is the standard empirical formula for wave loads on slender members where the cross-section diameter is small compared to the wavelength (typically D < λ/5). Targets include fixed jacket platform legs and braces, conductors, risers, subsea pipelines and offshore wind monopiles. For large-diameter gravity-based structures (GBS) or semi-submersible pontoons whose diameter approaches the wavelength, diffraction is no longer negligible and the load must instead be obtained from diffraction theory such as MacCamy-Fuchs or a full 3D diffraction analysis.

For rough circular cylinders (typical jacket legs) the API RP 2A WSD and DNV-OS-J101 design practice is C_D ≈ 1.0-1.2 and C_M ≈ 2.0. Smooth cylinders drop to roughly C_D ≈ 0.65 and C_M ≈ 1.6. The values are functions of Reynolds number, Keulegan-Carpenter number and surface roughness, calibrated from the experiments of Sarpkaya, Isaacson and others. Marine growth (barnacles, mussels) can double C_D, so an additional aged-load case is normally checked for the second half of design life.

KC = u_max·T/D is the number of cylinder diameters a water particle travels during one wave period, and it indicates whether drag or inertia dominates. For KC < 5 the orbit is small, no significant separation occurs and the inertia (C_M) term dominates — this is the regime of large-diameter monopiles. For KC > 20-30 strong vortex shedding develops and the drag (C_D) term dominates — typical of slender risers and pipelines. The range 5 < KC < 20 is a transitional regime where both terms are of the same order and design care is highest.

The usual yardstick is the diameter-to-wavelength ratio D/λ. If D/λ < 0.2 (≈ 1/5), the slender-body approximation is valid and Morison is sufficient. With the tool's default inputs (H_s=6 m, T_p=12 s) the wavelength is ≈ 225 m, so a D=2 m jacket leg sits at D/λ ≈ 0.009 — well inside Morison's range. For large bodies with D/λ > 0.2 (GBS bases, storage tanks, very large 8-10 m monopiles), the structure perturbs the incident wave field; diffraction and reflection then matter, and you need the MacCamy-Fuchs analytical solution or a panel-method diffraction code such as WAMIT or HydroD.

Real-world applications

Fixed jacket platforms (Gulf of Mexico and North Sea): The majority of oil and gas production platforms operated by Shell, ExxonMobil and Equinor are steel jacket structures whose legs (1-2 m diameter) and horizontal/diagonal braces are the bread-and-butter use case for the Morison equation. The 100-year design wave (central North Sea: H_s ≈ 14 m, T_p ≈ 15 s) gives base shears and overturning moments computed with time-domain non-linear solvers such as SACS or USFOS, which size the jacket legs and pile foundations. Many platforms now 50+ years old see in-service waves more severe than the original design, and Morison-equation re-analysis underpins the life-extension assessment.

Offshore wind monopiles (North Sea and Japanese coast): Hornsea, Walney Extension and most commercial offshore wind farms worldwide use 7-10 m diameter steel monopiles. Their large diameter pushes them into the KC < 5 inertia-dominated regime, so the Morison equation is augmented with the MacCamy-Fuchs diffraction correction. The same modified-Morison approach is used in Japan's Choshi (Chiba) and Akita offshore wind farms under DNV-OS-J101 / IEC 61400-3. Coupled aero-hydro-elastic simulations in FAST or Bladed combine these wave loads with the rotor aerodynamic loads.

Floating offshore wind and spar-type FPSO mooring: Hywind Scotland (the world's first commercial floating wind farm) and Japan's Fukushima floating demonstrator place wind turbines on spar or semi-submersible hulls. The mooring lines (100-150 mm chain, polyester rope) themselves see Morison loads from waves and currents, used in dynamic mooring tension analysis by tools such as OrcaFlex. Companies like Vestas and MHI-Vestas are now scaling from demonstrators to commercial production, and Morison calculations feed directly into the fatigue-life assessment of mooring ropes.

Subsea pipelines and risers: Deep-water oil and gas pipelines (0.3-0.6 m diameter) and rigid risers see chronic Morison loads from bottom currents and internal waves. Their small diameter puts them firmly in the drag-dominated regime, and vortex-induced vibration (VIV) becomes the long-term fatigue driver. Dedicated tools such as Shear7 and VIVA solve the Morison-based modal problem, and helical strakes or tension adjustment are used as mitigation. Petrobras's deep-water Brazil fields are the canonical case.

Common misconceptions and design pitfalls

The biggest trap is treating C_D and C_M as fixed material constants. They are in fact strong functions of Reynolds number Re, KC number and relative roughness k/D. Sarpkaya's classic oscillating-flow tests show C_D wandering between 0.6 and 1.4 around KC = 10, settling near 0.8 at KC = 30. On a North Sea jacket reassessment, raising C_D from a 0.7 design value to a measured 1.1 increased the predicted wave load by 60%. Stick to the conservative values recommended by API RP 2A or DNV-OS-J101 unless you have project-specific tank tests or CFD.

Second, only computing the surface kinematics. This tool, like classic linear (Airy) theory, evaluates the peak at the still-water level z = 0. Real water particle velocity decays exponentially with depth (in deep water as exp(-kz)). For a jacket leg submerged to 50 m below the mean surface, u_max = 1.57 m/s at the top falls to about 0.43 m/s (= 1.57·exp(-0.028·30)) at 30 m depth. Design practice integrates the load along the depth; the tool's "Total wave load = peak·wetted length" is an upper-bound order-of-magnitude estimate. SACS and USFOS handle the depth integration automatically.

Finally, the misconception that the Morison equation works the same in deep and shallow water. This tool uses the deep-water approximation (k = ω²/g), which breaks down in shallow water (d/λ < 0.05). In the shallow limit, λ ≈ T·√(g·d) and the wavelength becomes a function of depth. For d = 15 m and T_p = 10 s, the deep-water formula gives λ = 156 m but the true value is ≈ 115 m (a 30% error), with similar errors in u_max. Shallow-and intermediate-water designs must instead solve the full dispersion relation ω² = g·k·tanh(k·d) iteratively (Newton-Raphson). Jetties, breakwaters and near-shore offshore wind are all in this regime.

Morison Equation Wave Loading on Offshore Structures Simulator