Morison Equation Simulator: Wave Forces on Offshore Structures

simulator intermediate ~10 min
Loading simulation...
F_max = 48.2 kN/m — 6m wave on 1.2m cylinder

A 6m wave with 10s period acting on a 1.2m diameter cylinder in 30m depth produces a maximum inline force of approximately 48.2 kN per meter of cylinder length, with drag and inertia contributing roughly equally at KC ≈ 10.

Formula

F = 0.5 × Cd × ρ × D × u|u| + Cm × ρ × (πD²/4) × ∂u/∂t
u = (πH/T) × cosh(k(z+d)) / sinh(kd) × cos(kx - ωt)
KC = u_max × T / D

Forces from the Sea

Offshore structures must withstand enormous hydrodynamic forces from ocean waves, currents, and storms. For slender cylindrical members — the legs, braces, and risers that form jacket platforms — the Morison equation provides the engineering tool for calculating these forces. Proposed by Morison, O'Brien, Johnson, and Schaaf in 1950, it decomposes the total force into two physically distinct components: drag and inertia.

Drag & Inertia Components

The drag force arises from flow separation and vortex shedding around the cylinder, proportional to the square of water particle velocity. The inertia force comes from the pressure gradient in the accelerating wave field, proportional to the local fluid acceleration. Their relative importance depends on the Keulegan-Carpenter number KC — a low KC means the flow barely moves past the cylinder (inertia dominates), while high KC means extensive vortex shedding (drag dominates).

Design Wave Analysis

Offshore platforms are designed for extreme waves — the 100-year return period wave height is the standard design criterion. In the North Sea, this can exceed 30 meters. Engineers compute wave kinematics using Airy (linear) or Stokes fifth-order theory, then apply the Morison equation at every structural member to determine total base shear and overturning moment. These govern foundation and structural sizing.

Beyond Morison

For large-diameter structures like gravity-based platforms, monopiles for wind turbines, and floating production vessels, the cylinder-to-wavelength ratio exceeds 0.2 and diffraction effects become significant. Here, potential flow panel methods solve the full wave-structure interaction problem. But for the slender-member world of jacket structures, the Morison equation remains the workhorse of offshore engineering seven decades after its publication.

FAQ

What is the Morison equation?

The Morison equation (1950) calculates wave forces on slender cylinders as the sum of a drag term (proportional to velocity squared) and an inertia term (proportional to acceleration). It applies when the cylinder diameter is small relative to the wavelength (D/L < 0.2), making it the standard tool for jacket platform design.

What is the Keulegan-Carpenter number?

KC = u_max × T / D is a dimensionless number that indicates whether drag or inertia dominates wave loading. For KC < 5, inertia dominates; for KC > 20, drag dominates; between 5 and 20, both are significant. It governs the choice of drag (Cd) and inertia (Cm) coefficients.

What are typical Cd and Cm values?

For smooth cylinders: Cd ≈ 0.65, Cm ≈ 1.6. For rough cylinders (marine growth): Cd ≈ 1.05, Cm ≈ 1.2. These values depend on KC number, Reynolds number, and surface roughness. Design codes like API RP 2A specify recommended values.

When does the Morison equation not apply?

When D/L > 0.2 (large structures relative to wavelength), diffraction effects become important and the Morison equation underestimates forces. Linear diffraction theory or panel methods (e.g., WAMIT) must be used instead for gravity-based structures, FPSOs, and large monopiles.

Sources

Embed

<iframe src="https://homo-deus.com/lab/ocean-engineering/wave-loading/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub