Real-time inertia, drag, and total force calculation using the Morison equation. Airy wave velocity field visualization, overturning moment, and KC number.
Wave & Structure Parameters
Presets
Wave Height H
m
Wave Period T
s
Water Depth d
m
Cylinder Diameter D
m
Current Velocity Uc
m/s
Inertia Coefficient Cm
Drag Coefficient Cd
Results
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Max Wave Force [kN]
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Max Inertia Force [kN]
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Max Drag Force [kN]
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Overturning Moment [MN·m]
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KC Number
Visualization
Wave Force Time Series (1 Wave Period)
Theory & Key Formulas
$$F = \underbrace{\rho C_m V \frac{du}{dt}}_{\text{Inertia Force}}+ \underbrace{\frac{1}{2}\rho C_d A \, u|u|}_{\text{Drag Force}}$$
What exactly is the Morison Equation? I see it's the main formula in this simulator, but why is it so important for offshore structures?
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Basically, it's the fundamental model for calculating the force from waves and currents on slender structures, like the cylindrical piles of an oil platform. It breaks the total force into two distinct parts: an inertia force from accelerating water and a drag force from flowing water. In practice, you can see their individual contributions change in real-time on the force graph when you adjust the wave period `T` or cylinder diameter `D` above.
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Wait, really? So which part dominates? If I make the waves faster by decreasing the period, does inertia or drag become more important?
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Great question! That's where the Keulegan–Carpenter (KC) number comes in. A low KC number (from short, fast waves) means inertia dominates because the water accelerates quickly. A high KC number (from tall, slow waves) means drag dominates. Try it: set a very short period like 5 seconds and watch the inertia force spike on the graph. Then, set a very long period like 15 seconds and see the drag force become more significant.
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That makes sense. But what about the current? The simulator has a `Uc` slider. How does a steady current change the forces compared to just waves?
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In practice, it adds a constant "push" that the wave oscillates around. For instance, a strong current from a tidal flow will create a permanent drag load on the structure. More importantly, it makes the total velocity `u` asymmetrical—faster in one direction than the other. Slide the `Uc` up to 2 m/s and watch how the drag force curve (the orange part) is no longer symmetrical above and below zero, creating a net mean load that's critical for fatigue analysis.
Physical Model & Key Equations
The total horizontal force per unit length on a cylindrical member is given by the Morison equation, which sums the inertia and drag components.
Where:
$\rho$ = water density, $C_m$ = inertia coefficient, $C_d$ = drag coefficient, $D$ = cylinder diameter.
$u(z,t)$ = total horizontal fluid velocity (wave + current), $\frac{\partial u}{\partial t}$ = local fluid acceleration.
The $u|u|$ term ensures the drag force always opposes the direction of flow.
The fluid kinematics (velocity $u$ and acceleration) are modeled using linear Airy wave theory combined with a uniform current.
Where:
$H$ = wave height, $T$ = wave period, $d$ = water depth, $k$ = wave number ($2\pi/L$), $\omega$ = wave angular frequency ($2\pi/T$), $U_c$ = current velocity.
The $\cosh/\sinh$ term describes how the wave motion decays with depth $z$ below the surface. This is why the forces are highest at the water surface in the visualization.
Frequently Asked Questions
The coefficients depend on the member shape, Reynolds number, and KC number. For a cylinder, Cm is typically around 2.0, and Cd ranges from 0.7 to 1.2. In this tool, the KC number is automatically calculated, so please adjust the coefficients based on design standards or experimental values using that value as a reference.
The shallower the water depth, the flatter the elliptical orbits of water particles become, and the velocity near the seabed decreases. Increasing the wave height increases both velocity and acceleration, leading to larger inertial and drag forces. The visualization allows you to intuitively understand where large forces act on the structure.
First, try reducing the member diameter or length to decrease the projected area and displaced volume, or lower the wave conditions (wave height and period). Additionally, revising the drag coefficient Cd to a smaller value within an appropriate range can be effective. Increasing the installation water depth of the structure reduces the flow velocity near the seabed, thereby lowering the overturning moment.
When the KC number is small (less than 20), drag becomes dominant, and the risk of vortex-induced vibration increases. In this tool, the KC number is automatically calculated, so it can be used as an indicator to determine the dominant force mode during design.
Real-World Applications
Jacket Platform Design: Engineers use this exact analysis, per standards like DNV-OS-J101, to calculate the extreme wave and current loads on the tubular legs and braces of offshore jackets. The overturning moment calculated by the simulator is a direct input for the foundation design to prevent the platform from toppling.
CFD Model Validation: The coefficients $C_m$ and $C_d$ are not constants—they depend on surface roughness, Reynolds number, and KC number. High-fidelity OpenFOAM CFD simulations of flow around cylinders are run to calibrate these coefficients, which are then used in the simpler, faster Morison equation for full structural analysis.
FEM Load Input: The distributed force profile $F(z,t)$ calculated here is applied as an external load in Finite Element Analysis (FEA) software like ANSYS or ABAQUS. This allows engineers to compute the resulting stresses, deflections, and fatigue life in every weld and joint of the offshore structure.
Mooring Line & Riser Analysis: For floating platforms, the forces on vertical risers (pipes bringing oil to the surface) and mooring lines are assessed using the Morison equation. Understanding the relative magnitude of inertia vs. drag force is crucial for predicting vortex-induced vibrations (VIV) that can cause metal fatigue.
Common Misconceptions and Points to Note
First and foremost, please do not think of the Morison equation as a universal tool that can calculate anything. It is targeted at slender, cylindrical members (slender bodies). For example, calculating the motion of a large floating structure (like a ship or a semi-submersible platform) in waves requires a different theory (potential flow theory). Another crucial point is that the "drag coefficient Cd" and the "inertia coefficient Cm" are not constants. While existing explanations mention the Keulegan–Carpenter (KC) number determining dominance, in reality, these coefficients vary significantly due to surface roughness (e.g., marine growth), flow turbulence, and the arrangement of columns (group effect). For instance, experimental values show that even for the same cylinder, Cd can vary from 0.6 to over 1.2. While tweaking coefficients in a tool is good for building intuition, for actual design, it is essential to refer to experimental values or code-recommended values appropriate for your specific environment.
Another point is to understand the limitations of Airy wave theory. This theory assumes "small-amplitude waves" where wave height is small compared to wavelength. This means it cannot strictly represent huge, steep waves with strong nonlinearity, like those during typhoons. For example, it might still be applicable for a wave height of 5m and wavelength of 50m (height/length = 0.1), but for a wave height of 10m and wavelength of 60m (ratio 0.17), consideration of higher-order theories like Stokes wave theory becomes necessary. Don't take the tool's output at face value; always get into the habit of asking yourself, "Does the input wave satisfy the small-amplitude wave assumption?"