係留系解析
Theory and Physics
Catenary Theory
Professor, where should I start with mooring analysis?
First, understand catenary theory. The shape of a chain or wire hanging under its own weight is a catenary curve. The relationship between tension $T$ and submerged weight per unit length $w$ is
Here, $T_H$ is the horizontal tension component. When including the part that touches the seabed (catenary touchdown point), the touchdown length, mooring radius, and tension angle at the anchor become design parameters.
When is dynamic mooring analysis necessary?
When the motion of the floating body due to waves and currents is significant, quasi-static catenary calculations ignore the inertial forces and fluid drag forces of the mooring line. For deep-sea FPSOs (Floating Production Storage and Offloading units) or floating offshore wind turbines, dynamic analysis that discretizes the mooring line using FEM (Finite Element Method) or Lumped Mass models and solves for the unsteady response under wave loads is essential.
Major Analysis Tools
What software is used for mooring analysis?
| Tool | Developer | Features |
|---|---|---|
| OrcaFlex | Orcina | Lumped Mass method based. Industry standard dynamic mooring analysis tool. |
| ANSYS AQWA | Ansys | Potential theory + mooring coupling. Strong in floating body motion response analysis. |
| OpenFAST (MoorDyn) | NREL | For floating offshore wind. Open source. |
| ProteusDS | DSA | Deep-sea mooring & riser analysis. FEM+CFD coupling. |
| STAR-CCM+ | Siemens | CFD+6DOF+mooring direct coupling. |
Can STAR-CCM+ solve mooring problems too?
Using STAR-CCM+'s DFBI (Dynamic Fluid Body Interaction) motion and Catenary Coupling, you can simultaneously solve for floating body motion in VOF waves and mooring tension. However, the computational cost is very high, so dedicated tools like OrcaFlex are more efficient in the initial design stages.
Mooring Theory Basics — The Catenary Equation for Mooring Lines and Its History (1691)
The "catenary (suspension line) equation" describing the shape of a mooring line (mooring line) was a problem for which Leibniz, Huygens, and Jacob Bernoulli independently derived solutions in 1691. The solution y=a·cosh(x/a) for a chain of uniform density hanging under its own weight is still directly used in offshore mooring design 300 years later. Particularly for deepwater mooring, the horizontal stiffness of "Catenary Mooring" depends on water depth, leading to the fundamental constraint that stiffness decreases and offset (steady-state deviation) increases as depth increases. For ultra-deepwater beyond 3000m, "Tension Leg Platform (TLP)" type mooring is adopted, but even here, the starting point for the static design of the mooring line is the catenary equation.
Physical Meaning of Each Term
- Time Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, splashing manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the time term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled"—meaning setting this term to zero. This significantly reduces computational cost, so trying a steady-state solution first is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the air, as a "carrier," transports heat via convection. Here's the interesting part—this term includes "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelms, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference creates the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. When switching to compressible analysis, if results become strange, it might be due to confusion between absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length).
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models).
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock wave capturing required), free surface flow (VOF/Level Set, etc., required).
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units. |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis. |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C. |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s]. |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for Laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Lumped Mass Method vs FEM
What methods are there for discretizing mooring lines?
The Lumped Mass method represents a mooring line as a chain of point masses and springs. It's simple to implement and fast to compute. OrcaFlex and MoorDyn use this method. FEM (Finite Element Method) is more precise and can handle bending stiffness and torsion. FEM is suitable for analyzing risers and umbilical cables.
How are wave loads applied?
It is standard to evaluate the fluid forces acting on a mooring line using the Morison equation.
$$ f = \frac{1}{2}\rho C_D D |u - \dot{x}|(u - \dot{x}) + \rho C_M \frac{\pi D^2}{4} \dot{u} - \rho (C_M - 1) \frac{\pi D^2}{4} \ddot{x} $$
What methods are there for discretizing mooring lines?
The Lumped Mass method represents a mooring line as a chain of point masses and springs. It's simple to implement and fast to compute. OrcaFlex and MoorDyn use this method. FEM (Finite Element Method) is more precise and can handle bending stiffness and torsion. FEM is suitable for analyzing risers and umbilical cables.
How are wave loads applied?
It is standard to evaluate the fluid forces acting on a mooring line using the Morison equation.
The first term is drag force, the second is Froude-Krylov force + added mass force, and the third is the reaction force of added mass. $C_D$ and $C_M$ are empirical coefficients; for chain, $C_D \approx 2.4$, and for wire, $C_D \approx 1.2$ are typical guidelines.
Fatigue Assessment and Extreme Value Analysis
What are the design standards for mooring systems?
DNV (Det Norske Veritas) DNV-OS-E301 and API RP-2SK (Station Keeping) are representative design standards. (1) Extreme Value Analysis: Maximum tension under environmental conditions with a 100-year return period must be below the breaking load divided by a safety factor. (2) Fatigue Assessment: Cumulative fatigue damage over the design life must be ≤ 1.0 (based on T-N curve). (3) Damaged Condition Analysis: The floating body must not drift away even if one mooring line is severed.
In what scenarios is coupling with CFD used?
CFD is needed for evaluating nonlinear phenomena that potential theory cannot handle, such as green water (deck overtopping) or slamming loads under extreme wave conditions (100-year wave), or assessing the impact of vortex-induced vibration (VIV) around the floating body on the mooring system. Approaches using STAR-CCM+ or OpenFOAM (waves2Foam / olaFlow) to directly simulate floating body motion in waves and obtain time series of mooring tension are increasing.
FEM Modeling of Mooring Chains — The "How Fine to Mesh" Problem
The first dilemma in FEM modeling of mooring lines is the number of element divisions. Modeling each actual chain link would explode the element count, so many practical analyses approximate the entire line with tens to hundreds of beam elements. The problem arises when wave loads are coupled; when wave periods are short (5-8 seconds), dynamic effects become significant, and the difference from tensions obtained by static analysis can reach 20-30%. Using OrcaFlex's mode analysis function to check natural periods beforehand is a standard practice in the industry.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is typically judged when residuals for Continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Pressure: 0.2-0.3, Velocity: 0.5-0.7 are common initial values. Reduce the factor if divergence occurs. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5-20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively calculated (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it reflects the physics that upstream information determines downstream conditions. Although it's first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
Mooring Design for Floating Offshore Wind
How is mooring for floating offshore wind different from traditional O&G (Oil & Gas)?
There are three major differences. (1) Wind loads are dominant; in addition to wave loads, the turbine's thrust force acts steadily. (2) Cost constraints are strict; large chains and anchors like in O&G are too expensive. (3) Mass production (tens to hundreds of units) is assumed, so standardization is crucial.
What mooring configurations are used?
| Mooring Configuration | Composition | Features |
|---|---|---|
| Catenary | Chain + Wire + Anchor | Well-established track record. Weight increases with depth. |
| Tension Leg | Vertical Tendon | For TLP (Tension Leg Platform). Restrains heave. |
| Semi-Taut | Synthetic Fiber Rope + Anchor | Lightweight. Polyester or HMPE ropes. |
| Shared Anchor | Multiple floating bodies share one anchor | Cost reduction. Note mutual interference. |
Is analyzing synthetic fiber ropes different from steel chains?
They are significantly different. Synthetic fiber
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