Improved HydroDyn strip-theory hydrodynamic load model for circular members floating on or just beneath the free surface

[luwang00]

This PR is NOT ready to be merged. The core new functionality has been implemented and tested, but it is still a work in progress. Also depends on merging PR #3487 first.

• Merge PR Clean-up, refactor, streamlining, and upgrades for HydroDyn strip-theory implementation #3487

Feature or improvement description
This PR provides an important upgrade to HydroDyn's strip-theory formulation by introducing a more accurate hydrodynamic load model for circular members on or near the free surface. It corrects the distributed hydrodynamic loads along the member both when the member is surface piercing (in a ship-like configuration) or at shallow submergence when the member is at a shallow inclination angle relative to the free surface. It also includes corrections to the joint/endplate loads to account for partial wetting.

HydroDyn already computes nonlinear hydrostatic loads on strip-theory (Morison) members by directly integrating the hydrostatic pressure on the instantaneous wetted surface (MHstLMod=2). This method is general and correctly evaluates the hydrostatic loads for any member/free-surface intersection. It also correctly handles partially wetted end plates.

However, the strip-theory hydrodynamic load model was still greatly simplified. More specifically, if the center of a section is below the wave free surface when wave stretching is enabled (WaveStMod>0 in SeaState) or below the still water level without wave stretching (WaveStMod=0 in SeaState), the hydrodynamic loads on this section were computed as if the entire section is submerged. Alternatively, if the section center is above the free surface or still water level, the section was treated as completely dry with zero hydrodynamic load. As a result, HydroDyn strip-theory members could not provide realistic hydrodynamic loads if the member is on and nearly parallel to the free surface with the section centers going in and out of water. The joints/endplates were handled in a similar fashion, being treated as either fully submerged or fully dry with no in-between states.

To address the fidelity gap between hydrodynamic and hydrostatic loads for near-surface/surface-piercing members with shallow inclination relative to the free surface, this PR adds improved strip-theory hydrodynamic load models for circular members specifically for these situations. The key changes relative to the baseline strip-theory load models, which are retained for members at a steeper angle relative to the free surface, are:

• Time-varying section added mass based on instantaneous section submergence, including partial submergence, for section added-mass and wave-scattering forces.
• The submergence-dependent transverse added-mass coefficients are based on the infinite-frequency added mass of Taylor (1930) and Greenhow and Ahn (1988) with newly developed piecewise polynomial representations and asymptotic approximations to facilitate fast and reliable evaluation at runtime. Note that fully submerged sections will also have corrected/time-dependent added mass if the submergence is shallow (center depth less than 5x section radii in this particular implementation).
• The submergence-dependent axial added mass is simply obtained by scaling the fully submerged axial added mass by the fraction of the circumference that is submerged.
• In addition to the standard Morison equation terms, also added a water entry/exit force following the Generalized von Kármán Model. This term is derived from momentum conservation and is related to the time derivative of section added mass. In the present implementation, the von Kármán water entry/exit force is generalized for oblique entry in the transverse direction and also generalized to the axial direction.
• With this enhancement, we do not intend to model the slamming impact force exactly due to the excessive constraints it puts on the time step and the structural solver. Instead, we simply include the water entry/exit force to ensure the hydrodynamic load is momentum conserving as the waterline rises and falls across the section. Therefore, the discontinuous slamming load at the point of water impact and exit is smoothed out over 10% of the section diameter to minimize the high-frequency excitation that can cause numerical difficulties with the structural solver. Nevertheless, dropping the member from the air into water at great height will still lead to a smooth but very high quasi "slamming" load on the structure. Note that the load smoothing slightly breaks momentum conservation during initial impact.
• The Froude-Krylov force is generalized to handle partial wetting of the section circumference. The section load in the transverse direction is integrated analytically based on local pressure gradient over the wetted arc for performance. In the axial direction, we equivalently scale the fully submerged axial force by the fraction of the circumference that is submerged.
• The axial and transverse drag forces are also corrected for partial wetting. In the transverse direction, the reference width of the member is the instantaneous waterline width if less than half of the section is submerged or fixed to the section diameter if more than half of the section is submerged. In the axial direction, the drag is simply scaled by the fraction of the circumference that is submerged. Right now, the distributed transverse and axial drag coefficients are still constant with only corrections to the reference width/wetted circumference. Submergence-dependent drag coefficients can be implemented in the future.
• Smooth blending between the new shallow-inclination section load model and the existing baseline model appropriate for members at a steeper angle relative to the free surface. The blending is performed on a nodal basis (without assuming straight undeflected member) based on the instantaneous angle between the section axial direction and the local free-surface normal over the range of 55 deg to 65 deg. The existing baseline model is used exclusively for 0 to 55 deg, while the new model is used exclusively for 65 to 90 deg. This is equivalent to using the new model when the section axial direction is inclined from the free surface by 0 to 25 degrees and using the existing baseline model when the inclination is between 35 to 90 degrees, with smooth blending between 25 and 35 degrees.
• For the joint/endplate hydrodynamic loads, a unified formulation is adopted that always accounts for partial wetting, except when the endplate is nearly parallel to the free surface (within 5°), where it is treated as either fully wet or fully dry (the previous behavior). The forces are scaled by the wetted area fraction r. More specifically, the added-mass and wave-scattering force are scaled by r^(3/2) while the Froude-Krylov force and drag are scaled by r. If velocity filtering is used, the normal relative velocity filtered is also scaled by r. The von Kármán water entry/exit force is also generalized for the joints and is included in the model.
• With the smooth blending of the new and existing distributed load models and the unified joint load model, the new capability currently does not require any new user inputs. It will automatically take effect with circular members when appropriate. (Rectangular members still have the old behavior.) However, new inputs might be added in the future if there is a strong need for the user to fine-tune model behavior, such as the smoothing length as a fraction of section radius for the slamming force or the inclination angle range for model blending.

Currently, the above new capabilities are only implemented for circular members. There is no plan to generalize it to rectangular members due to the greatly increased complexity and a lack of reference data.

The core feature has been implemented and preliminarily tested and verified, including a stress test where a light horizontal rigid cylinder (the fully submerged added mass is 4.6x the structural mass) is dropped from 10 m above water. After the initial impact, the member subsequently bounced back out of water before settling over time. This is a numerically highly challenging case because the initial Jacobian encodes zero added mass (member is in the air), and as the member hits the water, the added mass rapidly increases from 0 to 4x structural mass. Nevertheless, the glue code remains stable with the help of automatic Jacobian update and underrelaxation.

Related issue, if one exists
N/A

Impacted areas of the software
HydroDyn (Morison)

Additional supporting information
Formulation and implementation details to come.

Generative AI usage
Formulated by human. This PR page is also written by human for readability and focus.
Code implementation assisted by LLM:
Co-authored-by: Anthropic Claude claude@anthropic.com
Assisted by: GitHub Copilot support@github.com

Test results, if applicable
Existing r-tests are not affected because the scenario addressed by this improvement was purposely avoided in the past. New r-tests will be prepared and require merging.

• r-test branch merging required