# The Effect of a Linear Free Surface Boundary Condition on the Steady-State Wave-Making of Shallowly Submerged Underwater Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Boundary Element Method

#### 2.2. RANSE CFD

#### 2.3. Simulation Validation

## 3. Results and Discussion

#### 3.1. Forces and Moments

#### 3.2. Wave Pattern and Longitudinal Wave Profile

#### 3.3. Breaking Waves

#### 3.4. Viscous Effects

#### 3.5. Wave Spectrum

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A 2D visualization of the prolate spheroid and submergence depths used in simulations. Submergence depths are measured from the body axis of symmetry to the calm-water surface and given as a factor of the body diameter.

**Figure 2.**Domain extent sensitivity analysis for three different Froude numbers (reported on top of each graph) and a submergence depth of one diameter.

**Figure 3.**Mesh density sensitivity analysis for three different Froude numbers (reported on top of each graph). An increasing mesh multiple represents a smaller panel size.

**Figure 5.**A representation of the free surface and body discretization used in the BEM (

**left**) and CFD (

**right**) simulations.

**Figure 6.**Correspondence of BEM and CFD simulation data to experimental wave resistance values for an 8:1 prolate spheroid that is submerged 1 diameter from the surface. Small differences show that both the BEM and CFD simulations are capable of predicting accurate wave-resistance values with the CFD simulation performing marginally better.

**Figure 7.**x force vs. Froude number at several test submergences. The

**left**plot shows the value of normalized force for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 8.**Pitch moment vs. Froude number at several test submergences. The

**left**plot shows the normalized moment value for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 9.**z force vs. Froude number at several test submergences. The

**left**plot shows the value of normalized force for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 10.**Wave pattern and longitudinal wave cut for a spheroid traveling with a submergence of one diameter and a Froude number of 0.48.

**Figure 11.**Wave pattern and longitudinal wave cut for a spheroid traveling with a submergence of one diameter and a Froude number of 0.69.

**Figure 12.**x force vs. Froude number at several test submergences at which wave breaking occurs. The

**left**plot shows the value of normalized force for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 13.**Pitch moment vs. Froude number at several test submergences at which wave breaking occurs. The

**left**plot shows the normailzed moment value for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 14.**z force vs. Froude number at several test submergences at which wave breaking occurs. The

**left**plot shows the value of normalized force for both the BEM (solid) and RANSE CFD (dotted) simulations. The

**right**plot shows the difference (CFD minus BEM) between the simulations.

**Figure 15.**Wave pattern and longitudinal wave cut for a spheroid travelling with a submergence of 0.51 diameters and a Froude number of 0.48.

**Figure 16.**Zoomed-in view of a breaking wave on top of a spheroid travelling with a submergence of 0.75 diameters and a Froude number of 0.48.

**Figure 17.**A zoomed-in view of the simulation mesh around the aft portion of a prolate spheroid. The meshes contain different numbers of prism layers to achieve the displayed y+ value and help create a consistent boundary layer.

**Figure 18.**Force in the x direction vs. depth of submergence for 4 different forward velocities using viscous and inviscid RANSE CFD. The

**left**plot shows the value of normalized force for both simulations. The

**right**plot shows the difference (viscous minus inviscid) between the simulations.

**Figure 19.**Velocity magnitude contours around a shallowly submerged prolate spheroid in a viscous fluid. The separated wake (shown in blue) moves toward the surface as the body submergence decreases.

**Figure 20.**Free-wave spectrum data calculated using a transverse wave cut method for depths of submergence of 0.51, 1, 2, and 3 diameters.

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**MDPI and ACS Style**

Lambert, W.; Brizzolara, S.; Woolsey, C.
The Effect of a Linear Free Surface Boundary Condition on the Steady-State Wave-Making of Shallowly Submerged Underwater Vehicles. *J. Mar. Sci. Eng.* **2023**, *11*, 981.
https://doi.org/10.3390/jmse11050981

**AMA Style**

Lambert W, Brizzolara S, Woolsey C.
The Effect of a Linear Free Surface Boundary Condition on the Steady-State Wave-Making of Shallowly Submerged Underwater Vehicles. *Journal of Marine Science and Engineering*. 2023; 11(5):981.
https://doi.org/10.3390/jmse11050981

**Chicago/Turabian Style**

Lambert, William, Stefano Brizzolara, and Craig Woolsey.
2023. "The Effect of a Linear Free Surface Boundary Condition on the Steady-State Wave-Making of Shallowly Submerged Underwater Vehicles" *Journal of Marine Science and Engineering* 11, no. 5: 981.
https://doi.org/10.3390/jmse11050981